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abstract: 'Using THz spectroscopy in external magnetic fields we investigate the low-temperature charge dynamics of strained HgTe, a three dimensional topological insulator. From the Faraday rotation angle and ellipticity a complete characterization of the charge carriers is obtained, including the 2D density, the scattering rate and the Fermi velocity. The obtained value of the Fermi velocity provides further evidence for the Dirac character of the carriers in the sample. In resonator experiments, we observe quantum Hall oscillations at THz frequencies. The 2D density estimated from the period of these oscillations agrees well with direct transport experiments on the topological surface state. Our findings open new avenues for the studies of the finite-frequency quantum Hall effect in topological insulators.'
author:
- 'A. M. Shuvaev'
- 'G. V. Astakhov'
- 'G. Tkachov'
- 'C. Brüne'
- 'H. Buhmann'
- 'L. W. Molenkamp'
- 'A. Pimenov'
bibliography:
- 'lit\_HgTe.bib'
title: Terahertz Quantum Hall Effect in a Topological Insulator
---
Three dimensional topological insulators [@hasan_rmp_2010; @qi_prb_2008] have attracted much interest recently, as they exhibit a number of unusual and non-trivial properties, such as protected conducting states on the surfaces of the sample. Unusual electrodynamics, such as a universal Faraday effect and an anomalous Kerr rotation have been predicted [@tse_prl_2010; @tse_prb_2011; @maciejko_prl_2010; @tkachov_prb_2011] for these surface states, their observation is still outstanding. We showed recently that strained HgTe, where the strain lifts the light-hole–heavy-hole degeneracy that normally is present in bulk HgTe, is a very promising 3D topological insulator [@brune_prl_2011]. This is because at low temperatures parasitic effects due to bulk carriers are practically absent. In static transport experiments a strained 70 nm thick HgTe layer [@brune_prl_2011] exhibits a quantum Hall effect (QHE), yielding direct evidence that the charge carriers in these layers are confined to the topological two dimensional (2D) surface states of the material. These findings are further corroborated by recent Faraday rotation data [@hancock_prl_2011] in a similar layer, which have been obtained using a terahertz time-domain technique.
In this work, we present the results of low temperature terahertz Faraday cw transmission experiments on another strained HgTe film. The carrier density, Fermi velocity and the scattering rate can be reliably determined from these data. In particular, we obtain the Fermi velocity $v_F = 0.52 \cdot 10^6$ m/s, which is in excellent agreement with the Faraday rotation experiments [@hancock_prl_2011] and the dc Shubnikov-de Haas measurements [@brune_prl_2011] on 70-nm-thick strained HgTe films as well as with band-structure calculations for the surface states in 3D topological insulators (see e.g. Ref. [@liu_prb_2010]). In the same sample we observe quantum Hall-induced oscillations at terahertz frequencies, providing further evidence for the 2D character of the conductivity. In the case of topological insulators, no finite frequency QHE has been reported up to now. The sample studied in this work is a coherently strained 52-nm-thick nominally undoped HgTe layer, grown by molecular beam epitaxy on an insulating CdTe substrate [@becker_pss_2007]. Transmittance experiments at terahertz frequencies (100 GHz $< \nu <$ 800 GHz) have been carried out in a Mach-Zehnder interferometer arrangement [@volkov_infrared_1985; @pimenov_prb_2005] which allows measurement of the amplitude and phase shift of the electromagnetic radiation in a geometry with controlled polarization. Using wire grid polarizers, the complex transmission coefficient can be measured both in parallel and crossed polarizers geometry. Static magnetic fields, up to 8 Tesla, have been applied to the sample using a split-coil superconducting magnet. To interpret the experimental data we use the ac conductivity tensor $\hat{\sigma} (\omega)$ obtained in the classical (Drude) limit from the Kubo conductivity of topological surface states (see e.g. Ref. [@tse_prb_2011]). The diagonal, $\sigma_{xx}
(\omega)$, and Hall, $\sigma_{xy} (\omega)$, components of the conductivity tensor as functions of THz frequency $\omega$ can be written as: $$\begin{aligned}
&& \sigma_{xx} (\omega)=\sigma_{yy} (\omega) =
\frac{1-i \omega \tau}{(1-i \omega \tau)^2 +(\Omega_c \tau)^2} \sigma_0
\,, \label{sxx}\\
&& \sigma_{xy} (\omega)=-\sigma_{yx} (\omega)= \frac{\Omega_c
\tau}{(1-i \omega \tau)^2 +(\Omega_c \tau)^2} \sigma_0 \,.
\label{sxy}\end{aligned}$$ Here, $\Omega_c = eBv_F/\hbar k_F$ is the cyclotron frequency, $\sigma_0$ is the dc conductivity, $B$ is the magnetic field, $v_F$, $k_F$, $e$, and $\tau$ are the Fermi velocity, Fermi wave-number, charge, and scattering time of the carriers, respectively. For the Dirac spin-helical surface states the Fermi wave-number depends on the 2D carrier density, $n_{2D}$, through relation $k_F=\sqrt{4\pi n_{2D}}$, with no spin degeneracy.
The transmission spectra can then be calculated using a transfer matrix formalism [@berreman_josa_1972; @shuvaev_epjb_2011; @shuvaev_prl_2011] which takes multiple reflection within the substrate into account. The electrodynamic properties of the CdTe substrate have been obtained in a separate experiment on a bare substrate. Further details of the fitting procedure can be found in the Supplementary information to Ref. [@shuvaev_prl_2011]. Neglecting any substrate effects, the complex transmission coefficients in parallel ($t_p$) and crossed ($t_c$) polarizers geometry can be written as: $$\begin{aligned}
&& t_p =\frac{4+2\Sigma_{xx}}
{4+4\Sigma_{xx}+\Sigma_{xx}^2+\Sigma_{xy}^2} \,, \label{tp}\\
&& t_c =\frac{2\Sigma_{xy}}
{4+4\Sigma_{xx}+\Sigma_{xx}^2+\Sigma_{xy}^2} \,. \label{tc}\end{aligned}$$ Here $\Sigma_{xx}$ and $\Sigma_{xy}$ are effective dimensionless 2D conductivities, defined as: $\Sigma_{xx}=\sigma_{xx}dZ_0$ and $\Sigma_{xy}=\sigma_{xy}dZ_0$ with the HgTe film thickness $d=52$nm and the vacuum impedance $Z_0 \approx 377\,\Omega$. In order to self-consistently obtain the parameters of the quasiparticles, the field-dependent complex transmission $t_p(B)$ and $t_c(B)$ for $\nu =$0.17 THz, 0.35 THz and 0.75 THz and the zero-field transmittance spectra $|t_c(\omega)|^2$ have been fitted simultaneously.
![*Magnetic field dependence of the transmission in strained HgTe.* (a-c) Transmission amplitude in parallel polarizers ($t_p$) geometry, showing cyclotron resonance at the positions indicated by the arrows. The frequency of the experiments is indicated in the panels. The inset shows the frequency dependent transmittance in zero external magnetic field, $|t_p(B=0)|^2$. Symbols: experiment, solid lines: simultaneous fit of all data with the Drude model as described in the text.[]{data-label="ftran"}](ftran.eps){width="0.6\linewidth"}
The inset in Fig. \[ftran\] shows the transmittance spectrum of the HgTe film at zero magnetic field. The characteristic oscillations in the spectrum, with a period of about 58 GHz, are due to Fabry-Pérot type interferences within the CdTe substrate. The absolute transmittance in the interference maxima is close to 95%, which reflects the low effective conductance of our HgTe film, $\Sigma_{xx} \ll 1$. At low frequencies, the maximum transmittance decreases and approaches $|t_p|^2 \simeq 0.7$ in the zero frequency limit. Such a behavior is typical for Drude carriers with a scattering rate in the frequency region of the experiment. Indeed, the solid line in the transmission spectra represents a Drude fit with the parameters given in the first row of Tab. \[tab\].
From the fits we obtain the Fermi velocity $v_F = 0.52 \cdot 10^6$ m/s. This value is very close both to $v_F = (0.51 \div 0.58) \cdot
10^6$ m/s as determined in the Faraday rotation experiments on a 70-nm-thick strained HgTe film [@hancock_prl_2011] and to $v_F =
0.42 \cdot 10^6$ m/s as extracted from dc Shubnikov-de Haas measurements on a patterned 70-nm-thick strained HgTe layer [@brune_prl_2011]. The obtained value of the Fermi velocity is also in very good agreement with the band-structure-theory result $v_F = 0.51 \cdot 10^6$ m/s for the linear (Dirac) part of the surface-state spectrum in topological insulators (see e.g. Ref. [@liu_prb_2010]). As an additional check of the 2D surface carrier dynamics in our sample, we have analyzed the terahertz transmission data of Ref. [@shuvaev_prl_2011] for a 70-nm-thick strained HgTe film at high temperature $T=200$ K and for a bulk (1000-nm-thick) unstrained HgTe sample. In both cases, the electrodynamics is governed by massive bulk carriers, for which the values of $v_F$ turn out to be much larger than the Dirac surface-state velocity, i.e., $v_F \approx 0.5 \cdot 10^6$ m/s (Tab. \[tab\]).
-----------------------------------------------------------------------------------------------------------------------------------------------------
Sample $T$(K) $n_{2D}$(cm$^{-2}$) $v_F$(ms$^{-1}$) $1/2\pi $G_{2D}=\sigma_0 \cdot d\ (\Omega^{-1})$
\tau$ (GHz)
------------------------------------------ -------- --------------------- ------------------ ------------- ------------------------------------------
52 nm (strained) \[this work\] 2 $1.08\cdot10^{11}$ $0.52\cdot 10^6$ 250 $7.6\cdot 10^{-4}$
70 nm (strained) [@shuvaev_prl_2011] 4 $4.8\cdot10^{10}$ $0.38\cdot 10^6$ 210 $4.3\cdot 10^{-4}$
200 $1.5\cdot10^{12}$ $1.63\cdot 10^6$ 360 $5.3\cdot 10^{-3}$
1000 nm (unstrained) [@shuvaev_prl_2011] 3 $4.2\cdot10^{11}$ $0.99\cdot 10^6$ 240 $2.8\cdot 10^{-3}$
200 $4.9\cdot10^{13}$ $9.36\cdot 10^6$ 360 $1.9\cdot 10^{-1}$
-----------------------------------------------------------------------------------------------------------------------------------------------------
![*Complex Faraday angle $\theta + i \eta$ in HgTe.* Bottom panels: Faraday rotation, top panels: ellipticity for the same frequencies as in Fig. \[ftran\]. The inset sketches the definitions of the Faraday rotation $\theta$ and ellipticity $\eta$. Symbols: experiment, solid lines: simultaneous fit of all data with the Drude model as described in the text. Angular units are radians.[]{data-label="fang"}](fang.eps){width="0.9\linewidth"}
Figure \[ftran\] shows the magnetic field dependent transmittance of the HgTe film in Faraday geometry and for parallel orientation of polarizer and analyzer. According to Eq. (\[tp\]), the transmittance in parallel polarizers ($t_p$) depends mainly on $\Sigma_{xx}$. For all three frequencies two clear minima in the transmitted signal are observed in the range below $\pm 1$T. The minima in $|t_p|$ roughly correspond to the cyclotron resonance energy and scale with magnetic field. This may be understood taking into account that in our case $\Sigma \ll 1$ and Eqs. (\[tp\],\[tc\]) simplify to: $$\label{trsimple}
t_p \simeq 1- \Sigma_{xx}/2; \quad t_c \simeq \Sigma_{xy}/2 \ .$$ In the limit $\omega\tau \gg 1$, Eq. (\[sxx\]) may be approximated by $$\label{sxx1}
\sigma_{xx} \simeq \frac{1-i \omega \tau}{(\Omega_c ^2-\omega ^2)\tau^2}
\sigma_0 \ ,$$ which leads to a resonance like feature for $\Omega_c=\omega$. Thus, the positions and widths of the minima in Fig. \[ftran\] are directly connected with the parameter $v_F/k_F$ and the scattering rate $\tau^{-1}$ of the charge carriers.
Figure \[fang\] shows the complex Faraday angle $\theta + i \eta$ as obtained at the same frequencies as in Fig. \[ftran\]. The polarization rotation $\theta$ and the ellipticity $\eta$ are obtained from the transmission data using: $$\begin{aligned}
&& \tan(2\theta)=2\Re(\chi)/(1-|\chi|^2)\ , \\
&& \sin(2\eta)=2\Im(\chi)/(1+|\chi|^2)\ .\end{aligned}$$ Here $\chi=t_c/t_p$ and the definitions of $\theta + i
\eta$ are shown graphically in the inset to Fig. \[fang\]. A direct interpretation of the complex Faraday angle is in general not possible because of the interplay of $\sigma_{xx}$ and $\sigma_{xy}$ in the data.
In the low frequency limit, $\omega\tau \ll 1$ Eq. (\[sxy\]) simplifies to the static result $\sigma_{xy} = \Omega_c \tau\sigma_0
/(1 +(\Omega_c \tau)^2) $. The last expression has a maximum at $\Omega_c (B) = \tau^{-1} $, which leads to maxima in $t_c$ and $\theta$ at about the same field value. Therefore, the Faraday angle provides a direct and an independent way of obtaining the scattering rate $1/\tau$. The solid line in Fig. \[fang\] are the fits which have been done simultaneously for all results presented above. In total, the parameters of the charge carriers have been obtained by simultaneously fitting ten data sets. The quite reasonable fit of all results proves that a single type of charge carriers dominates the electrodynamics in the range of frequencies and magnetic fields used in these experiments.
![*Faraday rotation in HgTe within resonator geometry.* (a) - Ellipticity, (b) - Faraday angle. Symbols - experiment, lines - fits according to Eqs. (1-4). Upper inset shows the experimental geometry within a Copper meshes resonator. Lower inset shows a magnified view of the Faraday angle demonstrating QHE oscillations.[]{data-label="fres"}](fres.eps){width="0.7\linewidth"}
Very solid evidence for the two dimensional character of the carriers probed in the Faraday rotation experiments would be the observation of quantum Hall plateaus, similar to the observation of the QHE in [@brune_prl_2011]. However, the accuracy of the experiments shown above does not allow to observe the QHE. In order to solve this problem, we have performed further Faraday transmission experiments on the same sample, now using a resonator geometry as shown in the inset of Fig. \[fres\].
![*Terahertz quantum Hall effect in HgTe.* (a) Two dimensional conductance: $G_{xx}$, (b) derivative of $G_{xy}$ ($dG_{xy}/dB^{-1}$). The data have been obtained within a resonator geometry and are plotted as a function of inverse magnetic field. The experimental data are shown as solid lines for frequencies as indicated. Dashes in the bottom panel marks the minima for negative magnetic fields. (c) Numbered positions of the minima in $G_{xx}$ and in the derivative of $G_{xy}$ for 0.14 THz and 0.19 THz. Straight lines yield interpolation to the origin. []{data-label="fqhe"}](fqhe.eps){width="0.95\linewidth"}
In these experiments, the sample is placed in the middle of a Fabry-Pérot resonator defined by metallic meshes. We have utilized Cu meshes with a 200 $\mu$m period. The distance between adjacent maxima of the resonator is $\simeq
51$GHz. In the frequency range between 100 and 200 GHz the quality factor of the loaded resonator is about $Q \sim 10$. This indicates that, effectively, the radiation passes about ten times through the sample before reaching the detector, which effectively increases the sensitivity to fine details by roughly the same value. As shown in Fig. \[fres\], in the resonator experiments the field dependence of the Faraday rotation and the ellipticity appears qualitatively similar to that in Fig. \[fang\]. An exact calculation of the complex transmission coefficients within a resonator is complicated because of the increased number of parameters. Therefore, in this case we utilize the simple equations Eqs. (\[sxx\])-(\[tc\]) which neglect the effect of the substrate and the resonator completely. Nevertheless, as clearly seen in Fig. \[fres\], the fits based on the simplified expressions reproduce the experimental results reasonably well. Fitting of the signals for parallel and crossed polarizers yields within experimental accuracy the same parameters as in the experiments without a resonator. The only parameter which differs from the results without a resonator is the absolute value of the conductivity. This is of course expected, and results from multiple transmission in the resonator and the influence of the substrate.
The main advantage of the resonator experiments is a higher sensitivity to the details of the field-dependent transmission. In addition to an overall field dependence similar to that in Figs. \[ftran\] and \[fang\], a tiny modulation of the signal can now be observed. To convert this modulation to a conventional presentation, we have inverted the transmittance curves into the 2D conductivity, using Eqs. (\[tp\] and \[tc\]). Because the absolute transmittance is not well-defined in the resonator experiments, we have scaled the absolute 2D conductance to agree with the data without a resonator. The final results expressed in form of the effective 2D conductance $G_{xx,xy}=\Sigma_{xx,xy}/Z_0$ are shown in Fig. \[fqhe\].
Fig. \[fqhe\]a shows the real part of the two dimensional conductance $G_{xx}$ as a function of inverse magnetic field. Clear oscillations in the conductance can be observed in this presentation. In general, the phenomenology of the QHE at terahertz frequencies is not well understood [@hols_prl_2002; @ikebe_prl_2010]. Existing experiments are generally limited to frequencies below 100 GHz and they are analyzed using scaling exponents [@sondi_rmp_1997; @hols_prl_2002]. In the resonator experiments, the field dependent oscillations can be observed both with parallel and crossed polarizers. Contrary to $G_{xx}$, the off-diagonal conductance $G_{xy}$ shows a substantial field dependence even in high magnetic fields. Therefore, no clear QHE signal can be directly detected in $G_{xy}$. In order to extract the QHE information from these data, we have plotted the derivative of the $G_{xy}$ as a function of an inverse magnetic field ($dG_{xy}/dB^{-1}$) in Fig. \[fqhe\]b. The derivative has the advantage of being insensitive to any residual slowly varying signals, and, importantly, the expected steps in $G_{xy}$ are transformed into the minima of the derivative. Finally, in order to analyze the quantum Hall effect, both the minima in $G_{xx}$ and in the derivative of $G_{xy}$ have been taken into account. In Fig. \[fqhe\]a,b the results at finite frequencies are compared with dc QHE on the same sample. The periodicity of the oscillations in the dc experiments is slightly different because of different carrier concentration at the sample surface, induced by exposure to photoresist and the presence of ohmic contacts.
The main results of the QHE experiments are represented in Fig. \[fqhe\]c demonstrating an approximate equidistant positioning of all minima (labeled by number $N$) in inverse magnetic fields $B^{-1}$ with the period of $\Delta B^{-1} = 0.18$ T$^{-1}$. This periodicity reflects the dependence of the number of the occupied Landau levels on $B^{-1}$. In a total, we detect the oscillations up to index number $\pm 10$; also the first oscillations with the Landau level index $\pm 1$ are clearly observed in the data. From the periodicity of these oscillations the effective 2D carrier density can be estimated according to the free electron expression $n_{2D}=e/(h\Delta B^{-1}) \simeq 1.4 \cdot 10^{11}$cm$^{-2}$. This value agrees reasonably well with the density $n_{2D}=1.08 \cdot
10^{11}$cm$^{-2}$ obtained directly from fitting the transmittance and the Faraday rotation on the basis of the Drude model (Tab. \[tab\]). Therefore, we may conclude that charge carriers which are responsible for the terahertz electrodynamics at low temperatures reveal 2D behavior. To further characterize the electron system in our sample we extrapolated the dependence $N(B^{-1})$ to the origin (see straight lines in Fig. \[fqhe\]c), which corresponds to the limit of very strong magnetic fields. At the origin we find a finite value $N \approx 1/2$ instead of $N=0$ as would be the case for the conventional QHE. Previously, similar extrapolated values were reported for graphene (see e.g. Ref. [@novoselov_nature_2005]), zero-gap HgTe quantum wells [@buttner_nphys_2011] and strained 70 nm-thick HgTe films [@brune_prl_2011], i.e. for materilas with 2D Dirac-like charge carriers encoding a nonzero Berry phase. We therefore believe that our teraherz QHE also indicates the 2D Dirac-like behavior.
In conclusion, we have analyzed the terahertz Faraday rotation in a strained HgTe film. From these data all relevant parameters of the charge carriers can be obtained. In addition, terahertz quantum Hall effect oscillations have been observed within the same experiment, which proved the two-dimensional character of the conductivity.
We thank E. M. Hankiewicz for valuable discussion. This work was supported by the by the German Research Foundation DFG (SPP 1285, FOR 1162) the joint DFG-JST Forschergruppe on ’Topological Electronics’, the ERC-AG project ’3-TOP’, and the Austrian Science Funds (I815-N16).
| ArXiv |
---
abstract: |
A new symplectic N-body integrator is introduced, one designed to calculate the global $360^\circ$ evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. This freely-available code is called [epi\_int]{}, and it is distinct from other such codes in its use of streamlines to calculate the effects of ring self-gravity. The great advantage of this approach is that the perturbing forces arise from smooth wires of ring matter rather than discreet particles, so there is very little gravitational scattering and so only a modest number of particles are needed to simulate, say, the scalloped edge of a resonantly confined ring or the propagation of spiral density waves.
The code is applied to the outer edge of Saturn’s B ring, and a comparison of Cassini measurements of the ring’s forced response to simulations of Mimas’ resonant perturbations reveals that the B ring’s surface density at its outer edge is $\sigma_0=195\pm 60$ gm/cm$^2$ which, if the same everywhere across the ring would mean that the B ring’s mass is about $90\%$ of Mimas’ mass.
Cassini observations show that the B ring-edge has several free normal modes, which are long-lived disturbances of the ring-edge that are not driven by any known satellite resonances. Although the mechanism that excites or sustains these normal modes is unknown, we can plant such a disturbance at a simulated ring’s edge, and find that these modes persist without any damping for more than $\sim10^5$ orbits or $\sim100$ yrs despite the simulated ring’s viscosity $\nu_s=100$ cm$^2$/sec. These simulations also indicate that impulsive disturbances at a ring can excite long-lived normal modes, which suggests that an impact in the recent past by perhaps a cloud of cometary debris might have excited these disturbances which are quite common to many of Saturn’s sharp-edged rings.
author:
- 'Joseph M. Hahn'
- 'Joseph N. Spitale'
- |
Submitted for publication in the\
[*Astrophysical Journal*]{} on December 28, 2012\
Revised April 26, 2013\
Accepted June 1, 2013
bibliography:
- 'biblio.bib'
title: |
An N-body Integrator for Gravitating Planetary Rings,\
and the Outer Edge of Saturn’s B Ring
---
Introduction {#intro_section}
============
A planetary ring is often coupled dynamically to a satellite via orbital resonances. The ring’s response to resonant perturbations varies with the forcing, and if the ring is for instance composed of low optical depth dust, then the ring’s response will vary with the satellite’s mass and its proximity. But in an optically thick planetary ring, such as Saturn’s main A and B rings or its many dense narrow ringlets, the ring is also interacting with itself via self gravity, so its response is also sensitive to the ring’s mass surface density $\sigma_0$ [@S84; @MB05; @HSP09]. So by measuring a dense ring’s response to satellite perturbations, and comparing that measurement to a model for the ring-satellite system, one can then infer the ring’s physical properties, such as its surface density $\sigma_0$, and perhaps other quantities too [@MB05; @TBN07; @HSP09]. Recently [@HSP09] developed a semi-analytic model of the outer edge of Saturn’s B ring, which is confined by an $m=2$ inner Lindblad resonance with the satellite Mimas. The resonance index $m$ also describes the ring’s anticipated equilibrium shape, with the ring-edge’s deviations from circular motion expected to have an azimuthal wavenumber of $m=2$. So the B ring’s expected shape is a planet-centered ellipse, which has $m=2$ alternating inward and outward excursions. The model of [@HSP09] also calculates the ring’s equilibrium $m=2$ response excited by Mimas, but that comparison between theory and observation was done during the early days of the Cassini mission when that spacecraft’s measurement of the ring-edge’s semimajor axis $a_{\mbox{\scriptsize edge}}$ was still rather uncertain. It turns out that the ring’s inferred surface density is very sensitive to how far the B ring’s outer edge extends beyond the resonance, which was quite uncertain then due to the uncertainty in $a_{\mbox{\scriptsize edge}}$, so the uncertainty in the ring’s inferred $\sigma_0$ was also relatively large. Now however $a_{\mbox{\scriptsize edge}}$ is known with much greater precision, so a re-examination of this system is warranted.
Cassini’s monitoring of the B ring also reveals that the ring’s outer edge exhibits several normal modes, which are unforced disturbances that are not associated with any known satellite resonances. Figure \[Bring\_fig\] illustrates this phenomenon with a mosaic of images that Cassini acquired of the B ring’s edge on 28 January 2008. [@SP10] have also fit a kinematic model to four years worth of Cassini images of the B ring; that model is composed of four normal modes having azimuthal wavenumbers $m=1,2,2,3$ that steadily rotate over time at distinct rates. In the best-fitting kinematic model there are two $m=2$ modes, one that is forced by and corotating with Mimas, as well as a free $m=2$ mode that rotates slightly faster. The amplitudes and orientations of all the modes as they appear in the 28 January 2008 data is also shown in Fig. \[Bring\_fit\_fig\]. Note that although the B ring’s outer edge, as seen in Fig. \[Bring\_fig\], might actually resemble a simple $m=2$ shape on 28 January 2008, at other times the ring-edge’s shape is much more complicated than a simple $m=2$ configuration, yet at other times the ring-edge is relatively smooth and nearly circular; see for example Fig. 1 of [@SP10]. This behavior is due to the superposition of the normal modes that are rotating relative to each other, which causes the B ring’s edge to evolve over time. Since this system is not in simple equilibrium, a time-dependent model of the ring that does not assume equilibrium is appropriate here.
So the following develops a new N-body method that is designed specifically to track the time evolution of a self-gravitating planetary ring, and that model is then applied to the latest Cassini results. Section \[method\_section\] describes in detail the N-body model that can simulate all $360^\circ$ of a narrow annulus in a self-gravitating planetary ring using a very modest number of particles. Section \[B ring\] then shows results from several simulations of the outer edge of Saturn’s B ring, and demonstrates how a ring’s observed epicyclic amplitudes and pattern speeds can be compared to N-body simulations to determine the ring’s physical properties. Results are then summarized in Section \[summary\].
Numerical method {#method_section}
================
The following briefly summarizes the theory of the symplectic integrator that [@DLL98] use in their [SYMBA]{} code and [@C99] use in the [MERCURY]{} integrator to calculate the motion of objects in nearly Keplerian orbits about a point-mass star. That numerical method is adapted here so that one can study the evolution of a self-gravitating planetary ring that is in orbit about an oblate planet.
symplectic integrators {#symplectic}
----------------------
The Hamiltonian for a system of N bodies in orbit about a central planet is $$\begin{aligned}
H &=& \sum_{i=0}^{N}\frac{p_i^2}{2m_i} + \sum_{i=0}^{N}\sum_{j>i}^{N} V_{ij},\end{aligned}$$ where body $i$ has mass $m_i$ and momentum $\mathbf{p}_i = m_i\mathbf{v}_i$ where $\mathbf{v}_i=\mathbf{\dot{r}}_i$ is its velocity and $V_{ij}$ is the potential such that $\mathbf{f}_{ij}=-\nabla_{\mathbf{r}_i}V_{ij}$ is the force on $i$ due to body $j$ where $\nabla_{\mathbf{r}_i}$ is the gradient with respect to coordinate $\mathbf{r}_i$, and the index $i=0$ is reserved for the central planet whose mass is $m_0$. Next choose a coordinate system such that all velocities are measured with respect to the system’s barycenter, so $\mathbf{p}_0 = -\sum_{j=1}^N\mathbf{p}_j$, and the Hamiltonian becomes $$\begin{aligned}
H &=& \sum_{i=1}^{N}\left(\frac{p_i^2}{2m_i} + V_{i0}\right)
+ \sum_{i=1}^{N}\sum_{j>i}^NV_{ij}
+ \frac{1}{2m_0}\left(\sum_{i=1}^N\mathbf{p}_i\right)^2
\equiv H_A + H_B + H_C\end{aligned}$$ since $V_{ij} = V_{ji}$. This Hamiltonian has three parts,
\[H\] $$\begin{aligned}
H_A &=& \sum_{i=1}^N\left(\frac{p_i^2}{2m_i} + V_{i0}\right)\\
H_B &=& \sum_{i=1}^N\sum_{j>i}^NV_{ij}\\
H_C &=& \frac{1}{2m_0}\left(\sum_{i=1}^N\mathbf{p}_i\right)^2,
\end{aligned}$$
and the following will employ spatial coordinates such that all $\mathbf{r}_i$ are measured relative to the central planet. This combination of planetocentric coordinates and barycentric velocities is referred to as ‘democratic-heliocentric’ coordinates in [@DLL98] and ‘mixed-center’ coordinates in [@C99]. In the above, $H_A$ is the sum of two-body Hamiltonians, $H_B$ represents the particles’ mutual interactions, and $H_C$ accounts for the additional forces that arise in this particular coordinate system that are due to the central planet’s motion about the barycenter.
Hamilton’s equations for the evolution of the coordinates $\mathbf{r}_i$ and momenta $\mathbf{p}_i$ for particle $i\ge1$ are $\mathbf{\dot{r}}_i = \nabla_{\mathbf{p}_i}H$ and $\mathbf{\dot{p}}_i = -\nabla_{\mathbf{r}_i}H$. So a particle that is subject only to Hamiltonian $H_B$ during short time interval $\delta t$ would experience the velocity kick $$\begin{aligned}
\label{dv}
\mathbf{\delta v}_i &=& \frac{{\mathbf{\dot{p}}}_i \delta t}{m_i} =
-\nabla_{\mathbf{r}_i}H_B\frac{\delta t}{m_i}
= \frac{\delta t}{m_i}\sum_{j=1}^N\mathbf{f}_{ij},\end{aligned}$$ which of course is $i$’s response to the forces exerted by all the other small particles in the system. And since $H_C$ is a function of momenta only, a particle subject to $H_C$ during time $\delta t$ will see its spatial coordinate kicked by $$\begin{aligned}
\label{dx}
\mathbf{\delta r}_i &=& \frac{\delta t}{m_0}\sum_{j=1}^{N} \mathbf{p}_j\end{aligned}$$ due to the planet’s motion about the barycenter.
Now let $\xi_i(t)$ represent any of particle $i$’s coordinates $x_i$ or momenta $p_i$; that quantity evolves at the rate [@G80] $$\begin{aligned}
\label{eom}
\frac{d\xi_i}{dt} &=& [\xi_i, H] = [\xi_i, H_A + H_B + H_C]
=(A + B + C) \xi_i\end{aligned}$$ where the brackets are a Poisson bracket, and the operator $A$ is defined such that $A\xi_i = [\xi_i, H_A]$, with operators $B$ and $C$ defined similarly. The solution to Eqn. (\[eom\]) for $\xi_i$ evaluated at the later time $t + \Delta t$ is formally $$\begin{aligned}
\label{eom_soln}
\xi_i(t + \Delta t) &=& e^{(A + B + C)\Delta t}\xi_i(t)\end{aligned}$$ [@G80], but this exact expression is in general not analytic and not in a useful form. However [@DLL98] and [@C99] show that the above is approximately $$\begin{aligned}
\label{eom_soln_approx}
\xi_i(t + \Delta t) &\simeq& e^{B\Delta t/2}e^{C\Delta t/2}e^{A\Delta t}
e^{C\Delta t/2}e^{B\Delta t/2}\xi_i(t),\end{aligned}$$ which indicates that five actions that are to occur as the system of orbiting bodies are advanced one timestep $\Delta t$ by the integrator. First ([*i.*]{}) the operator $e^{B\Delta t/2}$ acts on $\xi_i(t)$, which increments ([*i.e.*]{} kicks) particle $i$’s velocity $\mathbf{v}_i$ by Eqn. (\[dv\]) due to the system’s interparticle forces with $\delta t=\Delta t/2$. Then ([*ii.*]{}) the $e^{C\Delta t/2}$ operator acts on the result of substep ([*i.*]{}) and kicks the particle’s spatial coordinates $\mathbf{r}_i$ according to Eqn. (\[dx\]) due to the central planet’s motion about the barycenter. Then in substep ([*iii.*]{}) the $e^{A\Delta t}$ operation advances the particle along its unperturbed epicyclic orbit about the central planet during a full timestep $\Delta t$, with this substep is referred to below as the orbital ‘drift’ step. Step ([*iv.*]{}) is another coordinate kick $\delta\mathbf{r}_i$ and the last step ([*v.*]{}) is the final velocity kick.
In a traditional symplectic N-body integrator the planet’s oblateness is treated as a perturbation whose effect would be accounted for during steps ([*i.*]{}) and ([*v.*]{}) which provide an extra kick to a particle’s velocity every timestep. Those kicks cause a particle in a circular orbit to have a tangential speed that is faster than the Keplerian speed by the fractional amount that is of order $\sim J_2(R/r)^2\sim3\times10^{-3}$ where $J_2\simeq0.016$ is Saturn’s second zonal harmonic and $r/R\sim2$ is a B ring particle’s orbit radius $r$ in units of Saturn’s radius $R$. The particle’s circular speed is super-Keplerian, and if its coordinates and velocities were to be converted to Keplerian orbit elements, its Keplerian eccentricity would also be of order $e\sim3\times10^{-3}$. This putative eccentricity should be compared to the observed eccentricity of Saturn’s B ring, which is the focus of this study and is of order $e\sim10^{-4}$, about 30 times smaller than the particle’s Keplerian eccentricity. The main point is, that one does not want to use Keplerian orbit elements when describing a particle’s nearly circular motions about an oblate planet because the Keplerian eccentricity is dominated by planetary oblateness whose effects obscures the ring’s much smaller forced motions.
To sidestep this problem, the following algorithm uses the [*epicyclic*]{} orbit elements of [@BL94] which provide a more accurate representation of an unperturbed particle’s orbit about an oblate planet. Note that this use of epicyclic orbit elements effectively takes the effects of oblateness out of the integrator’s velocity kick steps ([*i.*]{}) and ([*v.*]{}) and places oblateness effects in the integrator’s drift step ([*iii.*]{}), which is preferable because the forces in the B ring that are due to oblateness are about $\sim10^4$ times larger than any satellite perturbation. The following details how these epicyclic orbit elements are calculated and are used to evolve the particle along its unperturbed orbit during the drift substep.
epicyclic drift {#drift}
---------------
This 2D model will track a particle’s motions in the ring plane, so the particle’s position and velocity relative to the central planet can be described by four epicyclic orbit elements: semimajor axis $a$, eccentricity $e$, longitude of periapse $\tilde{\omega}$, and mean anomaly $M$. For a particle in a low eccentricity orbit about an oblate planet, the relationship between the particle’s epicyclic orbit elements and its cylindrical coordinates $r, \theta$ and velocities $v_r, v_\theta$ are
\[rv\] $$\begin{aligned}
\label{r}
r &=& a\left[1 - e\cos M +
\left(\frac{\eta_0}{\kappa_0}\right)^2(2 - \cos^2M)e^2 \right] \\
\theta &=& \tilde{\omega} + M +
\frac{\Omega_0}{\kappa_0}\left\{2e\sin M + \left[\frac{3}{2} +
\left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2\sin M\cos M\right\}\\
v_r &=& a\kappa_0\left[e\sin M +
2\left(\frac{\eta_0}{\kappa_0}\right)^2e^2\sin M\cos M\right]\\
\label{v_t}
v_\theta &=& a\Omega_0\left\{1 + e\cos M -
2\left(\frac{\eta_0}{\kappa_0}\right)^2e^2 +
\left[1 + \left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2\cos^2 M\right\},
\end{aligned}$$
which are adapted from Eqns. (47-55) of [@BL94]. These equations are accurate to order ${\cal O}(e^2)$ and require $e\ll1$. Here $\Omega_0(a)$ is the angular velocity of a particle in a circular orbit while $\kappa_0(a)$ is its epicyclic frequency and the frequency $\eta_0(a)$ is defined below, all of which are functions of the particle’s semimajor axis $a$. Also keep in mind that when the following refers to the particle’s orbit elements, it is the [*epicyclic*]{} orbit elements that are intended[^1], which are distinct from the [*osculating*]{} orbit elements that describe pure Keplerian motion around a spherical planet. But these distinctions disappear in the limit that the planet becomes spherical and the orbit frequencies $\Omega_0, \kappa_0$, and $\eta_0$ all converge on the mean motion $\sqrt{Gm_0/a^3}$, where $G$ is the gravitational constant and $m_0$ is the central planet’s mass; in that case, Eqns. (\[rv\]) recover a Keplerian orbit to order ${\cal O}(e^2)$.
The three orbit frequencies $\Omega_0$, $\kappa_0$, and $\eta_0$ appearing in Eqns. (\[rv\]) are obtained from spatial derivatives of the oblate planet’s gravitational potential $\Phi$, which is $$\begin{aligned}
\Phi(r) &=& -\frac{Gm_0}{r} +
\frac{Gm_0}{r}\sum_{k=1}^{\infty}J_{2k}P_{2k}(0)
\left(\frac{R_p}{r}\right)^{2k}\end{aligned}$$ where $R_p$ is the planet’s effective radius, $J_{2k}$ is one of the oblate planet’s zonal harmonics, and $P_{2k}(0)$ is a Legendre polynomial with zero argument. For reasons that will be evident shortly, these calculations will only preserve the $J_{2}$ term in the above sum, so $$\begin{aligned}
\Phi(r) &=& -\frac{Gm_0}{r}\left[ 1 +
\frac{1}{2}J_2\left(\frac{R_p}{r}\right)^2\right]\end{aligned}$$ and the orbital frequencies are
\[orbit\_frequencies\] $$\begin{aligned}
\label{Omega^2}
\Omega_0^2(a) &=& \left.\frac{1}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a}
= \frac{Gm_0}{a^3}\left[1 + \frac{3}{2}J_2\left(\frac{R_p}{a}\right)^2 \right]\\
\label{kappa^2}
\kappa_0^2(a) &=& \left.\frac{3}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a}
+ \left.\frac{\partial^2\Phi}{\partial r^2}\right|_{r=a}
= \frac{Gm_0}{a^3}\left[1 - \frac{3}{2}J_2\left(\frac{R_p}{a}\right)^2 \right]\\
\eta_0^2(a) &=& \left.\frac{2}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a}
- \left.\frac{r}{6}\frac{\partial^3\Phi}{\partial r^3}\right|_{r=a}
= \frac{Gm_0}{a^3}\left[1 - 2J_2\left(\frac{R_p}{a}\right)^2 \right]\\
\beta_0^2(a) &=& - \left.\frac{r^4}{24}\frac{\partial^4\Phi}{\partial r^4}\right|_{r=a}
= \frac{Gm_0}{a^3}\left[1 + \frac{15}{2}J_2\left(\frac{R_p}{a}\right)^2 \right]
\end{aligned}$$
where the additional frequency $\beta_0(a)$ is needed below.
During the particle’s unperturbed epicyclic drift phase its angular orbit elements $M$ and $\tilde{\omega}$ advance during timestep $\Delta t$ by amount
\[dM\] $$\begin{aligned}
\Delta M &=& \kappa\Delta t\\
\Delta \tilde{\omega} &=& (\Omega - \kappa)\Delta t
\end{aligned}$$
where the frequencies $\Omega$ and $\kappa$ in Eqns. (\[dM\]) differ slightly from Eqns. (\[orbit\_frequencies\]) due to additional corrections that are of order ${\cal O}(e^2)$:
\[Omega\_kappa\] $$\begin{aligned}
\Omega(a,e) &=& \Omega_0\left\{1 +
3\left[\frac{1}{2} - \left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2 \right\}\\
\kappa(a,e) &=& \kappa_0\left(1 +
\left\{\frac{15}{4}\left[\left(\frac{\Omega_0}{\kappa_0}\right)^2
- \left(\frac{\eta_0}{\kappa_0}\right)^4\right] -
\frac{3}{2}\left(\frac{\beta_0}{\kappa_0}\right)^2\right\}e^2 \right)
\end{aligned}$$
[@BL94].
[@BL94] also show that the above equations have three integrals of the motion: the particle’s specific energy $E$, its specific angular momentum $h$, and its epicyclic energy $I_3$. Those integrals are
\[integrals\] $$\begin{aligned}
E &=& \frac{1}{2}(v_r^2 + v_\theta^2) + \Phi(r) = \frac{1}{2}(a\Omega_0)^2 + \Phi(a)
+ \frac{1}{2}(a\kappa_0)^2 e^2 + {\cal O}(e^4)\\
h &=& r v_\theta = a^2\Omega_0 + {\cal O}(e^4)\\
\label{I3}
\mbox{and}\quad I_3 &=& \frac{1}{2}[v_r^2 + \kappa_0^2(r-a)^2] - \eta_0^2(r-a)^3/a
= \frac{1}{2}(a\kappa_0e)^2 + {\cal O}(e^4).
\end{aligned}$$
Advancing the particle along its epicyclic orbit require converting its cylindrical coordinates and velocities into epicyclic orbit elements. To obtain the particle’s semimajor axis, solve the angular momentum integral $h(a)=a^2\Omega_0$, which is quadratic in $a$ so $$\begin{aligned}
\label{a}
a &=&g\left(1 + \sqrt{1 - \frac{3J_2}{2g^2}}\right) R_p\end{aligned}$$ where $g=(rv_\theta)^2/2Gm_0R_p$. Note though that if the $J_4$ and higher oblateness terms had been preserved in the planet’s potential, then the angular momentum polynomial would be of degree 4 and higher in $a$, for which there is no known analytic solution. That equation could still be solved numerically, but that step would have to be performed for all particles at every timestep, which would slow the N-body algorithm so much as to make it useless. So only the $J_2$ term is preserved here, which nonetheless accounts for the effects of planetary oblateness in a way that is sufficiently realistic.
To calculate the particle’s remaining orbit elements, use Eqn. (\[I3\]) to obtain the $I_3$ integral which then provides its eccentricity via $$\begin{aligned}
\label{e}
e &=& \frac{\sqrt{2I_3}}{a\kappa_0}.\end{aligned}$$ Then set $x=e\cos M$ and $y=e\cos M$ and solve Eqns. (\[r\]) and (\[v\_t\]) for $x$ and $y$:
\[xy\] $$\begin{aligned}
x &=& \left(\frac{\eta_0}{\kappa_0}\right)^2
\left[2(1+e^2) - \frac{v_\theta}{a\Omega_0} - \frac{r}{a}\right]
+ 1 - \frac{r}{a}\\
\mbox{and}\quad y &=& \frac{v_r/a\kappa_0}{1 + 2(\eta_0/\kappa_0)^2x},
\end{aligned}$$
which then provides the mean anomaly via $\tan M=y/x$.
To summarize, the epicyclic drift step uses Eqns. (\[integrals\]–\[xy\]) to convert each particle’s cylindrical coordinates into epicyclic orbit elements. The particles’ orbit frequencies $\Omega(a,e)$ and $\kappa(a,e)$ are obtained via Eqns. (\[orbit\_frequencies\]) and (\[Omega\_kappa\]), and Eqns. (\[dM\]) are then used to advance each particle’s orbit elements $M$ and $\tilde{\omega}$ during timestep $\Delta t$, with Eqns. (\[rv\]) used to convert the particles’ orbit elements back into cylindrical coordinates.
velocity kicks due to the ring’s internal forces {#kicks}
------------------------------------------------
The N-body code developed here is designed to follow the dynamical evolution of all $360^\circ$ of a narrow annulus within a planetary ring, and it is intended to deliver accurate results quickly using a desktop PC. The most time consuming part of this algorithm is the calculation of the accelerations that the gravitating ring exerts on all of its particles, so the principal goal here is to design an algorithm that will calculate these accelerations with sufficient accuracy while using the fewest possible number of simulated particles.
### streamlines
The dominant internal force in a dense planetary ring is its self gravity, and the representation of the ring’s full $360^\circ$ extent via a modest number of [*streamlines*]{} provides a practical way to calculate rapidly the acceleration that the entire ring exerts on any one particle. A streamline is the closed path through the ring that is traced by those particles that share a common initial semimajor axis $a$. The simulated portion of the planetary ring will be comprised of $N_r$ discreet streamlines that are spaced evenly in semimajor axis $a$, with each streamline comprised of $N_\theta$ particles on each streamline, so a model ring consists of $N_rN_\theta$ particles. Simulations typically employ $N_r\sim100$ streamlines with $N_\theta\sim50$ particles along each streamline, so a typical ring simulation uses about five thousand particles. Note though that the assignment of particles to a given streamline is merely labeling; particles are still free to wander over time in response to the ring’s internal forces: gravity, pressure, and viscosity. But as the following will show, the simulated ring stays coherent and highly organized throughout the run, in the sense that particles on the same streamline do not pass each other longitudinally, nor do adjacent streamlines cross. Because the simulated ring stays so highly organized, there is no radial or transverse mixing of the ring particles, and the particles will preserve over time membership in their streamline[^2].
### ring self gravity {#ring_gravity}
The concept of gravitating streamlines is widely used in analytic studies of ring dynamics [@GT79; @BGT83aj; @BGT86; @LR95; @HSP09], and the concept is easily implemented in an N-body code. Because the simulated portion of the ring is narrow, its streamlines are all close in the radial sense. Consequently the gravitational pull that one streamline exerts on a particle is dominated by the nearest part of the streamline, with that acceleration being quite insensitive to the fact that the more distant and unimportant parts of the perturbing streamline are curved. So the perturbing streamline can be regarded as a straight and infinitely long wire of matter whose linear density is $\lambda\simeq m_pN_\theta /2\pi a$ to lowest order in the streamline’s small eccentricity $e$, where $m_p$ is the mass of a single particle. The gravitational acceleration that a wire of matter exerts on the particle is $$\begin{aligned}
\label{Ag}
A_g &=& \frac{2G\lambda}{\Delta}\end{aligned}$$ where $\Delta$ is the separation between the particle and the streamline. However the particles in that streamline only provide $N_\theta$ discreet samplings of a streamline that is after all slightly curved over larger spatial scales. So to find the distance to nearest part of the perturbing streamline, the code identifies at every timestep the three perturbing particles that are nearest in longitude to the perturbed particle. A second-degree Lagrange polynomial is then used to fit a smooth continuous curve through those three particles [@KS], and this polynomial provides a convenient method for extrapolating the perturbing streamline’s distance $\Delta$ from the perturbed particle. This procedure is also illustrated in Fig. \[streamline\_fig\], which shows that the radial and tangential components of that acceleration are
\[Ag\_r\_theta\] $$\begin{aligned}
A_{g,r} &\simeq& A_g\\
\mbox{and}\qquad A_{g,\theta} &\simeq& -A_g v_r'/v_\theta'
\end{aligned}$$
to lowest order in the perturbing streamline’s eccentricity $e'$, where $v_r'$ and $v_\theta'$ are the radial and tangential velocity components of that streamline. Equation (\[Ag\_r\_theta\]) is then summed to obtain the gravitational acceleration that all other streamlines exerts on the particle.
To obtain the gravity that is exerted by the streamline that the particle inhabits, treat the particle as if it resides in a gap in that streamline that extends midway to the adjacent particles. The nearby portions of that streamline can be regarded as two straight and semi-infinite lines of matter pointed at the particle whose net gravitational acceleration is $$\begin{aligned}
\label{Ag_streamline}
A_g &=& 2G\lambda\left(\frac{1}{\Delta_+} - \frac{1}{\Delta_-}\right)\end{aligned}$$ where $\Delta_+$ and $\Delta_-$ are the particle’s distance from its neighbors in the leading (+) and trailing (-) directions. The radial and tangential components of that streamline’s gravity are
\[Ag\_r\_theta\_streamline\] $$\begin{aligned}
A_{g,r} &\simeq& A_g v_r/v_\theta\\
\mbox{and}\qquad A_{g,\theta} &\simeq& A_g
\end{aligned}$$
where $v_r, v_\theta$ are the perturbed particle’s velocity components.
A major benefit of using Eqn. (\[Ag\]) to calculate the ring’s gravitational acceleration is that there is no artificial gravitational stirring. This is in contrast to a traditional N-body model that would use discreet point masses to represent what is really a continuous ribbon of densely-packed ring matter. Those gravitating point masses then tug on each other in amounts that very rapidly in magnitude and direction as they drift past each other in longitude, and those rapidly varying tugs will quickly excite the simulated particles’ dispersion velocity. As a result, the particles’ unphysical random motions tend to wash out the ring’s large-scale coherent forced motions, which is usually the quantity that is of interest. So, although Eqn. (\[Ag\]) is only approximate because it does not account for the streamline’s curvature that occurs far away from a perturbed ring particle, Eqn. (\[Ag\]) is still much more realistic and accurate than the force law that would be employed in a traditional global N-body simulation of a planetary ring, which out of computational necessity would treat a continuous stream of ring matter as discreet clumps of overly massive gravitating particles.
### ring pressure {#pressure}
A planetary ring is very flat and its vertical structure will be unresolved in this model, so a 1D pressure $p$ is employed here. That pressure $p$ is the rate-per-length that a streamline segment communicates linear momentum to the adjacent streamline orbiting just exterior to it, with that momentum exchange being due to collisions occurring among particles on adjacent streamlines. So for a small streamline segment of length $\delta\ell$ that resides somewhere in the ring’s interior, the net force on that segment due to ring pressure is $\delta f = [p(r-\Delta) - p(r)]\delta\ell$ since $p(r-\Delta)$ is the pressure or force-per-length exerted by the streamline that lies just interior and a distance $\Delta$ away from segment $\delta\ell$, and $p(r)$ is the force-per-length that segment $\delta\ell$ exerts on the exterior streamline. And since force $\delta f = A_p \delta m$ where $\delta m=\lambda\delta\ell$ is the segment’s mass, the acceleration on a particle due to ring pressure is $$\begin{aligned}
\label{Ap}
A_p &=& \frac{\delta f}{\delta m} = \frac{p(r-\Delta) - p(r)}{\lambda}
\simeq-\frac{\Delta}{\lambda}\frac{\partial p}{\partial r}
=-\frac{1}{\sigma}\frac{\partial p}{\partial r}\end{aligned}$$ since the ring’s surface density $\sigma=\lambda/\Delta$.
Formulating the acceleration in terms of pressure differences across adjacent streamlines is handy because the model can then easily account for the large pressure drop that occurs at a planetary ring’s edge, which can be quite abrupt when the ring’s edge is sharp. For a particle orbiting at the ring’s innermost streamline, the acceleration there is simply $A_p=-p(r)/\lambda$ since there is no ring matter orbiting interior to it so $p(r-\Delta)=0$ there; likewise the acceleration of a particle in the ring’s outermost streamline is $A_p = p(r-\Delta)/\lambda$. Pressure is exerted perpendicular to the streamline and hence it is predominantly a radial force, so by the geometry of Fig. \[streamline\_fig\] the radial component of the acceleration due to pressure is $A_{p,r}\simeq A_p$ while the tangential component $A_{p,\theta}\simeq-A_pv_r/v_\theta$ is smaller by a factor of $e$, where $v_r$ and $v_\theta$ are the perturbed particle’s radial and tangential velocities. This accounts for the pressure on the particle due to adjacent streamlines.
The acceleration on the particle due to pressure gradients in the particle’s streamline is simply $A_p=-(\partial p/\partial\theta)/(r\sigma)$. This acceleration points in the direction of the particle’s motion, so the radial and tangential components of that acceleration are $A_{p,r}\simeq A_p v_r/v_\theta$ and $A_{p,\theta}\simeq A_p$.
Acceleration due to pressure requires selecting an equation of state (EOS) that relates the pressure $p$ to the ring’s other properties, and this study will treat the ring as a dilute gas of colliding particles for which the 1D pressure is $p=c^2\sigma$ where $c$ is the particles dispersion velocity. However alternate EOS exist for planetary rings, and that possibility is discussed in Section \[EOS\].
A simple finite difference scheme is used to calculate the pressure gradient in Eqn. (\[Ap\]) in the vicinity of particle $i$ in streamline $j$ that lies at at longitude $\theta_{i,j}$. Lagrange polynomials are again used to evaluate the adjacent streamlines’ planetocentric distances $r_{i, j-1}$ and $r_{i, j+1}$ along the particle’s longitude $\theta_{i,j}$, so the pressure gradient at particle $i$ in streamline $j$ is $$\begin{aligned}
\label{dp_dr}
\left.\frac{\partial p}{\partial r}\right|_{i,j} &\simeq&
\frac{p_{i,j+1} - p_{i, j-1}}{r_{i, j+1} - r_{i, j-1}}\end{aligned}$$ where the pressures in the adjacent streamlines $p_{i, j+1}$ and $p_{i, j-1}$ are also determined by interpolating those quantities to the perturbed particle’s longitude $\theta_{i,j}$.
The surface density $\sigma_{i,j}$ in the vicinity of particle $i$ in streamline $j$ is determined by centering a box about that particle whose radial extent spans half the distance to the neighboring streamlines, so $$\begin{aligned}
\label{sigma}
\sigma_{i,j} &=& \frac{2\lambda_j}{r_{i,j+1} - r_{i,j-1}}.\end{aligned}$$ If however streamline $j$ lies at the ring’s inner edge where $j=0$ then the surface density there is $\sigma_{i,0}=\lambda_0/(r_{i,1} - r_{i,0})$ while the surface density at the outermost $j=N_r -1$ streamline is $\sigma_{i, N_r - 1}=\lambda_{N_r - 1}/(r_{i, N_r - 1} - r_{i, N_r - 2})$.
### ring viscosity {#viscosity}
Viscosity has two types, shear viscosity and bulk viscosity. Shear viscosity is the friction that results as particles on adjacent streamlines collide as they flow past each other. The friction due to this shearing motion causes adjacent streamlines to torque each other, so shear viscosity communicates a radial flux of angular momentum through the ring. A particle on a streamline experiences a net torque and hence a tangential acceleration when there is a radial gradient in that angular momentum flux.
And if there are additional spatial gradients in the ring’s velocities that cause ring particles to converge towards or diverge away from each other, then these relative motions will cause ring particles to bump each other as they flow past, which transmits momentum through the ring via the pressure forces discussed above. However the ring particles’ viscous bulk friction tends to retard those relative motions, and that friction results in an additional flux of linear momentum through the ring. Any radial gradients in that linear momentum flux then results in a radial acceleration on a ring particle.
The 1D radial flux of the $z$ component of angular momentum due to the ring’s shear viscosity is derived in Appendix \[shear\_appendix\]: $$\begin{aligned}
\label{F_shear}
F &=& -\nu_s\sigma r^2\frac{\partial\dot{\theta}}{\partial r}\end{aligned}$$ (see Eqn. \[F\_app\]) where $\nu_s$ is the ring’s kinematic shear viscosity and $\dot{\theta}=v_\theta/r$ is the angular velocity. The quantity $F$ is the rate-per-length that one streamline segment communicates angular momentum to the adjacent streamline orbiting just exterior, so the net torque on a streamline segment of length $\delta\ell$ is $\delta\tau=[F(r-\Delta) - F(r)]\delta\ell$ but $\delta\tau=rA_{\nu,\theta}\delta m$ where $\delta m = \lambda\delta\ell$ so the tangential acceleration due to the ring’s shear viscosity is $$\begin{aligned}
\label{A_vs}
A_{\nu,\theta} &=& \frac{F(r-\Delta) - F(r)}{\lambda r}
=-\frac{1}{\sigma r}\frac{\partial F}{\partial r}.\end{aligned}$$ Again this differencing approach is useful because it easily accounts for the large viscous torque that occurs at a ring’s sharp edge since $A_{\nu,\theta}=- F(r)/\lambda r$ at the ring’s inner edge and $A_{\nu,\theta}=F(r - \Delta)/\lambda r$ at the ring’s outer edge.
Appendix \[bulk\_appendix\] shows that the radial flux of linear momentum due to the ring’s shear and bulk viscosity is $$\begin{aligned}
\label{G}
G &=& -\left(\frac{4}{3}\nu_s + \nu_b\right)\sigma\frac{\partial v_r}{\partial r}
- \left(\nu_b - \frac{2}{3}\nu_s\right)\frac{\sigma v_r}{r}\end{aligned}$$ (Eqn. \[G\_appendix2\]) where $\nu_b$ is the ring’s bulk viscosity. This quantity is analogous to a 1D pressure so the corresponding acceleration is $$\begin{aligned}
\label{A_vb}
A_{\nu,r} &=& \frac{G(r-\Delta) - G(r)}{\lambda}
=-\frac{1}{\sigma}\frac{\partial G}{\partial r}\end{aligned}$$ in the ring’s interior and $A_{\nu,r}=-G(r)/\lambda$ or $A_{\nu,r}=G(r-\Delta)/\lambda$ along the ring’s inner or outer edges.
To evaluate the partial derivatives that appear in the flux equations (\[F\_shear\]) and (\[G\]), Lagrange polynomials are again used to determine the angular and radial velocities $\dot{\theta}$ and $v_r$ in the adjacent streamlines, interpolated at the perturbed particle’s longitude, with finite differences used to calculate the radial gradients in those quantities.
### satellite gravity {#sat_gravity}
All ring particles are also subject to each satellite’s gravitational acceleration, $A_s=Gm_s/\Delta^2$, where $m_s$ is the satellite’s mass and $\Delta$ is the particle-satellite separation. Satellites also feel the gravity exerted by all the ring particles, as well as the satellites’ mutual gravitational attractions.
And once all of the accelerations of each ring particle and satellite are tallied, each body is then subject to the corresponding velocity kicks of steps ([*i.*]{}) and ([*v.*]{}) that are described just below Eqn. (\[eom\_soln\_approx\]).
tests of the code {#tests}
-----------------
The N-body integrator developed here is called [epi\_int]{}, which is shorthand for [*epicyclic integrator*]{}, and the following briefly describes the suite of simulations whose known outcomes are used to test all of the code’s key parts.
[**Forced motion at a Lindblad resonance:**]{} numerous massless particles are placed in circular orbits at Mimas’ $m=2$ inner Lindblad resonance. In this test, Mimas’ initially zero mass is slowly grown to its current mass over an exponential timescale $\tau_s=1.6\times10^4$ ring orbits, which excites adiabatically the ring particle’s forced eccentricities to levels that are in excellent agreement with the solution to the linearized equations of motion, Eqn. (42) of [@GT82]. Similar results are also obtained for the particle’s response to Janus’ $m=7$ inner Lindblad resonance, which is responsible for confining the outer edge of Saturn’s A ring. These simulations test the implementation of the integrator’s kick-step-drift scheme as well as the satellite’s forcing of the ring.
[**Precession due to oblateness:**]{} this simple test confirms that the orbits of massless particles in low eccentricity orbits precess at the expected rate, $\dot{\tilde{\omega}}(a)=\Omega - \kappa = \frac{3}{2}J_2(R_p/a)^2\Omega(a)$, due to planetary oblateness $J_2$.
[**Ringlet eccentricity gradient and libration:**]{} when a narrow eccentric ringlet is in orbit about an oblate planet, dynamical equilibrium requires the ringlet to have a certain eccentricity gradient so that differential precession due to self-gravity cancels that due to oblateness. And when the ringlet is composed of only two streamlines then this scenario is analytic, with the ringlet’s equilibrium eccentricity gradient given by Eqn. (28b) of [@BGT83]. So to test [epi\_int]{}’s treatment of ring self-gravity, we perform a suite of simulations of narrow eccentric ringlets that have surface densities $40<\sigma<1000$ gm/cm$^2$ with initial eccentricity gradients given by Eqn. (28b), and integrate over time to show that these pairs of streamlines do indeed precess in sync with no relative precession, as expected, over runtimes that exceed of the timescale for massless streamlines to precess differentially. And when we repeat these experiments with the ringlets displaced slightly from their equilibrium eccentricity gradients, we find that the simulated streamlines librate at the frequency given by Eqn. (30) of [@BGT83], as expected.
[**Density waves in a pressure-supported disk:**]{} this test examines the model’s treatment of disk pressure, and uses a satellite to launch a two-armed spiral density wave at its $m=2$ ILR in a non-gravitating pressure supported disk. The resulting pressure wave has a wavelength and amplitude that agrees with Eqn. (46) of [@W86], as expected.
[**Viscous spreading of a narrow ring:**]{} in this test [epi\_int]{} follows the radial evolution of an initially narrow ring as it spreads radially due to its viscosity, and the simulated ring’s surface density $\sigma(r,t)$ is in excellent agreement with the expected solution, Eqn. (2.13) of [@P81].
Simulations of the Outer Edge of Saturn’s B Ring {#B ring}
================================================
The semimajor axis of the outer edge of Saturn’s B ring is $a_{\mbox{\scriptsize edge}}=117568\pm4$ km, and that edge lies $\Delta a_2=12\pm4$ km exterior to Mimas’ $m=2$ inner Lindblad resonance (ILR) (@SP10, hereafter SP10). Evidently Mimas’ $m=2$ ILR is responsible for confining the B ring and preventing it from viscously diffusing outwards and into the Cassini Division. Mimas’ $m=2$ ILR excites a forced disturbance at the ring-edge whose radius–longitude relationship $r(\theta)$ is expected to have the form $r(\theta, t) = a_{\mbox{\scriptsize edge}} - R_m\cos m(\theta - \tilde{\omega}_m)$ where $R_m$ is the epicyclic amplitude of the mode whose azimuthal wavenumber is $m$ and whose orientation at time $t$ is given by the angle $\tilde{\omega}_m(t)$. This forced disturbance is expected to corotate with Mimas’ longitude, and such a pattern would have a pattern speed $\dot{\tilde{\omega}}_m=d\tilde{\omega}_m/dt$ that satisfies $\dot{\tilde{\omega}}_m=\Omega_s$ where $\Omega_s$ is satellite Mimas’ angular velocity.
SP10 have analyzed the many images of the B ring’s edge that have been collected by the Cassini spacecraft, and they show that this ring-edge does indeed have a forced $m=2$ shape that corotates with Mimas as expected. But they also show that the B ring’s edge has an additional [*free*]{} $m=2$ pattern that rotates slightly faster than the forced pattern. SP10 also detect two additional modes, a slowly rotating $m=1$ pattern as well as a rapidly rotating $m=3$ pattern. These findings are confirmed by stellar occulation observations of the B ring’s outer edge that also detect additional lower-amplitude $m=4$ and $m=5$ modes [@NFH12].
The following will use the N-body model to investigate the higher amplitude $m=1,2$, and 3 modes seen at the B ring’s edge. But keep in mind that only the $m=2$ forced pattern has a known driver, namely, Mimas’ $m=2$ ILR, while the nature of the perturbation that launched the other three free modes in the B ring is quite unknown. So to study the B ring’s behavior when those free modes are present, an admittedly ad hoc method is used. Specifically, the simulated ring particles’ initial conditions are constructed in a way that plants a free $m=1,2$, or 3 pattern at the simulated ring’s edge at time $t=0$. The N-body integrator then advances the system over time, which then reveals how those free patterns evolve over time. And to elucidate those findings most simply, the following subsections first consider the B ring’s $m=1$, 2, and 3 patterns in isolation.
All simulations use a timestep $\Delta t=0.2/2\pi=0.0318$ orbit periods, so there are 31.4 timesteps per orbit of the simulated B ring, and nearly all simulations use oblateness $J_2=0.01629071$, which is the same value we used in previous work [@HSP09].
And lastly, these simulations also zero the viscous acceleration that is exerted at the simulated B ring’s innermost and outermost streamlines, to prevent them from drifting radially due to the ring’s viscous torque. This is in fact appropriate for the simulation’s innermost streamline, since in reality the viscous torque from the unmodeled part of the B ring should deliver to the inner streamline a constant angular momentum flux $F$ that it then communicates to the adjacent streamline, so the viscous acceleration $A_{\nu,\theta}\propto \partial F/\partial r$ at the simulation’s inner edge really should be zero. But zeroing the viscous acceleration of outer streamline might seem like a slight-of-hand since it should be $A_{\nu,\theta}=F/\lambda r$ according Section \[viscosity\]. But setting $A_{\nu,\theta}=0$ is done because, if not, then the outermost streamline will slowly but steadily drifts radially outwards past Mimas’ $m=2$ ILR, which also causes that streamline’s forced eccentricity to slowly and steadily grow as the streamline migrates. This happens because the model does not settle into a balance where the ring’s positive viscous torque on its outermost streamline is opposed by a negative torque exerted by the satellite’s gravity. We also note that the semi-analytic model of this resonant ring-edge, which is described in [@HSP09], also had the same difficulty in finding a torque balance. So to sidestep this difficulty, this model zeros the viscous acceleration at the outermost streamline, which keeps its semimajor axis static as if it were in the expected torque balance. This then allows us to compare simulations to the B ring’s forced $m=2$ pattern to that measured by the Cassini spacecraft. The validity of this approximation is also assessed below in Section \[force\].
the forced and free patterns {#m=2}
----------------------------
SP10 detect a forced $m=2$ pattern at the B ring’s outer edge that has an epicyclic amplitude $R_2=34.6\pm0.4$ km, and that forced pattern corotates with the satellite Mimas. They also detect a free pattern whose epicyclic amplitude is $2.7$ km larger, so the forced and free patterns are nearly equal in amplitude, and the free pattern rotates slightly faster than the forced pattern by $\Delta\dot{\tilde{\omega}}_2=0.0896\pm0.0007$ degrees/day (SP10). The radius-longitude relationship for a ring-edge that experiences these two modes can be written $$\begin{aligned}
\label{dr_2}
r(\theta, t) &=& a - R_2\cos m(\theta - \theta_s) -
\tilde{R}_2\cos m(\theta - \tilde{\omega}_2)\end{aligned}$$ where $R_2$ is the epicyclic amplitude of the forced pattern that corotates with Mimas whose longitude is $\theta_s(t)$ at time $t$, and $\tilde{R}_2$ is the epicyclic amplitude of the free pattern with $\tilde{\omega}_2(t)$ being the free pattern’s longitude.
The N-body integrator [epi\_int]{} is used to simulate the forced and free $m=2$ patterns that are seen at the outer edge of the B ring, for simulated rings having a variety of initial surface densities $\sigma_0$. These simulations use $N_r=130$ streamlines that are distributed uniformly in the radial direction with spacings $\Delta a=5.13$ km, so the radial width of the simulated portion of the B ring is $w=(N_r - 1)\Delta a=662$ km. Each streamline is populated with $N_\theta=50$ particles that are initially distributed uniformly in longitude $\theta$ and in circular coplanar orbits. These simulations use a total of $N_rN_\theta=6500$ particles, which is more than sufficient to resolve the $m=2$ patterns seen here. These systems are evolved for $t=41.5$ years, which corresponds to $3.2\times10^4$ orbits, and is sufficient time to see the simulation’s slightly faster free $m=2$ pattern lap the forced $m=2$ pattern several times. The execution time for these high resolution, publication-quality simulations is 1.5 days on a desktop PC, but sufficiently useful preliminary results from lower-resolution simulations can be obtained in just a few hours.
The B ring’s viscosity is unknown, so these simulations will employ a value for the kinematic shear viscosity $\nu_s$ and bulk viscosity $\nu_b$ that are typical of Saturn’s A ring, $\nu_s=\nu_b=100$ cm$^2$/sec [@TBN07]. The simulated particles’ dispersion velocity $c$ is also chosen so that the ring’s gravitational stability parameter $Q=c\kappa/\pi G\sigma_0=2$, since Saturn’s main rings likely have $1\lesssim Q\lesssim2$ [@S95]. Mimas’ mass is $m_s=6.5994\times10^{-8}$ Saturn masses, and its semimajor axis $a_s$ is chosen so that its $m=2$ inner Lindblad resonance lies $\Delta a_{\mbox{\scriptsize res}}=12.2$ km interior to the simulated B ring’s outer edge. This model only accounts for the $J_2=0.01629071$ part of Saturn’s oblateness, so the constraint on the resonance location puts the simulated Mimas at $a_s=185577.0$ km, which is 38 km exterior to its real position.
Starting the ring particles in circular orbits provides an easy way to plant equal-amplitude free and forced $m=2$ patterns in the ring. This creates a free $m=2$ pattern that at time $t=0$ nulls perfectly the forced $m=2$ pattern due to Mimas. However the free pattern rotates slightly faster than the forced pattern, so the ring’s epicyclic amplitude varies between near zero and $\sim2R_2$ as the rotating patterns interfere constructively or destructively over time. This behavior is illustrated in Fig. \[m=2\_fig\] which shows results from a simulation of a B ring whose undisturbed surface density is $\sigma_0=280$ gm/cm$^2$. The wire diagrams show the ring’s streamlines via radius versus longitude plots, with dots indicating individual particles, and the adjacent grayscale map shows the ring’s surface density at that instant. Figure \[m=2\_fig\] shows snapshots of the system at five distinct times that span one cycle of the ring’s circulation: at time $t=26.4$ yr when the ring’s outermost streamline is nearly circular due to the forced and free patterns being out of phase by nearly $180^\circ/m=90^\circ$ and interfering destructively, to time $t=28.2$ yr when the forced and free patterns are in phase and interfere constructively, to nearly circular again at time $t=30.0$ yr.
The circulation cycle seen in Fig. \[m=2\_fig\] repeats for the duration of the integration, which spans about 10 cycles. The gray lines in Fig. \[m=2\_fig\] show the semimajor axes $a$ of all particles on each streamline; note that all particles on a given streamline preserve a common semimajor axes, and this is also true of their eccentricities $e$. In the simulations shown here, the two orbit elements $a$ and $e$ do not vary with the particle’s longitude $\theta$. This however is distinct from the particles’ angular orbit elements $M$ and $\tilde{\omega}$, which do vary linearly with longitude $\theta$ along each streamline. Recall that the [epi\_int]{} code does not in any way force or require particles to inhabit a given streamline. The streamline concept is only used when calculating the forces that all of the ring’s streamlines exert on each particle, which the symplectic integrator then uses to advance these particles forwards in time. Although a particle’s $e$ and $a$ are in principle free to drift away from that of the other streamline-members, that does not happen in the simulations shown here; evidently the particles’ $a$ and $e$ evolve slowly in the orbit-averaged sense, with that time-averaged evolution being independent of longitude $\theta$. This accounts for why all particles on the same streamline have the same evolution in $a$ and $e$. This time-averaged evolution is also a standard assumption that is routinely invoked in analytic models of planetary rings (see [*cf*]{}. @GT79 [@BGT86; @HSP09]), and the simulations shown here confirm the validity of that assumption.
A suite of seven B ring simulations is performed for rings whose undisturbed surface densities range over $120\le\sigma_0\le360$ gm/cm$^2$. Results are summarized in Fig. \[R2\_fig\] which shows the forced epicyclic amplitude $R_2$ (solid curve) and the free epicyclic amplitude $\tilde{R}_2$ (dashed curve) from each simulation. These amplitudes are obtained by fitting Eqn. (\[dr\_2\]) to the simulated B ring’s outermost streamline assuming that the free pattern there rotates at a constant rate, $\tilde{\omega}_2(t) = \tilde{\omega}_0 + \dot{\tilde{\omega}}_2t$ where $\tilde{\omega}_0$ is the free pattern’s angular offset at time $t=0$ and $\dot{\tilde{\omega}}_2$ is the free mode’s pattern speed. Equation (\[dr\_2\]) provides an excellent representation of the ring-edge’s behavior over time, and that equation has four parameters $R_2, \tilde{R}_2, \tilde{\omega}_0$, and $\dot{\tilde{\omega}}_2$ that are determined by least squares fitting. The observed epicyclic amplitude of the B ring’s forced $m=2$ component is $R_2=34.6\pm0.4$ km (SP10), and the gray bar in Fig. \[R2\_fig\] indicates that the outer edge of the B ring has a surface density of about $\sigma_0=195$ gm/cm$^2$. And if we naively assume that the ring’s surface density is everywhere the same, then its total mass of Saturn’s B ring is about $90\%$ of Mimas’ mass.
Figure \[R2\_fig\] also shows that the amplitude of the forced pattern $R_2$ gets larger for rings that have a smaller surface density $\sigma_0$, due to the ring’s lower inertia, with the forced response varying roughly as $R_2\propto\sigma_0^{-0.67}$. This also makes lighter rings more difficult to simulate, because their larger epicyclic amplitudes also causes the ring’s streamlines to get more bunched up at periapse. For instance in the $\sigma_0=280$ gm/cm$^2$ simulation of Fig. \[m=2\_fig\], the ring’s edge at longitudes $\theta=\theta_s$ and $\theta=\theta_s\pm\pi$ are overdense by a factor of 3 at time $t=28.2$ yr, which is when the force and free patterns add constructively. Streamline bunching in lighter rings is even more extreme, which is also more problematic, because streamlines that are too compressed can at times cross in these overdense sites, and the simulated ring’s subsequent evolution becomes unreliable.
To avoid the streamline crossing that occurs in simulations of lower surface density, the model also grows the mass of Mimas exponentially over the timescale $\tau_s$ that takes values of $0.41\le \tau_s\le6.2$ years, with faster satellite growth ($\tau_s=0.41$ yrs or 320 B ring orbits) occurring in simulations of a heavy B ring having $\sigma_0\ge280$ gm/cm$^2$ and slower growth ($\tau_s=6.2$ yrs or 4800 B ring orbits) for the lighter $\sigma_0\le240$ gm/cm$^2$ ring simulations. The satellite growth timescale $\tau_s$ controls the amplitude of the free pattern $\tilde{R}_2$, with the ring having a smaller free epicyclic amplitude $\tilde{R}_2$ when $\tau_s$ is larger; see the dashed curve in Fig. \[R2\_fig\]. Indeed, when the satellite grows over a timescale $\tau_s\gg6.2$ yrs ([*i.e.*]{} $\tau_s\gg4800$ orbits), the ring responds adiabatically to forcing by the slowly growing Mimas, and shows only a forced $m=2$ pattern that corotates with Mimas, with the free $m=2$ pattern having a negligible amplitude. Consequently, only the $\sigma_0=280, 320$, and $360$ gm/cm$^2$ simulations in Fig. \[R2\_fig\] are faithful in their attempt to reproduce a B ring whose free epicyclic amplitude $\tilde{R}_2$ is slightly larger than the forced amplitude $R_2$. However the lower-surface density simulations have free patterns whose amplitudes are smaller than the forced patterns, and these simulated rings have outer edges whose longitude of periapse librate about Mimas’ longitude, rather than circulate.
Also of interest here is the so-called radial depth of the $m=2$ disturbance, $\Delta a_{e/10}$, which is defined as the semimajor axis separation between the ring’s outer edge and the streamline whose mean eccentricity is one-tenth that of the edge’s eccentricity. For these $m=2$ simulations the radial depth is $\Delta a_{e/10}=154$km, so the radial width of the simulated part of the ring is $w=4.3\Delta a_{e/10}$.
### sensitivity to resonance location and other factors {#edge_sensitivity}
The surface density $\sigma_0$ that is inferred from the amplitude of the ring’s forced motion $R_2$ is very sensitive to the uncertainty in the ring’s semimajor axis, which is $\delta a_{\mbox{\scriptsize edge}}$. For example, when the B ring is simulated again but with its outer edge instead extending further out by $\delta a_{\mbox{\scriptsize edge}}=4$ km, those simulations show that the ring’s forced amplitude $R_2$ is larger by about 6 km, which requires increasing $\sigma_0$ by $\delta\sigma_0=60$ gm/cm$^2$ so that the simulated $R_2$ is in agreement with the observed value. Similarly, when the simulated ring’s edge is moved inwards by $\delta a_{\mbox{\scriptsize edge}}=4$ km, the forced amplitude $R_2$ is smaller and the ring’s surface density $\sigma_0$ must be decreased by $\delta\sigma_0$ to compensate. So the surface density of the B ring-edge is $\sigma_0=195\pm60$ gm/cm$^2$, and this value represents the mean surface density of outer $\Delta a_{e/10}\simeq150$km that is most strongly disturbed by Mimas’ $m=2$ resonance. These results are also in excellent agreement with the semi-analytic model of [@HSP09], which calculated only the ring’s forced motion.
However these results are very insensitive to the model’s other main unknown, the ring’s viscosity $\nu$. For instance, when we re-run the $\sigma_0=200$ gm/cm$^2$ simulation with the ring’s shear and bulk viscosities increases as well as decreased by a factor of 10, we obtain the same forced response $R_2$. So these findings are insensitive to range of ring viscosities considered here, $10<\nu<1000$ cm$^2$/sec.
### free $m=2$ pattern {#free m=2}
The dotted curve in Fig. \[m2\_fig\] shows the simulations’ free $m=2$ pattern speeds $\dot{\tilde{\omega}}_2$, which is also sensitive to the ring’s undisturbed surface density $\sigma_0$. The purpose of this subsection is to illustrate how a free normal mode can also be used to determine the ring’s surface density. Although these result will not be as definitive as the value of $\sigma_0$ that was inferred from the ring’s forced pattern, due to a greater sensitivity to the observational uncertainties, the following illustrates an alternate technique that in principle can be used to infer the surface density of other rings, such as the many narrow ringlets orbiting Saturn that also exhibit free normal modes.
But first note the models’ large discrepancy with the observed free $m=2$ pattern speed reported in SP10, which is the upper horizontal bar in Fig. \[m2\_fig\]. This discrepancy is [*not*]{} due to the $\delta a_{\mbox{\scriptsize edge}}=\pm4$km uncertainty in the ring-edge’s semimajor axis, which makes the simulated ring particles’ mean angular velocity uncertain by the fraction $\delta\Omega/\Omega=1.5\delta a_{\mbox{\scriptsize edge}}/a_{\mbox{\scriptsize edge}}
\simeq 0.005\%$. We find empirically that the simulations’ pattern speeds are also uncertain by this fraction, so $\delta\dot{\tilde{\omega}}_2\simeq0.02$ deg/day, which is the vertical extent of the gray band around the simulated data in Fig. \[m2\_fig\].
Rather, this discrepancy is indirectly due to the absence of the $J_4$ and higher terms from the N-body simulations. To demonstrate this, repeat the $\sigma_0=200$ gm/cm$^2$ simulation with $J_2$ boosted slightly by factor $f^\star=1.0395013$ so that the second zonal harmonic is $J_2^\star=f^\star J_2=0.016934294$. This increases the simulated B ring-edge’s angular velocity slightly to $\Omega_{\mbox{\scriptsize edge}}=758.8824$ deg/day, which is in fact the ring’s true angular velocity at $a=a_{\mbox{\scriptsize edge}}$ when the higher order $J_4$ and $J_6$ terms are also accounted for[^3]. And since Saturn’s gravitational force there is $a_{\mbox{\scriptsize edge}}\Omega_{\mbox{\scriptsize edge}}^2$, this means that Saturn’s gravity on the simulated particles at $r=a_{\mbox{\scriptsize edge}}$ is in fact the true value. Note that boosting $J_2$ to the slightly larger value $J_2^\star$ also requires bringing the simulated Mimas inwards and just interior to its true semimajor axis by 2km. Which speeds up both the forced and free pattern speeds, and is why this simulation’s free $m=2$ pattern speed $\dot{\tilde{\omega}}_2$, which is the cross in Fig. \[m2\_fig\], is in better agreement with the observed pattern speed. So the discrepancy between all the other simulated and observed pattern speeds $\dot{\tilde{\omega}}_2$ is due to those models’ not accounting for the additional gravity that is due to the $J_4$ and higher terms in Saturn’s oblate figure. Compensating for the absence of those oblateness effects requires altering the simulated satellite’s orbits slightly, which in turn alters the forced and free pattern speeds slightly. But the following will show that these two patterns’ [*relative*]{} speeds are quite insensitive to the particular value of $J_2$ and the absence of the $J_4$ and higher terms.
The best way to compare simulated to observed free $m=2$ patterns is to consider the free $m=2$ pattern speed relative to the forced pattern speed, which is the satellite’s mean angular velocity $\Omega_{\mbox{\scriptsize sat}}$. That frequency difference is $\Delta\dot{\tilde{\omega}}_2=\dot{\tilde{\omega}}_2-\Omega_{\mbox{\scriptsize sat}}$, and is plotted versus ring surface density $\sigma_0$ in Fig. \[m2\_rel\_fig\]. Black dots are from the simulation and the light gray band indicates the $\delta\dot{\tilde{\omega}}_2\simeq0.02$ deg/day spread that results from the $\delta a_{\mbox{\scriptsize edge}}=\pm4$ km uncertainty in the ring-edge’s semimajor axis. The relatively large uncertainty in $a_{\mbox{\scriptsize edge}}$ means that one can only conclude from Fig. \[m2\_rel\_fig\] that $\sigma_0\lesssim210$ gm/cm$^2$. If however the uncertainty in $a_{\mbox{\scriptsize edge}}$ were instead $\delta a_{\mbox{\scriptsize edge}}=\pm1$ km, then the uncertainty in $\Delta\dot{\tilde{\omega}}_2$ would be 4 times smaller (darker gray band), which would have allowed us to determine the ring surface density with a much smaller uncertainty of only $\pm20$ gm/cm$^2$. The lesson here is that if one wishes to use models of free patterns to infer $\sigma_0$ in, say, narrow ringlets, then one will likely need to know the ring-edge’s semimajor axis with a precision of $\delta a_{\mbox{\scriptsize edge}}\simeq\pm1$ km.
The cross in Fig. \[m2\_rel\_fig\] indicates that the the free $m=2$ pattern speed relative to the forced is unchanged when Saturn’s oblateness is boosted to $J_2^\star$. And to demonstrate that this kind of plot is rather insensitive to oblateness effects, the white dot in Fig. \[m2\_rel\_fig\] shows that these relative pattern speeds change only very slightly even when $J_2$ is set to zero.
Note though that there will be instances where there is no forced mode with which to compare pattern speeds. In that case it will be convenient to convert the free pattern speed $\dot{\tilde{\omega}}_m=\Omega_{ps}$ into a radius by solving the Lindblad resonance criterion $$\begin{aligned}
\label{resonance_eqn}
\kappa(r) &= \epsilon m[\Omega(r) - \Omega_{ps}]\end{aligned}$$ for the resonance radius $r=a_m$, where $\kappa(r)$ is the ring particles’ epicyclic frequency (Eqn. \[kappa\^2\]), and $\epsilon=+1 (-1)$ at an inner (outer) Lindblad resonance. So for the simulated B ring’s free $m=2$ mode, Eqn. (\[resonance\_eqn\]) is solved for the radius $r=\tilde{a}_2$ of the $\epsilon=+1$ inner Lindblad resonance that is associated with this mode. That quantity is to be compared to a nearby reference distance, which in this case would be the semimajor axis of the B ring’s outer edge $a_{\mbox{\scriptsize edge}}$. Results are shown in Fig. \[da2\_fig\], which shows the simulations’ distance from the B ring’s outer edge to the free $m=2$ pattern’s ILR , $\Delta a_2 = a_{\mbox{\scriptsize edge}} - \tilde{a}_2$, plotted versus ring surface density $\sigma_0$. Heavier rings have a faster pattern speeds (Fig. \[m2\_fig\]–\[m2\_rel\_fig\]), and so the pattern’s resonance resides at a higher orbital frequency $\Omega(r)$ and thus must lie further inwards of the ring’s outer edge in order to satisfy the resonance condition, Eqn. (\[resonance\_eqn\]). Figure \[da2\_fig\] has the same information content as Fig. \[m2\_rel\_fig\], which is why it also tells us that the B ring’s outer edge has $\sigma_0\lesssim210$ gm/cm$^2$. However a plot like Fig. \[da2\_fig\] will also provide the best way to interpret the B ring’s free $m=3$ mode, which is examined below in subsection \[m=3\].
Lastly, note that the free $m=2$ patterns seen in these simulations persist for $3\times10^4$ orbits or 40 years without any sign of damping, despite the ring’s viscosity $\nu=100$ cm$^2$/sec. This is illustrated in Fig. \[R\_epi\_fig\], which plots the ring-edge’s epicyclic amplitude over time for the nominal $\sigma_0=200$ gm/cm$^2$ simulation. Indeed we have also rerun this simulation using a viscosity that is ten times larger and still saw no damping. These experiments reveal a possibly surprising result, that a free pattern can persist at a ring-edge for a considerable length of time, likely hundreds of years or longer, and Section \[force\] will show that this longevity is due to the viscous forces being several orders or magnitude weaker than the ring’s other interval forces. So one possible interpretation of the free modes seen at the B ring and at other ring edges is that they are relics from past disturbances in Saturn’s ring that may have happened hundreds or more years ago. This possibility is discussed further in Section \[impulse\].
the free pattern {#m=3}
----------------
The B ring’s free $m=3$ mode has an epicyclic amplitude of $\tilde{R}_3=11.8\pm0.2$ km, a pattern speed $\dot{\tilde{\omega}}_3=507.700\pm0.001$ deg/day, and the inner Lindblad resonance associated with this pattern speed lies $\Delta a_3=24\pm4$ km interior to the B ring’s outer edge (SP10).
To excite a free $m=3$ pattern at the ring-edge, place a fictitious satellite in an orbit that has an $m=3$ inner Lindblad resonance $\Delta a_3=24$ km interior to the ring’s outer edge. Noting that the satellite Janus happens to have an $m=3$ resonance in the vicinity, about 2000 km inwards of the B ring’s edge, these simulations use a Janus-mass satellite to perturb the simulated ring for about $1650$ orbits (about 2 years), which excites an $m=3$ pattern at the ring’s outer edge. The satellite is then removed from the system, which converts the pattern into a free normal mode, and [epi\_int]{} is then used to evolve the now unperturbed ring for another $1.8\times10^4$ orbits (about 23 years). Figure \[m3\_fig\] plots the ring-edge’s epicyclic amplitude, where it is shown that the free mode persists at the B ring’s outer edge, undamped over time, despite the simulated ring’s viscosity of $\nu=100$ cm$^2$/sec.
A suite of such B ring simulations is performed, with ring surface densities $120\le\sigma_0\le360$ gm/cm$^2$ and all other parameters identical to the nominal model of Section \[m=2\] except where noted in Fig. \[da3\_fig\] caption. The pattern speed $\Omega_{ps}=\dot{\tilde{\omega}}_3$ of the $m=3$ normal mode is then extracted from each simulation, with those speeds again being slightly faster in the heavier rings. Those pattern speeds are then inserted into Eqn. (\[resonance\_eqn\]) which is solved for the radius of the inner Lindblad resonance $\tilde{a}_3$, each of which lies a distance $\Delta a_3 = a_{\mbox{\scriptsize edge}} - \tilde{a}_3$ inwards of the ring’s outer edge, and those distances are plotted in Fig. \[da3\_fig\] versus ring surface density $\sigma_0$. The simulated distances $\Delta a_3$ are compared to the observed edge-resonance distance reported in SP10, which indicates a ring surface density $160\le\sigma_0\le310$ gm/cm$^2$. This finding is consistent with the the results from the $m=2$ patterns, but this constraint on $\sigma_0$ is again rather loose due to the $\delta a_{\mbox{\scriptsize edge}}=\pm4$ km uncertainty in the ring-edge’s semimajor axis. But our purpose here is to show how one might use models of free normal modes to infer the surface density of other rings and narrow ringlets, which again will likely require knowing the ring-edge’s semimajor axis to $\pm1$ km or better.
Also note that the radial depth of this $m=3$ disturbance is $\Delta a_{e/10}=50$ km, about three times smaller than the radial depth of the $m=2$ disturbance.
the free pattern {#m=1}
----------------
The B ring’s free $m=1$ mode has an epicyclic amplitude of $\tilde{R}_1=20.9\pm0.4$ km and a pattern speed $\dot{\tilde{\omega}}_1=5.098\pm0.003$ deg/day that is slightly faster than the local precession rate, and the inner Lindblad resonance that is associated with this pattern speed lies $\Delta a_1=253\pm4$ km interior to the B ring’s outer edge (SP10). Several simulations of the B ring’s $m=1$ pattern are evolved for model rings having surface densities of $120\le\sigma_0\le360$ gm/cm$^2$. To excite the $m=1$ pattern at the simulated ring’s edge, again arrange a fictitious satellite’s orbit so that its $m=1$ ILR lies $\Delta a_1=253$ km interior to the B ring’s edge, which is the site where the resonance condition (Eqn. \[resonance\_eqn\]) is satisfied when the satellite’s mean angular velocity matches the ring particles’ precession rate, $\Omega_s = \dot{\tilde{\omega}}=\Omega_{ps}$. The simulated ring is perturbed by a satellite whose mass is about 20% that of Mimas, for $1.6\times10^4$ orbits or 21 years, which excites a forced $m=1$ pattern at the ring’s edge that corotates with the satellite. The satellite is then removed, which converts the forced $m=1$ pattern into a free pattern, and the ring is evolved for another $6.4\times10^4$ orbits or 83 years. For each simulation the free pattern speed is measured, and Eqn. (\[resonance\_eqn\]) is then used to convert the free pattern speed into a resonance radius $\tilde{a}_1$, which is displayed in Fig. \[da1\_fig\] that shows that resonance’s distance from the ring’s outer edge, $\Delta a_1=a_{\mbox{\scriptsize edge}} - \tilde{a}_1$. As the figure shows, the free $m=1$ pattern rotates slightly faster in the heavier ring and thus the associated $m=1$ ILR must lie further inwards in order to satisfy the resonance condition $\Omega_{ps}=\dot{\tilde{\omega}}=\frac{3}{2}J_2(R_p/a)^2\Omega$. Again there is no damping of the free $m=1$ pattern, which stays localized at the ring’s outer edge over the simulation’s 83 yr timespan, despite the simulated ring’s viscosity $\nu=100$ cm$^2$/sec.
The radial depth of this $m=1$ disturbance is much greater than the others, $\Delta a_{e/10}=614$ km, which is about four times larger than the $m=2$ disturbance. Comparing Fig. \[da1\_fig\] to Figs. \[da2\_fig\] and \[da3\_fig\] also shows that the LR associated with the $m=1$ disturbance lies about 10 times further from the ring-edge than the $m=1$ and $m=2$ resonances. Which is why the $m=1$ simulation uses streamlines whose width $\Delta a$ is $\sim10\times$ larger, since a wider portion of the B ring-edge must be simulated in order to capture this disturbances’ deeper reach into the B ring.
Note also that the $\pm4$ km uncertainty in this resonance’s position relative to the B ring edge, which is entirely due to the uncertainty in the B ring-edge’s semimajor axis, is in this case relatively small. Which is why the ring’s free $m=1$ mode can also be used to probe its surface density with some precision (unlike the free $m=2$ and $m=3$ modes), and is consistent with a B ring surface density of $\sigma_0\simeq200$ gm/cm$^2$,
convergence tests {#convergence}
-----------------
A number of simulations have also been performed, which repeat the ring simulations using various particle numbers $N_r$ and $N_\theta$ and various widths $w$ of the simulated ring. We find that the results reported here do not change significantly when the simulated ring is populated densely with enough particles, and when the radial width of the simulated B ring is sufficiently wide. Those convergence tests reveal that the number of particles along each streamline must satisfy $N_\theta\ge 20m$, that the radial width of each streamline should satisfy $\Delta a\le 0.04\Delta a_{e/10}$, and that the total width of the simulated ring should satisfy $w>4\Delta a_{e/10}$. All of the simulations reported here satisfy these requirements.
Discussion
==========
This section re-examines the model’s treatment of viscous effects at the ring’s edge, and also describes related topics that will be considered in followup work.
the ring’s internal forces {#force}
--------------------------
Figure \[forces\_fig\] plots the accelerations that the ring’s internal forces—gravity, pressure, and viscosity—exert on each ring particle. These accelerations are from the nominal $\sigma_0=200$ gm/cm$^2$ simulation that is described in Figs. \[R2\_fig\]–\[R\_epi\_fig\], and these accelerations are plotted versus each particle’s distance from the ring’s edge, so those forces obviously get larger closer to the ring’s disturbed outer edge. But the main point of Fig. \[forces\_fig\] is that the ring’s self gravity is the dominant internal force in the ring, exceeding the pressure force by a factor of $\sim100$ at the ring’s outer edge and by a larger factor elsewhere. Those pressure forces are also about $\sim10\times$ larger than the ring’s viscous forces. But recall that those simulations had zeroed the viscous acceleration that the ring exerts on its outermost streamline (Section \[B ring\]), when that acceleration should instead be $A_{\nu,\theta}=F/\lambda r$ as indicated by the large blue dot at the right edge of Fig. \[forces\_fig\]. Note though that the neglected viscous acceleration of the ring’s edge is still about $\sim1000\times$ smaller than that due to ring gravity and $\sim10\times$ smaller than that due to ring pressure. So this justifies neglecting, at least for the short-term $t\sim100$ yr simulations considered here, the much smaller viscous forces at the ring’s outer edge.
Nonetheless this study’s neglect of the small viscous force at the ring’s outer edge implies that this model does not yet account for the B ring’s radial confinement by Mimas’ $m=2$ ILR. So there appears to be some missing physics that will be necessary if one is interested in the ring’s resonant confinement or the ring’s long-term evolution over $t\gg100$ yr timescales. The suspected missing physics is described below.
### unmodeled effects: the viscous heating of a resonantly confined ring-edge {#viscous heating}
The model’s inability to confine the B ring’s outer edge at Mimas’ $m=2$ ILR may be a consequence of the ring’s kinematic viscosity $\nu$ being treated here as a constant parameter everywhere in the simulated ring. Although treating $\nu$ as a constant is a simple and plausible way to model the effects of the ring’s viscous friction, it might not be adequate or accurate if one wishes to simulate the resonant confinement of a planetary ring. This is because the ring’s viscosity transports both energy and angular momentum radially outwards through the ring. So if the ring’s outer edge is to be confined by a satellite’s $m^{\mbox{\scriptsize th}}$ Lindblad resonance, the satellite must absorb the ring’s outward angular momentum flux, which it can do by exerting a negative gravitational torque at the ring’s edge. But [@BGT82] show via a simple Jacobi-integral argument that resonant interactions only allow the satellite to absorb but a fraction of the energy that viscosity delivers to the ring-edge. Consequently the ring’s viscous friction still delivers some orbital energy to the ring-edge where it accumulates and heats up the ring particles’ random velocities $c$. And if collisions among particles are the main source of the ring’s viscosity, then $\nu_s\simeq c^2\tau/2\Omega(1+\tau^2)$ where $\tau\propto\sigma$ is the ring’s optical depth [@GT82]. In this case viscous heating would increase $c$ as well as $\nu_s$ at the ring’s edge. The enhanced dissipation there should also increase the angular lag $\phi$ between the ring-edge’s forced pattern and the satellite’s longitude (see Eqn. 83b of @HSP09). Which will also be important because the gravitational torque that the satellite exerts on the ring-edge varies as $\sin\phi$ [@HSP09], and that torque needs to be boosted if the satellite is to confine the spreading ring.
To model this phenomenon properly, the [epi\_int]{} code also needs to employ an energy equation, one that accounts for how viscous heating tends to increase the ring particles’ dispersion velocity $c$ and viscosity $\nu_s$ nearer the ring’s edge. The increased dissipation and the resulting orbital lag will allow the satellite to exert a greater torque on the ring which, we suspect, will enable the satellite to resonantly confine the simulated ring’s outer edge. The derivation of this energy equation and its implementation in [epi\_int]{} are ongoing, and those results will be reported on in a followup study.
an alternate equation of state {#EOS}
------------------------------
The EOS adopted here is appropriate for a dilute gas of colliding ring particles whose mutual separations greatly exceed their sizes. This should be regarded as a limiting case since ring particles can of course be packed close to each other in the ring. But [@BGT85] consider the other extreme limiting case, with close-packed particles that reside shoulder to shoulder in the ring. In that case the ring is expected to behave as an incompressible fluid whose volume density $\rho=\sigma/2h$ stays constant. So when some perturbation causes ring streamlines to bunch up and increases the ring’s surface density $\sigma$, the ring’s vertical scale height $h$ also increases as ring particles are forced to accumulate along the vertical direction. This in turn increases the ring’s pressure due to the larger gravitational force along the vertical direction.
[@BGT85] show that infinitesimal density waves in an incompressible disk are unstable and grow in amplitude over time. This phenomenon is related to the viscous overstability, and [@LR95] show that it can distort a narrow eccentric ringlet’s streamlines in a way that accounts for its $m=1$ shapes. [@BGT85] also suggest that unstable density waves can be trapped between a Lindblad resonance and the B ring’s outer edge, which might explain the normal modes seen there, and [@SP10] use this concept to estimate the ring’s surface density there.
But keep in mind that this instability only occurs when the ring particles are densely packed to the point of being incompressible, which requires the ring to be very thin and dynamically cold. We have shown here that the amplitude of the B ring’s forced motions indicates that the ring-edge has a surface density $\sigma\simeq200$ gm/cm$^2$. So if this ring is incompressible and composed of icy spheres having a mean volume density of $\rho=\sigma/2h\simeq0.5$ gm/cm$^3$, this then requires a B ring thickness of only $h\sim2$ meters, which is rather thin compared to other estimates [@CBC10]. Similarly the ring particles’ dispersion velocity $c$ must be small compared to that expected for a dilute particle gas, so $c\ll (h\Omega\sim0.3$ mm/sec), which again is cold compared to all other estimates for Saturn’s rings [@CBC10]. The upshot is that an incompressible EOS requires the ring to be very thin and dynamically cold, likely much colder and thinner than is generally thought. Consequently we are optimistic that the compressible EOS used here, $p=\sigma c^2$, is the appropriate choice for simulations of the outer edge of Saturn’s B ring. Nonetheless in a followup investigation we do intend to encode the incompressible EOS into [epi\_int]{}, to see if the BGT instability can account for the higher $m\ge2$ free modes that are seen at the outer edge of the B ring and in many other narrow ringlets.
impulse origin for normal modes {#impulse}
-------------------------------
The simulations of Section \[B ring\] used a fictitious temporary satellite to excite the free modes that occur at many Saturnian ring edges. These simulations used an admittedly ad hoc method—the sudden appearance and disappearance of a satellite—to excite these modes. Nonetheless these models demonstrate that transient and impulsive events can excite normal modes at ring edges, and those simulations show that normal modes can persist at the ring’s edge for hundreds of years after the disturbance has occurred. Which suggests that an impulsive event in the recent past, perhaps an impact into Saturn’s rings, might be responsible for exciting the normal modes that are seen at the outer edge of the B ring, as well as the normal modes that are also seen along the edges of several narrower ringlets [@FMR10; @HNB10; @FNC11; @NFH12]
The possibility that normal modes are due to an impact is motivated by the discovery of vertical corrugations in Saturn’s C and D rings [@HBS07; @HBE11] and in Jupiter’s main dust ring [@SHB11]. These vertical structures are spirals that span a large swath of each ring, and they are observed to wind up over time due to the central planet’s oblateness. Evolving the vertical corrugations backwards in time also unwinds their spiral pattern until some moment when the affected region is a single tilted plane. Unwinding the Jovian corrugation shows that that disturbance occurred very close to the date when the tidally disrupted comet Shoemaker-Levy 9 impacted Jupiter in 1994, which suggests an impact from a tidally disrupted comet as the origin of these ring-tilts [@SHB11]. However a single sub-km comet fragment cannot tilt a large $\sim2\times10^5$ km-wide planetary ring. But a disrupted comet can produce an extended cloud of dust, and if that disrupted dust cloud returns to the planet with enough mass and momentum, then it might tilt a ring that at a later date would be observed as a spiral corrugation.
However the tidal disruption of comet about a low-density planet like Saturn is more problematic, because tidal disruption only occurs when the comet’s orbit is truly close to parabolic and not too hyperbolic, and with periapse just above the planet’s atmosphere [@ST92; @RBL98].
But it is easy to envision an alternate scenario that might be more likely, with a small km-sized comet originally in a heliocentric orbit coming close enough to Saturn to instead strike the main A or B rings. This scenario is more probable because the cross-section available to orbits impacting the main rings is significantly larger than those resulting in tidal disruption. The impacting comet’s considerably greater momentum will nonetheless carry the impactor through the dense A or B rings, but the collision itself is likely energetic enough to shatter the comet. And if that collision is sufficiently dissipative, then the resulting cometary debris will then stay bound to Saturn, and in an orbit that will return that debris back into the ring system on its next orbit. Small differences among the orbits of individual debris particles’ means that, when the debris encounters the rings again, that impacting debris will be spread across a much larger footprint on the ring, which presumably will allow any dense rings or ringlets to absorb the debris’ mass and momentum in a way that effectively gives the ring particles there a sudden velocity kick $\mathbf{\Delta v}$ in proportion to the comet debris density $\rho$ and velocity $\mathbf{v}_r$ relative to the ring matter. But if comet Shoemake-Levy 9’s (SL9) impact with Jupiter is any guide, then impact by a cloud of comet debris could last as long week of time, which might tend to smear this effect out due to the ring’s orbital motion. But that effect would be offset if the debris train’s dust cloud is also rather clumpy, like the SL9 debris train was. Indeed, it is possible that this scenario might also account for the spiral corrugations of Saturn’s C and D ring. It is also conceivable that an inclined cloud of impacting comet debris might also excite the vertical analog of normal modes—long-lived vertical oscillations of a ring’s edge. This admittedly speculative scenario will be pursued in a followup study, to determine whether debris from an impact-disrupted comet can excite the normal modes seen at ring edges, and to determine the mass of the progenitor comet that would be needed to account for these modes’ observed amplitudes.
Summary of results {#summary}
==================
We have developed a new N-body integrator that calculates the global evolution of a self-gravitating planetary ring as it orbits an oblate planet. The code is called [epi\_int]{}, and it uses the same kick-drift-step algorithm as is used in other symplectic integrators such as [SYMBA]{} and [MERCURY]{}. However the velocity kicks that are due to ring gravity are computed via an alternate method that assumes that all particles inhabit a discreet number of streamlines in the ring. The use of streamlines to calculate ring self gravity has been used in analytic studies of rings [@GT79; @BGT83aj; @BGT86], and the streamline concept is easily implemented in an N-body code. A streamline is the closed path through the ring that is traced by particles having a common semimajor axis. All streamlines are radially close to each other, so the gravitation acceleration due to a streamline is simply that due to a long wire, $A=2G\lambda/\Delta$ where $\lambda$ is the streamline’s linear density and $\Delta$ is the particle’s distance from the streamline. Which is very useful since particles are responding to the pull of smooth wires rather than discreet clumps of ring matter so there is no gravitational scattering. Which means that only a modest number of particles are needed, typically a few thousand, to simulate all $360^\circ$ of a scalloped ring like the outer edges of Saturn’s A and B ring. Only a few thousand particles are also needed to simulate linear as well as nonlinear spiral density waves, and execution times are just a few hours on a desktop PC.
Another distinction occurs during the particles’ unperturbed drift step when particles follow the epicyclic orbit of [@LR95] about an oblate planet, rather than the usual Keplerian orbit about a spherical planet. This effectively moves the perturbation due to the planet’s oblate figure out of the integrator’s kick step and into the drift step. The code also employs hydrodynamic pressure and viscosity to account for the transport of linear and angular momentum through the ring that arises from collisions among ring particles. Another convenience of the streamline formulation is that it easily accounts for the large pressure drop that occurs at a ring’s sharp edge, as well as the large viscous torque that the ring exerts there. The model also accounts for the mutual gravitational perturbations that the ring and the satellites exert on each other. The [epi\_int]{} code is written in IDL, and the source code is available for download at [http://gemelli.spacescience.org/‘hahnjm/software.html](http://gemelli.spacescience.org/~hahnjm/software.html).
This integrator is used to simulate the forced response that the satellite Mimas excites at its $m=2$ inner Lindblad resonance (ILR) that lies near the outer edge of Saturn’s B ring. That resonance lies $\Delta a_2=12 \pm4$ km inwards of the ring’s edge, and simulations show that the ring’s forced epicyclic amplitude varies with the ring’s surface density $\sigma_0$ as $R_2\propto\sigma_0^{0.67}$. Good agreement with Cassini measurements of $R_2$ occurs when the simulated ring has a surface density of $\sigma_0=195\pm 60$ gm/cm$^2$ (see Fig. \[R2\_fig\]), where the uncertainty in $\sigma_0$ is dominated by the $\delta a_{\mbox{\scriptsize edge}}=4$ km uncertainty that [@SP10] find in the ring-edge’s semimajor axis. This $\sigma_0$ is the mean surface density over that part of the B ring that is disturbed by this resonance, whose influence in the ring extends to a radial distance of $\Delta a_{e/10}\sim150$ km from the B ring’s outer edge. And if we naively assume that this surface density is the same everywhere across Saturn’s B ring, then its total mass is about $90\%$ of Mimas’ mass.
Cassini observations reveal that the outer edge of Saturn’s B ring also has several free normal modes that are not excited by any known satellite resonances. Although the mechanism that excites these free modes is uncertain, we are nonetheless able to excite free modes in a simulated ring via various ad-hoc methods. For instance, a fictitious satellite’s $m^{\mbox{\scriptsize th}}$ Lindblad resonance is used to excite a forced pattern at the ring edge. Removing that satellite then converts the forced patten into a free normal mode that persists in these simulations for up to $\sim100$ years or $\sim10^5$ orbits without any damping, despite the simulated ring having a kinematic viscosity of $\nu=100$ cm$^2$/sec; see Fig. \[m3\_fig\] for one example.
Alternatively, starting the ring particles in circular orbits while subject to Mimas’ $m=2$ gravitational perturbation excites both a forced and a free $m=2$ pattern that initially null each other precisely at the start of the simulation. But the forced patten corotates with Mimas’ longitude while the free pattern rotates slightly faster in a heavier ring, which suggests that a free mode’s pattern speed can also be used to infer a ring’s surface density $\sigma_0$. However the free pattern speed is also influenced by the $J_4$ and higher terms in the oblate planet’s gravity field, which are absent from this model which only accounts for the $J_2$ component. So the simulated pattern speed cannot be compared directly to the observed pattern speed; see Fig. \[m2\_fig\]. To avoid this difficulty, the resonance condition (Eqn. \[resonance\_eqn\]) is used to calculate the radius of the Lindblad resonance that is associated with the free normal mode. Plotting the distances of the simulated and observed resonances from the B ring’s edge (Figs. \[da2\_fig\], \[da3\_fig\], and \[da1\_fig\]) then provides a convenient way to compare simulations to observations of free modes in a way that is insensitive to the planet’s oblateness.
Simulations of the B ring’s free $m=2$ and $m=3$ patterns are consistent with Cassini measurements of the B ring’s normal modes when the simulated ring-edge again has a surface density of $\sigma_0\sim200$ gm/cm$^2$, which is a nice consistency check. But these particular measurements do not provide tight constraint on the ring’s $\sigma_0$, due to the fact that the $m=2$ and $m=3$ Lindblad resonances only lie $\Delta a_m\sim25$ km from the outer edge of a ring whose semimajor axis $a$ is uncertain by $\delta a_{\mbox{\scriptsize edge}}=4$ km. However the B ring’s free $m=1$ normal mode does lie much deeper in the ring’s interior, $\Delta a_1=253\pm4$, so the uncertainly in its location is fractionally much smaller, and this normal mode does confirm the $\sigma_0\simeq200$ gm/cm$^2$ value that was inferred from simulations of the B ring’s forced response $R_2$.
One of the goals of this study is to determine whether simulations of free modes can be used to determine the surface density and mass of a narrow ringlet. Such ringlets show a broad spectrum of free normal models over $0\le m\le 5$ [@FMR10; @HNB10; @FNC11; @NFH12], and the answer appears to be yes since free pattern speeds do increase with $\sigma_0$. However Section \[free m=2\] shows that the semimajor axes of the ringlet’s edges likely need to be known to a precision of $\delta a_{\mbox{\scriptsize edge}}\sim1$ km in order for a free mode to provide a useful measurement of the ringlet’s $\sigma_0$.
The origin of these free modes, which are quite common along the edges of Saturn’s broad rings and its many narrow ringlets, is uncertain. [@BGT85] show that, if a planetary ring’s particles are packed shoulder to shoulder such that the ring behaves like an incompressible fluid, then that ring is unstable to the growth of density waves, a phenomenon also termed viscous overstability, and they suggest that the B ring’s normal modes might be due to unstable waves that are trapped between a Lindblad resonance and the ring’s edge. To study this further, we will in a followup study adapt [epi\_int]{} to employ an incompressible equation of state, to see if the viscous overstability can in fact account for the free normal modes seen along the Saturnian ring edges.
Although the current version of [epi\_int]{} does not account for the origin of these free modes, one can still plant a free mode along the edge of a simulated ring by temporarily perturbing a ring at a fictitious satellite’s Lindblad resonance, and then removing that satellite, which creates an unforced mode that persists undamped at the ring-edge for more than $\sim10^5$ orbits or $\sim100$ yrs despite the simulated ring having a kinematic viscosity of $\nu=100$ cm$^2$/sec. Because this forcing is suddenly turned on and off, this suggests that any sudden or impulsive disturbance of the ring can excite normal modes, with those disturbances possibly persisting for hundreds or maybe thousands of years. And in Section \[impulse\] we suggest that the Saturnian normal modes might be excited by an impact with a collisionally disrupted cloud of comet dust. This is a slight variation of the scenario that [@HBS07] and [@SHB11] propose for the origin of corrugated planetary rings, and in a followup investigation we intend to determine whether such impacts can also account for the normal modes seen in Saturn’s rings.
And lastly, we find that [epi\_int]{}’s treatment of ring viscosity has difficulty accounting for the radial confinement of the B ring’s outer edge by Mimas’ $m=2$ inner Lindblad resonance. This model employs a kinematic shear viscosity $\nu_s$ that is everywhere a constant, which causes the simulation’s outermost streamline to slowly but steadily drift radially outwards. Which in turn causes the ring’s forced epicyclic amplitude $R_2$ to slowly grow over time, and makes difficult any comparison to Cassini’s measurement of $R_2$. To sidestep this difficulty, the model zeros the torque that the simulated ring exerts on its outermost streamline, which does allow the ring to settle into a static configuration that can be compared to Cassini observations and yields a measurement of the ring’s surface density $\sigma_0$. This approximate treatment is also examined in in Section \[force\], which shows that the viscous acceleration of the ring-edge, had it been included in the simulation, is still orders of magnitude smaller than that due to ring self gravity. So this study of the dynamics of the B ring’s forced and free modes is not adversely impacted by this approximate treatment. But this does mean that the B ring’s radial confinement is still an unsolved problem, and Section \[viscous heating\] suggests that this might be a consequence of treating $\nu_s$ as a constant. [@BGT82] show that viscosity’s outward transport of energy should also heat the ring’s outer edge and increase the ring particles’ dispersion velocity $c$ there. And if collisions among ring particles are the dominant source of ring viscosity, then $\nu_s\propto c^2$ and viscous dissipation would be enhanced at the ring edge, which in turn would increase the angular lag between the ring’s forced response and the Mimas’ longitude. That then would increase the gravitational torque that that satellite exerts on the ring-edge. So in a followup study we will modify [epi\_int]{} to address this problem in a fully self-consistent way, to see if enhanced dissipation at the ring-edge also increases Mimas’ gravitational torque there sufficiently to prevent the B ring’s outer edge from flowing viscously beyond that satellite’s $m=2$ inner Lindblad resonance.
[**Acknowledgments**]{}
J. Hahn’s contribution to this work was supported by grant NNX09AU24G issued by NASA’s Science Mission Directorate via its Outer Planets Research Program. The authors thank Denise Edgington of the University of Texas’ Center for Space Research (CSR) for composing Fig. \[streamline\_fig\], and J. Hahn thanks Byron Tapley for graciously providing office space and the use of the facilities at CSR. The authors are also grateful for the helpful suggestions provided by an anonymous reviewer.
Appendix \[shear\_appendix\] {#shear_appendix}
============================
The following calculates the flux of angular momentum that is communicated via a disk’s viscosity. The disk is flat and thin and has a vertical halfwidth $h$ and constant volume density $\rho$ that is related to its surface density $\sigma$ via $\rho=\sigma/2h$. The disk is assumed viscous, and its gravity is ignored here since this Appendix is only interested in the angular momentum flux that is transported solely by viscosity.
The density of angular momentum in the disk is $\bm{\ell} = \mathbf{r}\times\rho\mathbf{v}$, and the vertical component along the $z=x_3$ axis is $\ell_3 = x_1\rho v_2 - x_2\rho v_1$ in Cartesian coordinates $x=x_1$ and $y=x_2$ where $\rho$ and $v_i$ are functions of position and time, so the time rate of change of $\ell_3$ is $$\begin{aligned}
\label{dl3/dt}
\frac{\partial\ell_3}{\partial t} &= x_1\displaystyle\frac{\partial}{\partial t}(\rho v_2)
- x_2\displaystyle\frac{\partial}{\partial t}(\rho v_1).\end{aligned}$$ The time derivatives in the above are Euler’s equation, $$\begin{aligned}
\label{EEqn}
\frac{\partial}{\partial t}(\rho v_i) &=-\displaystyle\sum_{k=1}^3
\frac{\partial\Pi_{ik}}{\partial x_k}\end{aligned}$$ where the $\Pi_{ik}$ are the elements of the momentum flux density tensor $$\begin{aligned}
\label{Pi}
\Pi_{ik} &= \rho v_iv_k + \delta_{ik}p - \sigma'_{ik}\end{aligned}$$ where $p$ is the pressure and $\sigma'_{ik}$ are the elements of the viscous stress tensor [@LL87]. Inserting Eqn. (\[Pi\]) into (\[dl3/dt\]) yields $$\begin{aligned}
\label{dl3/dt_v2}
\frac{\partial\ell_3}{\partial t} &= -x_1\nabla\cdot\bm{\Pi}_2 + x_2\nabla\cdot\bm{\Pi}_1\end{aligned}$$ where the vector $$\begin{aligned}
\label{Pi_vector}
\bm{\Pi}_i=\displaystyle\sum_{k=1}^3\Pi_{ik}\bm{\hat{x}}_k\end{aligned}$$ is the flux density of the $i$ component of linear momentum and $\bm{\hat{x}}_k$ is the unit vector along the $x_k$ axis. Equation (\[dl3/dt\_v2\]) can be rewritten $$\begin{aligned}
\label{dl3/dt_v3}
\frac{\partial\ell_3}{\partial t} &= -\nabla\cdot(x_1\bm{\Pi}_2 - x_2\bm{\Pi}_1)
+\bm{\Pi}_2\cdot\nabla x_1 - \bm{\Pi}_1\cdot\nabla x_2\end{aligned}$$ but note that $\bm{\Pi}_1\cdot\nabla x_2 - \bm{\Pi}_2\cdot\nabla x_1=\Pi_{21}-\Pi_{12}=
\sigma'_{12} - \sigma'_{21}=0$ since the viscous stress tensor is symmetric (Eqn. \[sigma’\]), so $$\begin{aligned}
\label{dl3/dt_v4}
\frac{\partial\ell_3}{\partial t} &= -\nabla\cdot \bm{F}_3\end{aligned}$$ where $$\begin{aligned}
\label{F3}
\bm{F}_3 &= x_1\bm{\Pi}_2 - x_2\bm{\Pi}_1.\end{aligned}$$ Integrating Eqn. (\[dl3/dt\_v4\]) over some volume $V$ that is bounded by area $A$ yields $$\begin{aligned}
\label{F3_flux}
\frac{\partial}{\partial t}\int_V \ell_3 dV &= -\int_V\nabla\cdot \bm{F}_3 dV =
-\int_A \bm{F}_3\cdot \bm{dA}\end{aligned}$$ by the divergence theorem, so Eqn. (\[F3\_flux\]) indicates that $\bm{F}_3$ is the flux of the $x_3$ component of angular momentum out of volume $V$ that is being transported by advection, pressure, and viscous effects.
This Appendix is interested in the part of $\bm{F}_3$ that is due to viscous effects, which will be identified as $\bm{F}'_3$ and is obtained by replacing $\Pi_{ik}$ in Eqn. (\[Pi\]) with $-\sigma'_{ik}$ so $$\begin{aligned}
\label{F3'}
\bm{F}'_3 &= (x_2\sigma'_{11} - x_1\sigma'_{21})\bm{\hat{x}}_1
+ (x_2\sigma'_{12} - x_1\sigma'_{22})\bm{\hat{x}}_2.\end{aligned}$$ This is the 2D flux of the $x_3$ component of angular momentum that is transported by the disk’s viscosity whose horizontal components in Cartesian coordinates are $\bm{F}'_3=F'_1\bm{\hat{x}}_1 + F'_2\bm{\hat{x}}_2$ where $F'_1=x_2\sigma'_{11} - x_1\sigma'_{21}$ and $F'_2=x_2\sigma'_{12} - x_1\sigma'_{22}$. However this Appendix desires the radial component of $\bm{F}'_3$ are some site $r,\theta$ in the disk, which is $F'_r=F'_1\cos\theta + F'_2\sin\theta$.
The elements of the viscous stress tensor are [@LL87] $$\begin{aligned}
\label{sigma'}
\sigma'_{ik} &= \eta\left(\displaystyle\frac{\partial v_i}{\partial x_k} +
\frac{\partial v_k}{\partial x_i}\right)
+ (\zeta - \frac{2}{3}\eta)\delta_{ik}\nabla\cdot\bm{v}\end{aligned}$$ where $\eta$ is the shear viscosity, $\zeta$ is the bulk viscosity, and $\delta_{ik}$ is the Kronecker delta. Inserting this into $F'_r$ and replacing $x_1=r\cos\theta$ and $x_2=r\sin\theta$ then yields $$\begin{aligned}
\label{F'_r}
F'_r &=-\eta r\left(\displaystyle\frac{\partial v_1}{\partial x_2}
+ \frac{\partial v_2}{\partial x_1}\right)\cos2\theta +
\eta r\left(\displaystyle\frac{\partial v_1}{\partial x_1}
- \frac{\partial v_2}{\partial x_2}\right)\sin2\theta.\end{aligned}$$ The horizontal velocities are $v_1=v_r\cos\theta - v_\theta\sin\theta $ and $v_2 = v_r\sin\theta + v_\theta\cos\theta $ when written in terms of their radial component $v_r$ and tangential component $v_\theta=r\dot{\theta}$. The derivatives in Eqn. (\[F’\_r\]) are $$\begin{aligned}
\label{dv/dx}
\begin{split}
\frac{\partial v_1}{\partial x_1} &=
\left(\cos\theta\frac{\partial}{\partial r}
-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)v_1\\
&= \cos^2\theta\frac{\partial v_r}{\partial r}
- \sin\theta\cos\theta r\frac{\partial\dot{\theta}}{\partial r}
+ \frac{\sin^2\theta}{r}v_r
- \frac{\sin\theta\cos\theta}{r}\frac{\partial v_r}{\partial\theta}
+ \frac{\sin^2\theta}{r}\frac{\partial v_\theta}{\partial\theta}\\
\frac{\partial v_2}{\partial x_2} &=
\left(\sin\theta\frac{\partial}{\partial r}
+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)v_2\\
&= \sin^2\theta\frac{\partial v_r}{\partial r}
+ \sin\theta\cos\theta r\frac{\partial\dot{\theta}}{\partial r}
+ \frac{\cos^2\theta}{r}v_r
+ \frac{\sin\theta\cos\theta}{r}\frac{\partial v_r}{\partial\theta}
+ \frac{\cos^2\theta}{r}\frac{\partial v_\theta}{\partial\theta}\\
\frac{\partial v_1}{\partial x_2} &=
\left(\sin\theta\frac{\partial}{\partial r}
+\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)v_1\\
\frac{\partial v_2}{\partial x_1} &=
\left(\cos\theta\frac{\partial}{\partial r}
-\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)v_2
\end{split}\end{aligned}$$ when written in terms of cylindrical coordinates, and the combinations of derivatives in Eqn. (\[F’\_r\]) are
\[dv/dx +- dv/dx\] $$\begin{aligned}
\label{dv1/dx2 + dv2/dx1}
\frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1}
&=& \left(\frac{\partial v_r}{\partial r}
-\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}
- \frac{v_r}{r}\right)\sin2\theta
+ \left(\frac{\partial v_\theta}{\partial r}
+\frac{1}{r}\frac{\partial v_r}{\partial \theta}
- \frac{v_\theta}{r}\right)\cos2\theta\\
\label{dv1/dx1 - dv2/dx2}
\frac{\partial v_1}{\partial x_1} - \frac{\partial v_2}{\partial x_2}
&=& \left(\frac{\partial v_r}{\partial r}
-\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}
- \frac{v_r}{r}\right)\cos2\theta
- \left(\frac{\partial v_\theta}{\partial r}
+\frac{1}{r}\frac{\partial v_r}{\partial \theta}
- \frac{v_\theta}{r}\right)\sin2\theta,
\end{aligned}$$
Inserting these into Eqn. (\[F’\_r\]) then yields a result that is thankfully much more compact, $$\begin{aligned}
\label{F'_r2}
F'_r &=-\eta\left( \displaystyle r^2\frac{\partial\dot{\theta}}{\partial r} +
\frac{\partial v_r}{\partial \theta}\right)
\simeq -\eta r^2\displaystyle \frac{\partial\dot{\theta}}{\partial r},\end{aligned}$$ noting that the second term in Eqn. (\[F’\_r2\]) may be neglected since the azimuthal gradient is much smaller than the radial gradient for the disks considered here. This is the radial component of the disk’s 2D viscous angular momentum flux density, so the 1D viscous angular momentum flux density is Eqn. (\[F’\_r2\]) integrated through the disk’s vertical cross section: $$\begin{aligned}
\label{F_app}
F &= \int_{-h}^{h} F'_rdx_3=
-\nu_s\sigma r^2\displaystyle \frac{\partial\dot{\theta}}{\partial r}\end{aligned}$$ where $\nu_s=\eta/\rho$ is the disk’s kinematic shear viscosity.
Appendix \[bulk\_appendix\] {#bulk_appendix}
===========================
The flux density of $x_1$-type momentum is $\bm{\Pi}_1$ (see Eqn. \[Pi\_vector\]) while the flux density of $x_2$-type momentum is $\bm{\Pi}_2$, so the flux density of radial momentum is $\bm{G} = \cos\theta\bm{\Pi}_1 + \sin\theta\bm{\Pi}_2$ and the radial component of this momentum flux density is $$\begin{aligned}
\label{G_appendix}
G_r &= \bm{G\cdot\hat{r}} = (\cos\theta\Pi_{11} + \sin\theta\Pi_{21})
\bm{\hat{x_1}\cdot\hat{r}}
+ (\cos\theta\Pi_{12} + \sin\theta\Pi_{22})\bm{\hat{x_2}\cdot\hat{r}}\\
&= \cos^2\theta\Pi_{11} + \sin\theta\cos\theta(\Pi_{12} + \Pi_{21}) + \sin^2\theta\Pi_{22}\end{aligned}$$ where $\bm{\hat{r}}$ is the unit vector in the radial direction. The part of that momentum flux that is transported solely by viscous effects will be called $G_r'$ and is again obtained by replacing the $\Pi_{ik}$ in the above with $-\sigma'_{ik}$: $$\begin{aligned}
\label{G'_r}
G_r'&=& -\cos^2\theta\sigma_{11} - \sin\theta\cos\theta(\sigma'_{12} + \sigma'_{21})
- \sin^2\theta\sigma'_{22}\\
&=& - 2\eta\left[ \displaystyle\cos^2\theta\frac{\partial v_1}{\partial x_1}
+ \sin^2\theta\frac{\partial v_2}{\partial x_2}
+ \sin\theta\cos\theta \left(
\frac{\partial v_1}{\partial x_2}
+ \frac{\partial v_2}{\partial x_1}\right)\right]
- (\zeta - \frac{2}{3}\eta)\bm{\nabla\cdot v}.\end{aligned}$$
Equations (\[dv/dx\]) provide the combination $$\begin{aligned}
\begin{split}
\cos^2\theta\frac{\partial v_1}{\partial x_1}
+ \sin^2\theta\frac{\partial v_2}{\partial x_2} &=
\left(\frac{3}{4} + \frac{1}{4}\cos4\theta\right)\frac{\partial v_r}{\partial r}
-\frac{1}{4}\sin4\theta r\frac{\partial \dot{\theta}}{\partial r}
+\frac{1}{2r}\sin^22\theta v_r
-\frac{1}{4r}\sin4\theta \frac{\partial v_r}{\partial \theta}\\
&+\frac{1}{2r}\sin^22\theta \frac{\partial v_\theta}{\partial \theta},
\end{split}\end{aligned}$$ and inserting this plus Eqn. (\[dv1/dx2 + dv2/dx1\]) into Eqn. (\[G’\_r\]) then yields $$\begin{aligned}
\label{G'_r2}
G_r' &= -\displaystyle\left(\frac{4}{3}\eta +\zeta\right)\frac{\partial v_r}{\partial r}
-\left(\zeta-\frac{2}{3}\eta\right)
\left(\frac{v_r}{r} + \frac{1}{r}\frac{\partial v_\theta}{\partial \theta}\right)\end{aligned}$$ but the $\partial v_\theta/\partial \theta$ term is again neglected in the streamline approximation. This is the 2D radial momentum flux due to viscous transport, so the vertically integrated linear momentum flux due to viscosity is $$\begin{aligned}
\label{G_appendix2}
G &=& \int_{-h}^{h} G'_rdx_3=
-\displaystyle\left(\frac{4}{3}\nu_s +\nu_b\right)\sigma\frac{\partial v_r}{\partial r}
-\left(\nu_b-\frac{2}{3}\nu_s\right)\frac{\sigma v_r}{r}.\end{aligned}$$
[^1]: Actually what we identify here as the semimajor axis $a$ is called $r_0$ in [@BL94], which differs slightly from what they identify as the epicyclic semimajor axis $a_e$ where $a_e=r_0(1 + e^2)$.
[^2]: But if the simulated ring is instead initialized with all particles on a given streamline having distinct (rather than common) values for $a$ and $e$, then the resulting streamlines can appear ragged in longitude $\theta$. And if that initial ring is sufficiently ragged or non-smooth, then that raggedness can grow over time as the particles $a$’s and $e$’s evolve independently. The main point is that the streamline model employed here succeeds when all streamlines are sufficiently smooth, and that is accomplished by initializing all particles in a given streamline with commmon $a,e$.
[^3]: This mean angular velocity is obtained using the physical constants given in the 25 August 2011 Cassini SPICE kernel file: $Gm_0=37940585.47323534$ km$^3$/sec$^2$, $J_2=0.016290787119$, $J_4=-0.000934741301$, and $J_6=0.000089240275$.
| ArXiv |
---
abstract: |
The reason why smart home remains not popularized lies in bad product user experience, purchasing cost, and compatibility, and a lack of industry standard[@avgerinakis2013recognition]. Echoing problems above, and having relentless devoted to software and hardware innovation and practice, we have independently developed a set of solution which is based on innovation and integration of router technology, mobile Internet technology, Internet of things technology, communication technology, digital-to-analog conversion and codec technology, and P2P technology among others. We have also established relevant protocols (without the application of protocols abroad). By doing this, we managed to establish a system with low and moderate price, superior performance, all-inclusive functions, easy installation, convenient portability, real-time reliability, security encryption, and the capability to manage home furnitures in an intelligent way. Only a new smart home system like this can inject new idea and energy into smart home industry and thus vigorously promote the establishment of smart home industry standard.\
author:
-
bibliography:
- 'GG.bib'
title: Promote the Industry Standard of Smart Home in China by Intelligent Router Technology
---
Smart home, router technology, industry standard
Introduction
============
Since this year, the waves of smart home are on its rise. This October, a company that sells intelligence temperature controllers NEST of Google acquired Revolv, a smart home central control equipment start-up. Xiaomi released its smart plug, smart camera, among four smart end new products. Enterprises at home and abroad one after another plunge into the big cake of smart home. Smart home is opening new vista and space for Internet and household appliance industry. All View Consulting forecasts that by 2020, the ecological product of domestic smart home appliance in China will reach one trillion yuan. SAIF Partners predict that the scale of smart home industry by traditional definition in China will reach 5.5 billion yuan in 2014, and that number will soar to 7.5 billion yuan in 2015. However, three stumbling blocks are in the advancing way of smart home industry: user experience, purchasing cost, and lousy compatibility[@albuquerque2014solution]. In response, we have independently developed a set of solution based on smart router technology. There are many problems existing in routers in the market. Firstly, wireless control based on 315M, 433M and other frequency ranges has no network protocol and can only send simple control command. Collision occurs when there are over three connected devices, which renders the process more difficult to succeed. Secondly, control network based on ZigBee registers small range, poor through-the-wall performance, complex protocol, inordinate price, and at the same time is exclusive and incompatible to devices existing in the market. The third one is control network based on WI-FI. WI-FI boasts a small control range and thus is limited to only a few connected devices. Normally when household router is connected to over ten devices the network would drop or other instabilities would happen. Having taken characteristics above into consideration, echoing the needs for transmission distance, stability, and controlled quantity, our system has utilized control network based on 433M frequency independently developed a self-organized protocol based on the control network which could bear dynamic networking functions similar to that of Zigbee and boast a connected devices number of over 100. As a result, our system has managed the networking capabilities of Zigbee with high connected devices number and long distance transmission. Also the protocol of 433M frequency is an open one and can be compatible with smart devices currently in the market. Therefore, smart router will change the landscape of smart home market and high-end router market and lays a solid foundation for establishment of industry standard for China smart home industry.
A set of solution aiming at promoting establishment of industry standard for China smart home market {#SEC: A set of solution aiming at promoting establishment of industry standard for China smart home market}
====================================================================================================
The set of solution includes {#SSEC: The set of solution includes}
----------------------------
Smart router,cloud server, mobile terminal, and intelligent terminal. Meanwhile, the system is a intelligence development platform which enables programmers worldwide to carry out secondary development on this platform and thus an ecological chain with sound circulation takes shape (similar to AppStore of Apple).
Feature of the whole set of solution {#SSEC: Feature of the whole set of solution}
------------------------------------
As a household smart center, the smart router is able to administer all of the connected devices, control smart devices in houses, keep houses in security under surveillance, monitor household environment in terms of temperature, humidity, PM 2.5 etc., alarm the police when household accidents happen such as smog or gas, enable users to be remotely connected to the smart center through devices such as cell phone and PAD at any time anywhere to observe and supervise household appliances, and at the same time promptly watch surveillance video in the house. To speak of, the cost of this device is only a small percentage of a traditional smart home product.
Users can know at first hand the environment of household through cell phones in many ways such as gas leakage, burglar break-in, illegal opening of doors or windows, abnormal temperature and humidity, smoke alarm, touching of valuable things, real-time temperature and humidity detection, PM2.5 detection. It can be seen as a safety housekeeper. At the same time, users can be promptly aware of the surveillance video image in the house through cell phones, which could provide a more perceptual and intuitive supervision to surrounding environment. Moreover, the router can control household appliances such as television, air conditioner as well as all the other household appliances with remote control. Last but not least, the router is also a cloud service center where users can put personal data in family cloud.
Introduction to specific functions {#SSEC: Introduction to specific functions}
----------------------------------
### Functions of router {#SSSEC: Functions of router}
Just like other ordinary routers, the router can visit the Internet and distribute WI-FI data[@zualkernan2009infopods].
### Intellisense {#SSSEC: Intellisense}
Users can receive alarm in houses through cell phones. Smart router can be adaptive to every alarm apparatus. When there is an alarm from apparatuses, smart router will promptly send alarm information to cell phones of users and provide security for users. Alarm apparatus spans gas leakage, burglar break-in, illegal opening of doors or windows, abnormal temperature and humidity, smoke alarm, touching of valuable things. Meanwhile, users can promptly get to know the temperature, humidity, and PM2.5 of houses.
### Intelligent surveillance {#SSSEC: Intelligent surveillance}
Smart router can be installed with USB camera of low price as well as wireless camera, which is convenient for users to obtain real-time image through cell phones. Wireless camera can be based on codec of H264, which makes image clear and smooth.
### Intelligent plug {#SSSEC: Intelligent plug}
Cell phones can remotely control the on and off of plug, which means controlling the appliances connected to the plug.
### Intelligent cloud service {#SSSEC: Intelligent cloud service}
Smart router can serve as a cloud service center that provides personal data management for family members and boasts a good level of privacy.
### Intelligence remote control {#SSSEC: Intelligence remote control}
X-Router enables users to put aside all remote controls at home and control all the household appliances through X-Router. For example, users can control lights, curtains, plugs, television, air conditioners, DVD, and STB through cell phones.
Main technologies {#SSEC: Main technologies}
-----------------
1\) Communication protocol to control establishment and implementation of protocol.
2\) Transfer function of communication protocol to achieve intelligence transfer of transmission and P2P.
3\) P2P technological research to achieve P2P of low flow.
4\) Establishment and implementation of camera protocol with original server protocol integrated.
5\) development of cell phone software and server side software. Establishment and implementation of chat protocol between software.
6\) Device terminal networking.
7\) establishment and implementation of connection control protocol of device terminal and router.
8\) Establishment and implementation of control protocol of device terminal of each types. Protocols for different types of devices are different and are individually carried out.
9\) Development and production of communication printed circuit board of communication device terminal.
10\) Build software and hardware platform for routers.
11\) Printed circuit board hardware circuit diagram design with low consumption, high simultaneous access, and long duration.
Technical index {#SSEC: Technical index}
---------------
### Low consumption {#SSSEC: Low consumption}
the transmitted power is only around 1mW. It also uses sleep mode with low power dissipation, making the device use much less electricity. According to estimates, the device can endure a continuous, active period of six months to two years just by two AA batteries, which other wireless devices can hardly match.
### Low cost {#SSSEC: Low cost}
the cost for the whole set of solution is around 100 yuan.
### Short time delay {#SSSEC: Short time delay}
communication delay and delay period for activation from sleep mode are both very short. Delay for a typical search equipment is 30ms, 15ms for activation from sleep mode, 15ms for active devices to join via channels. Thus the technology is best suited for application in wireless control that is highly commanding in time delay (such as industry control).
### Self-organized network technique {#SSSEC:Self-organized network technique}
a star schema has the maximum capacity for 254 slave units and one primary device. And the network is flexible.
### Reliability {#SSSEC: Reliability}
Strategy to avoid collision is employed. Specialized slot time is conserved for communication business in need of stable bandwidth, so that competition and conflict for sending data are avoided. The MAC layer employs a completely definite data transmission mode where each sent data package must wait for the confirmation from the recipient. If any problem occurs in the transmission process, the data can be sent again.
### Security {#SSSEC: Security}
HTTPS encryption that supports authentication and certification and uses encrypted algorithm.
### P2P technology {#SSSEC: P2P technology}
a technology that deals with NAT gateway or firewall penetration.
### Low power dissipation technology of cell phone clients {#SSSEC: Low power dissipation technology of cell phone clients}
a real-time technology that researches low power dissipation.
Explanation of key technologies of each terminal {#SSEC: Explanation of key technologies of each terminal}
------------------------------------------------
### Server terminal {#SSSEC: Server terminal}
Server terminal is the bridge for the whole system to connect where cell phones and routers build data and exchange data, and carry out P2P (peer-to-peer) communication. The burden of servers is reduced. Meanwhile servers provide login and registration as well as user information management for users. Servers can expand and form a cluster according to workload dynamic, thus increasing the processing capacity for cell phones and routers[@min2013design].
We have pulled up a protocol of our own, which could identify which type of connection employed judging the type of request (Normally small data is transmitted through server UDP, and large data uses P2P connection). It connects the data path from cell phones to routers and make it possible to transparently transmit data from cell phones to router terminals.
### Smart router terminal {#SSSEC: Smart router terminal}
Smart router terminal is the core part of our project, on which we build software and hard ware platform of routers. Firstly, it is a router with a high level of performance by which common PC and cell phones can visit the Internet. Secondly, it is our data processing center that achieves our core interaction protocol, and code and decode all the control alarm data and then process them. Thirdly, its radio frequency identification function enables it to send and receive radio frequency,and use the protocol we develop to code and decode to process digital signal and analog signal.
By virtue of the protocol, it connects the data path from routers to device terminals. With the aid of servers, it makes possible transparent transmission of data from cell phones to device terminals.
### Cell phone {#SSSEC: Cell phone}
As long as cell phones of users can be connected to 2G/3G/4G/wifi Internet, users can be conveniently connected to servers via software of cell phones, transparently communicate with smart routers and device terminals through the protocol.
Users can register by cell phone software and log on the server, modify their personal information by HTTP communication protocol, add friends by chat protocol, and chat with friends. Cell phones can send router administer command through (the path between cell phones and routers) introduced above and thus set up for switches of routers, wireless switches, and PPPOE.
When cell phones terminal want to control device terminal, a control protocol will be generated and be sent to the device terminal through the (path from cell phone to device terminal) introduced above. The device terminal receives the decoded information of the control protocol and obtains the specific command and implement it. Device terminal needs to report its state or generate the corresponding reporting command and send it to cell phones terminal through the path introduced above when it receives the alarm information which needs to be reported to users. The cell phones terminal receives the reported decoded protocol, obtains the specific command, and then registers its state on the UI of the application software or alarms the police.
### Device terminal {#SSSEC: Device terminal}
Device terminal in effect realizes the communication protocol of the solution and can be seamlessly and smoothly extended. Any protocol that realizes the solution can join the router and be part of the networking to control the router.
Device terminal spans all the alarm apparatuses, plug, bulb, power supply, sensor, and household appliance.
### Termianl SDK {#SSSEC:Termianl SDK}
If we can connect data channels between each device, we can do a lot of things on it. Our solution is open to SDK of device communication. By virtue of SDK, the third party can control devices that it develops through its application and achieve platformization purpose.
### 6)Camera cloud serive cluster and camera {#SSSEC: 6)Camera cloud serive cluster and camera}
Considering features of camera such as strong performance of processor, large data throughput, and large amount of data of images, as we need to ensure that camera does not affect the stability of other devices[@yuan2014accountability], we make our camera separate image data and control data. Transmission and storage of image data are solely handled by server clusters.
Communication of camera terminal and cell phones can be realized by server transfer, or by P2P connection through hole punching technique of servers and cell phones.
When cell phone terminals need to be connected to cameras, they will also be connected to camera cloud service cluster. They firstly make a request for P2P, if the path does not support it, it will commence server transfer. Control operation of cell phones such as rotating the camera, shifting up, down, to left, or right, is carried out through the path to the main servers. Cell phone terminals can local-save image date of cameras, or directly save the data on remote Yunfile such as Yun Baidu and Kuaipan via users’ Yunfile accounts.
Content of innovation technology {#SSEC: Content of innovation technology}
--------------------------------
### Combination of radio frequency technology and router {#SSSEC: Combination of radio frequency technology and router}
Tradition home gateway master control systems all employ specialized processors and communication protocol abroad with only single function and inordinate price[@nejad2014operation]. With the development of chip technology, the function of chips of household routers are powerful enough to carry out control over smart home appliances[@razminia2011chaotic]. Therefore, we design this smart router, add radio frequency module to routers, write programmes for control and communication protocol, thus achieve a master control system with multi-functions, low cost, and high stability, which can also be used as general routers.
### Home gateway communication system {#SSSEC:Home gateway communication system}
Home gateway communication system utilizes exclusive custom protocol which can not be compatible with third party system and makes it difficult to upgrade[@garcia2012smart]. The system hopes to maintain open and universal in its application and integrates elements of Jingle protocol used in social network. The system also adds certification, control, and device discovery functions to the protocol, thus the system is robust in its expansiveness, can interact with any server that supports Jingle protocol, and communicate with any client that supports Jingle protocol. Meanwhile, the system supports device control and social interaction. As a result, our system can not only control smart devices, but also enable family members to communicate via social network.
### 3£©Control network protocol {#SSSEC:3£©Control network protocol}
There are three types of control network by traditional definition: The first one is wireless control based on 315M, 433M and other frequency ranges. The benefit of such control network is a high level of accuracy and a far-reaching control range as much as 1000 meters if power reaches its maximum. On the flip side, there is no network protocol and it can only send simple control command. Collision occurs when there are over three connected devices, which renders the process more difficult to succeed[@han2014generating].
The second one is control network based on ZigBee. Its shortcomings lie in small range,poor through-the-wall performance, complex protocol,inordinate price,and at the same time it is exclusive and incompatible to devices existing in the market[@garcia2013multi].
The third one is control network based on WI-FI. WI-FI boasts a small control range and thus is limited to only a few connected devices. Normally when household router is connected to over ten devices the network would drop or other instabilities would happen[@kammerer2012router]. Having taken characteristics above into consideration, echoing the needs for transmission distance, stability, and controlled quantity, our system has utilized control network based on 433M frequency independently developed a self-organized protocol based on the control network which could bear dynamic networking functions similar to that of Zigbee and boast a connected devices number of over 100. As a result, our system has managed the networking capabilities of Zigbee with high connected devices number and long distance transmission. Also the protocol of 433M frequency is an open one and can be compatible with smart devices currently in the market, such as 433M infrared body detecting alarm.
### High-speed router {#SSSEC:High-speed router}
The transmission rate of traditional router is only 150m/300M. Our router employs AC technology and thus the transmission rate can reach as much as 900M.
![\[fig:2-1\]Product concept structure graph.](luyouqi.pdf "fig:"){height="29.00000%" width="50.00000%"}\
Theoretical basis (including data transmission) {#SSEC:Theoretical basis (including data transmission)}
-----------------------------------------------
### TCP/IP protocol {#SSSEC:TCP/IP protocol}
TCP/IP protocol is the abbreviation for Transmission Control Protocol/Internet Protocol. It is also known as network communication protocol. It is the fundamental protocol of Internet and the foundation of international Internet network, comprised of IP protocol at network layer and TCP protocol at transport layer. TCP/IP defines the standard for how electrical devices are connected to Internet and how data transmits between them. The protocol employs a hierarchical structure of four layers and each layer calls the protocol provided by its following layer to fulfil its need. To put it in blank words, TCP is in charge of spotting problems occurring in transmission and once problem occurs it sends our signals to command a new transmission until all the data is safely and correctly sent to the destination. While IP set an address for each connected device of Internet.
The communication of all the terminals and devices in this project is based on TCP/IP protocol.
### Jingle/XMPP protocol {#SSSEC:Jingle/XMPP protocol}
XMPP(Extensible Messaging Presence Protocol) is a protocol based on extensible markup language (XML) applied in instant messages (IM) and online presence detection.It promotes the on-time and instant operation between servers. The protocol is likely to ultimately allow Internet users to send instant messages to others on the Internet even if the operating systems and browsers are different.
The P2P connection and transmission in this solution are both achieved based on revised Jingle XMPP protocol.
Experimental basis {#SSEC:Experimental basis}
------------------
The router structure as follows is the ultimate version utilized by this project after multiple practices.
![\[fig:3-1\]The Router Frame.](jiegou.pdf "fig:"){height="29.00000%" width="50.00000%"}\
Product index {#SSEC:Product index}
-------------
### Smart route {#SSSEC:Smart route}
1)size: 23£ª23£ª5
2)transmission standard: IEEE 802.11ac/b/g/n
3)wireless transmission rate: 300M+700M
4)antenna gain: 5dbi
5)wired Internet access: one WAN, four LAN, 1000/100/10subject to adjustment and configuration
6)safety standard: 64/128 WEP encryption technology, WPA/WPA2 encryption, supports WPS WI-FI Protected Setup function
7)transmission band: 2.4GHz and 5.0GHz
8)reset key: 1
9)WPS key: 1
10)supported connected device terminals: 255
11)capable of responding to command of cell phone clients and each device terminals in three seconds.
### Server {#SSSEC:Server}
1)One server can simultaneously support 6000 users.
2)7\*24h smooth operation
### Wireless device {#SSSEC:Wireless device}
1)effective range: outdoors 1500M£¬indoors 300M
2)frequency range: 240-930MHz
3)FSK, GFSK and OOK modulation mode
4)maximum output power: 20dBm
5)sensitivity: -121dBm
6)low power dissipation: 18.5mA (reception); 85mA@+20dBm (transmission)
7)data transmission rate: 0.123-256kbps
Conclusion {#SEC: Conclusion}
==========
After relentless research and practice, our final smart router has greatly overcame the three big shortcomings of smart home, user experience, purchasing cost, and bad compatibility. We bring a revolution to the smart home market and make it accessible to common people. We enable common people to experience a life of intelligence and high quality with a moderate price, thus making smart home more popularized in China market.
The appearance of smart routers also brings new vista to the smart home market and high-end router market, bring fresh vibrancy and business opportunities to China market. Also industry standard of China smart home will achieve breakthroughs with further research and development of smart router technology.
In researching and developing the technology of smart router, we have come across some inevitable problems. For example, during research and development, although we harbour a strong awareness of protection to the environment and carried out protection measures, the external environment is still affected. But we believe that when products are mature enough to be massively produced, relevant technologies will surely be applied in avoiding environment pollution risk from the source or reducing it.
Acknowledgement {#SEC: Acknowledgement}
================
The research subject was supported by the department of Civil Engineering and the department of Computer Science$\&$Engineering Jinjiang College, Sichuan University. I would like to express my thanks to Prof. Bingfa Lee¡¯s suggestions and guidance, as well as Guanguuan Yang$\&$Zhuo Li and Hui Zhang whose books give me a lot of inspiration.
| ArXiv |
---
author:
- 'M. H. D. van der Wiel'
- 'D. A. Naylor'
- 'G. Makiwa'
- 'M. Satta'
- 'A. Abergel'
bibliography:
- '../../literature/allreferences.bib'
title: 'Three-dimensional distribution of hydrogen fluoride gas toward NGC6334I and I(N)[^1]'
---
[ The HF molecule has been proposed as a sensitive tracer of diffuse interstellar gas, while at higher densities its abundance could be influenced heavily by freeze-out onto dust grains. ]{} [ We investigate the spatial distribution of a collection of absorbing gas clouds, some associated with the dense, massive star-forming core NGC6334I, and others with diffuse foreground clouds elsewhere along the line of sight. For the former category, we aim to study the dynamical properties of the clouds in order to assess their potential to feed the accreting protostellar cores. ]{} [ We use far-infrared spectral imaging from the SPIRE iFTS to construct a map of HF absorption at 243 in a 6$\times$35 region surrounding NGC6334 I and I(N). ]{} [ The combination of new, spatially fully sampled, but spectrally unresolved mapping with a previous, single-pointing, spectrally resolved HF signature yields a three-dimensional picture of absorbing gas clouds in the direction of NGC6334. Toward core I, the HF equivalent width matches that of the spectrally resolved observation. At angular separations $\gtrsim$20 from core I, the HF absorption becomes weaker, consistent with three of the seven components being associated with this dense star-forming envelope. Of the remaining four components, two disappear beyond $\sim$1 distance from the NGC6334 filament, suggesting that these clouds are spatially associated with the star-forming complex. Our data also implies a lack of gas phase HF in the envelope of core I(N). Using a simple description of adsorption onto and desorption from dust grain surfaces, we show that the overall lower temperature of the envelope of source I(N) is consistent with freeze-out of HF, while it remains in the gas phase in source I. ]{} [ We use the HF molecule as a tracer of column density in diffuse gas ($n_\mathrm{H}$$\approx$$10^2$–$10^3$ ), and find that it may uniquely trace a relatively low density portion of the gas reservoir available for star formation that otherwise escapes detection. At higher densities prevailing in protostellar envelopes ($\gtrsim$$10^4$ ), we find evidence of HF depletion from the gas phase under sufficiently cold conditions. ]{}
Introduction {#sec:intro}
============
The hydrogen fluoride molecule, HF, was first observed in the interstellar medium by @neufeld1997b with the [*Infrared Space Observatory*]{} [[*ISO*]{}, @kessler1996]. While [*ISO*]{} had a wavelength range that encompassed only the $J$=2–1 rotational transition of HF, the next observatory able to observe HF – the [*Herschel*]{} Space Observatory [@pilbratt2010] – covered longer far-infrared wavelengths, and it thus opened up access to the ground-state rotational transition, $J$=1–0, at 1232.48 GHz (243.24 ). [*Herschel*]{} has observed HF in absorption along many lines of sight, both inside the Galaxy [@neufeld2010b; @sonnentrucker2010; @sonnentrucker2015; @philips2010; @kirk2010; @monje2011a; @emprechtinger2012; @lopez-sepulcre2013a; @goicoechea2013] and in nearby extragalactic objects [@rangwala2011; @kamenetzky2012; @rosenberg2014a; @monje2014]. HF absorption has even been detected with ground-based observatories: @monje2011c have made use of the substantial redshift of the Cloverleaf quasar at $z$=2.56 shifting the HF 1–0 line into the submillimeter window attainable with the CSO on Mauna Kea, and @kawaguchi2016 detect it in the $z$=0.89 absorber toward PKS1830$-$211, using ALMA in the Chilean Atacama desert. Because of its large dipole moment and high Einstein $A$ coefficient for radiative decay, rotational states $J$$\neq$0 of HF only become significantly populated in highly energetic conditions. It is for this reason that HF has been clearly detected in emission in a mere handful of cases: in the inner region of an AGB star’s envelope [IRC+10216, @agundez2011], in the Orion Bar photodissociation region [@vandertak2012a], and in an external galaxy harboring an actively accreting black hole [Mrk231, @vanderwerf2010]. Atomic fluoride, F, has a unique place in the interstellar chemistry of simple molecules. It is the only element which, simultaneously, (1) is mainly neutral because of its ionization potential $>$13.6 eV, (2) reacts exothermically with – unlike *any* other neutral atom – to form its neutral diatomic hydride HF, and (3) lacks an efficient chemical pathway to produce its hydride cation HF$^+$ due to the strongly endothermic nature of the reaction with H${_3}^+$. We refer to @neufeld2009b, references therein, and the comprehensive review by @gerin2016 for more details on the chemistry of HF and a comparison with other hydride molecules. For the reasons listed above, chemical models predict that essentially all interstellar F is locked in HF molecules [@zhu2002; @neufeld2005], which has been confirmed by observations across a wide range of atomic and molecular ISM conditions [e.g., @sonnentrucker2010; @sonnentrucker2015]. With recent experimental results by @tizniti2014 showing that, especially at low temperatures approaching 10 K, the reaction F + $\rightarrow$ HF + H proceeds somewhat slower than earlier assumptions, chemical models are now able to reproduce HF/ ratios of $\sim$, measured most directly by @indriolo2013a, and observed to be rather stable across different sightlines. Interferometric observations show that CF$^+$, the next most abundant F-bearing species after HF, has an abundance roughly two orders of magnitude lower than HF, both inside our Galaxy [@liszt2015b] and in an extragalactic absorber [@muller2016]. As for destruction of HF, the most efficient processes are UV photodissociation and reactions with C$^+$, but both of these are unable to drive the majority of fluoride out of HF, due to shielding, already at modest depths of $A_V>0.1$ [@neufeld2005]. Because of the constant HF/ abundance ratio and the high probability that HF molecules are in the rotational ground state, measurements of HF $J$=0$\rightarrow$1 absorption provide a straightforward proxy of column density. This has led to the suggestion that, at least in diffuse gas, HF absorption is a more reliable tracer of total gas column density than the widely used carbon monoxide (CO) rotational *emission* lines, and is more sensitive than CH or absorption [e.g., @gerin2016]. Apart from the uncertain and variable CO abundance, local excitation conditions have a profound effect on the level populations of CO, complicating the conversion from observed line strength of a particular CO transition to column density [@bolatto2013]. The greatest gas-phase CO abundance variations occur in dense, cold regions where CO freezes out onto surfaces of dust grains, proven by observed CO abundances decreasing in the gas phase and increasing in the ice phase as conditions get colder [e.g., @jorgensen2005a; @pontoppidan2005a]. In addition, the particular fraction of the neutral ISM that is in the diffuse/translucent phase is inconspicuous in CO [@bolatto2013], but is detectable using hydride absorption lines. Of course, for absorption line studies, one relies on lines of sight with sufficiently strong continuum background, for example those toward dense star-forming clouds. Such restrictions do not apply for emission line tracers. Besides CO rotational lines, fine structure line emission due to atomic C and the C$^+$ and N$^+$ ions has been used as a tracer of (diffuse) gas throughout the Galaxy [e.g., @langer2014a; @velusamy2014b; @gerin2015a; @goicoechea2015b; @goldsmith2015]. For all these tracers, however, the conversion to column density depends strongly on physical properties such as ionization fraction and excitation conditions.
Based on the above arguments, HF absorption measurements are a good tracer of overall gas column density. However, as addressed for example by @philips2010 and @emprechtinger2012, HF itself may suffer from freeze-out effects as occurs with other interstellar molecules. While studies have been done on the interaction of with HF as a polluting agent in the Earth’s atmosphere [@girardet2001], the density and temperature conditions needed for HF adsorption onto dust grains have not been studied in astrophysical contexts so far. Any freeze-out of interstellar HF will obfuscate the direct connection between HF absorption depth and column density described above. The well-known progression of pre- and protostellar stages for stars with masses similar to the Sun [@shu1977] is not applicable for high-mass stars ($\gtrsim8$ ). In the latter category, protostellar hydrogen fusion starts while accretion from the surrounding gas envelope is still ongoing [@palla1993]. In the ‘competitive accretion’ scenario, multiple massive protostars in a clustered environment are fed from the same gas reservoir [@bonnell2001b]. For high-mass protostars, material can continuously be added to the gravitationally bound circumstellar envelope which provides the reservoir for further accretion onto the protostar. It is therefore important, particularly for regions of *high-mass* star formation, to study not just the gravitationally bound circumstellar envelopes, but also the dynamical properties of surrounding gas clouds. Especially for the latter component, simple hydride molecules have the potential to reveal gas reservoirs to which emission lines of ‘traditional’ tracer species, such as CO, , and CS, are insensitive due to their relatively high critical densities. In this paper, we investigate two envelopes of (clusters of) protostars embedded in the molecular cloud as well as lower density clouds surrounding the dense complex.
The filamentary, star-forming cloud NGC 6334, at a distance of 1.35 kpc [@wu2014], harbors a string of dense cores, identified in the far-infrared by roman numerals I through VI [@mcbreen1979], with an additional source identified $\sim$2 north of source I, later named ‘I(N)’ [@gezari1982]. The larger scale NGC6334 filament has an column density of $>$ even at positions away from the embedded cores [@russeil2013]. @zernickel2013 observed the velocity structure of NGC6334 at 0.15 pc resolution. These authors explain the velocity profile along the filament with a cylindrical model collapsing along its longest axis under the influence of gravity. In this paper we study specifically the region of $\sim$2.4$\times$1.6 pc surrounding the embedded cores and . Source I is host to an ultra-compact region, designated source ‘F’ in a 6 cm radio image of the cloud [@rodriguez1982]. Based on multi-wavelength dust continuum measurements, studies by @sandell2000 and @vandertak2013a have independently determined that the mass of source I(N) exceeds that of sister source I by a factor of $\sim$2–5, but the ratio of their bolometric luminosities is 30–140 in favor of source I, due to the markedly lower temperature for source I(N). As expected for a warm (up to $\sim$100 K), dense, massive star-forming core, NGC6334 I is extremely rich in molecular lines, spectacularly demonstrated by the 4300 lines detected in the 480–1907 GHz spectral survey by @zernickel2012[^2]. The differences between the two neighboring cores all suggest that core I is in a more evolved stage of star formation than core I(N). Both cores have been studied with radio and (sub)millimeter interferometer observatories, showing that each separates into several subcores at arcsecond resolution [i.e., at scales $\lesssim$0.01 pc, @hunter2006; @hunter2014; @brogan2009]. To probe gas clouds in front of the NGC6334 complex, absorption measurements have been obtained in lines of several hydrides. Spatially extended OH hyperfine line absorption at was observed toward the NGC6334 filament by @brooks2001. The spectrally resolved mapping observations from the Australia Telescope Compact Array allowed these authors to ascribe particular velocity components of the absorption to a foreground cloud close to NGC6334 and other components to clouds with even larger angular extent. @vanderwiel2010 used the Heterodyne Instrument for the Far-Infrared [HIFI, @degraauw2010] onboard [*Herschel*]{} to study the spectral profile of the rotational ground state lines of CH at 532 and 537 GHz, and found four distinct absorption components overlapping with the velocity range of OH absorption, and one single emission component emanating in core NGC6334 I itself. At 1232.5 GHz in the same spectral survey, @emprechtinger2012 find the exact same four absorbing clouds in the HF rotational ground state, and invoke three velocity components to explain the hot core component. While the hot core component(s) appear in emission in CH, they are in absorption in HF, because CH 1$\rightarrow$0 has a lower Einstein $A$ coefficient than HF 1$\rightarrow$0 (see above).
---------------- ------------- --------------- ----------------- ------------- ------------ --------- -------
Observation Target name Right Ascension Declination Jiggle Gain
observation ID date (J2000) (J2000) pattern mode
1342214827 2011-02-26 NGC6334 I 17205415 $-$3547074 4$\times$4 Nominal 11491
1342214841 2011-02-27 NGC6334 I(N) 17205609 $-$3545073 4$\times$4 Nominal 11491
1342251326 2012-09-24 NGC6334 I 17205415 $-$3547074 4$\times$4 Bright 9605
1342214828 2011-02-26 NGC6334 ‘INT’ 17204810 $-$3545429 Stare Nominal 915
1342214829 2011-02-26 NGC6334 ‘OFF’ 17203904 $-$3543435 Stare Nominal 915
---------------- ------------- --------------- ----------------- ------------- ------------ --------- -------
The CH and HF signatures were observed toward NGC6334 I with the high spectral resolution spectrometer HIFI in single point mode [@degraauw2010]; its single pixel receiver did not provide any spatial information. In a *map* of CH or HF absorption covering the region surrounding source I, one would expect to see a disentanglement of the different spatial extent of each velocity component as illustrated in Fig. \[fig:HFabscomp\]. Toward core I, the velocity resolved HF absorption signature, with a total equivalent width, $\int (1-I_\mathrm{norm}) \mathrm{d}V$, of 16 , was modeled with seven components. At positions away from , but still on the NGC6334 filament, the equivalent width should diminish to 11 representing the four foreground clouds, while at positions off of the cores and the filament, only the two foreground components that are more extended than the dense molecular cloud should be visible, and the equivalent width should drop to 3 .
This paper presents results from [*Herschel*]{} SPIRE iFTS spectral mapping observations toward a 6$\times$35 region surrounding cores I and I(N) in the NGC6334 star-forming complex. The observations are described in Sect. \[sec:obs\] and the resulting map of HF absorption depth is discussed in Sect. \[sec:obsresults\]. The signal is interpreted in Sect. \[sec:analysis\], both in the context of foreground clouds and in that of freeze-out conditions in the dense cores. Conclusions are summarized in Sect. \[sec:conclusions\].
Observations and data reduction {#sec:obs}
===============================
The spectral mapping observations used in this work were obtained as part of the ‘evolution of interstellar dust’ guaranteed time program [@abergel2010] with the Spectral and Photometric Imaging Receiver [SPIRE, @griffin2010] on board the [*Herschel*]{} space observatory [@pilbratt2010]. SPIRE’s imaging Fourier Transform Spectrometer (iFTS) provides a jiggling observing mode that uses its 54 detectors to obtain Nyquist sampled spatial maps, covering the entire frequency range of the Spectrometer Long Wavelength (SLW, 447–1018 GHz) and the Spectrometer Short Wavelength (SSW, 944–1568 GHz) bands. The spectral resolution of 1.2 GHz corresponds to a resolving power $\nu/\Delta\nu\approx10^3$, roughly 300 at the frequency of the HF 1–0 transition, 1232.5 GHz (243 ).
Three partly overlapping, fully sampled SPIRE iFTS 4$\times$4 jiggle observations were performed in a total of nine hours of observing time, two centered on NGC6334 I and the third on NGC6334 I(N). A fourth, sparsely sampled observation, centered $\sim$2 northwest of core I, has considerable overlap with the combined area covered by the three other observations. This fourth observation is treated as an extra jiggle position in the gridding process described below. Finally, a fifth observation, also sparsely sampled, is centered 45 northwest of core I and its footprint therefore has no overlap with our mapped area. At each jiggle position, four repeated scans of the FTS mechanism were executed in high spectral resolution mode. Details of the observations are summarized in Table \[t:obs\]. The placement of the different pointings described here is shown in Fig. \[fig:FTSfootprints\] in the Appendix.
After inspection of the initial observations from February 2011, some detectors were found to suffer from saturation due to the bright emission toward source I. The observation toward source I was therefore repeated in ‘bright source mode’ [@lu2014b] in September 2012 to obtain well calibrated spectra toward the brightest position. The majority of detector/jiggle combinations in the original observation point toward less bright regions and are therefore still useful in constructing the final map.
The above SPIRE iFTS observations are processed with the ‘extended source’ pipeline in HIPE 12.1.0 and the `spire_cal_12_3` calibration tree, which includes the outer ring of partly vignetted detectors [@fulton2016]. The pipeline is interrupted at the pre-cube stage, before spectra from individual jiggle positions are gridded onto a rectangular spatial pattern. The spectrum for each jiggle position and each detector is visually inspected. Discarding all spectra that show excessive noise and/or irregular continuum shape (resulting from partial saturation) results in filtering 28 of the total of 919 SLW spectra (3%) and 135 of the 1696 SSW spectra (8%). The final processing step is the combination of the individual positions from each of the 19 (SLW) or 37 (SSW) detectors of each of the 55 jiggle positions into spectral cubes with square spatial pixels in Right Ascension and Declination coordinates. Due to the complex frequency dependence of the beam size and shape [@makiwa2013], the native angular resolution of SPIRE iFTS observations varies non-monotonously across its frequency range. We use the convolution gridding scheme, which weighs each input value according to the distance from the target pixel center by means of a differential Gaussian kernel, with the aim of obtaining a cube with a constant effective reference beam of 43 FWHM. The pixel grid is identical for SLW and SSW, with square pixels measuring 175$\times$175. Gaussian convolution only results in a completely Gaussian target beam if the original beam is also well represented by a Gaussian shape. Such is the case for the entire SSW band and for low frequencies in SLW, but not for SLW frequencies between $\sim$700 and 1018 GHz [@makiwa2013].
Results {#sec:obsresults}
=======
The spectral cubes, as constructed in Sect. \[sec:obs\], show a smooth dust continuum superposed with spectrally unresolved lines. The dominant line signal in the cube arises from the ladder of rotational transitions of CO ($J$=4–3 to 13–12). Early versions of the CO intensity maps of NGC 6334, based on subsets of the SPIRE iFTS observations used here, were presented in @naylor2013 and @makiwa2013osafts. To retain the highest possible spectral resolution, we use unapodized FTS data, in which any unresolved spectral line has a Sinc-shaped profile. It is therefore important that one carefully fits and subtracts the Sinc-shaped profiles of nearby bright lines, to avoid any remaining sidelobes of strong lines affecting the apparent profile of the weak absorption line under study. This work focuses on absorption signatures of two hydride molecules, for which we use two spectral sections of the data cubes: 1132–1332 GHz from SSW and 760–935 GHz from SLW, chosen specifically to include the two CO emission lines closest to HF 1–0 at 1232.48 GHz [rotational spectroscopy by @nolt1987] and 1–0 at 835.08 GHz [rotational transition frequency measured by @pearson2006]. We construct a script in HIPE [@ott2010], derived from one of the post-pipeline analysis scripts provided by the SPIRE iFTS working group [@polehampton2015], to fit a third order polynomial for the continuum simultaneously with the following lines: CO at 1152.0 and 1267.0 GHz (SSW), and CO at 806.7 and 921.8 and \[\] at 809.3 GHz (SLW). Knowing that the intrinsic width of CO lines in this region is only a few [@zernickel2012], lines are spectrally unresolved by SPIRE iFTS, and we adopt for each spectral line a Sinc profile with a fixed peak-to-first-zero-crossing width of 1.18 GHz. We then divide the observed spectrum by the fit of continuum and two/three Sinc lines to obtain a continuum-normalized spectrum, $I_\mathrm{norm}$. This process, illustrated in the top panel of Fig. \[fig:fittingprocess\], is repeated for each spatial pixel in the cube.
In the resulting continuum-normalized cube around 1232 GHz, we fit two Sinc functions to the absorption profile of HF 0$\leftarrow$1 at 1232.48 GHz and the nearby emission line of 2$_{2,0}$$\rightarrow$2$_{1,1}$ at 1228.79 GHz (Fig. \[fig:fittingprocess\], middle panel). In the continuum-normalized cube around 835 GHz, we fit three Sinc functions to the absorption profile of 0$\leftarrow$1 at 835.08 GHz and emission lines at 771.18 and 881.27 GHz. The maps of equivalent width of the HF and absorption depth, in units of Hz, are multiplied by the ratio of the speed of light, c, and the observed frequency, $\nu_\mathrm{obs}$, to convert to units: c/$\nu_\mathrm{obs}$= Hz$^{-1}$ for HF, and Hz$^{-1}$ for . Line fits are rejected if the signal-to-noise ratio is lower than 2 and/or the fitted line center is more than half a resolution element from the line’s expected frequency at $-8.3$ [@vanderwiel2010]. The resulting map of HF equivalent width in Fig. \[fig:HFabsmap\] reveals the spatial distribution of the HF absorption feature, detected in 81% (signal-to-noise$>$2) or 72% (signal-to-noise$>$3) of all the pixels in the 6$\times$35 map coverage. See also the signal-to-noise map in Fig. \[fig:SNmapHF\].
Uncertainties are calculated from the spectral rms noise in the continuum-normalized cubes in two 20 GHz ranges surrounding the HF absorption line. For , the frequency ranges for calculating noise are composed of a 10 GHz section below and a 30 GHz section above the frequency, to avoid incorporating residual from the fit to the blended CO 7–6 and \[\] $^3$P$_2$–$^3$P$_1$ lines at 806 and 809 GHz. The noise on the equivalent width is obtained by multiplying the unitless spectral rms with the instrumental line width of 1.18 GHz $\times$ c/$\nu_\mathrm{obs}$. Noise values are variable across the maps, in the range 1.2–3.4 for HF and 1.4–7 for . We do not include the following contributions to the uncertainty. Firstly, any multiplicative effects such as those of the absolute intensity calibration [@benielli2014; @swinyard2014] are divided out by normalizing the spectra to the local continuum. Secondly, additive uncertainties in the continuum level offsets are $\sim$ for SLW and $\sim$ for SSW [@swinyard2014; @hopwood2015]. These values are negligible compared to the brightness of the continuum in our cube, which exceeds the offset uncertainty by a factor of a few hundred even in the faintest outer regions.
Spectra from the ‘OFF’ observation, pointed just northwest of the mapped area shown in Fig. \[fig:HFabsmap\], are also inspected at the HF frequency, but no convincing detections are found. The spectra from individual detectors in the OFF observation exhibit rms noise values between 4 and 10 , with a median of 6 . This noise is considerably higher than that in the cube pixels based on the other four observations combined, in which each pixel encompasses at least four, but typically more than eight individual detector pointings. The lack of HF absorption detections in the OFF position is thus consistent at the 2-$\sigma$ level with an HF absorption depth $\lesssim$12 in the area 3–6 northwest of source I, i.e., absorption depths could be anywhere in the range shown in our mapped area, except the central 40 around source I itself where the strongest absorption is seen.
To rule out contamination of the HF signature by other spectral lines within SPIRE’s spectral resolution element, we inspect the high spectral resolution spectrum toward the position of the chemically rich core NGC6334 I [@zernickel2012], observed as part of the spectral survey key program CHESS [@ceccarelli2010] using the HIFI spectrometer [@degraauw2010; @roelfsema2012]. The only spectral lines detected by HIFI in a frequency span of $\pm$2 GHz around the HF frequency are four marginally detected methanol emission lines (A. Zernickel, private communication, Jun. 2014) together amounting to $<$0.4 in equivalent width. The possible methanol contaminations for the measured HF absorption depth are therefore contained within the uncertainty for our HF equivalent widths quoted above. The effect of the emission line at 1229 GHz (see also above) could be more significant: at the brightest position, toward core I, the water line is as bright as one third of the deepest HF absorption. As described above and shown in Fig. \[fig:fittingprocess\], the effects of the water line on the HF absorption profile are taken into account by applying a simultaneous fit of these two lines, separated in frequency by three times the SPIRE instrumental line width.
We also detect the signature of 1$\leftarrow$0 absorption at 835.08 GHz in the spectral map from the SLW array. We refrain from interpreting its signal here for the following reasons. Firstly, the signal-to-noise ratio in the absorption map is much lower than that in the HF map (see Fig. \[fig:SNmapHF\] and \[fig:SNmapCHplus\]), resulting in signal-to-noise$>$2 detections in only 46% of the mapped pixels. For completeness, the distribution of detected absorption is shown in the Appendix in Fig. \[fig:SNmapCHplus\]. Importantly, around the position of source I, there is no confident detection to be compared with heterodyne observations from @zernickel2012. Only a few isolated pixels near that position have detections of , but at a signal-to-noise of $<$3. Secondly, the spectrally resolved HIFI spectrum toward source I (A. Zernickel, private communication, Jun. 2014) show seven distinct emission line features due to methanol at frequencies within 0.6 GHz of the line, i.e., half of the SPIRE spectral resolution. The combined intensity of these lines is sufficient to compensate more than half of the absorption observed by HIFI, and they therefore severely contaminate the spectrally unresolved profile of absorption in the SPIRE spectrum. In fact, these methanol emission lines could be the cause of the weak detection at the position of core I with SPIRE’s modest spectral resolution. Thirdly, the transition falls in a frequency range in which the beam profile of the SPIRE iFTS is non-Gaussian in shape [@makiwa2013], complicating the map convolution and the interpretation of any spatial structure.
Our SPIRE spectroscopic data also show evidence for detections of NH$_2$ at 952.6 and 959.5 GHz, and NH at 974.5 GHz and at 1000 GHz. However, compared to HF, it is more challenging for astrochemical models to explain the observed abundances of N-bearing hydrides [e.g., @persson_cm2012], and the analysis of line features is complicated by hyperfine structure within rotational transitions [cf. spectrally resolved detections with HIFI by @zernickel2012]. For these two reasons, we refrain from interpreting the NH and NH$_2$ absorption depth maps in this paper, but for completeness they are shown in Figs. \[fig:SNmapNH\] and \[fig:SNmapNH2\]. The OH$^+$ doublet at 909.05 and 909.16 GHz is also within the frequency range covered by the SPIRE iFTS, but the bright methanol emission line at 909.07 GHz apparent in the HIFI spectrum published by @zernickel2012 would make analysis of the blended OH$^+$ signature in the SPIRE spectrum impossible.
Analysis and discussion {#sec:analysis}
=======================
HF optical depth, equivalent width, and column density {#sec:eqwidth}
------------------------------------------------------
An absorption line is saturated when its depth reaches zero, i.e., absorbing all continuum photons in any specific spectral channel. HIFI observations seem to show that the combined absorption feature of HF due to the NGC6334 I envelope, hot cores, and the foreground cloud at $-3.0$ is saturated between of $-7$ and $-4$ (Fig. \[fig:HFabscomp\]a). The individual components, however, do not reach 100% absorption. Of all seven components, the cloud at $+6.5$ comes closest to having saturated absorption in HF. As already noted by @emprechtinger2012, even this component is only marginally optically thick, evidence for which is provided by the line width of the same component in the optically thin tracer CH [@vanderwiel2010] which is the same (1.5 ) as for HF. All other components are believed to be optically thin [@emprechtinger2012].
Absorption line depth is converted into optical depth using $I_\mathrm{norm}=\mathrm{e}^{-\tau_\nu}$. With the caution of one of the seven components being marginally saturated, we take the optically thin limit to relate optical depth integrated over the line profile, , to column density, $N_\mathrm{HF}$, following @neufeld2010b: $$\label{eq:tautocolumn}
\int \tau_{\nu,\mathrm{HF}} \mathrm{d}V = \frac{A_\mathrm{ul} g_\mathrm{u} \lambda^3}{8\pi g_\mathrm{l}} N_\mathrm{HF} = \powm{4.16}{-13} \, [\mathrm{cm}^2 \, \mathrm{km}\, \mathrm{s}^{-1}]\ N_\mathrm{HF} ,$$ where $A_\mathrm{ul}$ is the Einstein $A$ coefficient for spontaneous emission, s$^{-1}$ for HF 1–0 [@pickett1998], $g_\mathrm{u}$=3 and $g_\mathrm{l}$=1 are the statistical weights of rotational levels $J$=1 and $J$=0, respectively, and $\lambda$ is the wavelength of the transition, 243.24 . In addition to the optically thin limit, Eq. (\[eq:tautocolumn\]) assumes that all HF molecules are in the rotational ground state, a fair assumption given its high $A_\mathrm{ul}$.
The conversion from $I_\mathrm{norm}$ to $\tau_\nu$ follows a linear relation for low values of $\tau_\nu$: $$\begin{aligned}
\nonumber
\tau_\nu & = & -\ln(I_\mathrm{norm}) \\
& \approx & 1-I_\mathrm{norm} \qquad [\mathrm{for}\ I_\mathrm{norm} \approx 1]. \end{aligned}$$ This relation holds to within $\sim$10% for $\tau_\nu<0.2$ (line absorbs up to 20% of the continuum), but $\tau_\nu/(1-I_\mathrm{norm})$ is already 1.2 at 40% absorption, rising to 2 at 80% absorption. The integrated optical depth is therefore systematically underestimated for a spectrally unresolved line that is smeared out over a velocity range wider than its intrinsic profile.[^3] Since the HF line is spectrally unresolved in our SPIRE spectra, a column density derived from these observations would merely constitute a lower limit to the true column density. Contrary to the optical depth, the absorption depth integrated over the line profile, i.e., the equivalent width of the absorbed ‘area’ below $I_\mathrm{norm}$=1, is conserved regardless of the spectral resolution. This is confirmed by the matching equivalent width values measured by HIFI and SPIRE toward core I: is 15.7 and 16.4 , respectively, with uncertainty margins of $\sim$1 in both cases. In the remainder of this paper, we analyze HF absorption depth measured with SPIRE based exclusively on the conserved quantity, equivalent width, . When deriving optical depths and column densities, we rely exclusively on spectrally resolved profiles such as that in @emprechtinger2012.
Distribution of HF absorbing clouds toward NGC6334 {#sec:absclouds}
--------------------------------------------------
The range of HF equivalent width values in the map shown in Fig. \[fig:HFabsmap\] can be divided into three regimes: (a) $>$12 , only occurring toward the position of core I; (b) 8–12 , spatially consistent with the larger scale filament in which cores I and I(N) are embedded; and (c) $\lesssim$5 , exclusively localized at projected distances $>$0.6 pc from the cores and the connecting filament. The non-detection of HF in the ‘OFF’ observation (see Sect. \[sec:obsresults\]), sparsely sampling the area just northwest of our map, is consistent with regimes (c) or (b).
### Distinguishing foreground from dense star-forming gas {#sec:distinguishforeground}
We interpret the three regimes in the context of the velocity resolved HF spectrum published by @emprechtinger2012, who identified seven individual physical components responsible for the HF absorption toward NGC6334 I: the dense envelope at =$-6.5$ , two compact subcores at $-6.0$ and $-8.0$ , and four foreground layers at $-3.0$, $0.0$, $+6.5$, and $+8.0$ . The spectral signature of each of these components is reproduced in our Fig. \[fig:HFabscomp\]a. Regime (a) requires all seven components to explain the total equivalent width of HF. The two other panels in Fig. \[fig:HFabscomp\] represent adaptations of the model from @emprechtinger2012 with progressively fewer absorption components taken into account. In Fig. \[fig:HFabscomp\]b, regime (b) is explained by the superposition of four absorbing foreground clouds, discarding the components associated with the envelope and subcores of NGC6334 I. We highlight that the HF absorption depth observed toward the dense star-forming envelope I(N) is consistent with regime (b). Differences in HF content between the dense envelopes I and I(N) are discussed in more detail in Sect. \[sec:freezeout\]. Finally, in Fig. \[fig:HFabscomp\]c, we show that regime (c) is consistent with a model composed of just two specific foreground clouds, namely those at $+6.5$ and $+8.0$ [@vanderwiel2010; @emprechtinger2012].
In contrast with the detailed study of the HF profile in @emprechtinger2012 and in this work, the spectral survey paper by @zernickel2012, analyzing $\sim$4300 individual spectral line features, uses a simplified model which explains the HF absorption with only three components. The two approaches are not inconsistent, but the latter paper groups components together as: (1) the NGC6334 I envelope and two subcores, (2) two foreground clouds that are kinematically close to the NGC6334 complex, and (3) two other foreground clouds with larger offsets [cf. Table 1 in @emprechtinger2012]. In this three-component model, regime (a) would be explained by groups (1)+(2)+(3), regime (b) by groups (2)+(3), and regime (c) by only group (3).
The combination of our HF absorption depth map with the previous, single-pointing, velocity resolved HF spectrum now reveals a three-dimensional picture of the layers of absorbing gas toward the NGC6334 complex. Our interpretation of the relative positions of the foreground layers is sketched in Fig. \[fig:geometrysketch\]. With the exception of the direction toward core I, the HF absorption depth at all other positions in the map can be explained by (a subset of) the four extended foreground clouds.
### Relation of foreground clouds to NGC6334 {#sec:foregroundrelation}
@vanderwiel2010 used a single-pointing [*Herschel*]{} HIFI observation of CH rotational line absorption, coupled with results from OH hyperfine line absorption measurements at radio wavelengths [@brooks2001], to suggest that the clouds at =$-3.0$ and $+0.0$ are associated with the NGC6334 complex, while the remaining two velocity components, at $+6.5$ and $+8.0$ , originate in foreground clouds farther away from the NGC6334 complex. This interpretation is consistent with the first two components being seen exclusively in regime (b) of the HF equivalent width map and the latter two components being spread over a more extended area of regimes (b) and (c) combined. Therefore, the spatial distribution of HF absorption measured with SPIRE iFTS supports the previous hypothesis that the $-3.0$ and $+0.0$ clouds are associated with NGC6334, since they have a spatial morphology that closely follows the dense molecular cloud traced by the 250 dust emission (Fig. \[fig:HFabsmap\]), i.e., a region roughly in east-west extent and stretching along the north-south direction. The vertical extent of the ‘related’ foreground clouds depicted in Fig. \[fig:geometrysketch\] is drawn directly from the observed east-west extent of these component in the plane of the sky.
Besides the diatomic hydride species HF, CH, and OH, a subset of our absorbing clouds also exhibit absorption lines due to [@emprechtinger2010; @vandertak2013a], [@ossenkopf2010b], and H$_2$Cl$^+$ [@lis2010a]. Absorption components in OH$^+$ and detected toward both cores by @indriolo2015a peak at =$-2$ and $+3$ . The former could be a blend of the $-3.0$ and $+0.0$ clouds seen in CH and HF, but the latter is inconsistent with any of our components. These lines were all detected in observations with the single-pixel, high spectral resolution [*Herschel*]{} HIFI spectrometer, in some cases toward both individual protostellar cores. It is interesting that ionized species, and H$_2$Cl$^+$, are only detected at values that match the $-3$ and $+0$ clouds. Chemical models tailored to halogen hydrides [@neufeld2009b] indicate that cation species become abundant under the influence of strong UV radiation. We hypothesize that the presence of and H$_2$Cl$^+$ is further evidence for the physical proximity of these two clouds to the massive protostars and the region embedded in one of the dense cores.
Another ionized species that has been detected in spectrally resolved observations toward NGC6334 is . While a velocity resolved observation of HF exists only toward the position of core I, 1–0 has been observed with HIFI toward both cores I and I(N), the latter as part of the WISH program [@vandishoeck2011; @benz2016]. Toward source I, the profile looks very similar to the HF profile, amounting to a total equivalent width of $\sim$20 . In comparison, the observation toward core I(N)[^4] reveals an equivalent width of only $\sim$13 . The majority (85%) of the reduced absorption toward core I(N) relative to core I falls in the $[-15,5]$ range, as expected if the missing components are the envelope and subcore components at =$-6.5$, $-6.0$, and $-8.0$ (cf. Fig. \[fig:HFabscomp\]), i.e., those components that occur only in regime (a) in our HF absorption map. As mentioned above in Sect. \[sec:obsresults\], the SPIRE iFTS spectral cube has too low signal-to-noise near 835 GHz to make a meaningful comparison with the signature detected by this instrument.
The foreground clouds at =$+6.5$ and $+8.0$ , supposedly unrelated to the NGC6334 dense filament [@brooks2001; @vanderwiel2010], have a combined HF equivalent width that is entirely consistent with the observation in this work that regime (c) is spatially extended beyond the dense filament. An unrelated set of foreground cloud(s) (the two leftmost clouds in Fig. \[fig:geometrysketch\]) are likely to have a larger angular extent than the background continuum source (dashed lines in Fig. \[fig:geometrysketch\]), despite a possibly modest linear size. This explains why a minimum level of HF equivalent width of $\sim$3 is detected not just toward the NGC6334 filament, but throughout the extent of our 6$\times$35 map (Figs. \[fig:HFabsmap\] and \[fig:SNmapHF\]).
All four foreground clouds detected in HF and CH are redshifted ($\geq$$-3$ ) with respect to the of the part of NGC6334 cloud near cores I and I(N) (around $-5$ , based on observations by @zernickel2013). Thus, the absorbing gas clouds are moving *toward* NGC6334, instead of following the Galactic rotation, which at $\ell=351\degr$ yields exclusively negative line-of-sight velocities for sources between Sun and NGC6334, i.e., approaching the standard of rest of stars in the Solar neighborhood. We therefore conclude that, in addition to the gas flows within the dense gas *along* the filament’s long axis [@zernickel2013], gas may also be accreting onto the filament in the perpendicular direction. We also note that, whereas @indriolo2015a [their Sect. 3.7 and Table 5] put the and absorption clouds toward both cores I and I(N) at the distance of 1.35 kpc of the NGC6334 cloud, their location along the sight line toward NGC6334 is in fact poorly constrained. Recognizing that there must be peculiar motions at play, deviating from the ‘rigid’ Galactic rotation curve, these clouds could in fact be anywhere between the local arm of the Milky Way and the Sagittarius arm that harbors the NGC6334 complex.
For the two foreground clouds related to NGC6334, at $-3$ and $+0$ , we calculate their total mass by multiplying the sum of their column densities with a rough estimate of the area covered by this component of 160$\times$270 (regime (b) in Fig. \[fig:HFabsmap\]), corresponding to 1.5 pc$^2$ at the distance of NGC6334. Depending on the choice of fiducial tracer, either taking $N_\mathrm{CH}$ from @vanderwiel2010 and $N_\mathrm{CH}$/$N_\mathrm{H_2}$= from @sheffer2008, or $N$(HF) from @emprechtinger2012 and $N_\mathrm{HF}$/$N_\mathrm{H_2}$= from @indriolo2013a, this calculation yields a mass in the range 37–98 or 69–191 , respectively. From symmetry arguments, a similar reservoir of additional gas is expected to lie behind the NGC6334 dense cloud. This means that a significant total mass of several hundred could be on its way to accreting onto the dense cloud NGC6334 near ($<$0.3 pc) the embedded cores I and I(N). This gas reservoir has escaped detection so far, because it does not appear in traditional gas tracers such as CO, , and CS. While there is evidence that the $-3.0$ and $+0.0$ foreground clouds are closer to NGC6334 than the $+6.5$ and $+8.0$ clouds, there is no direct metric of the geometrical distance along the line of sight from each cloud to the dense filament and cores. Therefore, we refrain from speculating about accretion time scales of even the ‘related’ clouds, since this would rely on unsupported assumptions on relative distances.
### Spatial distribution of HF toward other Galactic sight lines
The only other published work of a spatial map of HF absorption so far is that toward by @etxaluze2013, using data also obtained with [*Herschel*]{} SPIRE iFTS. A direct comparison of measured absorption line depths is complicated by the choice of @etxaluze2013 to present signal in terms of integrated optical depth, apparently without taking into account the systematic underestimation of optical depths derived from spectrally unresolved measurements, as discussed in Sect. \[sec:eqwidth\]. Nonetheless, it appears that the total HF absorption toward Sgr B2(M) is about an order of magnitude stronger than toward NGC6334 I. @etxaluze2013 find a variation of HF absorption depth of only a factor $\sim$2 across the $\sim$25 mapped area, significantly less variation than in our fully sampled map toward NGC6334 I. Any intrinsic variation may have been partially smoothed by the interpolation process that was applied in @etxaluze2013 to construct a map from the spatially undersampled SPIRE iFTS observation. More importantly, the line of sight toward Sgr B2, close to the Galactic Center, crosses many more spiral arms than that toward NGC6334. Evidence of this is found for example by @qin2010, who detect a total of 31 individual velocity components in CH rotational ground state absorption toward Sgr B2(M). The 30 foreground clouds, associated with the various intervening Galactic arms, amount to a total CH column density of , with Sgr B2(M) itself adding a component of only [@qin2010]. This is consistent with many sheets of foreground gas together creating a roughly uniform cover of absorbing gas spanning at least a few arcminutes on the sky. Comparatively little additional absorption is contributed by the massive molecular cloud Sgr B2 and the embedded cores in the background, which explains the lack of variation in HF absorption depth seen toward Sgr B2 by @etxaluze2013. In contrast, our map of HF absorption toward NGC6334 reveals a mix of components due to foreground clouds and the star-forming envelope and cores within NGC6334, which we are able to disentangle owing to the complementary, velocity resolved spectra of CH and HF obtained with [*Herschel*]{} HIFI [@vanderwiel2010; @emprechtinger2012].
In addition, we highlight the discovery by @lopez-sepulcre2013a of a foreground cloud toward the intermediate-mass star-forming core , based on single-pointing HIFI spectra of HF and other hydride molecules. Several oxygen-bearing hydrides show absorption exclusively at a blueshifted velocity relative to the background protostellar core. The same foreground cloud was later also identified in H$_2$Cl$^+$ by @kama2015a. The HF profile shows absorption at the same blueshifted velocity, but shows additional evidence for a second absorption component that matches the of the protostellar core [@lopez-sepulcre2013a]. Instead of kinematical and morphological arguments such as those used in this work to determine the physical location of foreground clouds toward NGC6334, @lopez-sepulcre2013a use detailed photochemical modeling to infer proximity of their OMC-2 foreground gas to a source of copious far-UV radiation. With that radiation source assumed to be the trapezium cluster of OB stars, it is concluded that the absorbing slab is physically connected to OMC-1. In an attempt to study the spatial distribution of this absorbing OMC-1 ‘fossil’ slab, we have searched archival SPIRE iFTS data toward ([*Herschel*]{} observation ID 1342214847) for signatures of HF at 1232.5 GHz (and at 835.1 GHz), but find no detections in any of the 37 (and 19) SSW (and SLW) detectors in the $\sim$3 footprint.
Freeze-out of HF in envelopes of dense cores {#sec:freezeout}
--------------------------------------------
The HF equivalent width of 15.9$\pm$1.4 measured at the position of NGC6334 I in our SPIRE map is explained in Sect. \[sec:absclouds\] by invoking the superposition of four absorption components in the foreground and three associated to the dense core I itself [see Fig. \[fig:HFabscomp\], and @emprechtinger2012]. A striking feature of our HF absorption map is the lack of additional absorption toward the position of core I(N), where the HF equivalent width of only 10.9$\pm$1.1 can be explained by the four foreground clouds alone, adding up to 10.5 (Fig. \[fig:HFabscomp\]b). In contrast, adding even just an envelope component similar to that of core I sums up to 13.7 (not counting the two subcore components), which is inconsistent with the observation toward core I(N). Since the envelope of core I(N) is more massive than that of core I, but has a similar size [see model fits in @vandertak2013a], the total gas column density toward core I(N) should be higher. Therefore, the lack of HF absorption associated to the I(N) core is not due to the difference in total () column. Instead, we hypothesize that HF is primarily frozen out onto dust grains in core I(N), while HF is in the gas phase in core I.
To support the hypothesis of HF being depleted from the gas phase in core I(N), we set up a rudimentary model based on the following ingredients. We take the spherically symmetric physical structure, i.e., radial profiles of density and temperature, of the envelopes of NGC6334 I and I(N) as fitted to submillimeter dust continuum maps and the far-infrared / submillimeter spectral energy distribution [@vandertak2013a]. We then calculate, at every radial point, the timescales for adsorption (freeze-out) and desorption (evaporation) of HF molecules onto dust grains. Following, e.g., @rodgers2003 and @jorgensen2005a, we assume that thermal desorption is the dominant mechanism that drives molecules from the grain surface back into the gas phase, and are left with the balance between adsorption rate: $$\lambda(n_\mathrm{H}, T_\mathrm{gas}) = \powm{4.55}{-18} \left( \frac{T_\mathrm{gas}}{m_\mathrm{HF}} \right)^{0.5} n_\mathrm{H} \qquad [\mathrm{s}^{-1}] ,
\label{eq:freezerate}$$ and desorption rate: $$\xi(T_\mathrm{dust}) = \nu_\mathrm{vib} \exp\left( -\frac{E_\mathrm{b,HF}}{k\,T_\mathrm{dust}} \right) \qquad [\mathrm{s}^{-1}] .
\label{eq:desorbrate}$$ Here, and are the temperatures of gas and dust, assumed to be equal as in the modeling of @vandertak2013a, $m_\mathrm{HF}$ is the molecular weight of HF (20), $n_\mathrm{H}$ is the density of hydrogen nuclei, $\nu_\mathrm{vib}$ is the vibrational frequency of the HF molecule in its binding site, for which we adopt $10^{13}$ s$^{-1}$, $k$ is the Boltzmann constant, and $E_\mathrm{b,HF}$ is the binding energy of HF to the dust grain surface. It has previously been inferred by @philips2010 that a density of $\sim$$10^5$ allows HF to condense onto dust grains, whereas densities of $\sim$$10^3$ more typical for diffuse gas are too low for HF freeze-out to occur. In our case, the density $n_\mathrm{H}$ – which incidentally exceeds $10^5$ at almost all radii in the envelopes of I and I(N) – enters directly into Equation \[eq:freezerate\] to govern the adsorption rate.
In this work we consider multiple versions of the desorption timescale, because the binding energy in the exponent of Equation \[eq:desorbrate\] is heavily dependent on the type of grain surface. Unlike for more common molecular species such as CO [e.g., @bisschop2006; @noble2012], the desorption behavior of HF from astrophysically relevant grain surfaces has not been studied experimentally, so we rely on theoretical calculations. Typical interstellar dust grains, especially those embedded in cold, star-forming regions, are covered in one or multiple layers of ice consisting of various molecules, mainly , CO, and [for a recent review, see @boogert2015]. We collect binding energy values for several types of grain surfaces in Table \[t:Ebinding\]. For CO and ice covered grains we adopt calculated binding energies from the literature [@chen2006; @rivera-rivera2012], while for hydrogenated bare silicate grains and ice covered grains, these values result from original ab initio chemical calculations performed for this work.
-------------------------------- ------------------- --------------------- -------
Type of grain surface $E_\mathrm{b,HF}$ $E_\mathrm{b,HF}/k$ Ref.
(kJ/mol) ($10^3$ K)
Hydrogenated crystaline silica 9.2 1.1 \[1\]
ice on amorphous silica 53 6.3 \[1\]
CO ice on amorphous silica 8.9 1.07 \[2\]
ice on amorphous silica 9.3 1.12 \[3\]
-------------------------------- ------------------- --------------------- -------
: Binding energies for HF onto various surfaces.[]{data-label="t:Ebinding"}
To calculate the HF binding energies in the first two rows of Table \[t:Ebinding\] (with a bare grain and with ice), we carry out quantum calculations within the Kohn-Sham implementation of Density Functional Theory using the Quantum Espresso Simulation Package [@giannozzi2009]. Perdew-Burke-Ernzerhof exchange-correlation functional ultrasoft pseudopotentials are used. KS valence states are expanded in a plane-wave basis set with a cutoff at 340 eV for the kinetic energy. The self-consistency of the electron density is obtained with the energy threshold set to $10^{-5}$ eV. Calculations are performed using the primitive unit cell containing a total number of 46 atoms for bare hydrogenated silica, and 54 atoms for hydrogenated silica covered with one layer of ice. The geometry optimization is used within the conjugate gradients scheme, with a threshold of 0.01 eV$\AA^{-1}$ on the Hellmann-Feynman forces on all atoms; the Si atoms of the bottom layers are fixed at their bulk values. The binding energy of HF with the SiH terminus of hydrogenated crystalline silica is based on calculations for the hydroxylated alpha-quartz (001) surface. The binding energy of HF with one layer of ice on amorphous hydrogenated silicate is estimated by assuming that the the most common structure in this case is the HF molecule interacting with a molecule bonded to silanol (SiOH), which is the most abundant surface group in amorphous silica [@ewing2014].
The binding energies of HF with CO and ice (last two rows of Table \[t:Ebinding\]) are taken from calculations by @rivera-rivera2012 and @chen2006, respectively. These authors performed calculations for molecules in the gas phase. We consider the gas phase binding energies of HF with CO and to be similar to those of HF with CO and ices adsorbed on an inert surface such as that of hydroxylated amorphous silica. This approximation is based on the weak interactions of these ices with hydroxylated silica and within the CO and molecular solids, so that the electronic density of CO and in solid form is not significantly altered with respect to their state in the gas phase. Hence, for the aim of the present work, the binding energy of the HF molecule with CO or as calculated in the gas phase is applicable for the condensed phase. The situation is notably different for interactions with in the gas or adsorbed form, because of its stronger interaction with silica and HF. For HF interacting with ice, we use binding energies from our own calculations described above.
With the physical structure of both envelopes, the equations for the adsorption/desorption balance, and the binding energy values, the ‘freeze-out’ region within each envelope is calculated in Fig. \[fig:HFphasediagram\]. Defining $t_\mathrm{freeze} = 1/\lambda$ and $t_\mathrm{desorb}=1/\xi$, HF molecules will deplete from the gas phase in the region of the envelope wherever $t_\mathrm{desorb}$ > $t_\mathrm{freeze}$. Numerical simulations by @das2016 suggest that, within a mixed-composition ice layer, the abundances of CO and are enhanced compared to in high-density ($\gtrsim10^5$ ) environments. At such densities applicable for the protostellar envelopes studied here, we thus expect the HF binding energy to lie close to, but slightly above that of pure CO or ice, and the true desorption time scale line in Fig. \[fig:HFphasediagram\] therefore somewhat to the left of the dash-dotted line for CO ice. In this case, HF is expected to stay frozen onto grain surfaces at a broad range of radii in core I(N): cumulative mass $\sim$0.1 to 1, i.e., 90% of the mass, where the temperature is $\lesssim$20 K. Core I(N) is overall colder than core I both in the envelope (blue solid lines in Fig. \[fig:HFphasediagram\]) and in the embedded subcores [@hunter2006]. For the comparatively warmer envelope of core I, the lines for the desorption timescale of HF from CO/ ice fall off the scale on the right hand side of the axes, leaving no freeze-out zone in this envelope. This could explain why HF is seen in the gas phase in core I, but not in core I(N).
In an alternative scenario in which the dust grains are covered in pure ice – or the desorption characteristics of a mixed mantle are dominated by that of ice [@collings2004] – the binding energy of HF with the ice would be greatly increased (see Table \[t:Ebinding\]). In this case, the HF freeze-out zone would expand to cover $>$98% of the mass for *both* envelopes, and our observations should have revealed no gas phase HF in either of the cores. Our observations are therefore inconsistent with pure ice coating on the grains. Instead, our interpretation relies on a significant part of the ice coating to consist of CO and/or molecules. In principle, the composition of ice coatings do not need to be the same in the two neighboring envelopes. Particularly, given the lower temperatures of core I(N), there is higher probability that a significant amount of is frozen out in that envelope. This, again, would enhance the binding energy of HF onto the ice-covered dust grains in the envelope of core I(N), which may help to explain the lack of gas-phase HF observed toward source I(N). Conversely, if additional mechanisms for desorption, e.g., induced by (UV) photons or cosmic rays, would be taken into account, $\xi$ would increase, and the freeze-out region would be pushed to larger radii in the envelopes. Particularly UV photodesorption could have a different effect in one envelope compared to the other, because core I is more evolved and contains an region. An increased total desorption rate would have no effect on HF freeze-out in the envelope of core I, in which HF is already completely in the gas phase, but would reduce the size of the freeze-out zone for envelope I(N). If, however, thermal desorption as expressed in Equation \[eq:desorbrate\] is the dominant desorption mechanism and the binding energy of HF onto the ice surface is close to that of CO or ice (Table \[t:Ebinding\]), our model predicts significant freeze-out of HF in core I(N), and none in core I. It is important to note that, indeed, the temperature and density conditions under which HF remains frozen onto dust grains depend greatly on the exact composition and mixing of the ice mantle and therefore on the chemical history. It has already been recognized for example for the CO molecule that the freeze-out temperature ‘threshold’ can vary considerably from one object to another [@qi2015b].
Conclusions {#sec:conclusions}
===========
In this work we present a map of HF absorption toward the northern end of the molecular cloud NGC6334, containing two well studied massive star-forming cores I and I(N). Although in the original definition of the observing program it was not anticipated that hydride absorption lines would be found within these data, the discovery space provided by the enormous frequency coverage of the [*Herschel*]{} SPIRE iFTS instrument has made this study possible. Such wide coverage in the far-infrared/submillimeter is only attainable with broadband FTS spectrometers [see @naylor2013 for a review]. The absorption line of HF is detected in 80% of our mapped area, although it is spectrally unresolved by SPIRE. By complementing the new SPIRE iFTS data with existing, single pointing, high spectral resolution spectra from the [*Herschel*]{} HIFI instrument [@vanderwiel2010; @emprechtinger2012], we construct a three-dimensional picture of gas clouds in front of and inside the massive star-forming filament NGC6334.
We find that our observations are consistent with a scenario of four individual foreground clouds on the line of sight toward NGC6334 I and I(N), two of which are unrelated to the star-forming complex (Sect. \[sec:distinguishforeground\]). The other two clouds are posited to be close to the dense molecular filament based on their spatial morphology. Their velocities are such that they are moving toward the star-forming cloud and could be adding several hundreds of solar masses of gas to the dense filament and the embedded cores in which massive star formation is already ongoing (Sect. \[sec:foregroundrelation\]). This component of gas is detected in rotational lines of diatomic hydride molecules, but had been unseen in studies of traditional dense gas tracers. In fact, using such a tracer, , @zernickel2013 have inferred that the roughly cylindrically shaped NGC6334 filament is collapsing along its longest axis. Our work now indicates that accretion may also be ongoing in the perpendicular (radial) direction. Future studies of (competitively) accreting high-mass star-forming cores may need to take into account this additional low-density phase of the gas reservoir.
Finally, in Sect. \[sec:freezeout\] we explain why HF is observed in the gas phase toward core I, but appears completely absent in core I(N). For this purpose, we use a simple description of adsorption and desorption time scales for HF interacting with dust grain surfaces, depending on the (radially variable) density and temperature. Since interactions of HF with interstellar-like dust grains have not been studied in the laboratory, we adopt binding energy values for different types of grain surfaces from theoretical calculations from the literature as well as from original work first presented in this paper. The conclusion is that the lower temperature of core I(N) compared to core I could lead to freeze-out of HF exclusively in the former, but only if the binding energy of HF onto the grain surface is governed by that of CO or ice on a silicate surface. In this case, at the densities relevant in the envelope of source I(N) ($>$ ), we find that HF freezes out in the region of the envelope where the temperature is below $\sim$20 K, rather similar to the freeze-out temperature often adopted for CO. In contrast, if is the dominant constituent in the ice mantles, our model predicts that HF should have been frozen out at all radii in the envelopes of both sources I(N) and I. Since we observe a significant amount of HF in the gas phase in source I, this scenario is inconsistent with our data.
Summarizing, this work uses HF as a sensitive tracer for (molecular) gas at relatively low densities that may be contributing mass to star forming cores. The HF signature reveals a gas reservoir that is inconspicuous in traditional dense gas tracers such as CO. In addition, we show that gas phase HF in higher density environments ($>$$10^5$ ) is extremely sensitive to interactions with dust grains and will be depleted significantly at low dust temperatures.
The research of MHDvdW at the University of Lethbridge was supported by the Canadian Space Agency (CSA) and the Natural Sciences and Engineering Research Council of Canada (NSERC), and at the University of Copenhagen by the Lundbeck Foundation. Research at the Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen’s programme of excellence.\
SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA). HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.\
This research has made use of NASA’s Astrophysics Data System Bibliographic Services. The graphical representations of the results in this paper were created using APLpy, an open-source plotting package for Python hosted at , Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration [-@astropy2013]), and the matplotlib plotting library [@matplotlib2007].\
The authors are grateful to Alexander Zernickel for providing and discussing excerpts of the CHESS spectral survey data, to Raquel Monje for providing the HF spectral model component profiles, and to Floris van der Tak for providing the physical structure models of the two envelopes in electronic table format. We thank Tommaso Grassi, Wing-Fai Thi, Jes Jørgensen, Søren Frimann, and Mihkel Kama for discussions.
Complementary figures
=====================
Maps displaying the signal-to-noise ratio of the detections of the HF and lines in each spatial pixel of our spectral cube (Sect. \[sec:obsresults\]) are shown in Fig. \[fig:SNmapHF\] and \[fig:SNmapCHplus\]. The colored contours in Fig. \[fig:SNmapCHplus\] show that absorption is only confidently detected (signal-to-noise $>$ 5) in the northeastern section of the map.
In addition, although the signal from nitrogen species is not interpreted in this paper, maps of line absorption depth due to NH and NH$_2$ are shown in Figs. \[fig:SNmapNH\] and \[fig:SNmapNH2\]. The continuum-normalized cube for these lines is created – analogous to those for HF and in Sect. \[sec:obsresults\] – by fitting the continuum and the CO 9–8 line in the 945–1055 GHz section of the SSW cube. Absorption lines of NH (at 974.47 and 999.98 GHz) and two of NH$_2$ (at 952.57 and 959.50 GHz) are then fitted simultaneously with emission lines due to 9–8 at 991.3 GHz and at 987.9 GHz.
[^1]: [*Herschel*]{} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
[^2]: See the introduction section of this reference for a list of earlier spectral survey work on NGC6334 I.
[^3]: The logarithm operator gives a disproportionally higher weight to more strongly absorbed channels. For example, a 20 wide feature with constant 80% absorption ($I_\mathrm{norm}$=0.20) and a smeared-out version of 320 wide at 5% absorption ($I_\mathrm{norm}$=0.95) both have the same equivalent width, = 16 . The latter, however, yields a $\int \tau_\nu \mathrm{d}V$ that is smaller by a factor 2. The effect is larger yet for line profiles that come even closer to full absorption.
[^4]: Estimate based on the HIFI spectrum downloaded from the Science Archive, observation ID 1342214306, processed with HIPE pipeline version 13.
| ArXiv |
---
abstract: 'We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0\,,$$ where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb S}$. This is an extension of Gursky’s gap theorem for positive Einstein metrics on the four-sphere.'
address:
- 'Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan'
- 'Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan'
- 'Mathematics Department, Indian Institute of Science, 560012 Bangalore, India'
author:
- 'Kazuo Akutagawa${}^*$'
- 'Hisaaki Endo${}^{**}$'
- Harish Seshadri
date: 'January, 2018; February, 2018 (revised version).'
title: |
A gap theorem for positive Einstein metrics\
on the four-sphere
---
Introduction and main results
=============================
A smooth Riemannian metric $g$ is said to be [*Einstein*]{} if its Ricci tensor ${\rm Ric}_g$ is a constant multiple $\lambda$ of $g$: $${\rm Ric}_g = \lambda\,g\,.$$ When such a metric exists, it is natural to ask whether it is unique. However in dimension $n \geq 5$, there exist many examples of closed $n$-manifolds each of which has infinitely many non-homothetic Einstein metrics (cf.[@Besse]). In fact, there exists infinitely many non-homothetic Einstein metrics of positive sacalar curvature ([*positive Einstein*]{} for brevity) on $S^n$ when $5 \le n \le 9$ [@Bohm] (cf.[@Jensen], [@B-K]). There are no non-existence or uniqueness results known when $n \geq 5$.
When $n = 4$, there are necessary topological conditions for a closed $4$-manifold $M$ to admit an Einstein metric [@Thorpe], [@Hitchin-1], [@LeBrun-2]. Uniqueness is known in some special cases: when $M$ is a smooth compact quotient of real hyperbolic $4$-space ([resp.]{} complex-hyperbolic $4$-space), the standard negative Einstein metric is the unique Einstein metric (up to rescaling and isometry) [@BCG] ([resp.]{}[@LeBrun-1]). In the positive case, there are some partial rigidity results on the $4$-sphere $S^4$ and the complex projective plane $\mathbb{CP}^2$ [@GL], [@G], [@Y]. When $M = S^4$, the standard round metric $g_{\mathbb{S}}$ of constant curvature $1$ is, to date, the only known Einstein metric (up to rescaling and isometry). In this connection we have the following gap theorem due to M.Gursky (see [@ABKS] for the significance of the constant $\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$):
\[Gursky\] Let $g$ be a positive Einstein metric on $S^4$. If its Yamabe constant $Y(S^4, [g])$ satisfies the following inequality $$Y(S^4, [g]) \geq \frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$$ then, up to rescaling, $g$ is isometric to $g_{\mathbb{S}}$. Here, $[g]$ denotes the conformal class of $g$.
Note that $Y(S^4, [h]) \leq Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi$ for any Riemannian metric $h$ and that $Y(S^4, [g]) = R_g \sqrt{V_g}$ for any Einstein metric $g$, where $R_g$ and $V_g = {\rm Vol}(S^4, g)$ denote respectively the scalar curvature of $g$ and the volume of $(S^4, g)$.
Our main result in this paper is an extension of Theorem\[Gursky\]:
\[MainThm1\] There exists a universal positive constant $\varepsilon_0 > 0$ with the following property$:$ If $g$ is a positive Einstein metric on $S^4$ with Yamabe constant $$Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0,$$ then, up to rescaling, $g$ is isometric to $g_{\mathbb{S}}$.
This result can be restated in terms of the [*Weyl constant*]{} of $[g]$ (cf.[@ABKS]). Indeed, the Chern-Gauss-Bonnet theorem (see Remark\[ALE\]-(1)) implies that the lower bound on the Yamabe constant is equivalent to the following upper bound on the Weyl constant: $\int_M |W_g|^2 d \mu_g < \frac{32}{3} \pi^2 + \widetilde{\varepsilon}_0$, where $\widetilde{\varepsilon}_0 := \frac{\varepsilon_0}{24}(16\sqrt{2}\pi - \varepsilon_0) > 0$.
More generally, we obtain the following (note that $8\sqrt{2}\pi = \frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}])$):
\[MainThm2\] For $c > 0$, let $\mathcal{E}_{\geq c}(S^4)$ denote the space of all unit-volume positive Einstein metrics $g$ on $S^4$ with $c \leq Y(S^4, [g]) < 8\sqrt{2}\pi$. Then the number of connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is finite. In particular, $\{ Y(S^4, [g]) \in [c, 8\sqrt{2}\pi)\ |\ g \in \mathcal{E}_{\geq c}(S^4) \}$ is a finite set $($possible empty$)$.
Here $ \mathcal{M}_1(S^4)/{\rm Diff}(S^4)$ has the $C^\infty$-topology and $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is endowed with the subspace topology.
These theorems follow from the following crucial result:
\[MainProp\] Let $\{g_i\}$ be a sequence in $ \mathcal{E}_{\geq c}(S^4)$ for some positive constant $c > 0$. Then there exists a subsequence $\{j\} \subset \{i\}$, $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $\mathcal{M}_1(S^4)$.
[**Remark:**]{} TheoremD of [@Anderson] states that the same conclusion as the one in Proposition\[MainProp\] holds for any sequence $\{g_i\} \subset \mathcal{E}_{\geq c}(M)$ on any closed $4$-manifold $M$ with $1 \leq \chi(M) \leq 3$, where $\chi(M)$ denotes the Euler characteristic of $M$. Unfortunately, the proof appears to be incorrect. Specifically, Theorem D is based on Lemma 6.3, which asserts that a Ricci-flat ALE 4-space $X$ with $\chi(X)=1$ is necessarily isometric to the Euclidean $4$-space $({\mathbb R}^4, g_{\mathbb{E}})$. This is not true: the Ricci-flat ALE 4-space $X_1$ constructed by Eguchi-Hanson [@EH] has a free, isometric ${\mathbb Z}_2$-action whose quotient $X_2 = X_1/{\mathbb Z}_2$ is a Ricci-flat ALE $4$-space with $\chi(X_2)=1$. Note that $X_2$ is nonorientable. Even if we assume that $X$ is orientable in Lemma6.3, the topological argument in the proof still contains some gaps. Proposition3.10 of [@Anderson-GAFA] corrects a minor inaccuracy of Lemma6.3. However, the proof also contains some gaps in the topological argument (see Remark\[Counter\] in $\S$4 for details).
Gursky’s proof of Theorem\[Gursky\] involves a sophisticated Bochner technique, a modified scalar curvature and a conformal rescaling argument. The proof of Proposition1.4 is based on topological results about $S^3$-quotients embedded in $S^4$ and the convergence theory of Einstein metrics in four-dimensions. Given this proposition, we invoke Gursky’s result to prove Theorems\[MainThm1\] and\[MainThm2\].
In $\S$2, we recall some background material and prove Theorems\[MainThm1\] and\[MainThm2\], assuming Proposition1.4. In $\S$3, we review two key results needed for the proof of Proposition1.4. Finally, in $\S$4, we prove Proposition1.4.\
\
[**Acknowledgements.**]{} The authors would like to thank Anda Degeratu and Rafe Mazzeo for valuable discussions on the eta invariant, and Shouhei Honda for helpful discussions on convergence results of Riemannian manifolds with bounded Ricci curvature. They would also like to thank Matthew Gursky and Claude LeBrun for useful advices, and Gilles Carron for crucial comments.\
Preliminaries and proofs of Theorems 1.2 and 1.3
================================================
We first review the definitions of Yamabe constants and Yamabe metrics. Let $M^n$ be a closed $n$-manifold with $n \geq 3$. It is well known that a Riemannian metric on $M$ is Einstein if and only if it is a critical point of the normalized Einstein-Hilbert functional $I$ on the space $\mathcal{M}(M)$ of all Riemannian metrics on $M$ $$I : \mathcal{M}(M) \rightarrow \mathbb{R},\quad g \mapsto I(g) := \frac{\int_MR_gd\mu_g}{{\rm Vol}(M, g)^{(n-2)/n}},$$ where $d\mu_g$ denotes the volume form of $g$. The restriction of $I$ to any conformal class $[g] := \{ e^{2f}\,g\ |\ f \in C^{\infty}(M) \}$ is always bounded from below. Hence, we can consider the following conformal invariant $$Y(M, [g]) := \inf_{\widetilde{g} \in [g]}I(\widetilde{g}),$$ which is called the [*Yamabe constant*]{} of $(M, [g])$. A remarkable theorem of H.Yamabe, N.Trudinger, T.Aubin and R.Schoen asserts that each conformal class $[g]$ contains metrics $\check{g}$, called [*Yambe metrics*]{}, which realize the minimum (cf.[@LP], [@Sc-1]) $$Y(M, [g]) = I(\check{g}).$$ These metrics must have constant scalar curvature $$R_{\check{g}} = Y(M, [g])\cdot V_{\check{g}}^{-2/n},$$ where $V_{\check{g}} = {\rm Vol}(M, \check{g})$. Aubin proved that $$Y(M^n, C) \leq Y(S^n, [g_{\mathbb{S}}]) = n(n-1) V_{g_{\mathbb{S}}}^{2/n}$$ for any conformal class $C$ on $M$. Obata’s Theorem[@Obata] implies that [*any Einstein metric is a Yamabe metric*]{}. When $n = 4$, $$Y(M^4, [g]) = R_{\widehat{g}} \sqrt{V_{\widehat{g}}} \leq Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi$$ for any Einstein metric $\widehat{g} \in [g]$.
Assuming Proposition1.4, we can now prove Theorem\[MainThm1\].
Suppose that there exists a sequence $\{g_i\}$ of unit-volume Einstein metrics on $S^4$ satisfying $$Y(S^4, [g_i]) = R_{g_i} < 8\sqrt{2}\pi\ \ ({\rm for}\ \ \forall i),\quad Y(S^4, [g_i]) = R_{g_i} \nearrow 8\sqrt{2}\pi\ \ ({\rm as}\ \ i \to \infty).$$ By Proposition1.4, there exists a subsequence $\{j\} \subset \{i\}$, a sequence $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $S^4$. Then, we get $$\label{lll}
Y(S^4, [g_{\infty}]) = R_{g_{\infty}} = 8\sqrt{2}\pi.$$
On the other hand, Theorem\[Gursky\] implies that $(S^4, g_{\infty})$ is isometric to $(S^4, g_{\mathbb{S}})$. Hence, $$Y(S^4, [g_{\infty}]) = Y(S^4, [g_{\mathbb{S}}]) = 8\sqrt{6}\pi.$$ This contradicts (\[lll\]). Therefore, there exists a positive constant $\varepsilon_0 > 0$ such that any unit-volume positive Einstein metric $g$ on $S^4$ satisfying $$Y(S^4, [g]) > 8\sqrt{2}\pi - \varepsilon_0$$ is isometric to $g_{\mathbb{S}}$.
By the result of N. Koiso[@Koiso-2 Theorem3.1] and [@Besse Corollary12.52] (cf.[@Ebin Theorem7.1], [@Koiso-1 Theorem2.2]), we first remark that, for each $g \in \mathcal{E}_{\geq c}(S^4)$, the premoduli space $\mathcal{E}_{\geq c}(S^4)$ around $g$ is a real analytic subset of a finite dimensional real analytic submanifold in $\mathcal{M}_1(S^4) := \{ g \in \mathcal{M}(S^4)\ |\ V_g = 1 \}$, and the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is arcwise connected. Moreover, the Yamabe constant $Y(S^4, [\bullet])$ is a locally constant function and and it takes (at most) countably many values on $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$.
Suppose that there exist infinitely many connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ (see [@Besse Chapters4 and 12] for the topology on it). Then, there exists a sequence $\{g_i\}$ in $\mathcal{E}_{\geq c}(S^4)$ such that the equivalence classes of any two $g_{i_1}$ and $g_{i_2}$ are contained in different connected components of $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ if $i_1 \ne i_2$. Similar to the proof of Theorem\[MainThm1\], there exists a subsequence $\{j\} \subset \{i\}$, a sequence $\{\phi_j\} \subset {\rm Diff}(S^4)$ and a unit-volume positive Einstein metric $g_{\infty}$ on $S^4$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ with respect to the $C^{\infty}$-topology on $S^4$. We note here that the topology of the moduli space is induced from the one of the space $\mathcal{M}_1(S^4)$. Then, there exists a large positive integer $j_0$ such that the set $\{\phi_j^*g_j\}_{j \geq j_0}$ is contained in a connected component. This contradicts the choice of $\{g_i\}$. Hence, the number of connected components of the moduli space $\mathcal{E}_{\geq c}(S^4)/{\rm Diff}(S^4)$ is finite (possibly zero). In particular, the set of $\{ Y(S^4, [g]) \in [c, 8\sqrt{2}\pi)\ |\ g \in \mathcal{E}_{\geq c}(S^4) \}$ is a finite set $($possibly empty$)$.
A review of two key results
===========================
An embedding theorem:
---------------------
It will be necessary to know which quotients $S^3/\Gamma$ of $S^3$ embed smoothly in $S^4$. The theorem below gives a complete answer, which is one of the two key results for the proof of Proposition1.4.
\[Key-2\] Let $\Gamma \subset SO(4)$ be a finite subgroup such that $S^3/\Gamma$ is a smooth quotient of $S^3$. If $S^3/\Gamma$ can be smoothly embedded in $S^4$, then either $\Gamma = \{1\}$ or $\Gamma = Q_8$. Here, $Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}$ denotes the quaternion group.
Convergence of Einstein metrics:
--------------------------------
We first review the definition of the energy of metrics on $4$-manifolds.
$(1)$ For a closed Riemannian $4$-manifold $(M, g)$, the [*energy*]{} $\mathscr{E}(g)$ of $g$ (or $(M, g)$) is defined by $$\mathscr{E}(g) := \frac{1}{8\pi^2}\int_M|\mathscr{R}_g|^2d\mu_g,$$ where $\mathscr{R}_g = (R^i_{\ jk\ell})$ denotes the curvature tensor of $g$ and $|\mathscr{R}_g|^2 = \frac{1}{4}R^i_{\ jk\ell}R_i^{\ jk\ell}$.
$(2)$ If $(X,h)$ is an [*asymptotically locally Euclidean*]{} $4$-manifold of order $\tau > 0$ (ALE $4$-space for brevity, cf.[@BKN]), the energy $\mathscr{E}(h)$ of $h$ (or $(X, h)$) is again defined by $$\mathscr{E}(h) := \frac{1}{8\pi^2}\int_X|\mathscr{R}_h|^2d\mu_h < \infty.$$
\[ALE\] $(1)$ By the Chern-Gauss-Bonnet formula, $\mathscr{E}(g) = \chi(M)$ for any Einstein metric $g$ on a closed $4$-manifold $M$. Indeed, $$\begin{aligned}
\mathscr{E}(g) &= \frac{1}{8\pi^2}\int_M|\mathscr{R}_g|^2d\mu_g = \frac{1}{8\pi^2}\int_M(|W_g|^2 + \frac{1}{24}R_g^2 + \frac{1}{2}|E_g|^2)d\mu_g\\
&= \frac{1}{8\pi^2}\int_M(|W_g|^2 + \frac{1}{24}R_g^2 - \frac{1}{2}|E_g|^2)d\mu_g = \chi(M),\end{aligned}$$ where $W_g = (W^i_{\ jk\ell})$ and $E_g = (E_{ij})$ denote respectively the Weyl tensor and the trace-free Ricci tensor ${\rm Ric}_g - \frac{R_g}{4}g$ of $g$, and $|W_g|^2 = \frac{1}{4}W^i_{\ jk\ell}W_i^{\ jk\ell}$. In particular, $\mathscr{E}(g) = 2$ if $M = S^4$.\
$(2)$ The Chern-Gauss-Bonnet formula for $4$-manifolds with boundary implies the following (cf.[@Hitchin-2 formula(7)]): any Ricci-flat ALE $4$-space $(X, h)$ with end $S^3/\Gamma$ satisfies $$\chi(X) = \mathscr{E}(h) + \frac{1}{|\Gamma|},$$ where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $\mathbb{R}^4 - \{0\}$ and $|\Gamma|$ is the order of $\Gamma$. If $\chi(X) = 1$, we get, in particular, the following: $$\mathscr{E}(h) = 1 - \frac{1}{|\Gamma|}.$$ $(3)$ Bando-Kasue-Nakajima[@BKN] proved that any Ricci-flat ALE $4$-space $(X, h)$ is an ALE $4$-space of order of $4$. Moreover, when $(X, h)$ is [*asymptotically flat*]{} (AF for brevity, cf.[@Bartnik]), that is, $\Gamma = \{1\}$, this combined with a result of R. Bartnik[@Bartnik Theorem4.3] implies that the mass of $(X, h)$ is zero. The Positive Mass Theorem[@Sc-1 Theorem4.3] for AF manifolds then implies that $(X, h)$ is isometric to $(\mathbb{R}^4, g_{\mathbb{E}})$. Note that $\mathscr{E}(h) = \mathscr{E}(g_{\mathbb{E}}) = 0$.
Recall again that any Einstein metric $g$ on a closed $4$-manifold $M$ satisfies that $Y(M, [g]) = R_g\sqrt{V_g}$. Moreover, if $g$ is a unit-volume Einstein metric with $Y(M, [g]) \geq c\ (c > 0)$, then ${\rm Ric}_g \geq \frac{c}{4}g$. Hence, Myers’ diameter estimate gives $${\rm diam}(M, g) \leq \frac{2\sqrt{3}\pi}{\sqrt{c}}.$$
Using this fact and Remark3.3-(1), we can now state a modified version of the convergence theorem for Einstein metrics due to M. Anderson[@Anderson], H. Nakajima[@Nakajima-1] and Bando-Kasue-Nakajima[@BKN], which is the other of the two key results for the proof of Proposition1.4.
\[Key-1\] Let $M$ and $\{g_i\}$ be respectively a closed $4$-manifold and a sequence of unit-volume positive Einstein metrics on $M$ with $Y(M, [g]) \geq c$ for a fixed $c > 0$. Then, there exist a subsequence $\{j\} \subset \{i\}$ and a compact Einstein $4$-orbifold $(M_{\infty}, g_{\infty})$ with a finite singular points $\mathcal{S} = \{p_1, p_2, \cdots, p_{\ell}\} \subset M_{\infty}$ $($possibly empty$)$ and an orbifold structure group $\Gamma_a \subset O(4)$ around $p_a$ for which the following assertions hold$:$\
$(1)$ $(M, g_j)$ converges to $(M_{\infty}, g_{\infty})$ in the Gromov-Hausdorff distance.\
$(2)$ There exists a smooth embedding $\phi_j : M_{\infty} - \mathcal{S} \rightarrow M$ for each $j$ such that $\phi_j^*g_j$ converges to $g_{\infty}$ in the $C^{\infty}$-topology on $M_{\infty} - \mathcal{S}$. If $\mathcal{S}$ is empty, then each $\phi_j$ is a diffeomorphism from $M_{\infty}$ onto $M$.\
$(3)$ For each $p_a \in \mathcal{S}$ and $j$, there exists $p_{a, j} \in M$ and a positive number $r_j$ such that\
$(3.1)$ $B_{\delta}(p_{a, j}; g_j)$ converges to $B_{\delta}(p_a;g_{\infty})$ in the pointed Gromov-Hausdorff distance for all $\delta > 0$, where $B_{\delta}(p_{a, j}; g_j)$ denotes the geodesic ball of radius $\delta > 0$ centered at $p_{a, j}$ with respect to $g_j$.\
$(3.2)$ $\lim_{j \to \infty}r_j = 0$.\
$(3.3)$ $((M, r_j^{-2}g_j), p_{a, j})$ converges to $((X_a, h_a), x_{a, \infty})$ in the pointed Gromov-Hausdorff distance, where $(X_a, h_a)$ is a complete, non-compact, Ricci-flat, non-flat ALE $4$-space of order $4$ with $$0 < \int_{X_a}|\mathscr{R}_{h_a}|^2d\mu_{h_a} < \infty,$$ and $x_{a, \infty} \in X_a$.\
$(3.4)$ There exists smooth embeddings $\Phi_j : X_a \rightarrow M$ such that $\Phi_j^*(r_j^{-2}g_j)$ converges to $h_a$ in the $C^{\infty}$-topology on $X_a$.\
$(4)$ It holds that $$\lim_{j \to \infty}\int_M|\mathscr{R}_{g_j}|^2d\mu_{g_j} \geq \int_{M_{\infty}}|\mathscr{R}_{g_{\infty}}|^2d\mu_{g_{\infty}} + \sum_a\int_{X_a}|\mathscr{R}_{h_a}|^2d\mu_{h_a}.$$
\[Tree\] Since $S^3/\Gamma_a$ is smoothly embedded in $M_{\infty}$ around $p_a$ for each $a$, it is also smoothly embedded in $M$ and it separates $M$ into two components $V_a, W_a$, which are compact $4$-manifolds with boundary. More precisely, $M = V_a \cup W_a,\ S^3/\Gamma_a = \partial V_a = \partial W_a = V_a \cap W_a$. Here, we choose $V_a$ satisfying $V_a \subset M_{\infty}$. The infinity $X_a(\infty) \cong S^3/\widetilde{\Gamma}_a$ of $X_a$ is also smoothly embedded in $M$. By the existence of intermediate Ricci-flat ALE $4$-orbifolds in the bubbling tree arising from each singular point $p_a$, $\Gamma_a \ne \widetilde{\Gamma}_a$ in general (cf.[@Bando], [@Nakajima-3]).
Proof of Proposition1.4
=======================
Let $\{g_i\}$ be a sequence of positive Einstein metrics on $S^4$ with $\{g_i\} \subset \mathcal{E}_{\geq c}(S^4)$ for some $c > 0$. We apply Theorem\[Key-1\], with $M=S^4$, for the sequence $\{g_i\}$. Then, in order to prove Proposition1.4, by Theorem\[Key-1\]-(2), it is enough to show that the singular set $\mathcal{S}$ is empty.
From now on, , that is, $\ell \geq 1$. By a similar reason to Remark\[Tree\], the case of $\Gamma_a = \{1\}$ for some $a\ (1 \leq a \leq \ell)$ may be possible logically for a general closed $4$-manifold, particularly $\Gamma_a \ = \{1\}$ for all $a = 1, 2, \cdots, \ell$. However, at least in the case of $M = S^4$, the following holds. We use the notation of Theorem\[Key-1\] and Remark\[Tree\].
\[Subkey\] $\Gamma_{a_0} \ne \{1\}$ for some $a_0\ (1 \leq a_0 \leq \ell)$.
Suppose that $\Gamma_a = \{1\}$ for all $a = 1, 2, \cdots, \ell$. As mentioned in Remark\[Tree\], a smooth embedded $S^3$ around $p_1 \in S^4$ separates $S^4$ into two components $V_1, W_1$ of compact $4$-manifolds with boundary satisfying $$S^4 = V_1 \cup W_1,\quad S^3 = \partial V_1 = \partial W_1 = V_1 \cap W_1,\quad V_1 \subset S^4_{\infty}.$$ By the Mayer-Vietoris exact sequence of homology groups for $(S^4; V_1, W_1)$, one can get $$H_i(V_1; \mathbb{R}) = H_i(W_1; \mathbb{R}) = 0\qquad {\rm for}\ \ i = 1, 2, 3,$$ and hence $\chi(V_1) = \chi(W_1) = 1$. Let $S^4_1 := V_1 \cup_{S^3} \overline{D^4}$ be a closed smooth $4$-manifold obtained by gluing along $S^3 = \partial V_1 = \partial\overline{D^4}$, where $\overline{D^4}$ denotes the closed $4$-ball in $\mathbb{R}^4$. Note that $\chi(S^4_1) = 2$. Similar to the above, a smooth embedded $S^3$ around $p_2 \in S^4_1$ separates $S^4_1$ into two components $V'_2, W'_2$. Then, the closed smooth $4$-manifold $S^4_2 := V'_2 \cup_{S^3} \overline{D^4}$ also satisfies that $\chi(S^4_2) = 2$. Repeating a similar procedure up to $a = \ell$, we get finally a closed smooth $4$-manifold $S^4_{\ell} := V'_{\ell} \cup_{S^3} \overline{D^4}$ with $\chi(S^4_{\ell}) = 2$. By construction, $S^4_{\ell}$ is homeomorphic to $S^4_{\infty}$ which implies that $\chi(S^4_{\infty}) = 2$.
By the removable singularities theorem for Einstein metrics [@BKN Theorem5.1], we note that $(S^4_{\infty}, g_{\infty})$ is a closed [*smooth*]{} Einstein $4$-manifold. Combining this with $\chi(S^4_{\infty}) = 2$, we get that $\mathscr{E}(g_{\infty}) = 2$. However, each Ricci-flat ALE $4$-space $(X_a, h_a)$ bubbling out from $p_a$ has a positive energy $\mathscr{E}(h_a) > 0$. This, combined with Theorem\[Key-1\]-(4) leads to a contradiction: $$2 = \lim_{j \to \infty}\mathscr{E}(g_j) \geq \mathscr{E}(g_{\infty}) + \sum_a\mathscr{E}(h_a) > 2.$$ Therefore, $\Gamma_{a_0} \ne \{1\}$ for some $a_0\ (1 \leq a_0 \leq \ell)$.
We can now prove Proposition1.4.
For simplicity, we assume that $a_0 = 1$. It then follows from Theorem\[Key-2\] that $\Gamma_1 = Q_8$. By Remark\[ALE\]-(3), we also obtain that $\widetilde{\Gamma}_1 = Q_8$. Even if $\widetilde{\Gamma}_a = Q_8$ for some $a$, a similar Mayer-Vietoris argument to that in the proof of Lemma\[Subkey\] still holds, and so $\chi(X_1) = 1$. It then follows from Remark\[ALE\]-(2) that $$\mathscr{E}(h_1) = \chi(X_1) - \frac{1}{|Q_8|} = 1- \frac{1}{8} = \frac{7}{8}.$$
By the signature theorem for compact $4$-orbifolds (cf.[@Nakajima-2 (4.5)]) and the calculation of eta invariant $\eta_S(S^3/\Gamma)$ for the signature operator [@Hitchin-2 Section3], the compact Einstein $4$-orbifold $(S^4_{\infty}, g_{\infty})$ satisfies that $$\tau(S^4_{\infty}) = \frac{1}{12\pi^2}\int_{S^4_{\infty}}\Big{(} |W_{g_{\infty}}^+|^2 - |W_{g_{\infty}}^-|^2 \Big{)}d\mu_{g_{\infty}}
- \sum_{a=1}^{\ell}\eta_S(S^3/\Gamma_a),\quad \eta_S(S^3/Q_8) = \frac{3}{4},$$ where $\tau(S^4_{\infty})$ denotes the signature of $S^4_{\infty}$. Since $H_2(S^4_{\infty}; \mathbb{R}) = 0$, we have that $H^2(S^4_{\infty}; \mathbb{R}) = 0$, and so $\tau(S^4_{\infty}) = 0$. Combining that $R_{g_{\infty}} \geq c > 0$ and $\eta_S(S^3/\Gamma_a) \geq 0$ with the above and Theorem\[Key-1\]-(4), we then obtain that $$\frac{9}{8} = 2 - \frac{7}{8} \geq \mathcal{E}(g_{\infty}) = \frac{1}{8\pi^2}\int_{S^4_{\infty}}\Big{(} |W_{g_{\infty}}|^2 + \frac{R_{g_{\infty}}^2}{24} \Big{)}d\mu_{g_{\infty}}
> \frac{1}{8\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}|^2d\mu_{g_{\infty}}$$ $$\qquad \quad \geq \frac{1}{8\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}^+|^2d\mu_{g_{\infty}} \geq \frac{3}{2}\Big{(} \frac{3}{4}
+ \frac{1}{12\pi^2}\int_{S^4_{\infty}}|W_{g_{\infty}}^-|^2d\mu_{g_{\infty}} \Big{)} \geq \frac{9}{8},$$ and hence it leads a contradiction. Therefore, $\mathcal{S} = \emptyset$.
As mentioned in Remark of $\S$1, we describe some details on the topological argument.
\[Counter\] Let $N_2$ be the nonorientable disk bundle over the the real projective plane $\mathbb{RP}^2$ with Euler number $2$. Let $T_4$ be the disk bundle of the complex line bundle over $S^2$ of degree $4$. Then, the natural double cover $T_4 \rightarrow N_2$ is the universal cover of $N_2$. Note that $S^4 = N_2 \cup_{\partial N_2}N_2$ and $\partial N_2 = S^3/Q_8$ (see [@Lawson] for details). Note also that $N_2$ is orientable since $N_2$ can be smoothly embedded in $S^4$ as a compact $4$-submanifold. Moreover, we have the following: $$\ H_1(N_2; \mathbb{Z}) = \mathbb{Z}_2,\quad H_i(N_2; \mathbb{Z}) = 0\ \ (i = 2, 3, 4),$$ $$H_2(T_4; \mathbb{Z}) = \mathbb{Z},\quad \ \ H_i(T_4; \mathbb{Z}) = 0\ \ (i = 1, 3, 4).$$ We do not know whether the orientable open $4$-manifold ${\rm Int}(N_2)$ admits a Ricci-flat ALE metric or not. (We have proved here only that such a metric never appears as a bubbling off Ricci-flat ALE metric from a sequence in $\mathscr{E}_{\geq c}(S^4)$.) However, $N_2$ becomes an orientable counterexample to the topological arguments in the proofs of [@Anderson Lemma6.3] and [@Anderson-GAFA Proposition3.10].
[99]{} K.Akutagawa, B.Botvinnik, O.Kobayashi and H.Seshadri, [*The Weyl functional near the Yamabe invariant*]{}, [**J. Geom. Anal. 13**]{} (2003), 1–20. M.Anderson, [*Ricci curvature bounds and Einstein metrics on compact manifolds*]{}, [**J. Amer. Math.Soc. 2**]{} (1989), 455-490. M.Anderson, [*Einstein metrics with prescribed conformal infinity on $4$-manifolds*]{}, [**Geom. Fanct. Anal. 18**]{} (2008), 305–366. S.Bando, [*Bubbling out of Einstein manifolds*]{}, [**Tohoku Math. J. 42**]{} (1990), 205–216; Correction and addition, [**Tohoku Math. J. 42**]{} (1990), 587-588. S.Bando, A.Kasue and H.Nakajima, [*On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth*]{}, [**Invent. Math. 97**]{} (1989), 313–349. R.Bartnik, [*The mass of an asymptotically flat manifold*]{}, [**Comm. Pure Appl. Math. 39**]{} (1986), 661–693. A.Besse, [*Einstein Manifolds*]{}, Springer, 1987. G.Besson, G.Courtois and S.Gallot, [*Entropies et rigidités des espaces localement symétriques de courbure strictement négative*]{}, [**Geom. Funct. Anal. 5**]{} (1995), 731–799. C.Böhm, [*Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces*]{}, [**Invent. Mtah. 134**]{} (1998), 145–176. J.-P.Bourguignon and H.Karcher, [*Curvature operators: pinching estimates and geometric examples*]{}, [**Ann. Sci. École Norm. Sup. 11**]{} (1978), 71–92. J.S.Crisp and J.A.Hillman, [*Embedding Seifert fibred and $3$-manifolds and ${\rm Sol}^3$-manifolds in $4$-space*]{}, [**Proc. London Math. Soc. 76**]{} (1998), 685–710. D.Ebin, [*The manifold of Riemannian metrics*]{}, Global Analysis, [**Proc. Symp. Pure Math. 15**]{} (1968), 11–40. T.Eguchi and A.J.Hanson, [*Self-dual solutions to Euclidean gravity*]{}, [**Ann. of Phys. 120**]{} (1979), 82–106. M.Gursky, [*Four-manifolds with $\delta W^+ = 0$ and Einstein constants on the sphere*]{}, [**Math. Ann. 318**]{} (2000), 417–43. M.Gursky and C.LeBrun, [*On Einstein manifolds of positive sectional curvature*]{}, [**Ann. Global Anal. Geom. 17**]{} (1999), 315–328. N.Hitchin, [*Compact four-dimensional Einstein manifolds*]{}, [**J. Differential Geom. 9**]{} (1974), 435–441. N.Hitchin, [*Einstein metrics and the eta-invariant*]{}, [**Bullettino U. M. I. 11-B**]{} (1997), 92–105. G.R.Jensen, [*Einstein metrics on principal fibre bundles*]{}, [**J. Differential Geom. 8**]{} (1973), 599–614. N.Koiso, [*Non-deformability of Einstein metrics*]{}, [**Osaka J. Math. 15**]{} (1978), 419–433. N.Koiso, [*Einstein metrics and complex structures*]{}, [**Invent. Math. 73**]{} (1983), 71–106. T.Lawson, [*Splitting $S^4$ on $\mathbb{RP}^2$ via the branched cover of $\mathbb{CP}^2$ over $S^4$*]{}, [**Proc. Amer. Math. Soc. 86**]{} (1982), 328–330. C.LeBrun, [*Einstein metrics and Mostow rigidity*]{}, [**Math. Res. Lett. 2**]{} (1995), 1–8. C.LeBrun, [*Four-manifolds without Einstein metrics*]{}, [**Math. Res. Lett. 3**]{} (1996), 133–147. J.Lee and T.Parker, [*The Yamabe problem*]{}, [**Bull. Amer. Math. Soc. 17**]{} (1987), 37–81. H.Nakajima, [*Hausdorff convergence of Einstein $4$-manifolds*]{}, [**J. Fac. Sci. Univ. Tokyo 35**]{} (1988), 411–424. H.Nakajima, [*Self-duality of ALE Ricci-flat $4$-manifolds and positive mass theorem*]{}, in Recent Topics in Differential and Analytic Geometry, [**Advanced Studies in Pure Math. 18-I**]{} (1990), 313–349. H.Nakajima, [*A convergence theorem for Einstein metrics and the ALE spaces*]{}, [**Amer. Math. Soc. Transl. 160**]{} (1994) 79–94. M.Obata, [*The conjectures on conformal transformations of Riemannian manifolds*]{}, [**J. Differential Geom. 6**]{} (1972), 247–258. R.Schoen, [*Variational theory for the total scalar curvature functional for Riemannian metrics and related topics*]{}, Topics in Calculus of Variations, [**Lect. Notes in Math. 1365**]{}, 121–154, Springer, 1989. J.Thorpe, [*Some remarks on the Gauss-Bonnet integral*]{}, [**J. Math. Mech. 18**]{} (1969), 779–786. D.Yang, [*Rigidity of Einstein $4$-manifolds with positive curvature*]{}, [**Invent. Math. 142**]{} (2000),435–450.
| ArXiv |
---
abstract: |
A pair is a holomorphic map from a Riemann surface to $S^2$ with additional properties. A dessin d’enfants is a bipartite graph with additional structure. It is well know that there is a bijection between pairs and dessins d’enfants.
Vassiliev has defined a filtration on formal sums of isotopy classes of knots. Motivated by this, we define a filtration on formal sums of Belyĭ pairs, and another on dessin d’enfants. We ask if the two definitions give the same filtration.
author:
- |
Jonathan Fine\
Milton Keynes\
England\
`[email protected]`
date: 28 September 2009
title: A filtration question on Belyĭ pairs and dessins
---
Introduction
============
First, we recall some definitions [@Belyui; @Dessins]. A *Belyĭ pair* is a Riemann surface $C$ together with a holomorphic map $f:C \to S^2 = \C \cup \{\infty\}$ to the Riemann sphere, such that $f'(p)$ is non-zero provided $f(p)$ is not $0$, $1$ or $\infty$. (Belyĭ proved that given $C$ such an $f$ can be found iff $C$ can be defined as an algebraic curve over the algebraic numbers.)
A *dessin d’enfants*, or *dessin* for short, is a graph $G$ together with a cyclic order of the edges at each vertex, and also a partition of the vertices $V$ into two sets $V_0$ and $V_1$ such that every edge joins $V_0$ to $V_1$. Necessarily, $G$ must be a bipartite graph. Traditionally, the vertices in $V_0$ and $V_1$ are coloured black and white respectively.
It is easy to see that a Belyĭ pair gives rise to a dessin, where $V_0=f^{-1}(0)$, $V_1 = f^{-1}(1)$, and the edges are the components of the inverse image $f^{-1}([0,1])$ of the unit interval in $\C$. The cyclic order arise from local monodromy around the vertices.
A much harder result, upon which our definitions rely, is that up to isomorphism every dessin arises from exactly one Belyĭ pair, or in other words that there is a bijection between isomorphism classes of Belyĭ pairs and dessins.
Definitions
===========
A *Belyĭ object* $B$ consists of $((B_C, B_f), B_D)$ where $(B_C, B_f)$ is a Belyĭ pair and $B_D$ is the associated dessin (or vice versa for the dessin and the pair).
The *Vassiliev space* $V=V_\C$ (for Belyĭ objects) is the vector space over $\C$ which has as basis the isomorphism classes of Belyĭ objects.
Clearly, when an edge is removed from a dessin then it is still a dessin. Suppose $D$ is a dessin, and $T$ is a subset of its edges. We will use $D \setminus T$ to denote the dessin so obtained. This same operation can also be applied to a Belyĭ object $B$, even though computing the associated curve $(B\setminus T)_C$ from $B_D$ and $T$ might be hard.
We will now define one or two filtrations of $V$.
Let $D$ be dessin and $S$ a $d$-element subset of $D$. Each subset $T$ of $S$ determines a dessin $S\setminus T$ and hence a object $B_{S\setminus T}$. Let $|T|$ denote the number of edges in $T$. Use $$B_S = \sum\nolimits _{T\subseteq S} (-1)^{|T|}B_{S \setminus T}$$ to define a vector $B_S$ in $V$, which we call *the expansion of a dessin with $d$ optional edges*.
Let $V_{D,d}$ be the span of the expansions of all dessins with $d$ optional edges. The sequence $$V =
V_{D, 0} \supseteq
V_{D, 1} \supseteq
V_{D, 2} \supseteq
V_{D, 3} \ldots$$ is the *dessin filtration* of $V$.
We can also think of a Belyĭ object as a map $f:C\to S^2$ (with special properties). Let $(C_1, f_1)$ and $(C_2, f_2)$ be Belyĭ pairs. Then there is of course a map $$g: C_1 \times C_2 \to S^2 \times S^2 \>.$$
Let $\Delta \subset S^2 \times S^2$ denote the diagonal, and let $C$ denote $g^{-1}(\Delta)$, and $f$ the restriction of $g$ to $C$. In general $$f: C \to \Delta \cong S^2$$ will not be a Belyĭ pair. There are two possible problems. The first is that $C\subset C_1\times C_2$ might have self intersections or be otherwise singular. If this happens, we replace $C$ by its resolution, which is unique.
The second problem is more interesting. It might be that $f$ has critical points not lying above the special points $0$, $1$ and $\infty$. This problem cannot be avoided. However, the above discussion does show that there is product, which we will denote by ‘$\circ$’, on holomorphic branched covers of $S^2$.
Let $W$ be the vector space with basis isomorphism classes of branched covers of $S^2$. We set $W_n$ to be the span of all products of the form $$(A_1 - B_1) \circ
(A_2 - B_2) \circ
\ldots \circ
(A_n - B_n)$$ for $A_i$ and $B_i$ basis vectors of $W$. Clearly, the $W_n$ provide a filtration of $W$.
The induced filtration of $V$ defined by $V_{B,n} = W_n \cap V$ is called the *filtration* of $V$.
Questions
=========
Are the two filtrations $V_D$ and $V_B$ equal?
If so, then we have also answered the next two questions.
The absolute Galois group acts on pairs, and preserves the filtration. Does this action also preserve the dessin filtration?
Because the dessins with $d$ edges, all of which are optional, span $V_d/V_{d+1}$, the dessin filtration has finite dimensional quotients. Does the filtration have finite dimensional quotients?
Investigating the last two questions might help us answer the first. They might also be of interest in their own right.
[9]{}
D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423–472
G. V. , Another proof of the three points theorem, Subornik: Mathematics 193 (2002), 329–32.
Leila Schneps, ed, The Grothendieck Theory of Dessins d’Enfants, London Math. Soc. Lecture Note Ser., vol 200, Cambridge Univ. Press 1994.
| ArXiv |
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abstract: 'We report the study of far-IR sizes of submillimeter galaxies (SMGs) in relation to their dust-obscured star formation rate (SFR) and active galactic nuclei (AGN) presence, determined using mid-IR photometry. We determined the millimeter-wave ($\lambda_{\rm obs}=1100\,\mu$m) sizes of 69 ALMA-identified SMGs, selected with $\geq10$$\sigma$ confidence on ALMA images ($F_{\rm 1100 \mu m}=1.7$–7.4mJy). We found that all the SMGs are located above an avoidance region in the millimeter size-flux plane, as expected by the Eddington limit for star formation. In order to understand what drives the different millimeter-wave sizes in SMGs, we investigated the relation between millimeter-wave size and AGN fraction for 25 of our SMGs at $z=1$–3. We found that the SMGs for which the mid-IR emission is dominated by star formation or AGN have extended millimeter-sizes, with respective median $R_{\rm c,e} = 1.6^{+0.34}_{-0.21}$ and 1.5$^{+0.93}_{-0.24}$kpc. Instead, the SMGs for which the mid-IR emission corresponds to star-forming/AGN composites have more compact millimeter-wave sizes, with median $R_{\rm c,e}=1.0^{+0.20}_{-0.20}$kpc. The relation between millimeter-wave size and AGN fraction suggests that this size may be related to the evolutionary stage of the SMG. The very compact sizes for composite star-forming/AGN systems could be explained by supermassive black holes growing rapidly during the SMG coalescing, star-formation phase.'
author:
- Soh Ikarashi
- 'KarinaI. Caputi'
- Kouji Ohta
- 'R.J. Ivison'
- 'ClaudiaD. P. Lagos'
- Laura Bisigello
- Bunyo Hatsukade
- Itziar Aretxaga
- 'JamesS. Dunlop'
- 'DavidH. Hughes'
- Daisuke Iono
- Takuma Izumi
- Nobunari Kashikawa
- Yusei Koyama
- Ryohei Kawabe
- Kotaro Kohno
- Kentaro Motohara
- Kouichiro Nakanishi
- Yoichi Tamura
- Hideki Umehata
- 'GrantW. Wilson'
- Kiyoto Yabe
- 'MinS. Yun'
title: 'Very compact millimeter sizes for composite star-forming/AGN submillimeter galaxies'
---
Introduction
============
The morphology and size of star-forming regions in submillimeter galaxies (SMGs) are important properties with which we can address the nature of their prodigious, dust-obscured star formation, and consequently the formation and evolution of the most massive galaxies. The Atacama Large Millimeter/submillimeter Array (ALMA) is enabling astronomers to image high-redshift SMGs with angular resolutions of $\lesssim0''$.3. Some ALMA studies have reported effective radii ($R_{\rm e}$) of $\sim0.3$–3kpc [e.g. @ika15; @sim15; @hod16]. These radii are small compared with what astronomers expected from studies of SMG sizes based on radio continuum and CO emission [e.g. @tac06; @big08; @ivi11]. These new results represent a new milestone in our understanding of star formation in SMGs, suggesting that these galaxies plausibly evolve to compact quiescent galaxies [e.g. @tof14; @ika15; @sim15].
As a next step, it would be useful to test the hypothesis that SMGs are connected to the formation of the most massive galaxies, being triggered by major mergers, and then evolving into compact quiescent galaxies via quenching in a QSO phase [e.g. @san88; @hop08; @tof14]. The compact submillimeter sizes of SMGs, including recent reports of the existence of subkilopersec-scale starburst cores [@ion16; @ika17; @ote17], suggests that the intense star-formation activity might be quenched by active galactic nuclei (AGN), as observed in some luminous QSOs [e.g. @mai12; @car16]. The link between SMGs and QSOs is still unclear, though. However, previous X-ray [e.g. @ale05; @wan13] and mid-IR [e.g. @ivi04; @cop10; @ser10] studies indicate that some SMGs do harbor AGN.
In this letter, we report a millimeter-wave size study of 69 ALMA-identified AzTEC SMGs. Firstly, we study the empirical relation between the ALMA continuum flux densities and the millimeter-wave sizes of SMGs. Secondly, we investigate the relationship between millimeter-wave sizes and the presence of AGN in SMGs at $z=1$–3, as determined from mid-IR data. We adopt throughout a cosmology with $H_{\rm 0}=70$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm M}=0.3$, and $\Omega_{\rm \Lambda}=0.7$.
ALMA Observations and samples
=============================
The sample used in this paper comes from our ALMA 1100-$\mu$m continuum imaging survey of 144 bright AzTEC/ASTE sources with $F_{\rm 1100 \mu m,\,AzTEC}\geq 2.4$mJy in the Subaru/[*XMM-Newton*]{} Deep Field [SXDF; @fur08]. The SXDF survey was conducted in the ALMA Cycles 2 and 3 (2013.1.00781, 2015.1.00442.S: PI. Hatsukade; B.Hatsukadeetal.2017, in preparation).
The ALMA observations in Cycle 2 were carried out with the array configurations C34-5 and C34-7, with 37–38 working 12-m antennas covering up to a $uv$ distance of $\sim 1500$k$\lambda$. In Cycle 3, the observations were executed in array configuration C40-4, covering up to a $uv$ distance of $\sim 1000$k$\lambda$. On-source integration times per source in each cycle were 0.6min. The typical synthesized beam size for our ALMA continuum images is $\sim 0.''30 \times 0.''23$ ($\rm PA \sim 56^{\circ}$), after combining the Cycle 2 and 3 data. The average r.m.s. noise level is 120$\mu$Jybeam$^{-1}$. The images were generated with Briggs weighting, using a robust parameter of 0.3.
The ALMA continuum maps yielded 70 ALMA-identified AzTEC SMGs (hereafter ASXDF SMGs) with $S_{\rm peak}/N\geq 10$ detections, suitable for reliable ALMA millimeter-wave size measurements [e.g. @ika15]. We removed one lensed SMG [ASXDF1100.001; @ika11], leaving 69 SMGs. ALMA fluxes were re-measured in tapered ALMA images with a synthesized beam of $\sim0{''}.6$, which is larger than the measured mm-wave sizes of SMGs in this paper, using the IMFIT task in CASA.
For 51 ASXDF SMGs, we obtained well-constrained photometric redshifts, with a median error $\delta z= 0.13\pm0.02$, based on the individual 1-$\sigma$ errors estimated by [*Le Phare*]{} [e.g. @ilb06] in spectral energy distribution (SED) model fitting using the $B$, $V$, $Rc$, $i'$, $z'$, $J$, $H$, $Ks$, 3.6 and 4.5$\mu$m data (S.Ikarashi et al. 2017, in preparation). The remaining SMGs lie outside the coverage of the optical/near-IR images, or have individual 1-$\sigma$ errors of $>1$. Photometric and spectroscopic redshifts from the literature are listed in Table \[tbl-1\].
ALMA millimeter-wave source size measurements
=============================================
We measured millimeter-wave sizes as circularized effective radii ($R_{\rm c,e}$) for the 69 ASXDF SMGs with ALMA visibility data, in the same manner as @ika15. We used $uv$-distance versus amplitude plots (hereafter $uv$-amp plots) for our measurements. Although the ALMA data cover $uv$ distances up to $\sim 1500$k$\lambda$, we used only data at $\leq 500$k$\lambda$, which corresponds to a scale of $\sim0.''2$. Adopting this cutoff for the longest $uv$ distance is the equivalent of smoothing with a larger size kernel in the image plane. We aim to mitigate the effects of possible clumpy structures in the size measurements and to measure $R_{\rm c,e}$ robustly. For the sources detected with $\geq10\sigma$ in the ALMA Cycle-2 images alone, we measured their sizes using only Cycle-2 data, to avoid effects due to any systematic absolute flux calibration offsets between our Cycle 2 and 3 data [^1]. We measured sizes by fitting a Gaussian model to the observed data in the $uv$-amp plots. When we measure the size, the other sources ($\geq5\sigma$) in each ALMA image were removed from the visibility data based on simple source properties derived by IMFIT task.
In order to estimate possible systematics in the size measurements, we injected mock sources into ALMA noise visibility images, generated from the actual ALMA data as in @ika15. Briefly we injected a symmetric Gaussian component for a range of source sizes and flux densities that cover the putative parameter range of our ASXDF sources with uniform probability. As tested in @ika15, our method can measure circularized effective radii correctly even if a source has an asymmetric Gaussian profile. We corrected our raw measured sizes based on the results of the simulations for the data used in this paper. As a result, we obtained ALMA millimeter-wave sizes of 0$''.08$–0$''.68$ (FWHM) for the 69 ASXDF SMGs. Note that ASXDF1100.009.1 has two distinct millimeter-wave components with a separation of $\sim$0$''$.6, sharing a host galaxy at $z_{\rm spec}=0.9$.
Relation between millimeter sizes and fluxes
============================================
Fig. \[fig:sizeflux\] (left panel) shows all 69 ASXDF SMGs in an ALMA 1100-$\mu$m vs. millimeter-wave size plot. Additionally, we plot 13 ALMA-identified, fainter SMGs at $z\gtrsim 3$ from @ika15. ASXDF SMGs are absent from the top-left and the bottom-right corners of this plot. The expected source selection limit for $\geq10\sigma$ continuum detection based on simple Gaussian models explains the absence of SMGs in the top-left corner. The bottom-right corner, instead, is free from any such selection biases, so the absence of SMGs requires an explanation.
The absence of SMGs in the bottom-right corner of Fig. \[fig:sizeflux\] can be interpreted as the influence of Eddington-limited star formation [@mur05]. According to @you08, which reported pioneering studies of maximum star formation in bright SMGs, a maximum star-formation rate is given by $$SFR_{\rm max} = 480\sigma^2_{400}D_{\rm kpc}\,\kappa^{-1}_{100} M_{\odot} yr^{-1},$$ where $D_{\rm kpc}$ is the characteristic physical scale of the starburst region in kpc, $\sigma_{\rm 400}$ is the line-of-sight gas velocity dispersion in units of 400kms$^{-1}$, and $\kappa_{\rm 100}$ is the dust opacity in units of 100cm$^2$g$^{-1}$. Here we adopt a Chabrier initial mass function [@cha03]; $\kappa_{\rm 100}=1$, as in @you08; and a median gas velocity dispersion of 510kms$^{-1}$ from CO line observations of SCUBA SMGs [@bot13]. We also adopt 2$\times$ FWHM or 4$\times R_{\rm c,e}$, which is expected to include 94% of the total light, as $D_{\rm kpc}$. The derived $SFR_{\rm max}$ was corrected with this factor of 0.94.
In order to plot the relation between SFR and physical scale described by Equation 1 on Fig. \[fig:sizeflux\] (the left panel), we assume a fixed redshift $z=2$. The conversion factors from ALMA fluxes to SFRs were derived by bootstrapping given a dust temperature ($T_d$) distribution for lensed 1.3mm-selected galaxies [@wei13] and an SED library with $T_d$ information compiled in @swi14. For these assumptions, we obtain a possible range of Eddington-limited star formation rates.
For a more direct comparison of the millimeter fluxes and sizes of SMGs with Eddington-limited star formation, we re-plot 51 of the 69 SMGs at $z=0.7$–6.8 with optical/near-IR photometric or spectroscopic redshifts on the SFR–physical size plane (Fig. \[fig:sizeflux\], right panel). The SFRs are derived from $F_{\rm 1100 \mu m}$, given the range of possible dust temperatures $T_d$ and SEDs noted above. We assume that the AGN contribution to the submillimeter flux is negligible [see references in @ros12]. In order to visualize the coverage of the size-SFR plane produced by the large SFR uncertainties (due to the unknown dust SED temperatures), we show the full SFR probability density distribution (rather than a single value) for each SMG. The results in both panels of Fig. \[fig:sizeflux\] show that the SMGs avoid the SFR region around the Eddington limit, suggesting that the minimum possible millimeter-wave sizes for bright SMGs are given by the Eddington limited star formation.
The empirical relation between flux and size can explain the apparent discrepancy between the reported (sub)millimeter-wave (median) sizes of $0.''20^{+0.''03}_{-0.''05}$ by @ika15 and $0.''3\pm0.''04$ by @sim15. Given $F_{\rm 870 \mu m}/F_{\rm 1100 \mu m}=2$ for conversion of the observed fluxes, @sim15 covered $F_{\rm 1100 \mu m} \gtrsim 2.5$mJy. In this regime, our ASXDF SMGs have a median size of$0.''31^{+0.''03}_{-0.''03}$.
\[fig:sizeflux\]
Relation between millimeter sizes and AGN {#sec:agn}
=========================================
We present our studies of the connection between the millimeter-wave sizes and AGN in SMGs, based on a mid-IR AGN diagnostic. We consider 25 ALMA-identified SMGs with $1<z_{\rm phot\,or\,spec}<3$, which are detected in all IRAC and MIPS 24$\mu$m images. All SMGs here have redshift information and a single component at $\sim$0$''$.2 resolution. More than 15 out of the 25 are located above $4\times$ the main sequence at $z\sim2$ in the stellar mass vs. SFR plane (Fig. \[fig:masssfr\]), indicating that the majority of the sample are starbursts [@bis18]. Note that among the 29 SMGs with $z=1$–3, four are not considered in our analysis: two SMGs are not detected at 24$\mu$m and the other two are blended in the IRAC maps.
Mid-IR AGN diagnostic
---------------------
A 4.5$\mu$m/8$\mu$m/24$\mu$m color-color plot has often been used as an AGN diagnostic for high-redshift, dusty infrared-luminous galaxies, such as SMGs and DOGs at $z\sim2$ [e.g. @ivi04; @ivi07; @pop08a; @pop08b]. We refer the reader to @kir15, who presented a detailed study of mid-IR SED evolution versus AGN fraction for high-$z$ galaxies. Empirical SED templates (top left panel in Fig. \[fig:sizeagn\]) suggest that high-redshift galaxies dominated by star formation or AGN in mid-IR light can be segregated from each other in the mid-IR color-color plane. The position of our 25 SMGs in this color-color plot shows that some of them do not follow either the model tracks for star-formation-dominated or AGN-dominated galaxies.
We generated the expected mid-IR colors of galaxies that are a composite of SF and AGN by combining SEDs of SF and AGN with various SF/AGN ratios. This ‘toy’ color prediction reproduces the colors of ‘composite SMGs’ which are likely to be dominated by neither an AGN nor a starburst in the mid-IR (top right panel in Fig. \[fig:sizeagn\]).
We divided the 25 SMGs into four sub-groups based on their 4.5/8/24-$\mu$m colors: star-forming, composite, AGN-dominant and ‘no class’. The criteria are:
- $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}<1.15$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}\geq 5$ (star-forming)
- $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}\geq1.15$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}\geq 5$ (composite)
- $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}\geq1.50$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}< 5$ (AGN)
- $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}<1.50$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}< 5$ (no class).
The model colors (top, Fig. \[fig:sizeagn\]) indicate that the SMGs categorized as ‘no class’ could be in the star-forming or composite classes. Due to this ambiguity, we consider the ‘no class’ separately.
Note that, In our diagnostic, the star-forming class and AGN dominant class are defined first. We choose $F_{\rm 8.0 \mu m}/F_{\rm 4.5 \mu m}=$1.15 as criterion for separation, as this ensures that all galaxies that satisfy neither an AGN criteria by @don12 nor another criteria by @ste05 also lie on the star-forming region of the colour-colour diagram. The predicted 24$\mu$m/8$\mu$m color evolution with redshift, as derived by public empirical mid-IR SED templates for high-$z$ star-forming galaxies, composite galaxies, and AGN dominant galaxies [@kir15], are shown along with our sample SMGs (bottom left, Fig. \[fig:sizeagn\]). For these templates, the respective mid-IR AGN fractions of each sample are $<$20, 20–80, and $\geq$80%. In this plot we averaged the public SEDs in each AGN class, after scaling all fluxes at $\lambda_{\rm rest}=8$$\mu$m. The predictions based on the Kirkpatrick et al. SED templates suggest that our criteria for 24$\mu$m/8$\mu$m color can work to select an AGN-dominant class, and show that our composite type is expected to have typically AGN fractions of around $\sim$50%, consistently with our ’toy’ models.
Results
-------
In the millimeter-wave physical size vs. SFR plot (bottom right panel in Fig. \[fig:sizeagn\]), all SMGs with composite mid-IR components are evidently more compact and located closer to the Eddington limit than the other SMGs with star-forming dominant or AGN dominant mid-IR components.
The respective median $R_{\rm c,e}$ for the SMGs classified as star-forming dominant, composites, and AGN dominant are 1.6$^{+0.34}_{-0.21}$, 1.0$^{+0.20}_{-0.20}$, and 1.5$^{+0.93}_{-0.24}$kpc. The size difference between the SMGs with composite and star-forming mid-IR components, and the difference between the SMGs with composite and AGN-dominant mid-IR components are real, with a significance level of $>99$%, according to the Kolmogorov-Smirnov test. This indicates that the composite type galaxies are characterized by more compact star-forming regions than those of the star-forming or AGN-dominant types.
We also explored the relation between size and stellar mass in our sample and found that the size differences are not a consequence of different stellar masses. Composite SMGs are the most compact of the three types, even at fixed stellar mass.
None of our ALMA-identified AzTEC SMGs are detected in the existing [*XMM-Newton*]{} X-ray maps [@ued08], probably because these maps are too shallow. Nevertheless, we can compare our results with the sizes derived for the host galaxies of five high-$z$, X-ray-selected AGN ($L_{\rm 2-8keV}=10^{42.1-43.6}$ergs$^{-1}$) by @har16. These authors reported a size distribution for their AGN hosts similar to the SMG sizes in @sim15. The most X-ray luminous source in their sample (with $L_{\rm 2-8keV}=10^{43.6}$ergs$^{-1}$) has an extended size, and the remaining four ($L_{\rm 2-8keV}=10^{42.1-43.4}$ergs$^{-1}$) have compact sizes, which are comparable to those of our composite type here (Fig. \[fig:sizeagn\], bottom right).
AGN growth during a very compact star-forming phase?
----------------------------------------------------
The very compact millimeter-wave sizes of the SMGs with composite mid-IR components suggest that a central supermassive black hole could be growing in a compact and coalescing star-forming phase, which is consistent with the predictions of @spr05 for galaxy major mergers. The extended millimeter-wave sizes of the SMGs of the star-forming dominant class can be explained by a mid-stage merger as seen in, e.g., VV114 [@sai15]. Actually ASXDF1100.055.1 with the star-forming dominant class shows merger-like near-IR morphology (Fig.\[fig:hst\]). Instead, the extended sizes of the SMGs with the AGN-dominant class are puzzling. In line with the evolutionary scenarios of, e.g., @san88 [@hop08; @tof14] that SMGs evolve into QSOs, these extended sizes may be explained by positive AGN feedback by a growing supermassive black hole in the phase of star-formation quenching, as it is suggested by simulations for luminous AGN/QSOs [e.g. @ish12; @zub13] and considered for some luminous QSOs [e.g. @car16]. In fact, ASXDF1100.057.1 with the AGN dominant class has a QSO-like near-IR morphology (Fig.\[fig:hst\]). However, no significant near-IR morphological difference between AGN-host and non-AGN-host galaxies, that are not submillimeter selected, is reported [e.g. @koc12]. The extended submillimeter sizes in our SMGs may come from the nature of their host galaxies.
[ l c c c c c c c c c ]{} ID & R.A. & Dec. &SNR& $F_{\rm 1100 \mu m}$ & $z_{\rm photo}$ & SFR & mm-wave size & mm-wave size & AGN\
& (J2000) & (J2000) & & (mJy) & & ($M_{\odot}$yr$^{-1}$) & (FWHM; arcsec) & ($R_{\rm c,e}$; kpc) & (mid-IR)\
ASXDF1100.002.1 & 2:17:30.63 & -4:59:36.8 & 15 & 4.81$\pm$0.43 & 3.3$^{+0.07}_{-0.87}$ & 990$^{+720}_{-340}$ & 0.42$^{+0.06}_{-0.02}$ & 1.6$^{+0.2}_{-0.1}$ &\
ASXDF1100.004.1 & 2:18:05.65 & -5:10:49.7 & 14 & 4.39$\pm$0.56 & 3.5$^{+0.35}_{-0.16}$ & 880$^{+420}_{-290}$ & 0.40$^{+0.06}_{-0.04}$ & 1.5$^{+0.2}_{-0.1}$ &\
ASXDF1100.005.1 & 2:17:30.45 & -5:19:22.5 & 25 & 7.24$\pm$0.45 & 0.7$^{+0.01}_{-0.01}$ & 1200$^{+990}_{-420}$ & 0.34$^{+0.04}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\
ASXDF1100.006.1 & 2:17:27.32 & -5:06:42.8 & 10 & 5.11$\pm$0.50 & 4.5$^{+0.18}_{-0.15}$ & 930$^{+340}_{-330}$ & 0.68$^{+0.06}_{-0.06}$ & 2.2$^{+0.2}_{-0.2}$ &\
ASXDF1100.007.1 & 2:18:03.01 & -5:28:42.0 & 20 & 6.26$\pm$0.53 & 3.2$^{+0.28}_{-0.22}$ & 1300$^{+930}_{-450}$ & 0.32$^{+0.04}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\
ASXDF1100.008.1 & 2:16:47.93 & -5:01:29.9 & 12 & 6.45$\pm$0.59 & 2.2$^{+0.02}_{-0.08}$ & 1500$^{+950}_{-460}$ & 0.62$^{+0.06}_{-0.06}$ & 2.6$^{+0.2}_{-0.2}$ & AGN\
ASXDF1100.009.1A & 2:17:42.11 & -4:56:27.6 & 19 & 4.68$\pm$0.40 &(0.5)$^a$ & 550$^{+430}_{-190}$ & 0.30$^{+0.02}_{-0.04}$ & 0.9$^{+0.1}_{-0.1}$ &\
ASXDF1100.009.1B & 2:17:42.16 & -4:56:28.5 & 11 & 1.16$\pm$0.12 &(0.5)$^a$ & 140$^{+110}_{-50}$ & 0.10$^{+0.08}_{-0.06}$ & 0.6$^{+0.5}_{-0.4}$ &\
ASXDF1100.011.1 & 2:17:50.59 & -5:30:59.2 & 13 & 4.22$\pm$0.41 & 5.5$^{+0.08}_{-0.63}$ & 730$^{+440}_{-260}$ & 0.38$^{+0.04}_{-0.04}$ & 1.1$^{+0.1}_{-0.1}$ &\
ASXDF1100.014.1$^{\dagger}$ & 2:17:29.77 & -5:03:18.6 & 11 & 3.12$\pm$0.17 & 2.2$^{+0.04}_{-0.03}$ & 690$^{+270}_{-210}$ & 0.50$^{+0.06}_{-0.08}$ & 2.1$^{+0.2}_{-0.3}$ & SF\
ASXDF1100.016.1 & 2:16:41.11 & -5:03:51.4 & 19 & 4.79$\pm$0.35 & 5.0$^{+0.54}_{-0.06}$ & 850$^{+390}_{-240}$ & 0.24$^{+0.02}_{-0.04}$ & 0.8$^{+0.1}_{-0.1}$ &\
ASXDF1100.018.1 & 2:18:13.83 & -4:57:43.5 & 14 & 3.47$\pm$0.32 & 1.7$^{+0.09}_{-0.02}$ & 850$^{+650}_{-280}$ & 0.26$^{+0.04}_{-0.04}$ & 1.1$^{+0.2}_{-0.2}$ & NO\
ASXDF1100.020.1$^{\bullet}$ & 2:18:23.73 & -5:11:38.5 & 13 & 4.94$\pm$0.43 & 2.7$^{+0.01}_{-0.01}$ & 1100$^{+460}_{-380}$ & 0.30$^{+0.04}_{-0.02}$ & 1.2$^{+0.2}_{-0.1}$ &\
ASXDF1100.021.1 & 2:18:16.49 & -4:55:08.8 & 16 & 4.03$\pm$0.28 & 2.3$^{+0.03}_{-0.04}$ & 920$^{+720}_{-310}$ & 0.28$^{+0.02}_{-0.04}$ & 1.1$^{+0.1}_{-0.2}$ & COM\
ASXDF1100.022.1 & 2:18:42.68 & -4:59:32.1 & 15 & 3.09$\pm$0.31 & 2.3$^{+0.01}_{-0.06}$ & 710$^{+550}_{-240}$ & 0.20$^{+0.04}_{-0.04}$ & 0.8$^{+0.2}_{-0.2}$ & COM\
ASXDF1100.023.2 & 2:18:20.40 & -5:31:43.2 & 10 & 2.17$\pm$0.27 & 2.5$^{+0.10}_{-0.12}$ & 480$^{+350}_{-160}$ & 0.16$^{+0.10}_{-0.06}$ & 0.6$^{+0.4}_{-0.2}$ &\
ASXDF1100.025.2$^{\dagger}$ & 2:17:32.59 & -4:50:26.4 & 13 & 2.34$\pm$0.12 & 3.4$^{+0.16}_{-0.07}$ & 470$^{+320}_{-150}$ & 0.34$^{+0.06}_{-0.04}$ & 1.3$^{+0.2}_{-0.1}$ &\
ASXDF1100.029.1$^{\dagger}$ & 2:17:20.80 & -4:49:49.5 & 11 & 2.67$\pm$0.21 & 2.8$^{+0.16}_{-0.17}$ & 570$^{+360}_{-180}$ & 0.46$^{+0.08}_{-0.10}$ & 1.8$^{+0.3}_{-0.4}$ & AGN\
ASXDF1100.031.1$^{\dagger}$ & 2:17:37.24 & -4:47:53.0 & 13 & 2.09$\pm$0.15 & 2.5$^{+0.18}_{-0.12}$ & 480$^{+380}_{-170}$ & 0.28$^{+0.04}_{-0.06}$ & 1.1$^{+0.2}_{-0.2}$ & COM\
ASXDF1100.033.1 & 2:18:03.56 & -4:55:27.3 & 15 & 4.86$\pm$0.33 & (2.6)$^c$ & 1100$^{+860}_{-350}$ & 0.34$^{+0.04}_{-0.02}$ & 1.4$^{+0.2}_{-0.1}$ & COM\
ASXDF1100.034.1 & 2:17:59.32 & -5:05:04.6 & 11 & 2.84$\pm$0.32 & (1.6)$^b$& 680$^{+640}_{-220}$ & 0.16$^{+0.08}_{-0.06}$ & 0.7$^{+0.3}_{-0.3}$ &\
ASXDF1100.035.1$^{\dagger,\bullet}$ & 2:17:35.37 & -5:28:37.3 & 12 & 2.09$\pm$0.12 & 2.7$^{+0.07}_{-0.11}$ & 450$^{+360}_{-150}$ & 0.52$^{+0.08}_{-0.08}$ & 2.1$^{+0.3}_{-0.3}$ &\
ASXDF1100.041.1 & 2:17:53.87 & -5:26:35.7 & 10 & 2.91$\pm$0.29 & 0.8$^{+0.00}_{-0.00}$ & 520$^{+260}_{-180}$ & 0.42$^{+0.06}_{-0.10}$ & 1.6$^{+0.2}_{-0.4}$ &\
ASXDF1100.042.1 & 2:18:38.29 & -5:03:18.3 & 12 & 3.26$\pm$0.40 & 3.2$^{+0.02}_{-0.01}$ & 680$^{+440}_{-240}$ & 0.42$^{+0.04}_{-0.06}$ & 1.6$^{+0.1}_{-0.2}$ &\
ASXDF1100.044.1 & 2:17:45.85 & -5:00:56.7 & 12 & 1.93$\pm$0.26 & 6.8$^{+0.20}_{-0.72}$ & 330$^{+210}_{-84}$ & 0.09$^{+0.07}_{-0.05}$ & 0.2$^{+0.2}_{-0.1}$ &\
ASXDF1100.046.1 & 2:17:13.34 & -4:58:57.4 & 16 & 4.00$\pm$0.32 & 3.5$^{+0.01}_{-0.10}$ & 810$^{+620}_{-280}$ & 0.28$^{+0.04}_{-0.04}$ & 1.0$^{+0.1}_{-0.1}$ &\
ASXDF1100.047.1$^{\dagger}$ & 2:17:56.73 & -4:52:39.0 & 11 & 2.25$\pm$0.17 & 2.2$^{+0.01}_{-0.02}$ & 500$^{+400}_{-160}$ & 0.40$^{+0.08}_{-0.06}$ & 1.6$^{+0.3}_{-0.2}$ & SF\
ASXDF1100.048.1$^{\dagger}$ & 2:17:46.16 & -4:47:47.2 & 14 & 2.55$\pm$0.11 & 2.5$^{+0.21}_{-0.12}$ & 570$^{+460}_{-200}$ & 0.40$^{+0.06}_{-0.04}$ & 1.6$^{+0.2}_{-0.2}$ & NO\
ASXDF1100.050.1$^{\star}$ & 2:18:22.30 & -5:07:37.0 & 11 & 3.32$\pm$0.40 & 3.0$^{+0.15}_{-0.15}$ & 700$^{+360}_{-240}$ & 0.24$^{+0.08}_{-0.08}$ & 0.9$^{+0.3}_{-0.3}$ &\
ASXDF1100.051.1$^{\dagger}$ & 2:18:23.96 & -5:32:07.8 & 12 & 2.63$\pm$0.23 & 0.7$^{+0.00}_{-0.04}$ & 430$^{+270}_{-150}$ & 0.08$^{+0.06}_{-0.04}$ & 0.3$^{+0.2}_{-0.1}$ &\
ASXDF1100.051.2$^{\dagger}$ & 2:18:24.59 & -5:31:48.5 & 11 & 2.88$\pm$0.23 & 4.7$^{+0.24}_{-0.15}$ & 520$^{+270}_{-160}$ & 0.30$^{+0.10}_{-0.06}$ & 1.0$^{+0.3}_{-0.2}$ &\
ASXDF1100.052.1$^{\dagger}$ & 2:17:33.17 & -5:01:54.5 & 11 & 2.05$\pm$0.14 & 2.8$^{+0.25}_{-0.65}$ & 440$^{+340}_{-150}$ & 0.34$^{+0.04}_{-0.06}$ & 1.3$^{+0.2}_{-0.2}$ & AGN\
ASXDF1100.055.1$^{\dagger}$ & 2:17:20.03 & -5:13:05.8 & 13 & 2.54$\pm$0.15 & 2.1$^{+0.02}_{-0.24}$ & 570$^{+290}_{-180}$ & 0.34$^{+0.06}_{-0.06}$ & 1.4$^{+0.2}_{-0.2}$ & SF\
ASXDF1100.057.1 & 2:17:32.41 & -5:12:50.9 & 12 & 3.54$\pm$0.38 & 1.9$^{+0.04}_{-0.11}$ & 820$^{+360}_{-260}$ & 0.34$^{+0.04}_{-0.06}$ & 1.4$^{+0.2}_{-0.3}$ & AGN\
ASXDF1100.076.1 & 2:16:41.04 & -5:01:12.5 & 13 & 4.13$\pm$0.55 & 4.8$^{+0.13}_{-0.41}$ & 750$^{+550}_{-230}$ & 0.34$^{+0.04}_{-0.06}$ & 1.1$^{+0.1}_{-0.2}$ &\
ASXDF1100.077.1$^{\dagger}$ & 2:18:11.00 & -4:49:51.9 & 12 & 1.69$\pm$0.20 & 4.1$^{+0.02}_{-0.12}$ & 320$^{+190}_{-110}$ & 0.22$^{+0.08}_{-0.08}$ & 0.8$^{+0.3}_{-0.3}$ &\
ASXDF1100.089.1 & 2:18:10.64 & -5:34:53.6 & 21 & 4.73$\pm$0.30 & 5.4$^{+0.11}_{-0.09}$ & 830$^{+600}_{-200}$ & 0.24$^{+0.04}_{-0.02}$ & 0.7$^{+0.1}_{-0.1}$ &\
ASXDF1100.095.1$^{\dagger}$ & 2:17:12.97 & -5:14:12.2 & 10 & 1.91$\pm$0.19 & 2.2$^{+0.11}_{-0.08}$ & 440$^{+320}_{-150}$ & 0.32$^{+0.08}_{-0.08}$ & 1.3$^{+0.3}_{-0.3}$ & AGN\
ASXDF1100.100.1 & 2:17:53.25 & -4:49:51.5 & 13 & 2.84$\pm$0.29 & 2.2$^{+0.16}_{-0.08}$ & 670$^{+550}_{-210}$ & 0.24$^{+0.04}_{-0.04}$ & 1.0$^{+0.2}_{-0.2}$ & COM\
ASXDF1100.105.1 & 2:18:02.86 & -5:00:31.6 & 13 & 2.86$\pm$0.30 & (1.1)$^b$ & 630$^{+460}_{-220}$ & 0.24$^{+0.06}_{-0.08}$ & 1.0$^{+0.2}_{-0.3}$ & COM\
ASXDF1100.107.1$^{\dagger}$ & 2:18:07.85 & -5:25:49.3 & 11 & 1.67$\pm$0.16 & 4.6$^{+0.18}_{-0.86}$ & 310$^{+190}_{-80}$ & 0.34$^{+0.06}_{-0.06}$ & 1.1$^{+0.2}_{-0.2}$ &\
ASXDF1100.115.1 & 2:16:59.42 & -5:10:55.8 & 12 & 4.23$\pm$0.33 & (0.6)$^a$ & 600$^{+500}_{-220}$ & 0.50$^{+0.06}_{-0.06}$ & 1.7$^{+0.2}_{-0.2}$ &\
ASXDF1100.134.1 & 2:17:54.80 & -5:23:23.8 & 15 & 3.27$\pm$0.27 & 2.5$^{+0.16}_{-0.05}$ & 740$^{+500}_{-260}$ & 0.24$^{+0.06}_{-0.04}$ & 1.0$^{+0.2}_{-0.2}$ & COM\
ASXDF1100.156.1 & 2:16:38.33 & -5:01:21.5 & 11 & 3.33$\pm$0.31 & 1.8$^{+0.04}_{-0.10}$ & 810$^{+630}_{-260}$ & 0.34$^{+0.06}_{-0.06}$ & 1.4$^{+0.3}_{-0.3}$ & SF\
ASXDF1100.188.1$^{\dagger,\star}$ & 2:16:41.94 & -5:07:04.3 & 10 & 2.42$\pm$0.18 & 2.6$^{+0.28}_{-0.20}$ & 530$^{+450}_{-180}$ & 0.22$^{+0.10}_{-0.08}$ & 0.9$^{+0.4}_{-0.3}$ &\
ASXDF1100.203.1$^{\dagger}$ & 2:18:23.15 & -5:27:02.0 & 11 & 1.90$\pm$0.12 & 2.5$^{+0.03}_{-0.15}$ & 440$^{+330}_{-150}$ & 0.34$^{+0.10}_{-0.10}$ & 1.4$^{+0.4}_{-0.4}$ & NO\
ASXDF1100.227.1 & 2:17:44.27 & -5:20:08.6 & 24 & 7.42$\pm$0.57 & 3.7$^{+0.35}_{-0.14}$ & 1400$^{+760}_{-510}$ & 0.34$^{+0.02}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\
ASXDF1100.228.1 & 2:18:09.66 & -5:18:43.1 & 12 & 3.11$\pm$0.34 & 1.9$^{+0.05}_{-0.14}$ & 740$^{+610}_{-240}$ & 0.38$^{+0.06}_{-0.06}$ & 1.6$^{+0.3}_{-0.2}$ & SF\
ASXDF1100.229.1 & 2:18:18.84 & -4:50:29.9 & 11 & 3.60$\pm$0.36 & 2.3$^{+0.05}_{-0.11}$ & 820$^{+620}_{-270}$ & 0.26$^{+0.06}_{-0.08}$ & 1.1$^{+0.2}_{-0.3}$ & COM\
ASXDF1100.235.1 & 2:17:36.00 & -5:20:34.4 & 13 & 4.64$\pm$0.40 & 2.3$^{+0.04}_{-0.14}$ & 1100$^{+820}_{-370}$ & 0.26$^{+0.06}_{-0.04}$ & 1.1$^{+0.2}_{-0.2}$ & COM\
ASXDF1100.236.1$^{\dagger}$ & 2:17:21.54 & -5:19:07.7 & 11 & 1.65$\pm$0.14 & 2.4$^{+0.02}_{-0.02}$ & 370$^{+250}_{-120}$ & 0.15$^{+0.09}_{-0.09}$ & 0.6$^{+0.4}_{-0.4}$ & COM\
ASXDF1100.247.1$^{\dagger}$ & 2:16:33.85 & -5:02:42.7 & 11 & 1.87$\pm$0.18 & 2.6$^{+0.11}_{-0.14}$ & 410$^{+260}_{-140}$ & 0.24$^{+0.08}_{-0.10}$ & 1.0$^{+0.3}_{-0.4}$ & COM\
ASXDF1100.003.1$^{\dagger}$ & 2:16:44.48 & -5:02:21.6 &15 &2.85$\pm$0.13 && &0.36$^{+0.04}_{-0.04}$ &&\
ASXDF1100.010.1 & 2:17:39.79 & -5:29:19.2 &24 &5.94$\pm$0.37 && &0.28$^{+0.02}_{-0.02}$ &&\
ASXDF1100.026.1$^{\dagger}$ & 2:17:42.55 & -5:29:00.3 &11 &1.69$\pm$0.17 && & 0.18$^{+0.06}_{-0.12}$&&\
ASXDF1100.040.1 & 2:17:55.24 & -5:06:45.1 &15 &3.14$\pm$0.35 && &0.20$^{+0.06}_{-0.04}$&&\
ASXDF1100.053.1 & 2:16:48.20 & -4:58:59.6 &10 &4.02$\pm$0.51 && &0.42$^{+0.06}_{-0.06}$ &&\
ASXDF1100.054.1 & 2:17:15.41 & -4:57:55.6 &11 &4.12$\pm$0.38 && &0.38$^{+0.06}_{-0.06}$ &&\
ASXDF1100.068.1 & 2:17:42.17 & -5:25:46.8 &12 &3.24$\pm$0.30 && &0.24$^{+0.04}_{-0.06}$&&\
ASXDF1100.070.1$^{\dagger}$ & 2:18:46.15 & -5:04:12.5 &12 &2.17$\pm$0.13 && &0.30$^{+0.04}_{-0.06}$&&\
ASXDF1100.074.1 & 2:18:33.31 & -4:58:07.0 &10 &2.77$\pm$0.33 && &0.32$^{+0.06}_{-0.06}$ &&\
ASXDF1100.097.1 & 2:18:18.54 & -5:34:34.7 &11 &2.53$\pm$0.26 && &0.20$^{+0.08}_{-0.06}$ & &\
ASXDF1100.097.2$^{\dagger}$ & 2:18:17.61 & -5:34:27.9 &10 &2.14$\pm$0.26 && &0.32$^{+0.08}_{-0.10}$& &\
ASXDF1100.133.1 & 2:18:05.51 & -5:35:46.5 &11 &2.25$\pm$0.26 && &0.08$^{+0.08}_{-0.04}$ & &\
ASXDF1100.161.1$^{\dagger}$ & 2:18:13.76 & -5:37:27.3 &12 &2.68$\pm$0.20 && &0.44$^{+0.06}_{-0.06}$ & &\
ASXDF1100.168.1 & 2:18:04.37 & -5:34:03.5 &11 &1.79$\pm$0.21 && &0.16$^{+0.08}_{-0.06}$& &\
ASXDF1100.213.1$^{\dagger}$ & 2:18:44.02 & -5:35:31.3 &12 &2.90$\pm$0.28 && &0.16$^{+0.08}_{-0.08}$& &\
ASXDF1100.231.1 & 2:17:59.65 & -4:46:49.8 &12 &2.88$\pm$0.36 && &0.28$^{+0.08}_{-0.08}$ &&\
ASXDF1100.243.1$^{\dagger}$ & 2:16:50.43 & -5:10:16.2 &10 &2.09$\pm$0.20 && &0.37$^{+0.09}_{-0.11}$&&\
ASXDF1100.252.1 & 2:17:05.65 & -5:15:04.9 &12 &2.62$\pm$0.25 && &0.24$^{+0.06}_{-0.08}$ &&\
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[^1]: Comparisons of the fluxes of ASXDF sources in our Cycle-1, 2 and 3 data indicated that the fluxes in the Cycle-3 data are systematically $\sim$20% smaller. Therefore, we corrected the primary flux calibration for this effect.
| ArXiv |
---
abstract: 'We study for the first time a three-dimensional octahedron constellation for a space-based gravitational wave detector, which we call the Octahedral Gravitational Observatory (OGO). With six spacecraft the constellation is able to remove laser frequency noise and acceleration disturbances from the gravitational wave signal without needing LISA-like drag-free control, thereby simplifying the payloads and placing less stringent demands on the thrusters. We generalize LISA’s time-delay interferometry to displacement-noise free interferometry (DFI) by deriving a set of generators for those combinations of the data streams that cancel laser and acceleration noise. However, the three-dimensional configuration makes orbit selection complicated. So far, only a halo orbit near the Lagrangian point L1 has been found to be stable enough, and this allows only short arms up to 1400km. We derive the sensitivity curve of OGO with this arm length, resulting in a peak sensitivity of about 2$\times10^{-23}\,\mathrm{Hz}^{-1/2}$ near 100Hz. We compare this version of OGO to the present generation of ground-based detectors and to some future detectors. We also investigate the scientific potentials of such a detector, which include observing gravitational waves from compact binary coalescences, the stochastic background and pulsars as well as the possibility to test alternative theories of gravity. We find a mediocre performance level for this short-arm-length detector, between those of initial and advanced ground-based detectors. Thus, actually building a space-based detector of this specific configuration does not seem very efficient. However, when alternative orbits that allow for longer detector arms can be found, a detector with much improved science output could be constructed using the octahedron configuration and DFI solutions demonstrated in this paper. Also, since the sensitivity of a DFI detector is limited mainly by shot noise, we discuss how the overall sensitivity could be improved by using advanced technologies that reduce this particular noise source.'
author:
- Yan Wang
- David Keitel
- Stanislav Babak
- Antoine Petiteau
- Markus Otto
- Simon Barke
- Fumiko Kawazoe
- Alexander Khalaidovski
- Vitali Müller
- Daniel Schütze
- Holger Wittel
- Karsten Danzmann
- 'Bernard F. Schutz'
title: 'Octahedron configuration for a displacement noise-canceling gravitational wave detector in space'
---
Introduction
============
The search for gravitational waves (GWs) has been carried out for more than a decade by ground-based detectors. Currently, the LIGO and Virgo detectors are being upgraded using advanced technologies [@aLIGO; @adVIRGO]. The ground-based detectors are sensitive in quite a broad band from about 10Hz to a few kHz. In this band possible GW sources include stellar-mass compact coalescing binaries [@Abadie2010b], asymmetric core collapse of evolved heavy stars [@FryerNew2011], neutron stars with a nonzero ellipticity [@Owen2009] and, probably, a stochastic GW background from the early Universe or from a network of cosmic strings [@Allen99; @Maggiore00].
In addition, the launch of a space-based GW observatory is expected in the next decade, such as the classic LISA mission concept [@LISA] (or its recent modification known as evolved LISA (eLISA) / NGO [@eLISA]), and DECIGO [@Ando2010]. LISA has become a mission concept for any heliocentric drag-free configuration that uses laser interferometry for detecting GWs. The most likely first GW observatory in space will be the eLISA mission, which has an arm length of $10^9$m and two arms, with one “mother” and two “daughter” spacecraft exchanging laser light in a V-shaped configuration to sense the variation of the metric due to passing GWs.
The eLISA mission aims at mHz frequencies, targeting other sources than ground-based detectors, most importantly supermassive black hole binaries. In a more ambitious concept, DECIGO is supposed to consist of a set of four smaller triangles (12 spacecraft in total) in a common orbit, leading to a very good sensitivity in the intermediate frequency region between LISA and advanced LIGO (aLIGO).
Here we want to present a concept for another space-based project with quite a different configuration from what has been considered before. The concept was inspired by a three-dimensional interferometer configuration in the form of an octahedron, first suggested in Ref. [@chen2006] for a ground-based detector, based on two Mach-Zehnder interferometers.
The main advantage of this setup is the cancellation of timing, laser frequency and displacement noise by combining multiple measurement channels. We have transformed this detector into a space-borne observatory by placing one LISA-like spacecraft (but with four telescopes and a single test mass) in each of the six corners of the octahedron, as shown in Fig. \[F:orbit\]. Therefore, we call this project the *Octahedral Gravitational Observatory* (OGO).
Before going into the mathematical details of displacement-noise free interferometry (DFI), we first consider possible orbits for a three-dimensional octahedron constellation in Sec. \[S:Orbit\]. As we will find later on, the best sensitivities of an OGO-like detector are expected at very long arm lengths. However, the most realistic orbits we found that can sustain the three-dimensional configuration with stable distances between adjacent spacecraft for a sufficiently long time are so-called “halo” and “quasihalo” orbits around the Lagrange point L1 in the Sun-Earth system.
These orbits are rather close to Earth, making a mission potentially cheaper in terms of fuel and communication, and corrections to maintain the formation seem to be reasonably low. On the other hand, a constellation radius of only 1000km can be supported, corresponding to a spacecraft-to-spacecraft arm length of approximately 1400km.
We will discuss this as the standard configuration proposal for OGO in the following, but ultimately we still aim at using much longer arm lengths. As a candidate, we will also discuss OGO orbits with $2\times10^9\,$m arm lengths in Sec. \[S:Orbit\]. However, such orbits might have significantly varying separations and would require further study of the DFI technique in such circumstances.
{width="\textwidth"}\
The octahedron configuration gives us 24 laser links, each corresponding to a science measurement channel of the distance (photon flight-time) variation between the test masses on adjacent spacecraft. The main idea is to use a sophisticated algorithm called *displacement-noise free interferometry* (DFI, [@kawamura2004; @chen2006; @chenkawa2006]), which proceeds beyond conventional Time-Delay Interferometry techniques (TDI, [@TintoDhurandhar; @otto2012]), and in the right circumstances can improve upon them.
It can cancel both timing noise and acceleration noise when there are more measurements than noise sources. In three dimensions, the minimum number of spacecraft for DFI is 6, which we therefore use for OGO: this gives $6-1$ relative timing (clock) noise sources and $3\times 6 = 18$ components of the acceleration noise, so that $24 > 5+18$ and the DFI requirement is fulfilled. On the one hand, this required number of links increases the complexity of the detector. On the other hand, it provides some redundancy in the number of shot-noise-only configurations, which could be very useful if one or several links between spacecraft are interrupted.
After applying DFI, we assume that the dominant remaining noise will be shot noise. For the case of an equal-arm-length three-dimensional constellation, we analytically find a set of generators for the measurement channel combinations that cancel simultaneously all timing and acceleration noise. We assume that all deviations from the equal-arm configuration are small and can be absorbed into a low-frequency part of the acceleration noise. We describe the procedure of building DFI combinations in Sec. \[S:TDI\]. This will also allow us to quantify the redundancy inherent in the six-spacecraft configuration. The technical details of the derivation can be found in Appendix \[S:Appendix\].
In Sec. \[S:Sens\], we compute the response functions of the octahedron DFI configuration and derive the sensitivity curve of the detector. We assume the conservative 1400km arm length, a laser power of 10W and a telescope diameter of 1m, while identical strain sensitivity is achievable for smaller telescopes and higher power.
Unfortunately, those combinations that cancel acceleration and timing noise also suppress the GW signal at low frequencies. This effect shows up as a rather steep slope $\sim f^{2}$ in the response function.
We present sensitivity curves for single DFI combinations and find that there are in principle 12 such noise-uncorrelated combinations (corresponding to the number of independent links) with similar sensitivity, leading to an improved network sensitivity of the full OGO detector. We find that the best sensitivity is achieved around 78Hz, in a range similar to that of ground-based detectors. The network sensitivity of OGO is better than that of initial LIGO at this frequency, but becomes better than that of aLIGO only below 10Hz. The details of these calculations are presented in Sec. \[sec:transfer\_function\].
At this point, in Sec. \[sec:performance\], we briefly revisit the alternative orbits with a longer arm length, which would result in a sensitivity closer to the frequency band of interest for LISA and DECIGO. For this variant of OGO, we assume LISA-like noise contributions (but without spacecraft jitter) and compare the sensitivity of an octahedron detector using DFI with one using TDI, thus directly comparing the effects of these measurement techniques.
Actually, we find that the $2\times10^9\,$m arm length is close to the point of equal sensitivity of DFI and TDI detectors in the limit of vanishing jitter. This implies that DFI would be preferred for even longer arm lengths, but might already become competitive at moderate arm lengths if part of the jitter couples into the displacement noise in such a way that it can also be canceled.
A major advantage of the OGO concept lies in its rather moderate requirement on acceleration noise, as detailed in Sec. \[sec:feasibility\]. For other detectors, this limits the overall performance, but in this concept it gets canceled out by the DFI combinations. Assuming some improvements in subdominant noise sources, our final sensitivity thus depends only on the shot-noise level in each link.
Hence, we can improve the detector performance over all frequencies by reducing solely the shot noise. This could be achieved, for example, by increasing the power of each laser, by introducing cavities (similar to DECIGO), or with nonclassical (squeezed) states of light. We briefly discuss these possibilities in Sec. \[sec:shot\_noise\_reduction\].
In Sec. \[S:Sources\], we discuss the scientific potentials OGO would have even using the conservative short-arm-length orbits. First, as a main target, the detection rates for inspiraling binaries are higher than for initial LIGO, but fall short of aLIGO expectations. However, joint detections with OGO and aLIGO could yield some events with greatly improved angular resolution. Second, due to the large number of measurement channels, OGO is good for probing the stochastic background. Furthermore, the three-dimensional configuration allows us to test alternative theories of gravity by searching for additional GW polarization modes. In addition, we briefly consider other source types such as pulsars, intermediate mass ($10^2 < M/M_{\odot} < 10^4$) black hole (IMBH) binaries and supernovae.
Finally, in Sec. \[S:Summary\], we summarize the description and abilities of the Octahedral Gravitational Observatory and mention additional hypothetical improvements. In this article, we use geometric units, $c = G = 1$, unless stated otherwise.
Orbits {#S:Orbit}
======
The realization of an octahedral constellation of spacecraft depends on the existence of suitable orbits. Driving factors, apart from separation stability, are assumed to be (i) fuel costs in terms of velocity $\Delta v$ necessary to deploy and maintain the constellation of six spacecraft, and (ii) a short constellation-to-Earth distance, required for a communication link with sufficient bandwidth to send data back to Earth. As described in the introduction, OGO features a three-dimensional satellite constellation. Therefore, using heliocentric orbits with a semimajor axis $a = 1$AU similar to LISA would cause a significant drift of radially separated spacecraft and is in our opinion not feasible.
However, in the last decades orbits in the nonlinear regime of Sun/Earth-Moon libration points L1 and L2 have been exploited, which can be reached relatively cheaply in terms of fuel [@Gomez1993]. A circular constellation can be deployed on a torus around a halo L1 orbit. The radius is limited by the amount of thrust needed for keeping the orbit stable. A realistic $\Delta v$ for orbit maintenance allows a nominal constellation radius of $r=1000$km [@Howell1999]. We assume the spacecraft B, C, E and F in Fig. \[F:orbit\] to be placed on such a torus, whereby the out-of-plane spacecraft A and D will head and trail on the inner halo. The octahedron formation then has a base length $L=\sqrt{2}\,r\approx1400$km. The halo and quasihalo orbits have an orbital period of roughly 180 days and the whole constellation rotates around the A-D line.
We already note at this point that a longer baseline would significantly improve the detector strain sensitivity. Therefore, we also propose an alternative configuration with an approximate average side length of $2\times10^9\,$m, where spacecraft A and D are placed on a small halo or Lissajous orbit around L1 and L2, respectively. The remaining spacecraft are arranged evenly on a (very) large halo orbit around either L1 or L2. However, simulations using natural reference trajectories showed that this formation is slightly asymmetric and that the variations in the arm lengths (and therefore in the angles between the links) are quite large. Nevertheless, we will revisit this alternative in Sec. \[sec:performance\] and do a rough estimation of its sensitivity. To warrant a full scientific study of such a long-arm-length detector would first require a more detailed study of these orbits.
Hence, we assume the 1400 km constellation to be a more realistic baseline, especially since the similarity of the spacecraft orbits is advantageous for the formation deployment, because large (and expensive) propulsion modules for each satellite are not required as proposed in the LISA/NGO mission [@NGOYellowBook; @LisaYellow]. The $2\times10^9\,$m formation will be stressed only to show the improvement of the detector sensitivity with longer arms.
Formation flight in the vicinity of Lagrange points L1 and L2 is still an ongoing research topic [@Folta2004]. Detailed (numerical) simulations have to be performed to validate these orbit options and to figure out appropriate orbit and formation control strategies. In particular the suppression of constellation deformations using non-natural orbits with correction maneuvers and required $\Delta v$ and fuel consumption needs to be investigated. Remaining deformations and resizing of the constellation will likely require a beam or telescope steering mechanism on the spacecraft.
In addition, the formation will have a varying Sun-incidence angle, leading to further issues for power supply, thermal shielding and blinding of interferometer arms. These points need to be targeted at a later stage of the OGO concept development as well as the effect of unequal arms on the DFI scheme.
Measurements and noise-canceling combinations {#S:TDI}
=============================================
In this section we will show how to combine the available measurement channels of the OGO detector to cancel laser and acceleration noise.
Each spacecraft of OGO is located at a corner of the octahedron, as shown in Fig. \[F:orbit\], and it exchanges laser light with four adjacent spacecraft. We consider interference between the beam emitted by spacecraft $I$ and received by spacecraft $J$ with the local beam in $J$, where $I,J = \{\mathrm{A,B,C,D,E,F}\}$ refer to the labels in Fig. \[F:orbit\]. For the sake of simplicity, we assume a rigid and nonrotating constellation. In other words, all arm lengths in terms of light travel time are equal, constant in time and independent of the direction in which the light is exchanged between two spacecraft. This is analogous to the first generation TDI assumptions [@TintoDhurandhar]. If the expected deviations from the equal arm configuration are small, then they can be absorbed into the low-frequency part of the acceleration noise. This imposes some restrictions on the orbits and on the orbit correction maneuvers. We also want to note that the overall breathing of the constellation (scaling of the arm length) is not important if the breathing time scale is significantly larger than the time required for the DFI formation, which is usually true. All calculations below are valid if we take the arm length at the instance of DFI formation, which is the value that affects the sensitivity of the detector.
The measurement of the fractional frequency change for each link is then given by $$s^{\mathrm{tot}}_{IJ} = h_{IJ} + b_{IJ} + \mathcal{D} p_{I} - p_{J} + \mathcal{D} \left( \vec{a}_{I}\cdot\hat{n}_{IJ} \right) - \left( \vec{a}_{J}\cdot\hat{n}_{IJ} \right) \, ,
\label{E:Mes1Gen}$$ where we have neglected the factors to convert displacement noise to optical frequency shifts. Here, we have the following:
(i) $h_{IJ}$ is the influence of gravitational waves on the link $I \rightarrow J$,
(ii) $b_{IJ}$ is the shot noise (and other similar noise sources at the photo detector and phase meter of spacecraft $J$) along the link $I \rightarrow J$.
(iii) $p_{I}$ is the laser noise of spacecraft $I$.
(iv) $\vec{a}_{I}$ is the acceleration noise of spacecraft $I$.
(v) $\hat{n}_{IJ} = ( \vec{x}_{J} - \vec{x}_{I}) / L$ is the unit vector along the arm $I \rightarrow J$ (with length $L$). Hence, the scalar product $\vec{a}_I \cdot \hat{n}_{IJ}$ is the acceleration noise of spacecraft $I$ projected onto the arm characterized by the unit vector $\hat{n}_{IJ}$.
This is similar to TDI considerations, but in addition to canceling the laser noise $p_I$, we also want to eliminate the influence of the acceleration noise, that is all terms containing $a_I$. Following Ref. [@TintoDhurandhar], we have introduced a delay operator $\mathcal{D}$, which acts as $$\mathcal{D} y(t) = y( t - L) \, .
\label{E:Delay}$$ Note that we use a coordinate frame associated with the center of the octahedron, as depicted in Fig. \[F:orbit\].
The basic idea is to find combinations of the individual measurements (Eq. \[E:Mes1Gen\]) which are free of acceleration noise $\vec{a}_{I}$ and laser noise $p_{I}$. In other words, we want to find solutions to the following equation: $$\sum_{\textrm{all} \ IJ \ \textrm{links}} q_{IJ} \ s_{IJ} = 0\,.
\label{E:GenCondNoiseNull}$$ In Eq. (\[E:GenCondNoiseNull\]), $q_{IJ}$ denotes an unknown function of delays $\mathcal{D}$ and $ s_{IJ}$ contains only the noise we want to cancel: $$\begin{aligned}
s_{IJ} &\equiv& s^{\mathrm{tot}}_{IJ}(b_{IJ} = h_{IJ} = 0) \nonumber \\
&=& \mathcal{D} p_{I} - p_{J} + \mathcal{D} \left( \vec{a}_{I}\cdot\hat{n}_{IJ} \right) - \left( \vec{a}_{J}\cdot\hat{n}_{IJ} \right).
\label{E:MesN}
\end{aligned}$$ If a given $q_{IJ}$ is a solution, then $f(\mathcal{D})q_{IJ}$ is also a solution, where $f(\mathcal{D})$ is a polynomial function (of arbitrary order) of delays. The general method for finding generators of the solutions for this equation is described in Ref. [@TintoDhurandhar] and we will follow it closely.
Before we proceed to a general solution for Eq. (\[E:GenCondNoiseNull\]), we can check that the solution corresponding to Mach-Zehnder interferometers suggested in Ref. [@chen2006] also satisfies Eq. (\[E:GenCondNoiseNull\]):
$$\begin{aligned}
Y_1 &= [ \,( s_{CD} + \mathcal{D} s_{AC} ) - ( s_{CA} + \mathcal{D} s_{DC} ) + ( s_{FD} + \mathcal{D} s_{AF} ) \nonumber \\
&- ( s_{FA} + \mathcal{D} s_{DF} ) \, ] - [ \, ( s_{BD} + \mathcal{D} s_{AB} ) - ( s_{BA} + \mathcal{D} s_{DB} ) \nonumber \\
& + ( s_{ED} + \mathcal{D} s_{AE} ) - ( s_{EA} + \mathcal{D} s_{DE} )\, ] \,.
\end{aligned}$$
Using the symmetries of an octahedron, we can write down two other solutions: $$\begin{aligned}
Y_2 &= [\, (s_{CE} + \mathcal{D} s_{BC}) - (s_{CB} + \mathcal{D} s_{EC}) + (s_{FE} + \mathcal{D} s_{BF}) \nonumber \\
&- (s_{FB} + \mathcal{D} s_{EF})\,] - [\,(s_{AE} + \mathcal{D} s_{BA}) - (s_{AB} + \mathcal{D} s_{EA})\nonumber\\
& + (s_{DE} + \mathcal{D} s_{BD}) - (s_{DB} + \mathcal{D} s_{ED}) \,]\, , \\
& \nonumber \\
Y_3 &= [\,(s_{DF} + \mathcal{D} s_{CD}) - (s_{DC} + \mathcal{D} s_{FD}) + (s_{AF} + \mathcal{D} s_{CA}) \nonumber\\
&- (s_{AC} + \mathcal{D} s_{FA})\,] - [\,((s_{EF} + \mathcal{D} s_{CE}) - (s_{EC} + \mathcal{D} s_{FE})\nonumber\\
&+ (s_{BF} + \mathcal{D} s_{CB}) - (s_{BC} + \mathcal{D} s_{FB}) \,]\,.
\end{aligned}$$
We can represent these solutions as 24-tuples of coefficients for the delay functions $q_{IJ}$:
$$\begin{aligned}
q_1 &= & \{ 1, 1, -1, -1, -1, -1, 1, 1, -\mathcal{D}, \mathcal{D}, 0, 0, -\mathcal{D},
\mathcal{D}, 0, 0, \nonumber\\
& & \mathcal{D},-\mathcal{D}, 0, 0, \mathcal{D}, -\mathcal{D}, 0, 0 \} \, , \\
q_2 &= & \{ -\mathcal{D} , \mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D} , 0 , 0 , 1 , 1 , -1 , -1 , -1 , -1 , 1 , 1, \nonumber\\
& & 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , \mathcal{D} , -\mathcal{D} \} \, , \\
q_3 &= & \{ 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D}, \nonumber\\
& & -1 , -1 , 1 , 1 , 1 , 1 , -1 , -1 \} \, .
\end{aligned}$$
The order used in the 24-tuples is $\{ BA$, $EA$, $CA$, $FA$, $BD$, $ED$, $CD$, $FD$, $AB$, $DB$, $CB$, $FB$, $AE$, $DE$, $CE$, $FE$, $AC$, $DC$, $BC$, $EC$, $AF$, $DF$, $BF$, $EF \}$, so that, for example, the first entry in $q_1$ represents the $s_{BA}$ coefficient in the $Y_1$ equation.
These particular solutions illustrate that not all links are used in producing a DFI stream. Multiple zeros in the equations for $q_1, q_2, q_3$ above indicate those links which do not contribute to the final result, and each time we use only 16 links. We will come back to the issue of “lost links” when we discuss the network sensitivity.
In the following, we will find generators of all solutions. The first step is to use Gaussian elimination (without division by delay operators) in Eq. (\[E:GenCondNoiseNull\]), and as a result, we end up with a single (master) equation which we need to solve: $$\begin{aligned}
0 &=& (\mathcal{D}-1)^2 q_{BC} + (\mathcal{D}-1)\mathcal{D} q_{CE} + (1-\mathcal{D})(\mathcal{D}-1)\mathcal{D} q_{DB} \nonumber \\
&+& (\mathcal{D}-1)((1-\mathcal{D})\mathcal{D}-1) q_{DC} \nonumber\\
&+& (\mathcal{D}-1) q_{DF} + (\mathcal{D}-1) q_{EF} \, . \label{E:TDIAcc_FinalEq}
\end{aligned}$$ In the next step, we want to find the so-called “reduced generators” of Eq. (\[E:TDIAcc\_FinalEq\]), which correspond to the reduced set $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. For this we need to compute the Gröbner basis [@Buchberger1970], a set generating the polynomial ideals $q_{IJ}$. Roughly speaking, the Gröbner basis is comparable to the greatest common divisor of $q_{IJ}$. Following the procedure from Ref. [@TintoDhurandhar], we obtain seven generators:
$$\begin{aligned}
\label{E:S1}
S_1 &=& \{ 0, \mathcal{D}^2 + \mathcal{D}, 0, - \mathcal{D} - \mathcal{D}^2, 1 - \mathcal{D},\mathcal{D}^2 + 1, -1 + \mathcal{D}, -1 - \mathcal{D}^2, \mathcal{D} - \mathcal{D}^2, 0, -\mathcal{D},\mathcal{D}^2, -\mathcal{D}^2 - 1, -\mathcal{D} - 1, 1, \nonumber \\
& & 1 + \mathcal{D} + \mathcal{D}^2, -\mathcal{D} + \mathcal{D}^2,0, \mathcal{D}, -\mathcal{D}^2, \mathcal{D}^2 + 1, 1 + \mathcal{D}, -1, -\mathcal{D} - \mathcal{D}^2 - 1 \},\\
& &\nonumber \\
S_2 &=& \{ \mathcal{D} + 1, \mathcal{D} + 1, -\mathcal{D} -1,-\mathcal{D}-1, -1+\mathcal{D},\mathcal{D}-1, 1-\mathcal{D}, 1 - \mathcal{D}, -2\mathcal{D},0,\mathcal{D},\mathcal{D},-2\mathcal{D},0, \mathcal{D},\mathcal{D}, 2\mathcal{D}, 0, -\mathcal{D}, \nonumber \\
& & -\mathcal{D},2\mathcal{D},0,-\mathcal{D},-\mathcal{D}\} , \\
& &\nonumber \\
S_3 &=& \{ 0, \mathcal{D}, -\mathcal{D}, 0, - 1, \mathcal{D} - 1, 1- \mathcal{D},1, 1 - \mathcal{D}, 1, -1 + \mathcal{D}, -1, -\mathcal{D},0, \mathcal{D}, 0, \mathcal{D}, 0, 0, -\mathcal{D}, \mathcal{D} -1,-1, 1, -\mathcal{D} + 1 \},\\
& &\nonumber \\
S_4 &=& \{ \mathcal{D}, -\mathcal{D} + \mathcal{D}^2, \mathcal{D}, -\mathcal{D}-\mathcal{D}^2, 2, -2\mathcal{D}+\mathcal{D}^2+2, -2+2\mathcal{D}, -2-\mathcal{D}^2,2\mathcal{D} - 2 -\mathcal{D}^2,-2,2 - 2\mathcal{D},2+\mathcal{D}^2,\mathcal{D} - \mathcal{D}^2, \nonumber \\
& & -\mathcal{D},-\mathcal{D},\mathcal{D} + \mathcal{D}^2,-2\mathcal{D} + \mathcal{D}^2,0,0,2\mathcal{D} - \mathcal{D}^2,-\mathcal{D} + \mathcal{D}^2 + 2, 2 + \mathcal{D},-2-\mathcal{D},\mathcal{D} - \mathcal{D}^2 - 2\} , \\
& &\nonumber \\
S_5 &=& \{ 0, \mathcal{D}^2 + \mathcal{D}, -\mathcal{D}^2, - \mathcal{D}, 1 - \mathcal{D}, \mathcal{D}^2 + 1, \mathcal{D} - \mathcal{D}^2 - 1, -1, \mathcal{D} - \mathcal{D}^2, 0, -\mathcal{D} + \mathcal{D}^2, 0, -1 - \mathcal{D}^2, -\mathcal{D} - 1, 1 + \mathcal{D}^2, \nonumber \\
& & 1 + \mathcal{D}, \mathcal{D}^2, \mathcal{D}, 0, -\mathcal{D}^2 - \mathcal{D}, -\mathcal{D} + \mathcal{D}^2 +1, 1, \mathcal{D} - 1, -1 - \mathcal{D}^2\},\\
& & \nonumber \\
S_6 &=& \{ \mathcal{D} + 2 + \mathcal{D}^2,\mathcal{D}+\mathcal{D}^3+2,-\mathcal{D} + \mathcal{D}^2 - 2, -\mathcal{D} - 2 - 2\mathcal{D}^2 - \mathcal{D}^3,-2 + 2\mathcal{D},2\mathcal{D} - \mathcal{D}^2 + \mathcal{D}^3 - 2,\nonumber \\
& & -2\mathcal{D} + 2\mathcal{D}^2 + 2,2 - 2\mathcal{D} - \mathcal{D}^2 - \mathcal{D}^3,\mathcal{D}^2 - 4\mathcal{D} - \mathcal{D}^3, 0, 2\mathcal{D} -2\mathcal{D}^2,2\mathcal{D} + \mathcal{D}^2 + \mathcal{D}^3,-3\mathcal{D} - \mathcal{D}^3,\mathcal{D} - \mathcal{D}^2, \nonumber \\
& &\mathcal{D} - \mathcal{D}^2, 2\mathcal{D}^2 + \mathcal{D} + \mathcal{D}^3, -\mathcal{D}^2 + 2\mathcal{D} + \mathcal{D}^3, -2\mathcal{D},0,\mathcal{D}^2 - \mathcal{D}^3,5\mathcal{D} + \mathcal{D}^3,\mathcal{D} + \mathcal{D}^2,-3\mathcal{D} - \mathcal{D}^2,-3\mathcal{D} - \mathcal{D}^3 \} , \\
& &\nonumber \\
S_7 &=& \{ 1, 1 + \mathcal{D}, -1, -1 - \mathcal{D}, 0, \mathcal{D}, 0, -\mathcal{D}, -\mathcal{D}, 0, 0, \mathcal{D}, -1 - \mathcal{D}, -1, 1, 1 + \mathcal{D}, \mathcal{D}, 0, 0, -\mathcal{D}, 1 + \mathcal{D}, 1, -1, -1 - \mathcal{D} \}.
\label{E:S7}
\end{aligned}$$
As before, these operators have to be applied to $s_{IJ}$, using the same ordering as given above. All other solutions can be constructed from these generators. A detailed derivation of expressions (\[E:S1\])–(\[E:S7\]) is given in Appendix \[S:Appendix\].
Before we proceed, let us make several remarks. The generators found here are not unique, just like in the case of TDI [@TintoDhurandhar]. The set of generators does not necessarily form a minimal set, and we can only guarantee that the found set of generators gives us a module of syzygies and can be used to generate other solutions. The combinations $S_1$ to $S_7$ applied on 24 raw measurements $s_{IJ}^{\mathrm{tot}}$ eliminate both laser and displacement noise while mostly preserving the gravitational wave signal. Note that again in those expressions we do not use all links – for example, if the link $BA$ is lost due to some reasons, we still can use $S_1, S_3, S_5$ to produce DFI streams.
Response functions and sensitivity {#S:Sens}
==================================
In the previous section we have found generators that produce data streams free of acceleration and laser noise. Now we need to apply these combinations to the shot noise and to the GW signal to compute the corresponding response functions.
Shot noise level and noise transfer function
--------------------------------------------
We will assume that the shot noise is independent (uncorrelated) in each link and equal in power spectral density, based on identical laser sources and telescopes on each spacecraft. We denote the power spectral density of the shot noise by $\widetilde{S}_{\rm sn}$. A lengthy but straightforward computation shows that the spectral noise $\tilde{S}_{\mathrm{n},i}$ corresponding to the seven combinations $S_i$, $i=1,\ldots,7$ from Eqs. (\[E:S1\]–\[E:S7\]) is given by
$$\begin{aligned}
\widetilde{S}_{\rm n, 1} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2\,( 9 + 2\cos2\epsilon + 3\cos4\epsilon)\,,\\
\widetilde{S}_{\rm n, 2} &=& 160\, \widetilde{S}_{\rm sn} \,\epsilon^2\,,\\
\widetilde{S}_{\rm n, 3} &=& \0 48\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,( 2 - \cos2\epsilon)\,,\\
\widetilde{S}_{\rm n, 4} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(24 -13\cos2\epsilon + 6\cos4\epsilon)\,,\\
\widetilde{S}_{\rm n, 5} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2(\, 9 - 2\cos 2\epsilon + 3\cos 4\epsilon)\,,\\
\widetilde{S}_{\rm n, 6} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(45 -6\cos2\epsilon+17\cos4\epsilon)\,,\\
\widetilde{S}_{\rm n, 7} &=& \048\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(2 + \cos2\epsilon)\,,
\end{aligned}$$
where $\epsilon \equiv \omega L/2$, with the GW frequency $\omega$. In the low frequency limit ($\epsilon \ll 1$), the noise $\widetilde{S}_{{\rm n }, i}$ for each combination $S_i$ is proportional to $\epsilon^2$.
Let us now compute the shot noise in a single link. We consider for OGO a configuration with LISA-like receiver-transponder links and the following parameters: spacecraft separation $L=1414\,$km, laser wavelength $\lambda=532$nm, laser power $P=10\,$W and telescope diameter $D=1\,$m. For this arm length and telescope size, almost all of the laser power from the remote spacecraft is received by the local spacecraft. Hence, the shot-noise calculation for OGO is different from the LISA case, where an overwhelming fraction of the laser beam misses the telescope [@LisaYellow].
For a Michelson interferometer, the sensitivity to shot noise is usually expressed as [@Maggiore_book] $$\sqrt{\widetilde{S}_h(f)}=\frac{1}{2L}\sqrt{\frac{\hbar c \lambda}{\pi P}}\,\,\, [1/\sqrt{\rm Hz}] \, ,$$ where we have temporarily restored the speed of light $c$ and the reduced Planck constant $\hbar$. Notice that the effect of the GW transfer function is not included here yet. For a single link $I \rightarrow J$ of OGO as opposed to a full two-arm Michelson with dual links, $\sqrt{\widetilde{S}_{h,IJ}}$ is a factor of 4 larger. However, our design allows the following two improvements: (i) Since there is a local laser in $J$ with power similar to the received laser power from $I$, the power at the beam splitter is actually $2P$, giving an improvement of $1/\sqrt{2}$. This is also different from LISA, where due to the longer arm length the received power is much smaller than the local laser power. (ii) If we assume that the arm length is stable enough to operate at the dark fringe, then we gain another factor of $1/\sqrt{2}$.
So, we arrive at the following shot-noise-only sensitivity for a single link: $$\label{E:shotnoise1}
\sqrt{\widetilde{S}_{h ,IJ}(f)}=\frac{1}{L}\sqrt{\frac{\hbar c \lambda}{\pi P}}\,\,\, [1/\sqrt{\rm Hz}] \, .$$
GW signal transfer function and sensitivity {#sec:transfer_function}
-------------------------------------------
Next, we will compute the detector response to a gravitational wave signal. We assume a GW source located in the direction $\hat{n} = -\hat{k} = \left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right)$ as seen from the detector frame. We choose unit vectors $$\hat{u} = \left[
\begin {array}{c}
\cos \theta \cos \phi \\
\cos \theta \sin \phi \\
-\sin \theta
\end {array} \right],\;\;\;\;\;
\hat{v} = \left[
\begin {array}{c}
\sin \phi \\
-\cos \phi \\
0
\end {array} \right]$$ orthogonal to $\hat{k}$ pointing tangentially along the $\theta$ and $\phi$ coordinate lines to form a polarization basis. This basis can be described by polarization tensors $\mathbf{e}_+$ and $\mathbf{e}_\times$, given by $$\mathbf{e}_+ \equiv \hat{u}\otimes \hat{u} - \hat{v}\otimes \hat{v}\,, \ \ \
\mathbf{e}_\times \equiv \hat{u}\otimes \hat{v} + \hat{v}\otimes \hat{u}\,.
\label{E:polbas}$$ The single arm fractional frequency response to a GW is [@EW1975] $$\begin{aligned}
h_{IJ} = \frac{H_{IJ} (t - \hat{k}\cdot\vec{x}_{I} - L) - H_{IJ} (t - \hat{k}\cdot\vec{x}_{J} )}{2 \left( 1 - \hat{k}\cdot\hat{n}_{IJ} \right)} ,
\label{E:E:GWlink}
\end{aligned}$$ where $\vec{x}_I$ is the position vector of the $I$-th spacecraft, $L$ the (constant) distance between two spacecraft and $$H_{IJ } (t) \equiv h_{+} (t) \ \xi_{+} (\hat{u}, \hat{v}, \hat{n}_{IJ}) + h_{\times} (t) \ \xi_{\times} (\hat{u}, \hat{v}, \hat{n}_{IJ}) \, .$$ Here $h_{+,\times}(t)$ are two GW polarizations in the basis (\[E:polbas\]) and
$$\begin{aligned}
\xi_{+}(\hat{u}, \hat{v}, \hat{n}_{IJ}) &\equiv \hat{n}_{IJ}^\textsf{T}\mathbf{e}_+ \hat{n}_{IJ} =
{\left(\hat{u}\cdot \hat{n}_{IJ} \right)}^{2} - {\left(\hat{v}\cdot \hat{n}_{IJ} \right)}^{2} \, , \nonumber\\
\xi_{\times}(\hat{u}, \hat{v}, \hat{n}_{IJ}) &\equiv \hat{n}_{IJ}^\textsf{T}\mathbf{e}_\times
\hat{n}_{IJ} = 2 \left(\hat{u}\cdot \hat{n}_{IJ} \right)\left(\hat{v}\cdot \hat{n}_{IJ} \right) \, .
\label{E:E:GWStrain}\end{aligned}$$
In order to find the arm response for arbitrary incident GWs, we can compute the single arm response to a monochromatic GW with Eq. (\[E:E:GWlink\]) and then deduce the following general response in the frequency domain, $$\begin{aligned}
h_{IJ}(f) & =& \epsilon \, {\rm sinc}\left[\epsilon(1-\hat{k}\cdot\hat{n}_{IJ})\right]
\mathrm{e}^{-\mathrm{i}\epsilon[\hat{k}\cdot(\vec{x}_{I} + \vec{x}_{J})/L +1]} \nonumber \\
& & \ \ \ \times \left[\xi_{+}(\hat{n}_{IJ}) h_{+}(f) + \xi_{\times}(\hat{n}_{IJ}) h_{\times}(f)\right] \,,
\end{aligned}$$ where we used the normalized sinc function, conventionally used in signal processing: ${\rm sinc}(x) := {\sin(\pi x)}/(\pi x)$.
Hence, the transfer function for a GW signal is $$\begin{aligned}
\mathcal{T}_{IJ+,\times}^{\rm GW}(f) &=& \epsilon \, {\rm sinc}\left[\epsilon(1-\hat{k}\cdot\hat{n}_{IJ})\right] \nonumber\\
& & \ \ \ \times\, \mathrm{e}^{-\mathrm{i}\epsilon[\hat{k}\cdot(\vec{x}_{I} + \vec{x}_{J})/L +1]}
\xi_{+, \times}(\hat{n}_{IJ}) \, .
\label{Eq:TF}
\end{aligned}$$ For the sake of simplicity, we will from now on assume that the GW has “+” polarization only. This simplification will not affect our qualitative end result. Substituting the transfer function for a single arm response into the above 7 generators \[Eqs. (\[E:S1\])-(\[E:S7\])\], we can get the transfer function $\mathcal{T}_{i}^{\rm GW}$ for each combination. The final expressions are very lengthy and not needed here explicitly.
Having obtained the transfer function, we can compute the sensitivity for each combination $i=1,\ldots,7$ as $$\label{E:sensitivity1}
\sqrt{\widetilde{S}_{h,i}} = \sqrt{\frac{\widetilde{S}_{{\rm n}, i}}{\langle(\mathcal{T}^{\rm GW}_i)^2\rangle}} \ ,$$ where the triangular brackets imply averaging over polarization and source sky location.
We expect up to 12 independent round trip measurements, corresponding to the number of back-and-forth links between spacecraft. It is out of the scope of this work to explicitly find all noise-uncorrelated combinations (similar to the optimal channels $A, E, T$ in the case of LISA [@TintoDhurandhar]). However, if we assume approximately equal sensitivity for each combination (which is almost the case for the combinations $S_1,\ldots,S_7$), we expect an improvement in the sensitivity of the whole network by a factor $1/\sqrt{12}$.
![Sensitivities for two single DFI combinations ($S_1$, *blue crosses* and $S_5$, *green plus signs*) of OGO (with $L\approx1400$km) and for the full OGO network sensitivity (scaled from $S_5$, *red solid line*). For comparison, the dashed lines show sensitivities for initial LIGO (H1 during science run S6, from Ref. [@Abadie2010a], *cyan dashed line*) and aLIGO (design sensitivity for high-power, zero detuning configuration, from Ref. [@aLIGO_sensitivity], *magenta dash-dotted line*).[]{data-label="F:Sens"}](ogo_sensitivity "fig:"){width="\columnwidth"}\
Therefore, we simply approximate the network sensitivity of the full detector as $\sqrt{\widetilde{S}_{h, {\rm net}}} = \sqrt{\widetilde{S}_{h,5}/12}$. Note that the potential loss of some links would imply that not all generators can be formed. We can lose up to 6 links and still be able to form a DFI stream (but probably only one). The number of lost links (and which links are lost exactly) will affect the network sensitivity. In our estimations below we deal with the idealized situation and assume that no links are lost.
We plot the sensitivity curves for individual combinations and the network sensitivity in Fig. \[F:Sens\]. For comparison we also show the design sensitivity curves of initial LIGO (S6 science run [@Abadie2010a]) and advanced LIGO (high laser power configuration with zero detuning of the signal recycling mirror [@aLIGO_sensitivity]). Indeed one can see that the sensitivities of the individual OGO configurations are similar to each other and close to initial LIGO. The network sensitivity of OGO lies between LIGO and aLIGO sensitivities. OGO as expected outperforms aLIGO below 10Hz, where the seismic noise on the ground becomes strongly dominant.
General performance of the DFI scheme {#sec:performance}
-------------------------------------
![Network sensitivities, scaled from $S_5$, of standard OGO (with DFI, arm length 1414 km, *red solid line*) compared to an OGO-like detector with spacecraft separation of $2\cdot10^9$ m, with either full DFI scheme (*blue crosses*) or standard TDI only (*green plus signs*). Also shown for comparison are (classic) LISA ($5\cdot10^9$ m, network sensitivity, *magenta dashed line*, from Ref. [@Larson2000]) and DECIGO (using the fitting formula Eq. (20) from Ref. [@Yagi2013], *cyan dash-dotted line*).[]{data-label="F:ogo_TDI_comparison_2Mkm"}](ogo_TDI_comparison_2Mkm_nojitter "fig:"){width="\columnwidth"}\
Having derived the full sensitivity curve of the OGO mission design with $L\approx1400$km as an exemplary implementation of the three-dimensional DFI scheme in space, let us take a step back and analyze the general performance of a DFI-enabled detector. These features are also what led us to consider the octahedron configuration in the first place.
Specifically, let us look in more detail at the low frequency asymptotic behavior of the transfer functions and sensitivity curves. We consider a LISA-like configuration with two laser noise free combinations: an unequal arm Michelson () and a Sagnac combination (). Let us assume for a moment that the only noise source is shot noise, which at low frequencies ($\epsilon \ll 1$) scales as $\sqrt{\widetilde{S}_{\mathrm{n},X}} \sim \epsilon^2$ and $\sqrt{\widetilde{S}_{\mathrm{n},\alpha}} \sim \epsilon^1$ for those two combinations, respectively.
The GW transfer function, for both TDI combinations, scales as $\mathcal{T}_\alpha,\mathcal{T}_X \sim \epsilon^2$; therefore, the sensitivity curves scale as $\sqrt{\widetilde{S}_{h,\alpha}} \sim \sqrt{\widetilde{S}_{\mathrm{n},\alpha}}/\mathcal{T}_\alpha \sim \epsilon^{-1}$ for and $\sqrt{\widetilde{S}_{h,X}} \sim \sqrt{\widetilde{S}_{\mathrm{n},X}}/\mathcal{T}_X \sim \epsilon^{0}$ for . We see that a LISA-like -combination has a *flat* shot-noise spectrum at low frequencies, corresponding to a flat total detector sensitivity if all other dominant noise sources can be canceled – which looks extremely attractive.
Thus, a naive analysis suggests that the acceleration and laser noise free combinations for an octahedron detector could yield a flat sensitivity curve at low frequencies. Checking this preliminary result with a more careful analysis was the main motivation for the research presented in this article.
In fact, as we have seen in Sec. \[sec:transfer\_function\], the full derivation delivers transfer functions that, in leading order of $\epsilon$, go as $\mathcal{T}_{1,2,\ldots,7} \sim \epsilon^3$. This implies that the sensitivity for laser and acceleration noise free combinations behaves as $\sqrt{\widetilde{S}_{h,1,2,\ldots,7}}/\mathcal{T}_{1,2,\ldots,,7} \sim \epsilon^{-2}$, which is similar to the behavior of acceleration noise. In other words, the combinations eliminating the acceleration noise also cancel a significant part of the GW signal at low frequencies.
In fact, we find that a standard LISA-like TDI-enabled detector of the same arm length and optical configuration as OGO could achieve a similar low-frequency sensitivity (at few to tens of Hz) with an acceleration noise requirement of only $\sim10^{-12}$ m/s${}^2\,\sqrt{\mathrm{Hz}}$. This assumes negligible spacecraft jitter and that no other noise sources (phase-meter noise, sideband noise, thermal noise) limit the sensitivity, which at this frequency band would behave differently than in the LISA band. In fact, the GOCE mission [@Drinkwater2003] has already demonstrated such acceleration noise levels at mHz frequencies [@Sechi2011], and therefore this seems a rather modest requirement at OGO frequencies. We therefore see that such a short-arm-length OGO would actually only be a more complicated alternative to other feasible mission designs.
In addition, it is hard to see from just the comparison with ground-based detectors in Fig. \[F:Sens\] how exactly the DFI method itself influences the final noise curve of OGO, and how much of its shape is instead determined by the geometrical and technical parameters of the mission concept (arm length, laser power, telescope size). Also, the secondary technological noise sources of a space mission in the comparatively high-frequency band of this exemplary OGO implementation are somewhat different from more well-studied missions like LISA and DECIGO.
Therefore, to disentangle these effects, we will now tentatively study a different version of OGO based on the alternative orbit with an average arm length of $2\cdot10^9$m, as mentioned in Sec. \[S:Orbit\]. It requires further study to determine whether a stable octahedron constellation and the DFI scheme are possible on such an orbit, but assuming they are, we can compute its sensitivity as before.
In Fig. \[F:ogo\_TDI\_comparison\_2Mkm\], we then compare this longer-baseline DFI detector with another detector with the same geometry and optical components, but without the DFI technique, using instead conventional TDI measurements. Here, we are in a similar frequency range as LISA and therefore assume similar values for the acceleration noise of $3 \cdot 10^{-15}$ m/s${}^2\,\sqrt{\mathrm{Hz}}$ [@LisaYellow] and secondary noise sources (phase meter, thermal noise, etc.; see Sec. \[sec:feasibility\]).
However, there is another noise source, spacecraft jitter, which is considered subdominant for LISA, but might become relevant for both the TDI and DFI versions of the $2\cdot10^9$m OGO-like detector. Jitter corresponds to the rotational degrees of freedom between spacecraft, and its coupling into measurement noise is not fully understood. We have therefore computed both sensitivities without any jitter. It seems possible that at least the part of jitter that couples linearly into displacement noise could also be canceled by DFI, or that an extension of DFI (e.g. more links) could take better care of this, and therefore that the full OGO with DFI would look more favorable compared to the TDI version when nonvanishing jitter is taken into account.
Generally, as one goes for longer arm lengths, the DFI scheme will perform better in comparison to the TDI scheme. At the high-frequency end of the sensitivity curves, both schemes are limited by shot noise and the respective GW transfer functions. Since the shot-noise level does not depend on the arm length, it remains the same for all relevant frequencies. Therefore, as the arm length increases, the high-frequency part of the sensitivity curves moves to the low-frequency regime in parallel (i.e. the corner frequency of the transfer function is proportional to $1/L$). This is the same for both schemes.
On the other hand, in the low-frequency regime of the sensitivity curves the two schemes perform very differently. For TDI, the low-frequency behavior is limited by acceleration noise, while for DFI this part is again limited by shot noise and the GW transfer function. When the arm length increases, the low-frequency part of the sensitivity curve in the TDI scheme moves to lower frequencies in proportion to $1/\sqrt{L}$; while for DFI, it moves in proportion to $1/L$.
Graphically, when the arm length increases, the high-frequency parts of the sensitivity curves in both schemes move toward the lower-frequency regime in parallel, while the low-frequency part of the sensitivity curve for DFI moves faster than for TDI.
Under the assumptions given above, we find that an arm length of $2\cdot10^9$m is close to the transition point where the sensitivities of TDI and DFI are almost equal, as shown in Fig. \[F:ogo\_TDI\_comparison\_2Mkm\]. At even longer arm lengths, employing DFI would become clearly advantageous.
Of course, these considerations show that a longer-baseline detector with good sensitivity in the standard space-based detector frequency band of interest would make a scientifically much more interesting case than the default short-arm OGO which we presented first. However, as no study on the required orbits has been done so far, we consider such a detector variant to be highly hypothetical and not worthy of a detailed study of technological feasibility and scientific potential yet. Instead, for the remainder of this paper, we concentrate again on the conservative 1400km version of OGO. Although the sensitivity curve in Fig. \[F:Sens\] already demonstrates its limited potential, we will attempt to neutrally assess its advantages, limitations and scientific reach.
Technological feasibility {#sec:feasibility}
-------------------------
Employing DFI requires a large number of spacecraft but on the other hand allows us to relax many of the very strict technological requirements of other space-based GW detector proposals such as (e)LISA and DECIGO. Specifically, the clock noise is canceled by design, so there is no need for a complicated clock tone transfer chain [@barke2010]. Furthermore, OGO does not require a drag-free technology, and the configuration has to be stabilized only as much as required for the equal arm length assumption to hold. This strongly reduces the requirements on the spacecraft thrusters. Also, for the end mirrors, which have to be mounted on the same monolithic structure for all four laser links per spacecraft, it is not required that they are freefalling. Instead, they can be fixed to the spacecraft.
Still, to reach the shot-noise-only limited sensitivity shown in Fig. \[F:Sens\], the secondary noise contributions from all components of the measurement system must be significantly below the shot-noise level. Considering a shot-noise level of about $2\cdot 10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ – which is in agreement with the value derived earlier for the 1400km version of OGO – this might be challenging.
When actively controlling the spacecraft position and hence stabilizing the distance and relative velocity between the spacecraft, we will be able to lower the heterodyne frequency of the laser beat notes drastically. Where LISA will have a beat note frequency in the tens of MHz, with OGO’s short arm length we could be speaking of kHz or less and might even consider a homodyne detection scheme as in LIGO. This might in the end enable us to build a phase meter capable of detecting relative distance fluctuations with a sensitivity of $10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ or below as required by OGO.
As mentioned before, temperature noise might be a relevant noise source for OGO: The relative distance fluctuations on the optical benches due to temperature fluctuations and the test mass thermal noise must be significantly reduced in comparison to LISA. But even though the LISA constellation is set in an environment which is naturally more temperature stable, stabilization should be easier for the higher-frequency OGO measurement band. A requirement of $10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ could be reached by actively stabilizing the temperature down to values of 1nK/$\sqrt{\mathrm{Hz}}$ at the corner frequency.
Assuming future technological progress, optimization of the optical bench layout could also contribute to mitigating this constraint, as could the invention of thermally more stable materials for the optical bench. Most likely, this challenge can be solved only with a combination of the mentioned approaches.
The same is true for the optical path length stability of the telescopes. We estimate the required pointing stability to be roughly similar to the LISA mission requirements.
Shot-noise reduction {#sec:shot_noise_reduction}
--------------------
Assuming the requirements from the previous section can be met, the timing and acceleration noise free combinations of the OGO detector are dominated by shot noise, and any means of reducing the shot noise will lead to a sensitivity improvement over all frequencies. In this subsection, we discuss possible ways to achieve such a reduction.
The most obvious solution is to increase laser power, with an achievable sensitivity improvement that scales with $\sqrt{P}$. However, the available laser power is limited by the power supplies available on a spacecraft. Stronger lasers are also heavier and take more place, making the launch of the mission more difficult. Therefore, there is a limit to simply increasing laser power, and we want to shortly discuss more advanced methods of shot-noise reduction.
One such hypothetical possibility is to build cavities along the links between spacecraft, similar to the DECIGO design [@Ando2010]. The shot noise would be decreased due to an increase of the effective power stored in the cavity. Effectively, this also results in an increase of the arm length. Note, however, that the sensitivity of OGO with cavities cannot simply be computed by inserting effective power and arm length into our previously derived equations. Instead, a rederivation of the full transfer function along the lines of Ref. [@Rakhmanov2005] is necessary.
Alternatively, squeezed light [@Schnabel10] is a way to directly reduce the quantum measurement noise, which has already been demonstrated in ground-based detectors [@SqueezedGEO; @Khalaidovski12]. However, squeezing in a space-based detector is challenging in many aspects due to the very sensitive procedure and would require further development.
Scientific perspectives {#S:Sources}
=======================
In this section, we will discuss the science case for our octahedral GW detector (with an arm length of 1400km) by considering the most important potential astrophysical sources in its band of sensitivity. Using the full network sensitivity, as derived above, the best performance of OGO is at 78Hz, between the best achieved performance of initial LIGO during its S6 science run and the anticipated sensitivity for advanced LIGO. OGO outperforms the advanced ground-based detectors below 10Hz, where the seismic noise strongly dominates. In this analysis, we will therefore consider sources emitting GWs with frequencies between 1Hz and 1kHz, concentrating on the low end of this range.
Basically, those are the same sources as for ground-based detectors, which include compact binaries coalescences (CBCs), asymmetric single neutron stars (continuous waves, CWs), binaries containing intermediate-mass black holes (IMBHs), burst sources (unmodeled short-duration transient signals), and a cosmological stochastic background.
We will go briefly through each class of sources and consider perspectives of their detection. As was to be expected from the sensitivity curve in Fig. \[F:Sens\], in most categories OGO performs better than initial ground-based detectors, but does not even reach the potential of the advanced generation currently under commissioning.
Therefore, this section should be understood not as an endorsement of actually building and flying an OGO-like mission, but just as an assessment of its (limited, but existing) potentials. This demonstrates that an octahedral GW detector employing DFI in space is in principle capable of scientifically interesting observations, even though improving its performance to actually surpass existing detectors or more mature mission proposals still remains a subject of further study.
In addition, we put a special focus on areas where OGO’s design offers some specific advantages. These include the triangulation of CBCs through joint detection with ground-based detectors as well as searching for a stochastic GW background and for additional GW modes.
Note that the hypothetical $2\cdot10^9$m variant of OGO (see Secs. \[S:Orbit\] and \[sec:performance\]) would have a very different target population of astrophysical sources due to its sensitivity shift to lower frequencies. Such a detector would still be sensitive to CBCs, IMBHs, and stochastic backgrounds, probably much more so. But instead of high-frequency sources like CW pulsars and supernova bursts, it would start targeting supermassive black holes, investigating the merging history of galaxies over cosmological scales.
However, as this detector concept relies on an orbit hypothesis not studied in any detail, we do not consider it mature enough to warrant a study of potential detection rates in any detail, and we therefore only refer to established reviews of the astrophysical potential in the frequency band of LISA and DECIGO, e.g. Ref. [@Sathya2009].
Coalescing compact binaries
---------------------------
Heavy stars in binary systems will end up as compact objects (such as NSs or BHs) inspiralling around each other, losing orbital energy and angular momentum through gravitational radiation. Depending on the proximity of the source and the detector’s sensitivity, we could detect GWs from such a system a few seconds up to a day before the merger and the formation of a single spinning object. These CBCs are expected to be the strongest sources of GWs in the frequency band of current GW detectors.
To estimate the event rates for various binary systems, we will follow the calculations outlined in Ref. [@Abadie2010b]. To compare with predictions for initial and advanced LIGO (presented in Ref. [@Abadie2010b]), we also use only the inspiral part of the coalescence to estimate the *horizon distance* (the maximum distance to which we can observe a given system with a given signal-to-noise ratio (SNR)). We use here the same detection threshold on signal-to-noise ratio, a SNR of $\rho=8$, as in Ref. [@Abadie2010b] and consider the same fiducial binary systems: NS-NS (with $1.4 \msun$ each), BH-NS (BH mass $10 \msun$, NS with $1.4 \msun$), and BH-BH ($10 \msun$ each).
For a binary of given masses, the sky-averaged horizon distance is given by $$D_{h} = \frac{4 \sqrt{5} \, G^{\frac{5}{6}} \, \mu^{\frac{1}{2}} \, M^{\frac{1}{3}}}{\sqrt{96} \, \pi^{\frac{2}{3}} \, c^{\frac{3}{2}} \, \rho}
\sqrt{\int_{f_{\rm min}}^{f_{\rm ISCO}} \frac{f^{-\frac{7}{3}}}{\widetilde{S}_{\rm h}(f)}\,\mathrm{d}f} \; .
\label{eq:Dhorizon}$$ Here, $M=M_1+M_2$ is the total mass and $\mu={M_1M_2}/{M}$ is the reduced mass of the system. We have used a lower cutoff of $f_{\rm min} = 1$Hz, and at the upper end the frequency of the innermost stable circular orbit is $f_{\rm ISCO} = c^3/(6^{3/2}\pi\ G\ M)$ Hz, which conventionally is taken as the end of the inspiral.
Now, for any given type of binary (as characterized by the component masses), we obtain the observed event rate (per year) using $\dot{N}=R \cdot N_{\mathrm{G}}$, where we have adopted the approximation for the number of galaxies inside the visible volume from Eq. (5) of Ref. [@Abadie2010b]: $$\label{NG}
N_{\mathrm{G}}=\frac{4}{3}\pi \left(\frac{D_{\mathrm{h}}}{\mathrm{Mpc}}\right)^3 (2.26)^{-3} \cdot 0.0116 \, ,$$ and the intrinsic coalescence rates $R$ per Milky-Way-type galaxy are given in Table 2 of Ref. [@Abadie2010b].
A single DFI combination $S_i$ has annual rates similar to initial LIGO, and the results for the network sensitivity of full OGO are summarized in Table \[T:cbc\_rates\]. For each binary, we give three numbers following the uncertainties in the intrinsic event rate (“pessimistic”, “realistic”, “optimistic”) as introduced in Ref. [@Abadie2010b].
NS-NS NS-BH BH-BH
------- ------------------- -------------------- --------------------
OGO (0.002, 0.2, 2.2) (0.001, 0.06, 2.0) (0.003, 0.1, 9)
LIGO (2e-4, 0.02, 0.2) (7e-5, 0.004, 0.1) (2e-4, 0.007, 0.5)
aLIGO (0.4, 40, 400) (0.2, 10, 300) (0.4, 20, 1000)
: Estimated yearly detection rates for CBC events, given in triplets of the form (lower limit, realistic value, upper limit) as defined in Ref. [@Abadie2010b].[]{data-label="T:cbc_rates"}
From this, we see that OGO achieves detection rates an order of magnitude better than initial LIGO. But we still expect to have only one event in about three years of observation assuming “realistic” intrinsic coalescence rates. The sensitivity of aLIGO is much better than for OGO above 10 Hz, and the absence of seismic noise does not help OGO much because the absolute sensitivities below 10 Hz are quite poor and only a very small fraction of SNR is contributed from the lower frequencies. This is the reason why OGO cannot compete directly with aLIGO in terms of total CBC detection rates, which are about two orders of magnitude lower.
However, OGO does present an interesting scientific opportunity when run in parallel with aLIGO. If OGO indeed detects a few events over its mission lifetime, as the realistic predictions allow, it can give a very large improvement to the sky localization of these sources. Parameter estimation by aLIGO alone typically cannot localize signals enough for efficient electromagnetic follow-up identification. However, in a joint detection by OGO and aLIGO, triangulation over the long baseline between space-based OGO and ground-based aLIGO would yield a fantastic angular resolution. As signals found by OGO are very likely to be picked up by aLIGO as well, such joint detections indeed seem promising. Additionally, the three-dimensional configuration and independent channels of OGO potentially allow a more accurate parameter estimation than a network of two or three simple L-shaped interferometers could achieve.
Stochastic background
---------------------
There are mainly two kinds of stochastic GW backgrounds [@Allen99; @Maggiore00]: The first is the astrophysical background (sometimes also called astrophysical foreground), arising from unresolved astrophysical sources such as compact binaries [@Farmer03] and core-collapse supernovae [@Ferrari99]. It provides important statistical information about distribution of the sources and their parameters. The second is the cosmological background which was generated by various mechanisms in the early Universe [@Brustein95; @Turner97; @Ananda07]. It carries unique information about the very beginning of the Universe ($\sim 10^{-28}$s). Thus, the detection of the GW stochastic background is of great interest.
Currently, there are two ways to detect the stochastic GW background. One of them [@Hogan01] takes advantage of the null stream (e.g. the Sagnac combination of LISA). By definition, the null stream is insensitive to gravitational radiation, while it suffers from the same noise sources as the normal data stream. A comparison of the energy contained in the null stream and the normal data stream allows us to determine whether the GW stochastic background is present or not. The other way of detection is by cross-correlation [@Allen99; @Seto06] of measurements taken by different detectors. In our language, this uses the GW background signal measured by one channel as the template for the other channel. In this sense, the cross-correlation can be viewed as matched filtering. Both ways require redundancy, i.e. more than one channel observing the same GW signal with independent noise.
Luckily, the octahedron detector has plenty of redundancy, which potentially allows precise background detection. There are in total 12 dual-way laser links between spacecraft, forming 8 LISA-like triangular constellations. Any pair of two such LISA-like triangles that does not share common links can be used as an independent correlation. There are 16 such pairs within the octahedron detector. Within each pair, we can correlate the orthogonal TDI variables A, E and T (as they are denoted in LISA [@TintoDhurandhar]). Altogether, there are $16\times3^2=144$ cross-correlations.
And we have yet more information encoded by the detector, which we can access by considering that any two connected links form a Michelson interferometer, thus providing a Michelson-TDI variable. Any two of these variables that do not share common links can be correlated. There are in total 36 such variables, forming 450 cross-correlations, from which we can construct the optimal total sensitivity.
Furthermore, each of these is sensitive to a different direction on the sky. So the octahedron detector has the potential to detect anisotropy of the stochastic background. However, describing an approach for the detection of anisotropy is beyond the scope of this feasibility study.
Instead, we will present here only an order of magnitude estimation of the total cross-correlation SNR. Usually, it can be expressed as $$\mathrm{SNR} = \frac{3H_0^2}{10\pi^2}\sqrt{T_{\mathrm{obs}}}\left[ 2 \sum_{k,l}\int_0^\infty {\rm d}f\frac{\gamma_{kl}^2(f)\Omega_{\rm gw}^2(f)} {f^6\widetilde{S}_{{\rm h}, k}(f)\widetilde{S}_{{\rm h}, l}(f)} \right]^{\frac{1}{2}}\,,$$ where $T_{\mathrm{obs}}$ is the observation time, $\Omega_{\rm gw}$ is the fractional energy-density of the Universe in a GW background, $H_0$ the *Hubble constant*, and $\widetilde{S}_{{\rm h}, k}(f)$ is the effective sensitivity of the $k$-th channel. $\gamma_{kl}(f)$ denotes the *overlap reduction function* between the $k$-th and $l$-th channels, introduced by Flanagan [@Flanagan93]. $$\gamma_{kl}(f) = \frac{5}{8\pi}\sum_{p=+,\times}\int {\rm d}\hat{\Omega}\,\mathrm{e}^{2\pi {\rm i} f \hat{\Omega}\cdot \Delta \mathbf{x}/c}F_k^p(\hat{\Omega})F_l^p(\hat{\Omega})\, ,$$ where $F_k^p(\hat{\Omega})$ is the antenna pattern function. As mentioned in the previous section, there might be 12 independent DFI solutions. These DFI solutions can form $12\times 11/2 = 66$ cross-correlations. According to Ref. [@Allen99], we know $\gamma_{kl}^2(f)$ varies between $0$ and $1$. As a rough estimate, we approximate $\sum_{k,l}\gamma_{kl}^2(f)\sim 10$; hence, we get the following result for OGO: $${\rm SNR} = 2.57\left(\frac{H_0}{72\,\frac{{\rm km} \, {\rm s^{-1}}}{{\rm Mpc}}}\right)^2\left(\frac{\Omega_{\rm gw}}{10^{-9}}\right)\left(\frac{T_{\mathrm{obs}}}{10\,{\rm yr}}\right)^{\frac{1}{2}}\,.$$ Initial LIGO has set an upper limit of $6.9\cdot 10^{-6}$ on $\Omega_{\rm gw}$ [@nature09], and aLIGO will be able to detect the stochastic background at the $1\cdot 10^{-9}$ level [@nature09]. Hence, our naive estimate of OGO’s sensitivity to the GW stochastic background is similar to that of aLIGO. Actually, an optimal combination of all the previously-mentioned possible cross-correlations would potentially result in an even better detection ability for OGO.
Testing alternative theories of gravity
---------------------------------------
![Relative sensitivity of the full OGO network (scaled from S5 combination) to alternative polarizations: $+$ mode (*blue solid line*), $x$ mode (*red crosses*), vector-$x$ mode (*green dash-dotted line*), vector-$y$ mode (*black stars*), longitudinal mode (*magenta dashed line*), and breathing mode (*cyan plus signs*). []{data-label="F:altern_polar"}](ogo_polar_modes_all){width="\columnwidth"}
In this section we will consider OGO’s ability to test predictions of General Relativity against alternative theories. In particular, we will estimate the sensitivity of the proposed detector to all six polarization modes that could be present in (alternative) metric theories of gravitation [@Hohmann2012]. We refer to Ref. [@Eardley1973] for a discussion on polarization states, which are (i) two transverse-traceless (tensorial) polarizations usually denoted as $+$ and $\times$, (ii) two scalar modes called breathing (or common) and longitudinal and (iii) two vectorial modes. We also refer to Refs. [@Will_LR; @Gair_LR] for reviews on alternative theories of gravity.
We have followed the procedure for computing the sensitivity of OGO, as outlined above, for the four modes not present in General Relativity, and we compare those sensitivities to the results for the $+, \times$ modes as presented in Fig. \[F:Sens\]. The generalization of the transfer function used in this paper \[Eq. \[Eq:TF\]\] for other polarization modes is given in Ref. [@Chamberlin_2012].
We have found that all seven generators show similar sensitivity for each mode. OGO is not sensitive to the common (breathing) mode, which is not surprising as it can be attributed to a common displacement noise, which we have removed by our procedure. The sensitivity to the second (longitudinal) scalar mode scales as $\epsilon^{-4}$ at low frequencies and is much worse than the sensitivity to the $+, \times$ polarizations below 200Hz. However, OGO is more sensitive to the longitudinal mode (by about an order of magnitude) above 500Hz. The sensitivity of OGO to vectorial modes is overall similar to the $+, \times$ modes: it is by a few factors less sensitive to vectorial modes below 200Hz and by similar factors more sensitive above 300Hz. These sensitivities are shown in Fig. \[F:altern\_polar\].
Pulsars – Continuous Waves
--------------------------
CWs are expected from spinning neutron stars with nonaxisymmetric deformations. Spinning NSs are already observed as radio and gamma-ray pulsars. Since CW emission is powered by the spindown of the pulsar, the strongest emitters are the pulsars with high spindowns, which usually are young pulsars at rather high frequencies. Note that the standard emission model [@Jaranowski1998] predicts a gravitational wave frequency $f_{\mathrm{gw}}=2f$, while alternative models like free precession [@Jones2001] and $r$-modes [@Andersson1998] also allow emission at $f_{\mathrm{gw}}=f$ and $f_{\mathrm{gw}}=\tfrac{4}{3}f$, where $f$ is the NS spin frequency.
OGO has better sensitivity than initial LIGO below 133Hz, has its best sensitivity around 78Hz, and is better than aLIGO below 9Hz. This actually fits well with the current radio census of the galactic pulsar population, as given by the ATNF catalog [@ATNF]. As shown in Fig. \[fig:pulsars\_atnf\], the bulk of the population is below $\sim$ 10Hz, and also contains many low-frequency pulsars with decent spindown values, even including a few down to $\sim$ 0.1Hz.
We estimate the abilities of OGO to detect CW emission from known pulsars following the procedure outlined in Ref. [@Abadie2011] for analysis of the Vela pulsar. The GW strain for a source at distance $D$ is given as $$h_0 = \frac{4 \pi^2 G I_{zz} \epsilon f^2}{c^4 D} \, ,$$ where $\epsilon$ is the ellipticity of the neutron star and we assume a canonical momentum of inertia $I_{zz} = 10^{38}$kgm$^2$. After an observation time $T_{\rm obs}$, we could detect a strain amplitude $$h_0 = \Theta\sqrt{\frac{S_\mathrm{h}}{T_{\mathrm{obs}}}} \, .$$ The statistical factor is $\Theta\approx11.4$ for a fully coherent targeted search with the canonical values of 1% and 10% for false alarm and false dismissal probabilities, respectively [@Abbott2004]. We find that, for the Vela pulsar (at a distance of 290pc and a frequency of $f_{\mathrm{Vela,gw}}=2 \cdot 11.19$Hz), with $T_{\mathrm{obs}}=30$ days of observation, we could probe ellipticities as low as $\epsilon \sim 5\cdot10^{-4}$ with the network OGO configuration. Several known low-frequency pulsars outside the aLIGO band would also be promising objectives for OGO targeted searches.
![Population of currently known pulsars in the frequency-spindown plain ($f$-$\dot{f}$). OGO could beat initial LIGO left of the red solid line and Advanced LIGO left of the green dashed line. Data for this plot were taken from Ref. [@ATNF] on March 2, 2012.[]{data-label="fig:pulsars_atnf"}](pulsars_atnf_f_fdot){width="\columnwidth"}
All-sky searches for unknown pulsars with OGO would focus on the low-frequency range not accessible to aLIGO with a search setup comparable to current Einstein@Home LIGO searches [@Aasi2013]. As seen above, the sensitivity estimate factors into a search setup related part $\Theta / \sqrt{T_{\mathrm{obs}}}$ and the sensitivity $\sqrt{S_\mathrm{h}}$. Therefore, scaling a search with parameters identical to the Einstein@Home S5 runs to OGO’s best sensitivity at 76Hz would reach a sensitivity of $h_0 \approx 3 \cdot 10^{-25}$. This would, for example, correspond to a neutron star ellipticity of $\epsilon \sim 4.9 \cdot 10^{-5}$ at a distance of 1kpc. Since the computational cost of such searches scales with $f^2$, low-frequency searches are actually much more efficient and would allow very deep searches of the OGO data, further increasing the competitiveness. Note, however, that for low-frequency pulsars the ellipticities required to achieve detectable GW signals can be very high, possibly mostly in the unphysical regime. On the other hand, for “transient CW”-type signals [@Prix2011], low-frequency pulsars might be the strongest emitters, even with realistic ellipticities.
Other sources
-------------
Many (indirect) observational evidences exist for stellar mass BHs, which are the end stages of heavy star evolution, as well as for supermassive BHs, the result of accretion and galactic mergers throughout the cosmic evolution, in galactic nuclei. On the other hand, there is no convincing evidence so far for a BH of an intermediate mass in the range of $10^2-10^4\msun$. These IMBHs might, however, still exist in dense stellar clusters [@Miller2004; @Pasquato2010]. Moreover, stellar clusters could be formed as large, gravitationally bound groups, and collision of two clusters would produce inspiralling binaries of IMBHs [@IMBH_pau2006; @IMBH_pau2009].
The ISCO frequency of the second orbital harmonic for a $300 \msun$-$300 \msun$ system is about 7Hz, which is outside the sensitivity range of aLIGO. Still, those sources could show up through the higher harmonics (the systems are expected to have non-negligible eccentricity) and through the merger and ring-down gravitational radiation [@Fregeau2006; @Mandel2008; @Yagi2012]. The ground-based LIGO and VIRGO detectors have already carried out a first search for IMBH signals in the $100\msun$ to $450\msun$ mass range [@Abadie2012_imbh].
With its better low-frequency sensitivity, OGO can be expected to detect a GW signal from the inspiral of a $300 \msun$-$300 \msun$ system in a quasicircular orbit up to a distance of approximately 245Mpc, again using Eq. (\[eq:Dhorizon\]). This gives the potential for discovery of such systems and for estimating their physical parameters.
As for other advanced detectors, unmodeled searches (as opposed to the matched-filter CBC and CW searches; see Ref. [@Abadie2012_burst] for a LIGO example) of OGO data have the potential for detecting many other types of gravitational wave sources, including, but not limited to, supernovae and cosmic string cusps. However, as in the case for IMBHs, the quantitative predictions are hard to produce due to uncertainties in the models.
Summary and Outlook {#S:Summary}
===================
In this paper, we have presented for the first time a three-dimensional gravitational wave detector in space, called the Octahedral Gravitational wave Observatory (OGO). The detector concept employs displacement-noise free interferometry (DFI), which is able to cancel some of the dominant noise sources of conventional GW detectors. Adopting the octahedron shape introduced in Ref. [@chen2006], we put spacecraft in each corner of the octahedron. We considered a LISA-like receiver-transponder configuration and found multiple combinations of measurement channels, which allow us to cancel both laser frequency and acceleration noise. This new three-dimensional result generalizes the Mach-Zehnder interferometer considered in Ref. [@chen2006].
We have identified a possible halolike orbit around the Lagrange point L1 in the Sun-Earth system that would allow the octahedron constellation to be stable enough. However, this orbit limits the detector to an arm length of $\approx1400$ km.
Much better sensitivity and a richer astrophysical potential are expected for longer arm lengths. Therefore, we also looked for alternative orbits and found a possible alternative allowing for $\approx 2\cdot10^{9}$ m arms, but is is not clear yet if this would be stable enough. Future studies are required to relax the equal-arm-length assumption of our DFI solutions, or to determine a stable, long-arm-length constellation.
Next, we have computed the sensitivity of OGO-like detectors – and have shown that the noise-cancelling combinations also cancel a large fraction of the GW signal at low frequencies. The sensitivity curve therefore has a characteristic slope of $f^{-2}$ at the low-frequency end.
However, the beauty of this detector is that it is limited by a single noise source at all frequencies: shot noise. Thus, any reduction of shot noise alone would improve the overall sensitivity. This could, in principle, be achieved with DECIGO-like cavities, squeezing or other advanced technologies. Also, OGO does not require drag-free technology and has moderate requirements on other components so that it could be realized with technology already developed for LISA Pathfinder and eLISA.
When comparing a DFI-enabled OGO with a detector of similar design, but with standard TDI, we find that at $\approx1400$ km, the same sensitivity could be reached by a TDI detector with very modest acceleration noise requirements.
However, at longer arm lengths DFI becomes more advantageous, reaching the same sensitivity as TDI under LISA requirements but without drag-free technology and clock transfer, at $\approx 2\cdot10^{9}$ m. Such a DFI detector would have its best frequency range between LISA and DECIGO, with peak sensitivity better than LISA and approaching DECIGO without the latter mission concept’s tight acceleration noise requirements and with no need for cavities.
Finally, we have assessed the scientific potentials of OGO, concentrating on the less promising, but more mature short-arm-length version. We estimated the event rates for coalescing binaries, finding that OGO is better than initial LIGO, but does not reach the level of advanced LIGO. Any binary detected with both OGO and aLIGO could be localized in the sky with very high accuracy.
Also, the three-dimensional satellite constellation and number of independent links makes OGO an interesting mission for detection of the stochastic GW background or hypothetical additional GW polarizations. Further astrophysically interesting sources such as low-frequency pulsars and IMBH binaries also lie within the sensitive band of OGO, but again the sensitivity does not reach that of aLIGO.
However, we point out that the improvement in the low-frequency sensitivity with increasing arm length happens faster for DFI as compared to the standard TDI. Therefore, searching for stable three-dimensional (octahedron) long-baseline orbits could lead to an astrophysically much more interesting mission.
Regarding possible improvements of the presented setup, there are several possibilities to extend and improve the first-order DFI scheme presented here. One more spacecraft could be added in the middle, increasing the number of usable links. Breaking the symmetry of the octahedron could modify the steep response function at low frequencies. This should be an interesting topic for future investigations.
In principle, the low-frequency behavior of OGO-like detectors could also be improved by more advanced DFI techniques such as introducing artificial time delays [@Somiya2007a; @Somiya2007b]. This would result in a three-part power law less steep than the shape derived in Sec. \[sec:transfer\_function\]. On the other hand, this would also introduce a new source of time delay noise. Therefore, such a modification requires careful investigation.
Acknowledgments
===============
We would like to thank Gerhard Heinzel for very fruitful discussions, Albrecht Rüdiger for carefully reading through the paper and helpful comments, Sergey Tarabrin for discussions on the optical layout, Masaki Ando for kindly sharing DECIGO simulation tools and Guido Müller for helpful comments on the final manuscript. Moreover, Berit Behnke, Benjamin Knispel, Badri Krishnan, Reinhard Prix, Pablo Rosado, Francesco Salemi, Miroslav Shaltev and others helped us with their knowledge regarding the astrophysical sources. We would also like to thank the anonymous referee for very insightful and detailed comments on the original manuscript. The work of the participating students was supported by the International Max-Planck Research School for Gravitational Waves (IMPRS-GW) grant. The work of S. B. and Y. W. was partially supported by DFG Grant No. SFB/TR 7 Gravitational Wave Astronomy and DLR (Deutsches Zentrum für Luft- und Raumfahrt). Furthermore, we want to thank the Deutsche Forschungsgemeinschaft (DFG) for funding the Cluster of Excellence QUEST – Centre for Quantum Engineering and Spacetime Research. We thank the LIGO Scientific Collaboration (LSC) for supplying the LIGO and aLIGO sensitivity curves. Finally, we would like to emphasize that the idea of a three-dimensional GW detector in space is the result of a student project from an IMPRS-GW lecture week. This document has been assigned LIGO document number and LIGO-P1300074 and AEI-preprint number AEI-2013-261.
Details on calculating the displacement and laser noise free combinations {#S:Appendix}
=========================================================================
Here we will give details on building the displacement (acceleration) and laser noise free configurations. The derivations closely follow the method outlined in [@TintoDhurandhar]. We want to find the generators solving Eq. (\[E:TDIAcc\_FinalEq\]), so called reduced generators because they correspond to the reduced set $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. We start with building the ideal $Z$: $$\begin{aligned}
Z = \left\{
\begin{array}{rcl}
f_1 &=& (\mathcal{D}-1)^2 \\
f_2 &=& (\mathcal{D}-1)\mathcal{D} \\
f_3 &=& (1-\mathcal{D})(\mathcal{D}-1) \\
f_4 &=& (\mathcal{D}-1)((1-\mathcal{D})\mathcal{D}-1) \\
f_5 &=& \mathcal{D}-1 \\
f_6 &=& \mathcal{D}-1 \\
\end{array}
\right.\,.
\label{E:TDIAcc_ideal}
\end{aligned}$$ The corresponding Gröbner basis to this ideal is: $$\mathcal{G}=\{ g_1 = \mathcal{D} - 1 \}.
\label{E:E:TDIAcc_Groebner}$$
The connection between $f_i$ and $g_j$ is defined by two transformation matrices $$\begin{aligned}
d & =& \left(
\begin{array}{c}
\mathcal{D}-1 \\
\mathcal{D} \\
1-\mathcal{D} \\
(1-\mathcal{D})\mathcal{D}-1 \\
1 \\
1 \\
\end{array}
\right)
\end{aligned}$$ and $c$ with (at least) two possible solutions $$c^{(1)} = \left( 0 \ 0 \ 0 \ 0 \ 1 \ 0 \right)\; \textrm{or} \; c^{(2)} = \left( 0 \ 0 \ 0 \ 0 \ 0 \ 1 \right).$$ The resulting basis is not unique and not necessarily independent. The first 6 reduced generators are given by the row vectors of the matrix $A^{(1)} = a^{(1)}_i = I - d\cdot c^{(1)}$ :
$$\begin{aligned}
a^{(1)}_1 & = \left\{ 1 , 0 , 0 , 0 , 0 , 1-\mathcal{D} \right\}, \\
a^{(1)}_2 & = \left\{ 0 , 1 , 0 , 0 , 0 , -\mathcal{D} \right\}, \\
a^{(1)}_3 & = \left\{ 0 , 0 , 1 , 0 , 0 , (\mathcal{D}-1)\mathcal{D} \right\}, \\
a^{(1)}_4 & = \left\{ 0 , 0 , 0 , 1 , 0 , 1+(\mathcal{D}-1)\mathcal{D} \right\}, \\
a^{(1)}_5 & = \left\{ 0 , 0 , 0 , 0 , 1 , -1 \right\}, \\
a^{(1)}_6 & = \left\{ 0 , 0 , 0 , 0 , 0 , 0 \right\}.
\end{aligned}$$
These reduced generators correspond directly to values for $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. As the Gröbner basis contains only one element, we cannot form other generator from $S$-polynomial.
We can form 6 other generators using $c^{(2)}$ instead of $c^{(1)}$. After applying those generators we have the following acceleration-free combinations:
$$\begin{aligned}
a^{(1)}_1 s^{n} & = 2 (p_{B} - p_{C} + p_{E} - p_{F} + \mathcal{D}(-p_{A} + p_{B} - p_{D} + p_{E} \nonumber \\ &
+ (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA})), \\
a^{(1)}_2 s^{n} & = -2 \mathcal{D} (p_{A} + p_{D} + p_{C} (-1 + q_{BA}) + p_{F} (-1 + q_{BA}) \nonumber \\ &
- (p_{B} + p_{E}) q_{BA}), \\
a^{(1)}_3 s^{n} & = 2 \mathcal{D} ((1 + \mathcal{D}) p_{A} + p_{D} - p_{E} - \mathcal{D} (p_{C} - p_{D} + p_{F}) \nonumber \\ &
+ p_{B} (-1 + q_{BA}) - (p_{C} - p_{E} + p_{F}) q_{BA}), \\
a^{(1)}_4 s^{n} & = 2 (p_{B} - p_{C} + p_{E} + \mathcal{D}^2 (p_{A} - p_{C} + p_{D} - p_{F}) \nonumber \\ &
- p_{F} + \mathcal{D} (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA}), \\
a^{(1)}_5 s^{n} & = 2 \mathcal{D} (p_{A} + p_{D} + p_{B} (-1 + q_{BA}) + p_{E} (-1 + q_{BA}) \nonumber \\ &
- (p_{C} + p_{F}) q_{BA}), \\
a^{(1)}_6 s^{n} & = 2 \mathcal{D} (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA},
\end{aligned}$$
where $s^{n}_{IJ}$ are given by Eq. (\[E:MesN\]). Note that we have a free (polynomial) function of delay $q_{BA}$ which we can choose arbitrary. We will omit subscripts $BA$ and use $q\equiv q_{BA}$. The arbitrariness of this function implies that terms which contain $q$ and terms free of $q$ are two independent sets of generators. We will keep $q$ until we obtain laser noise free combinations, and then split each generator in two. After some analysis only two out of six acceleration free generators are independent, so we can rewrite them as
$$\begin{aligned}
s_1 &= y_{12} + \mathcal{D}(y_{13} + qy_{12}),\\
s_3 &= -y_{13} + \mathcal{D}(y_{12} -y_{13}) + qy_{12},\\
s_4 &= y_{12} + \mathcal{D}q y_{12} + \mathcal{D}^2(y_{12} - y_{13}),\\
s_2+s_5 &= y_{12} - 2y_{13},\label{E:s2ps5}\\
s_2-s_5 &= (2q-1)y_{12},\label{E:s2ms5}\\
s_6 &= q y_{12},
\end{aligned}$$
where $$\begin{aligned}
s_1 &=& \frac{a^{(1)}_1 s^{n}}{2}, s_2 = -\frac{\mathcal{D}^{-1}( a^{(1)}_2 s^{n})}{2}, s_3 = \frac{\mathcal{D}^{-1}( a^{(1)}_3 s^{n})}{2} \nonumber \\
s_4 &=& \frac{a^{(1)}_4 s^{n}}{2}, s_5 = \frac{\mathcal{D}^{-1}( a^{(1)}_5 s^{n})}{2} , s_6 = \frac{\mathcal{D}^{-1}( a^{(1)}_6 s^{n})}{2}\end{aligned}$$ and $$y_{12} = p_B+p_E-p_C-p_F,\, y_{13} = p_B+p_E-p_A-p_D\,.$$ We have introduced the inverse delay operator, $ \mathcal{D}^{-1}$, for mathematical convenience, which obeys $\mathcal{D}\mathcal{D}^{-1} = \mathds{1}$. One can easily get rid of it by applying the delay operator on both sides. The final result will not contain the operator $\mathcal{D}^{-1}$. Next we use Eqs. (\[E:s2ps5\]) and (\[E:s2ms5\]) to express $y_{12}, y_{13}$ and eliminate them from the other equations. The resulting combinations that eliminate both acceleration and laser noise are
$$\begin{aligned}
&(1-2q)s_1 + (-1-2\mathcal{D}q)s_2 + (1+\mathcal{D})s_5\\
&(1-2q)s_3 + \mathcal{D}(q-1)s_2 + (-1+2q+q\mathcal{D})s_5\\
&(1-2q)s_4 - (1+ \mathcal{D}q)(s_2-s_5) - \mathcal{D}^2((1-q)s_2 - qs_5)\\
&(1-2q)s_6 - q(s_2 - s_5).
\end{aligned}$$
Out of these solutions we obtain seven independent generators which we have rewritten in the final form similar to the $Y$-equations from Sec. \[S:TDI\]. They are explicitly given by Eqs. (\[E:S1\])–(\[E:S7\]).
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| ArXiv |
---
author:
- 'Xián O. Camanho'
title: Phase transitions in general gravity theories
---
Introduction {#intro}
============
Higher-curvature corrections to the Einstein-Hilbert (EH) action appear in any sensible theory of quantum gravity as next-to-leading orders in the effective action and some, [*e.g.*]{} the Lanczos-Gauss-Bonnet (LGB) action [@Lanczos], also appear in realizations of string theory [@GBstrings1]. This quadratic combination is particularly important as any quadratic term can be brought to the LGB form, $\mathcal{R}^2=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-R_{\mu\nu}R^{\mu\nu}+R^2$, via field redefinitions.
Due to the non-linearity of the equations of motion, these theories generally admit more than one maximally symmetric solution, $R_{\mu \nu \alpha\beta}=\Lambda_i(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha})$; (A)dS vacua with effective cosmological constants $\Lambda_{i}$, whose values are determined by a polynomial equation [@BoulwareDeser], -2mm $$\Upsilon [\Lambda] \equiv \sum_{k=0}^{K}c_{k}\,\Lambda^{k} = c_{K}\prod_{i=1}^{K}\left( \Lambda -\Lambda _{i}\right) =0 ~.
\label{cc-algebraic}$$ $K$ being the highest power of curvature (without derivatives) in the field equations. $c_0=1/L^2$ and $c_1=1$ give canonically normalized cosmological and EH terms, $c_{k\geq 2}$ are the LGB and higher order couplings (see [@JDEere] for details).
Any vacua is [*a priori*]{} suitable in order to define boundary conditions for the gravity theory we are interested in; [*i.e.*]{} we can define sectors of the theory as classes of solutions that asymptote to a given vacuum [@CE]. In that way, each branch has associated static solutions, representing either black holes or naked singularities, -2mm $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}\ d\Omega_{d-2}^{2} ~, \qquad \qquad f,g \xrightarrow{r\rightarrow \infty} -\Lambda_i r^2 ~,
\label{bhansatz}$$ and other solutions with the same asymptotics. The main motivation of the present work is that of studying transitions between different branches of solutions. This is important in order to investigate whether a new type of instability involving non-perturbative solutions occurs in the theory. This new kind of phase transitions have been recently investigated in the context of LGB [@Camanho2012] and Lovelock gravities [@comingsoon].
Higher order free particle
==========================
The existence of branch transitions in higher curvature gravity theories is a concrete expression of the multivaluedness problem of these theories. In general the canonical momenta, $\pi_{ij}$, are not invertible functions of the velocities, $\dot{g}^{ij}$ [@Teitelboim1987]. An analogous situation may be illustrated by means of a simple one-dimensional example [@Henneaux1987b]. Consider a free particle lagrangian containing higher powers of velocities, -1mm $$L(\dot{x})=\frac{1}{2}\dot{x}^2-\frac13\dot{x}^3+\frac1{17}\dot{x}^4
\label{paction}$$ In the hamiltonian formulation the equation of motion just implies the constancy of the conjugate momentum, $\frac{d}{dt}p=0$. However, being this multivalued (also the hamiltonian), the solution is not unique. Fixing boundary conditions $x(t_{1,2})=x_{1,2}$, an obvious solution would be constant speed $
\dot{x}=(x_2-x_1)/(t_2-t_1)\equiv v
$ but we may also have jumping solutions with constant momentum and the same mean velocity.
![Lagrangian and momentum for the action (\[paction\]). For the same mean velocity $v$, the action is lower for jumps between $v_\pm$ (big dot) than for constant speed, the minimum action corresponding to the value on the dashed line ([*effective*]{} Lagrangian).[]{data-label="fig:1"}](L-v2.eps "fig:") ![Lagrangian and momentum for the action (\[paction\]). For the same mean velocity $v$, the action is lower for jumps between $v_\pm$ (big dot) than for constant speed, the minimum action corresponding to the value on the dashed line ([*effective*]{} Lagrangian).[]{data-label="fig:1"}](p-v2.eps "fig:")
In our example, for mean velocities corresponding to multivalued momentum (see figure \[fig:1\]) solutions are infinitely degenerate as the jumps may occur at any time and unboundedly in number as long as the mean velocity is the same. Nevertheless, this degeneracy is lifted once the value of the action is taken into account. The minimal action path is the naive one for mean velocities outside the range covered by the dashed line whereas in that interval it corresponds to arbitrary jumps between the velocities of the two extrema. The [*effective*]{} Lagrangian (dashed line) is a convex function of the velocities and the effective momentum dependence corresponds to the analogous of the Maxwell construction from thermodynamics (see [@comingsoon] for a detailed explanation of this one-dimensional example).
Generalized Hawking-Page transitions
====================================
In the context of General Relativity in asymptotically AdS spacetimes, the Hawking-Page phase transition [@HawkingPage] is the realization that above certain temperature the dominant saddle in the gravitational partition function comes from a black hole, whereas for lower temperatures it corresponds to the thermal vacuum. The [*classical*]{} solution is the one with least Euclidean action among those with a smooth Euclidean section.
When one deals with higher curvature gravity there is a crucial difference that has been overlooked in the literature. In addition to the usual continuous and differentiable metrics (\[bhansatz\]), one may construct distributional metrics by gluing two solutions corresponding to different branches across a spherical shell or [*bubble*]{} [@wormholes; @wormholes2]. The resulting solution will be continuous at the bubble –with discontinuous derivatives, even in absence of matter. The higher curvature terms can be thought of as a sort of matter source for the Einstein tensor. The existence of such [*jump*]{} metrics, as for the one-dimensional example, is due to the multivaluedness of momenta in the theory.
In the gravitational context, continuity of momenta is equivalent to the junction conditions that need to be imposed on the bubble. In the EH case, Israel junction conditions [@Israel1967], being linear in velocities, also imply the continuity of derivatives of the metric. The generalization of these conditions for higher curvature gravity contain higher powers of velocities, thus allowing for more general situations.
Static bubble configurations, when they exist, have a smooth Euclidean continuation. It is then possible to calculate the value of the action and compare it to all other solutions with the same asymptotics and temperature. This analysis has been performed for the LGB action [@Camanho2012] for unstable boundary conditions [@BoulwareDeser]. The result suggests a possible resolution of the instability through bubble nucleation.
In the case of LGB gravity there are just two possible static spacetimes to be considered in the analysis for the chosen boundary conditions; the thermal vacuum and the static bubble metric, the usual spherically symmetric solution (\[bhansatz\]) displaying a naked singularity. For low temperatures the thermal vacuum is the preferred solution whereas at high temperatures the bubble will form, as indicated by the change of sign on the relative free energy. The bubble pops out in an unstable position and may expand reaching the boundary in a finite time thus changing the asymptotics and charges of the solution, from the initial to the inner ones.
Still, if the free energy is positive the system is metastable. It decays by nucleating bubbles with a probability given, in the semiclassical approximation, by $e^{-F/T}$. Therefore, after enough time, the system will always end up in the stable horizonful branch of solutions, the only one usually considered as relevant. This is then a natural mechanism that selects the general relativistic vacuum among all the possible ones, the stable branch being the endpoint of the initial instability.
Discussion
==========
The phenomenon described here is quite general. It occurs also for general Lovelock gravities [@comingsoon] and presumably for more general classes of theories. In the generic case, however, the possible situations one may encounter are much more diverse. We may have for instance stable bubble configurations as opposed to the unstable ones discussed above or even bubbles that being unstable cannot reach the boundary of the spacetime. Other generalizations may include transitions between positive and negative values of $\Lambda_i$ and even non-static bubble configurations.
Another situation one may think of is that of having different gravity theories on different sides of the bubble. This has a straightforward physical interpretation if we consider the higher order terms as sourced by other fields that vary accross the bubble. For masses above $m^2>\|\Lambda_{\pm}\|$ a bubble made of these fields will be well approximated by a thin wall and we may integrate out the fields for the purpose of discussing the thermodynamics. If those fields have several possible vacuum expectation values leading to different theories we may construct interpolating solutions in essentially the same way discussed above. In this case the energy carried by the bubble can be interpreted as the energy of the fields we have integrated out.
The author thanks A. Gomberoff for most interesting discussions, and the Front of pro-Galician Scientists for encouragement. He is supported by a spanish FPU fellowship. This work is supported in part by MICINN and FEDER (grant FPA2011-22594), by Xunta de Galicia (Consellería de Educación and grant PGIDIT10PXIB206075PR), and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042).
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| ArXiv |
---
abstract: |
Observations of radio halos and relics in galaxy clusters indicate efficient electron acceleration. Protons should likewise be accelerated and, on account of weak energy losses, can accumulate, suggesting that clusters may also be sources of very high-energy (VHE; $E>100$ GeV) gamma-ray emission. We report here on VHE gamma-ray observations of the Coma galaxy cluster with the VERITAS array of imaging Cherenkov telescopes, with complementing -LAT observations at GeV energies. No significant gamma-ray emission from the Coma cluster was detected. Integral flux upper limits at the 99% confidence level were measured to be on the order of $(2-5)\times
10^{-8}\ {\rm ph.\,m^{-2}\,s^{-1}}$ (VERITAS, $>220\ {\rm GeV}$) and $\sim 2\times 10^{-6}\ {\rm
ph.\,m^{-2}\, s^{-1}}$ (, $1-3\ {\rm GeV}$), respectively. We use the gamma-ray upper limits to constrain CRs and magnetic fields in Coma. Using an analytical approach, the CR-to-thermal pressure ratio is constrained to be $< 16\%$ from VERITAS data and $< 1.7\%$ from data (averaged within the virial radius). [These upper limits are starting to constrain the CR physics in self-consistent cosmological cluster simulations and cap the maximum CR acceleration efficiency at structure formation shocks to be $<50\%$. Alternatively, this may argue for non-negligible CR transport processes such as CR streaming and diffusion into the outer cluster regions. ]{} Assuming that the radio-emitting electrons of the Coma halo result from hadronic CR interactions, the observations imply a lower limit on the central magnetic field in Coma of $\sim (2 - 5.5)\,\mu{\rm G}$, depending on the radial magnetic-field profile and on the gamma-ray spectral index. Since these values are below those inferred by Faraday rotation measurements in Coma (for most of the parameter space), this [renders]{} the hadronic model a very plausible explanation of the Coma radio halo. Finally, since galaxy clusters are dark-matter (DM) dominated, the VERITAS upper limits have been used to place constraints on the thermally-averaged product of the total self-annihilation cross section and the relative velocity of the DM particles, ${\left\langle \sigma v \right\rangle}$.
author:
- 'T. Arlen, T. Aune, M. Beilicke, W. Benbow, A. Bouvier, J. H. Buckley, V. Bugaev, K. Byrum, A. Cannon, A. Cesarini, L. Ciupik, E. Collins-Hughes, M. P. Connolly, W. Cui, R. Dickherber, J. Dumm, A. Falcone, S. Federici, Q. Feng, J. P. Finley, G. Finnegan, L. Fortson, A. Furniss, N. Galante, D. Gall, S. Godambe, S. Griffin, J. Grube, G. Gyuk, J. Holder, H. Huan, G. Hughes, T. B. Humensky, A. Imran, P. Kaaret, N. Karlsson, M. Kertzman, Y. Khassen, D. Kieda, H. Krawczynski, F. Krennrich, K. Lee, A. S Madhavan, G. Maier, P. Majumdar, S. McArthur, A. McCann, P. Moriarty, R. Mukherjee, T. Nelson, A. O’Faoláin de Bhróithe, R. A. Ong, M. Orr, A. N. Otte, N. Park, J. S. Perkins, M. Pohl H. Prokoph, J. Quinn, K. Ragan, L. C. Reyes, P. T. Reynolds, E. Roache, J. Ruppel, D. B. Saxon, M. Schroedter, G. H. Sembroski, C. Skole, A. W. Smith, I. Telezhinsky, G. Tešić, M. Theiling, S. Thibadeau, K. Tsurusaki, A. Varlotta, M. Vivier, S. P. Wakely, J. E. Ward, A. Weinstein, R. Welsing, D. A. Williams, B. Zitzer'
- 'C. Pfrommer, A. Pinzke'
bibliography:
- 'refs.bib'
title: 'Constraints on Cosmic Rays, Magnetic Fields, and Dark Matter from Gamma-Ray Observations of the Coma Cluster of Galaxies with VERITAS and '
---
Introduction
============
Clusters of galaxies are the largest virialized objects in the Universe, with typical sizes of a few Mpc and masses on the order of $10^{14}$ to $10^{15} M_{\odot}$. According to the currently favored hierarchical model of cosmic structure formation, larger objects formed through successive mergers of smaller objects with galaxy clusters sitting on top of this mass hierarchy [see @article:Voit:2005 for a review]. Most of the mass ($\sim$80%) in a cluster is dark matter (DM), as indicated by galaxy dynamics and gravitational lensing [@article:DiaferioSchindlerDolag:2008]. Baryonic gas making up the intra-cluster medium (ICM) contributes about 15% of the total cluster mass and individual galaxies account for the remainder (about 5%). The ICM gas mass also comprises a significant fraction of the observable (baryonic) matter in the Universe.
The ICM is a hot ($T\sim 10^{8}$ K) plasma emitting thermal bremsstrahlung in the soft X-ray regime [see, e.g., @article:Petrosian:2001]. This plasma has been heated primarily through collisionless structure-formation shocks that form as a result of the hierarchical merging and accretion processes. Such shocks and turbulence in the ICM gas in combination with intra-cluster magnetic fields also provide a means to accelerate particles efficiently [see, e.g., @article:ColafrancescoBlasi:1998; @article:Ryu_etal:2003]. Many clusters feature megaparsec scale halos of nonthermal radio emission, indicative of a population of relativistic electrons and magnetic fields permeating the ICM [@article:Cassano_etal:2010]. There are two competing theories to explain radio halos. In the “hadronic model”, the radio-emitting electrons and positrons are produced in inelastic collisions of cosmic-ray (CR) ions with the thermal gas of the ICM [@article:Dennison:1980; @article:EnsslinPfrommerMiniatiSubramanian:2011]. In the “re-acceleration model”, a long-lived pool of 100-MeV electrons—previously accelerated by formation shocks, galactic winds, or jets of active galactic nuclei (AGN)—interacts with plasma waves that are excited during states of strong ICM turbulence, e.g., after a cluster merger. This may result in second order Fermi acceleration and may produce energetic electrons ($\sim 10$ GeV) sufficient to explain the observable radio emission [@article:SchlickeiserSieversThiemann:1987; @article:BrunettiLazarian:2010]. Observations of possibly nonthermal emission from clusters in the extreme ultraviolet [EUV; @article:SarazinLieu:1998] and hard X-rays [@article:RephaeliGruber:2002; @article:Fusco-Femiano_etal:2004; @article:Eckert_etal:2007] may provide further indication of relativistic particle populations in clusters, although the interpretation of these observations as nonthermal diffuse emission has been disputed on the basis of more sensitive observations [see, e.g., @article:Ajello_etal:2009; @article:Ajello_etal:2010; @article:Wik_etal:2009].
Galaxy clusters have, for many years, been proposed as sources of gamma rays. If shock acceleration in the ICM is an efficient process, a population of highly relativistic CR protons and heavy ions is to be expected in the ICM. The main energy-loss mechanism for CR hadrons at high energies is pion production through the interaction of CRs with nuclei in the ICM. Pions are short lived and decay. The decay of neutral pions produces gamma rays and the decay of charged pions produces muons, which then decay to electrons and positrons. Due to the low density of the ICM ($n_{\mathrm{ICM}}\sim
10^{-3}$ cm$^{-3}$), the large size and the volume-filling magnetic fields in the ICM, CR hadrons will be confined in the cluster on timescales comparable to, or longer than, the Hubble time [@article:Volk_etal:1996; @article:Berezinsky_etal:1997] and they can therefore accumulate. For a given CR distribution function, the hadronically induced gamma-ray flux is directly proportional to the CR-to-thermal pressure fraction, $X_{\mathrm{CR}}={\left\langle P_{{\mathrm{CR}}} \right\rangle}/
{\left\langle P_{\mathrm{th}} \right\rangle}$ [see, e.g., @article:EnsslinPfrommerSpringelJubelgas:2007], where the brackets indicates volume averages. A very modest $X_{{\mathrm{CR}}}$ of a few percent implies an observable flux of gamma rays [e.g., @article:PfrommerEnsslin:2004b].
Hydrostatic estimates of cluster masses, which are determined by balancing the thermal pressure force and the gravitational force, are biased low by the presence of any substantial nonthermal pressure component, including a CR pressure contribution. Similarly, a substantial CR pressure can bias the temperature decrement of the cosmic microwave background (CMB) due to the Sunyaev-Zel’dovich effect in the direction of a galaxy cluster. This could then severely jeopardize the use of clusters to determine cosmological parameters. Comparing X-ray and optical potential profiles in the centers of galaxy clusters yields an upper limit of 20-30% of nonthermal pressure (that can be composed of CRs, magnetic fields or turbulence) relative to the thermal gas pressure [@article:Churazov_etal:2008; @article:Churazov_etal:2010]. An analysis that compares spatially resolved weak gravitational lensing and hydrostatic X-ray masses for a sample of 18 galaxy clusters detects a deficit of the hydrostatic mass estimate compared to the lensing mass of $20\%$ at $R_{500}$ – the radius within which the mean density is 500 times the critical density of the Universe – suggesting again a substantial nonthermal pressure contribution on large scales [@article:Mahdavi_etal:2008]. Observing gamma-ray emission is a complementary method of constraining the pressure contribution of CRs that is most sensitive to the cluster core region. However, it assumes that the CR component is fully mixed with the ICM and may not allow for a detection of a two-phase structure of CRs and the thermal ICM. An $X_{\mathrm{CR}}$ of only a few percent is required in order to produce a gamma-ray flux observable with the current generation of gamma-ray telescopes, rendering this technique at least as sensitive as the dynamical and hydrostatic methods (which are more general in that they are sensitive to any nonthermal pressure component).
Gamma-ray emission can also be produced by Compton up-scattering of ambient photons, for example CMB photons, on ultra-relativistic electrons. Those electrons can either be secondaries from the CR interactions mentioned above, or injected into the ICM by powerful cluster members and further accelerated by diffusive shock acceleration or turbulent reacceleration processes [@article:SchlickeiserSieversThiemann:1987 and references therein].
A third mechanism for gamma-ray production in a galaxy cluster could be self-annihilation of a DM particle, e.g., a weakly interacting massive particle (WIMP). As already mentioned, about 80% of the cluster mass is in the form of dark matter, which makes galaxy clusters interesting targets for DM searches [@article:EvansFerrerSarkar:2004; @article:BergstromHooper:2006; @article:PinzkePfrommerBergstrom2009; @article:Cuesta_etal:2011] despite their large distances compared to other common targets for DM searches, such as dwarf spheroidal galaxies [@article:Strigari_etal:2007; @article:Acciari_etal:2010; @article:Aliu_etal:2009] or the Galactic Center [@article:Kosack_etal:2004; @article:Aharonian_etal:2006; @article:Aharonian_etal:2009b; @article:Abramowski_etal:2011].
While several observations of clusters of galaxies have been made with satellite-borne and ground-based gamma-ray telescopes, a detection of gamma-ray emission from a cluster has yet to be made. Observations with EGRET [@article:Sreekumar_etal:1996; @article:Reimer_etal:2003] and the Large Area Telescope (LAT) on board the Gamma-ray Space Telescope [@article:Ackermann_etal:2010] have provided upper limits on the gamma-ray fluxes (typically $\sim10^{-9}$ ph cm$^{2}$ s$^{-1}$ for -LAT observations) for several galaxy clusters in the MeV to GeV band. Upper limits on the very-high-energy (VHE) gamma-ray flux from a small sample of clusters, including the Coma cluster, have been provided by observations with ground-based imaging atmospheric Cherenkov telescopes [IACTs; @article:Perkins_etal:2006; @inproc:Perkins_etal:2008; @article:Aharonian_etal:2009a; @article:Aleksic_etal:2010; @article:Aleksic_etal:2012].
The Coma cluster of galaxies (ACO 1656) is one of the most thoroughly studied clusters across all wavelengths [@article:Voges_etal:1999]. Located at a distance of about 100 Mpc [$z=0.023$; @article:StrubleRood:1999], it is one of the closest massive clusters [$M
\sim10^{15}M_{\odot}$; @article:Smith:1983; @article:Kubo_etal:2008]. It hosts both a giant radio halo [@article:Giovannini_etal:1993; @article:Thierbach_etal:2003] and peripheral radio relic, which appears connected to the radio halo with a “diffuse” bridge [see discussion in @article:BrownRudnick:2010]. It has been suggested [@article:Ensslin_etal:1998] and successively demonstrated by cosmological simulations which model the nonthermal emission processes [@article:PfrommerEnsslinSpringel:2008; @article:Pfrommer:2008; @article:Battaglia_etal:2009; @article:Skillman_etal:2011], that the relic could well be an infall shock. Extended soft thermal X-ray (SXR) emission is evident from the ROSAT all-sky survey in the 0.1 to 2.4 keV band [@article:BrielHenryBohringer:1992]. Observations with XMM-Newton [@article:Briel_etal:2001] revealed substructure in the X-ray halo supported by substantial turbulent pressure of at least $\sim 10 \%$ of the total pressure [@article:Schuecker_etal:2004]. The Coma cluster is a natural candidate for gamma-ray observations.
In this article, results from the VERITAS observations of the Coma cluster of galaxies are reported, with complementing analysis of available data from the Large Area Telescope (LAT) on board the Gamma-ray Space Telescope. The VERITAS and -LAT data have been used to place constraints on cosmic-ray particle populations, magnetic fields, and dark matter in the cluster. Throughout the analyses, a present day Hubble constant of $H_{0} = 100h$ km s$^{-1}$ Mpc$^{-1}$ with $h=0.7$ has been used.
VERITAS Observations, Analysis, and Results
===========================================
The VERITAS gamma-ray detector [@article:Weekes_etal:2002] is an array of four 12 m-diameter imaging atmospheric Cherenkov telescopes [@article:Holder_etal:2006] located at an altitude of $\sim$1250 m a.s.l. at the Fred Lawrence Whipple Observatory in southern Arizona (31$^{\circ}$ 40 30 N, 110$^{\circ}$ 57 07 W). Each of the telescopes is equipped with a 499-pixel camera covering a 3.5$^{\circ}$ field of view. The array, completed in the fall of 2007, is designed to detect gamma-ray emission from astrophysical objects in the energy range from 100 GeV to more than 30 TeV. Depending on the zenith angle and quality selection criteria imposed during the data analysis, the effective energy range may be narrower than that. The energy resolution is $\sim 15$% and the angular resolution (68% containment) is $\sim 0.1^{\circ}$ per event at 1 TeV and slightly larger at low energy. At the time of the Coma cluster observations, the sensitivity of the array allowed for detection of a point source with a flux of 1% of the steady Crab Nebula flux above 300 GeV at the confidence level of five standard deviations ($5\sigma$) in under 45 hours.[^1]
The Coma cluster was observed with VERITAS between March and May in 2008 with all four telescopes fully operational. The total exposure amounts to 18.6 hours of quality-selected live time, i.e., time periods of astronomical darkness with clear sky conditions and no technical problems with the array. The center of the cluster was tracked in *wobble* mode, where the expected source location is offset from the center of the field of view by 0.5 degrees, to allow for simultaneous background estimation [@article:Fomin_etal:1994]. All of the observations were made in a small range with average zenith angle $\sim 21^{\circ}$.
The data analysis was performed following the standard VERITAS procedures described in @inproc:Cogan_etal:2007 and @inproc:Daniel_etal:2007. Prior to event reconstruction and selection, all shower images are calibrated and cleaned. Showers are then reconstructed for events with at least two telescopes contributing images that pass the following quality selection criteria: more than four participating pixels in the camera, number of photoelectrons in the image larger than 75, and the distance from the image centroid to the center of the camera less than $1.43^{\circ}$. These quality selection criteria impose an energy threshold[^2] of about 220 GeV. In addition, events for which only images from the two closest-spaced telescopes[^3] survive quality selection are rejected, as they introduce an irreducible high background rate due to local muons, degrading the instrument sensitivity [@article:MaierKnapp:2007].
Gamma-ray-like events are separated from the CR background by imposing selection criteria (cuts) on the mean-scaled length and width parameters [@article:Aharonian_etal:1997; @article:Krawczynski_etal:2006] calculated from a parametrized moment analysis of the shower images [@inproc:Hillas:1985]. These parameters are averages over the four telescopes weighted with the total amplitude of the images, that measure the image moments width and length scaled with values expected for gamma rays. In this analysis, events with a mean-scaled length in the range 0.05-1.19 and a mean-scaled width in the range 0.05-1.08 are selected as gamma-ray-like events. These ranges for the gamma-hadron separation cuts were optimized [*a priori*]{} for a weak point source (3% Crab Nebula flux level) and a differential spectral index of 2.4, using data taken on the Crab Nebula during the same epoch. Because the VHE gamma-ray spectrum for the Coma cluster is expected to be a power-law function with an index of about 2.3 [@article:PinzkePfrommer:2010], these cuts are suitable for the analysis of the Coma cluster data set. It is noted that slightly varying the spectral index ($\pm$ 0.2) does not significantly impact the cuts used for quality selection and gamma-hadron separation in this work.
The Coma cluster is a very rich cluster of galaxies with many plausible sites for gamma-ray emission: the core region, the peripheral radio relic, and individual powerful cluster member galaxies. VERITAS has a large enough field of view to allow investigation of several of these scenarios. In this work, the focus has been on the core region and three cluster members. The core region is treated as either a point source or a mildly extended source, a uniform disk with intrinsic radius $0.2^{\circ}$ or $0.4^{\circ}$, similar to the extension of the thermal soft X-ray emission from the core. There is evidence of a recent merger event between the two central galaxies NGC 4889 and NGC 4874 [@article:Tribble:1993]. There is also evidence for an excess of nonthermal X-ray emission from these galaxies as well as from the galaxy NGC 4921 [@article:Neumann_etal:2003]. Therefore, searches for point-like VHE gamma-ray emission have been conducted at the locations of these galaxies. The regions of interest considered in this work are summarized in Table \[table:roi\].
The ring-background model [@article:Aharonian_etal:2001] is used to estimate the background due to CRs misinterpreted as gamma rays (the cuts described above reject more than 99% of all CRs). The total number of events in a given region of interest is then compared to the estimated background from the off-source region scaled by the ratio of the solid angles to produce a final excess or deficit. The VHE gamma-ray significance is then calculated according to Formula 17 in @article:LiMa:1983. Significance skymaps over the VERITAS field of view produced with a $0.2^{\circ}$ integration radius are shown in Figure \[fig:skymaps\] with overlaid X-ray and radio contours from the ROSAT all-sky survey [@article:BrielHenryBohringer:1992] and GBT 1.4 GHz observations [@article:BrownRudnick:2010] respectively.
Depending on the assumed extent of the source and the point-spread function, we can define an ON region, into which a defined fraction of the source photons should fall. No significant excess of VHE gamma rays from the Coma cluster was detected with VERITAS, as illustrated by the $\theta^{2}$ distribution shown in Figure \[fig:thetasq\], in which source events would pile up at small values of $\theta^2$ for a point source and fall into a somewhat wider range of $\theta^2$ values for an extended source. The $\theta^{2}$ distribution is a plot of event density versus the square of the angular separation from a given location. It permits a comparison of the ON-source event distribution with that of other locations, in this case a ring-shaped region, into which only background events should fall, the so-called OFF-source region. The $\theta^{2}$ distribution extends out to 0.42 square degrees to cover both the case of point-like and extended emission from the core of the Coma cluster. The $\theta^{2}$ distributions for the member galaxies also considered in this work are very similar to that in Figure \[fig:thetasq\] and show no excess of gamma rays. A 99% confidence level upper limit is calculated for each region of interest using events from the ON-source and OFF-source regions and the method described by @article:Rolke_etal:2005 assuming a Gaussian-distributed background. A lower bound of zero is imposed on the gamma-ray flux from the Coma cluster, which prevents artificially low flux upper limits in the case that the best-fit source flux is formally negative. Figure \[fig:sigdist\] shows the distribution of significances over the VERITAS skymap, which is well fit by a Gaussian with a mean close to zero and a standard deviation within a few percent of unity.
Table \[table:results\] lists the upper limits for the selected regions of interest shown in Table \[table:roi\]. These upper-limit calculations depend on the gamma-ray spectrum, which in this work is assumed to be a power law in energy, $dN/dE\propto E^{-\alpha}$, where the spectral index $\alpha$ was allowed to have a value of 2.1, 2.3, or 2.5.
-LAT Analysis and Results
=========================
LAT on board has observed the Coma cluster in all-sky survey mode since its launch in June 2008. -LAT is sensitive to gamma rays in the 20 MeV to $\sim300$ GeV energy range and is complementary to the VERITAS observations. @article:Ackermann_etal:2010 reported on the search for gamma-ray emission from thirty-three galaxy clusters in the data from the first 18 months, including the Coma cluster, for which an upper limit of $4.58\times10^{-9}$ ph cm$^{-2}$ s$^{-1}$ in the 0.2 to 100 GeV energy band was reported. This limit is expected to improve as the exposure is increased. In this work an updated analysis is presented as a complement to the VERITAS results which includes data taken between August 5, 2008 and April 17, 2012.
The LAT-data analysis of this work follows the same procedure as described in detail in @2012ApJS..199...31N and was performed with the Fermi Science Tools version 9.23.1. To only include events with high probability of being photons, the P7SOURCE class and the corresponding P7SOURCE\_V6 instrument-response functions were used throughout this work.
A zenith-angle cut of 100$^\circ$ was applied to eliminate albedo gamma rays from the Earth’s limb, excluding time intervals during which any part of the region of interest (ROI) was outside the field of view. In addition, time intervals were removed during which the observatory was transiting the Southern Atlantic Anomaly or the rocking angle exceeded 52$^\circ$.
The ROI is defined to be a square region of the sky measuring $14^{\circ}$ on a side and centered on $\alpha_{J2000}=194.953$ and $\delta_{J2000}=27.9806$, the nominal position of the Coma cluster.
Only photons with reconstructed energy greater than 1 GeV are considered, for which the 68%-containment radius of the point-spread function (PSF) is narrower than $\sim0.8^{\circ}$. The Fermi-LAT collaboration estimates the systematic uncertainties on the effective area at 10 GeV to be around 10% [^4].
The background emission in the ROI was modeled using fourteen point sources listed in the second LAT source catalog [@2012ApJS..199...31N], the LAT standard Galactic diffuse emission component (`gal_2yearp7v6_v0.fit`), and the corresponding isotropic template (`iso_p7v6source.txt`) that accounts for extragalactic emission and residual cosmic-ray contamination. Due to the large tails of the PSF at low energy, further fourteen point sources, lying $\sim4^\circ$ outside the ROI, were included in the source model.
The energy spectra of twenty-four sources are described by a power law. The remaining four sources[^5], being bright sources, are modeled with additional degrees of freedom using the log-normal representation, which is typically used for modeling Blazar spectra.
The analysis is performed in three energy bins: 1–3 GeV, 3–10 GeV, and 10–30 GeV. To find the best fit spectral parameters, a binned maximum-likelihood analysis [@1996ApJ...461..396M] is performed for each energy bin on a map with $0.1^{\circ}$ pixel size in gnomonic (TAN) projection, covering the entire ROI. To determine the significance of the sources, and in particular that of the Coma cluster, the analysis tool uses the likelihood-ratio test statistic [@1996ApJ...461..396M] defined as, $${\rm TS}=-2\left(\ln L_0-\ln L\right),$$ where $L_0$ is the maximum likelihood value for the null hypothesis and $L$ is the maximum likelihood with the additional source at a given position on the sky.
In the likelihood analysis the spatial parameters of the sources were kept fixed at the values given in the catalog, whereas the spectral parameters of the point sources in the ROI, along with the normalization of the diffuse components, were allowed to freely vary. We analyzed three cases in which the gamma-ray emission from the Coma cluster was assumed to follow a power-law spectrum with a photon index $\alpha=2.1$, 2.3, and 2.5. The spectral indices of all point sources were permitted to freely vary between $\alpha=0$ and $\alpha=5$. We considered the emission as being caused both by a point-like and a spatially extended source (a uniform disk) with radius $r=0.2^\circ$ or $r=0.4^\circ$, as in the VERITAS analysis.
No significant gamma-ray signal was detected. For one free parameter, the flux from the Coma cluster, the detection significance is computed as the square root of the test statistic (TS follows a $\chi_1^2$ distribution). The highest test statistic was obtained for the high-energy band, where TS $\sim0.8$ for the point source model, TS $\sim0.7$ for the disk model with $r=0.2^{\circ}$, and TS $\sim2$ for the disk model with $r=0.4^{\circ}$.
We therefore used the profile likelihood method [@article:Rolke_etal:2005] to derive flux upper limits at the 99% confidence level in the energy range 1–30 GeV, assuming both an unresolved, point-like or spatially extended emission, as shown in Table \[table:fermi\].
Gamma Ray Emission from Cosmic Rays
===================================
We decided to adopt a multifaceted approach to constrain the CR-to-thermal pressure distribution in the Coma cluster using the upper limits derived from the VERITAS and -LAT data in this work. This approach includes (1) a simplified multi-frequency analytical model that assumes a constant CR-to-thermal energy density and a power-law spectrum in momentum, (2) an analytic model derived from cosmological hydrodynamical simulations of the formation of galaxy clusters, and (3) a model that uses the observed intensity profile of the giant radio halo in Coma to place a lower limit on the expected gamma-ray flux in the hadronic model – where the radio-emitting electrons are secondaries from CR interactions and which is independent of the magnetic field distribution. This last approach translates into a minimum CR pressure which, if challenged by tight gamma-ray limits/detections, permits scrutiny of the hadronic interaction model of the formation of giant radio halos. Alternatively, realizing a spatial CR distribution that is consistent with the flux upper limits, and requiring the model to match the observed radio data, enables us to derive a lower limit on the magnetic field distribution. We stress again that this approach assumes the validity of the hadronic interaction model. Modeling the CR distribution through different techniques enables us to bracket our lack of understanding about the underlying plasma physics that shapes the CR distribution hence to reflect the Bayesian priors that are imposed on the modeling [see @article:PinzkePfrommerBergstrom for a discussion].
Simplified analytical model {#sec:simple}
---------------------------
We start by adopting a simplified analytical model that assumes a power-law CR spectrum and a constant CR-to-thermal pressure ratio, i.e., we adopt the isobaric model of CRs following the approach of @article:PfrommerEnsslin:2004b. To be independent of additional assumptions and in line with earlier work in the literature, we do not impose a low-momentum cutoff, $q$, on the CR distribution function, i.e., we adopt $q=0$. Since, [*a priori*]{}, the CR spectral index is unconstrained,[^6] we vary it in the range $2.1<\alpha<2.5$, which is compatible with the radio spectral index of the giant radio halo of the Coma cluster after accounting for the spectral steepening at frequencies $\nu\sim5~{\mathrm}{GHz}$ due to the Sunyaev-Zel’dovich effect [@article:Ensslin:2002; @article:PfrommerEnsslin:2004b].[^7] To model the thermal pressure, we adopt the electron density profile for the Coma cluster that has been inferred from ROSAT X-ray observations [@article:BrielHenryBohringer:1992] and use a constant temperature of $kT= 8.25$ keV throughout the virial region.
Table \[table:constraints\_simple\] shows the resulting constraints on the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}} = {\left\langle P_{{\mathrm{CR}}} \right\rangle}/{\left\langle P_{\mathrm}{th} \right\rangle}$, averaged within the virial radius, $R_{\mathrm}{vir}=2.2$ Mpc, that we define as the radius of a sphere enclosing a mean density that is 200 times the critical density of the Universe. Constraints on $X_{\mathrm{CR}}$ with VERITAS flux upper limits (99% CL) strongly depend on $\alpha$. This is due to the comparably large energy range from GeV energies (that dominate the CR pressure, provided $\alpha>2$ and the CR population has a nonrelativistic low-momentum cutoff, i.e., $q<m_{p}c$, where $m_{p}$ is the proton mass) to energies at 220 GeV, where our quality selection criteria imposed the energy threshold. These gamma-ray energies correspond to 1.6 TeV CRs – an energy ratio of more than 3 orders of magnitude, which explains the sensitivity to small changes in $\alpha$. The flux measurements within 0.2$^{\circ}$ are the most constraining due to a competition between the integrated signal and the background as the integration radius increases. This yields limits on $X_{\mathrm{CR}}$ between 0.048 and 0.43 (for $\alpha$ varying between 2.1 and 2.5), with a constraint of $X_{\mathrm{CR}}<0.1$ for $\alpha=2.3$ (close to the spectral index predicted by the simulations of @article:PinzkePfrommer:2010 around 220 GeV). Constraints on $X_{\mathrm{CR}}$ with -LAT limits (99% CL) depend only weakly on $\alpha$ because GeV-band gamma rays are produced by CRs with energies near the relativistic transition, that dominantly contribute to the CR pressure. $X_{\mathrm{CR}}$-constraints with -LAT limits are most constraining for an aperture of 0.4$^{\circ}$; despite the slightly weaker flux upper limits in comparison to the smaller radii of integration, we expect a considerably larger gamma-ray luminosity due to the increasing volume in this model. The best limit of $X_{\mathrm{CR}}< 0.012$ is achieved for $\alpha=2.3$, while the limit for $\alpha=2.1$ is only slightly worse $(X_{\mathrm{CR}}<0.017)$.
Simulation-based approach {#sec:simulation}
-------------------------
We complement the simplified analytical analysis with a more realistic and predictive approach derived from cosmological hydrodynamical simulations. We adopt the universal spectral and spatial gamma-ray model developed by @article:PinzkePfrommer:2010 to estimate the emission from decaying neutral pions which in clusters dominates over the inverse-Compton (IC) emission above 100 MeV. Given a density profile as, e.g., inferred by cosmological simulations or X-ray observations, the analytic approach models the CR distribution and the associated radiative emission processes from radio to the gamma-ray band. This formalism was derived from high-resolution simulations of clusters of galaxies that included radiative hydrodynamics, star formation and supernova feedback, and it followed the CR physics by tracing the most important injection and loss processes self-consistently while accounting for the CR pressure in the equation of motion [@article:PfrommerSpringelEnsslinJubelgas; @article:EnsslinPfrommerSpringelJubelgas:2007; @article:JubelgasSpringelEnsslinPfrommer:2008]. The results are in line with earlier numerical results on some of the overall characteristics of the CR distribution and the associated radiative emission processes [@article:DolagEnsslin:2000; @article:MiniatiRyuKangJones:2001; @article:Miniati:2003; @article:Pfrommer_etal:2007; @article:PfrommerEnsslinSpringel:2008; @article:Pfrommer:2008].
The overall normalization of the CR and gamma-ray distribution scales nonlinearly with the acceleration efficiency at structure formation shocks. Following recent observations of supernova remnants [@article:Helder_etal:2009] as well as theoretical studies [@article:KangJones:2005], we adopt an optimistic but nevertheless realistic value of this parameter and assume that 50% of the dissipated energy at strong shocks is injected into CRs, with this efficiency decreasing rapidly for weaker shocks. Since the vast majority of internal formation shocks (merger and flow shocks) are weak shocks with Mach numbers $M\lesssim3$ [e.g., @article:Ryu_etal:2003], they do not contribute significantly to the CR population in clusters. Instead, strong shocks during the formation epoch of clusters and strong accretion shocks at the present time (at the boundary of voids and filaments/supercluster regions) dominate the acceleration of CRs which are adiabatically transported through the cluster. Hence, the model provides a plausible upper limit for the CR contribution from structure formation shocks in galaxy clusters which can be scaled with the effective acceleration efficiency. Other possible CR sources, such as AGN and starburst-driven galactic winds have been neglected for simplicity but could in principle increase the expected gamma-ray yield.
These cosmological simulations only consider advective transport of CRs by [bulk gas flows that inject a turbulent cascade, leading to]{} centrally-enhanced density profiles. However, other means of CR transport such as diffusion and streaming may flatten the CR radial profiles. [The CRs stream along magnetic field lines in the opposite direction of the CR number density gradient (at any energy). In the stratified cluster atmosphere, this implies a net flux of CRs towards larger radii, equalizing the CR number density with time if not counteracted by advective transport. It has been suggested that advection velocities only dominate over the CR streaming velocities for periods with trans- and supersonic cluster turbulence during a cluster merger and drop below the CR streaming velocities for relaxing clusters. As a consequence, a bimodality of the CR spatial distribution is expected to result; with merging (relaxed) clusters showing a centrally concentrated (flat) CR energy density profile [@article:EnsslinPfrommerMiniatiSubramanian:2011]. This translates into a bimodality of the expected diffuse radio and gamma-ray emission of clusters, since more centrally concentrated CRs will find higher target densities for hadronic CR proton interactions. As a result of this, relaxed clusters could have a reduced gamma-ray luminosity by up to a factor of five [@article:EnsslinPfrommerMiniatiSubramanian:2011].]{} Hence, tight upper limits on the gamma-ray emission can constrain a combination of acceleration physics and transport properties of CRs.
We adopt the density profile of thermal electrons as discussed in §\[sec:simple\] and model the temperature profile of the Coma cluster with a constant central temperature of $kT= 8.25$ keV and a characteristic decline toward the cluster periphery in accordance with a fit to the universal profile derived from cosmological cluster simulations [@article:PinzkePfrommer:2010; @article:Pfrommer_etal:2007] and the behavior of a nearby sample of deep [*Chandra*]{} cluster data [@article:Vikhlinin_etal:2005]. This enables us to adopt the spatial and spectral distribution of CRs according to the model by @article:PinzkePfrommer:2010 that neglects the contribution of supernova remnants, AGN, and cluster galaxies.
Figure \[fig:spectrum\] shows the expected integral spectral energy distribution of Coma within the virial radius (dotted line). This suggests a spectral index of $\alpha=2.1$ in the energy interval 1-3 GeV and $\alpha=2.3$ for energies probed by VERITAS ($>220$ GeV). Also shown are integrals of the differential spectrum for finite energy intervals across the angular apertures tested in this study ([dashed]{} lines). These model fluxes (summarized in Table \[table:constraints\_simple\]) are compared to and VERITAS flux upper limits for the same energy intervals. Constraints on $X_{\mathrm{CR}}$ with the gamma-ray flux limit of in the energy interval 1-3 GeV ($<0.4^\circ$) are most constraining, [since that combination of a specific energy interval and aperture minimizes the ratio of the upper limit to the expected model flux. In particular, this upper limit is 24% below]{} the model predictions that assume an optimistically large shock-acceleration efficiency and CR transport parameters as laid out above. [Hence this enables us to constrain a combination of maximum shock acceleration efficiency and CR transport parameters.]{} In our further analysis, we use the most constraining -LAT flux limits in the energy interval 1-3 GeV as well as the gamma-ray flux limits of VERITAS in the energy range above 220 GeV.
Figure \[fig:xcr\] shows the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}} =
{\left\langle P_{{\mathrm{CR}}} \right\rangle}/{\left\langle P_{\mathrm}{th} \right\rangle}$, as a function of radial distance, $R$, from the Coma cluster center and contained within $R$. All radii are shown in units of the virial radius, $R_{\mathrm}{vir}=2.2$ Mpc. To compute the CR pressure, we assume a low-momentum cutoff of the CR distribution at $q = 0.8\,m_{p}c$, where $m_{p}$ is the proton mass. [This is suggested by cosmological cluster simulations and reflects]{} the high Coulomb cooling rates at low CR energies. The CR-to-thermal pressure ratio rises toward the outer regions on account of the higher efficiency of CR acceleration at the peripheral accretion shocks compared to the weak central flow shocks. Adiabatic compression of a mixture of CRs and thermal gas disfavors the CR pressure relative to the thermal pressure on account of the softer equation of state of CRs. The weak increase of $X_{{\mathrm{CR}}}$ toward the core is due to the comparably fast thermal cooling of gas.
In the case of VERITAS, for the most constraining regions tested (within an aperture of radius 0.2$^{\circ}$), the predicted CR pressure is a factor of 7.2 below the inferred upper limits of VERITAS (see Table \[table:results\] and assuming a spectral index of $\alpha=2.3$ which matches the simulated one at energies $E_\gamma=200$ GeV). To first order, we can scale the averaged CR-to-thermal pressure ratio of our model by that factor, keep the spatial behavior and obtain an integrated limit of the CR-to-thermal pressure ratio of $X_{{\mathrm{CR}}}<0.112$ within 0.2$^{\circ}$ that translates to a limit within the cluster virial radius of $X_{\mathrm{CR}}<0.162$ (solid lines of Figure \[fig:xcr\]). This limit is less constraining by 50% in comparison to the simplified analytical model, which gives $X_{\mathrm{CR}}<0.1$. This difference is explained by the concavity of the simulated spectrum which therefore carries more pressure at GeV energies than a pure power-law spectrum with $\alpha=2.3$.
[As already aluded to,]{} the most constraining -LAT upper limit in the energy interval 1-3 GeV ($<0.4^\circ$) is a factor of [0.76 smaller]{} than our model predictions (assuming $\alpha=2.1$ which is very close to the simulated spectral index for the energy range 1-3 GeV). Scaling our integrated CR-to-thermal pressure profile yields a constraint of $X_{{\mathrm{CR}}}<0.012$ within 0.4$^{\circ}$ that translates to a limit within the cluster virial radius of $X_{\mathrm{CR}}<0.017$ (dashed lines of Figure \[fig:xcr\]). The $X_{\mathrm{CR}}$ constraint evaluated within the cluster virial radius is comparable to the constraint of $X_{\mathrm{CR}}<0.017$ in our simplified model. Naturally, with the -LAT limits we probe the region around GeV energies that dominate the CR pressure, and we do not expect any differences to the simplified power-law model in comparison to the universal CR spectrum with its concave CR spectrum found in the simulations.
Minimum gamma-ray flux {#sec:Fmin}
----------------------
For clusters that host radio halos, we can derive a minimum gamma-ray flux in the hadronic model of radio halos – where the radio-emitting electrons are secondaries from CR interactions. Hadronic interactions channel about the same power into secondary electrons and $\pi^{0}$-decay gamma rays. A stationary distribution of CR electrons loses all its energy to synchrotron radiation for strong magnetic fields ($B \gg B_{\mathrm}{CMB} \simeq (1+z)^2\,3.2 \mu{\rm G}$, where $B_{\mathrm}{CMB}$ is the equivalent magnetic field strength of the CMB so that $B_{\mathrm}{CMB}^2/8\pi$ equals the CMB energy density). Thus the ratio of gamma-ray to synchrotron flux becomes independent of the spatial distribution of CRs and thermal gas [@article:Voelk:1989; @article:Pohl:1994; @article:Pfrommer:2008], in particular with $\alpha_{\nu}\simeq 1$ as the observed synchrotron spectral index. Hence we can derive a minimum gamma-ray flux in the hadronic model $$\label{eq:Fmin}
F_{\gamma,{\mathrm}{min}} = \frac{{\displaystyle}A_{\gamma}}{{\displaystyle}A_{\nu}}\frac{{\displaystyle}L_{\nu}}{{\displaystyle}4\pi D_{{\mathrm}{lum}}^{2}},$$ where $L_{\nu}$ is the observed luminosity of the radio mini-halo, $D_{{\mathrm}{lum}}$ denotes the luminosity distance to the respective cluster, and $A_\gamma$ and $A_\nu$ are dimensional constants that depend on the hadronic physics of the interaction [@article:Pfrommer:2008; @Pfrommer_etal:2008]. Lowering the magnetic field would require an increase in the energy density of CR electrons to reproduce the observed synchrotron luminosity and thus increase the associated gamma-ray flux.
To derive a minimum gamma-ray flux that can be compared to the upper limits, we need to determine the radio flux within the corresponding angular regions. To this end, we fit the point-source-subtracted, azimuthally-averaged radio-halo profile at 1.38 GHz [@article:Deiss_etal:1997] with a $\beta$-model, $$\label{beta}
S_{\nu} (r_{\bot})= S_{0} \left[ 1 + \left( \frac{r_{\bot}}{r_{{\mathrm}{c}}}\right)^{2}\right]^{-3\beta + 1/2},$$ where $S_{0} = 1.1 \times 10^{-3}\,{\mathrm}{Jy\,arcmin}^{-2}$, $r_{{\mathrm}{c}} = 450$ kpc, and $\beta = 0.78$. Within the error bars, this profile is consistent with 326-MHz data taken by @article:Govoni_etal:2001 when scaled with a radio spectral index of 1.15.
The results for the minimum gamma-ray flux $F_{\gamma,{\mathrm}{min}}(>220~{\mathrm}{GeV})$ and the minimum CR-to-thermal pressure ratio $X_{{\mathrm{CR}},\,{\mathrm}{min}} = X_{\mathrm{CR}}F_{\gamma,{\mathrm}{min}}/F_{\gamma,{\mathrm}{iso}}$ are shown in Table \[table:constraints\], where $F_{\gamma,{\mathrm}{iso}}$ is the gamma-ray flux in the simplified model introduced in §\[sec:simple\]. [Even in the most constraining cases, and assuming $\alpha\leq 2.3$, these are a factor of $\sim 60$ below the VERITAS upper limits (for $\alpha=2.1$, $<0.2^{\circ}$) and a factor of $\sim 20$ below the -LAT upper limits (for $\alpha=2.3$, $<0.4^{\circ}$)]{}. Note that these minimum gamma-ray fluxes are sensitive to the variation of the CR proton spectral index with energy as a result of, for example, momentum-dependent diffusion. Assuming a plausible value for the central magnetic field of Coma of 5 $\mu$G [@article:Bonafede_etal:2010], the radio-halo emission at GHz frequencies is dominated by electrons with energy $E_{\mathrm}{e} \sim 2.5$ GeV (which corresponds to proton energies $E_{\mathrm}{p}
\sim 40$ GeV). Gamma rays with an energy of $200$ GeV are produced by CR protons with an energy of $E_{\mathrm}{p} \sim 1.6$ TeV – a factor of 50 higher than those probed by radio halo observations. A steepening of the CR proton spectral index of 0.2 between 40 GeV and 1.6 TeV would imply a decrease in the minimum gamma-ray flux by a factor of two.
Constraining the Magnetic Field {#sec:B}
-------------------------------
In the previous section, we have obtained an absolute lower limit on the gamma-ray emission in the hadronic model by assuming high magnetic fields, $B\gg B_{\mathrm}{CMB}$. We can turn the argument around and use our upper limit on the gamma-ray emission (and by extension on the CR pressure) to infer a lower limit on the magnetic field needed to explain the observed radio emission. This, again, assumes the validity of the hadronic model of radio halos, in which the radio-emitting electrons are secondaries from CR interactions. A stronger gamma-ray constraint will tighten the magnetic-field limit. In case of a conflict with magnetic field measurements by other methods, e.g., Faraday rotation measure (RM),[^8] the hadronic model of radio halos would be challenged. The method we use to constrain the magnetic field inherits a dependence on the assumed radial scaling which we parametrize as $$\label{eq:B}
B(r) = B_{0} \,\left(\frac{n_{{\mathrm}{e}}(r)}{n_{{\mathrm}{e}}(0)}\right)^{\alpha_B},$$ as suggested by Faraday RM studies and numerical magnetohydrodynamical (MHD) simulations [@article:Bonafede_etal:2010; @article:Bonafede_etal:2011 and references therein]. Here $n_{{\mathrm}{e}}$ denotes the Coma electron density profile [@article:BrielHenryBohringer:1992]. In fact, the magnetic field in the Coma cluster is among the best constrained, because its proximity permits RM observations of seven radio sources located at projected distances of 50 to 1500 kpc from the cluster center. The best-fit model yields $B_{0} = 4.7^{+0.7}_{-0.8}\,\mu$G and $\alpha_{B} =
0.5^{+0.2}_{-0.1}$ [@article:Bonafede_etal:2010]. We aim to constrain the central field strength, $B_{0}$, and we permit the magnetic decline, $\alpha_{B}$, to vary within a reasonable range of $\Delta\alpha_{B}=0.2$ as suggested by [those]{} Faraday RM studies. We proceed as follows:
1. Given a model for the magnetic field with $\alpha_B$ and an initial guess for $B_0$, we determine the profile of the CR-to-thermal pressure ratio, $X_{{\mathrm{CR}}}(r)$, by matching the hadronically-produced synchrotron emission to the observed radio-halo emission over the entire extent. To this end, we deproject the fit to the surface-brightness profile of Eq. \[beta\] (using an Abel integral equation, see Appendix of @article:PfrommerEnsslin:2004b) yielding the radio emissivity,
$$\label{eq:Coma:radio}
j_{\nu} (r) = \frac{S_{0}}{2\pi\, r_{{\mathrm}{c}}}\,
\frac{6\beta - 1}{\left(1 + r^{2}/r_{{\mathrm}{c}}^{2}\right)^{3 \beta}}\,
\mathcal{B}\left(\frac{1}{2}, 3\beta\right)
= j_{\nu,0} \left(1 + r^2/r_{{\mathrm}{c}}^{2}\right)^{-3 \beta},$$
where $\mathcal{B}$ denotes the beta function. It is generically true for weak magnetic fields ($B<B_{{\mathrm}{CMB}}$) in the outer parts of the Coma halo that the product $X_{{\mathrm{CR}}}(r)X_{B}(r)$ (where $X_B$ denotes the magnetic-to-thermal energy density ratio) has to increase by a factor of about 100 toward the radio-halo periphery to account for the observed extent. If we were to adopt a steeper magnetic decline such as $\alpha_{B}=0.5$ which produces a flat $X_{B}(r)$, the CR-to-thermal pressure ratio would have to rise accordingly by a factor of 100.
2. Given this realization for $X_{{\mathrm{CR}}}$, we compute the pion-decay gamma-ray surface-brightness profile, integrate the flux within a radius of $(0.13, 0.2, 0.4)$ degree, and scale the CR profile in order to match the corresponding VERITAS/flux upper limits. This scaling factor, $X_{{\mathrm{CR}},0}$, depends on the CR spectral index, $\alpha$, (assuming a power-law CR population for simplicity), the radial decline of the magnetic field, $\alpha_{B}$, and our initial guess for $B_{0}$.
3. We then solve for $B_{0}$ while matching the observed synchrotron profile and fixing the profile of $X_{{\mathrm{CR}}}(r)$ as determined through the previous two steps. Note that for $B_{0} \gg B_{{\mathrm}{CMB}}$ and a radio spectral index of $\alpha_{\nu}=1$, the solution would be degenerate since the luminosity of the radio halo scales as $$\label{eq:Lnu}
L_{\nu} \propto \int dV Q(E)\,\frac{B^{1+\alpha_\nu}}{B^2 + B_{{\mathrm}{CMB}}^2} \to \int dV Q(E),$$ where $Q(E)$ denotes the electron source function.
4. Inverse-Compton cooling of CR electrons on CMB photons introduces a characteristic scale of $B_{{\mathrm}{CMB}}\simeq 3.2\,\mu$G which imprints as a nonlinearity on the synchrotron emissivity as a function of magnetic field strength (see Eq. (\[eq:Lnu\])). Hence we have to iterate through the previous steps until our solution for the minimum magnetic field $B_{0}$ converges.
Table \[table:Bmin\] shows the resulting lower limit of the central magnetic field ranging from $B_{0} = 0.5$ to $1.4\,\mu$G in case the of VERITAS and from $B_{0} = 1.4$ to $5.5\,\mu$G in the case of -LAT.[^9] Since these lower limits on $B_{0}$ are below the values favored by Faraday RM for [most of the]{} parameter space spanned by $\alpha_{B}$ and $\alpha$ [(and never exceed the values for the phenomenological Faraday RM-inferred $B$-model)]{}, the hadronic model is a viable explanation of the Coma radio halo. [In fact, the -LAT upper limits start to rule out the parameter combination of $\alpha_{B}\gtrsim 0.7$ and $\alpha \gtrsim 2.5$ for the hadronic model of the Coma radio halo.]{} Future gamma-ray observations of the Coma cluster may put more stringent constraints on the parameters of the hadronic model.
A few remarks are in order. (1) For the VERITAS limits, the hardest CR spectral indices correspond to the tightest limits on $B_{0}$, because the CR flux is constrained around 1 TeV and a comparably small fraction of CRs at 100 GeV would be available to produce radio-emitting electrons. A high magnetic field would be required to match the observed synchrotron emission. The opposite is true for the upper limits at 1 GeV, which probe CRs around a pivot point of 8 GeV: a soft CR spectral index implies a comparably small fraction of CRs at 100 GeV and hence a strong magnetic field is needed to match the observed synchrotron flux. (2) For a steeper magnetic decline (larger $\alpha_{B}$), the CR number density needs to be larger to match the observed radio-emission profiles, which would yield a higher gamma-ray flux so that the upper limits are more constraining. This implies tighter lower limits for $B_{0}$. (3) Interestingly, in all cases, the 0.4$^{\circ}$-aperture limits are the most constraining. For a given magnetic realization, a substantially increasing CR-to-thermal pressure profile is needed to match the observed radio profiles, and therefore that CR realization produces a larger flux within 0.4$^{\circ}$ in comparison with the simplified CR model ($X_{{\mathrm{CR}}} = {\mathrm}{const.}$), for which the 0.2$^{\circ}$-aperture limits are more constraining in the case of VERITAS. Physically, the large CR pressure in the cluster periphery may arise from CR streaming into the large available phase space in the outer regions.
As a final word, in Table \[table:Bmin\] we show the corresponding values for the CR-to-thermal pressure ratio (at the largest emission radius at 1 Mpc) such that the model reproduces the observed radio surface-brightness profile.[^10] They should be interpreted as upper limits since they are derived from flux upper limits. For the -LAT upper limits, they range from 0.08 to 0.27; hence the $X_{{\mathrm{CR}}}$ profiles always obey the energy condition, i.e., $P_{{\mathrm{CR}}} <
P_{\mathrm{th}}$, over the entire range of the radio-halo emission ($< 1$ Mpc).[^11] The corresponding values for $X_{\mathrm{CR}}$ in the cluster center are smaller than 0.01 for the entire parameter space probed in this study. We conclude that the hadronic model is not challenged by current Faraday RM data and is a perfectly viable possibility in explaining the Coma radio-halo emission.
Emission from Dark Matter Annihilations
=======================================
As already mentioned in the introduction, most of the mass in a galaxy cluster is in the form of DM. While the nature of DM remains unknown, a compelling theoretical candidate is a WIMP. The self-annihilation of WIMPs can produce either monoenergetic gamma-ray lines or a continuum of secondary gamma rays that deviates significantly from the power-law spectra observed from most conventional astrophysical sources, with a sharp cut-off at the WIMP mass. These spectral features together with the expected difference in the intensity distribution compared to conventional astrophysical sources allow a clear, indirect detection of DM.
The expected gamma-ray flux due to self-annihilation of WIMPs in a dark-matter halo is given by $$\frac{d\Phi_{\gamma}(\Delta\Omega,E)}{dE}=
\frac{{\left\langle \sigma v \right\rangle}}{8\pi m_{\chi}^{2}}\,\frac{dN_{\gamma}}{dE}\, J(\Delta\Omega),
\label{eqn:WIMPflux}$$ where ${\left\langle \sigma v \right\rangle}$ is the thermally-averaged product of the total self-annihilation cross section and the relative WIMP velocity, $m_{\chi}$ is the WIMP mass, $\frac{dN_{\gamma}}{dE}$ is the differential gamma-ray yield per annihilation[^12], $\Delta\Omega$ is the observed solid angle, and $J$ is the so-called astrophysical factor – a factor which determines the DM annihilation rate and depends on the DM distribution.
Given the upper limit on the observed gamma-ray rate, defined as the ratio of the event number detected within the observing time $T_{\mathrm{obs}}$, $R_{\gamma}(99\%\ \mathrm{CL}) = N_{\gamma}(99\%\ \mathrm{CL}) / T_{\mathrm{obs}}$, we can place constraints on the WIMP parameter space $(m_{\chi}, {\left\langle \sigma v \right\rangle})$. Integrating Eq. (\[eqn:WIMPflux\]) over energy we find $${\left\langle \sigma v \right\rangle}(99\%\ \mathrm{CL}) <
R_{\gamma}(99\%\ \mathrm{CL})\, \frac{8\pi m_{\chi}^{2}}{J(\Delta\Omega)}\,
\left[\int^{m_{\chi}}_{0} dE\ A_{\mathrm{eff}}\,\frac{dN_\gamma (E)}{dE}\right]^{-1},$$ where $A_{\mathrm{eff}}$ is the effective area of the gamma-ray detector. Because the self-annihilation of a WIMP is a two-body process, the astrophysical factor $J(\Delta\Omega)$ is the line-of-sight integral of the DM density squared $$J(\Delta\Omega)=\int_{\Delta\Omega}d\Omega\int d\lambda\ \rho_{\chi}^{2}(\lambda,\Omega),$$ where $\lambda$ represents the line of sight. In this work, we have modeled the Coma DM distribution with a Navarro, Frenk and White (NFW) profile [@article:NavarroFrenkWhite:1997], $$\rho_{\chi}(r)=\rho_{s}\left(\frac{r}{r_{s}}\right)^{-1}\left(1+\frac{r}{r_{s}}\right)^{-2},$$ where $r_{s}$ is the scale radius and $\rho_{s}$ is the scale density. Using weak-lensing measurements of the virial mass in the Coma cluster and the DM halo mass-concentration relation derived from $N$-body simulation of structure formation [@article:Bullock_etal:2001], @article:Gavazzi_etal:2009 find, and list in their Table 1 (note that they define $R_{\rm vir}=R_{\rm 100})$, $M_{\rm vir}=M_{\rm 200} =9.7(+6.1/-3.5)\cdot 10^{14}\,
h^{-1}\ {\rm M_\odot}$ and $C_{\rm vir}=C_{\rm 200}=3.5(+1.1/-0.9)$, which we translate into the density-profile parameters $r_{s}=0.654$ Mpc and $\rho_{s}=4.4\times 10^{14}$ M$_{\odot}$/Mpc$^{3}$. Note that the uncertainties are not necessarily distributed as a Gaussian, and also arise from the choice of dark-matter profile. According to the latest high-resolution DM-only simulations of nine rich galaxy clusters, the inner regions of the smooth density profiles are quite well approximated by the NFW formula [@article:Gao_etal:2012]. However, gravitational interactions of DM with baryons may modify these predictions. This could give rise to either an increasing inner density slope due to adiabatic contraction of the DM component in response to cooling baryons in the central regions or a decreasing density slope due to violent baryonic feedback processes pushing gas out of the center by, e.g., energy injection through AGNs. However, on scales $r\gtrsim 0.45$ Mpc or more than 20% of $R_{\rm vir}$ (which are of relevance for the present work), different assumptions about the inner slope of the smooth DM density profile translate to uncertainties in the resulting astrophysical factor. Table \[table:astrofactor\] lists the astrophysical factors calculated for the different VERITAS apertures considered in this work. Table \[table:astrofactor\] also lists the astrophysical factor calculated for the background region, which is used to estimate the gamma-ray contamination from DM annihilation in the background region. As long as the DM contribution to the event number in the background region is negligible, the upper limits derived here directly scale with the astrophysical factor, ${\rm UL} (<\sigma v>)\propto J^{-1}$. The analysis uses a ring region to estimate the background in a ON region. We have to compute the expected level of gamma-ray emission from DM annihilation in the ring region in order to check that it is negligible with respect to the level of gamma-ray emission from DM annihilation in the ON region. This is equivalent to compute the astrophysical factor of the ON and OFF source region since this quantity is related to the rate of DM annihilations.
The resulting exclusion curves in the $({\left\langle \sigma v \right\rangle}, m_{\chi})$ parameter space are shown in Figure \[fig:dm\] for three different DM self-annihilation channels, W$^{+}$W$^{-}$, b$\bar{\mathrm{b}}$, and $\tau^{+}\tau^{-}$. Depending on the DM annihilation channel, the limits are on the order of $10^{-20}$ to $10^{-21}$ cm$^{3}$ s$^{-1}$. The minimum for each exclusion curve and corresponding DM particle mass is listed in Table \[table:DMlimits\]. We stress that these limits are derived with conservative estimates of the astrophysical factor $J$. They do not include any boost to the annihilation rate possibly due to DM substructures populating the Coma halo, which could scale down the present limits by a factor $O(1000)$ in the most optimistic cases [@article:PinzkePfrommerBergstrom; @article:Gao_etal:2012].
We also note that when the size of the integration region is increased, the limits on ${\left\langle \sigma v \right\rangle}$ result from a competition between the gain in the astrophysical factor ${\left\langle J \right\rangle}$ and the integrated background. For integration regions larger than 0.2$^{\circ}$ in radius, the astrophysical factors no longer compensate for the increased number of background events, and the signal-to-noise ratio deteriorates.
Discussion and Conclusions
==========================
We have reported on the observations of the Coma cluster of galaxies in VHE gamma rays with VERITAS and complementary observations with the *Fermi*-LAT. VERITAS observed the Coma cluster of galaxies for a total of 18.6 hours of high-quality live time between March and May in 2008. No significant excess of gamma rays was detected above an energy threshold of $\sim 220$ GeV. The *Fermi*-LAT has observed the Coma cluster in all-sky survey mode since its launch in June 2008. We have used all data available from launch to April 2012 for an updated analysis compared to published results [@article:Ackermann_etal:2010]. Again, no significant excess of gamma rays was detected. We have used the VERITAS and *Fermi*-LAT data to calculate flux upper limits at the 99% confidence level for the cluster core (considered as both a point-like source and a spatially-extended emission region) and for three member galaxies. The flux upper limits obtained were then used to constrain properties of the cluster.
We have employed various approaches to constrain the CR population and magnetic field distribution that are complementary in their assumptions and hence well suited to assessment of the underlying Bayesian priors in the models. (1) We used a simplified “isobaric CR model” that is characterized by a constant CR-to-thermal pressure fraction and has a power-law momentum spectrum. While this model is not physically justified [*a priori*]{}, it is simple and widely used in the literature and captures some aspects of more elaborate models such as (2) the simulation-based analytical approach of @article:PinzkePfrommer:2010. The latter is a “first-principle approach” that predicts the CR distribution spectrally and spatially for a given set of assumptions. It is powerful since it only requires the density profile as input due to the approximate universality of the CR distribution (when neglecting CR diffusion and streaming). Note, however, that inclusion of these CR transport processes may be necessary to explain the radio-halo bimodality. (3) Finally, we used a pragmatic approach which models the CR and magnetic distributions in order to reproduce the observed emission profile of the Coma radio halo. While this approach is also not physically justified, it is powerful because it shows what the “correct” model has to achieve and can point in the direction of the relevant physics.
Within this pragmatic approach, we employ two different methods. Firstly, adopting a high magnetic field everywhere in the cluster ($B\gg B_{\mathrm}{CMB}$) yields the minimum gamma-ray flux in the hadronic model of radio halos which we find to be a factor of 20 (60) below the most constraining flux upper limits of -LAT (VERITAS). Secondly, by matching the radio-emission profile (i.e., fixing the radial CR profile for a given magnetic field model) and by requiring the pion-decay gamma-ray flux to match the flux upper limits (i.e., fixing the normalization of the CR distribution), we obtain lower limits on the magnetic field distribution under consideration. Our limits for the central magnetic field range from $B_{0} = 0.5$ to $1.4\,\mu$G (for VERITAS flux limits) and from $B_{0} = 1.4$ to $5.5\,\mu$G (for -LAT flux limits). Since all [*(but one) of*]{} these lower limits on $B_0$ are below the values favored by Faraday RM, $B_{0} =
4.7^{+0.7}_{-0.8}\,\mu$G [@article:Bonafede_etal:2010], the hadronic model is a very attractive explanation of the Coma radio halo. [The -LAT upper limits start to rule out the parameter combination of $\alpha_{B}\gtrsim 0.7$ and $\alpha \gtrsim 2.5$ for the hadronic model of the Coma radio halo.]{}
Applying our simplified “isobaric CR model” to the most constraining VERITAS limits, we can constrain the CR-to-thermal pressure ratio, $X_{\mathrm{CR}}$, to be below 0.048–0.43 (for a CR or gamma-ray spectral index, $\alpha$, varying between 2.1 and 2.5). We obtain a constraint of $X_{\mathrm{CR}}<0.1$ for $\alpha=2.3$, the spectral index predicted by simulations at energies around 220 GeV. This limit is more constraining by a factor of 1.6 than that of the simulation-based model which gives $X_{\mathrm{CR}}<0.16$. This difference is due to the concave form of the simulated spectrum which provides more pressure at GeV energies in comparison to a pure power-law spectrum of $\alpha=2.3$.
The -LAT flux limits constrain $X_{\mathrm{CR}}$ to be below 0.012–0.017 (for $\alpha$ varying between 2.3 and 2.1), only weakly depending on the assumed CR spectral index. Assuming $\alpha=2.1$, which is very close to the simulated spectral index for the energy range of 1–3 GeV, we obtain a constraint which is identical to that from our simulation-based model within the virial radius of $X_{\mathrm{CR}}<0.017$. That constraint improves to $X_{{\mathrm{CR}}}<0.012$ for an aperture of 0.4$^\circ$ corresponding to a physical scale of $R \simeq R_{200}/3 \simeq 660$ kpc. [These upper limits are now starting to constrain the CR physics in self-consistent cosmological cluster simulations and cap the maximum CR acceleration efficiency at structure formation shocks to be $<50\%$. Alternatively, this may argue for non-negligible CR transport processes such as CR streaming and diffusion into the outer cluster regions [@article:Aleksic_etal:2012].]{} These are encouraging results in that we constrain the CR pressure (of a phase that is fully mixed with the ICM) to be at most a small fraction ($<0.017$) of the overall pressure. As a result, hydrostatic cluster masses and the total Comptonization parameter due to the Sunyaev-Zel’dovich effect suffer at most a very small bias due to CRs.
We have also used the flux upper limits obtained with VERITAS to constrain the thermally-averaged product of the total self-annihilation cross section and the relative velocity of DM particles. Modeling the Coma cluster DM halo with a NFW profile we derived limits on ${\left\langle \sigma v \right\rangle}$ to be on the order of $10^{-20}$ to $10^{-21}$ cm$^{-3}$ s$^{-1}$ depending on the chosen aperture. These limits are based on conservative estimates of the astrophysical factor, where a possible boost to the annihilation rate due to DM substructures in the cluster halo has been neglected. Including such a boost could scale down the present limits by a factor $O(1000)$ in the most optimistic cases.
This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748) and by STFC in the U.K. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.
The *Fermi* LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France.
C.P. gratefully acknowledges financial support of the Klaus Tschira Foundation. A.P. acknowledges NSF grant AST-0908480 for support.
[lcc]{} Core & [12$^{\mathrm{h}}$59$^{\mathrm{m}}$48.7$^{\mathrm{s}}$]{} & [+27$^{\circ}$5850.0]{}\
NGC 4889 & [13$^{\mathrm{h}}$00$^{\mathrm{m}}$08.13$^{\mathrm{s}}$]{} & [+27$^{\circ}$5837.03]{}\
NGC 4874 & [12$^{\mathrm{h}}$59$^{\mathrm{m}}$35.71$^{\mathrm{s}}$]{} & [+27$^{\circ}$5733.37]{}\
NGC 4921 & [13$^{\mathrm{h}}$01$^{\mathrm{m}}$26.12$^{\mathrm{s}}$]{} & [+27$^{\circ}$5309.59]{}\
\[table:roi\]
[lccccccccc]{} Core & 0 & 17 & 0.84 & 2.59 & (0.78%) & 2.78 & (0.83%) & 2.97 & (0.89%)\
& 0.2 & -41 & -1.0 & 1.96 & (0.59%) & 2.09 & (0.63%) & 2.21 & (0.66%)\
& 0.4 & -26 & -0.30 & 4.44 & (1.3%) & 4.74 & (1.4%) & 5.02 & (1.5%)\
NGC 4889 & 0 & 3 & 0.14 & - & - & 1.85 & (0.55%) & - & -\
NGC 4874 & 0 & -14 & -0.71 & - & - & 1.51 & (0.45%) & - & -\
NGC 4921 & 0 & -4 & -0.23 & - & - & 2.41 & (0.72%) & - & - \[table:results\]
[lccc]{} & 1.882 (0.000) & 0.759 (0.000) & 0.671 (0.830)\
& 2.109 (0.152) & 0.899 (0.000) & 0.719 (0.740)\
& 2.438 (0.201) & 1.232 (0.619) & 0.875 (1.387)\
& 1.946 (0.000) & 0.788 (0.000) & 0.667 (0.874)\
& 2.180 (0.169) & 0.941 (0.000) & 0.725 (0.828)\
& 2.524 (0.246) & 1.275 (0.742) & 0.869 (1.390)\
& 2.008 (0.000) & 0.816 (0.000) & 0.663 (0.915)\
& 2.246 (0.189) & 0.979 (0.020) & 0.720 (0.864)\
& 2.606 (0.291) & 1.313 (0.856) & 0.861 (1.387)\
\[table:fermi\]
[cccccc]{}\
0 & 0.1 & 0.23 & 0.97 & 1.9 & 14.8\
0.2 & 0.048 & 0.10 & 0.43 & 2.9 & 7.2\
0.4 & 0.067 & 0.15 & 0.62 & 4.4 & 10.8\
0 & 0.035 & 0.024 & 0.033 & 1.4 & 1.34\
0.2 & 0.024 & 0.017 & 0.022 & 2.1 & 1.00\
0.4 & 0.017 & 0.012 & 0.016 & 3.2 & 0.76\
\[table:constraints\_simple\]
[ccccccc]{}\
0 & 1.6 & 0.7 & 0.3 & 6.7 & 6.1 & 11\
0.2 & 3.1 & 1.4 & 0.6 & 7.8 & 7.2 & 13\
0.4 & 6.3 & 2.8 & 1.3 & 9.8 & 9.0 & 16\
0 & 3.5 & 4.8 & 6.4 & 6.7 & 6.1 & 11\
0.2 & 6.8 & 9.3 & 12.5 & 7.8 & 7.2 & 13\
0.4 & 13.5 & 18.6 & 25.0 & 9.8 & 9.0 & 16\
\[table:constraints\]
[ccccccc]{} 0.3 & 0.69 & 0.57 & 0.48 & 1.38 & 1.95 & 2.68\
0.5 & 0.97 & 0.80 & 0.68 & 1.94 & 2.74 & 3.78\
0.7 & 1.40 & 1.17 & 0.99 & 2.80 & 3.97 & 5.50\
&\
0.3 & 0.46 & 1.05 & 4.55 & 0.11 & 0.08 & 0.11\
0.5 & 0.74 & 1.70 & 7.47 & 0.18 & 0.13 & 0.17\
0.7 & 1.09 & 2.59 & 11.55 & 0.27 & 0.19 & 0.26\
\[table:Bmin\]
[ccc]{} 0 & $5.7\times 10^{16}$ & $1.3\times 10^{14}$ (negligible)\
0.2 & $8.1\times 10^{16}$ & $4.4\times 10^{14}$ ($<0.01{\left\langle J \right\rangle}_{{\mathrm}{signal}}$, negligible)\
0.4 & $9.4\times 10^{16}$ & $1.3\times 10^{15}$ ($\simeq0.01{\left\langle J \right\rangle}_{{\mathrm}{signal}}$, negligible) \[table:astrofactor\]
[lccc]{} W$^{+}$W$^{-}$ & 0 & 2000 & $1.1\times 10^{-20}$\
& 0.2 & 1900 & $4.3\times 10^{-21}$\
& 0.4 & 1900 & $8.4\times 10^{-21}$\
$b\bar{b}$ & 0 & 3500 & $1.2\times 10^{-20}$\
& 0.2 & 3400 & $4.4\times 10^{-21}$\
& 0.4 & 3500 & $8.7\times 10^{-21}$\
$\tau^{+}\tau^{-}$ & 0 & 670 & $2.4\times 10^{-21}$\
& 0.2 & 650 & $9.1\times 10^{-22}$\
& 0.4 & 660 & $1.8\times 10^{-21}$ \[table:DMlimits\]
[^1]: The integral flux sensitivity above 300 GeV was improved by about 30% with the relocation of one telescope in the summer of 2009.
[^2]: The energy threshold is defined as the energy corresponding to the maximum of the product function of the observed spectrum and the collection area. It does not vary significantly for the different source scenarios and assumed spectral indices reported in this work.
[^3]: In the array configuration prior to summer 2009, two telescopes had a separation of only 35 m.
[^4]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT\_caveats.html
[^5]: 2FGLJ1303.1+2435, 2FGLJ1310.6+3222, 2FGLJ1226.0+2953, and 2FGLJ1224.9+2122
[^6]: The hadronic interaction physics guarantees that the CR spectral index coincides with that of the resulting pion-decay gamma-ray emission at energies $E\gg 1$ GeV that are well above the pion bump [see discussion in @article:PfrommerEnsslin:2004b].
[^7]: Assuming a magnetic field of 1 $\mu$G, the CR protons responsible for the GHz radio emitting electrons have an energy of $\sim100$ GeV and are $\sim$ 20 times less energetic than those CR protons responsible for 200-GeV gamma-ray emission.
[^8]: Generally, Faraday RM analyses of the magnetic field strength by, e.g., background sources observed through clusters, are degenerate with the magnetic coherence scale and may be biased by the unknown correlation between magnetic and density fluctuations.
[^9]: Note that a central magnetic field of $3\,\mu$G corresponds in the Coma cluster to a magnetic-to-thermal energy density ratio of $X_B=0.005$.
[^10]: Note that in this section, we determine the radial behavior of $X_{\mathrm{CR}}$ by adopting a specific model for the magnetic field and requiring the modeled synchrotron surface-brightness profile to match the observed data of the Coma radio halo. This is in contrast to the simplified analytical CR model where $X_{\mathrm{CR}}$ is constant (§ \[sec:simple\]) and to the simulation-based model where $X_{\mathrm{CR}}(r)$ is derived from cosmological cluster simulations (§ \[sec:simulation\]).
[^11]: See Figure 3 in @article:PfrommerEnsslin:2004a for the entire parameter range assuming minimum energy conditions, and @article:PfrommerEnsslin:2004b, Figure 7 for a parametrization as adopted in this study. We caution, however, that the minimum-energy condition is violated at the outer radio-halo boundary for the range of minimum magnetic-field values inferred by this study.
[^12]: In this work, we have calculated the differential gamma-ray yield per annihilation using the Pythia Monte Carlo simulator.
| ArXiv |
---
abstract: 'We investigate the supersymmetry (SUSY) structures for inductor-capacitor circuit networks on a simple regular graph and its line graph. We show that their eigenspectra must coincide (except, possibly, for the highest eigenfrequency) due to SUSY, which is derived from the topological nature of the circuits. To observe this spectra correspondence in the high frequency range, we study spoof plasmons on metallic hexagonal and lattices. The band correspondence between them is predicted by a simulation. Using terahertz time-domain spectroscopy, we demonstrate the band correspondence of fabricated metallic hexagonal and lattices.'
author:
- Yosuke Nakata
- Yoshiro Urade
- Toshihiro Nakanishi
- Fumiaki Miyamaru
- Mitsuo Wada Takeda
- Masao Kitano
nocite: '[@*]'
title: ' Supersymmetric correspondence in spectra on a graph and its line graph: From circuit theory to spoof plasmons on metallic lattices '
---
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Introduction
============
Supersymmetry (SUSY) is a conjectured symmetry between fermions and bosons. Although the concept of SUSY was introduced in high-energy physics and remains to be experimentally confirmed, the underlying algebra is also found in quantum mechanics. When the SUSY algebra is applied to the field of quantum mechanics it is called supersymmetric quantum mechanics (SUSYQM) [@Cooper1994]. The algebraic relations of SUSY link two systems that at first glance might seem to be very different. The linkage through SUSY can be utilized to construct exact solutions for various systems in quantum mechanics. Recently, SUSYQM has been applied to construct quantum systems enabling exotic quantum wave propagations: reflectionless or invisible defects in tight-binding models [@Longhi2010] and complex crystals [@Longhi2013a], transparent interface between two isospectral one-dimensional crystals [@Longhi2013], reflectionless bent waveguides for matter-waves [@Campo2014], and disordered systems with Bloch-like eigenstates and band gaps [@Yu2015].
The SUSY structure was also found in other physics fields besides quantum mechanics, e.g., statistical physics through the Fokker-Planck equations [@Bernstein1984]. Through the similarity between quantum-mechanical probability waves and electromagnetic waves, the SUSY structure can be formulated for electromagnetic systems. Electromagnetic SUSY structures have been found in one-dimensional refractive index distributions [@Chumakov1994; @Miri2013], coupled discrete waveguides [@Longhi2010; @Miri2013], weakly guiding optical fibers with cylindrical symmetry [@Miri2013], planar waveguides with varying permittivity and permeability [@Laba2014], and non-uniform grating structures [@Longhi2015]. Even a quantum optical deformed oscillator with $\mathrm{SU}(1,1)$ group symmetry and its SUSY partner were constructed as a classical electromagnetic system [@Zuniga-Segundo2014].
The SUSY transformation generates new optical systems whose spectra coincide with those of the original system (except possibly for the highest eigenvalue of the fundamental mode of original or generated systems). The SUSY transformations have been utilized to synthesize mode filters [@Miri2013] and distributed-feedback filters with any desired number of resonances at the target frequencies [@Longhi2015]. The scattering properties of the optical systems paired by the SUSY transformation are related to each other [@Longhi2010; @Miri2013]. It is possible to design an optical system family with identical reflection and transmission characteristics by using the SUSY transformations [@Miri2014]. A reflectionless potential derived from the trivial system by SUSY transformation was applied to design transparent optical intersections [@Longhi2015a]. Moreover, SUSY has also been intensively investigated in non-Hermitian optical systems. If a system is invariant under the simultaneous operations of the space and time inversions, it is called $\mathcal{PT}$-symmetric. The SUSY transformation for the $\mathcal{PT}$-symmetric system allows for arbitrarily removing bound states from the spectrum [@Miri2013a]. In addition, non-Hermitian optical couplers can be designed [@Principe2015]. By using double SUSY transformations, the bound states in the continuum were also formulated in tight-binding lattices [@Longhi2014; @Longhi2014a] and continuous systems [@Correa2015]. The SUSY transformation in the $\mathcal{PT}$-symmetric system can also reduce the undesired reflection of one-way-invisible optical crystals [@Midya2014].
From an experimental perspective, it is still challenging to extract the full potential of electromagnetic SUSY because of fabrication difficulties. However, using dielectric coupled waveguides, researchers have realized a reflectionless potential [@Szameit2011], interpreted as a transformed potential derived from the trivial one by a SUSY transformation [@Longhi2010], and SUSY mode converters [@Heinrich2014]. The SUSY scattering properties of dielectric coupled waveguides have also been observed [@Heinrich2014a].
As we have described so far, many studies have been done for the electromagnetic SUSY, but their focusing point is mainly limited to dielectric structures. Recent progress of plasmonics [@AlexanderMaier2007] and metamaterials [@Solymar2009] using metals in optics demands further studies of SUSY for metallic systems. To design and analyze the characteristics of metallic structures, intuitive electrical circuit models are very useful, because they extract the nature of the phenomena despite reducing the degree of freedom for the problem [@Nakata2012a]. Actually, a circuit-theoretical design strategy called [*metactronics*]{} has been proposed even in the optical region [@Engheta2007] and the circuit theory for plasmons has also been developed [@Staffaroni2012]. If we could design circuit models enabling exotic phenomena, they open up new possibilities for application to higher frequency ranges due to the scale invariance of Maxwell equations. Thus, in this paper we develop how SUSY appears in inductor-capacitor circuit networks and demonstrate the SUSY correspondence in the high frequency region. In particular, we focus on the SUSY structure for inductor-capacitor circuit networks on a graph and its line graph.
This article is organized as follows. In Sec. \[sec:2\], we start by introducing the graph-theoretical concepts and formulate a general class of inductor-capacitor circuit network pairs related through SUSY, derived from the topological nature of the graphs representing the circuits. In Sec. \[sec:3\], we theoretically and experimentally demonstrate the SUSY eigenfrequency correspondence for paired metallic lattices in the terahertz frequency range. In Sec. \[sec:4\], we summarize and conclude the paper.
Theory \[sec:2\]
================
Eigenequation for inductor-capacitor circuit networks
------------------------------------------------------
We consider an inductor-capacitor circuit network on a simple directed graph $G=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ and $\mathcal{E}$ are the sets of vertices and directed edges, respectively. The modifier [*simple*]{} means that there are no multiple edges between any vertex pair and no edge (loop) that connects a vertex to itself. The number of the edges connected to a vertex $v$ of a graph is called the degree of $v$. A [*regular*]{} graph is a graph whose every vertex has the same degree. We assume that $G$ is an $m$-[*regular*]{} graph with all vertices having degree $m$. The capacitors, all with the same capacitance $C$, are connected between each vertex $v\in \mathcal{V}$ and the ground. Coils, all with the same inductance $L$, are loaded along all $e \in \mathcal{E}$. An example of $G$ and the inductor-capacitor circuit network on it are shown in Fig. \[fig:lc\_ladder\](a) and (b).
![\[fig:lc\_ladder\] (a) Example of simple $3$-regular directed graph. (b) Inductor-capacitor circuit network on the graph. (c) Line graph of the graph shown in (a). (d) Inductor-capacitor circuit network on the line graph (c). ](lc_ladder.eps){width="86mm"}
For $v\in \mathcal{V}$ and $e\in \mathcal{E}$, the incidence matrix $\mathsf{X}=[X_{ve}]$ of a directed graph $G$ is defined as follows: $X_{ve}=-1$ ($e$ enters $v$), $X_{ve}=1$ ($e$ leaves $v$), otherwise $X_{ve}=0$.
Using vector notation, we represent the current distribution $J_e$ flowing along $e\in \mathcal{E}$ as a column vector $\vct{J}=[J_e]^\mathrm{T}$. The charge distribution is denoted by $\vct{q}=[q_v]^\mathrm{T}$ with a stored charge $q_v$ at $v\in \mathcal{V}$. The charge conservation law is given by $$\dot{\vct{q}}=-\mathsf{X} \vct{J}, \label{eq:1}$$ where the time derivative is represented by the dot. The scalar potential $\Phi_v$ at $v \in {\mathcal V}$ must satisfy Faraday’s law of induction, so we have $$\dot{\vct{J}}=\frac{1}{L} \mathsf{X}^\mathrm{T}\vct{\Phi} , \label{eq:2}$$ with $\vct{\Phi}=[\Phi_v]^\mathrm{T}$. The scalar potential $\vct{\Phi}$ is written as $$\vct{\Phi}=\mathsf{P} \vct{q}, \label{eq:3}$$ with a potential matrix $\mathsf{P}$. In our case, $\mathsf{P}$ is given by $$\mathsf{P}=C^{-1}\mathsf{I}, \label{eq:4}$$ where we use the identity matrix $\mathsf{I}$.
From Eqs. (\[eq:1\])–(\[eq:4\]), we obtain $$\ddot{\vct{q}}=-{\omega_0}^2\mathsf{X}\mathsf{X}^\mathrm{T} \vct{q}, \nonumber$$ with $\omega_0=1/\sqrt{LC}$. Assuming $\vct{q}=\tilde{\vct{q}}\exp(-\ii \omega t)+\cc$, we have an eigenequation $$\mathsf{L}\tilde{\vct{q}}= \left(\frac{\omega}{\omega_0}\right)^2\tilde{\vct{q}} \label{eq:5}$$ with the Laplacian $\mathsf{L}=\mathsf{X}\mathsf{X}^\mathrm{T}$. We introduce an adjacency matrix $\mathsf{A}=[A_{vw}]$, where $A_{vw}$ is 1 if $v, w\in \mathcal{V}$ are connected by an edge, otherwise 0. From a direct calculation, we can write $\mathsf{L}$ by $\mathsf{A}$ as $$\mathsf{L}=\mathsf{X}\mathsf{X}^\mathrm{T}=-\mathsf{A}+m\mathsf{I}, \label{eq:6}$$ where $m$ is the degree of the vertex of $G$. Note that $\mathsf{L}$ is independent of the direction of the edges in $G$ because $\mathsf{L}$ is expressed in terms of $\mathsf{A}$ and $\mathsf{I}$.
The directed graph $G$ can be also regarded as an undirected graph. For $e\in \mathcal{E}$, we can make an undirected edge $\bar{e}$, where the bar operator ignores the direction of the edge. Then, we have $\bar{G}=(\mathcal{V},\bar{\mathcal{E}})$ with $\bar{\mathcal{E}}=\{\bar{e}|e\in \mathcal{E}\}$. We can also define the undirected incidence matrix $\bar{\mathsf{X}}=[\bar{X}_{ve}]$ as $\bar{X}_{ve}=1$ ($e$ and $v$ are connected), otherwise $\bar{X}_{ve}=0$. Using $\bar{\mathsf{X}}$, $\mathsf{A}$ is written as follows [@Biggs1994]: $$\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}=\mathsf{A}+m \mathsf{I}. \label{eq:7}$$ From Eqs. (\[eq:6\]) and (\[eq:7\]), we obtain $$\mathsf{L}=- \bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}+2m\mathsf{I}. \label{eq:8}$$
SUSY correspondence in spectra on a simple regular graph and its line graph
---------------------------------------------------------------------------
Next, we introduce the line graph concept [@Biggs1994]. The line graph $L(G)=(\mathcal{V}\sub{L}, \mathcal{E}\sub{L})$ of a directed graph $G$ is constructed as follows. Each edge in $G$ is considered to be a vertex of $L(G)$. Two vertices of $L(G)$ are connected if the corresponding edges in $G$ have a vertex in common. There are two possible choices for the direction of each edge in $L(G)$ and we adapt one of them. From here on, we only consider $L(G)$ of a simple $m$-regular graph $G$. In this case, the line graph $L(G)$ is a simple $m\sub{L}$-regular graph. The degree $m\sub{L}$ can be represented by $m$. For a vertex $v \in \mathcal{V}$ included in $e\in \mathcal{E}$, there are $m-1$ edges $e'\in \mathcal{E}\setminus\{e \}$ connected to $v$. Then, we obtain $$m\sub{L}=2(m-1). \label{eq:9}$$ Note that $(m/2)\#\mathcal{V}= \#\mathcal{E}=\#\mathcal{V}\sub{L}=(2/m\sub{L})\#\mathcal{E}\sub{L}$ is satisfied for a finite graph $G$, where $\#\mathcal{S}$ represents the numbers of the elements of the set $\mathcal{S}$. Figure \[fig:lc\_ladder\](c) is an example of the line graph of the graph $G$ shown in Fig. \[fig:lc\_ladder\](a). Figure \[fig:lc\_ladder\](d) is the inductor-capacitor circuit network on $L(G)$.
In the context of mathematics, it is known that the spectra of the Laplacians for the graph and its line graph are related to each other [@Shirai2000]. For the convenience of the readers, we rederive this property in a simple manner and apply it to the inductor-capacitor circuit networks. The Laplacian of $L(G)$ is written as $$\mathsf{L}\sub{L}=\mathsf{X}\sub{L}{\mathsf{X}\sub{L}}^\mathrm{T}=-\mathsf{A}\sub{L}+m\sub{L}\mathsf{I}\sub{L}, \label{eq:10}$$ with the identity matrix $\mathsf{I}\sub{L}$, the incidence matrix $\mathsf{X}\sub{L}$, and the adjacency matrix $\mathsf{A}\sub{L}$ for $L(G)$. The adjacency matrix of $L(G)$ is represented as follows [@Biggs1994]: $$\mathsf{A}\sub{L}=\bar{\mathsf{X}}^\mathrm{T} \bar{\mathsf{X}}-2\mathsf{I}\sub{L}. \label{eq:11}$$ From Eqs. (\[eq:10\]) and (\[eq:11\]), we have $$\mathsf{L}\sub{L}=-\bar{\mathsf{X}}^\mathrm{T} \bar{\mathsf{X}}+(m\sub{L}+2)\mathsf{I}\sub{L}.
\label{eq:12}$$ Now, we consider the composite system of $L(G)$ and $G$. Then the composite Laplacian $\mathcal{L}\sub{c}$ is given by $$\mathcal{L}\sub{c}=-\mathcal{K}\sub{c} +2m\mathcal{I}\sub{c}, \label{eq:13}$$ with $$\mathcal{K}\sub{c}=
\begin{bmatrix}
\bar{\mathsf{X}}^\mathrm{T}\bar{\mathsf{X}}&0\\
0&\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}
\end{bmatrix},\
\mathcal{I}\sub{c}=
\begin{bmatrix}
\mathsf{I}\sub{L}&0\\
0&\mathsf{I}
\end{bmatrix},
\nonumber$$ where we have used Eqs. (\[eq:8\]), (\[eq:9\]), and (\[eq:12\]). The composite operator $\mathcal{K}\sub{c}$ is written as $$\mathcal{K}\sub{c}=\mathcal{Q}\mathcal{Q}^\dagger+\mathcal{Q}^\dagger\mathcal{Q}, \label{eq:14}$$ where the symbol $\dagger$ represents the Hermitian conjugate, and we define the supercharge as $$\mathcal{Q} =
\begin{bmatrix}
0 & 0\\
\bar{\mathsf{X}} & 0
\end{bmatrix}.$$ These operators satisfy the superalgebra [@Cooper1994]: $$[\mathcal{K}\sub{c},\mathcal{Q}]=[\mathcal{K}\sub{c},\mathcal{Q}^\dagger]=0,$$ $$\{\mathcal{Q},\mathcal{Q}^\dagger\}=\mathcal{K}\sub{c},\ \{\mathcal{Q},\mathcal{Q}\}=\{\mathcal{Q}^\dagger,\mathcal{Q}^\dagger\}=0,$$ where $\{\mathcal{A},\mathcal{B}\}$ and $[\mathcal{A},\mathcal{B}]$ are the anticommutator and the commutator, respectively. Therefore, the eigenspectra of the inductor-capacitor circuit networks on the simple regular graph and its line graph must coincide except, possibly, for the highest eigenfrequency. Actually, if we have eigenvector $\vct{x}$ satisfying $\mathsf{L}\vct{x}=E\vct{x}$ with the eigenvalue $E$, we obtain $\bar{\mathsf{X}}^\mathrm{T}\vct{x}$, satisfying $\mathsf{L}\sub{L}(\bar{\mathsf{X}}^\mathrm{T}\vct{x})=E(\bar{\mathsf{X}}^\mathrm{T}\vct{x})$. Then, we have an eigenvector $\bar{\mathsf{X}}^\mathrm{T}\vct{x}$ for $\mathsf{L}\sub{L}$ when $\bar{\mathsf{X}}^\mathrm{T}\vct{x}\ne \vct{0}$ ($E\ne 2m$). The eigenvalue $E$ of $\mathsf{L}$ and $\mathsf{L}\sub{L}$ must satisfy $E\leq 2m$, because $\bar{\mathsf{X}}\bar{\mathsf{X}}^\mathrm{T}$ and $\bar{\mathsf{X}}^\mathrm{T}\bar{\mathsf{X}}$ are positive-semidefinite. For an eigenvector $\vct{y}_{2m}$ of $\mathsf{L}$ ($\mathsf{L}\sub{L}$) with eigenvalue $E=2m$, the partner mode cannot be obtained by multiplying $\bar{\mathsf{X}}^\mathrm{T}$ ($\bar{\mathsf{X}}$), because of $\bar{\mathsf{X}}^\mathrm{T}\vct{y}_{2m}=0$ ($\bar{\mathsf{X}}\vct{y}_{2m}=0$). The condition for complete spectral coincidence of the eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$ is discussed in Appendix \[sec:appA\]. Note that quantum tight-binding models represented by Eqs. (\[eq:8\]) and (\[eq:12\]) are isospectral except, possibly, for the highest eigenenergy, but an accurate tuning of the on-site potential satisfying Eqs. (\[eq:8\]) and (\[eq:12\]) is usually difficult to achieve. The significant point of SUSY for the inductor-capacitor circuit networks is that the on-site potential tuning is accomplished naturally.
If $G$ is a periodic graph with lattice vectors $\{\thrvct{a}_i\}$, we can show that the spectral coincidence except possibly for the highest eigenfrequency holds for [*each wave vector.* ]{} We define parallel translation with $\thrvct{a}_i$ as $\mathsf{P}^\mathcal{E}_i$ and $\mathsf{P}^\mathcal{V}_i$ for edges and vertices, respectively. From the translational symmetry, we have $\bar{\mathsf{X}}\mathsf{P}^\mathcal{E}_i=\mathsf{P}^\mathcal{V}_i\bar{\mathsf{X}}$. For a Bloch vector $\vct{x}_{\thrvct{k}}$ satisfying $\mathsf{P}^\mathcal{E}_i \vct{x}_{\thrvct{k}}=\exp({\ii\thrvct{k}\cdot\thrvct{a}_i}) \vct{x}_{\thrvct{k}}$, we have $\mathsf{P}^\mathcal{V}_i(\bar{\mathsf{X}}\vct{x}_{\thrvct{k}})=\exp(\ii\thrvct{k}\cdot\thrvct{a}_i)(\bar{\mathsf{X}}\vct{x}_{\thrvct{k}})$. This means that $\bar{\mathsf{X}}\vct{x}_{\thrvct{k}}$ is also a Bloch vector. Similar discussion can be applied to $\bar{\mathsf{X}}^\mathrm{T}$. Then, $\bar{\mathsf{X}}$ and $\bar{\mathsf{X}}^\mathrm{T}$ map Bloch vectors to Bloch vectors without changing $\thrvct{k}$. Therefore, the decomposition shown in Eq. (\[eq:13\]) is valid in the subspace of Bloch vectors with a wave vector $\thrvct{k}$. This means that the spectral coincidence except, possibly, for the highest eigenfrequency holds for each wave vector.
Examples \[subsec:c\]
---------------------
### finite case\[subsub:1\]
For the graph shown in Fig. \[fig:lc\_ladder\](a), we have $$\bar{\mathsf{X}}=\begin{bmatrix}
0 &1 &1 &1 &0 &0\\
1 &0 &1 &0 &1 &0\\
1 &1 &0 &0 &0 &1\\
0 &0 &0 &1 &1 &1
\end{bmatrix}\nonumber.$$ Then, we get $\omega=2\omega_0,\ 2\omega_0,\ 2\omega_0,\ 0$ for the inductor-capacitor circuit network on $G$. On the other hand, $\omega=\sqrt{6}\omega_0,\ \sqrt{6}\omega_0,\ 2\omega_0,\ 2\omega_0,\ 2\omega_0,\ 0$ are obtained for the inductor-capacitor circuit network on $L(G)$. We can see that all angular eigenfrequencies for $G$ are included in those for $L(G)$.
### Infinite case
![\[fig:kagome\_hex\] (color online) (a) Hexagonal lattice. (b) lattice. (c) Dispersion relation for a hexagonal inductor-capacitor circuit network. (d) Dispersion relation for a inductor-capacitor circuit network. (e) Eigenmodes of the higher band at the $\Gamma$ and $\mathrm{M}$ points for the hexagonal lattice. (f) Eigenmodes of the middle band at the $\Gamma$ and $\mathrm{M}$ points for the lattice. ](kagome_hex.eps){width="86mm"}
Here, we consider a hexagonal lattice as $G$ \[Fig. \[fig:kagome\_hex\](a)\]. The line graph $L(G)$ is a lattice \[Fig. \[fig:kagome\_hex\](b)\]. To see the spectra coincidence directly, we calculate the angular eigenfrequencies $\omega$. At first, we calculate the eigenvalue $\alpha$ and $\alpha\sub{L}$ for $\mathsf{A}$ and $\mathsf{A}\sub{L}$, respectively. Due to the Bloch theorem, it is enough to calculate them in the restricted space $\mathcal{W}_{\thrvct{k}_\parallel}$ of waves with wave vector $\thrvct{k}_\parallel$. For the hexagonal lattice, we have two vertices $v_p\in \mathcal{V}$ $ (p=1,2)$ in a unit cell. The vertex displaced from $v_p$, with $i\thrvct{a}_1+j\thrvct{a}_2$, is denoted by $v_p^{(i,j)}$ for $(i,j)\in \mathbb{Z}^2$, where $\thrvct{a}_1$ and $\thrvct{a}_2$ are lattice vectors for $G$. Now, we define $\vct{\Psi}_1 (\thrvct{k}_\parallel)=\sum_{(i,j)\in\mathbb{Z}^2} \exp(\ii \thrvct{k}_\parallel\cdot(i\thrvct{a}_1+j\thrvct{a}_2) ){\vct{v}_1^{(i,j)}}$ and $\vct{\Psi}_2 (\thrvct{k}_\parallel)=\sum_{(i,j)\in\mathbb{Z}^2} \exp(\ii \thrvct{k}_\parallel\cdot(i\thrvct{a}_1+j\thrvct{a}_2) ){\vct{v}_2^{(i,j)}}$, where $\{\vct{v}_{p}^{(i,j)}|(i,j)\in\mathbb{Z}^2, p=1,2\}\subset \mathcal{H}$ is a complete orthogonal basis of the Hilbert space $\mathcal{H}$. The vector subspace $\mathcal{W}_{\thrvct{k}_\parallel}$ is spanned by $\vct{\Psi}_1 (\thrvct{k}_\parallel)$ and $\vct{\Psi}_2 (\thrvct{k}_\parallel)$. The action of $\mathsf{A}$ in the restricted space $\mathcal{W}_{\thrvct{k}_\parallel}$ is represented by a $2\times 2$ matrix $\mathsf{A}({\thrvct{k}_\parallel})=[A_{ij}({\thrvct{k}_\parallel})]$, satisfying $\mathsf{A}\vct{\Psi}_i({\thrvct{k}_\parallel})= \sum_{j=1}^{2}\vct{\Psi}_j({\thrvct{k}_\parallel}) A_{ji}({\thrvct{k}_\parallel})$. Diagonalizing $\mathsf{A}({\thrvct{k}_\parallel})$ we have $$\alpha(\thrvct{k}_\parallel)=\pm\sqrt{3+2F(\thrvct{a}_1, \thrvct{a}_2;\thrvct{k}_\parallel)}, \label{eq:15}$$ with $F(\thrvct{u}_1, \thrvct{u}_2; \thrvct{k}_\parallel)=\cos (\thrvct{k}_\parallel\cdot\thrvct{u}_1)+\cos( \thrvct{k}_\parallel\cdot\thrvct{u}_2)+\cos \thrvct{k}_\parallel\cdot(\thrvct{u}_1-\thrvct{u}_2)$. By applying a similar calculation to the lattice, we obtain $$\alpha\sub{L}(\thrvct{k}_\parallel) =-2, 1\pm\sqrt{3+2F(\thrvct{a}_1\sur{L}, \thrvct{a}_2\sur{L};\thrvct{k}_\parallel)}, \label{eq:16}$$ with lattice vectors $\thrvct{a}_1\sur{L}$ and $\thrvct{a}_2\sur{L}$ for $L(G)$. Using Eqs. (\[eq:5\]), (\[eq:6\]), and (\[eq:15\]), we obtain $$\frac{\omega}{\omega_0}=\sqrt{3\pm\sqrt{3+2F(\thrvct{a}_1, \thrvct{a}_2;\thrvct{k}_\parallel)}} \label{eq:17}$$ for the hexagonal lattice. From Eqs. (\[eq:5\]), (\[eq:10\]), and (\[eq:16\]), the lattice also has the dispersion relation $$\frac{\omega}{\omega_0}=\sqrt{6}, \sqrt{3\pm\sqrt{3+2F(\thrvct{a}_1\sur{L}, \thrvct{a}_2\sur{L};\thrvct{k}_\parallel)}}. \label{eq:18}$$ The obtained dispersion relations are shown in Figs. \[fig:kagome\_hex\](c) and (d). The lower two bands are identical as we expected. Note that these bands are determined only by the product $LC$ and are independent of the ratio $L/C$. We also show examples of the eigenmodes for the hexagonal and lattices in Figs. \[fig:kagome\_hex\](e) and (f).
Band correspondence between metallic hexagonal and lattices \[sec:3\]
=====================================================================
In the previous section, we developed inductor-capacitor circuit networks that are related through SUSY. As an example, we saw that the bands of hexagonal and inductor-capacitor circuit networks are isospectral by SUSY, except for the highest band of the lattice. In this section, we examine this correspondence for realistic system. It is known that bar-disk resonators composed of metallic disks connected by metallic bars can be qualitatively modeled by the inductor-capacitor circuit networks discussed in Sec. \[sec:2\], because charges on the disks are coupled dominantly by the current flowing along the bars [@Nakata2012; @Kajiwara2016]. The modes on the bar-disk resonators are called spoof plasmons. Here, we study the spoof plasmons of metallic hexagonal and lattices whose designs are shown in Fig. \[fig:kagome\_hex\_lattice\_design\].
![\[fig:kagome\_hex\_lattice\_design\] Designs for (a) metallic hexagonal lattice and (b) metallic lattice. The following parameters are used: $d=10\,\U{\mu m}$, $r=150\,\U{\mu m}$, $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$.](kagome_hex_lattice_design.eps){width="86mm"}
Simulation\[sec:3-1\]
---------------------
![\[fig:hex\_kagome\_bands\] (color online) Dispersion relations obtained by simulation for a (a) metallic hexagonal lattice and (b) metallic lattice. The fitting parameters for the theoretical models discussed in Appendix \[sec:appB\] are given by $\omega_0\sur{hex}=2\pi \times 0.107\,\U{THz}$, $\eta\sur{hex}_0=0.0916$ for the hexagonal lattice and $\omega_0\sur{kag}=2\pi\times 0.101\,\U{THz}$ and $\eta\sur{kag}_0=0.142$ for the lattice. ](hex_kagome_band.eps){width="86mm"}
{width="172mm"}
We perform an eigenfrequency analysis for the metallic hexagonal and lattices by the finite element method solver (<span style="font-variant:small-caps;">Comsol Multiphysics</span>). The parameters of the structures for Fig. \[fig:kagome\_hex\_lattice\_design\] are as follows: bar width $d=10\,\U{\mu m}$, radius of disks $r=150\,\U{\mu m}$, distance between nearest disks $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$. In each simulation, the finite thickness metallic lattice parallel to $z=0$ is located in $z\in [-h/2,h/2]$. The unit cell in the $xy$ plane is the rhombus spanned by the lattice vectors and denoted by $U$. To reduce the degrees of freedom, we use the mirror symmetry with respect to $z=0$. A simulation domain with the material parameters of a vacuum is set in $U\times [0,6l]$ with $l=\sqrt{3} b=800\,\U{\mu m}$, and a perfect magnetic conductor condition imposed on the surface $z=0$. Half of the structure in $z\in [0,h/2]$ is engraved in the simulation domain and a perfect electric conductor (PEC) boundary condition is imposed on the structure surface. A perfect matched layer (PML) in $U\times [5l, 6l]$ with a PEC boundary at $z=6l$ is used to truncate the infinite effect. The periodic boundary condition with a phase shift (Floquet boundary conditions with a wave vector $\thrvct{k}_\parallel$) is applied to $\partial U \times [0,6l]$. Changing $\thrvct{k}_\parallel$ along the Brillouin zone boundary, we calculate the eigenfrequencies. To remove the modes which are not localized near the metallic surface [@Parisi2012], we select the modes with $$\xi=\frac{\int_{U\times [9l/2, 6l]} |\tilde{\thrvct{E}}|^2 \dd V}{\int_{U\times [h/2, l/2]} |\tilde{\thrvct{E}}|^2 \dd V}<1,\nonumber$$ where the complex amplitude of the electric field of the mode is denoted by $\tilde{\thrvct{E}}$.
The calculated eigenfrequencies for the metallic hexagonal and lattices are shown in Fig. \[fig:hex\_kagome\_bands\] as circles. Note that some points are missing because unphysical modes located near PML accidentally exist or couple with the modes. As we explained earlier, we eliminated such modes with $\xi\geq 1$. In Fig. \[fig:hex\_kagome\_bands\], we observe the lower two band correspondence between the metallic hexagonal and lattices. The bands of the metallic hexagonal lattice are about 5% higher than those of the lattice. However, we can say that the band correspondence is qualitatively established. The detailed theoretical models for fitting curves are discussed later in Appendix. \[sec:appB\]. Figure \[fig:eigenmodes\] shows the electric flux density $D_z$ on $z=h/2$ of the specific modes, where $D_z$ corresponds to the surface charge on the metal. These mode profiles agree with the theoretically calculated eigenmodes shown in Figs. \[fig:kagome\_hex\](e) and (f).
![\[fig:hdbr\_kdbr\] (color online) Microphotographs of the (a) metallic hexagonal and (b) metallic lattices.](hdbr_kdbr.eps){width="86mm"}
Experiment
----------
{width="172mm"}
To investigate the dispersion relation experimentally, we fabricated the metallic hexagonal and lattices by etching and performed transmission measurement on them using the terahertz time-domain spectroscopy technique. The samples made of stainless steel (SUS304) are shown in Fig. \[fig:hdbr\_kdbr\]. The geometrical parameters of these samples are the same as those for the simulation model in Sec. \[sec:3-1\]. The area where structures are patterned is $4\,\U{cm}\times 4\,\U{cm}$. The terahertz beam is generated by a spiral antenna and collimated by a combination of a hyper-hemispherical silicon lens and a Tsurupica$^\text{\textregistered}$ lens. The beam diameter is set to $13\, \U{mm}$ by an aperture. Wire-grid polarizers are located near the emitter and detector, which are adjusted so that the emitted and detected fields have the same linear polarization. The transmission spectrum $T(\omega)$ in the frequency domain is obtained from $T(\omega)=|\tilde{E}(\omega)/\tilde{E}\sub{ref}(\omega)|^2$, where $\tilde{E}(\omega)$ and $\tilde{E}\sub{ref}(\omega)$ are Fourier transformed electric fields with and without the sample. To scan the Brillouin zone, power transmission spectra are measured with changing incident angle $\theta$ from $\theta=0^\circ$ to $60^\circ$ with step $2.5^\circ$. Here, the magnitude of the in-plane wave vector $\thrvct{k}_\parallel$ is given by $k_\parallel = (\omega/c) \sin \theta$, where $c$ is the speed of light.
To observe the higher band of the hexagonal lattice and the middle band of the lattice, the incident waves are set as follows: (i) transverse electric (TE) modes in $\Gamma$–$\mathrm{K}$ scan and (ii) transverse magnetic (TM) modes in $\Gamma$–$\mathrm{M}$ scan. Figures \[fig:transmission\](a) and (b-1) show the power transmission spectra for these incident waves entering into the metallic hexagonal and lattices, respectively. The calculated eigenfrequencies are shown simultaneously as circles in Fig. \[fig:transmission\]. We can see that the transmission dips form a band from $0.15$ to $0.3\,\U{THz}$. The calculated eigenfrequencies are located around the experimental transmission dips. Thus, the SUSY band correspondence for the second band is experimentally demonstrated.
The highest band for the metallic lattice can be observed for differently polarized incident waves. Figure \[fig:transmission\](b-2) shows the transmission spectra for the metallic lattice, where the incident waves are set as (i) TM modes in the $\Gamma$–$\mathrm{K}$ scan and (ii) TE modes in the $\Gamma$–$\mathrm{M}$ scan. In Fig. \[fig:transmission\](b-2), we can see the flat band reported in Ref. . Note that the frequencies of the lowest band modes are under the light line, so it is impossible to excite them by free-space plane waves. To excite the lowest band modes, another method, e.g., attenuated total reflection measurement, is needed [@Kajiwara2016].
Discussion \[sec:3.3\]
----------------------
![\[fig:tuned\_kagome\] (color online) Comparison between eigenfrequencies for a tuned metallic lattice with $d=28\,\U{\mu m}$ and metallic hexagonal lattice with $d=10\,\U{\mu m}$. The other parameters are $r=150\,\U{\mu m}$, $b=800/\sqrt{3}\,\U{\mu m}$, and thickness $h=30\,\U{\mu m}$. ](tuned_kagome.eps){width="86mm"}
In the previous subsections, the band correspondence between the metallic hexagonal and lattices was demonstrated, but a $5\%$ discrepancy between the bands was also observed. Here, we investigate the possibility to compensate empirically for the discrepancy. We calculated the eigenfrequencies for the metallic lattice with the bar width $d=28\,\U{\mu m}$. The other parameters are the same as the previous one. The calculated results are shown in Fig. \[fig:tuned\_kagome\] as circles. To compare with the previous result, the eigenfrequencies for the metallic hexagonal lattice with $d=10\,\U{\mu m}$ are also plotted in Fig. \[fig:tuned\_kagome\]. We can see the improvement of band correspondence between the metallic hexagonal lattice and tuned metallic lattice.
Conclusion \[sec:4\]
====================
In this paper, we showed that the inductor-capacitor circuit networks on a simple regular graph and its line graph are related through SUSY, and their spectra must coincide (except possibly for the highest eigenfrequency). The SUSY structure for the circuits was derived from the topological nature of the graphs. To observe SUSY correspondence of the bands in the high frequency range, we investigated the metallic hexagonal and lattices. The band correspondence between them was predicted by a simulation. We performed terahertz time-domain spectroscopy for these metallic lattices and observed the band correspondence. Finally, we proposed an empirical tuning method to reduce the discrepancy of the corresponding bands of the metallic hexagonal and lattices.
The theoretical results are formulated for the inductor-capacitor circuit networks and independent of the implementations. Therefore, our results is also applicable to transmission-line systems such as microstrip. The SUSY correspondence in the spectra of inductor-capacitor circuit networks has the potential to extend mode filters [@Miri2013] and mode converters [@Heinrich2014] in two-dimensional (spoof) plasmonic systems.
The present research was supported by the JSPS KAKENHI Grant No. 25790065 and Grant-in-Aid for JSPS Fellows No. 13J04927. Two of the authors (Y. N. and Y. U.) were supported by JSPS Research Fellowships for Young Scientists.
Condition for complete spectral coincidence of eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$\[sec:appA\]
==============================================================================================================
In this Appendix, we consider condition for complete spectral coincidence of the eigenvalues of $\mathsf{L}$ and $\mathsf{L}\sub{L}$. Here, we mainly consider the finite graph cases. At first, we introduce a bipartite graph. A bipartite graph is a graph whose vertex set can be separated into two disjoint sets $\mathcal{V}_1$ and $\mathcal{V}_2$ such that every edge is connected between a vertex in $\mathcal{V}_1$ and that in $\mathcal{V}_2$. Using the concept of a bipartite graph, we obtain the following lemma:
Consider a connected graph $G=(\mathcal{V}, \mathcal{E})$ satisfying $\#\mathcal{V}>0$ and $\#\mathcal{E}>0$. Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. In this case, $\rank \bar{\mathsf X}< \#\mathcal{V}$ is satisfied if and only if $G$ is bipartite.
$\Leftarrow$: If the graph is bipartite, $\mathcal{V}$ is represented by the disjoint union of $\mathcal{V}_1$ and $\mathcal{V}_2$ as $\mathcal{V}=\mathcal{V}_1 \sqcup \mathcal{V}_2$. The $v_i$-component row vector of $\bar{\mathsf X}$ is denoted by $\bar{\vct{X}}_{i}$, where $v_i \in \mathcal{V}$ $(i=1,2,\cdots, n)$, and $n=\#\mathcal{V}$. Without loss of generality, we can assume $v_i \in \mathcal{V}_1$ $(i=1,2,\cdots, l)$ and $v_i\in \mathcal{V}_2$ $(i=l+1, l+2,\cdots, n)$. Because the graph is bipartite, $\sum_{i=1}^{l}\bar{\vct{X}}_{i}=\sum_{i=l+1}^{n}\bar{\vct{X}}_{i}$ is satisfied.\
$\Rightarrow$: If we assume $\rank \bar{\mathsf X}\ne \#\mathcal{V}$, $$\sum_{i=1}^{n}c_i \bar{\vct{X}}_{i}=0 \label{eq:19}$$ is satisfied for $c_i \in \mathbb{R}$ and not every $c_i=0$. At first we show $c_i\ne 0$ for all $i\in\{1,2,\cdots,n\}$. We assume $c_{q}=0$ for $q \in \{1,2,\cdots,n\}$. For arbitrary $r \in \{1,2,\cdots,n\}$, we consider a path $(v_{f(1)}, e_{1}, v_{f(2)}, e_2, \cdots, e_{p}, v_{f(p+1)})$ from $v_{q}$ to $v_{r}$, where $p$ is the path length, $f$ is a function from $\{1,2,\cdots, p+1\}$ to $\{1,2,\cdots, n\}$ satisfying $f(1)=q$ and $f(p+1)=r$, and the edge $e_{s}$ connects vertices $v_{f(s)}$ and $v_{f(s+1)}$ $(s=1,2,\cdots, p)$. Considering $e_{s}$-column of Eq. (\[eq:19\]), we have $c_{f(s)}=-c_{f(s+1)}$. Then, we obtain $c_i=0$ for all $i$. This leads a contradiction because we assumed not every $c_i=0$. Therefore, we have $c_i\ne0$ for all $i$. We assume $c_i>0$ $(i=1,2,\cdots, l)$, and $c_i<0$ $(i=l+1,l+2,\cdots, n)$, without loss of generality. For a given arbitrary $e\in \mathcal{E}$, we consider the column component about $e$ of Eq. (\[eq:19\]). Then, we find $e$ is connected between a vertex in $\mathcal{V}_1=\{v_1,v_2,\cdots,v_l\}$ and that in $\mathcal{V}_2=\{v_{l+1},v_{l+2},\cdots,v_{n}\}$. This shows $G$ is bipartite. This proof is based on Ref. .
From this lemma, we can prove the following theorem:
Consider a simple $m$-regular connected graph $G=(\mathcal{V}, \mathcal{E})$ with $\#\mathcal{E}>0$. There exists at least one mode with $\omega/\omega_0=\sqrt{2m}$ for the inductor-capacitor circuit network on $G$ if and only if $G$ is bipartite.
Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. $G$ is bipartite $\Leftrightarrow$ $\rank{\bar{\mathsf X}}<\#\mathcal{V}$ $\Leftrightarrow$ $\dim \ker{\bar{\mathsf{X}}^\mathrm{T}}=\#\mathcal{V}-\rank{\bar{\mathsf X}}>0$ $\Leftrightarrow$ $^\exists \vct{x}$ satisfying $\bar{\mathsf{X}}^\mathrm{T}\vct{x}=0$ $\Leftrightarrow$ $^\exists \vct{x}$ satisfying $\mathsf{L}\vct{x}=2m\vct{x}$.
On the other hand, we can formulate the condition for presence of eigenvalue $E=2m$ of $\mathsf{L}\sub{L}$ as follows:
Consider a simple $m$-regular graph $G=(\mathcal{V}, \mathcal{E})$ with $\#\mathcal{V}>0$. There exists at least one mode with $\omega/\omega_0=\sqrt{2m}$ for the inductor-capacitor circuit network on $L(G)$ if $m> 2$.
Let $\bar{\mathsf X}$ be an undirected incidence matrix of $\bar{G}$. We have $\dim \ker{\bar{\mathsf X}}=\#\mathcal{E}-\rank{\bar{\mathsf{X}}}\geq \#\mathcal{E}-\#\mathcal{V}=\frac{m-2}{2}\#\mathcal{V}>0$ for $m>2$. Then, $^\exists \vct{x}$ satisfying $\bar{\mathsf{X}}\vct{x}=0$. Finally, $^\exists \vct{x}$ satisfying $\mathsf{L}\sub{L}\vct{x}=2m\vct{x}$.
From these theorems, all spectra of the inductor-capacitor circuit networks on a simple $m$-regular ($m>2$) connected bipartite graph and its line graph completely coincide. On the other hand, we find that $\mathcal{L}\sub{c}$ have a non-degenerated eigenvector with eigenvalue $E=2m$ for a simple $m$-regular ($m>2$) connected non-bipartite graph $G$ (e.g. Sec. \[subsub:1\]). For $m=2$, all spectra of the inductor-capacitor circuit networks on a simple connected $m$-regular graph $G$ and its line graph $L(G)$ coincide because of $G=L(G)$. In this case, there are $\omega/\omega_0=\sqrt{2m}$ modes if and only if $\#\mathcal{V}$ is even.
To analyze an infinite periodic graph with lattice vectors $\{\thrvct{a}_i\}$, we take a supercell spanned by $\{N_i \thrvct{a}_i\}$, $N_i>1$. We impose a periodic boundary condition (called Born–von Karman boundary condition) on the sides of the supercell. Note that this boundary condition just leads to discretization of wave vectors in the Brillouin zone. Now, the infinite graph is reduced to a finite graph and we can use the theorems. For example, the hexagonal lattice is a simple 3-regular connected bipartite graph. Then, the spectra of the inductor-capacitor circuit networks on hexagonal and lattices include $\omega/\omega_0=\sqrt{6}$, simultaneously. Note that the spectra of the inductor-capacitor circuit networks on hexagonal and lattices completely coincide, but their dispersion relations do not.
Detailed theoretical model for the metallic hexagonal and lattices \[sec:appB\]
===============================================================================
In this appendix, we derive detailed theoretical models for fitting the eigenfrequencies of the metallic hexagonal and lattices. For the metallic hexagonal and lattices, the circuit models treated in Sec. \[sec:2\] are approximately valid. To improve the model accuracy, we have to take into account capacitive couplings between disks. Considering only nearest neighboring couplings, we modify Eq. (\[eq:4\]) as $\mathsf{P}=C^{-1}( \mathsf{I} + \eta \mathsf{A})$. Here, $C^{-1} \eta \mathsf{A}$ represents the capacitive coupling between the adjacent disks [@Kajiwara2016]. Then, we obtain $$\frac{\omega}{\omega_0}=\sqrt{ \big[m-\alpha(\thrvct{k}_\parallel)\big]\big[1+\eta \alpha(\thrvct{k}_\parallel)\big] }$$ for the metallic hexagonal lattice and $$\frac{\omega}{\omega_0}=\sqrt{ \big[m\sub{L}-\alpha\sub{L}(\thrvct{k}_\parallel)\big]\big[1+\eta \alpha\sub{L}(\thrvct{k}_\parallel)\big] }$$ for the metallic lattice. Generally, the coupling constant $\eta$ depends on $\omega$ as $\eta=\eta_0 \exp[\ii (\omega/c) b]$, where $b$ is the distance between the nearest disks [@Yeung2011]. The imaginary part of $\eta$ represents the resistive component. If we only focus on the real part of the eigenfrequencies, we may ignore the resistive term (note that we have already ignored the imaginary part of $L$ and $C$). Then, we assume $\eta=\eta_0 \cos[(\omega/c) b]$. Using the real part of the eigenvalues calculated by simulation, we numerically minimize the error $$\mathrm{err}\sur{hex}(\omega_0, \eta_0)=\sum_{(\omega,\thrvct{k}_\parallel) \in \text{\{data points\}} }g\big(\omega_0, \eta_0;\omega, m, \alpha(\thrvct{k}_\parallel)\big)^2$$ for the hexagonal lattice and $$\mathrm{err}\sur{kag}(\omega_0, \eta_0)=\sum_{(\omega,\thrvct{k}_\parallel) \in \text{\{data points\}} }g\big(\omega_0, \eta_0;\omega, m\sub{L}, \alpha\sub{L}(\thrvct{k}_\parallel)\big)^2$$ for the lattice, where we define $$g(\omega_0, \eta_0;\omega, m, \alpha)=\omega- \omega_0\sqrt{\left[m-\alpha\right]\left[1+\alpha \eta_0 \cos\left(\frac{\omega b}{c}\right)\right]}.$$ The obtained fitting parameters are as follows: $\omega_0=\omega_0\sur{hex}=2\pi \times 0.107\,\U{THz}$, $\eta_0=\eta\sur{hex}_0=0.0916$ for the hexagonal lattice, and $\omega_0=\omega_0\sur{kag}=2\pi\times 0.101\,\U{THz}$ and $\eta_0=\eta\sur{kag}_0=0.142$ for the lattice. Because the magnetic coupling and higher order effects (beyond the nearest capacitive coupling) are included in these parameters, the parameters for the hexagonal and lattice can be different. The dispersion curves with these fitting parameters are shown in Fig. \[fig:hex\_kagome\_bands\]. These curves agree with the simulated data despite the simplicity of the model.
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| ArXiv |
---
abstract: 'We recently generalised the lattice permutation condition for Young tableaux to Kronecker tableaux and hence calculated a large new class of stable Kronecker coefficients labelled by co-Pieri triples. In this extended abstract we discuss important families of co-Pieri triples for which our combinatorics simplifies drastically.'
author:
- 'C. Bowman'
- 'M. De Visscher'
- 'J. Enyang'
bibliography:
- 'bib.bib'
title: |
The lattice permutation condition\
for Kronecker tableaux\
(Extended abstract)
---
Introduction
============
Perhaps the last major open problem in the complex representation theory of symmetric groups is to describe the decomposition of a tensor product of two simple representations. The coefficients describing the decomposition of these tensor products are known as the [*Kronecker coefficients*]{} and they have been described as ‘perhaps the most challenging, deep and mysterious objects in algebraic combinatorics’. Much recent progress has focussed on the stability properties enjoyed by Kronecker coefficients.
Whilst a complete understanding of the Kronecker coefficients seems out of reach, the purpose of this work is to attempt to understand the [*stable*]{} Kronecker coefficients in terms of oscillating tableaux. Oscillating tableaux hold a distinguished position in the study of tensor product decompositions [@MR1035496; @MR3090983; @MR2264927] but surprisingly they have never before been used to calculate Kronecker coefficients of symmetric groups. In this work, we see that the oscillating tableaux defined as paths on the graph given in \[brancher\] (which we call Kronecker tableaux) provide bases of certain modules for the partition algebra, $P_s(n)$, which is closely related to the symmetric group. We hence add a new level of structure to the classical picture — this extra structure is the key to our main result: the co-Pieri rule for stable Kronecker coefficients.
$$\begin{tikzpicture}[scale=0.4]
\begin{scope} \draw (0,3) node { $\scalefont{0.5}\varnothing$ };
\draw (-3,0) node { $ \frac{1}{2}$ }; \draw (-3,3) node { $ 0$ };
\draw (-3,-3) node { $ 1$ }; \draw (-3,-6) node { $ 1\frac{1}{2}$ };
\draw (-3,-9) node { $ 2$ }; \draw (-3,-12) node { $ 2\frac{1}{2}$ };
\draw (-3,-15) node { $ 3$ };
\draw (0,0) node { $\scalefont{0.5}\varnothing$ };
\draw (0,-3) node { \text{ $\scalefont{0.5}\varnothing$ }} ; \draw (+3,-3) node {
$
\scalefont{0.4}\yng(1)$
} ;
\draw (0,-6) node { \text{ $\scalefont{0.5}\varnothing$ }};
\draw (3,-6) node { $ \,
\scalefont{0.5}\yng(1) $ };
\draw[<-] (0.0,1) -- (0,2); \draw[->] (0.0,-0.75) -- (0,-2.25);
\draw[->] (0.5,-0.75) -- (2.5,-2.25); \draw[->] (2.5,-3.75) -- (0.5,-5.25); \draw[->] (0,-3.75) -- (0,-5.25); \draw[->] (3,-3.75) -- (3,-5.25);
\draw[->] (0,-6.75) -- (0,-8.25); \draw[->] (03,-6.75) -- (3,-8.25);
\draw[->] (0.5,-6.75) -- (2.5,-8.25); \draw[->] (4,-6.75) -- (8,-8.25); \draw[->] (3.5,-6.75) -- (5,-8.25);
\draw (+0,-9) node { $\scalefont{0.5}\varnothing$ };
\draw (+3,-9) node { $
\scalefont{0.4}\yng(1) $ } ;
\draw (+6,-9) node { $
\scalefont{0.4}\yng(2)$ } ;
\draw (+9,-9) node { $
\scalefont{0.4}\yng(1,1)$ } ;
\draw[<-] (0,-11.25) -- (0,-9.75); \draw[<-] (03,-11.25) -- (3,-9.75); \draw[<-] (06,-11.25) -- (6,-9.75); \draw[<-] (9,-11.25) -- (9,-9.75);
\draw[<-] (0.5,-11.25) -- (2.5,-9.75); \draw[<-] (4,-11.25) -- (8,-9.75); \draw[<-] (3.5,-11.25) -- (5,-9.75);
\draw (+0,-12) node { $\scalefont{0.5}\varnothing$ };
\draw (+3,-12) node { $
\scalefont{0.4}\yng(1)$ } ;
\draw (+6,-12) node { $
\scalefont{0.4}\yng(2)$ } ;
\draw (+9,-12) node { $
\scalefont{0.4}\yng(1,1)$ } ;
\draw[->] (0,-12.75) -- (0,-14.25); \draw[->] (03,-12.75) -- (3,-14.25); \draw[->] (06,-12.75) -- (6,-14.25); \draw[->] (9,-12.75) -- (9,-14.25);
\draw[->] (0.5,-12.75) -- (2.5,-14.25); \draw[->] (4,-12.75) -- (8,-14.25); \draw[->] (3.5,-12.75) -- (5,-14.25);
\draw (+0,-15) node { $\scalefont{0.5}\varnothing$ };
\draw (+3,-15) node { $
\scalefont{0.4}\yng(1) $ } ;
\draw (+6,-15) node { $
\scalefont{0.4}\yng(2)$ } ;
\draw (+9,-15) node { $
\scalefont{0.4}\yng(1,1)$ } ;
\draw[->] (6.5,-12.75) -- (12,-14.25); \draw[->] (7,-12.75) -- (14.75,-14.25);
\draw[->] (9.5,-12.75) -- (15.25,-14.25); \draw[->] (10,-12.75) -- (17.7,-14.1);
\draw (12,-15) node { $
\scalefont{0.4}\yng(3)$ } ;
\draw (15,-15) node { $
\scalefont{0.4}\yng(2,1)$ } ;
\draw (18,-15) node { $
\scalefont{0.4}\yng(1,1,1)$ } ;
\end{scope}\end{tikzpicture}$$
\[brancher\]
A momentary glance at the graph given in \[oscillate\] reveals a very familiar subgraph: namely Young’s graph (with each level doubled up). The stable Kronecker coefficients labelled by triples from this subgraph are well-understood — the values of these coefficients can be calculated via a tableaux counting algorithm known as the Littlewood–Richardson rule [@LR34]. This rule has long served as the hallmark for our understanding of Kronecker coefficients. The Littlewood–Richardson rule was discovered as a rule of two halves (as we explain below). In [@BDE] we succeed in generalising one half of this rule to all Kronecker tableaux, and thus solve one half of the stable Kronecker problem. Our main result unifies and vastly generalises the work of Littlewood–Richardson [@LR34] and many other authors [@RW94; @Rosas01; @ROSAANDCO; @BWZ10; @MR2550164]. Most promisingly, our result counts explicit homomorphisms and thus works on a structural level above any description of a family of Kronecker coefficients since those first considered by Littlewood–Richardson [@LR34].
In more detail, given a triple of partitions $(\lambda,\nu, \mu)$ and with $|\mu|=s$, we have an associated skew $P_s(n)$-module spanned by the Kronecker tableaux from $\lambda$ to $\nu$ of length $s$, which we denote by $\Delta_s(\nu \setminus\lambda ) $. For $\lambda=\scalefont{0.5}\varnothing$ and $n{\geqslant}2s$ these modules provide a complete set of non-isomorphic $P_s(n)$-modules (and we drop the partition $\scalefont{0.5}\varnothing$ from the notation). The stable Kronecker coefficients are then interpreted as the dimensions, $$\label{dagger}\tag{$\dagger$}
\overline{g}(\lambda,\nu,\mu)
= \dim_{\mathbb{Q}}( {\operatorname{Hom}}_{ P_{s}(n)}( \Delta_{s}(\mu), \Delta_s(\nu \setminus\lambda ) ) )$$for $n{\geqslant}2s$. Restricting to the Young subgraph, or equivalently to a triple $(\lambda,\nu,\mu)$ of so-called [*maximal depth*]{} such that $|\lambda| + |\mu| = |\nu|$, these modules specialise to the usual simple and skew modules for symmetric groups; hence the multiplicities $\overline{g}(\lambda,\nu,\mu)$ are the Littlewood–Richardson coefficients. We hence recover the well-known fact that the Littlewood–Richardson coefficients appear as the subfamily of stable Kronecker coefficients labelled by triples of maximal depth. The tableaux counted by the Littlewood–Richardson rule satisfy 2 conditions: the [*semistandard*]{} and [*lattice permutation*]{} conditions. In [@BDE] we generalise the lattice permutation condition to Kronecker tableaux.
Let $(\lambda,\nu,\mu)$ be a [co-Pieri]{} triple or a triple of maximal depth. Then the stable Kronecker coefficient $\overline{g}(\lambda, \nu, \mu)$ is given by the number of semistandard Kronecker tableaux of shape $\nu\setminus\lambda$ and weight $\mu$ whose reverse reading word is a lattice permutation.
The observant reader will notice that the statement above describes the Littlewood–Richardson coefficients uniformly as part of a far broader family of stable Kronecker coefficients (and is the first result in the literature to do so). Whilst the classical Pieri rule (describing the semistandardness condition for Littlewood–Richardson tableaux) is elementary, it served as a first step towards understanding the full Littlewood–Richardson rule; indeed Knutson–Tao–Woodward have shown that the Littlewood–Richardson rule follows from the Pieri rule by associativity [@taoandco]. We hope that our generalisation of the co-Pieri rule (the lattice permutation condition for Kronecker tableaux) will prove equally useful in the study of stable Kronecker coefficients.
The definition of [semistandard Kronecker tableaux]{} naturally generalises the classical notion of semistandard Young tableaux as certain “orbits" of paths on the branching graph given in \[brancher\] (see Section 1.2 and Definition \[sstrd\]). The [lattice permutation condition]{} is identical to the classical case once we generalise the dominance order to all steps in the branching graph $\mathcal{Y}$ to define the reverse reading word of a semistandard Kronecker tableau (see \[sec:latticed\]).
**Examples of co-Pieri triples.** The definition of [*co-Pieri triples* ]{} is given in [@BDE Theorem 4.12] and can appear quite technical at first reading; we present a few special cases here.
- $\lambda$ and $\mu$ are one-row partitions and $\mu$ is arbitrary. This family has been extensively studied over the past thirty years and there are many distinct combinatorial descriptions of some or all of these coefficients [@RW94; @Rosas01; @ROSAANDCO; @BWZ10; @MR2550164], none of which generalises.
- the two skew partitions $\lambda \ominus (\lambda \cap \nu)$ and $\nu \ominus (\lambda \cap \nu)$ have no two boxes in the same column and $|\mu| = \max \{|\lambda \ominus (\lambda \cap \nu)| , |\nu \ominus (\lambda \cap \nu)|\}$. It is easy to see that if, in addition, $(\lambda, \nu, \mu)$ is a triple of maximal depth, then this case specialises to the classical co-Pieri triples.
- $\lambda = \nu = (dl,d(l-1), \ldots , 2d,d)$ for any $l,d{\geqslant}1$ and $|\mu| {\leqslant}d$.
In this extended abstract we have chosen to focus primarily on case $(i)$ as these triples carry many of the tropes of general co-Pieri triples (but with significant simplifications which serve to make this abstract more approachable) and because case $(i)$ should be familiar to many readers due to its many appearances in the literature.
The partition algebra and Kronecker tableaux {#sec2}
=============================================
\[sec:standard\]
The combinatorics underlying the representation theory of the partition algebras and symmetric groups is based on partitions. A [*partition*]{} $\lambda $ of $n$, denoted $\lambda \vdash n$, is defined to be a sequence of weakly decreasing non-negative integers which sum to $n$. We let $\varnothing$ denote the unique partition of 0. Given a partition, $\lambda=(\lambda _1,\lambda _2,\dots )$, the associated [*Young diagram*]{} is the set of nodes $[\lambda]=\left\{(i,j)\in\mathbb{Z}_{>0}^2\ \left|\ j{\leqslant}\lambda_i\right.\right\}.$ We define the length, $\ell(\lambda)$, of a partition $\lambda$, to be the number of non-zero parts. Given $\lambda = (\lambda_1,\lambda_2, \ldots,\lambda_{\ell} )$ a partition and $n$ an integer, define $\lambda_{[n]}=(n-|\lambda|, \lambda_1,\lambda_2, \ldots,\lambda_{\ell}).$ Given $\lambda_{[n]} $ a partition of $n$, we say that the partition has [*depth*]{} equal to $|\lambda|$.
The partition algebra is generated as an algebra by the elements $s_{k,k+1}$, $p_{k+1/2}$ ($1{\leqslant}k{\leqslant}r-1$) and $p_k$ ($1{\leqslant}k{\leqslant}r$) pictured below modulo a long list of relations. One can visualise any product in this algebra as simply being given by concatenation of diagrams, modulo some surgery to remove closed loops [@BDE]. $${s}_{k,k+1}=
\begin{minipage}{34mm}\scalefont{0.8}\begin{tikzpicture}[scale=0.45]
\draw (0,0) rectangle (6,3);
\foreach \x in {0.5,1.5,...,5.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\draw (2.5,-0.49) node {$k$};
\draw (2.5,+3.5) node {$\overline{k}$};
\begin{scope} \draw (0.5,3) -- (0.5,0);
\draw (5.5,3) -- (5.5,0);
\draw (2.5,3) -- (3.5,0);
\draw (4.5,3) -- (4.5,0);
\draw (3.5,3) -- (2.5,0);
\draw (1.5,3) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
p_{k+1/2}
=\begin{minipage}{34mm}\scalefont{0.8}\begin{tikzpicture}[scale=0.45]
\draw (0,0) rectangle (6,3);
\foreach \x in {0.5,1.5,...,5.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\draw (2.5,-0.49) node {$k$};
\draw (2.5,+3.5) node {$\overline{k}$};
\begin{scope} \draw (0.5,3) -- (0.5,0);
\draw (5.5,3) -- (5.5,0);
\draw (1.5,3) -- (1.5,0);
\draw (3.5,0) arc (0:180:.5 and 0.5);
\draw (2.5,3) arc (180:360:.5 and 0.5);
\draw (2.5,3) -- (2.5,0);
\draw (4.5,3) -- (4.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
p_k
=\begin{minipage}{34mm}\scalefont{0.8}\begin{tikzpicture}[scale=0.45]
\draw (0,0) rectangle (6,3);
\foreach \x in {0.5,1.5,...,5.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\draw (2.5,-0.49) node {$k$};
\draw (2.5,+3.5) node {$\overline{k}$};
\begin{scope} \draw (0.5,3) -- (0.5,0);
\draw (5.5,3) -- (5.5,0);
\draw (4.5,0) -- (4.5,3);
\draw (1.5,3) -- (1.5,0);
\draw (3.5,3) -- (3.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
$$
Define the branching graph $\mathcal{Y}$ as follows. For $k\in {{{\mathbb Z}_{{\geqslant}0}}}$, we denote by $\mathscr{P}_{{\leqslant}k}$ the set of partitions of degree less or equal to $k$. Now the set of vertices on the $k$th and $(k+1/2)$th levels of $\mathcal{Y}$ are given by $${\mathcal{Y}}_{k}= \{ (\lambda,k-|\lambda|) \mid \lambda \in \mathscr{P}_{{\leqslant}k}\}
\qquad
{\mathcal{Y}}_{k+1/2} =
\{ (\lambda,k-|\lambda|) \mid \lambda \in \mathscr{P}_{{\leqslant}k}\}.$$ The edges of $\mathcal{Y}$ are as follows,
- for $(\lambda,l) \in {\mathcal{Y}}_k$ and $(\mu,m) \in {\mathcal{Y}}_{k+1/2}$ there is an edge $(\lambda,l) \to(\mu,m)$ if $\mu = \lambda$, or if $\mu $ is obtained from $\lambda $ by removing a box in the $i$th row for some $i{\geqslant}1$; we write $\mu =\lambda- \varepsilon_0$ or $\mu =\lambda- \varepsilon_i$, respectively.
- for $(\lambda,l) \in {\mathcal{Y}}_{k+1/2}$ and $(\mu,m) \in {\mathcal{Y}}_{k+1}$ there is an edge $(\lambda,l) \to(\mu,m)$ if $\mu = \lambda$, or if $\mu $ is obtained from $\lambda $ by adding a box in the $i$th row for some $i{\geqslant}1$; we write $\mu =\lambda+ \varepsilon_0$ or $\mu =\lambda+ \varepsilon_i$, respectively.
When it is convenient, we decorate each edge with the index of the node that is added or removed when reading down the diagram. The first few levels of $\mathcal{Y}$ are given in Figure \[brancher\]. When no confusion is possible, we identify $(\lambda,l) \in \mathcal{Y}_{k}$ with the partition $\lambda$.
Given $\lambda \in \mathscr{P}_{r-s} \subseteq \mathcal{Y}_{r-s}$ and $\nu \in \mathscr{P}_ {{\leqslant}r} \subseteq \mathcal{Y}_{r}$, we define a [*standard Kronecker tableau*]{} of shape $ \nu \setminus \lambda $ and degree $s$ to be a path ${\mathsf{t}}$ of the form $$\label{genericpath}
\lambda = {\mathsf{t}}(0) \to {\mathsf{t}}(\tfrac{1}{2}) \to {\mathsf{t}}(1)\to \dots \to {\mathsf{t}}(s-\tfrac{1}{2})\to {\mathsf{t}}(s) = \nu,$$ in other words ${\mathsf{t}}$ is a path in $\mathcal{Y}$ which begins at $\lambda$ and terminates at $\nu$. We let ${\mathrm{Std}}_s(\nu \setminus \lambda)$ denote the set of all such paths. If $\lambda = \emptyset \in \mathcal{Y}_0$ then we write ${\mathrm{Std}}_r(\nu)$ instead of ${\mathrm{Std}}_r(\nu \setminus \emptyset)$. Given ${\mathsf{s}},{\mathsf{t}}$ two standard Kronecker tableaux of degree $s$, we write ${\mathsf{s}}\trianglerighteq {\mathsf{t}}$ if ${\mathsf{s}}(k)\trianglerighteq {\mathsf{t}}(k)$ for all $0{\leqslant}k{\leqslant}s$.
We can think of a path as either the sequence of partitions or the sequence of boxes removed and added. We usually prefer the latter case and record these boxes removed and added pairwise. For a pair $(-\varepsilon_p,+\varepsilon_q)$ we call this an add or remove step if $p=0$ or $q=0$ respectively (because the effect of this step is to add or remove a box) and we call this a dummy step if $p=q$ (as we end up at the same partition as we started); we write $a(q)$ or $r(p)$ for an add or remove step and $d(p)$ for a dummy step. Many examples are given below, in particular the reader should compare the paths of Example \[example3\] with those depicted in the central diagram in Figure \[maximaldepth\]. We let ${\mathsf{t}}^\lambda$ denote the most dominant element of ${\mathrm{Std}}_s(\lambda)$, namely that of the form: $$\underbrace{
d(0) \circ d(0) \circ
\dots
\circ d(0) }_{r-|\lambda|}
\circ
\underbrace{
a(1)
\circ
\dots\circ
a(1)}_{\lambda_1}\circ
\underbrace{
a(2)
\circ
\dots\circ
a(2)}_{\lambda_2}\circ
\cdots$$ Given $\lambda
\in\mathscr{P}_{r-s} \subseteq \mathcal{Y}_{r-s}$ and $\nu \in\mathscr{P}_{{\leqslant}r} \subseteq \mathcal{Y}_r$, define the [*skew cell module*]{} $$\Delta_s(\nu\setminus\lambda) = {\rm Span}\{ {\mathsf{t}}^\lambda \circ {\mathsf{s}}\mid {\mathsf{s}}\in {\mathrm{Std}}_s(\nu\setminus\lambda)\}$$ with the action of $P_s(n)\hookrightarrow P_{r-s}(n) \otimes P_s(n)\hookrightarrow P_r(n)$ given as in [@BDE Section 2.3]. If $\lambda =\varnothing$, then we simply denote this module by $\Delta_s(\nu)$. Let $\lambda\in \mathscr{P}_{r-s}$, $\mu\in \mathscr{P}_s$ and $\nu \in \mathscr{P}_{{\leqslant}r}$. Then we are able to define the stable Kronecker coefficients (even if this is not their usual definition) to be the multiplicities $$\overline{g}(\lambda,\nu,\mu)
= \dim_{\mathbb{Q}}( {\operatorname{Hom}}_{ P_{s}(n)}( \Delta_{s}(\mu), \Delta_s(\nu \setminus\lambda ) ) )$$ for all $n{\geqslant}2s$. When $s=|\nu|-|\lambda|$, the (skew) cell modules for partition algebras specialise to the usual Specht modules of the symmetric groups and we hence easily see that these stable coefficients coincide with the classical Littlewood–Richardson coefficients.
The action of the partition algebra
===================================
Understanding the action of the partition algebra on skew modules is difficult in general. In this section, we show that this can be done to some extent in the cases of interest to us. We have assumed that $|\mu|=s$, therefore the ideal $P_s(n)p_r P_s(n)\subset P_s(n)$ annihilates $\Delta_s(\mu)$ and this motivates the following definition.
We define the [Dvir radical]{} of the skew module $\Delta_s(\nu\setminus\lambda)$ by $${\sf DR}_s(\nu\setminus\lambda) = \Delta_s(\nu\setminus\lambda)P_s(n)p_rP_s(n)
\subseteq \Delta_s(\nu\setminus\lambda)$$ and set $$\Delta^0_s(\nu\setminus\lambda)=
\Delta_s(\nu\setminus\lambda) /{\sf DR}_s(\nu\setminus\lambda).$$ If $s=|\nu|-|\lambda|$, then set $ {\mathrm{Std}}^0_s(\nu\setminus\lambda) = {\mathrm{Std}}_s(\nu\setminus\lambda).
$ If $\lambda$ and $\nu$ are one-row partitions, then set $ {\mathrm{Std}}^0_s(\nu\setminus\lambda) \subseteq {\mathrm{Std}}_s(\nu\setminus\lambda)
$ to be the subset of paths, ${\mathsf{s}}$, whose steps are of the form $$r(1)=(-1,+0) \qquad d(1)=(-1,+1) \qquad a(1)=(-0,+1)$$ and such that the total number of boxes removed in ${\mathsf{s}}$ is less than or equal to $|\lambda|$.
Fix ${\mathsf{t}}\in {{\mathrm{Std}}}_r( \nu )$ and $1{\leqslant}k {\leqslant}r$ and suppose that $${\mathsf{t}}{(k-1)} \xrightarrow{-t} {\mathsf{t}}(k-\tfrac{1}{2}) \xrightarrow{+u} {\mathsf{t}}(k+1) \xrightarrow{-v} {\mathsf{t}}(k+\tfrac{1}{2}) \xrightarrow{+w} {\mathsf{t}}(k+1).$$ We define $ {\mathsf{t}}_{k \leftrightarrow k+1}\in {\mathrm{Std}}_r(\nu)$ to be the tableau, if it exists, determined by $ {\mathsf{t}}_{k \leftrightarrow k+1}(l) ={\mathsf{t}}(l) $ for $l\neq k, k \pm \tfrac{1}{2} $ and $${\mathsf{t}}_{k \leftrightarrow k+1} {(k-1)} \xrightarrow{-v} {\mathsf{t}}_{k \leftrightarrow k+1}{(k-\tfrac{1}{2})} \xrightarrow{+w}
{\mathsf{t}}_{k \leftrightarrow k+1}{(k)} \xrightarrow{-t} {\mathsf{t}}_{k \leftrightarrow k+1}(k+\tfrac{1}{2}) \xrightarrow{+u} {\mathsf{t}}_{k \leftrightarrow k+1}(k+1).$$ Let $(\lambda,\nu,s) $ be such that $s=|\nu|-|\lambda|$, or $\lambda$ and $\nu$ are both one-row partitions, then $\Delta^0_s(\nu\setminus\lambda)$ is free as a ${{\mathbb Z}}$-module with basis $$\{{\mathsf{t}}\mid {\mathsf{t}}\in {\mathrm{Std}}^0_s(\nu\setminus\lambda)\}$$ and the $P_s(n)$-action on $\Delta_s^0(\nu\setminus\lambda)$ is as follows: $$\label{co-case}
({\mathsf{t}}+ {\sf DR_s(\nu\setminus\lambda)} ) s_{k,k+1} =
\begin{cases}
{{\mathsf{t}}_{k\leftrightarrow k+1}} + {\sf DR_s(\nu\setminus\lambda)} &\text{if ${{\mathsf{t}}_{k\leftrightarrow k+1}}$ exists} \\
- {\mathsf{t}}+\sum_{{\mathsf{s}}\rhd {\mathsf{t}}} r_{{\mathsf{s}}{\mathsf{t}}}{\mathsf{s}}+ {\sf DR_s(\nu\setminus\lambda)} &\text{otherwise}
\end{cases}$$ for $1{\leqslant}k < s$ and $ ({\mathsf{t}}+ {\sf DR_s(\nu\setminus\lambda)} ) p_{k,k+1} =
0$ and $
( {\mathsf{t}}+ {\sf DR_s(\nu\setminus\lambda)} ) p_{k} =
0
$ for $1{\leqslant}k {\leqslant}s$. The coefficients $r_{{\mathsf{s}}{\mathsf{t}}}\in \mathbb{Q}$ are given in [@BDE Theorem 2.9].
The set ${\mathrm{Std}}^0((3,3)\setminus (2,1))$ has two elements $${\mathsf{t}}_1=a(1) \circ a(2)\circ a(2)
\qquad
{\mathsf{t}}_2=a(2) \circ a(1)\circ a(2).$$ These are depicted on the lefthand-side of \[actioner\]. We have that $$s_{1,2}=\left(\begin{array}{cc}0 & 1 \\1 & 0\end{array}\right)
\qquad
s_{2,3}=\left(\begin{array}{cc}1 & -1 \\0 & -1\end{array}\right).$$
$$\begin{tikzpicture}[scale=0.55]
\path (0,0)edge[decorate] node[left] {$-0$} (0,-2);
\path (0,-2)edge[decorate] node[right] {$+2$} (2,-4);
\path (0,-2)edge[decorate] node[left] {$+1$} (-2,-4);
\path (-2,-6)edge[decorate] node[left] {$-0$} (-2,-4);
\path (2,-6)edge[decorate] node[right] {$-0$} (2,-4);
\path (-2,-6)edge[decorate] node[left] {$+2$} (0,-8);
\path (2,-6)edge[decorate] node[right] {$+1$} (0,-8);
\path (0,0-8)edge[decorate] node[left] {$-0$} (0,-2-8);
\path (0,-2-8)edge[decorate] node[left] {$+2$} (0,-4-8);
\fill[white] (0,0) circle (17pt);
\begin{scope}
\fill[white] (0,0) circle (17pt);
\draw (0,0) node {$ \scalefont{0.4}\yng(2,1) $ };
\fill[white] (0,-2) circle (17pt); \draw (0,-2) node{$ \scalefont{0.4}\yng(2,1)$ };
\fill[white] (-2,-4) circle (17pt); \draw (-2,-4) node{$ \scalefont{0.4}\yng(3,1) $ };
\fill[white] (2,-4) circle (17pt); \draw (2,-4) node{$ \scalefont{0.4}\yng(2,2) $ };
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\fill[white] (0,-4-8) circle (17pt); \draw (0,-4-8) node{$ \scalefont{0.4}\yng(3,3) $ };
\end{scope}
\end{tikzpicture} \qquad\qquad
\begin{tikzpicture}[scale=0.55]
\path (0,0)edge[decorate] node[left] {$-1$} (-2,-2);
\path (0,0)edge[decorate] node[left] {$-0$} (2,-2);
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\fill[white] (0,0) circle (17pt);
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\[example3\] The set ${\mathrm{Std}}_3^0((4)\setminus(4))$ consists of the 7 oscillating tableaux $$\begin{array}{ccccc}
{\mathsf{s}}_1= r(1)\circ d(1)\circ a(1) &
{\mathsf{s}}_2= d(1)\circ r(1)\circ a(1) &
{\mathsf{s}}_3= r(1)\circ a(1) \circ d(1)
\\
{\mathsf{s}}_4= a(1)\circ r(1)\circ d(1) &
{\mathsf{s}}_5= d(1)\circ a(1) \circ r(1) &
{\mathsf{s}}_6= a(1)\circ d(1)\circ r(1) &
\\
& {\mathsf{s}}_7=d(1)\circ d(1) \circ d(1)
\end{array}$$ pictured in \[actioner\]. We have that $$s_{1,2}=
\left(\begin{array}{ccccccc}
\cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\
\cdot & \cdot & 1 & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & 1 & \cdot \\
\cdot & \cdot & \cdot & \cdot & 1 & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 \\\end{array}\right)
\qquad
s_{2,3}=
\left(\begin{array}{ccccccc}
\cdot & \cdot & 1 & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & 1 & \cdot & \cdot \\
1 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & 1 & \cdot \\
\cdot & 1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & 1 & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 1 \\\end{array}\right)$$ It is not difficult to see that this module decomposes as follows $$\Delta^0_3((4)\setminus(4))= 2 \Delta^0_3((3))
\oplus 2 \Delta^0_3((2,1))
\oplus \Delta^0_3((1^3)).$$
Semistandard Kronecker tableaux {#sec:semistandard}
===============================
For any $(\lambda,\nu,s) \in {\mathscr{P}_{r-s}}\times {\mathscr{P}_{{\leqslant}r}} \times {{\mathbb Z}}_{> 0} $ and any $\mu \vdash s$ we have $$\overline g( \lambda, \nu,\mu) = \dim_{\mathbb{Q}}{\operatorname{Hom}}_{P_s(n)}(\Delta_s(\mu), \Delta_s^0(\nu\setminus\lambda) ) = \dim_{{\mathbb{Q}}} {\operatorname{Hom}}_{{\mathbb{Q}}\mathfrak{S}_s}({\sf S}(\mu), \Delta_s^0(\nu \setminus \lambda)),$$ where ${\mathbb{Q}}\mathfrak{S}_s$ is viewed as the quotient of $P_s(n)$ by the ideal generated by $p_r$. Now for each $\mu = (\mu_1, \mu_2, \ldots , \mu_l) \vdash s$ we have an associated Young permutation module ${\sf M} (\mu) = {\mathbb{Q}}\otimes_{\mathfrak{S}_\mu} {\mathbb{Q}}\mathfrak{S}_s$ where $\mathfrak{S}_\mu = \mathfrak{S}_{\mu_1} \times \mathfrak{S}_{\mu_2}\times \dots\times \mathfrak{S}_{\mu_l} \subseteq \mathfrak{S}_s$. As a first step towards understanding the stable Kronecker coefficients, it is natural to consider $$\dim_{{\mathbb{Q}}} {\operatorname{Hom}}_{\mathfrak{S}_s}({\sf M} (\mu), \Delta^0_s(\nu \setminus \lambda) )$$ and to attempt to construct a basis in terms of semistandard (Kronecker) tableaux.
\[sstrd\] Let $(\lambda,\nu,s) \in \mathscr{P}_{r-s} \times \mathscr{P}_{{\leqslant}r} \times \mathbb{N}$ be a pair of one-row partitions or a triple of maximal depth. Let $\mu = (\mu_1, \mu_2, \ldots , \mu_l)\vdash s$ and let ${\mathsf{s}}, {\mathsf{t}}\in {\mathrm{Std}}^0_s(\nu \setminus \lambda)$.
1. For $1{\leqslant}k <s$ we write ${\mathsf{s}}\overset{k}{\sim} {\mathsf{t}}$ if ${\mathsf{s}}= {\mathsf{t}}_{k\leftrightarrow k+1}$.
2. We write ${\mathsf{s}}\overset{\mu}{\sim} {\mathsf{t}}$ if there exists a sequence of standard Kronecker tableaux ${\mathsf{t}}_1, {\mathsf{t}}_2, \ldots , {\mathsf{t}}_d \in {\mathrm{Std}}^0_s(\nu\setminus\lambda)$ such that $${\mathsf{s}}= {\mathsf{t}}_{1} \overset{k_1}{\sim} {\mathsf{t}}_{2} , \
{\mathsf{t}}_{2} \overset{k_2}{\sim} {\mathsf{t}}_{3} , \ \dots \ ,
{\mathsf{t}}_{d-1}\overset{k_{d-1}}{\sim} {\mathsf{t}}_{d}
={\mathsf{t}}$$ for some $k_1,\dots, k_{d-1}\in \{1, \ldots , s-1\} \setminus
\{ [\mu]_c \mid c = 1, \ldots , l-1 \}$. We define a [tableau of weight]{} $\mu$ to be an equivalence class of tableau under $\overset{\mu}{\sim} $, denoted $[{\mathsf{t}}]_\mu = \{ {\mathsf{s}}\in {\mathrm{Std}}^0_s(\nu\setminus \lambda) \, |\, {\mathsf{s}}\overset{\mu}{\sim} {\mathsf{t}}\}$.
3. We say that a Kronecker tableau, $[{\mathsf{t}}]_\mu$, of shape $\nu\setminus \lambda$ and weight $\mu$ is [semistandard]{} if for any ${\mathsf{s}}\in [{\mathsf{t}}]_\mu$ and any $ k \not \in \{[\mu_c] \mid c = 1, \ldots , l-1 \}$ the tableau ${\mathsf{s}}_{k\leftrightarrow k+1}$ exists. We let ${\mathrm{SStd}}_s^0(\nu\setminus \lambda, \mu)$ denote the set of semistandard Kronecker tableaux of shape $\nu\setminus \lambda$ and weight $\mu$.
To represent these semistandard Kronecker tableaux graphically, we will add frames’ corresponding to the composition $\mu$ on the set of paths ${\mathrm{Std}}_s^0(\nu \setminus \lambda)$ in $\mathcal{Y}$. For ${\mathsf{t}}=(-\varepsilon_{i_1}, + \varepsilon_{j_1}, \ldots , -\varepsilon_{i_s}, + \varepsilon_{j_s})$ we say that the integral step $(-\varepsilon_{i_k}, + \varepsilon_{j_k})$ belongs to the $c$th frame if $[\mu]_{c-1} < k{\leqslant}[\mu]_c$. Thus for ${\mathsf{s}}, {\mathsf{t}}\in {\mathrm{Std}}_s^0(\nu\setminus \lambda)$ we have that ${\mathsf{s}}\overset{\mu}{\sim} {\mathsf{t}}$ if and only if ${\mathsf{s}}$ is obtained from ${\mathsf{t}}$ by permuting integral steps within each frame (as in Figures \[anewfigforintro\] and \[maximaldepth\]).
\[YOUNGSRULE\] Let $(\lambda, \nu, s)$ be a co-Pieri triple and $\mu\vdash s$. We define $ \varphi_{\mathsf{T}}({{\mathsf{t}}^\mu}) = \sum_{{\mathsf{s}}\in {\mathsf{T}}}{\mathsf{s}}$ for ${\mathsf{T}}\in {\mathrm{SStd}}_s^0(\nu \setminus \lambda, \mu)$. Then $ {\operatorname{Hom}}_{\mathfrak{S}_s}({\sf M}(\mu), \Delta_s^0(\nu\setminus \lambda))$ has ${{\mathbb Z}}$-basis $ \{\varphi_{\mathsf{T}}\mid {\mathsf{T}}\in {\mathrm{SStd}}_s^0(\nu \setminus \lambda, \mu)\}$.
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(0,-18) circle (17pt); (0,-18.2) node[$ \scalefont{0.4}\yng(5,3,2)$ ]{}; (0,-20.2) circle (19pt); (0,-20.1) node[$ \scalefont{0.4}\yng(5,3,3)$ ]{};
\[semiexam2\] Let $\lambda =(4) $, $\nu =(4)$ and $s=5$ and $\mu=(2,2,1) \vdash {5}$. An example of a semistandard tableau, ${\mathsf{V}}$, of shape $\nu\setminus \lambda$ and weight $\mu$ is given by the rightmost diagram in Figure \[anewfigforintro\]. The semistandard tableau ${\mathsf{V}}$ is an orbit consisting of the following four standard tableaux $$\begin{aligned}
&{\mathsf{v}}_1= r(1) \circ d(1) \circ d(1) \circ a(1) \circ a(1)
\ \quad \
{\mathsf{v}}_2= d(1) \circ r(1) \circ d(1) \circ a(1) \circ a(1)
\ \quad \ \\
& {\mathsf{v}}_3= r(1) \circ d(1) \circ a(1) \circ d(1) \circ a(1)
\ \quad \
{\mathsf{v}}_4= d(1) \circ r(1) \circ a(1) \circ d(1) \circ a(1)
\end{aligned}$$ We have a corresponding homomorphism $
\varphi_{\mathsf{V}}\in {\operatorname{Hom}}_{\mathfrak{S}_s}({\sf M}(2,2,1), \Delta_s((4)\setminus (4))
$ given by $$\varphi_{\mathsf{T}}({\mathsf{t}}^{(2,2,1)})={\mathsf{v}}_1+{\mathsf{v}}_2+{\mathsf{v}}_3+{\mathsf{v}}_4.$$
The classical picture for semistandard Young tableaux
-----------------------------------------------------
We now wish to illustrate how our Definition \[sstrd\] and the familiar visualisation of a semistandard Young tableaux coincide for triples of maximal depth. Given $\lambda \vdash {r-s} , \nu \vdash {r}, \mu = (\mu_1, \mu_2, \ldots , \mu_\ell ) \vdash s $ such that $\lambda \subseteq \nu$ a Young tableau of shape $\nu\ominus \lambda$ and weight $\mu$ in the classical picture is visualised as a filling of the boxes of $[\nu\ominus \lambda]$ with the entries $$\underbrace{1, \dots, 1}_{\mu_1}, \underbrace{2,\dots, 2}_{\mu_2},
\ldots, \underbrace{\ell ,\dots, \ell }_{\mu_\ell }$$ so that they are weakly increasing along the rows and columns. One should think of this classical picture of a Young tableau of weight $\mu$ simply as a diagrammatic way of encoding an $\mathfrak{S}_\mu$-orbit of standard Young tableaux as follows. Let ${\mathsf{s}}$ be a standard Young tableau of shape $\nu\ominus \lambda$ and let $\mu$ be a partition. Then define $\mu({\mathsf{s}})$ to be the Young tableau of weight $\mu$ obtained from ${\mathsf{s}}$ by replacing each of the entries $[\mu]_{c-1} < i {\leqslant}[\mu]_c$ in ${\mathsf{s}}$ by the entry $c$ for $ c {\geqslant}1$. We identify a Young tableau, ${\mathsf{S}}$, of weight $\mu$ with the set of standard Young tableaux, $\mu^{-1}({\mathsf{S}})=\{{\mathsf{s}}\mid \mu({\mathsf{s}})={\mathsf{S}}\}$.
In either picture, a Young tableau of weight $\mu$ is merely a picture which encodes an $\mathfrak{S}_\mu$-orbit of standard Young tableaux. We picture a Young tableau, ${\mathsf{S}}$, of weight $\mu$ as the orbit of paths $\mu^{-1}({\mathsf{S}})$ in the branching graph with a frame to record the partition $\mu$.
A tableau of weight $\mu$ in the classical picture would be said to be semistandard if and only if the entries are strictly increasing along the columns. In our picture, this is equivalent to condition 3 of Definition \[sstrd\].
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\qquad\qquad
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\[semiexam1\] Let $\lambda =(2,1) $, $\nu =(3,3,2)$ and $s=5$. Then $(\lambda, \nu,s)$ is a triple of maximal depth. Take $\mu=(2,2,1) \vdash {5}$. The semistandard tableau ${\mathsf{U}}$ is an orbit consisting of the following four standard tableaux $$\begin{aligned}
&{\mathsf{u}}_1= a(1) \circ a(2) \circ a(2) \circ a(3) \circ a(3)
\ \quad \
{\mathsf{u}}_2= a(2) \circ a(1) \circ a(2) \circ a(3) \circ a(3)
\ \quad \ \\
& {\mathsf{u}}_3= a(1) \circ a(2) \circ a(3) \circ a(2) \circ a(3)
\ \quad \
{\mathsf{u}}_4= a(2) \circ a(1) \circ a(3) \circ a(2) \circ a(3)
\end{aligned}$$ pictured as follows $$\scalefont{0.9}
\Yboxdim{12pt}
\mu^{-1} \left( \; \Yvcentermath1 \young(\ \ 1,\ 12,23)\ \right) =
\left\{
\; \Yboxdim{12pt} \Yvcentermath1 \young(\ \ 1,\ 23,45)
\ \ , \ \
\Yboxdim{12pt} \Yvcentermath1 \young(\ \ 2,\ 13,45)
\ \ , \ \
\Yboxdim{12pt} \Yvcentermath1 \young(\ \ 2,\ 14,35)
\ \ , \ \
\Yboxdim{12pt} \Yvcentermath1 \young(\ \ 1,\ 23,45)
\ \right\}.
$$ We have a corresponding homomorphism $
\varphi_{\mathsf{U}}\in {\operatorname{Hom}}_{\mathfrak{S}_s}({\sf M}(2,2,1), \Delta_s((3,3,2)\setminus (2,1))
$ given by $$\varphi_{\mathsf{T}}({\mathsf{t}}^{(2,2,1)})={\mathsf{u}}_1+{\mathsf{u}}_2+{\mathsf{u}}_3+{\mathsf{u}}_4.$$ Compare this orbit sum over 4 tableaux with the picture in Figure \[maximaldepth\] and the statement of Theorem \[YOUNGSRULE\].
Let $\lambda =(2,1) $, $\nu =(3,3,2)$ and $s=5$. Then $(\lambda, \nu,s)$ is a triple of maximal depth. Take $\mu=(2,2,1) \vdash {5}$. The full list of semistandard tableaux (pictured in the classical fashion) are as follows $$\Yvcentermath1 \young(\ \ 1,\ 12,23)
\quad
\Yvcentermath1 \young(\ \ 1,\ 13,22)
\quad
\Yvcentermath1 \young(\ \ 1,\ 22,13)
\quad
\Yvcentermath1 \young(\ \ 2,\ 13,12)$$ The first two of these semistandard tableaux are pictured in our diagrammatic fashion in \[maximaldepth\].
Latticed Kronecker tableaux {#sec:latticed}
===========================
We now provide the main result of the paper, namely we combinatorially describe $$\overline{g}(\lambda, \nu, \mu) = \dim {\operatorname{Hom}}_{\mathfrak{S}_s}({\sf S}(\mu), \Delta_s^0(\nu\setminus \lambda))$$ for $(\lambda, \nu, \mu)$ a triple of maximal depth or such that $\lambda$ and $\nu$ are both one-row partitions. One can think of a path ${\mathsf{t}}\in {\mathrm{Std}}_s(\nu\setminus\lambda)$ as a sequence of partitions; or equivalently, as the sequence of boxes added and removed. We shall refer to a pair of steps, $(-\varepsilon_a,+\varepsilon_b)$, between consecutive integral levels of the branching graph as an [integral step]{} in the branching graph. We define [types]{} of integral step (move-up, dummy, move-down) in the branching graph of $P_r(n)$ and order them as follows, $$\begin{array}{ccccccccc}
&\text{move-up } & &\text{dummy } & &\text{move-down }
&
\\
& (-\varepsilon_p, + \varepsilon_q)&< & (-\varepsilon_t, + \varepsilon_t)
&< &(-\varepsilon_u, + \varepsilon_v)
\end{array}$$ for $p>q$ and $u< v$; we refine this to a total order as follows,
1. we order $(-\varepsilon_p, + \varepsilon_q)< (-\varepsilon_{p'}, + \varepsilon_{q'}) $ if $q<q'$ or $q=q'$ and $p>p'$;
2. we order $(-\varepsilon_t, + \varepsilon_t) < (-\varepsilon_{t'}, + \varepsilon_{t'}) $ if $t>t'$;
3. we order $(-\varepsilon_u, + \varepsilon_v)< (-\varepsilon_{u'}, + \varepsilon_{v'})$ if $u>u'$ or $u=u'$ and $v<v'$.
We sometimes let $a(i):={m{\downarrow}}(0,i)$ (respectively $r(i):={m{\uparrow}}(i,0)$) and think of this as [adding]{} (respectively [removing]{}) a box. We start with any standard tableau ${\mathsf{s}}\in {\mathrm{Std}}_s^0(\nu \setminus \lambda)$ and any $\mu = (\mu_1, \mu_2, \ldots , \mu_l)\vdash s$. Write $${\mathsf{s}}= (-\varepsilon_{i_1},
+\varepsilon_{j_1},
-\varepsilon_{i_2},
+\varepsilon_{j_2},
\dots
, -\varepsilon_{i_s},
+\varepsilon_{j_s}).$$ Recall from the previous section that, to each integral step $(-\varepsilon_{i_k}, + \varepsilon_{j_k})$ in ${\mathsf{s}}$, we associate its frame $c$, that is the unique positive integer such that $[\mu]_{c-1} < k {\leqslant}[\mu]_c.$
\[jdfhklssdhjhlashlfs\] We encode the integral steps of ${\mathsf{s}}$ and their frames in a $2\times s$ array, denoted by $\omega_\mu ({\mathsf{s}})$ (called the $\mu$-reverse reading word of ${\mathsf{s}}$) as follows. The first row of $\omega_\mu({\mathsf{s}})$ contains all the integral steps of ${\mathsf{s}}$ and the second row contains their corresponding frames. We order the columns of $\omega_\mu({\mathsf{s}})$ increasingly using the ordering on integral steps given in Definition 2.5. For two equal integral steps we order the columns so that the frame numbers are weakly decreasing. Given ${\mathsf{S}}\in {\mathrm{SStd}}_s^0(\nu\setminus \lambda, \mu)$, it is easy to see that $\omega_\mu ({\mathsf{s}})=\omega_\mu ({\mathsf{t}})$ for any pair ${\mathsf{s}},{\mathsf{t}}\in {\mathsf{S}}$ and so we define the $\mu$-reverse reading word, $\omega({\mathsf{S}})$, of ${\mathsf{S}}$ in the obvious fashion. For ${\mathsf{S}}\in {\mathrm{SStd}}_s^0(\nu\setminus \lambda, \mu)$ we write $$\omega({\mathsf{S}}) = (\omega_1({\mathsf{S}}), \omega_2({\mathsf{S}}))$$ where $\omega_1({\mathsf{S}})$ (respectively $\omega_2({\mathsf{S}})$) is the first (respectively second) row of $\omega({\mathsf{S}})$. Note that $\omega_2({\mathsf{S}})$ is a sequence of positive integers such that $i$ appears precisely $\mu_i$ times, for $i{\geqslant}1$.
\[semiexam3\] For $\lambda=(2,1)$ and $\nu=(3,3,2)$, the steps taken in the semistandard tableau ${\mathsf{U}}$ of Figure \[maximaldepth\] are $$a(1), a(2), a(2), a(3), a(3)$$ We record the steps according to the dominance ordering for the partition algebra ($a(1)< a(2) < a(3)$) and refine this by recording the frame in which these steps occur backwards, as follows $$\omega({\mathsf{U}})=
\left(\begin{array}{cccccccccccc}
a(1)&a(2)&a(2)&a(3) &a(3)
\\
1& 2& 1 & 3 &2
\end{array}\right).$$ For $\lambda=(4)$ and $\nu=(5)$, the steps taken in the semistandard tableau ${\mathsf{V}}$ on the right of Figure \[anewfigforintro\] are $$r(1), d(1), d(1), a(1), a(1).$$ We record the steps according to the dominance ordering for the partition algebra ($r(1)< d(1) < a(1)$) and we refine this by recording the frame in which these steps occur backwards, as follows $$\omega({\mathsf{V}})=\left(\begin{array}{cccccccccccc}
r(1)&d(1)&d(1)&a(1) &a(1)
\\
1& 2& 1 & 3 &2
\end{array}\right)$$ and notice that $\omega_2({\mathsf{U}})=\omega_2({\mathsf{V}})$. We leave it as an exercise for the reader to verify that the rightmost tableau depicted in Figure \[anewfigforintro\] has reading word $$\left(\begin{array}{cccccccccccc}
\ r(4) \ & \ r(1) \ & \ r(1) \ &m{\downarrow}(2,3) &m{\downarrow}(2,3)
\\
1& 2& 1 & 2&3
\end{array}\right).$$
For ${\mathsf{S}}\in {\mathrm{SStd}}_s^0(\nu\setminus \lambda, \mu)$ we say that its reverse reading word $\omega({\mathsf{S}})$ is a lattice permutation if $\omega_2({\mathsf{S}})$ is a string composed of positive integers, in which every prefix contains at least as many positive integers $i$ as integers $i+1$ for $i{\geqslant}1$. We define ${\mathrm{Latt}}_s^0(\nu \setminus \lambda, \mu)$ to be the set of all ${\mathsf{S}}\in {\mathrm{SStd}}_s^0(\nu\setminus \lambda, \mu)$ such that $\omega({\mathsf{S}})$ is a lattice permutation. For any co-Pieri triple $(\lambda, \nu, s)$ and any $\mu\vdash s$ we have $$\overline{g}(\lambda, \nu, \mu) = \dim_{{\mathbb{Q}}}{\operatorname{Hom}}_{\mathfrak{S}_s}({\sf S}(\mu), \Delta_s^0(\nu\setminus \lambda)) = |{\mathrm{Latt}}_s^0(\nu\setminus \lambda, \mu)|.$$
For example, we have that $$\overline{g}((2,1), (3,3,2), (2,2,1))= 1 =
\overline{g}((4),(4),(2,2,1))$$ and that the corresponding homomorphisms are constructed in Examples \[semiexam2\] and \[semiexam1\]. That these semistandard tableaux satisfy the lattice permutation property is checked in Example \[semiexam3\]. Verifying that these are the only semistandard tableaux satisfying the lattice permutation property is left as an exercise for the reader. Similarly, one can check that $ \overline{g}((7,5,1^2), (6,3,3), (2,2,1))= 1 $.
The (non-stable) Kronecker coefficients are also indexed by partitions. As we increase the size of the first row of each of the indexing partitions of the Kronecker coefficients, we obtain a weakly increasing sequence of coefficients; the limiting values of these sequences are the stable Kronecker coefficients which have been the focus of this paper. The non-stable Kronecker coefficients labelled by two 2-line partitions can be written as an alternating sum of at most 4 stable Kronecker coefficients labelled by two 1-line partitions [@BDE Proposition 7.6]. (In fact, any non-stable Kronecker coefficient can be written as an alternating sum of stable Kronecker coefficients.) This should be compared with the existing descriptions of Kronecker coefficients labelled by two 2-line partitions [@RW94; @Rosas01] which also involve alternating sums with at most 4 terms.
The advantages of our description are that $(1)$ ours is the first description that generalises to other stable Kronecker coefficients (and in particular the first description of any family of Kronecker coefficients subsuming the Littlewood–Richardson coefficients) and $(2)$ it counts explicit homomorphisms and therefore works on a higher structural level than all other descriptions of stable Kronecker coefficients since those first considered by Littlewood and Richardson [@LR34].
| ArXiv |
---
abstract: 'We discuss the underlying relativistic physics which causes neutron stars to compress and collapse in close binary systems as has recently been observed in numerical (3+1) dimensional general relativistic hydrodynamic simulations. We show that compression is driven by velocity-dependent relativistic hydrodynamic terms which increase the self gravity of the stars. They also produce fluid motion with respect to the corotating frame of the binary. We present numerical and analytic results which confirm that such terms are insignificant for uniform translation or when the hydrodynamics is constrained to rigid corotation. However, when the hydrodynamics is unconstrained, the neutron star fluid relaxes to a compressed nonsynchronized state of almost no net intrinsic spin with respect to a distant observer. We also show that tidal decompression effects are much less than the velocity-dependent compression terms. We discuss why several recent attempts to analyze this effect with constrained hydrodynamics or an analysis of tidal forces do not observe compression. We argue that an independent test of this must include unconstrained relativistic hydrodynamics to sufficiently high order that all relevant velocity-dependent terms and their possible cancellations are included.'
address:
- ' University of Notre Dame, Department of Physics, Notre Dame, IN 46556'
- ' University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550'
- ' University of Notre Dame, Department of Physics, Notre Dame, IN 46556'
author:
- 'G. J. Mathews and P. Marronetti'
- 'J. R. Wilson'
title: 'RELATIVISTIC HYDRODYNAMICS IN CLOSE BINARY SYSTEMS: ANALYSIS OF NEUTRON-STAR COLLAPSE'
---
INTRODUCTION {#sec:level1}
============
The physical processes occurring during the last orbits of a neutron-star binary are currently a subject of intense interest [@wm95]-[@sbs98]. In part, this recent surge in interest stems from relativistic numerical hydrodynamic simulations in which it has been noted [@wm95; @wmm96; @mw97] that as the stars approach each other their interior density increases. Indeed, for an appropriate equation of state, our numerical simulations indicate that binary neutron stars collapse individually toward black holes many seconds prior to merger. This compression effect would have a significant impact on the anticipated gravity-wave signal from merging neutron stars. It could also provide an energy source for cosmological gamma-ray bursts [@mw97].
In view of the unexpected nature of this neutron star compression effect and its possible repercussions, as well as the extreme complexity of strong field general relativistic hydrodynamics, it is of course imperative that there be an independent confirmation of the existence of neutron star compression before one can be convinced of its operation in binary systems. In view of this it is of concern that the initial numerical results reported in [@wm95; @wmm96; @mw97] have been called into question. A number of recent papers [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte; @sbs98] have not observed this effect in Newtonian tidal forces [@lai], first post-Newtonian (1PN) dynamics [@rs96; @wiseman; @shibata; @lombardi; @lw96; @sbs98], tidal expansions [@brady; @flanagan; @thorne97], or in binaries in which rigid corotation has been imposed [@baumgarte]. The purpose of this paper is to point out that none of these recent studies could or should have observed the compression effect which we observe in our calculations.
Moreover, this flurry of activity has caused some confusion as to the physics to which we attribute the effects observed in the numerical calculations. The present paper, therefore, summarizes our derivation of the physics which drives the collapse. We illustrate how such terms have been absent in some Newtonian or post-Newtonian approximations to the dynamics of the binary system. We also present numerical results and analytic expressions which demonstrate how the compression forces result in an orbiting dynamical system from the presence of fluid motion with respect to the corotating frame. As such, they could not appear in an an analysis of relativistic external tidal forces no matter how many orders are included in the tidal expansion parameter (e.g. [@flanagan; @thorne97]) unless self gravity from internal hydrodynamic motion is explicitly accounted for. The effect could also not arise in systems with uniform translation or rigid corotation.
The implication of the present study is that any attempt to confirm or deny the compression driving force requires an unconstrained, untruncated relativistic hydrodynamic treatment. At present, ours is still the only existing such calculation. Hence, despite claims to the contrary [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte; @sbs98], the neutron star compression effect has not yet been independently tested.
Another confusing aspect surrounding the numerical results has been our choice of a conformally flat spatial three-metric for the solution of the field equations. Indeed, it has been speculated that this approximate gauge choice (in which the gravitational radiation is not explicitly manifest) may have somehow led to spurious results. A second purpose of this paper, therefore, is to emphasize that the compression driving terms are a completely general result from the relativistic hydrodynamic equations of motion. The advantages of the conformally flat condition are that the algebraic form of the compression driving terms is easier to identify and that the solutions to the field equations obtain a simple form. It does not appear to be the case, however, that the imposition of a conformally flat metric drives the compression. It has been nicely demonstrated in the work of Baumgarte et al. [@baumgarte] that conformal flatness does not necessarily lead to neutron-star compression.
The Spatially Conformally Flat Condition
========================================
There has been some confusion in the literature as to the uncertainties introduced by imposing a conformally flat condition (henceforth [*CFC*]{}) on the spatial three-metric. Therefore we summarize here some attempts which we and others have made to quantify the nature of this approximation.
The only existing strong field numerical relativistic hydrodynamics results in three unrestricted spatial dimensions to date have been derived in the context of the [*CFC*]{} as described in detail in [@wm95; @wmm96; @mw97].
We begin with the usual ADM (3+1) metric [@adm62; @york79] in which there is a slicing of the spacetime into a one-parameter family of three-dimensional hypersurfaces separated by differential displacements in a timelike coordinate, $$ds^2 = -(\alpha^2 - \beta_i\beta^i) dt^2 +
2 \beta_i dx^i dt + \gamma_{ij}dx^i dx^j~~,
\label{metric}$$ where we take Latin indices to run over spatial coordinates and Greek indices to run over four coordinates. We also utilize geometrized units ($G = c = 1$) unless otherwise noted. The scalar $\alpha$ is called the lapse function, $\beta_i$ is the shift vector, and $\gamma_{i j}$ is the spatial three metric.
In what follows, we make use of the general relation between the determinant of the four metric $g_{\alpha \beta}$ and the ADM metric coefficients $$det(g_{\alpha \beta} ) = - \alpha^2 det({\gamma_{i j}}) \equiv \alpha^2 \gamma^2~~,$$ where $\gamma \equiv \sqrt{-det(\gamma_{i j})}$.
The conformally flat metric condition simply expresses the three metric of Eq. (\[metric\]) as a position dependent conformal factor $\phi^4$ times a flat-space Kronecker delta $$\gamma_{i j} = \phi^4 \delta_{ij}~~.$$
It is common practice (e.g. [@evans85; @cook93; @brugmann97]) to impose this condition when solving the initial value problem in numerical relativity. It is the natural choice for our three-dimensional quasiequilibrium orbit calculations [@wmm96] which in essence seek to identify a sequence of initial data configurations for neutron-star binaries.
The reason conformal flatness is chosen most frequently for the initial value problem is that it simplifies the solution of the hydrodynamics and field equations. The six independent components of the three metric are reduced to a single position dependent conformal factor.
Since conformal flatness implies no transverse traceless part of $\gamma_{i j}$ it can minimize the amount of initial gravitational radiation apparent in the initial configuration. However, in general the physical data still contain a small amount of preexisting gravitational radiation. This has been clearly demonstrated in numerical calculations of axisymmetric black-hole collisions [@smarr]. In exact numerical simulations, the gravitational radiation appears as the time derivatives of the spatial three metric ($\dot \gamma_{i j}$) and its conjugate (the extrinsic curvature $\dot K_{i j}$) are evolved. The immediate evolution of the fields from conformally flat initial data is characterized by the development of a weak gravity wave exiting the system.
An estimate of the radiation content of initial data slices for axisymmetric black hole collisions has been made by Abrahams [@abrahams]. Even for high values of momentum, the initial slice radiation is always less than about 10% of the maximum possible radiation energy (as estimated from the area theorem).
Two questions then are relevant to our application of the [*CFC*]{}. One is the validity of this metric choice for the initial value problem, and the other is the effect on the system of the “hidden” gravitational radiation in the physical data.
Regarding the validity of the [*CFC*]{} one has a great deal of freedom in choosing coordinates and initial conditions as long as the initial space is Riemannian and the metric coefficients satisfy the constraint equations of general relativity [@mtw]. Indeed, we have shown in [@wmm96] that exact solutions for the [*CFC*]{} metric coefficients can be obtained by imposing the ADM Hamiltonian and momentum constraint conditions. Nevertheless, in three dimensions a physical space is conformally flat if and only if the Cotton-York tensor vanishes [@kramer80; @eisenhart], $$C^{i j} = 2 \epsilon^{i k l}\biggl( R^j_{~k}
- {1 \over 4} \delta^j_{~k}R \biggr)_{;l}~~,
\label{cy}$$ where $R^j_{~k}$ is the Ricci tensor and $R$ is the Ricci scalar for the three space.
Equation (\[cy\]) vanishes by fiat for the three-space metric we have chosen. However, conformally flat solutions for physical problems have only been proven [@kramer80; @eisenhart] for spaces of special symmetry (e.g. constant curvature, spherical symmetry, time symmetry, Robertson-Walker, etc. [@kramer80]). Hence, the invocation of the CFC here and in other applications is an assumption. That is, it is a valid solution to the Einstein constraint equations, but does not necessarily describe a physical configuration to which two neutron stars will evolve. Nevertheless, this is a valid approximation as long as the nonconformal contributions from the $\dot \gamma_{i j}$ and $\dot K_{i j}$ equations in the exact two-neutron star problem remain small. Indeed, numerical tests for an axisymmetric rotating neutron star [@cookcfa] and a comparison of the [*CFC*]{} vs. an exact metric expansion for an equal-mass binary [@rs96] have indicated that conformal flatness is a good approximation when it can be tested.
As a related illustration, consider the Kerr solution for a rotating black hole. It is well known that the Kerr metric is not conformally flat. The close binaries we study have specific angular momentum only slightly greater than that of an extreme Kerr black hole. Also, they ultimately merge and collapse to a single Kerr black hole. Hence, an analysis of the Cotton-York tensor for a Kerr black hole is another indicator of the degree to which conformal flatness is a valid approximation for neutron-star binaries.
Figure \[fig1\]. gives the dimensionless scaled Cotton-York parameter $C^{\theta \phi} m^3$ for a maximally rotating Kerr black hole as a function of proper distance. For illustration, consider the decrease of this quantity as one moves away from the horizon at $m = r$ as a measure of the rate at which the metric becomes conformally flat. The maximally rotating ($a = m$) black hole of this example, however, is an extreme example of compactness and angular velocity relative to any of the neutron stars in our simulations.
It can be seen in figure \[fig1\] that, even for this extreme case, the dimensionless tensor coefficient $C^{\theta \phi} m^3$ diminishes rapidly away from the black hole. At the separation of interest for binary neutron stars approaching their final orbits ($r/m \sim 25$ where $m$ is the total binary mass and $r$ the separation between stars), this coefficient has already diminished to $\sim 10^{-3}$ of the value at the event horizon, ($r/m \sim 1$). Thus, the effect of either star on its companion is probably well approximated by conformal flatness. Regarding the interior of the neutron stars themselves, in our studies the stars are rotating so slowly (even when corotating) that the deviation from conformal flatness is probably negligible. Thus, it seems plausible that conformal flatness is a reasonable approximation for most physical aspects involving the spatial three-metric of binary neutron-star systems.
The next issue concerns the “hidden” radiation in the physical data. To address this we decompose the extrinsic curvature into longitudinal $K^{i j}_L$ and transverse $K^{i j}_T$ components as proposed by York [@york73], $$K^{ij} = K^{i j}_L + K^{i j}_T~~.
\label{kdecomp}$$ By definition the transverse part obeys $$D_i K^{i j}_T = 0~~,
\label{kt}$$ where $D_i$ are covariant derivatives. The longitudinal part can be derived from a properly symmetrized vector potential. We find $$D_i K^{i j}_L = 8 \pi S^i~~,
\label{kl}$$ where $S^i$ are spatial components of the contravariant four-momentum density.
The product $K^{i j}_T K_{Tij}$ is a measure of the hidden radiation energy density. To find $K^{i j}_T$ then from our numerical calculations, we first find $K_{i j}$ by choosing maximal slicing \[$Tr(K_{i j}) = 0$\] and requiring that the trace free part of the $\dot \gamma_{i j}$ equation vanish. This gives [@wmm96] $$2\alpha K_{ij} = (D_i \beta_j+D_j \beta_i -{2\over 3} \phi^{-4}
\delta_{ij} D_k \beta^k)~~.
\label{detweiler}$$ We then determine $K^{i j}_L$ from the equilibrium momentum density \[Eq. (\[kl\])\] and subtract $K^{i j}_L$ from $K^{i j}$.
We find that this measure of the “hidden” gravitational radiation energy density is a small fraction of the total gravitational mass energy of the system, $$\int K^{i j}_T K_{Tij} {dV\over 8 \pi} \approx 2 \times 10^{-5} ~
{\rm M_G}~~.$$ Hence, we conclude that the CFC is probably a good approximation to the initial data.
This should be an excellent approximation for the determination of stellar structure and stability. However, an unknown uncertainty enters if one attempts to reconstruct the time evolution of the system (e.g. the gravitational waveform) from this sequence of quasistatic initial conditions. At present we make this connection approximately via a multipole expansion [@thorne] for the gravitational radiation as described in [@wmm96].
An Electromagnetic Analogy
--------------------------
The meaning of imposing a conformally flat spatial metric can, perhaps, be qualitatively understood in an electromagnetic analogy. Both the ADM formulation of relativity and Maxwell’s equations can be written as two constraint equations plus two dynamical equations. In electromagnetism the constraint equations for electric and magnetic fields are embodied in the $\nabla \cdot E$ and $\nabla \cdot B$ equations, while the dynamical equations are contained in Ampere’s law and Faraday‘s law. In relativity the analogous constraint equations are the ADM momentum and Hamiltonian constraints. The dynamical equations are the ADM $\dot K_{i j}$ and $\dot \gamma_{i j}$ equations. In either electromagnetism or gravity, any field configuration which satisfies the constraint equations alone represents a valid initial value solution. However, one must analyze its physical meaning.
For example, consider two orbiting charges. One could construct an electric field which satisfies the constraint by simply summing over the electrostatic field from two point charges. Similarly, one can construct a static magnetic field from the charge current by imposing $\dot E = \dot B = 0$ in the dynamical equations. However, by forcing the dynamical equations to vanish, one has precluded the existence of electromagnetic radiation. In this field configuration, therefore one has unknowingly imposed ingoing radiation to cancel the outgoing electromagnetic waves.
Similarly, enforcing $\dot K_{i j}$ = $\dot \gamma_{i j} = 0$ might in part be thought of as implying the existence of ingoing gravitational radiation to cancel the outgoing gravity waves. Nevertheless, in both cases, this remains a good approximation to the physical system (with no ingoing wave) as long as the energy density contained in the radiation is small compared to the energy in orbital motion. Gravity waves enter in two ways: as estimated above there is an insignificant amount of “hidden” radiation induced by our choice of the CFC; there is also the emission of gravitational radiation by the orbiting binary system. The binary gravity-wave emission is estimated in our calculations by evaluating the multipole moments and using the appropriate formulas [@wmm96]. The fractional energy and angular momentum loss rate as determined by the multipole expansion method is quite small, e.g. $\dot J/\omega J \sim 10^{-4}$ in all of our calculations [@wmm96; @mw97]. Hence, it can be concluded that the energy in gravitational radiation is indeed small compared to the energy in orbital motion.
The emission of gravity waves also induces a reaction force which we have incorporated into our hydrodynamic equations by the quadrupole formula. The radiation reaction force is so small, however, that it is difficult to discern it in the numerical results. In most of our calculations we simply neglect the back reaction terms and thereby obtain quasistatic orbit solutions.
Solutions to Field Equations
----------------------------
With a conformally flat metric, the constraint equations for the field variables $\phi$, $\alpha$, and $\beta^i$ reduce to simple Poisson-like equations in flat space. The Hamiltonian constraint equation [@york79] for the conformal factor $\phi$ becomes [@wmm96; @evans85], $$\nabla^2{\phi} = -2\pi
\phi^5 \biggl[ (1+U^2) \sigma - P
+ {1 \over 16\pi} K_{ij}K^{ij}\biggr]~.
\label{phieq}$$ where $\sigma$ is the inertial mass-energy density $$\sigma \equiv \rho (1 + \epsilon) + P ~~,$$ and $\rho$ is the local proper baryon density which is simply related to the baryon number density $n$, $\rho = \mu m_\mu
n/N_A$, where $\mu$ is the mean molecular weight, $m_\mu$ the atomic mass unit, and $N_A$ is Avogadro’s number. [$\epsilon$]{} denotes the internal energy per unit mass of the fluid, and $P$ is the pressure. In analogy with special relativity we have also introduced a Lorentz-like variable $$\begin{aligned}
\biggl[ 1 + U^2\biggr]^{1/2} \equiv \alpha U^t &= &\biggl[ 1 + U^j U_j \biggr]^{1/2}
\nonumber \\
&&= \biggl[ 1 + \gamma^{i j} U_i U_j \biggr]^{1/2}~~,
\label{weq}\end{aligned}$$ where $U_i$ is the spatial part of the covariant four velocity. Here we explicitly write $U^2$ (in place of $W^2 - 1 $ used in [@wm95; @wmm96; @mw97]) because it emphasizes the extra velocity dependence here and in the equations of motion.
In the Newtonian limit, the r.h.s. of Eq. (\[phieq\]) is dominated [@wmm96] by the proper matter density $\rho$, but in relativistic neutron stars there are also contributions from the internal energy density $\epsilon$, pressure $P$, and extrinsic curvature. This Poisson source is also enhanced by the generalized curved-space Lorentz factor $(1+U^2)$. This velocity factor becomes important as the orbit decays deeper into the gravitational potential and the orbital kinetic energy of the binary increases.
It was pointed out in the appendix of [@mw97] that in analogy to the velocity-dependent enhancement of the source for Eq. (\[phieq\]), the Poisson source for the $v^4$ post-Newtonian correction to the effective potential also exhibits velocity-dependence. This appendix has been misinterpreted as a statement that we attribute the compression to a first post-Newtonian effect. We therefore wish to state clearly that the appendix in that paper was merely an illustration of how the effective gravity begins to show velocity dependence even in a post-Newtonian expansion. The velocity dependence of the post-Newtonian source is not the main compression driving force. The compression derives mostly from the hydrodynamic terms described herein. It is not obvious, however, at what post-Newtonian order the compression effect should be counted, since different authors have treated these terms differently. We return to this point below.
In a similar manner [@wmm96], the Hamiltonian constraint, together with the maximal slicing condition, provides an equation for the lapse function, $$\begin{aligned}
\nabla^2(\alpha\phi) = && 2 \pi
\alpha \phi^5 \biggl[ (3(U^2+1) \sigma \nonumber \\
&& - 2 \rho(1 + \epsilon) + 3 P
+ {7 \over 16\pi} K_{ij}K^{ij}\biggr]~~.
\label{alphaeq}\end{aligned}$$ Here again, the source is strengthened when the fluid is in motion through the presence of a $U^2+1$ factor and the $K_{ij}K^{ij}$ term.
The momentum constraints [@york79] provide an elliptic equation [@wmm96] for the shift vector, $$\nabla^2 \beta^i = {\partial \over \partial x^i} \biggl({1 \over 3}
\nabla \cdot \beta\biggr) + 4 \pi \rho_3^i~,
\label{wilson3}$$ $$\begin{aligned}
\rho_3^i & =& \biggl(4\alpha \phi^4 S_i - 4 \beta^i(U^2+1)\sigma \biggr)
\nonumber \\
&& {1 \over 4\pi} {\partial ln(\alpha/\phi^6) \over \partial x^j}
({\partial \over \partial x^j}\beta^i + {\partial \over \partial x^i} \beta^j
-{2\over 3}\delta_{ij} {\partial \over \partial x^k} \beta^k)~,
\label{betaeq}\end{aligned}$$ where we have introduced [@w79] the Lorentz contracted coordinate covariant momentum density, $$S_i = \sigma W U_i~~.
\label{momdef}$$
As noted previously and in Ref. [@wmm96], we only solve equation (\[wilson3\]) for the small residual frame drag after the dominant $\vec \omega \times \vec r$ contribution to $\vec \beta$ has been subtracted.
Relativistic Hydrodynamics
==========================
The techniques of general relativistic hydrodynamics have been in place and well studied for over 25 years [@w79]. The basic physical processes which induce compression can be traced to completely general terms in the hydrodynamic equations of motion. To illustrate this we first summarize the completely general derivation of the relativistic covariant momentum equation in Eulerian form and identify the terms which we believe to be the dominant contributors to the relativistic compression effect.
For hydrodynamic simulations it is convenient to explicitly consider two different spatial velocity fields. One is $U_i$, the spatial components of the covariant four velocity. The other is $V^i$, the contravariant coordinate matter three velocity, which is related to the four velocity $$V^i = { U^i \over U^t} =
{\gamma^{i j} U_j \over U^t} - \beta^i~~.
\label{threevel}$$ It is convenient to select the shift vector $\beta^i$ such that the coordinate three velocity vanishes when averaged over the star, $\langle V^i\rangle = 0$. This minimizes coordinate fluid motion with respect to the shifting ADM grid.
The perfect fluid energy-momentum tensor is $$T_{\mu \sigma} = \sigma U_\mu U_\sigma
+ P g_{\mu \sigma}~~.$$ However, it is convenient to derive the hydrodynamic equations of motion using the mixed form, $$T_\mu^{~\nu} =
g^{\sigma \nu} T_{\mu \sigma} = \sigma U_\mu U^\nu + P \delta_{\mu}^{~\nu}~~,$$
The vanishing of the spatial components of the divergence of the energy momentum tensor $$\biggr(T_i^{~\mu}\biggl)_{;\mu} = 0$$ leads to an evolution equation for the spatial components of the covariant four momentum, $$\begin{aligned}
{1\over\alpha \gamma}{\partial (S_i \gamma )\over\partial t} +&&
{1\over\alpha \gamma}{\partial (S_i V^j\gamma )\over\partial x^j}
+{\partial P\over \partial x^i} \nonumber \\
&& + {1 \over 2} {\partial g^{\alpha \beta}\over\partial
x^i} { S_\alpha S_\beta \over S^t} = 0
~~.
\label{divmom}\end{aligned}$$ The covariant momentum equation is used because of its close similarity with Newtonian hydrodynamics. The first two terms are advection terms familiar from Newtonian fluid mechanics. The latter two terms are the pressure and gravitational forces, respectively.
Expanding the gravitational acceleration into individual terms we have $$\begin{aligned}
{\dot S_i}& + & S_i{\dot \gamma \over\gamma}
+{1\over\gamma}{\partial\over\partial x^j}(S_iV^j\gamma)
+ {\alpha \partial P\over \partial x^i}
- S_j {\partial \beta^j \over \partial x^i}
\nonumber \\
& + & \sigma {\partial \alpha \over \partial x^i}
+ \sigma \alpha \biggl( U^2 {\partial \ln{\alpha} \over \partial x^i}
+ {U_j U_k \over 2 } {\partial \gamma^{j k}
\over \partial x^i}\biggr) = 0~~.
% - 2 {\partial \ln{\phi}\over\partial x^i} \biggr) = 0
%&+ & \sigma\biggl((1+U^2){\partial \alpha \over \partial x^i}
%+ \alpha {U_j U_k \over 2 } {\partial \gamma^{j k}
%\over \partial x^i}\biggr) = 0~~.
\label{hydromom}\end{aligned}$$
Similar forms can be derived for the condition of baryon conservation and the evolution of internal energy [@wmm96; @w79]. However, the above momentum equation is sufficient for the present discussion.
It is now worthwhile to consider the ”gravitational” forces embedded in the expanded terms of Equation (\[hydromom\]). These result from the affine connection terms $\Gamma^\mu_{\mu \lambda} T^{\mu \lambda}$ in the covariant differentiation of $ T^{\mu \nu}$.
The term containing $\partial \alpha/\partial x^i$ comes from the time-time component of the covariant derivative. It is of course the well known analog of the Newtonian gravitational force as can easily be seen in the Newtonian limit $\alpha \rightarrow 1 - Gm/r$.
The term $S_j (\partial \beta^j /\partial x^i)$ comes from the space-time covariant derivative. In an orbiting system it is convenient to allow $\beta^j$ to follow the orbital motion of the stars. In our specific application [@wmm96] we let $\vec \beta = \vec \omega \times
\vec R + \vec \beta_{resid}^{drag}$ where $\omega$ is chosen to minimize matter motion on the grid. Hence, $\vec \omega \times
\vec R$ includes the major part of rotation plus frame drag. The quantity $\vec \beta_{resid}^{drag}$ is the residual frame drag after subtraction of rotation and is very small for the almost nonrotating stars which result from our calculations. With $\beta^j$ dominated by $ \vec \omega \times
\vec R $, the term $S_j (\partial \beta^j /\partial x^i)$ is predominantly a centrifugal force.
The $U^2 \partial \ln{\alpha} / \partial x^i$ term arises from the time-time component of the affine connection piece of the covariant derivative. The $(U_j U_k /2 ) \partial \gamma^{j k}
/\partial x^i$ term similarly arises from the space-space components. They do not have a Newtonian analog. As we shall see, these terms cancel when a frame can be chosen such that the whole fluid is at rest with respect to the observer (or in the flat space limit). However, for a star with fluid motion in curved space, they describe additional velocity-dependent forces.
We identify the nonvanishing combination of these $U^2$-dependent force terms and the $S_j (\partial \beta^j /\partial x^i)$ term as the major contributors to the net compression driving force.
This suggests some useful test problems for our hydrodynamic simulations. For example, in simple uniform translation the effects of these terms must cancel to leave the stellar structure unchanged. Similarly, as discussed below, any fluid motion such that the four velocity can be taken as proportional to a simple Killing vector (e.g. rigid corotation) these force terms must cancel [@baumgarte; @kramer80]. However, for more general states of motion, e.g. noncorotating stars, differential rotation, meridional circulation, turbulent flow, etc., these forces do not obviously cancel, but must be evaluated numerically.
Indeed, as discussed below, the sign of these terms is such that a lower energy configuration for the stars than that of rigid corotation can be obtained by allowing the fluid to respond to these forces. As we shall see, the numerical relaxation of binary stars from corotation (or any other initial spin configuration) produces a nonsynchronous (approximately irrotational) state of almost no intrinsic neutron-star spin in which the central density and gravitational binding energy increase.
Conformally Flat Relativistic Hydrodynamics {#hydro}
-------------------------------------------
The practical implementation of conformal flatness means that, given a distribution of mass and momentum on some manifold, we first solve the constraint equations of general relativity at each time for a given distribution of mass-energy. We then evolve the hydrodynamic equations to the next time step. Thus, at each time slice we obtain a solution to the relativistic field equations and then can study the hydrodynamic response of the matter to these fields [@wmm96].
For the [*CFC*]{} metric, the relativistic momentum equation is derived by simply replacing $\gamma^{j k} \rightarrow \phi^{-4}
\delta^{j k}$ in Eq. (\[hydromom\]). $$\begin{aligned}
{\partial S_i\over\partial t}& +& 6 S_i{\partial \ln\phi\over\partial t}
+{1\over\phi^6}{\partial\over\partial x^j}\biggl(\phi^6S_iV^j\biggr)
+\alpha{\partial P\over \partial x^i}
- S_j {\partial \beta^j \over \partial x^i}
\nonumber \\
& + & \sigma {\partial \alpha \over \partial x^i}
+ \sigma \alpha U^2 \biggl( {\partial \ln{\alpha} \over \partial x^i}
- 2 {\partial \ln\phi\over\partial x^i} \biggr) = 0~~.
\label{hydromomcfa}\end{aligned}$$ Here as in Eq. (\[hydromom\]), the first term with ${\partial \alpha/\partial x^i}$ is the relativistic analog of the Newtonian gravitational force.
In Eq. (\[hydromomcfa\]) there are two ways in which the effective gravitational force might increase for finite $U^2$. One is that the matter contribution to the source densities for $\alpha$ or $\phi$ are increased by factors of $\sim 1 + U^2$ \[cf. Eqs. (\[phieq\]) and (\[alphaeq\])\]. The more dominant effect, however, is from the combination of the $S_j {\partial \beta^j /\partial x^i}$ term and the $U^2 [\partial \ln{\alpha} / \partial x^i -
2\partial \ln{\phi}/\partial x^i]$ terms in Eq. (\[hydromomcfa\]).
As noted previously, these compression driving terms result from the affine connection part $\Gamma^\mu_{\mu \lambda} T^{\mu \lambda}$ of the covariant differentiation of $ T^{\mu \nu}$. These terms have no Newtonian analog but describe a general relativistic increase in the gravitational force as $U^2$ increases. As noted in [@wmm96; @mw97] (see also Fig \[fig2\] below) for a binary, $U^2$ is approximately uniform over the stars, and the increase in central density due to these additional forces scales as $\approx U^4$. This scaling, however, is the net result from a nontrivial cancellation of terms and must be treated carefully. We shall return to this point below.
The proper way to determine the post-Newtonian order at which the compression driving terms enter would be to count the powers of $c^2$ which appear in the denominator of a term. For example, if we divide the last two terms in Eq. (\[hydromomcfa\]) by the gradient of the $\alpha$ term (the analog of the Newtonian gravitational force) we would obtain a ratio of order $U^2/c^2$ which would be manifestly first post Newtonian. However, in the first post-Newtonian treatment of Wiseman [@wiseman], these velocity terms were explicitly disregarded. Thus, the effects of these terms could not have been present in that calculation. It is no surprise, therefore that no effect was observed in Ref. [@wiseman].
Also note that the $2{\partial \ln{\phi}/\partial x^i}$ term in Eq. (\[hydromomcfa\]) enters with a sign such that the total $U^2$-dependent contribution is further increased by about twice that from the ${\partial \ln{\alpha}/\partial x^i}$ contribution alone. (The factor of 2 in front of the derivative comes from the requirement that $\phi^2 \sim (1/\alpha)$ in the Newtonian limit [@wmm96].)
A further increase of binding arises from the $K^{ij}K_{ij}$ terms in the field sources, but these terms are much smaller than the $U^2$ contributions for a binary system.
Comment on the Relativistic Bernoulli Equation
----------------------------------------------
For comparison with other work in the literature it is instructive to discuss the derivation of the relativistic Bernoulli equation from equation (\[hydromom\]). It has been pointed out (e.g. [@baumgarte]) that the hydrodynamics reduces to a simple equation for a fluid in which the velocity field can be represented by a Killing vector. In our notation this equation can be written, $$d \ln{(U^t)} = {dP \over \sigma}~~.
\label{bernoulli}$$
The demonstration that the relativistic Bernoulli equation (\[bernoulli\]) is exactly reproduced from Eq. (\[hydromom\]) and indeed for any case in which a Killing vector can be imposed, was recently brought to our attention by T. Nakamura [@nakamura]. We summarize the derivation here in the conformally flat metric both for clarity and to show that conformal flatness does not violate this important constraint.
To begin with, note that in the ADM formalism, the existence of a Killing vector is equivalent to being able to choose a shifted ADM grid such that $V^i = 0$ everywhere for the fluid. Next use Eq. (\[threevel\]) to solve for $\beta^i$ and divide by $\sigma$. The resulting equation for stationary motion is $${1 \over \sigma} {\partial P\over \partial x^i} = U^t U_i {\partial \over \partial x^i}
\biggl({U_i \over \phi^4 U^t}\biggr)
- (\alpha U^t)^2 {\partial ln{\alpha} \over \partial x^i}
+ 2 U^2 {\partial \ln{\phi} \over \partial x^i} ~~.
\label{bernoulli1}$$ The recovery of the relativistic Bernoulli equation requires that the r.h.s. $= \partial \ln{U^t}/\partial x^i$. With some straightforward algebraic manipulation it is possible to show that all of the terms on the r.h.s. cancel except for one term from the $\beta$ derivative, $-\phi^{-4} U^2 \partial \ln{U^t}/\partial x^i$. The completion of the proof is simply to note that this term is equal to $\partial \ln{U^t}/\partial x$ by Eq. (\[weq\]). The result is Eq. (\[bernoulli\]).
It is instructive to consider the change in the relativistic Bernoulli equation when there is no Killing vector, i.e. $V_i \ne 0$. Along the same lines of the derivation of Eq. (\[bernoulli1\]), It can be shown [@nakamura] that the momentum equation can be rewritten in our notation as, $$\begin{aligned}
{1 \over \alpha \sigma}\biggl[{\dot S_i}& + & S_i{\dot \gamma \over\gamma}
+{1\over\gamma}{\partial\over\partial x^j}(S_iV^j\gamma)\biggr]
+ U^t U_j {\partial V^j \over \partial x^i} \nonumber \\
& =&
-{1 \over \sigma} {\partial P\over \partial x^i}
+ {\partial ln U^t \over \partial x^i}~~.
\label{hydromomb}\end{aligned}$$ The r.h.s. is just the relativistic Bernoulli equation in the limit that the l.h.s. vanishes. In general fluid flow, however, the l.h.s. contains not only the advection terms (in brackets), but also an additional surviving part of the $\beta^j$ derivative.
It can be seen from this that imposing a Killing vector ($V^i = 0$) means that only simple hydrostatic equilibrium is obtained for stationary systems. However, when nontrivial hydrodynamic motion is allowed, the extra forces embodied in the l.h.s. of Eq. (\[hydromomb\]) are manifest. The presence of fluid motion not represented by a simple Killing vector, thus leads to a deviation from the simple relativistic Bernoulli solution. Any attempt to model this deviation requires a careful treatment of the dynamical properties of the fluid described by the l.h.s. of Eq. (\[hydromomb\]).
CONSTRAINED HYDRODYNAMICS
=========================
Further insight into the complexity of the physics contained in the relativistic equations of motion can be gained by considering some simple examples of constrained hydrodynamics for which the answer is known. These pose useful tests of our numerical scheme. Since some have proposed that the effect we observe may be an artifact of numerical resolution or approximation, we present here a summary of various test problems designed to illustrate the stability of the numerics and also to compare with some of the calculations in the literature. These calculations also demonstrate that the compression effect vanishes in the limiting cases which have been studied by others. Hence, they could not have been observed. They highlight the fact that the effect we observe only appears in a strong field dynamic treatment which accounts for internal motion of stellar material in response to the binary and its effect on the star’s self gravity. At present, ours may be the only existing result. This is consistent with the conclusion of [@shapiro] based upon test particle dynamics.
Bench Mark Calculations
-----------------------
To test for the presence of the compression driving forces we consider two bench-mark initial calculations. The bench mark of no compression is that of an isolated star. In our three dimensional hydrodynamic calculations, the single star structure is derived from Eq. (\[hydromom\]) in the limit $$S_i = U_i = V^i = \beta^i = 0~~.$$ The condition of hydrostatic equilibrium in isotropic coordinates is then trivially derived from Eq. (\[hydromom\]) $${\partial P\over \partial x^i} = - \sigma {\partial \ln{
\tilde \alpha} \over \partial x^i} ~~,
\label{static}$$ where the tilde denotes that the metric coefficients are evaluated in the fluid rest frame. The Newtonian limit of the right hand side is recovered as $\tilde \alpha \rightarrow 1 - G m/r$. Hence, we again identify the ${\partial \ln{\tilde \alpha} / \partial x^i}$ term with the relativistic analog of the Newtonian gravitational force. Eq. (\[static\]) also trivially reproduces to relativistic Bernoulli equation (\[bernoulli\]).
We have of course tested our three dimensional calculations for single isolated stars. A single star remains stable on the grid indefinitely, except when baryon mass exceeds the maximum stable mass allowed by the TOV equations. Above the maximum TOV mass the stars begin to collapse on a dynamical timescale as they should. We have also checked that the grid resolution used in our binary calculations is adequate to produce the correct central density, stellar radius, and gravitational mass of a single isolated star [@mw97]. Hence, it seems unlikely that inadequate grid resolution is the source of the compression effect as some have proposed.
In order to facilitate comparisons with the literature, and to avoid confusion over equation of state (EOS) issues, we have employed a simplistic $\Gamma=2$ polytropic EOS, $P = K\rho^\Gamma$, where $K= 1.8 \times 10^5$ erg cm$^3$ g$^{-2}$. This gives a maximum neutron-star mass of 1.82 $M_\odot$. The gravitational mass of a single $m_B = 1.625$ $M_\odot$ star in isolation is 1.51 $M_\odot$ and the central density is $\rho_c =5.84\times 10^{14}$ g cm$^{-3}$. The compaction ratio is $m/R = 0.15$, similar to one of the stars considered in [@baumgarte]. Note that this EOS leads to stars with a lower compaction ratio than the stars we considered in [@wmm96; @mw97] for which $m/R \approx 0.2$. Hence, the effects of tidal forces in the present calculations should be more evident.
The bench mark in which compression is present is that of two equal mass stars in a binary computed with unconstrained hydrodynamics. The binary stars have the same baryon mass ($m_B = 1.625$ $M_\odot$ each), the same EOS, and a fixed angular momentum $J = 2.5 \times 10^{11}$ cm$^2$ ($J/M_B^2 = 1.09$ where M$_B = 2 m_B$). For these conditions the binary stars have $U^2 = 0.025$ and are at a coordinate separation of $\approx 100$ km. The stars are stable but close to the collapse instability. Hence, they have experienced some compression which has increased their central density by 14% up to $\rho_c = 6.68\times 10^{14}$ g cm$^{-3}$.
The central densities of these two bench marks are summarized in the first and last entries of Table \[table1\]. To compare with these bench-mark calculations we have computed equilibrium configurations for stars under the various conditions outlined below. The test for the presence or absence of compression inducing forces will be the comparison of the numerically computed central density with that of a single isolated star or stars in a binary.
Stars in Uniform Translation {#linear}
----------------------------
As a first nontrivial test, now consider a star as seen from an observer in an inertial frame which is in uniform translation with respect to the fluid. Choosing motion along the $x$ coordinate, the fluid three velocity is, $${ U^x \over U^t} = V^x = {\rm Constant} ~~.$$ However, the observer is still free to choose the ADM shift vector such that the computational grid remains centered on the star. That is, although $S_i, U_i, \ne 0$, we can still choose $V^i = 0$. This gives a restriction on $\beta^i$ from Eq. (\[threevel\]), $$\beta^x = {\gamma^{x x} \alpha U_x \over W} ~~.
\label{betadef}$$ Note, that this is an ADM coordinate freedom. It is not equivalent to a coordinate boost. It is in fact a Killing vector which is convenient for numerical hydrodynamics. It allows the matter to remain centered on the grid even though the equations of motion are being solved for fluid which is not at rest with respect to the observer.
With $V^i = 0$, the $x$ component of the momentum equation (in equilibrium) becomes, $${\partial P\over \partial x} =
{S_x \over \alpha} {\partial \beta^x \over \partial x}
- \sigma \biggl[(U^2 + 1)
{\partial ln{\alpha} \over \partial x}
+ {U_j U_k \over 2 } {\partial {\gamma^{j k}} \over \partial x} \biggr]~~.$$ With a [*CFC*]{} metric this becomes, $${\partial P\over \partial x} = {S_x \over \alpha} {\partial \beta^x \over \partial x}
- \sigma \biggl[(U^2 + 1)
{\partial ln{\alpha} \over \partial x}
- 2 U^2 {\partial \ln{\phi} \over \partial x} \biggr]~~.
\label{hydrostatcfa}$$
There are now several differences between this expression and that for an observer in the fluid rest frame. For one, there is the shift vector derivative $\partial \beta^x /\partial x$. Even in uniform translation this derivative is nonzero due to the variations of the metric coefficients over the star (cf. Eq. \[betadef\]). Also, the effective gravity is enhanced by the $(U^2 + 1)$ velocity factor. The ${U_j U_k / 2 } {\partial {\gamma^{j k}} / \partial x^i}$ term also appears. In addition, the effective source terms (\[phieq\]) and (\[alphaeq\]) for the [*CFC*]{} metric coefficients are enhanced both by $(U^2 + 1)$ factors and the $K_{i j} K^{i j}$ term.
In spite of these differences, we nevertheless know that the locally determined pressure and inertial density must be the same as those determined for a star at rest. Indeed, since we can choose a Killing vector ($V^i = 0$) these equations must reduce to the relativistic Bernoulli equation (\[bernoulli\]).
Thus, this is an important numerical test problem. We solve the full hydrodynamic equations explicitly, e.g. Eq. (\[hydromomcfa\]), under the initial condition of nonzero $U_x$ for a single star. The cancellations embedded in the hydrodynamics are not obvious. Nevertheless, in the end, all of these effects must cancel to leave the stellar central density unchanged (except for a Lorentz contraction factor).
To solve the uniform translation problem numerically we have applied the Hamiltonian and momentum constraints to determine the metric coefficients. We then evolved the full hydrodynamic equations to equilibrium. Figure \[fig2\] shows the numerically evaluated central density for such translating stars as a function of $U^2$. These stars were calculated with the EOS of [@wmm96]. This is compared with the central density for binary stars evolved at the same $U^2$ value using the same EOS. One can see that the translating stars maintain a constant central density (within numerical error) as they should. In contrast, the central density of binary stars grows as $\approx U^4$. This growth is the nontrivial net result from the velocity dependent terms in Eq. (\[hydromomcfa\]). It is not obvious, however, to what post-Newtonian order this dependence corresponds.
As summarized in Table \[table1\], the central density for a uniformly translating $\Gamma=2$ star with $U^2 = 0.025$ (for comparison with the binary bench-mark calculation) is $5.90 \times 10^{14}$ g cm$^{-3}$. Within numerical accuracy, this central density is identical with that of an isolated star at rest.
This is at least indicative that our observed growth in central density may not be numerical error as some have suggested (e.g. [@brady; @thorne97; @shapiro]). Such an error would likely be apparent in this test case. We argue that the difference between simple translation and binary orbits relates to the physics of the binary system itself, in particular physics which is not apparent in uniform translation, an analysis of tidal forces, or a truncated expansion which does not contain sufficient terms to adequately describe the dynamical response of the fluid.
Tidal Forces
------------
It has been pointed out [@lai; @flanagan; @thorne97] that tidal forces are in the opposite sense to the compression driving forces discussed here. That is, tidal forces distort the stars and decrease the central density and therefore render the stars less susceptible to collapse. We have argued [@mw97] that although such stabilizing forces are present in our calculations they are much smaller in magnitude than the velocity-dependent compression driving terms. Nevertheless, the evolution of the matter fields in a calculation in which only tidal forces are present still represents a useful test of our numerical results. Stars in which only tidal forces act, should be stable and the central density should decrease rather than increase as the stars approach.
To test the effects of tidal forces alone we have constructed an artificial test calculation in which we place stars on the grid in a binary, but with no initial angular or linear momentum, i.e. $J = 0$ and $U^2 = 0$. This initial condition would normally evolve to an axisymmetric collision between the stars. However, after updating the matter fields, we artificially return the center of mass of the stars to the same fixed separation after each time step. We also reset to zero the mean velocity component directed along the line between centers. This sequence is repeated until the matter fields come to equilibrium. Since the velocity dependent forces eventually vanish, the only remaining forces are the pressure and static gravitational (including tidal) forces.
Results as a function of separation distance are shown in Table \[table2\] for the $\Gamma=2$ EOS and Table \[table3\] for the realistic EOS used in [@wmm96]. For the realistic EOS the central density indeed decreases as the stars approach consistent with the expectations from Newtonian and relativistic tidal analyses [@lai; @thorne97]. For the $\Gamma=2$ polytropic EOS, the central density also decreases as the stars approach and remain below the central density of an isolated star. The fact that this table is not monotonic at the innermost point, however, may be due to a limitation of this numerical approximation for tidal forces as the ratio of separation to neutron-star radius diminishes.
Although the tidal forces do indeed stabilize the stars, their effect on the central density is quite small ($\sim 0.2\%$ decrease) compared to the net increase in density caused by the compression forces present for the binary. This is consistent with the relative order-of-magnitude estimates for these effects described in [@mw97].
Stars in Rigid Corotation
-------------------------
As a next nontrivial example, consider stars in a binary system which are restricted to rigid corotation. In a recent series of papers, Baumgarte et al. [@baumgarte] have studied neutron-star binaries using the same conformally flat metric. Their work differs from ours in that rather than solving the hydrodynamic equations, they describe the four velocity field by a Killing vector whereby the stars are forced to corotate rigidly. They also impose spatial symmetry in the three Cartesian coordinate planes so that they can solve the problem in only one octant. One should keep in mind, however, that rigid corotation is not necessarily the lowest energy configuration or the most natural [@bc92] final state for two neutron stars near their final orbits. This assumption, though artificial, is nevertheless a means to constrain and simplify the fluid motion degrees of freedom. It is much easier to implement and therefore becomes an interesting test problem for codes seeking to explore the true hydrodynamic evolution of close binaries.
Indeed, it is possible to show [@kramer80] that in this limit, the neutron star hydrostatic equilibrium can be described by a simple Bernoulli equation in which the compression driving force terms are absent except for a weak velocity dependence. Analytically, the reason for this is trivially obvious from Eq. (\[hydromomb\]). The existence of a Killing vector is equivalent to setting $V^i = 0$ globally. Choosing the ADM coordinates to remain centered on the stars, in steady state the time derivatives vanish along with the rest of the l.h.s of Eq. (\[hydromomb\]). Only the relativistic Bernoulli equation (\[bernoulli\]) survives.
It is not surprising, therefore that in [@baumgarte] it has been demonstrated that in this special symmetry, the central density of the stars does not increase (within numerical error) as the stars approach the inner most stable circular orbit. In very close orbits the density actually decreases relative to the central density of stars at large separation. They also find that the orbit frequency remains close to the Newtonian frequency. Both of these results are interesting in that they confirm that the compression effect does not occur (as it should not) in this special symmetry. They also demonstrate that conformal flatness is not the source of the compression.
Accepting the results of [@baumgarte] as correct, this then becomes another important test of our calculations. That is, if we artificially impose rigid corotation, then the central density should remain nearly constant until the stars are close enough that tidal effects cause the central density to decrease rather than increase.
Imposing rigid corotation, however, is not a trivial test problem to implement without completely replacing the hydrodynamic equations with the corresponding Bernoulli solution of [@baumgarte; @kramer80]. (Indeed, we have done this [@marronetti] and reproduce the results of [@baumgarte] quite well). Moreover, we have found that directly modifying the hydrodynamic equations in an attempt to mimic a dynamically unstable configuration is difficult. One might think that the simplest way to implement corotation would be to impose a high fluid viscosity. Indeed high viscosity would resist the hydrodynamic forces described herein. However, a high fluid viscosity also resists the much weaker tidal forces and prevents the numerical relaxation to quasistatic equilibrium. It is thus difficult to achieve tidal locking by simply increasing the viscosity.
Instead, we introduce artificial forces on the fluid which continually drive the system toward a state of rigid corotation while allowing the system to at least somewhat respond hydrodynamically. To do this we define accelerations $(\dot U_i)_{\rm Rigid}$ necessary to achieve rigid rotation by $$(\dot U_i)_{\rm Rigid} \equiv {(\tilde U_i - U_i) \over \Delta t}$$ where $\tilde U_i$ are components of the rigidly corotating covariant four velocity in the $ x- y$ orbit plane. These are determined by requiring that $\beta^i = (\omega \times r)^i$ and setting $V^i = 0$ in Eq. (\[threevel\]). $$\tilde U_y = {\omega x \phi^4 \over \alpha
\sqrt{1 - \omega^2 R^2 \phi^4/\alpha^2 }}~~,$$ $$\tilde U_x = {-\omega y \phi^4 \over \alpha
\sqrt{1 - \omega^2 R^2 \phi^4/\alpha^2 }}~~,$$ where $R$ is the coordinate distance from the center of mass of the binary.
At each time step we then update the momentum density using a combination of the hydrodynamic and corotating acceleration terms, $$\dot U_i = f (\dot U_i)_{\rm Rigid} + (1 - f)(\dot U_i)_{\rm Hydro}$$ where $(\dot U_i)_{\rm Hydro}$ is the acceleration from the full hydrodynamic equation of motion \[Eq. (\[hydromomcfa\])\].
Numerically, we find that if $f$ is small ($< 0.2$) the hydrodynamic forces dominate and corotation is not obtained. On the other hand, for $f > 0.2$ the system is not stable, i.e. the stars deform and the velocities become erratic. We have therefore run with $f = 0.2$ which temporarily produces a velocity field which is close to rigid corotation. That is, the residual three velocities are damped to a fraction of the orbit speed. This is, perhaps, good enough to make qualitative comparisons with the expectations from a truly corotating system.
Starting from the unconstrained initial configuration, we find that when the stars have achieved approximate corotation the central density has decreased from $6.68 \times 10^{14}$ g cm$^{-3}$ to $5.90 \times 10^{14}$ g cm$^{-3}$ which is close to the value for stars in isolation ($5.84 \times 10^{14}$ g cm$^{-3}$). The calculated gravitational mass is slightly greater than that of the unconstrained binary. However, with the large artificial force terms needed to approximate corotation, gravitational mass is not a well defined quantity in this simulation. Also, the orbit frequency was not sufficiently converged for a meaningful comparison.
The Spin of Binary Stars
------------------------
As noted above our simulations indicate that neutron stars relax to a state of almost no intrinsic spin. In a separate paper [@wm98] we analyze the nature and formation of this state in more detail. For the present discussion, however, we summarize in Figure \[fig3\] a study of the relaxation to this state from states of arbitrary initial rigid rotation (including corotation).
As a means to distinguish the intrinsic spin motion of the fluid with respect to a non-orbiting distant observer, we define a quantity which is analogous to volume averaged intrinsic stellar spin in the orbit plane, $$J_S = \sum_{i=1,2}\int \biggl[(x - \tilde x_i)S_y - (y-\tilde y_i)S_x \biggr]
{\phi^2 \over \alpha} d V_i~~,$$ where $(\tilde x_i,\tilde y_i,\tilde z_i = 0)$ is the coordinate center of mass of each star.
In this study we have imposed an initial angular velocity $\omega_S$ in the corotating frame to obtain various initial rigidly rotating spin angular momenta (including corotation, $\omega_S = 0$), but for fixed total $J/M_B^2 = 1.4$. We have considered spin angular frequencies in the range $-900 < \omega_S < 900$ rad sec$^{-1}$, corresponding to $-0.03 < J_S/m_B^2 < 0.17$. We then let the system evolve hydrodynamically with the stars maintained at zero temperature.
In Ref. [@mw97] we showed that neutrino emission is sufficient to radiate away the released gravitational energy and keep the stars at near zero temperature until just before collapse. This is the reason that we have treated this as a relaxation problem. That is, unlike a true hydrodynamic calculation, the relaxation calculation presented here, assumes that the stars radiate efficiently and stay at zero temperature. Therefore, this evolution does not need to conserve energy or circulation. This relaxation assumption is the reason the stars can evolve to a different spin (lower energy) state without violating the circulation theorem.
Figure \[fig3\] shows the spin $J_S/m_B^2$ as a function of time for each initial condition. In each case, the system relaxed to a state of almost no net spin within about three sound crossing times ($t \sim 0.6$ msec). These calculations suggest that rapidly spinning neutron stars in close orbits are unstable. The true evolution time, however, would be much longer.
We also note that the quantity $\int \sigma [\sqrt{1+ U^2}
-1] dV$ decreased as the system evolved from rigid rotation to hydrodynamic equilibrium. Since this quantity is related to the kinetic energy of the binary, this indicates that the hydrodynamic lowest energy state is one of lower kinetic energy (for fixed total angular momentum) than that of rigid rotation.
As far as the compression effect is concerned, one wishes to know whether the response of the stars is simply due to that fact that they have no spin (and therefore no internal centrifugal force to support them against the compression forces), or whether more complex fluid motion within the star itself affects the stability. To test this, we have constructed stars of no spin ($J_S = 0$) by simply damping the residual motion to that of $J_{S} = 0$ after each update of the velocity fields.
Since this no-spin state is so close to the true hydrodynamic equilibrium, this produced stable $J_{S} = 0$ equilibrium stars for the binary. For this case, the central density converges to $\rho_c = 6.56 \times 10^{14}$ g cm$^{-3}$ which is very close to the high value for the unconstrained hydrodynamics. This result would seem to indicate that most of the increase in density can be attributed to the velocity with respect to the corotating frame generated by the fact that the stars have almost no spin.
DISCUSSION
==========
For clarity, we summarize in this section our conclusions regarding why the neutron-star compression effect was not observed in some other recent works.
First consider post-Newtonian expansions. In the work of Wiseman [@wiseman] the force terms containing $U^2$ were explicitly deleted from the computation of the stellar structure \[cf. Eq. (8) in that paper\]. Only the $dln{\alpha}/dx$ term was included. The recovery of simple hydrostatic equilibrium was thus unavoidable.
The PN orbiting ellipsoids of Shibata et al. [@shibata] included more terms. Indeed, it was noted that there are two effects at 1PN order. One is the self gravity of each star of the binary and the other is the gravity acting between the stars. In their calculations the self gravity dominates causing the stars to become more compact. This is consistent with the compression effect described here in the sense that relativistic corrections can dominate over Newtonian tidal forces. However, the self gravity terms in [@shibata] appear to only include the usual 1PN terms which would equally apply to stars in isolation. Hence, the velocity-dependent compression driving terms are probably not present.
Their results for stars in corotation are consistent with ours under the same constraint. They also note that approaches in which PN corrections to the gravity between the stars is included without also including the corrections to the self gravity (as in [@lw96]) can be misleading.
In the work of Lombardi et al. [@lombardi] both corotating and irrotational equilibria were computed. However, in their calculations it appears that the stars become less compact as they approach contrary to our results and the results of [@shibata]. It may be that the reason for this is that in Lombardi et al. the post-Newtonian corrections to self gravity were only computed for stars “instantaneously at rest”. The authors chose to “exclude the spin kinetic energy contribution to the self energy”. It is such terms, however, which we identify with the compression effect.
The conformally flat corotating equilibria computed by Baumgarte et al. [@baumgarte] are consistent with our results. Since their stars were restricted to rigid corotation, only the hydrostatic Bernoulli solution would result. They could not have observed the compression forces which result from fluid motion with respect to the corotating frame.
We have argued in this paper that if one wishes to explore this effect, it would be best to apply a complete unconstrained strong-field relativistic hydrodynamic treatment for stars which are not in corotation. In this regard, a recent paper [@sbs98] has come to our attention in which hydrodynamic simulations of both corotating and irrotational binaries have been studied in a first post-newtonian approximation to conformally-flat gravity but using the full relativistic hydrodynamics equation (\[hydromom\]). For both corotational and irrotational stars the central density is observed to oscillate about a value which is less than that of isolated stars. Hence, the authors conclude that no compression effect is present.
Since this calculation contains many of the higher order terms to which we attribute the compression effect, it is not immediately obvious why the compression effect was not observed. This may indeed be a real contradiction. We suggest, however, that this simulation did not observe the effect because of their use of an unrealistically soft $\Gamma = 1.4$ EOS. The authors chose this EOS because the stars become so extended that one can compute arbitrarily close binaries without encountering the relativistic inner orbit instability. For the irrotational stars (model $Bc$ in [@sbs98]), which is the only simulation that might have observed the compression effect, the compaction ratio is only $M/R = 0.023$. Hence, a 1.45 M$_\odot$ neutron star would have an unrealistic radius of 93 km.
However, since they have simulated very extended stars at very close separation, the tidal forces are much stronger relative to the relativistic compression driving terms than in any of the simulations which we have done.
The ratio of the stabilizing tidal correction $\Delta E_{tidal}$ to the destabilizing energy from compression $\Delta E_{comp}$ should scale [@mw97; @lai] as $${\Delta E_{tidal} \over \Delta E_{comp}} \propto \biggl({R \over r}\biggr)^6~~,$$ where $R$ is the neutron star radius and $r$ is the orbital separation. For model Bc in [@sbs98] we estimate that this ratio is $^>_\sim 200$ times greater than any of the binary stars we have considered. Hence, it is quite likely that the authors have simply chosen an unrealisticly soft equation of state for which the tidal forces dominate over compression. It might be very interesting to see the results from a similar study for stars with a realistic compaction ratio and several radii apart.
Concerning tidal expansions, in Brady & Hughes [@brady] an attempt was made to analyze the stability of a central star perturbed by an orbiting point particle. The metric and stress-energy were perturbed in terms of order $\epsilon = \mu/R$ where $\mu$ is the point particle mass and $R$ its coordinate distance from the central star. The Einstein equation was then linearized to terms of order $\epsilon$. The result of this linearization was that the only possible correction to the central density was a single monopole term of order $\mu/R \sim v^2$. However, in our numerical results as shown in Figure \[fig3\]. the central density is observed to increase as $v^4$. Hence, it may be that the expansion of Ref. [@brady] was truncated at too low order to observe the compression effect described here. The main reason that they could not observe the effect, however, is that the terms involving motion of the central star were discarded. We attribute the compression effect to an enhancement of the self gravity due to motion of the stars with respect to the corotating frame. Hence, the neglect of terms involving motion of the central star precludes the possibility of observing the effect.
We believe that the same conclusion is true in the treatments by Refs. [@flanagan; @thorne97]. The analysis of Flanagan [@flanagan] is based upon the method of matched asymptotic expansion. The metric is approximated $$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}^{NS} + h_{\mu \nu}^B~~,$$ where the superscript $NS$ refers to the self contribution from one star and $B$ refers to the contribution from a distant companion. The internal gravity of a static neutron star $h_{\mu \nu}^{NS}$ is expanded to all orders. The binary tidal contribution $h_{\mu \nu}^B$ is expanded in powers of the ratio of stellar radius to orbital separation.
First we suggest that such a decomposition may be questionable for a close neutron-star binary. In our metric one can write the metric perturbation as $$h_{i j} = (\phi^4 -1) \delta_{i j}~~.$$ The conformal factor $\phi$ is a solution to a Poisson equation involving source terms from the two stars. Between the stars, the only source of the fields arises from the $K_{i j} K^{i j}$ terms which are quite small. Hence, neglecting $K_{i j} K^{i j}$ terms, $\phi$ is additive in the “vacuum” between the stars, $$\phi = \phi_1 + \phi_2 = 1 +
{m_1 \over 2 \vert r-r_1 \vert}
+ {m_2 \over 2 \vert r-r_2 \vert} ~~.$$ Expanding $h_{i j}$ around star $1$ in the presence of a distant companion $2$ we have $$\begin{aligned}
h_{i j}& =& {4 \over 2}\biggl({m_1 \over \vert r-r_1 \vert} +
{m_2 \over \vert r-r_2 \vert}\biggr)\nonumber \\
& & + {6 \over 4}\biggl({m_1 \over \vert r-r_1 \vert} +
{m_2 \over \vert r-r_2 \vert}\biggr)^2 + \cdot \cdot \cdot \nonumber \\
&&\nonumber \\
&& = h_{\mu \nu}^{NS} + h_{\mu \nu}^B + {\rm cross~terms} ~~.\end{aligned}$$ However, for the binary systems we have considered, the cross terms are $\sim$ 15% to 20% of the sum $h_{\mu \nu}^{NS} + h_{\mu \nu}^B$. Hence, they can not be neglected. The errors associated with this decomposition may be part of the reason that the compression effects are not apparent in this work.
A related concern is with the expansion of the stress-energy tensor in [@flanagan]. We have noted that most of the compression arises from the net effect of velocity dependent terms in the covariant derivative of the stress-energy tensor. In [@flanagan] the stress energy is expanded is powers of the curvature $R^{-m}$. The author states [@flanagan] “We assume initial conditions of vanishing $T_{\mu \nu}^{(2)}$, so that the only source for perturbations is the external tidal field.” An analysis which only considers perturbations from the external tidal field (and not motions of the fluid) will not observe the compression effect. The result of [@flanagan] is that the central density is unchanged until tidal forces enter at $O(R^6)$. This is consistent with our results in the limit of only tidal perturbations acting on the stars. It is not clearto us, however, to what degree the velocity dependent terms are included or excluded by this expansion. A more careful recent revision (E. Flanagan, Priv. Comm.) shows an effect coming in a lower order, but not necessarily as strong as we have noted.
In the paper of Thorne [@thorne97], a similar tidal expansion is applied. In that work only the stabilizing effect of tidal forces was considered along with the stabilizing effect of rotation. However, the increased self gravity from velocity-dependent forces was not included. Hence, the conclusions of [@thorne97] are consistent with our results based upon tidal forces. So are the Newtonian tidal effects computed in [@lai].
CONCLUSIONS
===========
The results of this study (cf. Table \[table1\]) are that we see almost no difference between the central density of an isolated star and a binary star in which rigid corotation has been artificially imposed, or one in which only tidal effects are included. Indeed, in the case of tidal forces alone, the central density in our simulations actually decreases as stars approach, consistent with other works.
An increase in the central density is only apparent in our binary simulations for stars with fluid motion with respect to the corotating frame. (Specifically we considered stars of low intrinsic spin in a binary.) In such cases there is no simple Killing vector which can be imposed to cancel the compression driving forces. We have argued here and in [@mw97] that the main compression effect arises from the net result of velocity-dependent hydrodynamic terms [@fn1]. These terms arise from the affine connection part of the covariant differentiation of the stress-energy tensor.
We show here that the compression effect would not have been observed in a study of tidal forces or any model which artificially imposes rigid corotation of the fluid. A proper treatment must consider all of the force terms apparent in the momentum equation (\[hydromom\]) to sufficient order that their effects on the fluid self gravity survive. A similar conclusion has been reached in [@shapiro] based on test particle dynamics near a Schwarzschild black hole. In that work it is concluded that at least 2.5 post-Newtonian particle dynamics is necessary before a dynamical collapse instability is manifest.
We argue that the results of this study are thus consistent with results in a number of recent papers [@lai; @rs96; @wiseman; @shibata; @lombardi; @lw96; @brady; @flanagan; @thorne97; @baumgarte] which have analyzed the stability of binary stars in various approximations and limits and see no effect. Since we do not disagree with the lack of a compression effect in the limits which they have imposed, we conclude that the existence or absence of the neutron-star compression effect has not yet been independently tested.
Therefore, if one wishes to explore this effect, it would be best to apply a complete unconstrained strong-field relativistic hydrodynamic treatment employing an EOS which produces realistically compact neutron stars. Another alternative, however, might be to study the quasi-equilibrium structure of nonspinning irrotational binary stars at sufficiently high order. In this regard a recently proposed formalism [@bonazzola] to compute quasi-equilibria for nonsynchronous binaries may be of some use. We have begun calculations in this independent formalism. The results will be reported in a forthcoming paper.
Regarding the existence of this low spin state, we find that such a state represents the unconstrained hydrodynamic equilibrium for a close binary. In Newtonian theory, stars are driven to corotation by tidal forces. However in [@bc92] it has been shown that Newtonian tidal forces are insufficient to produce corotation before neutron-star merger unless the viscosity is unrealistically high. Nevertheless, in the absence of strong tidal forces, neutron stars stars gradually spin down. Therefore, even apart from the hydrodynamic effects described here, stars of low spin are likely to be members of close binaries. The hydrodynamic effects described herein, however, could hasten the spin down as stars approach their final orbits and cause the stars to become more compact.
Work at University of Notre Dame supported in part by DOE Nuclear Theory grant DE-FG02-95ER40934, NSF grant PHY-97-22086, and by NASA CGRO grant NAG5-3818. Work performed in part under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405-ENG-48 and NSF grant PHY-9401636.
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------------- --------------------- ----------------------------------
Environment Constraints $\rho_c$ ($10^{14}$ g cm$^{-3})$
Single Star Hydrostatic $5.84$
Single Star Uniform Translation $5.90$
Binary Tidal Only $5.82$
Binary Rigid Corotation $5.90$
Binary Rigid No Spin $6.56$
Binary Full Hydrodynamics $6.68$
\[table1\]
------------- --------------------- ----------------------------------
: Central density for m$_B = 1.625$ M$_\odot$ stars in various conditions using a $\Gamma = 2$ EOS.
----------------- ---------------------------------- --
Separation (km) $\rho_c$ ($10^{14}$ g cm$^{-3})$
41.8 $5.821$
50.6 $5.816$
81.8 $5.821$
$\infty$ $5.837$
\[table2\]
----------------- ---------------------------------- --
: Central density vs. coordinate separation between centers for m$_B = 1.625$ M$_\odot$ ($\Gamma=2$) stars in which only tidal forces are included. The neutron star radius (in isotropic coordinates) is 12 km.
----------------- ---------------------------------- --
Separation (km) $\rho_c$ ($10^{14}$ g cm$^{-3})$
31.2 $14.15$
37.4 $14.16$
64.8 $14.20$
103.8 $14.25$
$\infty$ $14.30$
\[table3\]
----------------- ---------------------------------- --
: Same as Table II but for the EOS of ref. \[2\]. The neutron star radius (in isotropic coordinates) is 6 km.
| ArXiv |
---
abstract: 'In this paper, we introduce Adaptive Cluster Lasso(ACL) method for variable selection in high dimensional sparse regression models with strongly correlated variables. To handle correlated variables, the concept of clustering or grouping variables and then pursuing model fitting is widely accepted. When the dimension is very high, finding an appropriate group structure is as difficult as the original problem. The ACL is a three-stage procedure where, at the first stage, we use the Lasso(or its adaptive or thresholded version) to do initial selection, then we also include those variables which are not selected by the Lasso but are strongly correlated with the variables selected by the Lasso. At the second stage we cluster the variables based on the reduced set of predictors and in the third stage we perform sparse estimation such as Lasso on cluster representatives or the group Lasso based on the structures generated by clustering procedure. We show that our procedure is consistent and efficient in finding true underlying population group structure(under assumption of irrepresentable and beta-min conditions). We also study the group selection consistency of our method and we support the theory using simulated and pseudo-real dataset examples.'
address: Indian Statistical Institute
author:
- Niharika Gauraha
- 'and Swapan K. Parui'
bibliography:
- 'niharika\_arXiv.bib'
title: 'Efficient Clustering of Correlated Variables and Variable Selection in High-Dimensional Linear Models'
---
High-Dimensional Data Analysis ,Correlated Variable Selection ,Adaptive Cluster Lasso ,Adaptive Cluster Representative Lasso ,Adaptive Cluster Group Lasso
Introduction
============
We consider the usual linear regression model $$\begin{aligned}
\label{eq:lr}
\textbf{Y} &= \textbf{X} \beta^{0}+\epsilon,\end{aligned}$$ with response vector $Y_{n \times 1}$, design matrix $X_{ n \times p}$, true underlying coefficient vector $\beta^0_{p \times 1}$ and error vector $\epsilon_{n\times 1}$. When the number of predictors (p) is much larger than the number of observations (n), $p >> n$, the ordinary least squares estimator is not unique and may over fit the data. The parameter vector $\beta^{0}$ can only be estimated based on given very few observations under assumption of sparsity in $\beta^{0}$. To infer the true active set $S_0 = \{j; \beta^{0}_{j} \neq 0\}$, the Lasso([@Tibshirani]), its variants and other regularized regression methods are mostly used for sparse estimation and variable selection. However, variable selection in situations involving high empirical correlation remains one of the most important issues. This problem is encountered in many applications such as in microarray analysis, a group of genes which participate in the same biological pathway tend to have highly correlated expression levels(see [@Segal]) and it is often required to consider all of them if they are related to the underlying biological process.
It has been proven that the design matrix must satisfy the following conditions for the Lasso to perform exact variable selection: irrepresentable(IR) condition([@Zhao]) and beta-min condition([@Buhlmann1]). Having highly correlated variables implies that the design matrix violates the IR condition. To deal with variable selection with correlated variables, mainly two approaches has been proposed: simultaneous clustering and model fitting and clustering followed by the sparse estimation (i.e. group lasso). The former approach imposes restrictive conditions on the design matrix. However, the time complexity for clustering of variables severely limits the dimension of data sets that can be processed by the later approach. Moreover, group selection in models with a larger number of groups is more difficult(see [@Wei]). To overcome these limitations we propose a three stage procedure, Adaptive Cluster Lasso method. Basically, we try to reduce the dimensions first using the Lasso(or its adaptive or thresholded version) before clustering of variables.
At a high level our method works as follows. At the first stage, the Lasso is used to do initial selection of variables then we also select those variables which are not selected by the Lasso but they are strongly correlated with the variables selected by the Lasso. If the design matrix satisfies the beta-min condition, then after the first stage, the selected set of variables contains the true active set and the dimensionality of the problem is reduced by a huge amount. At the second stage, we perform clustering of variables on the reduced model, so that strongly correlated variables are grouped together in disjoint clusters. In the third stage, we do group-wise sparse estimation based on the structures generated by clustering procedure. The second and third stages of ACL together is the same as Cluster Group Lasso(CGL) or Cluster Representative Lasso(CRL) with ordinary hierarchical correlation based clustering, defined in [@Buhlmann2]. Hence, ACL method is an extension of the clustering lasso methods proposed in [@Buhlmann2]. Mainly, there are two lines of thought, the one is to find an appropriate and efficient clustering of correlated variables and the other line of thought is to avoid the false negatives. With these thoughts in mind, we develop a computationally efficient variable selection procedure $\hat{S}_{ACL}$, which identifies the appropriate correlated group structures and selects all variables from a group of correlated variables where at least one of them is active. Assuming Group Irrepresentable(GIR) and group beta-min condition on the design matrix, we prove that the $\hat{S}_{ACL}$ selects the true model, with much less computational complexity. We show that the dimensionality reduction and subsequent clustering and CGL(or CRL) improves over the plain clustering and CGL(or CRL). We illustrate the proposed method and compare it with the methods proposed in [@Buhlmann2] by extensive simulation studies and we also apply it to a pseudo-real dataset. The rest of this paper is organized as follows. In section 2, we provide notations, assumptions, review of relevant work and we discuss our contribution. In section 3, we review mathematical theory of the Lasso and group Lasso. In section 4, we describe the proposed algorithm which mostly selects more adequate models in terms of model interpretation and prediction performance. In section 5, we study theoretical properties of the proposed method. We also show that the variable selection is consistent for high dimensional sparse problems. In section 6, we provide numerical results based on simulated and pseudo real dataset. Section 7 contains the computational details and we shall provide conclusion in section 8.
Background and Notations
========================
In this section, we state notations and assumptions, we define required concepts and we also provide review of the relevant work.
Notations and Assumptions
-------------------------
The following notations,assumptions and definitions are applied throughout this paper.
We consider the usual linear regression set up with univariate response variable $Y \in R$ and p-dimensional predictors $X_i \in R^p$: $$\begin{aligned}
\label{eq:lr2}
Y_i& = \sum_{j=1}^{p} X_{i}^{(j)}\beta^{0}_{j} + \epsilon_i \quad i=1,...,n \; \;j=1,...,p\end{aligned}$$ or, in matrix notation (as in Equation \[eq:lr\]) $$\begin{aligned}
\textbf{Y} &= \textbf{X} \beta^{0}+\epsilon\end{aligned}$$ where $\beta^{0} \in R^p$ are unknown true regression coefficients to be estimated, and the components of the noise vector $\epsilon \in R^n$ are i.i.d. $N(0, \sigma^2)$. The columns of the design matrix X are denoted by $X^{j}$. We assume that the design matrix **X** is fixed, the data is centred and the predictors are standardized, so that $\sum_{i=1}^{n} Y_i = 0$, $\sum_{i=1}^{n} X^{j}_{i} = 0$ and $\frac{1}{n} X^{j'}X^{j} = 1$ for all $j=1,...,p$.\
The $L_1$-norm is defined as: $$\begin{aligned}
\label{eq:l1}
\|\beta\|_1 = \textstyle \sum_{j=1}^p |\beta_j|\end{aligned}$$ $L_2$-norm squared is defined as: $$\begin{aligned}
\label{eq:l2}
\|\beta\|^{2}_2 = \textstyle \sum_{j=1}^p \beta^{2}_{j}\end{aligned}$$ The $L_\infty-$ norm is defined as: $$\begin{aligned}
\label{eq:linf}
\|\beta\|_{\infty} = \textstyle max_{1 \leq i \leq n |} |\beta_j|\end{aligned}$$ The true active set $S_0$ denotes the support of the subset selection solution($S_0 = supp(\beta_0)$) and defined as $$\begin{aligned}
\label{eq:s0}
S_0 &= \{j; \beta^{0}_{j} \neq 0\}\end{aligned}$$ The sign function is defined as: $$\begin{aligned}
sign(x) = \left\lbrace \begin{array}{ll}
-1 & \text{ if } x < 0 \\
0 & \text{ if } x = 0\\
1 & \text{ if } x > 0
\end{array} \right.\end{aligned}$$ The (scaled)Gram matrix(covariance matrix) is defined as $$\hat{\Sigma}= \frac{X'X}{n}$$ The $\beta_S$ has zeroes outside the set S,as $$\beta_S = \{ \beta_j I(j \in S) \}$$ and $\beta = \beta_S + \beta_{S^c}$.\
For the given $S \subset \{1,2,...,p \}$, the covariance matrix can be partitioned as: $$\begin{aligned}
\label{sigmaPart}
\Sigma = \left[ \begin{array}{cc}
\Sigma_{11} = \Sigma(S) & \Sigma_{12}(S)\\
\Sigma_{21}(S)\quad \quad &\Sigma_{22} = \Sigma(S^c)
\end{array} \right]\end{aligned}$$ Minimum eigenvalue of a matrix A is denotes as $\Lambda_{min}(A)$.
Clustering of Variables {#sub:clust}
-----------------------
We use correlation based, bottom-up agglomerative hierarchical clustering methods to cluster predictors, which forms groups of variables based on correlations between them. For further details on grouping of variables and determining the number of clusters, we refer to [@Buhlmann2].
The Lasso and the Group Lasso
-----------------------------
The Least Absolute Shrinkage and Selection Operator (Lasso) was introduced by Tibshirani [@Tibshirani]. It is a penalized least squares method that imposes an L1-penalty on the regression coefficients, which does both shrinkage and automatic variable selection simultaneously due to the nature of the L1-penalty. We denote $\hat{\beta}$, as a Lasso estimated parameter vector. Assume $\lambda$ is the regularization parameter, then then Lasso estimator is computed as: $$\begin{aligned}
\hat{\beta} \in \mathop{arg min}_{\beta \in \mathbb{R}^p} \{\frac{1}{n} \| \textbf{y} - \textbf{X} \beta \|_{2}^{2}+ \lambda \|\beta\|_1 \}\end{aligned}$$ and the estimated active set is denoted as $\hat{S}$ and defined as $$\begin{aligned}
\label{eq:s1}
\hat{S} = \{j; \hat{\beta}_{j} \neq 0\}\end{aligned}$$
The Lasso error vector is defined as $$\begin{aligned}
\Delta = \hat{\beta} - \beta^0\end{aligned}$$
One of the major disadvantages of the the lasso is that, the Lasso tends to select single or a few variables, from a group of highly correlated variables. When the distinct groups or clusters among the variables are known a priory and it is desirable to select or drop the whole group instead of single variables. Then the Group Lasso (see [@Yuan]) or its variants are used, that imposes an $L_2$-penalty on the coefficients within each group to achieve such group sparsity. Here we define some more notations and state assumptions for the group Lasso. We may interchangeably use $\beta^{0}$ and $\beta$ for the true regression coefficient vector, the later one is without the superscript. Let us assume that the parameter vector $\beta$ is structured into groups, $G = \{ G_1, . . . , G_q \}$, where $q < n $, denotes the number of groups. The partition $G$ basically builds a partition of the index set $\{ 1,...,p\}$ with $ \cup_{r=1}^{q} G_r = \{1,... , p\}$ and $G_r \cap G_l = \emptyset, \quad r\neq l$. The parameter vector $\beta$, then has the structure $\beta = \{ \beta_{G_1}, ..., \beta_{G_q} \}$ where $\beta_{G_j} = \{\beta_r: r \in G_j \}$.
The columns of the each group is represented by $X^{G_j}$. $$X = (X^{(1)}, ..., X^{(p)}) = (X^{(G_1)}, ... , X^{(G_q)})$$ The response vector Y can also be written as $$Y = \sum_{j=1}^{q} X^{(G_j)} \beta_{G_j} + \epsilon$$ where $X^{(G_j)} \beta_{G_j} = \sum_{k=1}^{m_k} X^{(G_j)}_k (\beta_{G_j})_k $. The loss function of the group Lasso is same as the loss function of the Lasso $\frac{1}{n}\|Y-X\beta \|_{2}^{2}$. The group Lasso penalty is defined as $$\|\beta\|_{2,1} = \textstyle \sum_{j=1}^q \| X^{G_j} \beta_{G_j} \|_2 \sqrt{\frac{m_j}{n}}$$ where $m_j = |G_j|$ is the group size. Since the penalty is invariant under parametrizations within-group. Therefore, without loss of generality, we can assume $\Sigma_{rr} = I$, the $m_r \times m_r$ identity matrix. Hence the group Lasso penalty can be written as $$\|\beta\|_{2,1} = \sum_{j=1}^q \sqrt{m_j} \|\beta_{G_j} \|_2$$
The Group Lasso estimator(with known q groups) is defined as $$\begin{aligned}
\label{eq:grpLasso}
\hat{\beta}_{grp}\in \mathop{argmin}_{\beta} \{ \frac{1}{n}\|Y-X\beta \|_{2}^{2} + \lambda \|\beta\|_{2,1} \}\end{aligned}$$
The group Lasso has the following properties:
- The group Lasso behaves like the lasso at the group level, depending on the value of the regularization parameter $\lambda$, the whole group of variables may drop out of the model.
- For singleton groups (when the group sizes are all one), it reduces exactly to the lasso.
- The group Lasso penalty is invariant under orthonormal transformation within the groups.
- The group Lasso estimator has similar oracle inequalities as the standard Lasso for prediction accuracy and estimation error. It has group wise variable selection property(We discuss mathematical theory in the section \[secLassoTheory\]).
Let W denote the actives group set , $W \subset \{ 1,...,q \}$, with cardinality $w = |W|$. Throughout the article, the following assumption are made for the group Lasso:
- The size of the each group is less than the number of observations. $$m_{max} < n$$.
- The number of active groups, w, is less than the number of observations (sparsity assumption).
### Cluster Group Lasso
When the group structure is not known then clusters $G_1, . . . , G_q$ are generated from the design matrix X( using correlation based method etc.). Then the group Lasso is applied to the resulting clusters. We denote the clusters selected by the group Lasso as $\hat{S}_{clust}$, and is defined as $$\begin{aligned}
\hat{S}_{clust} = \{r: \text{ cluster }G_r \text{ is selected, } r = 1,...,q\}\end{aligned}$$ The union of the selected clusters gives the selected set of variables. $$\begin{aligned}
\hat{S}_{CGL} = \cup_{r \in \hat{S}_{clust}} G_r\end{aligned}$$
### Cluster Representative Lasso
Similar to the CGL, the cluster representative Lasso, first identifies groups among the variables and then applies the lasso for cluster representatives (see [@Buhlmann2]). When sign of the regression coefficients within a group is the same then taking group representatives is advantageous, whereas when near cancellation among $\beta^{0}_{j}(j \in G_r)$ takes place then CGL is preferred.\
We define representative for each cluster as $$\begin{aligned}
\bar{X}^{(r)} = \frac{1}{|G_r|} \sum_{j \in G_r} X^{(j)}, \quad r = 1,...,q.\end{aligned}$$ The design matrix of cluster representatives is denoted as $\bar{X}_{n \times q}$. Then optimization problem for CRL is defined using response $Y$ and the design matrix of cluster representatives $\bar{X}$ as: $$\begin{aligned}
\hat{\beta}_{CRL} \in \arg\!\min_{\beta} ( \|{ \textbf{y} - \bar{\textbf{X}} \beta} \|^{2}_{2}+ \lambda_{CRL} \|\beta \|_1 )\end{aligned}$$
The selected clusters are then denoted as: $$\begin{aligned}
\hat{S}_{clust,CRL} = \{r; \hat{\beta}_{CRL,r} \neq 0, r = 1, . . . , q \}\end{aligned}$$
and the selected variables are obtained as the union of the selected clusters as: $$\begin{aligned}
\hat{S}_{CRL} = \cup_{r \in \hat{S}_{clust,CRL}} G_r\end{aligned}$$
Review of Relevant work and our Contribution
--------------------------------------------
Here, we provide a brief review of relevant work in this area, and we also show that how our proposal differs or extends the previous studies.
The Lasso can not do variable selection in the situations where predictors are highly correlated. As mentioned before, to handle correlated covariates in variable selection methods, two algorithmic approaches have been developed in the past: clustering of variables and model fitting either simultaneously or at two different stages. Examples of the methods that do clustering and model fitting simultaneously are Elastic Net([@Hui]), Fused LASSO([@Fused]), octagonal shrinkage and clustering algorithm for regression(OSCAR, [@oscar]) and Mnet([@Mnet] ) etc. The Elastic Net uses a combination of the $L_1$ and $L_2$ penalties, OSCAR uses a combination of $L_1$ norm and and $L_{\infty}$ norm and Mnet uses a combination of $L_2$ and Minimum Concave Penalty(MCP). We note that these methods use only combination of penalties, they do not use any specific information on the correlation pattern among the predictors and hence they do not reveal any group structure in the data. Now, we discuss a few methods that perform clustering and model fitting at different stages, i.e. Cluster Group Lasso(CGL, [@Buhlmann2]), Cluster representative Lasso(CRL,[@Buhlmann2]), Stability Feature Selection using Cluster Representative LASSO (SCRL, [@Niharika]) and sparse Laplacian shrinkage estimator(SLS, [@SLS]). CRL, CGL and SCRL use correlation based and canonical correlation methods to perform hierarchical clustering. SLS also considers the correlation patterns among predictors but requires that highly correlated variables should have similar predictive effects.The main disadvantage of this approach is mainly due to clustering in the presence of unstructured data or noise features. It is difficult to determine the exact group structures or the exact number of groups in high-dimensions and in the presence of noise features. Moreover, the CPU time taken by clustering algorithms is unacceptable when the number of predictors are huge. To address these problems, we propose to reduce the dimensionality before performing clustering which makes our proposal different from previous work.
Basically, our work can be viewed as an extension of the two stage procedure, Cluster Lasso Methods with correlation based clustering, proposed in [@Buhlmann2]. The extension is that we add a dimensionality reduction stage prior to performing the clustering, which leads to clustering of variables more accurately and efficiently and thus consistent group variable selection. In particular, we consider Adaptive Clustering Group Lasso(ACGL), where the Lasso is used as preprocessing step at the first stage, correlation based clustering at second stage and the group Lasso in the third stage(defined in section 4). We also consider the CGL method with ordinary hierarchical clustering, denoted by CGLcor, see [@Buhlmann2]. We compare ACGL and CGLcor in terms of predictive performance, variable selection and CPU time expended, in section 5. Our extensive simulation studies show that ACGL outperforms the CGLcor.
Mathematical Theory of the Lasso and the Group Lasso {#secLassoTheory}
====================================================
In this section, we review the results required for proving consistent variable selection( and group variable selection) in high dimensional linear models. For more details on the mathematical theory for the lasso and group lasso, we refer to: [@Sara2], [@Zhao], [@Wei] [@Buhlmann2] and [@Buhlmann1].
The Lasso compatibility condition holds for a fixed set $S \subset \{1,...,p\}$ with cardinality $s = |S|$, a constant $\phi_{comp}(S) > 0$ and if for all $ \| \Delta_{S^c}\|_1 \leq 3 \| \Delta_{S}\|_1 \neq 0$ the following holds $$\begin{aligned}
\| \Delta_{S}\|_{1}^{2} \leq \left\lbrace \frac{s\frac{1}{n} \| X \Delta \|_{2}^{2} }{ \phi^{2}(S)} \right\rbrace\end{aligned}$$ where $\phi_{comp}(S) $ is called the compatibility constant.
The constant 3 is due to the condition $\lambda \geq 2\lambda_0$, which is required to overrule the stochastic process part,(see [@Buhlmann1] for details). Without loss of generality we can assume $S = \{1,...,s \}$ and partition the covariance matrix in block-wise form as given in equation \[sigmaPart\]. Assuming $\Sigma^{-1}_{11}$ is invertible, the various form of Irrepresentable(IR) Conditions are defined as follows.
The strong irrepresentable condition is said to be met for the set S, with cardinality $s = |S|$, if the following holds: $$\begin{aligned}
\label{IR1}
\|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} < 1, \quad \forall \tau_S \in \mathbb{R}^s \; such \; that \;
\| \tau_S \|_{\infty} \leq 1\end{aligned}$$ The weak irrepresentable condition holds for a fixed $\tau_S \in \mathbb{R}^s$ if $$\begin{aligned}
\label{IR2}
\|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} \leq 1\end{aligned}$$ For some $0<\theta <1$, the $\theta$ uniform irrepresentable condition holds if $$\begin{aligned}
\label{IR3}
\mathop{max}_{\| \tau_S \|_{\infty} \leq 1} \|\Sigma_{12}(S) \Sigma^{-1}(S) \tau_S \|_{\infty} \leq \theta\end{aligned}$$
Sufficient conditions(eigenvalue and mutual incoherence) on design matrix to hold IR are discussed in [@Zhao] and [@Hastie].
the Beta Min Condition is met for the regression coefficient $\beta^0$, if $\min |\beta^0| \geq \frac{4 \lambda s_0}{\phi^{2}(S)}$
Under the following assumptions the Lasso selects the true active set $S_0$ with high probability:
- Irrepresentable Condition holds for $S_0$.
- beta-min condition holds for $\beta^0$.
The following inequality shows the bounds for prediction error and estimation error of the Lasso estimator.( for derivation and proof we refer to [@Buhlmann1]). $$\begin{aligned}
\frac{1}{n}\| X \Delta \|_{2}^{2} + \lambda \|\Delta \|_1 & \leq \frac{4\lambda^2 s}{ \phi^{2}_{comp}}
\end{aligned}$$ Our error analysis for the group Lasso is based on the pure active group and pure noise group assumptions, that is,\
$(A5):$ all variables are active variables within an active group and no variables are active in a noise group.\
We define the group Lasso error as $\Delta_{G_r} = \beta_{G_r} - \beta^{0}_{G_r} $, and also assume the following.\
$(A6)$: We assume that clustering process identifies the group structure correctly.
The group Lasso compatibility condition holds for a fixed set $W \subset \{1,...,q\}$ with cardinality $w = |W|$, a constant $\phi_{grp}(W) > 0$ and if for all $ \sum_{r \in W^c}\| \Delta_{G_r} \|_{2} \leq 3 \sum_{r \in W}\| \Delta_{G_r} \|_{2} \neq 0$ the following holds $$\begin{aligned}
(\sum_{r \in W}\| \Delta_{G_r} \|_{2})^2 \leq \left\lbrace \frac{ w\frac{1}{n} \| X \Delta \|_{2}^{2} }{\phi_{grp}^{2}(W)} \right\rbrace\end{aligned}$$ where $\phi_{grp}(W) $ is called the group Lasso compatibility constant.
The Lasso compatibility condition implies the group Lasso compatibility condition, it is explained by the following Lemma(See Lemma 8.2 of the book [@Buhlmann1], for the proof).
Let $W \subset \{ 1,...,q \}$ be a group index set, say, $W = \{ 1, ..., w\}$ Consider the full index set corresponding to W: $$S = \{ (1,1), ..., (1,m1),..., (w,1),...,(w,m_w) \} = \{ 1,...,s \}$$, where (i,j) denotes jth member of ith group and $s = \sum^{w}_{j=1} m_j$. If compatible condition holds for S with compatiblility constant $\phi(S)$ then the compatibility condition holds for the $\phi_{grp}(W)$, and $\phi_{grp}(W) \geq \phi(S)$
The group IR condition is met for the set W with a constant $0< \theta < 1$, if for all $\tau \in \mathbb{R}^s$ with $\|\tau \|_{2, \infty} = \mathop{max}_{1\leq r \leq q} \| \tau_{G_r} \|_2 \leq 1 $, the following holds $$\begin{aligned}
\frac{1}{m_r} \|(\Sigma_{21} \Sigma^{-1}_{11} K \tau)_{G_r} \| \quad\forall r \not\in W,\end{aligned}$$ where $K = diag(m_1 I_{m_1} , ..., m_w I_{m_w})$
We note that the GIR definition reduces to the Lasso IR condition for singleton groups(see [@Basu]).
The group beta-min Condition is met for $\beta^0$ , if $\|\beta^{G_r}\|_{\infty} \geq \frac{D \lambda \sqrt{m_r}}{n} \quad \forall r \in W $, where $D>0$ is a constant which depends on $\sigma, \phi_{grp}$ and other constants used in cone constraints and GIR conditions.
We note that, only one component of the $\beta^{G_r}, \forall r \in W$ has to be sufficiently large, because we aim to select groups as a whole, and not individual variables. For its exact form, we refer to [@Florentina].
Under the following assumptions the group Lasso selects the true active groups $W$ with high probability:
- GIR Condition holds for $W$.
- Group beta-min condition holds for $\beta^{G_r}, \forall r \in W$.
Next, we discuss sufficient condition for the GIR to hold. We denote $\Sigma_{r,l} = X^{G_r^{'}} X^{G_l}/n$, $r,l \in \{ 1,...,q\}$. We partition the covariance matrix group wise. (here we assume that each $\Sigma_{r,r}$ is non-singular, or we may use the pseudo inverse) $$\begin{aligned}
R_W =
\left[ \begin{array}{cccc}
I & \Sigma^{-1/2}_{11}\Sigma_{12}\Sigma^{-1/2}_{22} & ... & \Sigma^{-1/2}_{11}\Sigma_{1w}\Sigma^{-1/2}_{ww} \\
\Sigma^{-1/2}_{22}\Sigma_{21}\Sigma^{-1/2}_{11} & I & ...& \Sigma^{-1/2}_{22}\Sigma_{2w}\Sigma^{-1/2}_{ww} \\
\vdots & \vdots & \ddots & \vdots\\
\Sigma^{-1/2}_{ww}\Sigma_{w1}\Sigma^{-1/2}_{11} & \Sigma^{-1/2}_{ww}\Sigma_{w2}\Sigma^{-1/2}_{22} &... & I
\end{array} \right]\end{aligned}$$ We note that diagonal elements are $I_{m_r \times m_r}$ identity matrix due to parameterization invariance properties.
Now Suppose that $R_W$, has smallest eigenvalue $\Lambda_{min}(R_{W}) > 0$ and that canonical correlations between groups are small enough that intern implies the incoherence assumptions. Therefore, under the eigenvalue and incoherence condition of $R_W$, the group irrepresentable condition holds(see [@Buhlmann2]).
Now we prove that the Lasso IR condition implies the group Lasso IR(GIR) condition.
Let $W \subset \{ 1,...,q \}$ be a group index set, say, $W = \{ 1, ..., w\}$ Consider the full index set corresponding to W: $$S = \{ (1,1), ..., (1,m1),..., (w,1),...,(w,m_w) \} = \{ 1,...,s \}$$ where $s = \sum^{w}_{j=1} m_j$. If the Lasso IR condition holds for the set S then the group Lasso IR condition holds for the set W.
Proof is trivial, the IR condition on the set S implies that $ \Sigma_{11}$ is invertible , $\Lambda_{min}(\Sigma_{11})>0$, and correlation between variables in S and between variables in $S$ and $S^c$ are small enough. That implies small enough canonical correlations within the groups in active groups W, and between the groups in $W$ and $W^c$. The small enough canonical correlations between groups ensure the incoherence assumptions and therefore the GIR condition holds.
The following inequality shows the similar bounds for prediction error and estimation error of the group Lasso estimator([@Buhlmann2]). $$\begin{aligned}
\frac{1}{n}\| X \Delta \|_{2}^{2} + \lambda \sum_{r =1}^{q}\| \Delta_{G_r} \|_{2} & \leq \frac{24\lambda^2 \sum_{r \in W} m_r}{ \phi^{2}_{grp}(W)}
\end{aligned}$$
The Adaptive Cluster Lasso Methods
==================================
It is known that the Lasso tends to select one or few variables from the group of highly correlated variables, even though many or all of them belong to the active set. We aim to avoid false negatives and solve clustering problem efficiently and more accurately. To solve clustering problem efficiently, we propose a preprocessing step to reduce the dimensionality using the Lasso methods before clustering of variables. To avoid the false negative, we use the concept of clustering the correlated variables and then selecting or dropping the whole group instead of single variables same as the CGLcor(CRLcor) proposed in [@Buhlmann2]. The proposed procedure, ACL is a 3-stage procedure, where we can choose to use different methods at different stages depending on the nature of the problem. The different stages of the ACL procedure is explained as follows.
1. Dimensionality Reduction\
For selecting initial set of variables, we use the Lasso(or its adaptive or thresholded version). Since the Lasso tends to select one or a few variables from the group of strongly correlated variables, therefore we use the Lasso to select the group representative predictors. After we have selected the initial set of variables(group representative members), we get the rest of the group members by simple correlated screening. In section 3, we have shown that for highly correlated structures, the variables set selected by this approach always contains the true active set under assumption of GIR and GBM on the design matrix. Let the variables set selected by Lasso is given by $$\begin{aligned}
\hat{S}_{Lasso} &= \{j; \hat{\beta}_{Lasso,j}(\lambda_1) \neq 0\} .\end{aligned}$$ Then we select correlated variables as $$\begin{aligned}
\hat{S}_{corr} &= \{k; \quad k \in \{1,...,p \}\setminus \hat{S}_{Lasso}, j \in \hat{S}_{Lasso} \text{ and }corr(X_j, X_k) > \rho\},\end{aligned}$$
where $\lambda_1$ is the tuning parameter used by Lasso and $\rho > 0.7$ denotes the strong correlation between two variables. Then the selected set of variable are given by $$\begin{aligned}
\hat{S}_{1} &= \hat{S}_{Lasso} \cup \hat{S}_{corr}\end{aligned}$$
2. Clustering of Variables\
After first stage there may be huge amount of reduction in the dimensionality, we denote the reduced design matrix as **$X_{red} = \{X_j; j \in \hat{S}_{1}\}$**. On the reduced set of predictors, we apply correlation based clustering methods to group strongly correlated variables into disjoint groups. We denote the inferred clusters as $G_1, . . . , G_q $.
3. Supervised Selection of Clusters\
From the reduced design matrix **$X_{red}$**, and inferred clusters $G_1, . . . , G_{q}$ as described in previous stages, we select the variables in a group-wise fashion which involves selecting or dropping the group as a whole. Various methods have been proposed to achieve grouping effect in case of highly correlated variables, i. e. the group Lasso([@Yuan]), Group Square-Root Lasso([@Florentina]), Adaptive group Lasso([@Wei]) and Lasso on cluster representatives etc. Suppose, the selected set of groups are denoted by $$\hat{S}_{G} = \{r: \text{ group } G_r \text{ is selected}\}$$ The final selected set of variables is then the union of the selected groups. $$\hat{S}_{ACL} = \cup_{r} r \in \hat{S}_{G}$$
**Input:** dataset $(Y,X)$\
**Output:** $\hat{S}$:= set of selected variables\
**Steps:** Perform Lasso on data $(Y,X)$, Denote $\hat{S}_{Lasso}$ as variable set selected\
$S_1$ := $\hat{S}_{Lasso}$\
Let $X_{red} = X^{S_1}$ be the reduced design matrix\
Perform Clustering of variables on data $X_{red}$,\
Denote clusters as $G_1, ..., G_q$ and partition variable set as\
$\hat{S_{G_1}}$, ..., $\hat{S_{G_q}}$\
Perform group Lasso on $(Y,X_{red})$ with group information $G_1, ..., G_q$, denote the selected set of groups as\
$\hat{S}_{cluster} = \{r; \text{ cluster } G_r \text{is selected, } r = 1, . . . , q\}.$\
The union of the selected groups is then the selected set of variables\
$\hat{S}_{ACL} = \cup_{r} r \in \hat{S}_{cluster} $\
**return** $\hat{S}$
Complexity Analysis of the ACL Method
-------------------------------------
In this section, we compute time complexity of the ACL method at different stages.
1. First stage\
The time complexity of the first stage consists of the time complexity of the Lasso plus time required for variable screening. suppose $\hat{s} = |\hat{S}_{Lasso}|$, denotes the number of variables selected by Lasso, then time taken by variable screening is $\hat{s}*(p-\hat{s})$, which is $O(p\hat{s})$.\
2. Second stage\
Computationally, The second step can be completely avoided. Clustering can be done while screening correlated variables at the first stage itself. However, We opted to state the clustering method separately for transparency and for deriving its theoretical properties, in particular comparing it with other methods where clustering is performed at different stages i.e. CGLcor. But while implementing the algorithm, we can efficiently combine variable screening and clustering. So no extra computational cost is added at this stage.\
3. Third stage\
The same computational complexity as for the group Lasso, which depends on the number of groups and size of each group.
Hence the overall time complexity of the proposed method is dominated by the time required for the Lasso and the group Lasso, see [@Julien] for complexity analysis of the Lasso.
Theoretical Properties of the ACL Procedure
===========================================
In this section, we study the theoretical properties of the ACL methods, and we show that nothing is lost by using ACL methods instead of Clustering Lasso methods with correlation based clustering as proposed in [@Buhlmann2]. Particularly, under the GIR and group beta-min condition, the ACGL method has the same accuracy as the CGLcor, in terms of estimation, prediction and variable selection. The gain is in terms of computations since the ACGL performs clustering on the reduced set of predictors. We introduce the following theorems which are needed for proving variable selection consistency for the proposed algorithm.
Suppose that, the uniform-$\theta$ IR condition is met for the true active set $S := S_0$, which in turn implies that with large probability, the Lasso does not make false positive selection of variables. Then for any $S_1 \subset S $, the uniform-$\theta_1( \leq \theta)$ IR condition holds for the set $S_1$.
**Proof** We invoke the result given in the book [@Buhlmann1], Corollary 7.2. Since the uniform-$\theta$ IR condition holds for the set S, then the following inequality also holds. $$\frac{\sqrt{s} \mathop{max}_{j \not\in S} \sqrt{\sum_{k \in S} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S)) } \leq \theta$$ where $\sigma_{jk}$ denotes the $(jk)^{th}$ entry of $\Sigma$.\
Now we delete some variables from S, and denote the reduced subset as $S_1$ and corresponding partition of variance-covariance matrix is $\Sigma_{11}(S_1)$ and $\Sigma_{21}(S_1)$. Since $S1 \subset S$, the following inequality holds because any symmetric minor of $\Sigma_{11}(S)$ will have min-eigenvalues at least as big as $\Lambda_{min}(\Sigma_{11}(S))$. $$\Lambda_{min}(\Sigma_{11}(S_1)) \geq \Lambda_{min}(\Sigma_{11}(S))$$ Therefore $$\theta_1 = \frac{\sqrt{s_1} \; max_{j \not\in S_1} \sqrt{\sum_{k \in S_1} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S_1)) } \leq
\frac{\sqrt{s} \; max_{j \not\in S} \sqrt{\sum_{k \in S} \sigma^{2}_{jk}}} {\Lambda^{2}_{min}(\Sigma_{11}(S)) } \leq \theta$$ Hence the IR condition holds for the set $S_1 \subset S$.
The similar result holds for the group Lasso which is given in the following lemma.
Suppose that, the uniform-$\theta$ IR condition is met for the set of true active groups $W \subset \{1,...,q \}$, It implies that with large probability, the Lasso does not make false positive selection of groups. Then the uniform-$\theta_1( \leq \theta)$ GIR condition holds for the following cases:
- for any $W_1 \subset W $, when the number of groups are reduced.
- when $|W_1| = |W| $ but $\{G_r, r \in W_1\} \subset \{ G_r, r \in W\} $, group sizes are reduced for some groups.
- when group size as well as number of groups are reduced, $W_1 \subset W $ and $\{G_r: r \in W_1\} \subset \{ G_r: r \in W \} $.
Proof is trivial. Suppose that IR condition holds for the active set $S$. If w disjoint groups are formed within S such that $ S= \sum_{j \in W} m_j$, then IR for the Lasso imply IR condition for the group Lasso. Since the Reduced-set IR will hold for $S_1 \subset S$, where the reduced set can be interpreted as change in group structure in terms of reduced number of groups and/or reduced size of groups, as
$$\begin{aligned}
S_1 &= \sum_{r \in W_1} m_r, \quad W_1 \subset W % \\
%S_1 &= \sum_{r \in W} m_{r}, \quad |m_{r_1}| \leq |m_r| \\
%S_1 &= \sum_{r \in W_1} m_{r_1}, \quad W_1 \subset W , \quad |m_{r_1}| \leq |m_r| \end{aligned}$$
Therefore Reduced-group IR will hold for the set $S_1$.
Case Studies
------------
In this section, we illustrate the variable selection consistency of the proposed method using a couple of scenarios under assumption of GIR, group beta-min and no noise case.
### Orthonormal Case
The case $ \Sigma \approx I $ corresponds to uncorrelated variables and hence IR condition holds for any $S \subset \{ 1,...,p\}$ and we also assume that beta-min condition holds for a fixed S.
We claim that ACGL selects the true active set $\hat{S}_{ACGL} = S_0$ with high probability.\
Proof:\
First, we perform the Lasso operation on the pair (Y,X), to get the initial set of variables, say $\hat{S}_1$. Then with high probability $\hat{S_1} = S_0$ under assumption of IR and beta-min condition. Since variables are uncorrelated no additional variable will be pulled in when we do correlation screening. At the second stage, we perform clustering on the reduced set of predictors. Clustering process will report the singleton groups due to independence structure, and finally at the third step CGL will select the true active set $S_0$ again, due to reduced-set IR condition.
We proved that ACGL consistently selects the true active set for the orthonormal case. It is obvious that, there is no advantage of using AGCL or the plain CGL/CRL methods over the standard Lasso for this case. But ACL outperforms over CGL/CRL in terms of computations, since AGCL considers the reduced set of predictors for clustering and the group lasso is called for reduced number of groups, whereas plain CGL/CRL considers all p variables for clustering which requires huge computations when the dimension is ultra high.
### Block Diagonal Case
The case $ \Sigma \approx diag(\mathcal{T}_1, \mathcal{T}_2, ..., \mathcal{T}_q) $ corresponds to uncorrelated groups and hence group IR condition holds for any $W \subset \{ 1,...,q\}$ and we also assume that group beta-min condition holds. Each $\mathcal{T}_i$ is a ${m_i \times m_i}$ matrix with elements as $$(\mathcal{T}_i){j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
\rho, & else
\end{array} \right.$$ where $\rho > 0.7$, since variables are highly correlated within each group. Without loss of generality we can assume that all the variables are ordered in a way such that all active groups come first. $W \subset \{1,...,q\} $ then $W = \{1,...,w \}. $ and we also assume pure active or pure noise group. Let $$\begin{aligned}
S_0 &= \{ (1,1),..., (1,m_1), ..., (w,1),..., (w,m_w) \} = \{1,...,s_0 \},
%W^{c} &= \{1,...,p \}/W, \text{ be the noise group }\end{aligned}$$ be the true active set. The Lasso tends to select(depending on the amount of regularization) one variable from each active block. Without loss of generality we assume that the Lasso selects (j,1) variable from each $j \in W$, and the selected variable set is $\hat{S}_{Lasso} = \{ (1,1),(2,1),...,(w,1) \}$, Now we add all variables from $\{1,...,p \}/S_{Lasso}$ which are strongly correlated with atleast of the the variable from $S_{Lasso}$. Therefore we get $$\begin{aligned}
\hat{S}_1 &= \{1,...,s \}\\
\implies & \hat{S}_1 = S_0
\end{aligned}$$ Hence, after the first stage of dimensionality reduction, the selected set of variables contains the true active variables. Assuming that the clustering procedure correctly identifies the true underlying group structure, then the group Lasso at the third stage correctly selects all the w groups, due to the sub-group IR condition for the group Lasso. Hence, the proposed method consistently selects the true active set under the assumption of GIR and group beta-min condition for the block diagonal case as well.
One may argue that, there is no need for the second and the third stage. Specifically, when the Lasso selects one variable per active group then correlation screening will bring in those correlated variables which were not selected by the Lasso. We refer to the discussion in [@Junzhou], on using the group Lasso over Lasso.
Numerical Results
=================
In this section, we consider three simulation settings and a pseudo real data example in order to empirically compare the performance of the proposed method with the other existing methods. Since the comparison between the Lasso, CGL and CRL have already been studied in the paper [@Buhlmann2], here we only report the results for ACGL and CGLcor methods.
Simulation Study
----------------
Three examples are considered in this simulation. In each example, data is simulated from the linear model in (equation 1) with fixed design **X**. All the three examples are the same as simulation examples used in the paper [@Buhlmann2].
For each example, our simulated dataset consisted of a training and an independent validation set and 50 such datasets were simulated for each example. The models were fitted on the training data for each 50 datasets and the model with the lowest test set Mean Squared Error(MSPE) was selected as the final model. For model interpretation, we consider true positive rate as a measure of performance. We also measure the CPU time expended by each methods. The MSE and the true positive rate are defined as follows. $$\begin{aligned}
MSE &= \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2\\
TPR &= f(|\hat{S}|) = \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}\end{aligned}$$
### Block Diagonal Model
We generate covariates from $N_p(0,\Sigma_1)$, where $\Sigma_1$ consists of 100 block matrices $\mathcal{T}$, and $\mathcal{T} $ is a ${10 \times 10}$ matrix, defined as $$\mathcal{T}_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
.9, & else
\end{array} \right.$$ For the regression coefficient $\beta^0$ the following four configurations are considered:\
(E1.1) $S_0 = \{1, 2, . . . , 20\}$ and for any $j \in S_0$ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (a new for each simulation run). This set up has all the active variables in the first two blocks of highly correlated variables.\
(E1.2) $S_0 = \{1,2,11,12 . . . , 91,92\}$ and for any $j \in S_0$ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (a new for each simulation run). This set up has all the active variables in the first and the second variables of the first ten blocks.\
(E1.3) The $\beta^0$ has the same configuration as in (E1.1) but we change the sign of randomly chosen half of the active parameters (a new for each simulation run).\
(E1.4) The $\beta^0$ has the same configuration as in (E1.2) but we change the sign of randomly chosen half of the active parameters (a new for each simulation run).\
Simulation results are reported in table \[table:E11\](MSE and standard deviation), figure \[fig:E1\](TPR) and table \[table:E13\](CPU time).
$\sigma$ Method E1.1 E1.2 E1.3 E1.4
---------- -------- ---------------- ---------------- ---------------- ----------------
3 ACGL 12.46 (1.76) 21.95 (2.98) 9.308 (2.40) 20.90 (4.81)
CGLcor 14.97 (2.40) 37.05 (5.21) 13.34 (2.06) 24.31 (6.50)
12 ACGL 188.23 (23.32) 149.97 (28.98) 129.29 (22.93) 165.91 (22.36)
CGLcor 206.19 (29.97) 186.61 (25.69) 160.31 (23.04) 168.26 (24.70)
: MSE(sd) for Example block diagonal model[]{data-label="table:E11"}
![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for block diagonal model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E1"}](figure1)
$\sigma$ Method E2.1 E2.2 E2.2 E2.4
---------- -------- ------ ------ ------ ------
3 ACGL 4.25 5.04 2.09 6.46
CGLcor 2510 2510 2510 2510
12 ACGL 3.91 5.86 1.08 3.33
CGLcor 2510 2510 2510 2510
: CPU times(in seconds) for block diagonal model[]{data-label="table:E13"}
From table \[table:E11\], we see that the ACGL method has lower prediction error than the CGLcor for all four configurations and figure \[fig:E1\] shows that the ACGL has higher TPR. From table \[table:E13\] we see that ACGL is much efficient, the CPU time required for ACGL for all four configurations are much less than as compared to the CPU time expended by CGLcor. Please note that CPU time for CGLcor is approximately the same for all configurations.
### Single Block Design
We generate covariates from $N_p(0,\Sigma_2)$, where $\Sigma_2$ consisted of a single group of strongly correlated variables of size 30, it is defined as $$\Sigma_{2;j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
0.9& i, j \in \{1, . . . , 30\} \text{ and } i != j, \\
0& otherwise
\end{array} \right.$$ The remaining 970 variables are uncorrelated. For the regression coefficient $\beta^0$ we consider the following four configurations:\
(E2.1) $S_0 = \{1, 2, . . . , 15\} \cup \{31, 32, . . . , 35\}$ and for any $j \in S_0 $ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (new for each simulation run). The correlated block contains 15, the most of the active predictors and the remaining five active predictors are uncorrelated.\
(E2.2) $S_0 = \{1, 2, . . . , 5\} \cup \{31, 32, . . . , 45\}$ and for any $j \in S_0 $ we sample $\beta^0_{j}$ from the set $\{.1,.2,.3, . . . , 2\}$ without replacement (new for each simulation run). Here the correlated block contains only 5 active predictors, and the remaining 15 predictors are uncorrelated.\
(E2.3) The $\beta^0$ has the same configuration as in (E2.1) but we change the sign of randomly chosen half of the active parameters (new for each simulation run).\
(E2.4) The $\beta^0$ has the same configuration as in (E2.2) but we change the sign of randomly chosen half of the active parameters (new for each simulation run).
Simulation results are reported in table \[table:E21\], table \[table:E23\] and figure \[fig:E2\].
$\sigma$ Method E2.1 E2.2 E2.3 E2.4
---------- -------- ---------------- ---------------- ---------------- ----------------
3 ACGL 11.40 (4.2) 29.94 (5.34) 15.01 3.28) 27.03 (3.9)
CGLcor 247.52 (28.74) 54.73 (10.59) 21.37 (9.51) 31.58 (14.17)
12 ACGL 146.17(23.46) 192.64 (12.81) 127.91 (22.02) 159.62 (26.40)
CGLcor 384.78 (48.26) 191.26 (25.55) 159.40 (23.88) 174.49 (25.40)
: MSE(sd) for single block model[]{data-label="table:E21"}
![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for single block model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E2"}](figure2)
$\sigma$ Method E2.1 E2.2 E2.3 E2.4
---------- -------- ------ ------ ------ ------
3 ACGL 7.58 5.52 2.63 5.63
CGLcor 2463 2463 2463 2463
12 ACGL 3.17 4.92 2.37 2.47
CGLcor 2463 2463 2463 2463
: CPU times(in seconds) for single block model[]{data-label="table:E23"}
Table \[table:E21\] shows that the ACGL method has lower predictive performance than the CGLcor for all four configurations and figure \[fig:E2\] shows that in terms of variable selection, ACGL is clearly better than CGLcor. From table \[table:E21\], we see that CPU time required(in seconds) for ACGL for all four configurations are much less than as compared to the CGLcor. The CPU time for CGLcor is approximately the same for all configurations.
### Duo Block Model
We generate covariates from $N_p(0,\Sigma_3)$, where $\Sigma_3$ consists of 500 block matrices $\mathcal{T}, $ and $\mathcal{T} $ is a ${2 \times 2}$ matrix, defined as $$\mathcal{T}_{j,k} = \left\lbrace \begin{array}{cc}
1, & j=k\\
.9, & else
\end{array} \right.$$ For the regression coefficient $\beta_0$ we consider $S_0 = \{1, 2, . . . , 20\}$ and for any $j \in S_0$ $$\beta^0_{j} = \left\lbrace \begin{array}{cc}
2, & j \in \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\},\\
\frac{\frac{1}{3} \sqrt{\frac{\log p}{n}} \sigma} {1.9}, & j \in \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}
\end{array} \right.$$ In this setup, the $\beta^0$ is the same for all 50 simulation runs. The Lasso would not select the variables from $\{2, 4, 6, . . . , 20\}$, since they do not satisfy the beta-min condition but it would select the other from $\{1, 3, 5, . . . , 19\}$. The Table \[table:E31\], Table \[table:E32\] and Figure \[fig:E3\] show the simulation results for the duo block model.
$\sigma$ Method MSE(sd)
---------- -------- ----------------
3 ACGL 20.82 (5.94)
CGLcor 32.00 (6.50)
12 ACGL 179.11 (19.35)
CGLcor 193.97 (27.05)
: MSE(sd) for duo block model[]{data-label="table:E31"}
![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for duo block model. ACGL(green solid line) and CGLcor(blue dashed-dotted line)[]{data-label="fig:E3"}](figure3)
$\sigma$ Method CPU time
---------- -------- ----------
3 ACGL 7.69
CGLcor $>$ 2500
12 ACGL 2.94
CGLcor $>$ 2500
: CPU time(in seconds) for duo block model[]{data-label="table:E32"}
The results show that the ACGL performs better in terms of predictive performance, variable selection and CPU time required. (We stopped recording CPU time for CGLcor after 2500sec, here clustering of a thousand of variables and then the group lasso for $\approx$500 clusters make the process very slow.
Pseudo Real Data
----------------
We consider here a real dataset, riboflavin$(n = 71, p = 4088)$ data for the design matrix X with synthetic regression coefficients $\beta^0$ and simulated Gaussian errors $N_n(0, \sigma^2 I)$. See [@HDview] for the details on riboflavin dataset. We select the first thousand covariates which have largest empirical variances. We fix the size of the active set to $s_0 =10$. For the true active set, we randomly select a variable, say variable k(a new in each simulation), and then we select other nine variables which have highest absolute correlation to the variable k, and for each $j \in S_0$ we set $\beta_j = 1$. This configuration is exactly the same as pseudo real example used in [@Buhlmann2].
The performance measures are reported in table \[table:Ribo1\], figure \[fig:Ribo\] and table \[table:Ribo2\], based on 50 independent simulation runs. Here we compare CRLcor([@Buhlmann2]) with Adaptive Cluster Representative Lasso(ACRL) where the Lasso is used as preprocessing step at the first stage, correlation based clustering at second step and the Lasso for cluster representatives in the third stage. The Group Lasso is not appropriate for this setup, since k is chosen arbitrarily and the group size may exceed the number of observations.
$\sigma$ Method MSPE(std)
---------- -------- ---------------
3 ACRL 2.36 (0.52)
CRLcor 39.02 (25.15)
15 ACRL 25.02 (4.03)
CRLcor 50.40 (27.68)
: MSE(sd) for Riboflavin dataset[]{data-label="table:Ribo1"}
\[h!\]
![Plot of $ \frac{ | \hat{S} \bigcap S_0 |} {|S_0|}$ vs. $|\hat{S}|$ for Riboflavin dataset. ACRL(green solid line) and CRLcor(magenta dashed-dotted line)[]{data-label="fig:Ribo"}](figure4)
$\sigma$ Method E3
---------- -------- ------
3 ACGL 1.47
CRLcor 2347
12 ACGL 1.48
CRLcor 2471
: CPU times(in seconds) for Riboflavin dataset[]{data-label="table:Ribo2"}
The table \[table:Ribo1\], figure \[fig:Ribo\] and table \[table:Ribo2\] show that ACRL performs better than CRLcor in terms of prediction, variable selection and CPU time consumption.
Computational Details
=====================
Statistical analysis was performed in R 3.2.2. We used, the package “glmnet" for penalized regression methods(the Lasso and adaptive Lasso), the package “gglasso" to perform group Lasso and the package “ClustOfVar" for clustering of variables.
Conclusion and Future Work
==========================
In this article, we proposed a three stage procedure for variable selection for high-dimensional linear model with strongly correlated variables. Our procedure is an extension of the algorithms proposed in [@Buhlmann2]. A technical extension compared with [@Buhlmann2] is that we propose to reduce the dimension at the first stage using Lasso(or its adaptive or thresholded version) prior to clustering at the second stage and then supervised selection of clusters in the third stage. We proved that the variables selected by our algorithm contains the true active set consistently under GIR and group beta-min conditions. Our simulation studies show that reducing dimension improves the speed and accuracy of the clustering process and then considering correlation structure improves variable selection and predictive performance.
Since the theoretical results we developed for our algorithms are not restricted to the squared error loss, it can be extended to the generalized linear models, i.e, the preprocessing step of dimensionality reduction can be added to the group Lasso for the logistic regression([@Meier]), and this is our future work.
References {#references .unnumbered}
==========
| ArXiv |
---
author:
- 'D. Hutsemékers [^1]'
- 'J. Manfroid'
- 'E. Jehin'
- 'J.-M. Zucconi'
- 'C. Arpigny'
title: 'The [$^{16}$OH/$^{18}$OH]{} and [OD/OH]{} isotope ratios in comet C/2002 T7 (LINEAR) [^2]'
---
Introduction {#sect:intro}
============
The determination of the abundance ratios of the stable isotopes of the light elements in different objects of the Solar System provides important clues into the study of their origin and history. This is especially true for comets which carry the most valuable information regarding the material in the primitive solar nebula.
The [$^{16}$O/$^{18}$O]{} isotopic ratio has been measured from space missions in a few comets. In-situ measurements with the neutral and ion mass spectrometers onboard the Giotto spacecraft gave [$^{16}$O/$^{18}$O]{} = 495$\pm$37 for H$_2$O in comet 1P/Halley (Eberhardt et al. [@Eberhardt]). A deep integration of the spectrum of the bright comet 153P/2002 C1 (Ikeya-Zhang) with the sub-millimeter satellite Odin led to the detection of the H$_2^{18}$O line at 548 GHz (Lecacheux et al. [@Lecacheux]). Subsequent observations resulted in the determination of [$^{16}$O/$^{18}$O]{} = 530$\pm$60, 530$\pm$60, 550$\pm$75 and 508$\pm$33 in the Oort-Cloud comets Ikeya-Zhang, C/2001 Q4, C/2002 T7 and C/2004 Q2 respectively (Biver et al. [@Biver]). Within the error bars, these measurements are consistent with the terrestrial value ([$^{16}$O/$^{18}$O]{} [(SMOW[^3])]{} = 499), although marginally higher (Biver et al. [@Biver]). More recently, laboratory analyses of the silicate and oxide mineral grains from the Jupiter family comet 81P/Wild 2 returned by the Stardust space mission provided [$^{16}$O/$^{18}$O]{} ratios also in excellent agreement with the terrestrial value. Only one refractory grain appeared marginally depleted in $^{18}$O ([$^{16}$O/$^{18}$O]{} = 576$\pm$78) as observed in refractory inclusions in meteorites (McKeegan et al. [@McKeegan]).
The D/H ratio has been measured in four comets. In-situ measurements provided D/H = 3.16$\pm$0.34 10$^{-4}$ for H$_2$O in 1P/Halley (Eberhardt et al. [@Eberhardt], Balsiger et al. [@Balsiger]), a factor of two higher than the terrestrial value (D/H [(SMOW)]{} = 1.556 10$^{-4}$). The advent of powerful sub-millimeter telescopes, namely the Caltech Submillimeter Observatory and the James Clerck Maxwell telescope located in Hawaii, allowed the determination of the D/H ratio for two exceptionally bright comets. In comet C/1996 B2 (Hyakutake), D/H was found equal to 2.9$\pm$1.0 10$^{-4}$ in H$_2$O (Bockelée-Morvan et al. [@Bockelee]), while, in comet C/1995 O1 (Hale-Bopp), the ratios D/H = 3.3$\pm$0.8 10$^{-4}$ in H$_2$O and D/H = 2.3$\pm$0.4 10$^{-3}$ in HCN were measured (Meier et al. [@Meier1; @Meier2]), confirming the high D/H value in comets. Both Hyakutake and Hale-Bopp are Oort-Cloud comets. Finally, bulk fragments of 81P/Wild 2 grains returned by Stardust indicated moderate D/H enhancements with respect to the terrestrial value. Although D/H in 81P/Wild 2 cannot be ascribed to water, the measured values overlap the range of water D/H ratios determined in the other comets (McKeegan et al. [@McKeegan]).
Among a series of spectra obtained with UVES at the VLT to measure the [$^{14}$N/$^{15}$N]{} and [$^{12}$C/$^{13}$C]{} isotope ratios in various comets from the 3880$\,$ÅCN ultraviolet band (e.g. Arpigny et al. [@Arpigny], Hutsemékers et al. [@Hutsemekers], Jehin et al. [@Jehin2], Manfroid et al. [@Manfroid2]), we found that the spectrum of [C/2002 T7]{} appeared bright enough to detect the $^{18}$OH lines in the $A\,^{2}\Sigma^{+}
- X\,^{2}\Pi_{i}$ bands at 3100 Å allowing –for the first time– the determination of the [$^{16}$O/$^{18}$O]{} ratio from [*ground-based*]{} observations. We also realized that the signal-to-noise ratio of our data was sufficient to allow a reasonable estimate of the [OD/OH]{} ratio from the same bands.
The possibility of determining the [$^{16}$O/$^{18}$O]{} ratio from the OH ultraviolet bands has been emphasized by Kim ([@Kim]). Measurements of the OD/OH ratio were already attempted by A’Hearn et al. ([@AHearn]) using high resolution spectra from the International Ultraviolet Explorer and resulting in the upper limit D/H $< 4 \, 10^{-4}$ for comet C/1989 C1 (Austin). These observations now become feasible from the ground thanks to the high ultraviolet throughput of spectrographs like UVES at the VLT.
Observations and data analysis
==============================
Observations of comet [C/2002 T7]{} were carried out with UVES mounted on the 8.2m UT2 telescope of the European Southern Observatory VLT. Spectra in the wavelength range 3040$\,$Å–10420$\,$Å were secured in service mode during the period May 6, 2004 to June 12, 2004. The UVES settings 346+580 and 437+860 were used with dichroic \#1 and \#2 respectively. In the following, only the brighest ultraviolet spectra obtained on May 6, May 26 and May 28 are considered. The 0.44 $\times$ 10.0 arcsec slit provided a resolving power $R \simeq 80000$. The slit was oriented along the tail, centered on the nucleus on May 26, and off-set from the nucleus for the May 6 and May 28 observations. The observing circumstances are summarized in Table \[tab:obs\].
[lcccccr]{}\
Date & $r$ & $\dot{r}$ & $\Delta$ & Offset & $t$ & Airmass\
(2004) & (AU) & (km/s) & (AU) & (10$^{3}$ km) & (s) &\
\
May 6 & 0.68 & 15.8 & 0.61 & 1.3 & 1080 & 2.2-1.9\
May 26 & 0.94 & 25.6 & 0.41 & 0.0 & 2677 & 1.3-1.8\
May 26 & 0.94 & 25.6 & 0.41 & 0.0 & 1800 & 2.1-2.7\
May 28 & 0.97 & 25.9 & 0.48 & 10.0 &3600 & 1.3-1.7\
\
\
[$r$ and $\dot{r}$ are the comet heliocentric distance and radial velocity; $\Delta$ is the geocentric distance; $t$ is the exposure time; Airmass is given at the beginning and at the end of the exposure]{}
The spectra were reduced using the UVES pipeline (Ballester et al. [@Ballester]), modified to accurately merge the orders taking into account the two-dimensional nature of the spectra. The flat-fields were obtained with the deuterium lamp which is more powerful in the ultraviolet.
The data analysis and the isotopic ratio measurements were performed using the method designed to estimate the carbon and nitrogen isotopic ratios from the CN ultraviolet spectrum (Arpigny et al. [@Arpigny], Jehin et al. [@Jehin] and Manfroid et al. [@Manfroid]). Basically, we compute synthetic fluorescence spectra of the $^{16}$OH, $^{18}$OH and $^{16}$OD for the $A\,^{2}\Sigma^{+} - X\,^{2}\Pi_{i}$ (0,0) and (1,1) ultraviolet bands for each observing circumstance. Isotope ratios are then estimated by fitting the observed OH spectra with a linear combination of the synthetic spectra of the two species of interest.
The OH model
------------
We have developed a fluorescence model for OH similar to the one described by Schleicher and A’Hearn ([@Schleicher]). As lines of the OH(2-2) bands are clearly visible in our spectra we have included vibrational states up to $v=2$ in the A$^2\Sigma^+$ and X$^2\Pi_i$ electronic states. For each vibrational state rotational levels up to $J=11/2$ were included, leading to more than 900 electronic and vibration-rotation transitions. The system was then solved as described in Zucconi and Festou ([@Zucconi]).
Accurate OH wavelengths were computed using the spectroscopic constants of Colin et al. ([@Colin]) and Stark et al. ([@Stark]). OD wavelengths were computed using the spectroscopic constants of Abrams et al. ([@Abrams]) and Stark et al. ([@Stark]). $^{18}$OH wavelengths were derived from the $^{16}$OH ones using the standard isotopic shift formula; they are consistent with the measured values of Cheung et al. ([@Cheung]).
Electronic transition probabilities for OH and OD are given by Luque and Crosley ([@Luque1; @Luque2]). We used the dipole moments of OH and OD measured by Peterson et al. ([@Peterson]) to compute the rotational transition probabilities and the vibrational lifetimes computed by Mies ([@Mies]). Because of the very small difference in the structure of $^{18}$OH and $^{16}$OH the transition probabilities for $^{18}$OH and $^{16}$OH are the same.
The OH fluorescence spectrum is strongly affected by the solar Fraunhofer lines, especially in the 0-0 band, so a carefully calibrated solar atlas is required. We have used the Kurucz ([@Kurucz]) atlas above 2990 Å and the A’Hearn et al. ([@AHearn1]) atlas below.
The role of collisions in the OH emission, in particular those with charged particles inducing transitions in the $\Lambda$ doublet ground rotational state, was first pointed out by Despois et al. ([@Despois]) in the context of the 18 cm radio emission and then also considered in the UV emission by Schleicher ([@SchleicherPHD], Schleicher and A’Hearn [@Schleicher]). Modeling the effect of collisions may be done by adding the collision probability transition rate between any two levels, $i$ and $j$: $$C_{i,j} = \sum_c{n_c({\bf r})\,{\rm v}_c({\bf r})\,\sigma_c(i,j,{\rm v}_c)}$$ where the sum extends over all colliders. $n_c$ is the local density of the particles inducing the transition, ${\rm v}_c$ is the relative velocity of the particles and $\sigma_c$ is the collision cross section. It also depends on the energy of the collision i.e. of ${\rm
v}_c$. The reciprocal transition rates are obtained through detailed balance: $$C_{j,i} = C_{i,j}\frac{g_i}{g_j}\exp(E_{ij}/kT)$$ in which $g_i$ is the statistical weight and $E_{ij}$ is the energy separation between the states. In order to reduce the number of parameters required to model the collisions we have adopted a simplified expression of the form $C_{i,j} = q_\Lambda$ for the transition in the $\Lambda$ doublet ground state. In order to better fit the OH spectra we have also found necessary to take into account rotational excitation through a similar expression $C_{i,j} = q_{rot}$ with $q_{rot}$ different from 0 only for dipole transitions, i.e. when $\Delta J < 2$, which appeared to correctly fit the data. Furthermore, since OH and OD have similar dipole moments, we assumed that collisional cross-sections are identical for both molecules.
The model assumes that the $^{16}$OH lines are optically thin. This is verified by the fact that it correctly reproduces both the faint and strong OH emission lines.
[$^{16}$OH/$^{18}$OH]{}
-----------------------
Two $^{18}$OH lines at 3086.272 Å and 3091.046 Å are clearly detected in the (0,0) band. However these lines are strongly blended with the $\sim$ 500 times brighter $^{16}$OH emission lines and then not useful for an accurate flux estimate. In fact the (1,1) band at 3121 Å, while fainter, is better suited for the determination of [$^{16}$OH/$^{18}$OH]{} since (i) the wavelength separation between $^{18}$OH and $^{16}$OH is larger ($\simeq$ 0.3 Å instead of 0.1 Å), and (ii) the sensitivity of UVES rapidly increases towards longer wavelengths while the atmospheric extinction decreases, resulting in a better signal-to-noise ratio.
Fig. \[fig:fig1\] illustrates a part of the observed OH (1,1) band together with the synthetic spectrum from the model. Two $^{18}$OH lines are clearly identified.
To actually evaluate [$^{16}$OH/$^{18}$OH]{} we first select the 3 brighest and best separated $^{18}$OH lines at $\lambda$ = 3134.315$\,$Å, 3137.459$\,$Å and 3142.203$\,$Å. These lines are then doppler-shifted and co-added with proper weights to produce an average profile which is compared to the $^{16}$OH profile similarly treated (cf. Jehin et al. [@Jehin] for more details on the method). We verified that the $^{16}$OH faint wings and nearby prompt emission lines (analysed in detail in a forthcoming paper) do not contaminate the $^{18}$OH lines nor the measurement of the isotopic ratios. The ratio [$^{16}$OH/$^{18}$OH]{} is then derived through an iterative procedure which is repeated for each spectrum independently. For the spectra of May 6, 26 and 28 we respectively derive [$^{16}$OH/$^{18}$OH]{} = 410$\pm$60, 510$\pm$130 and 380$\pm$290. The uncertainties are estimated from the co-added spectra by considering the rms noise in spectral regions adjacent to the $^{18}$OH lines, and by evaluating errors in the positioning of the underlying pseudo-continuum (i.e. the dust continuum plus the faint wings of the strong lines). The weighted average of all measurements gives [$^{16}$OH/$^{18}$OH]{} = 425 $\pm$ 55.
Since OH is essentially produced from the dissociation of H$_2$O, [$^{16}$OH/$^{18}$OH]{} represents the [$^{16}$O/$^{18}$O]{} ratio in cometary water, with the reasonable assumption that photodissociation cross-sections are identical for H$_2$$^{18}$O and H$_2$$^{16}$O.
[OD/OH]{}
---------
The detection of OD lines is much more challenging since one may expect the OD lines to be a few thousand times fainter than the OH lines. Fortunately, the wavelength separation between OD and OH ($\gtrsim$ 10 Å) is much larger than between $^{18}$OH and $^{16}$OH such that both the (0,0) and (1,1) bands can be used with no OD/OH blending (apart from chance coincidences). Since no individual OD lines could be detected, we consider the 30 brighest OD lines (as predicted by the model) for co-addition. After removing 3 of them, blended with other emission lines, an average profile is built with careful Doppler-shifting and weighting as done for $^{18}$OH. Only our best spectra obtained on May 6 and May 26 are considered, noting that the (0,0) band –which dominates the co-addition– is best exposed on May 26 while the (1,1) band is best exposed on May 6, due to the difference in airmass. The resulting OD line profiles are illustrated in Fig. \[fig:fig3\] and \[fig:fig4\] and compared to a synthetic spectrum computed with [OD/OH]{} = 4 10$^{-4}$. OD is detected as a faint emission feature which is present [*at both epochs*]{}. From the measurement of the line intensities, we derive [OD/OH]{} = 3.3$\pm$1.1 10$^{-4}$ and 4.1$\pm$2.0 10$^{-4}$ for the spectra obtained on May 6 and 26 respectively. The weighted average is [OD/OH]{} = 3.5$\pm$1.0 10$^{-4}$. The difference in the lifetime of OD and OH (van Dishoeck and Dalgarno [@Vandishoeck]) does not significantly affect our results since the part of the coma sampled by the UVES slit is two orders of magnitude smaller than the typical OH scale-length. The uncertainties on [OD/OH]{} were estimated as for [$^{16}$OH/$^{18}$OH]{}. Possible errors on the isotopic ratios related to uncertainties on the collision coefficients were estimated via simulations and found to be negligible. Even in the hypothetical case that collisions differently affect OD and OH, errors are much smaller than the other uncertainties, as expected since the contribution of collisions is small with respect to the contribution due to pure fluorescence.
To estimate the cometary D/H ratio in water, HDO/H$_2$O must be evaluated. While the cross-section for photodissociation of HDO is similar to that of H$_2$O, the production of OD+H is favoured over OH+D by a factor around 2.5 (Zhang and Imre [@Zhang], Engel and Schinke [@Engel]). Assuming that the total branching ratio for HDO $\rightarrow$ OD + H plus HDO $\rightarrow$ OH + D is equal to that of H$_2$O $\rightarrow$ OH + H, we find HDO/H$_2$O $\simeq$ 1.4 OD/OH. With D/H = 0.5 HDO/H$_2$O, we finally derive D/H = 2.5$\pm$0.7 10$^{-4}$ in cometary water. The factor (OD+H)/(OH+D) = 2.5 adopted in computing the branching ratios for the photodissociation of HDO is an average value over the spectral region where the cross-sections peak. In fact (OD+H)/(OH+D) depends on the wavelength and roughly ranges between 2 and 3 over the spectral regions where absorption is significant (Engel and Schinke [@Engel], Zhang et al. [@Zhang2], Yi et al. [@Yi]). Fortunately, even if we adopt the extreme ratios (OD+H)/(OH+D) = 2 or (OD+H)/(OH+D) = 3 instead of 2.5, the value of the D/H isotopic ratio is not changed by more than 6%.
Discussion
==========
We have measured the oxygen isotopic ratio [$^{16}$O/$^{18}$O]{} = 425 $\pm$ 55 from the OH $A\,^{2}\Sigma^{+} - X\,^{2}\Pi_{i}$ ultraviolet bands in comet [C/2002 T7]{}. Although marginally smaller, our value do agree within the uncertainties with [$^{16}$O/$^{18}$O]{} = 550 $\pm$ 75 estimated from observations by the Odin satellite (Biver et al. [@Biver]), with the [$^{16}$O/$^{18}$O]{}ratios determined in other comets, and with the terrestrial value (Sect. \[sect:intro\]).
To explain the so-called “oxygen anomaly” i.e. the fact that oxygen isotope variations in meteorites cannot be explained by mass-dependent fractionation, models of the pre-solar nebula based on CO self-shielding were proposed, predicting enrichments, with respect to the [SMOW]{} value, of $^{18}$O in cometary water up to [$^{16}$O/$^{18}$O]{}$\sim$ 415 (Yurimoto & Kuramoto [@Yurimoto]). Recently, Sakamoto et al. ([@Sakamoto]) found evidence for such an enrichment in a primitive carbonaceous chondrite, supporting self-shielding models. The value of [$^{16}$O/$^{18}$O]{} we found in [C/2002 T7]{} is also marginally smaller than the terrestrial value and compatible with these predictions. On the other hand, the measurement of [$^{16}$O/$^{18}$O]{} = 440 $\pm$ 6 in the solar photosphere (Ayres et al. [@Ayres]; cf. Wiens et al. [@Wiens] for a review of other, less accurate, measurements) indicates that solar ratios may deviate from the terrestrial ratios by much larger factors than anticipated, requiring some revision of the models. More observations are then critically needed to get an accurate value of [$^{16}$O/$^{18}$O]{} in comets, assuming that cometary water is pristine enough and can be characterized by a small set of representative values. Namely, if self-shielding is important in the formation of the solar system, it is not excluded that significant variations can be observed between comets formed at different locations in the solar system, like Oort cloud and Jupiter-family comets.
We also detected OD and estimated D/H = 2.5 $\pm$ 0.7 10$^{-4}$ in water. Our measurement is compatible with other values of D/H in cometary water and marginally higher than the terrestrial value (Sect. \[sect:intro\]). Our observations were not optimized for the measurement of OD/OH (neither for [$^{16}$OH/$^{18}$OH]{}) and one of our best spectra was obtained at airmass $\sim$ 2 with less than 20 min of exposure time for a comet of heliocentric magnitude m$_r \simeq$ 5 (for comparison, comet Hale-Bopp reached m$_r \simeq$ $-$1). All these observing circumstances can be improved, including observations at negative heliocentric velocities to increase the OD/OH fluorescence efficiency ratio (cf. figure 1 of A’Hearn et al. [@AHearn]). This opens the possibility to routinely measure both the [$^{16}$O/$^{18}$O]{} and D/H ratios from the ground, together with the [$^{12}$C/$^{13}$C]{} and [$^{14}$N/$^{15}$N]{} ratios, for a statistically significant sample of comets of different types (e.g. Oort-cloud, Halley-type, and hopefully Jupiter-family comets although the latter are usually fainter). The measurement of D/H is especially important since it allows to limit the contribution of comets to the terrestrial water, the high D abundance implying that no more than about 10 to 30% of Earth’s water can be attributed to comets (e.g. Eberhardt et al. [@Eberhardt], Dauphas et al. [@Dauphas], Morbidelli et al. [@Morbidelli]). However, only a full census of D/H in comets could answer this question. In particular, if Jupiter-family comets, thought to have formed in farther and colder places in the Solar System, are characterized by an even higher D/H, closer to the ratio measured in the interstellar medium water, then the fraction of cometary H$_2$O brought onto the Earth could be even smaller.
We thank the referee, Dominique Bockelée-Morvan, for comments which helped to significantly improve the manuscript. We are also grateful to Paul Feldman and Hal Weaver for useful discussions.
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[^1]: DH is Senior Research Associate FNRS; JM is Research Director FNRS; and EJ is Research Associate FNRS
[^2]: Based on observations collected at the European Southern Observatory, Paranal, Chile (ESO Programme 073.C-0525).
[^3]: Standard Mean Ocean Water
| ArXiv |
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abstract: 'We use direct numerical simulations to investigate the interaction between the temperature field of a fluid and the temperature of small particles suspended in the flow, employing both one and two-way thermal coupling, in a statistically stationary, isotropic turbulent flow. Using statistical analysis, we investigate this variegated interaction at the different scales of the flow. We find that the variance of the fluid temperature gradients decreases as the thermal response time of the suspended particles is increased. The probability density function (PDF) of the fluid temperature gradients scales with its variance, while the PDF of the rate of change of the particle temperature, whose variance is associated with the thermal dissipation due to the particles, does not scale in such a self-similar way. The modification of the fluid temperature field due to the particles is examined by computing the particle concentration and particle heat fluxes conditioned on the magnitude of the local fluid temperature gradient. These statistics highlight that the particles cluster on the fluid temperature fronts, and the important role played by the alignments of the particle velocity and the local fluid temperature gradient. The temperature structure functions, which characterize the temperature fluctuations across the scales of the flow, clearly show that the fluctuations of the fluid temperature increments are monotonically suppressed in the two-way coupled regime as the particle thermal response time is increased. Thermal caustics dominate the particle temperature increments at small scales, that is, particles that come into contact are likely to have very large differences in their temperature. This is caused by the nonlocal thermal dynamics of the particles, and the scaling exponents of the inertial particle temperature structure functions in the dissipation range reveal very strong multifractal behavior. Further insight is provided by the PDFs of the two-point temperature increments and by the flux of temperature increments across the scales. All together, these results reveal a number of non-trivial effects, with a number of important practical consequences.'
author:
- 'M. Carbone, A. D. Bragg'
- 'M. Iovieno'
bibliography:
- 'JFM2018.bib'
title: 'Multiscale fluid–particle thermal interaction in isotropic turbulence'
---
Introduction
============
The interaction between inertial particles and scalar fields in turbulent flows plays a central role in many natural problems, ranging from cloud microphysics [@Pruppacher2010; @Grabowski2013] to the interactions between plankton and nutrients [@DeLillo2014], and dust particle flows in accretion disks [@Takeuchi2002]. In engineered systems, applications involve chemical reactors and combustion chambers, and more recently, microdispersed colloidal fluids where the enhanced thermal conductivity due to particle aggregations can give rise to non-trivial thermal behavior [@Prasher2006; @Momenifar2015], and which can be used in cooling devices for electronic equipment exposed to large heat fluxes [@Das2006].
In this work, we focus on the heat exchange between advected inertial particles and the fluid phase in a turbulent flow, with a parametric emphasis relevant to understanding particle-scalar interactions in cloud microphysics. Understanding the droplet growth in clouds requires to characterize the interaction between water droplets and the humidity and temperature fields. A major problem is to understand how the interaction between turbulence, heat exchange, condensational processes, and collisions can produce the rapid growth of water droplets that leads to rain initiation [@Pruppacher2010; @Grabowski2013]. While the study of the transport of scalar fields and particles in turbulent flows are well established research areas in both theoretical and applied fluid dynamics [@Kraichnan1994; @Taylor1922], the characterization of the interaction between scalars and particles in turbulent flows is a relatively new topic [@Bec2014], since the problem is hard to handle analytically, requires sophisticated experimental techniques, and is computationally demanding.
When temperature differences inside the fluid are sufficiently small, the temperature field behaves almost like a passive scalar, that is, the fluid temperature is advected and diffused by the fluid motion but has negligible dynamical effect on the flow. Even in this regime, the statistical properties of the passive scalar field are significantly different from those of the underlying velocity field that advects it. Different regimes take place according to the Reynolds number and the ratio between momentum and scalar diffusivities [@Shraiman2000; @Warhaft2000; @Watanabe2004].
Experiments, numerical simulations and analytical models show that a passive scalar field is always more intermittent than the velocity field, and passive scalars in turbulence are characterized by strong anomalous scaling [@Holzer1994]. This is due to the formation of ramp–cliff structures in the scalar field [@Celani2000; @Watanabe2004]: large regions in which the scalar field is almost constant are separated by thin regions in which the scalar abruptly changes. The regions in which the scalar mildly changes are referred to as Lagrangian coherent structures. The thin regions with large scalar gradient, where the diffusion of the scalar takes place, are referred to as fronts. It has been shown that the large scale forcing influences the passive scalar statistics at small scales [@Gotoh2015]. In particular, a mean scalar gradient forcing preserves universality of the statistics while a large scale Gaussian forcing does not. However, the ramp-cliff structure was observed with different types of forcing, implying that this structure is universal to scalar fields in turbulence [@Watanabe2004; @Bec2014]. Moreover, recent measurements of atmospheric turbulence have shown that external boundary conditions, such as the magnitude and sign of the sensible heat flux, have a significant impact on the fluid temperature dynamics within the inertial range, while for the same scales the fluid velocity increments are essentially independent of these large-scale conditions [@zorzetto18].
When a turbulent flow is seeded with inertial particles, the particles can sample the surrounding flow in a non-uniform and correlated manner [@Toschi2009]. Particle inertia in a turbulent flow is measured through the Stokes number ${\text{\textit{St}}}\equiv\tau_p/\tau_\eta$, which compares the particle response time to the Kolmogorov time scale. A striking feature of inertial particle motion in turbulent flows is that they spontaneously cluster even in incompressible flows [@maxey87; @wang93; @Bec2007; @Ireland2016]. This clustering can take place across a wide range of scales [@Bec2007; @Bragg2015b; @Ireland2016], and the small-scale clustering is maximum when ${\text{\textit{St}}}={\textit{O}\left( 1 \right)}$. A variety of mechanisms has been proposed to explain this phenomena: when ${\text{\textit{St}}}\ll1$ the clustering is caused by particles being centrifuged out of regions of strong rotation [@maxey87; @chun05], while for ${\text{\textit{St}}}\geq {\textit{O}\left( 1 \right)}$, a non-local mechanism generates the clustering, whose effect is related to the particles memory of its interaction with the flow along its path-history [@gustavsson11b; @gustavsson16; @bragg14b; @bragg2015a; @Bragg2015b]. Note that recent results on the clustering of settling inertial particles in turbulence have corroborated this picture, showing that strong clustering can occur even in a parameter regime where the centrifuge effect cannot be invoked as the explanation for the clustering, but is caused by a non-local mechanism [@ireland16b].
When particles have finite thermal inertia, they will not be in thermal equilibrium with the fluid temperature field, and this can give rise to non-trivial thermal coupling between the fluid and particles in a turbulent flow. A thermal response time $\tau_\theta$ can be defined so that the particle thermal inertia is parameterized by the thermal Stokes number ${\text{\textit{St}}}_\theta \equiv \tau_\theta/\tau_\eta$ [@Zaichik2009]. Since both the fluid temperature and particle phase-space dynamics depend upon the fluid velocity field, there can exist non-trivial correlations between the fluid and particle temperatures even in the absence of thermal coupling. Indeed, it was show by [@Bec2014] that inertial particles preferentially cluster on the fronts of the scalar field. Associated with this is that the particles preferentially sample the fluid temperature field, and when combined with the strong intermittency of temperature fields in turbulent flows, that can cause particles to experience very large temperature fluctuations along their trajectories.
Several works have considered aspects of the fluid-particle temperature coupling using numerical simulations. For example, [@Zonta2008] investigated a particle-laden channel flow, with a view to modeling the modification of heat transfer in micro–dispersed fluids. They considered both momentum and temperature two–way coupling and observed that, depending on the particle inertia, the heat flow at the wall can increase or decrease. [@Kuerten2011] considered a similar set-up with larger dispersed particles, and they observed a stronger modification of the fluid temperature statistics due to the particles. [@Zamansky2014; @Zamansky2016] considered turbulence induced by buoyancy, where the buoyancy was generated by heated particles. They observed that the resulting flow is driven by thermal plumes produced by the particles. As the particle inertia was increased, the inhomogeneity and the effect of the coupling were enhanced in agreement with the fact that inertial particles tend to cluster on the scalar fronts. [@Kumar2014] examined how the spatial distribution of droplets is affected by large scale inhomogeneities in the fluid temperature and supersaturation fields, considering the transition between homogeneous and inhomogeneous mixing. A similar flow configuration was also investigated by [@Gotzfried2017].
Each of these studies was primarily focused on the effect of the inertial particles on the large-scale statistics of the fluid temperature field. However, the results of [@Bec2014] imply that the effects of fluid-particle thermal coupling could be strong at the small scales, owing to the fact that they cluster on the fronts of the temperature field. Moreover, there is a need to understand and characterize the multiscale thermal properties of the particles themselves. In order to address these issues, we have conducted direct numerical simulations (DNS) to investigate the interaction between the scalar temperature field and the temperature of inertial particles suspended in the fluid, with one and two-way thermal coupling, in statistically stationary, isotropic turbulence. Using statistical analysis, we probe the multiscale aspects of the problem and consider the particular ways that the inertial particles contribute to the properties of the fluid temperature field in the two-way coupled regime.
The paper is organized as follows. In section \[sec:simul\] we present the physical model used in the DNS, and present the parameters in the system. In section \[sec:diss\] the statistics of the fluid temperature and time derivative of the particle temperature are considered, which allow us to quantify the contributions to the thermal dissipation in the system from the fluid and particles. In section \[sec:fluct\] we consider the statistics of the fluid and particle temperature. In section \[sec:FATT\] we consider the heat flux due to the particle motion conditioned on the local fluid temperature gradients in order to obtain insight into the details of the thermal coupling. In section \[sec:SF\] we consider the structure functions of the fluid and particle temperature increments, along with their scaling exponents. In section \[sec:2pPDF\] we consider the probability density functions (PDFs) of the fluid and particle temperature increments, along with the PDFs of the fluxes of the fluid and particle temperature increments across the scales of the flow. Finally, concluding remarks are given in section \[sec:concl\].
The physical model {#sec:simul}
==================
In this section we present the governing equations of the physical model which will be solved numerically to simulate the thermal coupling and behavior of a particle-laden turbulent flow.
Fluid phase
-----------
We consider a statistically stationary, homogeneous and isotropic turbulent flow, governed by the incompressible Navier-Stokes equations. The turbulent velocity field advects the fluid temperature field (assumed a passive scalar), together with the inertial particles. In this study, we account for two-way thermal coupling between the fluid and particles, but only one-way momentum coupling. Therefore, the governing equations for the fluid phase are $$\begin{aligned}
\bnabla\bcdot{ \mathbf{u} } &=& 0 \label{NScontinuity},\\
\partial_t { \mathbf{u} } + { \mathbf{u} }\bcdot\bnabla{ \mathbf{u} } &=& -\frac{1}{\rho_0}\bnabla p + \nu\nabla^2 { \mathbf{u} } + { \mathbf{f} } \label{NSmomentum},\\
\partial_t T + { \mathbf{u} }\bcdot\bnabla T &=& \kappa \nabla^2 T - C_T + f_T \label{NSscalar}.\end{aligned}$$ Here ${ \mathbf{u} }\left({ \mathbf{x} },t\right)$ is the velocity of the fluid, $p\left({ \mathbf{x} },t\right)$ is the pressure, $\rho_0$ is the density of the fluid, $\nu$ is its kinematic viscosity, $T\left({ \mathbf{x} },t\right)$ is the temperature of the fluid and $\kappa$ is the thermal diffusivity. The ratio between the the momentum diffusivity $\nu$ and the thermal diffusivity $\kappa$ defines the Prandtl number $\Pran\equiv\nu/\kappa$. In this work, we consider $\Pran=1$, leaving further exploration of its effect on the system to future work. The ${ \mathbf{f} }$ and $f_T$ terms in equations and represent the external forcing, and $C_T$ is the thermal feedback of the particles on the fluid temperature field, that is, the heat exchanged per unit time and unit volume between the fluid and particles at position ${ \mathbf{x} }$.
When the forcing is confined to sufficiently large scales, it is assumed that the details of the forcing do not influence the small-scale dynamics. Previous experimental evidence seems to confirm this [@Sreenivasan1996], leading to a universal behaviour of the small-scales. However, recent studies [@Gotoh2015] pointed out that this hypothesis of universality is partially violated by the advected scalar fields, whose inertial range statistics exhibit sensitivity to the details of the imposed forcing. Since we aim to characterize temperature and temperature gradient fluctuations in the dissipation range for different inertia of the suspended particles, we employ a forcing that imposes the same total dissipation rate for all the simulations. This produces results which can be meaningfully compared for different parameters of the suspended particles, since the response of the system to the same injected thermal power can be examined. Therefore, we employ a large scale forcing which imposes the average dissipation rate [@Kumar2014], that is $$\hat{{ \mathbf{f} }}({ \mathbf{k} }) = \varepsilon \frac{\hat{{ \mathbf{u} }}({ \mathbf{k} })}
{\sum_{{ \mathbf{k} }\in\mathcal{K}_f} \left\Vert \hat{{ \mathbf{u} }}({ \mathbf{k} }) \right\Vert^2 },
\quad
\hat{f}_T({ \mathbf{k} }) = \chi\frac{\hat{T}({ \mathbf{k} })}{\sum_{{ \mathbf{k} }\in\mathcal{K}_f} \left\vert \hat{T}({ \mathbf{k} }) \right\vert^2 },$$ in the wavenumber space (a hat indicates the Fourier transform and ${ \mathbf{k} }$ is the wavenumber). Here $\mathcal{K}_f$ is the set of forced wavenumbers while $\varepsilon$ and $\chi$ are the imposed dissipation rates of velocity and temperature variance, respectively. Since both the velocity and temperature statistics at large scales tend to be close to Gaussian, this forcing behaves similarly to a random Gaussian forcing. The value of the parameters relative to the fluid flow, employed in the simulations are in table \[tab:flow\].
------------------------------------------- ---------------- -------------
Kinematic viscosity $\nu$ 0.005
Prandtl number $\Pran$ 1
Velocity fluctuations dissipation rate $\varepsilon$ 0.27
Temperature fluctuations dissipation rate $\chi$ $0.1$
Kolmogorov time scale $\tau_\eta$ 0.136
Kolmogorov length scale $\eta$ 0.0261
Taylor micro-scale $\lambda$ 0.498
Integral length scale $\ell$ 1.4
Root mean square velocity $u'$ 0.88
Kolmogorov velocity scale $u_\eta$ 0.192
Small scale temperature $T_\eta$ $0.117$
Taylor Reynolds number $\Rey_\lambda$ 88
Integral scale Reynolds number $\Rey_l$ 244
Forced wavenumber $k_f$ $\sqrt{2}$
Number of Fourier modes $N$ $128$ (3/2)
Resolution $N\eta/2$ 1.67
------------------------------------------- ---------------- -------------
: Flow parameters in dimensionless code units. The characteristic parameters of the fluid flow are defined from its energy spectrum $E{\left( k \right)}\equiv\int_{{\left\Vert { \mathbf{k} } \right\Vert}=k}\left\Vert { \mathbf{\hat{u}} }{\left( { \mathbf{k} } \right)} \right\Vert^2 {\text{\textrm{d}}}{ \mathbf{k} }/2$. The dissipation rate of turbulent kinetic energy is: $\varepsilon \equiv2\nu\int k^2 E{\left( k \right)} {\text{\textrm{d}}}k$. The Kolmogorov length $\eta\equiv\left(\nu^3/\varepsilon\right)^{1/4}$, time scale $\tau_\eta\equiv\left(\nu/\varepsilon\right)^{1/2}$ and velocity scale $u_\eta \equiv \eta/\tau_\eta$. The Taylor micro-scale is: $\lambda\equiv u'/\sqrt{\langle\left|\bnabla { \mathbf{u} }\right|^2\rangle}$. The root mean square velocity is $u'\equiv\sqrt{(2/3)\int E{\left( k \right)} {\text{\textrm{d}}}k}$ and the integral length scale $\ell \equiv \left.\upi\middle/\left(2u'^2\right)\right. \int E{\left( k \right)}/k {\text{\textrm{d}}}k$. Similarly, the spectrum, root mean square value and dissipation rate of the scalar field are: $E_T{\left( k \right)}\equiv\int_{{\left\Vert { \mathbf{k} } \right\Vert}=k}\left\vert \hat{T}{\left( { \mathbf{k} } \right)} \right\vert^2 {\text{\textrm{d}}}{ \mathbf{k} }/2$, $T' \equiv\sqrt{(1/2)\int E_{T}{\left( k \right)} {\text{\textrm{d}}}k}$, $\chi \equiv 2\kappa \int k^2 E_T{\left( k \right)} {\text{\textrm{d}}}k$. The small scale temperature is determined by the viscosity and dissipation rate: $T_\eta\equiv\sqrt{\chi\tau_\eta}$. Since the Prandtl number is unitary the small scales of the scalar and the velocity field are of the same order.[]{data-label="tab:flow"}
Particle phase
--------------
We consider rigid, point-like particles which are heavy with respect to the fluid, and small with respect to any scale of the flow. In particular, the particle density $\rho_p$ is much larger than the fluid density $\rho_p\gg\rho_0$, and the particle radius $r_p$ is much smaller than the Kolmogorov length scale $r_p \ll \eta$. With these assumptions (and neglecting gravity) the particle acceleration is described by the Stokes drag law. Analogously, the rate of change of the particle temperature is described by Newton’s law for the heat conduction $$\begin{aligned}
{\frac{\textrm{d} { \mathbf{x} }_p}{\textrm{d} t}} &\equiv& { \mathbf{v} }_p,
\label{eq:partx}\\
{\frac{\textrm{d} { \mathbf{v} }_p}{\textrm{d} t}} &=& \frac{ { \mathbf{u} } \left({ \mathbf{x} }_{p},t \right) - { \mathbf{v} }_p }{\tau_p},
\label{eq:partv}\\
{\frac{\textrm{d} \theta_p}{\textrm{d} t}} &=& \frac{ T \left( { \mathbf{x} }_{p} , t \right) - \theta_{p} }{\tau_\theta}.
\label{eq:parttheta}\end{aligned}$$ Here $\tau_p \equiv\left.2\rho_pr_p^2\middle/\left(9\rho_0\nu\right)\right.$ is the particle momentum response time, $\tau_\theta \equiv \left. \rho_p c_p r_p^2\middle/\left(3\rho_0 c_0\kappa\right)\right.$ is the particle thermal response time, $c_p$ is the particle heat capacity, and $c_0$ is the fluid heat capacity at constant pressure. The Stokes number is defined as ${\text{\textit{St}}}\equiv \left.\tau_p\middle/\tau_\eta\right.$ ,and the thermal Stokes number is defined as ${\text{\textit{St}}}_\theta \equiv \left.\tau_\theta\middle/\tau_\eta\right.$, where $\tau_\eta$ is the Kolmogorov time scale.
Our simulations focus on a dilute suspension regime with particle volume fraction $\phi=4\times 10^{-4}$. While this volume fraction is large enough for two-way momentum coupling between the particles and fluid to be important [e.g. @elghobashi91], we ignore this in the present study. The motivation is that including both two-way momentum and two-way thermal coupling introduces too many competing effects that would compound a thorough understanding of the problem. In this study we therefore ignore momentum coupling, but account for two-way thermal coupling, and in a follow up study we will include the effects of two-way momentum coupling.
We consider nine values of ${\text{\textit{St}}}_\theta$ and three values of ${\text{\textit{St}}}$ in order to explore the behavior of the system over a range of parameter values. Since we are accounting for thermal coupling, each combination of ${\text{\textit{St}}}_\theta$ and ${\text{\textit{St}}}$ must be simulated separately, and when combined with the large number of particles in the flow domain, the set of simulations require considerable computational resources. Therefore, in the present study we restrict attention to $\Rey_\lambda =88$, but future explorations should consider larger $\Rey_\lambda$ in order to explore the behavior when there exists a well-defined inertial range in the flow.
In order to obtain deeper insight into the role of the two-way thermal coupling, we perform simulations with (denoted by S1) and without (denoted by S2) the thermal coupling. The particle parameters employed in the simulations are in table \[tab:particles\].
Thermal coupling
----------------
In the two-way thermal coupling regime, the thermal energy contained in the fluid is finite with respect to the thermal energy of the particles, therefore, when heat flows from the fluid to the particle the fluid loses thermal energy at the particle position. Due to the point-mass approximation, the feedback from the particles on the fluid temperature field is a superposition of Dirac delta functions, centered on the particles. Hence the coupling term in equation is given by $$C_T \left({ \mathbf{x} },t\right) = \frac{4}{3}\upi\frac{\rho_p}{\rho_0}\frac{c_p}{c_0} r_p^3 \sum_{p=1}^{N_P}{\frac{\textrm{d} \theta_p}{\textrm{d} t}}\delta\left({ \mathbf{x} } - { \mathbf{x} }_p\right).
\label{eq:CT}$$
--------------------------------- ------------------------------- ----------------------------------------------------
Particle phase volume fraction $\phi$ $0.0004$
Particle to fluid density ratio $\rho_p/\rho_0$ $1000$
Particle back reaction $C_T$ S1: included; S2: neglected.
Stokes number ${\text{\textit{St}}}$ $0.5$; $1$; $3$.
Thermal Stokes number ${\text{\textit{St}}}_\theta$ $0.2$; $0.5$; $1$; $1.5$; $2$; $3$; $4$; $5$; $6$.
Number of particles $N_P$ $12500992$; $4419584$; $847872$.
--------------------------------- ------------------------------- ----------------------------------------------------
: Particles parameters in dimensionless code units. The Stokes number is ${\text{\textit{St}}}\equiv\tau_p/\tau_\eta$ and thermal Stokes number ${\text{\textit{St}}}_\theta\equiv\tau_\theta/\tau_\eta$ and the particle response times are defined in the text. In the simulations, ${\text{\textit{St}}}_\theta$ is varied by varying the particle heat capacity. The different combinations of ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are simulated including the two-way thermal thermal coupling (simulations S1) and neglecting it (simulations S2).[]{data-label="tab:particles"}
![(*a*) Three-dimensional energy spectrum of the fluid velocity field (open squares) and temperature field (open circles). The temperature field is computed without any feedback from the particles on the fluid flow (simulations S2). (*b*) Second order longitudinal structure functions of the particle velocity for various Stokes numbers.[]{data-label="fig:flowstat"}]({figure1-crop}.pdf){width="\textwidth"}
Numerical method
----------------
We perform direct numerical simulation of incompressible, statistically steady and isotropic turbulence on a tri-periodic cubic domain. Equations , , and are solved by means of the pseudo-spectral Fourier method for the spatial discretization. The $3/2$ rule is employed for dealiasing [@Canuto1988], so that the maximum resolved wavenumber is $k_{\textrm{max}}=N/2$. The required Fourier transforms are executed in parallel using the P3DFFT library [@Pekurovsky2012]. Forcing is applied to a single scale, that is to all wavevectors satisfying $\left\Vert { \mathbf{k} } \right\Vert^2 = k_f$, with $k_f=2$, and the equations for the fluid velocity and temperature Fourier coefficients are evolved in time by means of a second order Runge-Kutta exponential integrator [@Hochbruck2010]. This method has been preferred to the standard integrating factor because of its higher accuracy and, above all, because of its consistency. Indeed, in order to obtain an accurate representation of small scale temperature fluctuations, it is critical that the numerical solution conserves thermal energy. The same time integration scheme is used to solve particle equations , and , thus providing overall consistency, since the system formed by fluid and particles is evolved in time as a whole.
The fluid velocity and temperature are interpolated at the particle position by means of fourth order B-spline interpolation. The interpolation is implemented as a backward Non Uniform Fourier Transform with B-spline basis: the fluid field is projected onto the B-spline basis in Fourier space through a deconvolution, than transformed into the physical space by means of a inverse Fast Fourier Transform (FFT). A convolution provides the interpolated field at particle position [@Beylkin1995]. Since B-splines have a compact support in physical space and deconvolution in Fourier space reduces to a division, this provide an efficient way to obtain high order interpolation. This guarantees smooth and accurate interpolation and its efficient implementation is suitable for pseudo-spectral methods [@vanHinsberg2012]. Moreover, the same method is used to obtain the spectral representation of the coupling term . The coupling term has to be projected on the Cartesian grid used to represent the fields. This is performed by means of the forward Nonuniform Fast Fourier Transform (NUFFT) with B-spline basis [@Beylkin1995]. Briefly, the algorithm works as follows. The convolution of the distribution $C_T{\left( { \mathbf{x} },t \right)}$ with the B-spline polynomial basis $B{\left( { \mathbf{x} } \right)}$ is computed in physical space, so that it can be effectively represented on the Cartesian grid $$\widetilde{C}\left({ \mathbf{x} },t \right) = \int C_T{\left( { \mathbf{x} },t \right)} B{\left( { \mathbf{x} }-{ \mathbf{y} } \right)}{\text{\textrm{d}}}{ \mathbf{y} }.$$ Then, the regularized field $\widetilde{C}$ is transformed to Fourier space and the convolution with the B-spline basis is efficiently removed: $$\widehat{{C}}{\left( { \mathbf{k} },t \right)} = \frac{\widehat{\widetilde{C}}{\left( { \mathbf{k} },t \right)}}{\widehat{B}{\left( { \mathbf{k} } \right)}}.$$ This algorithm allows an efficient and accurate spectral representation of the particle back-reaction [@Carbone2018]. Indeed, the NUFFT satisfies the constraints for interpolation schemes [@Sundaram1996]: the backward and forward transformations are symmetric and the non locality, introduced in physical space due to the convolution, is removed in Fourier space. For these reasons this technique is preferred to shape regularization functions, [@Maxey1997].
Characterization of the thermal dissipation rate {#sec:diss}
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In the flow under consideration, the total dissipation rate of the temperature field $\chi$ is constant due to the forcing term $f_T$. The total dissipation has a contribution from the fluid and particle phases and is given by [@Sundaram1996] $$\chi = \kappa {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + \frac{\phi}{\tau_\theta}\frac{\rho_p c_p}{\rho_0 c_0} {\left\langle \left( T {\left( { \mathbf{x} }_p,t \right)} - \theta_p \right)^2 \right\rangle}.\label{eq:dissbal0}$$ We indicate with $\chi_f$ the dissipation due to the fluid temperature gradient and with $\chi_p$ the dissipation due to the particles, the two terms in the right hand side of equation , so that $\chi=\chi_f+\chi_p$. Note that both contributions to the dissipation rate are proportional to the kinematic thermal conductivity of the fluid since $\tau_\theta\propto 1/\kappa$, and hence both the dissipation mechanisms are due to molecular diffusivity.
A characteristic length of the dissipation due to the particles can be defined as $$\eta_p \equiv \frac{r_p}{\sqrt{3 \phi }}\label{eq:disslen}$$ and using this, the balance of the dissipation of the temperature fluctuations can be written as $$\chi = \kappa \left[ {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + {\left\langle \left(\frac{ T {\left( { \mathbf{x} }_p,t \right)} - \theta_p}{\eta_p}\right)^2 \right\rangle} \right].\label{eq:dissbal}$$ In these simulations the volume fraction $\phi$ is constant, so the characteristic length of the dissipation due to the particles is proportional to the particle radius.
The portion of temperature fluctuations dissipated by the two different mechanisms depends on the statistics of the differences between the particle and local fluid temperatures. In the limit ${\text{\textit{St}}}_\theta\to 0$ we have $T {\left( { \mathbf{x} }_p,t \right)}=\theta_p$, such that all of the dissipation is associated with the fluid. In the general case, the statistics of $T {\left( { \mathbf{x} }_p,t \right)}-\theta_p$ depend not only on ${\text{\textit{St}}}_\theta$, but also implicitly upon ${\text{\textit{St}}}$, with the statistics of $T {\left( { \mathbf{x} }_p,t \right)}$ depending on the spatial clustering of the particles. This coupling between the particle momentum and temperature dynamics can lead to non-trivial effects of particle inertia on $\chi_p$.
Thermal dissipation due to the temperature gradients
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Since the flow is isotropic, $\chi_f$ is given by $$\chi_f = 3\kappa{\left\langle {\left( \partial_x T \right)}^2 \right\rangle}$$ We consider fixed Reynolds number and $\Pran=1$, thus $\kappa$ is the same in all the presented simulations, and so $\langle(\partial_x T)^2\rangle$ fully characterizes $\chi_f$. Moreover, given the expected structure of the field $\partial_x T$, it is instructive to consider its full Probability Density Function (PDF), in addition to its moments in order to know how different regions of the flow contribute to the average dissipation rate $\chi_f$.
![PDF of the fluid temperature gradient $\partial_x T$ from simulations S1, for ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*), and for various ${\text{\textit{St}}}_\theta$. (*c*) Dissipation rate $\chi_f$ of the fluid temperature fluctuations, for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*d*) Kurtosis of the fluid temperature gradient PDF.[]{data-label="fig:PDFgrT"}]({figure2-crop}.pdf){width="\textwidth"}
Figures \[fig:PDFgrT\](a-b) show the normalized PDFs of $\partial_x T$ for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$ respectively, and for various ${\text{\textit{St}}}_\theta$, where the PDFs are normalized using the standard deviation of the distribution, $\sigma_{\partial_x T}$. The distribution is almost symmetric and it displays elongated exponential tails. The largest temperature gradients exceed the standard deviation by an order of magnitude [@Overholt1996]. Remarkably, the shape of the PDF shows a very weak dependence on ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, such that the PDF shape scales with $\sigma_{\partial_x T}$.
The variance of the fluid temperature gradient is proportional to the actual dissipation rate of the temperature fluctuation (the proportionality factor being $3\kappa$, fixed in our simulations). In contrast to the PDF shape, the suspended particles have a strong impact on $\chi_f$, as shown in figure \[fig:PDFgrT\]. As ${\text{\textit{St}}}_\theta$ is increased, $\chi_f$ decreases. However, this is mainly due to the fact that as ${\text{\textit{St}}}_\theta$ is increased, $\chi_p$ increases, and so $\chi_f$ must decrease since $\chi=\chi_f+\chi_p$ is fixed. The influence of the Stokes number on $\chi_p$ is very small in the range of parameters considered.
The kurtosis of the fluid temperature gradients is shown in figure \[fig:PDFgrT\](d), as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$. The kurtosis is approximately constant, and much larger than the value of a Gaussian distribution. The behavior of the kurtosis confirms that the fluid temperature gradient PDF is approximately self-similar.
Thermal dissipation due to the particle dynamics
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The dissipation rate due to the particles, $\chi_p$, depends on the difference between the particle temperature and the fluid temperature at the particle position $$\chi_p = \kappa{\left\langle \left(\frac{ T {\left( { \mathbf{x} }_p,t \right)} - \theta_p}{\eta_p}\right)^2 \right\rangle}$$ For notational simplicity, we define $\varphi_p \equiv \left.\left(T {\left( { \mathbf{x} }_p,t \right)} - \theta_p\right)\middle/\eta_p\right.$. When $\varphi_p$ is normalized by its standard deviation, we can relate this to the rate of change of the particle temperature using equation $$\frac{ \dot{\theta}_p }{ \sigma_{ \dot{\theta}_p } } = \frac{\varphi_p}{\sigma_{\varphi_p}}.$$
![PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*), and for various ${\text{\textit{St}}}_\theta$. Plots (*a-c*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d*) are from simulations S2, in which the two-way coupling is neglected. (*e*) Dissipation rate $\chi_p$ of the temperature fluctuations due to the particles, for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*f*) Kurtosis of the PDF of $\dot{\theta}_p$.[]{data-label="fig:PDFdeT"}]({figure3-crop}.pdf){width="\textwidth"}
The normalized PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$, and for various ${\text{\textit{St}}}_\theta$ is shown in figure \[fig:PDFdeT\]. Figure \[fig:PDFdeT\](a) shows the normalized PDF of $\dot{\theta}_p$, for ${\text{\textit{St}}}=1$ for the set of simulations S1, in which the two-way thermal coupling is taken to account. Figure \[fig:PDFdeT\](b) shows the corresponding results for simulations S2, in which the two-way thermal coupling is neglected. The normalized PDF of $\dot{\theta}_p$ for ${\text{\textit{St}}}=3$, with and without the two-way thermal coupling, is shown in figures \[fig:PDFdeT\](c-d).
![(a) Variance of the particle temperature rate of change as a function of the thermal Stokes number for different Stokes numbers. The dotted lines represent the expected asymptotic behaviour for ${\text{\textit{St}}}_\theta\ll1$ and ${\text{\textit{St}}}_\theta\gg1$. (b) Normalized PDF of the particle temperature rate of change, $\dot{\theta_p}$ at ${\text{\textit{St}}}_\theta=1$ for various Stokes number, ${\text{\textit{St}}}<0.5$. The dotted line shows a Gaussian PDF for reference. Results obtained neglecting the particle thermal feedback.[]{data-label="fig:compare"}]({figure4-crop}.pdf){width="\textwidth"}
In contrast to the fluid temperature gradient PDFs, the shape of the PDF of $\dot{\theta}_p$ is not self-similar with respect to its variance. As ${\text{\textit{St}}}_\theta$ is increased, the normalized PDF becomes narrower. This is due to the fact that as ${\text{\textit{St}}}_\theta$ is increased, the particles respond more slowly to changes in the fluid temperature field, analogous to the “filtering” effect for inertial particle velocities in turbulence [@salazar12a; @Ireland2016]. The PDF shapes are mildly affected by ${\text{\textit{St}}}$, and for larger ${\text{\textit{St}}}_\theta$, extreme fluid temperature-particle temperature differences are suppressed when the two-way thermal coupling is neglected.
The variance of $\dot{\theta}_p$ is proportional to the particle dissipation rate $\chi_p$, and the results for this are shown in figure \[fig:PDFdeT\](e), for various ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, and for simulations S1 and S2. The results show that as ${\text{\textit{St}}}_\theta$ is increased, $\chi_p$ increases. This is mainly because as ${\text{\textit{St}}}_\theta$ is increased, the thermal memory of the particle increases, and the particle temperature depends strongly on its encounter with the fluid temperature field along its trajectory history for times up to ${\textit{O}\left( \tau_\theta \right)}$ in the past. As a result, the particle temperature can differ strongly from the local fluid temperature. The results also show that $\chi_p$ is dramatically suppressed when two-way thermal coupling is accounted for. One reason for this is that as shown earlier, two-way thermal coupling leads to a suppression in the fluid temperature gradients. As these gradients are suppressed, the fluid temperature along the particle trajectory history differs less from the local fluid temperature than it would have in the absence of two-way thermal coupling, and as a result $\chi_p$ is decreased.
The results for kurtosis of $\dot{\theta}_p$, as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$ are shown in figure \[fig:PDFdeT\](f). The results show that the kurtosis decreases with increasing ${\text{\textit{St}}}_\theta$. This is mainly due to the filtering effect mentioned earlier, wherein as ${\text{\textit{St}}}_\theta$ is increased, the particles are less able to respond to rapid fluctuations in the fluid temperature along their trajectory. Further, the kurtosis is typically larger when the two-way thermal coupling is taken into account (simulations S1), and is maximum for ${\text{\textit{St}}}=1$. This is due to the particle clustering on the fronts of the fluid temperature field, as will be discussed in section \[sec:FATT\].
Our results for the PDF of $\dot{\theta}_p$ and its moments differ somewhat from those in [@Bec2014]. This is in part due to the difference in the forcing methods employed by [@Bec2014] and that in our study. The solution of may be written as [@Bec2014] $${\left\langle \dot{\theta_p}^2 \right\rangle} = \frac{1}{2\tau_\theta^3}\int_0^\infty {\left\langle \Big( \delta_t T_p(t)\Big)^2 \right\rangle}\exp{\left( -\frac{t}{\tau_\theta} \right)}{\text{\textrm{d}}}t,
\label{eq:solthetalag}$$ where $\delta_t T_p(t)\equiv T{\left( { \mathbf{x} }_p{\left( t \right)},t \right)}-T{\left( { \mathbf{x} }_p{\left( 0 \right)},0 \right)}$.
In the regime ${\text{\textit{St}}}_\theta\ll1$, the exponential in decays very fast in time so that the main contribution to the integral comes from $\delta_t T_p$ for infinitesimal $t$, with $\delta_t T_p \sim t^n$ for $t\to 0$. Substituting $\delta_t T_p \sim t^n$ into we obtain the leading order behavior$${\left\langle \dot{\theta_p}^2 \right\rangle} \sim \frac{1}{2\tau_\theta^3}\int_0^\infty t^{2n} \exp{\left( -\frac{t}{\tau_\theta} \right)}{\text{\textrm{d}}}t \sim {\text{\textit{St}}}_\theta^{2n-2},\;{\text{\textit{St}}}_\theta\ll 1.$$ [@Bec2014] used a white in time forcing for the fluid scalar field, giving $n=1/2$, and yielding $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-1}$ for ${\text{\textit{St}}}_\theta\ll1$. However, the forcing scheme that we have employed generates a field $T({ \mathbf{x} },t)$ that evolves smoothly in time, so $n=1$ and $\langle\dot{\theta_p}^2\rangle\sim$ constant for ${\text{\textit{St}}}_\theta\ll1$.
For ${\text{\textit{St}}}_\theta\gg 1$, the integral in is dominated by uncorrelated temperature increments, $\delta_t T \sim t^0$, such that $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-2}$. The comparison between figure \[fig:compare\](a) and figure 5 of [@Bec2014] highlights the different asymptotic behavior of $\sigma_{\dot{\theta_p}}^2\equiv \langle\dot{\theta_p}^2\rangle$ for ${\text{\textit{St}}}_\theta\ll 1$, but the same behavior $\langle\dot{\theta_p}^2\rangle \sim {\text{\textit{St}}}_\theta^{-2}$ for ${\text{\textit{St}}}_\theta\gg 1$. Further, as expected, our DNS data approaches these asymptotic regimes for both the cases with and without two-way thermal coupling.
Another difference is that in the results of [@Bec2014], the tails of the PDFs of $\dot{\theta}_p$ for ${\text{\textit{St}}}_\theta=1$ become heavier as ${\text{\textit{St}}}$ is increased, whereas our results in figure \[fig:PDFdeT\] show that while the kurtosis of these PDFs increases from ${\text{\textit{St}}}=0.5$ to ${\text{\textit{St}}}=1$, it then decreases from ${\text{\textit{St}}}=1$ to ${\text{\textit{St}}}=3$. In order to examine this further, we performed simulations (without two-way thermal coupling) for ${\text{\textit{St}}}_\theta=1$ and ${\text{\textit{St}}}\leq 0.4$. The results are shown in figure \[fig:compare\](b), and in this regime we do in fact observe that the tails of the PDFs of $\dot{\theta}_p$ become increasingly wider as ${\text{\textit{St}}}$ is increased. Taken together with the results in figure \[fig:PDFdeT\], this implies that in our simulations, the tails of the PDFs of $\dot{\theta}_p$ become increasingly wider as ${\text{\textit{St}}}$ is increased until ${\text{\textit{St}}}\approx 1$, where this behavior then saturates, and upon further increase of ${\text{\textit{St}}}$ the tails start to narrow. This non-monotonic behavior is due to the particle clustering in the fronts of the temperature field, which is strongest for ${\text{\textit{St}}}\approx 1$ (see §\[sec:FATT\]). While the results in [@Bec2014] over the range ${\text{\textit{St}}}\leq 3.7$ do not show the tails of the PDFs of $\dot{\theta}_p$ becoming narrower, their results clearly show that the widening of the tails saturates (see inset of figure 5 in [@Bec2014]). It is possible that if they had considered larger ${\text{\textit{St}}}$, they would have also began to observe a narrowing of the tails as ${\text{\textit{St}}}$ was further increased. Possible reasons why the widening of the tails saturates at a lower value of ${\text{\textit{St}}}$ in our DNS than it does in theirs include is the effect of Reynolds number ($\Rey_\lambda=315$ in their DNS, whereas in our DNS $\Rey_\lambda=88$), and differences in the scalar forcing method.
Characterization of the temperature fluctuations {#sec:fluct}
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This section consists of a short overview of the one-point temperature statistics. Note that due to the large scale forcing used in the DNS, the one-point statistics of the flow are affected by the forcing method employed.
Fluid temperature fluctuations
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![PDF of the fluid temperature for ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*), and for various ${\text{\textit{St}}}_\theta$. (*c*) Variance of the fluid temperature fluctuations for different ${\text{\textit{St}}}$ as a function of ${\text{\textit{St}}}_\theta$. (*d*) Kurtosis of the fluid temperature PDF. These results are from simulations S1 in which the two-way thermal coupling is considered.[]{data-label="fig:PDFT"}]({figure5_106-crop}.pdf){width="\textwidth"}
Figures \[fig:PDFT\](a-b) show the normalized one-point PDF of the fluid temperature for ${\text{\textit{St}}}=1$ and ${\text{\textit{St}}}=3$, respectively, and for various ${\text{\textit{St}}}_\theta$. The PDFs are normalized with the standard deviation of the distribution $\sigma_{T}$. The PDFs are almost Gaussian for low ${\text{\textit{St}}}_\theta$, while the tails become wider as ${\text{\textit{St}}}_\theta$ is increased. However, we are unable to explain the cause of this enhanced non-Gaussianity. The temperature PDFs are also not symmetric, and display a bump in the right tail. This behavior was also reported by [@Overholt1996] for the case without particles, and it appears to be a low Reynolds number effect that is also dependent on the forcing method employed.
The effect of ${\text{\textit{St}}}$ on $\sigma_T$ is striking, whereas we saw earlier in figure \[fig:PDFgrT\](c) that $\chi_f$ only weakly depends on ${\text{\textit{St}}}$. To explain the dependence upon the Stokes number we note that the energy balance can be rewritten as $$\chi = \kappa \left[ {\left\langle \left\Vert \bnabla T \right\Vert^2 \right\rangle} + \frac{2}{3}\frac{\phi}{\tau_\eta}\frac{\rho_p}{\rho_0}\frac{1}{{\text{\textit{St}}}}{\left\langle \left( T {\left( { \mathbf{x} }_p,t \right)} - \theta_p\right)^2 \right\rangle} \right]\label{eq:balSt}.$$ The factor $\phi\rho_p/\left(\rho_0\tau_\eta\right)$ is constant in our simulations. Therefore, since our DNS data suggest that $\chi_f$ is a function of ${\text{\textit{St}}}_\theta $ only (see figure \[fig:PDFgrT\](c)), from and we obtain $${\left\langle T{\left( { \mathbf{x} }_p,t \right)}^2 \right\rangle} - {\left\langle \theta_p^2 \right\rangle} \propto {\text{\textit{St}}}f{\left( {\text{\textit{St}}}_\theta \right)}.$$
The kurtosis of the fluid temperature fluctuation is shown in figure \[fig:PDFT\](d), as a function of ${\text{\textit{St}}}_\theta$ and for various ${\text{\textit{St}}}$. For small ${\text{\textit{St}}}_\theta$, the kurtosis of the fluid temperature fluctuation is close to the value for a Gaussian PDF, namely $3$. However, as ${\text{\textit{St}}}_\theta$ is increased, the kurtosis increases. Furthermore, the kurtosis decreases with increasing ${\text{\textit{St}}}$ for the range considered in our simulations. The explanation of these trends in the kurtosis is unclear.
Particle temperature fluctuations
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![PDF of the particle temperature for ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*), for various ${\text{\textit{St}}}_\theta$. Plots (*a-c*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d*) are from simulations S2, in which the two-way coupling is neglected. (*e*) Variance of the particle temperature fluctuations for different ${\text{\textit{St}}}$ numbers as a function of ${\text{\textit{St}}}_\theta$. (*f*) Kurtosis of the particle temperature distribution.[]{data-label="fig:PDFtheta"}]({figure6_106-crop}.pdf){width="\textwidth"}
Figures \[fig:PDFtheta\](a-b) show the normalized one-point PDF of the particle temperature with ${\text{\textit{St}}}=1$, for various ${\text{\textit{St}}}_\theta$, and for simulations S1 and S2. Figures \[fig:PDFtheta\](c-d) show the corresponding results for ${\text{\textit{St}}}=3$, and the PDFs are normalized by their standard deviations. When the two-way thermal coupling is accounted for, the tails of the particle temperature distribution tend to become wider as ${\text{\textit{St}}}_\theta$ is increased. On the other hand, when the two-way coupling is neglected, the PDF of the particle temperature is very close to Gaussian, and its shape is not sensitive to either ${\text{\textit{St}}}$ or ${\text{\textit{St}}}_\theta$.
The variance of the particle temperature fluctuations monotonically decrease with increasing ${\text{\textit{St}}}_\theta$, as shown in figure \[fig:PDFtheta\](e). The results also show a strong dependence on ${\text{\textit{St}}}$, but most interestingly, the dependence on ${\text{\textit{St}}}$ is the opposite for the cases with and without two-way coupling. To understand this we note that using the formal solution to the equation for $\dot{\theta}_p(t)$ (ignoring initial conditions) we may construct the result $$\Big\langle \theta^2_p(t)\Big\rangle=\frac{1}{\tau_\theta^2}\int^t_0\int^t_0 \Big\langle T({ \mathbf{x} }_p(s),s)T({ \mathbf{x} }_p(s'),s')\Big\rangle e^{-(2t-s-s')/\tau_\theta}\,ds\,ds'.$$ If we now substitute into this the exponential approximation $$\langle T({ \mathbf{x} }_p(s),s)T({ \mathbf{x} }_p(s'),s')\rangle \approx \langle T^2({ \mathbf{x} }_p(t),t)\rangle \exp[-|s-s'|/\tau_T],$$ where $\tau_T$ is the timescale of $T({ \mathbf{x} }_p(t),t)$, then we obtain $$\Big\langle \theta^2_p(t)\Big\rangle= \frac{\langle T^2({ \mathbf{x} }_p(t),t)\rangle}{1+\tau_\theta/\tau_T} .$$ This result reveals that the particle temperature variance is influenced by ${\text{\textit{St}}}$ in two ways. First, $\langle T^2({ \mathbf{x} }_p(t),t)\rangle$ depends upon the spatial clustering of the inertial particles, and this depends essentially upon ${\text{\textit{St}}}$. Second, the timescale $\tau_T$ is the timescale of the fluid temperature field measured along the inertial particle trajectories, and hence depends upon ${\text{\textit{St}}}$. For isotropic turbulence, this timescale is expected to decrease as ${\text{\textit{St}}}$ is increased, which would lead to $\langle \theta^2_p(t)\rangle$ decreasing as ${\text{\textit{St}}}$ increases, which is the behavior observed in figure \[fig:PDFtheta\](e). In the presence of two-way coupling, however, $\langle T^2({ \mathbf{x} },t)\rangle$ increases with increasing ${\text{\textit{St}}}$, as shown earlier. In the two-way coupled regime this increase in $\langle T^2({ \mathbf{x} },t)\rangle$ leads to an increase in $\langle T^2({ \mathbf{x} }_p(t),t)\rangle$ that dominates over the decrease of $\tau_T$ with increasing ${\text{\textit{St}}}$, and as a result $\langle \theta^2_p(t)\rangle$ increases with increasing ${\text{\textit{St}}}$.
The kurtosis of the particle temperature increases with increasing ${\text{\textit{St}}}_\theta$ when the two-way thermal coupling is accounted for, as shown in figure \[fig:PDFtheta\](f) (simulations S1, filled symbols). Conversely, the kurtosis of the particle temperature remains constant as ${\text{\textit{St}}}_\theta$ is increased when the two-way thermal coupling is ignored (simulations S2, open symbols).
Statistics conditioned on the local fluid temperature gradients {#sec:FATT}
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In this section we consider additional quantities to obtain deeper insight into the one-point particle to fluid heat flux. In particular, we explore the relationship between this heat flux and the local fluid temperature gradients.
Particle clustering on the temperature fronts
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![(*a*) Radial distribution function (RDF) as a function of the separation $r/\eta$ for various ${\text{\textit{St}}}$. (*b*) Particle number density conditioned on the magnitude of the fluid temperature gradient at the particle position, for various ${\text{\textit{St}}}$. These results are from simulations S2, in which the two-way thermal coupling is neglected.[]{data-label="fig:FATTconc"}]({figure7-crop}.pdf){width="\textwidth"}
It is well known that inertial particles in turbulence form clusters [@Bec2007], which may be quantified using the radial distribution function (RDF). As shown in figure \[fig:FATTconc\](a), the particle number density in our simulations at small separations is a order of magnitude larger than the mean density when ${\text{\textit{St}}}=\textit{O}{\left( 1 \right)}$. [@Bec2014] showed that inertial particles also exhibit a tendency to preferentially cluster in the fluid temperature fronts where the temperature gradients are large. To demonstrate this, they measured the temperature dissipation rate at the particle positions and showed that this was higher than the Eulerian dissipation rate of the fluid temperature fluctuations. Alternatively, we may quantify this tendency for inertial particles to cluster in the fluid temperature fronts by computing the particle number density conditioned on the magnitude of the fluid temperature gradient $$n_P{\left( \left\Vert\bnabla T\right\Vert \right)} = \frac{\sum_p\int_V \delta{\left( { \mathbf{x} }-{ \mathbf{x} }_p \right)}{\text{\textrm{d}}}{ \mathbf{x} }}{N_P},\; V=\left\{ { \mathbf{x} }: \left\Vert\bnabla T{\left( { \mathbf{x} } \right)}\right\Vert = \left\Vert\bnabla T\right\Vert\right\}.$$ Defining $\|\bnabla T\|_{rms}$ as the rms value of $\|\bnabla T\|$, small values of $\|\bnabla T\|/\|{ \mathbf{\bnabla} } T\|_{rms}$ may be interpreted as corresponding to the large scales, and are associated with the Lagrangian coherent structures in which the temperature field is almost constant. Large values of $\|\bnabla T\|/\|\bnabla T\|_{rms}$ may be interpreted as corresponding to the small scales, and are associated with fronts in the fluid temperature field.
The results for $n_P$ are shown in figure \[fig:FATTconc\](b), corresponding to simulations without two-way thermal coupling (the results show only a weak dependence on ${\text{\textit{St}}}_\theta$ when the two-way coupling is included). For fluid particles, $n_P$ decays almost exponentially with increasing $\|\bnabla T\|$. For values of ${\text{\textit{St}}}$ at which the maximum particle clustering takes place, $n_P$ is an order of magnitude larger than the value for fluid particles in regions of strong temperature gradients. These results therefore support the conclusions of [@Bec2014] that inertial particles preferentially cluster in the fronts of the fluid temperature field where $\|\bnabla T\|/\|\bnabla T\|_{rms}$ is large.
Particle motion across the temperature fronts
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To obtain further insight into the thermal coupling between the particles and fluid we consider the properties of the particle heat flux conditioned on $\|\bnabla T\|$. In particular, we consider the following quantity $$q_n {\left( \left\Vert \bnabla T \right\Vert \right)} \equiv {{\left( T{\left( { \mathbf{x} }_p \right)}-\theta_p \right)}^n{ \mathbf{v} }_p\bcdot { \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)}}\Big\vert_{\left\Vert \bnabla T \right\Vert},$$ where $\mathbf{n}_T$ is the normalized temperature gradient $$\mathbf{n}_T{\left( { \mathbf{x} }_p \right)} \equiv \frac{\bnabla T{\left( { \mathbf{x} }_p \right)}}{\left\Vert \bnabla T{\left( { \mathbf{x} }_p \right)} \right\Vert}.$$
![(*a*) Results for ${\left\langle \left\vert q_0(\|\bnabla T\|)\right\vert \right\rangle}/u_\eta$, for various ${\text{\textit{St}}}$. (*b*) Results for $\langle|\cos \alpha_p|\rangle$ as a function of $\|\bnabla T\|$, for various ${\text{\textit{St}}}$. These results are from simulations S2, in which the two-way thermal coupling is neglected.[]{data-label="fig:FATTq0"}]({figure8-crop}.pdf){width="\textwidth"}
The statistics of $q_n$ provide a way to quantify the relationship between the particle heat flux and the local temperature gradients in the fluid. Understanding this relationship is key to understanding how the particles modify the properties of the fluid temperature and temperature gradient fields.
The efficiency with which the particles cross the fronts in the fluid temperature field is quantified by ${\left\langle \left\vert q_0\right\vert \right\rangle}$, and our results for this quantity are shown in figure \[fig:FATTq0\](a). The curves are normalized with the Kolmogorov velocity scale $u_\eta$. The results show that as ${\text{\textit{St}}}$ is increased, the particles move across the fronts with increasingly large velocities. This behavior is non-trivial since it is known that the kinetic energy of an inertial particle decreases with increasing ${\text{\textit{St}}}$ [@Zaichik2009; @Ireland2016].
It is also important to consider whether the reduction of ${\left\langle \left\vert q_0\right\vert \right\rangle}$ as $\|\bnabla T\|$ increases is due to the reduction of the norm of the particle velocity or to the lack of alignment between the particle velocity and the fluid temperature gradient at the particle position. Figure \[fig:FATTq0\](b) displays the average of the absolute value of the cosine of the angle between the particle velocity and temperature gradient $$\cos\alpha_p \equiv \frac{{ \mathbf{v} }_p}{\left\Vert { \mathbf{v} }_p \right\Vert} \bcdot \frac{\bnabla T {\left( { \mathbf{x} }_p \right)}}{\left\Vert \bnabla T {\left( { \mathbf{x} }_p \right)}\right\Vert},$$ conditioned on $\|{ \mathbf{\bnabla} }T\|$.
The results show that as $\|\bnabla T\|$ is increased, the particle motion becomes misaligned with the local fluid temperature gradient. This then shows that the reduction of ${\left\langle \left\vert q_0\right\vert \right\rangle}$ as $\|\bnabla T\|$ increases is due to non-trivial statistical geometry in the system. The results also show that as ${\text{\textit{St}}}$ is increased, the cosine of the angle between the fluid temperature gradient and the particle velocity becomes almost independent of $\|\bnabla T\|$, and ${\left\langle \left\vert\cos\alpha_p\right\vert \right\rangle}\approx 1/2$, the value corresponding to $\cos{\left( \alpha_p \right)}$ being a uniform random variable. This shows that as ${\text{\textit{St}}}$ is increased, the correlation between the direction of the particle velocity and the local fluid velocity gradient vanishes.
Heat flux due to the particle motion across the fronts
------------------------------------------------------
![Results for ${\left\langle q_1{\left( {\left\Vert \bnabla T \right\Vert} \right)} \right\rangle}/{\left( u_\eta T_\eta \right)}$ for ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*) and ${\text{\textit{St}}}=3$ (*e-f*), and for various ${\text{\textit{St}}}_\theta$. Plots (*a-c-e*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d-f*) are from simulations S2, in which the two-way coupling is neglected.[]{data-label="fig:FATTq1"}]({figure9-crop}.pdf){width="\textwidth"}
We now turn to consider the quantity ${\left\langle q_1 \right\rangle}$. When the particle moves from a cold to a warm region of the fluid, the component of the particle velocity along the temperature gradient is positive, ${ \mathbf{v} }_p\bcdot { \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)} > 0$. If the particle is also cooler than the local fluid so that $T{\left( { \mathbf{x} }_p \right)}-\theta_p>0$, then as it moves into the region where the fluid is warmer, $q_1>0$ meaning that the particle will absorb heat from the fluid, and will therefore tend to reduce the local fluid temperature gradient. When the particle moves from a warm to a cold region of the flow, if $T{\left( { \mathbf{x} }_p \right)}-\theta_p<0$ then $q_1$ is also positive, so that again the particle will act to reduce the local temperature gradient in the fluid. Therefore, $q_1>0$ indicates that the action of the inertial particles is to smooth out the fluid temperature field, reducing the magnitude of its temperature gradients, and $q_1<0$ implies the particles enhance the temperature gradients.
The results for ${\left\langle q_1 \right\rangle}$ are shown in figure \[fig:FATTq1\] for various ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, including (simulations S1) and neglecting (simulations S2) the two-way thermal coupling. On average we observe ${\left\langle q_1 \right\rangle}\ge 0$, such that the particles tend to make the fluid temperature field more uniform. The results show that ${\left\langle q_1 \right\rangle}$ tends to zero as $\|\bnabla T\|\to0$. This indicates that the particles spend enough time in the Lagrangian coherent structures to adjust to the temperature of the fluid. However, ${\left\langle q_1 \right\rangle}$ increases significantly as $\|\bnabla T\|$ increases, suggesting that inertial particles can carry large temperature differences across the fronts. In the limit ${\text{\textit{St}}}_\theta\to 0$, ${\left\langle q_1 \right\rangle}\to0$ reflecting the thermal equilibrium between the particles and the fluid. As ${\text{\textit{St}}}_\theta$ is increased, the heat-flux becomes finite, however, if ${\text{\textit{St}}}_\theta$ is too large, the particle temperature decorrelates from the fluid temperature and the heat exchange is not effective. Hence, ${\left\langle q_1 \right\rangle}$ can saturate with increasing ${\text{\textit{St}}}_\theta$. The results show that ${\left\langle q_1 \right\rangle}$ increases with increasing ${\text{\textit{St}}}$, associated with the decoupling of ${ \mathbf{v} }_p$ and ${ \mathbf{n} }_T{\left( { \mathbf{x} }_p \right)}$ discussed earlier. Finally, the results also show that two-way thermal coupling reduces ${\left\langle q_1 \right\rangle}$. This is simply a reflection of the fact that since the particles tend to smooth out the fluid temperature gradients, the disequilibrium between the particle and local fluid temperature is reduced, which in turn reduces the heat flux due to the particles.
Temperature structure functions {#sec:SF}
===============================
We now turn to consider two-point quantities in order to understand how the two-way thermal coupling affects the system at the small scales.
Fluid temperature structure functions
-------------------------------------
The $n$-th order structure function of the fluid temperature field is defined as $$S^n_T{\left( r \right)} \equiv {\left\langle \left\vert \Delta T(r,t)\right\vert^n \right\rangle}$$ where $\Delta T(r,t)$ it the difference in the temperature field at two points separated by the distance $r$ (the “temperature increment”). The results for $S^2_T$, with different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are shown in figure \[fig:SFT\].
![Results for $S^2_T$ for different ${\text{\textit{St}}}_\theta$, for ${\text{\textit{St}}}=0.5$ (*a*), ${\text{\textit{St}}}=1$ (*b*) and ${\text{\textit{St}}}=3$ (*c*). (*d*) Scaling exponents of the fluid temperature structure functions at small separation, $r\le2\eta$, at ${\text{\textit{St}}}=1$. The data is from simulations S1 in which the two-way thermal coupling is considered.[]{data-label="fig:SFT"}]({figure10-crop}.pdf){width="\textwidth"}
The results show that $S^2_T$ decreases monotonically with increasing ${\text{\textit{St}}}_\theta$ at all scales when the two-way thermal coupling is taken to account. In the dissipation range, $S^2_T$ is directly connected to the dissipation rate, and is suppressed in the same way for the three different ${\text{\textit{St}}}$ considered. Conversely, the suppression of the large scale fluctuations is stronger as ${\text{\textit{St}}}$ is reduced, at least for the range of ${\text{\textit{St}}}$ considered here.
The scaling exponents of the structure functions of the temperature field $$\zeta^n_T \equiv \frac{{\text{\textrm{d}}}\log S^n_T {\left( r \right)}}{{\text{\textrm{d}}}\log r}$$ are shown in figure \[fig:SFT\](d) for $r\le2\eta$. The results show that the fluid temperature field remains smooth (to within numerical uncertainty) even when suspended particles are suspended in the flow. This is not trivial since the contribution from the particle to fluid coupling term $C_T \left({ \mathbf{x} }+{ \mathbf{r} },t\right)-C_T \left({ \mathbf{x} },t\right)$ need not be a smooth function of $r$.
Particle temperature structure functions
----------------------------------------
The $n$-th order structure function of the particle temperature $\theta_p{\left( t \right)}$ is defined as $$S^n_{\theta}{\left( r \right)} \equiv {\left\langle \left\vert \Delta \theta_{p}\right\vert^n \right\rangle}_{ r}$$ where $\Delta \theta_{p}(t)$ is the difference in the temperature of the two particles, and the brackets denote an ensemble average, conditioned on the two particles having separation $r$. The results for $S^2_{\theta}$ for different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, with and without two-way thermal coupling, are shown in figure \[fig:SFtheta\].
![Results for $S^2_\theta$ for different ${\text{\textit{St}}}_\theta$, for ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*) and ${\text{\textit{St}}}=3$ (*e-f*). Plots (*a-c-e*) are from simulations S1, in which the two-way thermal coupling is considered, while plots (*b-d-f*) are from simulations S2, in which the two-way coupling is neglected.[]{data-label="fig:SFtheta"}]({figure11-crop}.pdf){width="\textwidth"}
The results show that $S^2_{\theta}$ depends on ${\text{\textit{St}}}_\theta$ in much the same way as the inertial particle relative velocity structure functions depend on ${\text{\textit{St}}}$ [@Ireland2016]. This is not surprising since the equation governing $\dot{\theta}_p$ is structurally identical to the equation governing the particle acceleration. However, important differences are that $\dot{\theta}_p$ depends on both ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, and also that the fluid temperature field is structurally different from the fluid velocity field, with the temperature field exhibiting the well-known ramp-cliff structure.
To obtain further insight into the behavior of $S^2_{\theta}$ and $S^n_{\theta}$ in general, we note that the formal solution for $\Delta \theta_{p}(t)$ is given by (ignoring initial conditions) $$\Delta\theta_p{\left( t \right)} =\frac{1}{\tau_\theta} \int_{0}^t \Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)} \exp{\left( -\frac{t-s}{\tau_\theta} \right)} {\text{\textrm{d}}}s,
\label{eq:soltheta}$$ where $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ is the difference in the fluid temperature at the two particle positions ${ \mathbf{x} }_p{\left( s \right)}$ and ${ \mathbf{x} }_p{\left( s \right)}+{ \mathbf{r} }_p{\left( s \right)}$. Equation shows that $\Delta\theta_p{\left( t \right)}$ depends upon $\Delta T$ along the path-history of the particles, and $\Delta\theta_p{\left( t \right)}$ is therefore a non-local quantity. The role of the path-history increases as ${\text{\textit{St}}}_\theta$ is increased since the exponential kernel in the convolution integral decays more slowly as $\tau_\theta$ is increased. Since the statistics of $\Delta T$ increase with increasing separation, particle-pairs at small separations are able to be influenced by larger values of $\Delta T$ along their path-history, such that $\Delta\theta_p{\left( t \right)}$ can significantly exceed the local fluid temperature increment $\Delta T{\left( { \mathbf{x} }_p{\left( t \right)},{ \mathbf{r} }_p{\left( t \right)},t \right)}$. This then causes $S^2_{\theta}$ to increase with increasing ${\text{\textit{St}}}_\theta$, as shown in figure \[fig:SFtheta\]. This effect is directly analogous to the phenomena of caustics that occur in the relative velocity distributions of inertial particles at the small scales of turbulence [@Wilkinson2005], and which occur because the inertial particle relative velocities depend non-locally on the fluid velocity increments experienced along their trajectory history [@bragg14c]. In analogy, we may therefore refer to the effect as “thermal caustics”, and they may be of particular importance for particle-laden turbulent flows where particles in close proximity thermally interact.
The results in figure \[fig:SFtheta\] also reveal a strong effect of ${\text{\textit{St}}}$, and one way that ${\text{\textit{St}}}$ affects these results is through the spatial clustering and preferential sampling of the fluid temperature field by the inertial particles. There is, however, another mechanism through which ${\text{\textit{St}}}$ can affect $S^2_{\theta}$. In particular, since, due to caustics, the relative velocity of the particles increases with increasing ${\text{\textit{St}}}$ at the small scales, then the values of $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ that may contribute to $\Delta\theta_p{\left( t \right)}$ become larger. This follows since if their relative velocities are larger, then over the time span $t-s\leq {\textit{O}\left( \tau_\eta \right)}$ the particle-pair can come from even larger scales where (statistically) $\Delta T{\left( { \mathbf{x} }_p{\left( s \right)},{ \mathbf{r} }_p{\left( s \right)},s \right)}$ is bigger. This effect would cause $S^2_{\theta}$ to increase with ${\text{\textit{St}}}$ for a given ${\text{\textit{St}}}_\theta$, further enhancing the thermal caustics, which is exactly what is observed in figure \[fig:SFtheta\]. The results also show that the thermal caustics are stronger for ${\text{\textit{St}}}_\theta\geq {\textit{O}\left( 1 \right)}$ when the two-way thermal coupling is ignored. This is mainly due to the reduction in the fluid temperature gradients due to the two-way thermal coupling described earlier, noting that in the limit of vanishing fluid temperature gradients, the thermal caustics necessarily disappear.
At larger scales where the statistics of $\Delta T$ vary more weakly with $r$, the non-local effect weakens, the thermal caustics disappear, and a filtering mechanism takes over which causes $S^2_{\theta}$ to decrease with increasing ${\text{\textit{St}}}_\theta$. This filtering effect is directly analogous to that dominating the large-scale velocities of inertial particles in isotropic turbulence, and is associated with the sluggish response of the particles to the large scale flow fluctuations due to their inertia [@Ireland2016].
![Scaling exponent of the structure functions of the particle temperature at small separation, $r\le2\eta$, for various thermal Stokes numbers ${\text{\textit{St}}}_\theta$, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*).[]{data-label="fig:scalexpSFtheta"}]({figure12-crop}.pdf){width="\textwidth"}
In the dissipation range our results show that $S^n_{\theta}$ behave as power laws, and the associated scaling exponents $\zeta^n_\theta$ are shown in figure \[fig:scalexpSFtheta\]. To reduce statistical noise, we estimate $\zeta^n_\theta$ by fitting the data for $S^n_{\theta}$ over the range $r\leq 2\eta$. Over this range, $S^n_{\theta}$ do not strictly behave as power laws, and hence the exponents measured are understood as average exponents. The results in figure \[fig:scalexpSFtheta\] reveal that particle temperature increments exhibit a strong multifractal behaviour. This multifractility is due to the non-local thermal dynamics of the particles and the formation of thermal caustics, described earlier. In particular, there exists a finite probability to find inertial particle-pairs that are very close but have large temperature differences because they experienced very different fluid temperatures along their trajectory histories. As with the thermal caustics, the multifractility is enhanced as ${\text{\textit{St}}}$ is increased.
Most interestingly, the results for $\zeta^n_\theta$ are only weakly affected by the two-way thermal coupling, despite the fact that we observed a significant effect of the coupling on $S^2_{\theta}$. This suggests that the two-way coupling affects the strength of the thermal caustics, but only weakly affects the scaling of the structure functions in the dissipation range.
Mixed structure functions
-------------------------
We turn to consider the behaviour of the flux of the temperature increments across the scales of the flow, which is associated with the mixed structure functions $$S_Q (r)\equiv {\left\langle \left(\Delta T(r,t)\right)^2 \Delta u_\parallel(r,t) \right\rangle}$$ where $\Delta u_\parallel$ is the longitudinal relative velocity difference. The results for $S_Q$, for different ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ are shown in figure \[fig:mixedSFT\]. Just as we observed for the fluid temperature structure functions, $-S_Q$ decreases monotonically with increasing ${\text{\textit{St}}}_\theta$, as was also observed for the fluid temperature dissipation rate $\chi_f$.
![Second order mixed structure functions of the fluid temperature field, for different thermal Stokes numbers of the suspended particles, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*). The data refer to the set of simulations S1, with thermal particle back-reaction included.[]{data-label="fig:mixedSFT"}]({figure13-crop}.pdf){width="\textwidth"}
![Second order mixed structure functions of the particle temperature, for different thermal Stokes numbers, at ${\text{\textit{St}}}=0.5$ (*a-b*) and ${\text{\textit{St}}}=1$ (*c-d*). The plots on the left (*a-c*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected.[]{data-label="fig:mixedSFtheta"}]({figure14-crop}.pdf){width="\textwidth"}
To consider the flux of the particle temperature increments, we begin by considering the exact equation that can be constructed for $S^n_\theta$ using PDF transport equations. In particular, if we introduce the PDF $\mathcal{P}({ \mathbf{r} },\Delta\theta,t)\equiv\langle \delta({ \mathbf{r} }_p(t)-{ \mathbf{r} })\delta(\Delta\theta_p(t)-\Delta\theta)\rangle$ and the associated marginal PDF $\varrho({ \mathbf{r} },t)\equiv \int\mathcal{P}\,d\Delta\theta$, where ${ \mathbf{r} }$ and $\Delta\theta$ are time-independent phase-space coordinates, then we may derive for a statistically stationary system the result (see [@bragg14b; @bragg2015a] for details on how to derive such results) $$\Big\langle [\Delta\theta_p(t)]^2\Big\rangle_{ \mathbf{r} }=\Big\langle \Delta T({ \mathbf{x} }_p(t){ \mathbf{r} }_p(t),t)\Delta\theta_p(t)\Big\rangle_{ \mathbf{r} }-\frac{\tau_\theta}{2\varrho}\frac{\partial}{\partial{ \mathbf{r} }}\bcdot\varrho \Big\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\Big\rangle_{ \mathbf{r} },$$ where ${ \mathbf{w} }_p(t)\equiv \partial_t{ \mathbf{r} }_p(t)$. The first term on the right-hand side is the local contribution that remains when there exist no fluxes across the scales, and this term determines the behavior of $\langle [\Delta\theta_p(t)]^2\rangle_{ \mathbf{r} }$ at the large scales of homogeneous turbulence where the statistics are independent of ${ \mathbf{r} }$. The second term on the right-hand side is the non-local contribution that arises for ${\text{\textit{St}}}_\theta>0$, and it is this term that is responsible for the thermal caustics discussed earlier. It depends on the spatial clustering of the particles through $\varrho$ (which is proportional to the RDF), and the flux $\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\rangle_{ \mathbf{r} }$ which, for an isotropic system, is determined by the longitudinal component $$S_{Q_p} (r)\equiv \frac{{ \mathbf{r} }}{r}\bcdot\Big\langle [\Delta\theta_p(t)]^2{ \mathbf{w} }_p(t)\Big\rangle_{r}.$$ The results for $S_{Q_p}$ from our simulations are shown in figure \[fig:mixedSFtheta\], and they show that without two-way coupling, $-S_{Q_p}$ monotonically increases with increasing ${\text{\textit{St}}}_\theta$ at the smallest scales. However, with two-way coupling, $-S_{Q_p}$ is maximum for intermediate values of ${\text{\textit{St}}}_\theta$, and this occurs because as shown earlier, as ${\text{\textit{St}}}_\theta$ is increased, the fluid temperature fluctuations are suppressed across the scales.
Distribution of the temperature increments and fluxes {#sec:2pPDF}
=====================================================
In this section we look at the distribution of the fluid and particle temperature increments in the dissipation range.
![Probability density function in normal form of the fluid temperature increments at small separations, $r=10\eta$, at ${\text{\textit{St}}}=1$ (*a*) and ${\text{\textit{St}}}=3$ (*b*). The data refer to the set of simulations S1, with thermal feedback included. (*c*) Standard deviation of the fluid temperature increments at small separation. (*d*) Kurtosis of the distribution of the fluid temperature increments at small separation.[]{data-label="fig:PDFdT"}]({figure15-crop}.pdf){width="\textwidth"}
Temperature increments in the dissipation range
-----------------------------------------------
The normalized PDF of the fluid temperature increments at separations $r=10\eta$ are shown in figure \[fig:PDFdT\] (for the fluid temperature field, we do not consider the PDFs of the velocity increments for $r\leq {\textit{O}\left( \eta \right)}$ since these are essentially identical to the PDFs of the fluid temperature gradients that were considered earlier). Just as we observed earlier for the PDFs of the fluid temperature gradients, the results in figure \[fig:PDFdT\] show that at larger separations the PDFs of the fluid temperature increments are also self similar and approximately collapse when scaled by their standard deviation. The standard deviation and kurtosis of the PDF, also shown in \[fig:PDFdT\], show that while the kurtosis is almost independent of ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$, the variance decreases with increasing ${\text{\textit{St}}}_\theta$, and increases with increasing ${\text{\textit{St}}}$. This latter result differs significantly from the behavior of the variance of the fluid temperature gradients which were almost independent of ${\text{\textit{St}}}$.
![Probability density function in normal form of the particle temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=1$ (*a-b*) and ${\text{\textit{St}}}=3$ (*c-d*). The plots on the left (*a-c*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected. (*e*) Standard deviation of the particle temperature increments at small separation. (*f*) Kurtosis of the distribution of the particle temperature increments at small separation.[]{data-label="fig:PDFdtheta"}]({figure16-crop}.pdf){width="\textwidth"}
The PDF of the particle temperature increments, along with its variance and kurtosis are shown in figure \[fig:PDFdtheta\]. The results show that while the variance of the PDF monotonically increases with increasing ${\text{\textit{St}}}_\theta$, the kurtosis can increase slightly with increasing ${\text{\textit{St}}}_\theta$ when ${\text{\textit{St}}}_\theta$ is below some threshold, after which the kurtosis monotonically decreases with increasing ${\text{\textit{St}}}_\theta$. However, across the parameter range studied, the PDFs are strongly non-Gaussian, with a maximum kurtosis value of $\approx 13$. The kurtosis values are also strongly and non-monotonically dependent on ${\text{\textit{St}}}$, with the largest values tending to occur for ${\text{\textit{St}}}=1$. This may be due to the clustering of the particles in the fronts of the temperature field, leading to large particle temperature differences. It may also be due to the non-local mechanisms described earlier since although the non-local effects can enhance non-Gaussianity in certain regimes, in the regime where the behavior is entirely non-local (e.g. for ${\text{\textit{St}}}\gg1$), the behavior becomes ballistic and $\Delta\theta_p(t)$ is governed by a central limit theorem and the PDF of $\Delta\theta_p(t)$ approaches a Gaussian distribution.
Flux of temperature increments in the dissipation range
-------------------------------------------------------
We finally turn to consider the PDFs of the fluid temperature flux $Q=\left(\Delta T(r,t)\right)^2 \Delta u_\parallel(r,t)$ and particle temperature flux $Q=[\Delta\theta_p(t)]^2{w}_\parallel(t)$, where ${w}_\parallel(t)$ is the parallel component of the particle-pair relative velocity.
![Probability density function in normal form of the flux of fluid temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=0.5$ (*a*) and ${\text{\textit{St}}}=1$ (*b*). The data refer to the set of simulations S1, with thermal feedback included.[]{data-label="fig:PDFfluxdT"}]({figure17-crop}.pdf){width="\textwidth"}
![Probability density function in normal form of the flux of particle temperature increments at small separations, $r\le2\eta$, at ${\text{\textit{St}}}=0.5$ (*a-b*), ${\text{\textit{St}}}=1$ (*c-d*), ${\text{\textit{St}}}=3$ (*e-f*). The plots on the left (*a-c-e*) refer to the set of simulations S1, in which the thermal particle back-reaction is included. The plots on the right (*b-d-f*) refer to the set of simulations S2, in which the thermal particle back-reaction is neglected.[]{data-label="fig:PDFfluxdtheta"}]({figure18-crop}.pdf){width="\textwidth"}
The PDF of the fluid temperature flux is plotted in normal form for $r\le2\eta$ in figure \[fig:PDFfluxdT\]. These normalized PDFs collapse onto each other for all ${\text{\textit{St}}}$ and ${\text{\textit{St}}}_\theta$ values considered. Thus, the fluid temperature flux simply scales with its variance in the dissipation range, and the variance of the flux is modulated by the particles but the shape of the distribution is not affected by the particle dynamics. The PDF are strongly negatively skewed and have a negative mean value, associated with the mean flux of thermal fluctuations from large to small scales in the flow.
The PDF of the particle temperature flux is plotted in normal form for $r\le2\eta$ in figure \[fig:PDFfluxdtheta\]. The PDF of the particle temperature flux across the scales is not self-similar with respect to its variance. Furthermore, the PDF becomes more symmetric as ${\text{\textit{St}}}_\theta$ is increased. This is associated with the increasingly non-local thermal dynamics of the particles, which allows the particle-pairs to traverse many scales of the flow with minimal changes in their temperature difference.
Conclusions {#sec:concl}
===========
Using direct numerical simulations, we have investigated the interaction between the scalar temperature field and the temperature of inertial particles suspended in the fluid, with one and two-way thermal coupling, in statistically stationary, isotropic turbulence.
We found that the shape of the probability density function (PDF) of the fluid temperature gradients is not affected by the presence of the particles when two-way thermal coupling is considered, and scales with its variance. On the other hand, the variance of the fluid temperature gradients decreases with increasing ${\text{\textit{St}}}_\theta$, while ${\text{\textit{St}}}$ plays a negligible role. The PDF of the rate of change of the particle temperature, whose variance is associated with the thermal dissipation due to the particles, does not scale in a self-similar way with respect to its variance, and its kurtosis decreases with increasing ${\text{\textit{St}}}_\theta$. The particle temperature PDFs and their moments exhibit qualitatively different dependencies on ${\text{\textit{St}}}$ for the case with and without two-way thermal coupling.
To obtain further insight into the fluid-particle thermal coupling, we computed the number density of particles conditioned on the magnitude of the local fluid temperature. In agreement with [@Bec2014], we observed that the particles cluster in the fronts of the temperature field. We also computed quantities related to moments of the particle heat flux conditioned on the magnitude of the local fluid temperature. These results showed how the particles tend to decrease the fluid temperature gradients, and that it is associated with the statistical alignments of the particle velocity and the local fluid temperature gradient field.
The two-point temperature statistics were then examined to understand the properties of the temperature fluctuations across the scales of the flow. By computing the structure functions, we observed that the fluctuations of the fluid temperature increments are monotonically suppressed as ${\text{\textit{St}}}_\theta$ increases in the two-way coupled regime. The structure functions of the particle temperatures revealed the dominance of thermal caustics at the small scales, wherein the particle temperature differences at small separations rapidly increase as ${\text{\textit{St}}}_\theta$ and ${\text{\textit{St}}}$ are increased. This allows particles to come into contact with very large temperature differences, which has a number of important practical implications. The scaling exponents of the inertial particle temperature structure functions in the dissipation range revealed strongly multifractal behavior. PDFs of the fluid temperature increments at different separations were found to scale in a self-similar way with their variance, just as was found for the temperature gradients. However, PDFs of the particle temperature increments do not exhibit this self-similarity, and their non-Gaussianity is much stronger than that for the fluid.
Finally, the flux of fluid temperature increments across the scales was found to decrease monotonically with increasing ${\text{\textit{St}}}_\theta$. The PDFs of this flux are strongly negatively skewed and have a negative mean value, indicating that the flux is predominately from the large to the smallest scales of the flow. In the two-way coupled regime, the presence of the inertial particles does not change the shape of the PDF. The PDF of the flux of particle temperature increments in the dissipation range becomes more and more symmetric as ${\text{\textit{St}}}_\theta$ is increased, associated with the increasingly non-local thermal dynamics of the particles.
The results presented have revealed a number of non-trivial effects and behavior of the particle temperature statistics. In future work it will be important to consider the role of gravitation settling and coupling with water vapor fields, both of which are important for the cloud droplet problem. Moreover, it will be interesting to include the two-way momentum coupling and to consider the non-dilute regime.
Acknowledgments
===============
This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562 [@xsede]. Specifically, the Comet cluster was used under allocation CTS170009. The authors also acknowledge the computational resources provided by LaPalma Supercomputer at the Instituto de Astrofísica de Canarias through the Red Española de Supercomputación (project FI-2018-1-0044).
| ArXiv |
---
abstract: '**Majorana fermions, quantum particles that are their own anti-particles, are not only of fundamental importance in elementary particle physics and dark matter, but also building blocks for fault-tolerant quantum computation. Recently Majorana fermions have been intensively studied in solid state and cold atomic systems. These studies are generally based on superconducting pairing with zero total momentum. On the other hand, finite total momentum Cooper pairings, known as Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, were widely studied in many branches of physics. However, whether FFLO superconductors can support Majorana fermions has not been explored. Here we show that Majorana fermions can exist in certain types of gapped FFLO states, yielding a new quantum matter: topological FFLO superfluids/superconductors. We demonstrate the existence of such topological FFLO superfluids and the associated Majorana fermions using spin-orbit coupled degenerate Fermi gases and derive their parameter regions. The implementation of topological FFLO superconductors in semiconductor/superconductor heterostructures are also discussed.**'
author:
- 'Chunlei Qu$^{1}$'
- 'Zhen Zheng$^{2}$'
- 'Ming Gong$^{3}$'
- 'Yong Xu$^{1}$'
- 'Li Mao$^{1}$'
- 'Xubo Zou$^{2}$'
- 'Guangcan Guo$^{2}$'
- 'Chuanwei Zhang$^{1}$'
title: Topological Superfluids with Finite Momentum Pairing and Majorana Fermions
---
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Topological superconductors and superfluids are exotic quantum matters that host topological protected excitations, such as robust edge modes and Majorana Fermions (MFs) with non-Abelian exchange statistics [@Wilczek]. MFs are important not only because of their fundamental role in elementary particle physics and dark matters [@Hisano], but also their potential applications in fault-tolerant topological quantum computation [@TQC]. Recently some exotic systems, such as $\nu =5/2$ fractional quantum Hall states [@TQC], chiral *p*-wave superconductors/superfluids [TQC]{}, heterostructure composed of $s$-wave superconductors and semiconductor nanowires (nanofilms) or topological insulators [@Fu; @JSau; @Roman; @Oreg; @Alicea; @Lee; @Mao], etc., have been proposed as systems supporting MFs. Following the theoretical proposals, exciting experimental progress for the observation of MFs has been made recently in semiconductor [@Mourik; @Deng; @Das; @Rokhinson] or topological insulator heterostructures [@YCui], although unambiguous experimental evidence for MFs is still lacked.
These theoretical and experimental studies are based on the superconducting Cooper pairing ($s$-wave or chiral $p$-wave) with zero total momentum, that is, the pairing is between two fermions with opposite momenta $\mathbf{k}$ and $\mathbf{-k}$ (denoted as BCS pairing hereafter). On the other hand, the superconducting pairing can also occur between fermions with finite total momenta (pairing between $\mathbf{k}$ and $\mathbf{-k+Q}$) in the presence of a Zeeman field, leading to spatially modulated superconducting order parameters in real space, known as FFLO states. The FFLO states were first predicted in 1960s [@FF64; @LO64], and now are a central concept for understanding exotic phenomena in many different systems [FFLOreview,FFLO2,FFLO3,FFLO1,Parish, HuPRA]{}. A natural question to ask is whether MFs can also exist in a FFLO superconductor or superfluid?
In this Letter, we propose that FFLO superconductors/superfluids may support MFs if they possess two crucial elements: gapped bulk quasi-particle excitations and nontrivial Fermi surface topology. These new quantum states are topological FFLO superconductors/superfluids. In this context, traditional gapless FFLO states induced by a large Zeeman field do not fall into this category. Here we propose a possible platform for the realization of topological FFLO superfluids using two-dimensional (2D) or one-dimensional (1D) spin-orbit (SO) coupled degenerate Fermi gases subject to in-plane and out-of-plane Zeeman fields. Recently, the SO coupling and Zeeman fields for cold atoms have already been realized in experiments [Ian, Pan, Peter, Zhang, Martin]{}, which provide a completely new avenue for studying topological superfluid physics. It is known that SO coupled degenerate Fermi gases with an out-of-plane Zeeman field support MFs with zero total momentum pairing [@CW; @Jiang; @Gong; @Melo]. We find in suitable parameter regions the in-plane Zeeman field can induce the finite total momentum pairing [@Zheng; @WYi; @Hui; @Lee2], while still keeps the superfluid gapped and preserves its Fermi surface topology. The region for topological FFLO superfluids depends not only on the chemical potential, pairing strength, but also on the SO coupling strength, total momentum and effective mass of the Cooper pair, as well as the orientation and magnitude of the Zeeman field, thus greatly increases the tunability in experiments. Finally, the potential implementation of the proposal in semiconductor/superconductor heterostructures is also discussed.
[**Results**]{}
**System and Hamiltonian**: Consider a SO coupled Fermi gas in the $xy$ plane with the effective Hamiltonian $$H=\sum_{\mathbf{k}\sigma \sigma ^{\prime }}c_{\mathbf{k},\sigma }^{\dagger
}H_{0}^{\sigma \sigma ^{\prime }}c_{\mathbf{k},\sigma ^{\prime }}+V_{\text{%
int}} \label{eq-H}$$where $H_{0}=\frac{\mathbf{k}^{2}}{2m}-\mu +\alpha \mathbf{k}\times \vec{%
\sigma}\cdot {\hat{e}_{z}}-\mathbf{h}\cdot \vec{\sigma}$, $\mathbf{k}%
=(k_{x},k_{y})$, $\alpha $ is the Rashba SO coupling strength, $\mathbf{h}%
=(h_{x},0,h_{z})$ is the Zeeman field and ${\vec{\sigma}}$ is the Pauli matrices. $V_{\text{int}}=g{\sum }c_{\mathbf{k}_{1},\uparrow }^{\dagger }c_{%
\mathbf{k}_{2},\downarrow }^{\dagger }c_{\mathbf{k}_{3},\downarrow }c_{%
\mathbf{k}_{4},\uparrow }$ describes the $s$-wave scattering interaction, where $g=-(\sum_{\mathbf{k}}\left( \mathbf{k}^{2}/m+E_{b}\right) ^{-1}{)}%
^{-1}$ is the scattering interaction strength, $E_{b}$ is the binding energy, and $\mathbf{k}_{1}+\mathbf{k}_{2}=\mathbf{k}_{3}+\mathbf{k}_{4}$ due to the momentum conservation. Without in-plane Zeeman field $h_{x}$, the Fermi surface is symmetric around $\mathbf{k}=0$, and the superfluid pairing is between atoms with opposite momenta $\mathbf{k}$ and $-\mathbf{k}$. While with both $h_{x}$ and SO coupling, the Fermi surface becomes asymmetric along the $y$ direction (see Fig. \[fig-FS\]a), and the pairing can occur between atoms with momenta $\mathbf{k}$ and $-\mathbf{k}+\mathbf{Q}$. In real space, such a finite total momentum pairing leads to a FF-type order parameter $\Delta (\mathbf{x})=\Delta e^{i\mathbf{Q}\cdot \mathbf{x}}$, where $\mathbf{Q}=(0,Q_{y})$ is parallel to the deformation direction of the Fermi surface [@Zheng; @WYi; @Hui]. Notice that the energies of the superfluids with total momentum $\mathbf{Q}$ and $\mathbf{-Q}$ are nondegenerate, therefore FF phase with a single $\mathbf{Q}$, instead of LO phase ($\Delta (\mathbf{x})=\Delta \cos (\mathbf{Q}\cdot \mathbf{x})$) where pairing occurs at both $\pm \mathbf{Q}$, is considered here. Hereafter, if not specified, FFLO superfluids refer to FF superfluids.
The dynamics of the system can be described by the following Bogliubov-de Gennes (BdG) Hamiltonian in the mean-field level, $$H_{\text{BdG}}(\mathbf{k})=%
\begin{pmatrix}
H_{0}({\frac{\mathbf{Q}}{2}}+\mathbf{k}) & \Delta \\
\Delta & -\sigma _{y}H_{0}^{\ast }({\frac{\mathbf{Q}}{2}}-\mathbf{k})\sigma
_{y}%
\end{pmatrix}%
, \label{eq-bdg}$$where the Nambu basis is chosen as $(c_{\mathbf{k}+\mathbf{Q}/2,\uparrow
},c_{\mathbf{k}+\mathbf{Q}/2,\downarrow },c_{-\mathbf{k}+\mathbf{Q}%
/2,\downarrow }^{\dagger },-c_{-\mathbf{k}+\mathbf{Q}/2,\uparrow }^{\dagger
})^{T}$. The gap, number and momentum equations are solved self-consistently to obtain $\Delta $, $\mu $ and $\mathbf{Q}$, see Methods, through which we determine different phases.
![**Single particle band structure and Berry curvature**. (a) Energy dispersion of the lower band. The green arrows represent the momenta of a Cooper pair of two atoms on the asymmetric Fermi surface. The red arrow represents the total finite momentum of the paring, which is along the deformation direction of the Fermi surface. (b) Berry curvature of the lower band, whose peak is shifted from the origin by $h_{x}/\protect\alpha $ along the $k_{y}$ direction. []{data-label="fig-FS"}](fig1.eps){width="3.2in"}
**Physical mechanism for topological FFLO phase**: Without SO coupling, the orientation of the Zeeman field does not induce any different physics due to SU(2) symmetry. The presence of both $h_{x}$ and SO coupling breaks this SU(2) symmetry, leading to a Fermi surface without inversion symmetry, see Fig. \[fig-FS\]a. Here, $h_{x}$ deforms the Fermi surface, leading to FFLO Cooper pairings; while $h_{z}$ opens a gap between the two SO bands, making it possible for the chemical potential to cut a single Fermi surface for the topological FFLO phase. The Berry curvature of the lower band reads as $$\Omega _{\mathbf{k}}={\frac{\alpha ^{2}h_{z}}{2(\alpha ^{2}k_{x}^{2}+(\alpha
k_{y}+h_{x})^{2}+h_{z}^{2})^{3/2}}}.$$Note that $h_{x}$ shifts the peak of Berry curvature from $\mathbf{k}=0$ to $%
(0,-h_{x}/\alpha )$ (denoted by arrow in Fig. \[fig-FS\]b). When atoms scatter from $\mathbf{k}$ to $\mathbf{k}^{{\prime }}$ on the Fermi surface, they pick up a Berry phase, whose accumulation around the Fermi surface $%
\theta =\int d^{2}\mathbf{k}\Omega _{\mathbf{k}}\approx \pi $. Such Berry phase modifies the effective interaction from $s$-wave ($V_{\mathbf{k}%
\mathbf{k^{\prime }}}\sim g$ is a constant) to $s$-wave plus asymmetric $p$-wave $$V_{\mathbf{k}\mathbf{k^{\prime }}}\sim g\left( ke^{-i\theta _{\mathbf{k}}}+{%
\frac{h_{x}}{\alpha }}\right) \left( k^{\prime }e^{i\theta _{\mathbf{k}%
^{\prime }}}-{\frac{h_{x}}{\alpha }}\right)$$on the Fermi surface. Here we recover the well-known chiral $p_{x}+ip_{y}$ pairing [@CW] in the limit $h_{x}=0$. The in-plane Zeeman field here creates an effective $s$-wave pairing component (although still hosts MFs), and the effective pairing is reminiscent to the ($s$+$p$)-wave pairing in some solid materials [@Yuan].
**Parameter region for MFs**: The BdG Hamiltonian (\[eq-bdg\]) satisfies the particle-hole symmetry $\Xi =\Lambda \mathcal{K}$, where $%
\Lambda =i\sigma _{y}\tau _{y}$, $\mathcal{K}$ is the complex conjugate operator, and $\Xi ^{2}=1$. The parameter region for the MFs is determined by the topological index $\mathcal{M}=\text{sign}(\text{Pf}\{\Gamma \})$, where Pf is the Pfaffian of the skew matrix $\Gamma =H_{\text{BdG}%
}(0)\Lambda $. $\mathcal{M}=-1(+1)$ corresponds to the topologically nontrivial (trivial) phase [@Parag]. The topological phase exists when $$h_{z}^{2}+\bar{h}_{x}^{2}>\bar{\mu}^{2}+\Delta ^{2}\text{,}\quad \alpha
h_{z}\Delta \neq 0\text{,}\quad E_{g}>0, \label{eq-parameter}$$where $\bar{h}_{x}=h_{x}+\alpha Q_{y}/2$ and $\bar{\mu}=\mu -Q_{y}^{2}/8m$. $%
E_{g}=\text{min}(E_{\mathbf{k},s})$ defines the bulk quasi-particle excitation gap of the system with $E_{\mathbf{k},s}$ as the particle branches of the BdG Hamiltonian (\[eq-bdg\]). The first condition reduces to the well-known $h_{z}^{2}>\Delta ^{2}+\mu ^{2}$ in BCS topological superfluids [@Mourik; @Deng; @Das; @Rokhinson; @Roman; @Oreg; @Gong]. The last condition ensures the bulk quasi-particle excitations are gapped to protect the zero energy MFs in the topological regime. The SO coupling and the FFLO vector shift the effective in-plane Zeeman field and the chemical potential. In contrast, in the BCS topological superfluids, the SO coupling strength, although required, does not determine the topological boundaries. Our system therefore provides more knobs for tuning the topological phase transition. To further verify condition (\[eq-parameter\]), we calculate the Chern number in the hole branches $\mathcal{C}=$ $\sum_{n}\mathcal{C}_{n}$ in the gapped superfluids [@Parag], and confirm $\mathcal{C}=+1$ when Eq. (5) is satisfied and $\mathcal{C}=0$ otherwise. Here $\mathcal{C}_{n}=\frac{1}{%
2\pi }\int d^{2}k\Gamma _{n}$ is the Chern number, $\Gamma _{n}=-2$Im$%
\left\langle \frac{\partial \Psi _{n}}{\partial k_{x}}|\frac{\partial \Psi
_{n}}{\partial k_{y}}\right\rangle $ is the Berry curvature [@Xiao], and $\left\vert \Psi _{n}\right\rangle $ is the eigenstate of two hole bands of the BdG Hamiltonian (\[eq-bdg\]).
![**The order parameter $\Delta $, chemical potential $\protect%
\mu $, bulk quasi-particle gap $E_{g}$, and FFLO vector $Q_{y}$ as a function of Zeeman fields**. In (b) and (d), the dashed lines are the best fitting with quadratic and linear functions in the small Zeeman field regime, respectively. In (a) and (b), $h_{x}=0.2E_{F}$, while in (c) - (d), $%
h_{Z}=0.2E_{F}$. Other parameters are $E_{b}=0.4E_{F}$, $\protect\alpha %
K_{F}=1.0E_{F}$. The vertical lines mark the points where the Pfaffian changes the sign.[]{data-label="fig-Delta"}](becbcs.eps){width="3in"}
The transition from non-topological to topological phases defined by Eq. (\[eq-parameter\]) can be better understood by observing the close and reopen of the excitation gap $E_{g}$, which is necessary to change the topology of Fermi surface. In Fig. \[fig-Delta\], we plot the change of $%
E_{g} $ along with the order parameter $|\Delta |$, the chemical potential $%
\mu $, and the FF vector $\mathbf{Q}$ as a function of Zeeman fields. For a fixed $h_{x}$ but increasing $h_{z}$, $E_{g}$ may first close and then reopen (Fig. \[fig-Delta\]a), signalling the transition from non-topological to topological gapped FFLO superfluids ($Q_{y}$ is finite for all $h_{z}$, see Fig. \[fig-Delta\]b). For a fixed $h_{z}$, the superfluid is gapped and $Q_{y}\propto h_{x}$ for a small $h_{x}$ (see Fig. \[fig-Delta\]d), thus any small $h_{x}$ can transfer the gapped BCS superfluids at $h_{x}=0$ to FFLO superfluids. However, such a small $h_{x}$ does not destroy the bulk gap of BCS superfluids (topological or non-topological), making gapped topological FFLO superfluids possible when the system is initially in topological BCS superfluids without $h_{x}$. With increasing $h_{x}$ (Fig. \[fig-Delta\]c), $E_{g}$ may first close but does not reopen immediately, signalling the transition from gapped FFLO superfluids to gapless FFLO superfluids. For a small $h_{z}=0.2E_{F}$, further increasing $h_{x}$ to $\sim 0.78E_{F}$, $E_{g}$ reopens again (Fig. \[fig-Delta\]c), signalling the transition from gapless FFLO to gapped topological FFLO superfluids. In this regime, $Q_{y}\sim 0.6K_{F}$, which is not small. For a strong enough Zeeman field, the pairing may be destroyed and the system becomes a normal gas.
The complete phase diagrams are presented in Fig. \[fig-Phases\]. Since $%
Q_{y}$ and $h_{x}$ have the same sign, the phase diagram show perfect symmetry in the $h_{x}-h_{z}$ plane. The BCS superfluids can only be observed at $h_{x}=0$, hence are not depicted. With increasing SO coupling strength, the topological FFLO phase is greatly enlarged through the expansion to the normal gas phase. For a small SO coupling (Fig. [fig-Phases]{}a), a finite $h_{z}$ is always required to create the topological FFLO phase; In the intermediate regime (Fig. \[fig-Phases\]b) we find an interesting parameter regime where the topological FFLO phase can be reached with an extremely small $h_{z}$ around $h_{x}\sim 0.8E_{F}$. However, the topological FFLO phase can never be observed at $h_{z}=0$, as analyzed before from the Berry curvature and Chern number. From Fig. [fig-Phases]{}a-b we see that the topological gapped FFLO phase can be mathematically regarded as an adiabatic deformation of the topological BCS superfluids by an in-plane Zeeman field, although their physical meaning are totally different. In Fig. \[fig-Phases\]c-d, we see that the gapless FFLO phase can be observed at small binding energy and small $h_{z}$, while for large enough binding energy, the system can be either topological or non-topological gapped phases. In this regime, $E_{g}\sim \sqrt{\mu
^{2}+\Delta ^{2}}-\sqrt{h_{x}^{2}+h_{z}^{2}}$, where $\mu \sim E_{F}-E_{b}/2$, and $\Delta ^{2}\sim 2E_{F}E_{b}$, thus $h_{z}\propto E_{b}$ is required to close and reopen $E_{g}$ (see Fig. \[fig-Phases\]c-d).
![(Color online). **Phase diagram of the FFLO superfluid**. The phases are labelled with different colors: topological gapped FFLO superfluid (red), non-topological gapped FFLO superfluid (yellow), gapless FFLO superfluid (blue) and normal gas (white). Other parameters are: (a) $%
E_{b}=0.4E_{F}$, $\protect\alpha k_{F}=0.5E_{F}$; (b) $E_{b}=0.4E_{F}$, $%
\protect\alpha k_{F}=1.0E_{F}$; (c) $h_{x}=0.5E_{F}$, $\protect\alpha %
k_{F}=0.5E_{F}$; (d) $h_{x}=0.5E_{F}$, $\protect\alpha k_{F}=1.0E_{F}$. The symbols in each panel are the tricritical points.[]{data-label="fig-Phases"}](phase_diagram.eps){width="3in"}
The tricritical points marked by symbols in Fig. \[fig-Phases\] are essential for understanding the basic structure of the phase diagram. Along the $h_{z}$ axis, the system only supports gapped BCS superfluids (topological or non-topological) and normal gas [@Gong], while along the $h_{x}$ axis the system only supports trivial FFLO superfluids and normal gas [@Zheng; @WYi; @Hui]. So the adiabatic connection between the topological BCS superfluids and trivial FFLO phases is impossible, and there should be some points to separate different phases, which are exactly the tricritical points. In our model the transition between different phases is of first-order process. The existence of tricritical point here should be in stark contrast to the tricritical point at finite temperature in the same system without SO coupling, which arises from the accidental intersection of first and second order transition lines [@Parish]. Therefore the tricritical points in Fig. \[fig-Phases\] cannot be removed, although their specific positions vary with the system parameters.
**Chiral edge modes**: The topological FFLO superfluids support exotic chiral edge modes. To see the basic features more clear, we consider the same model in a square lattice with the following tight-binding Hamiltonian, $$H_{\text{L}}=H_{0}+H_{\text{Z}}+H_{\text{so}}+V_{\text{int}}, \label{eq-TB}$$where $H_{0}=-t\sum_{\langle i,j\rangle ,\sigma }c_{i\sigma }^{\dagger
}c_{j\sigma }-\mu \sum_{i\sigma }n_{i\sigma }$, $H_{\text{Z}%
}=-h_{x}\sum_{i}(c_{i\uparrow }^{\dagger }c_{i\downarrow }+c_{i\downarrow
}^{\dagger }c_{i\uparrow })-h_{z}\sum_{i}(n_{i\uparrow }-n_{i\downarrow })$, $H_{\text{so}}=-\frac{\alpha }{2}\sum_{i}(c_{i-\hat{x}\downarrow }^{\dagger
}c_{i\uparrow }-c_{i+\hat{x}\downarrow }^{\dagger }c_{i\uparrow }+ic_{i-\hat{%
y}\downarrow }^{\dagger }c_{i\uparrow }-ic_{i+\hat{y}\downarrow }^{\dagger
}c_{i\uparrow }+\text{H.C})$, and $V_{\text{int}}=-U\sum_{i}n_{i\uparrow
}n_{i\downarrow }=\sum_{i}\Delta _{i}^{\ast }c_{i\downarrow }c_{i\uparrow
}+\Delta _{i}c_{i\uparrow }^{\dagger }c_{i\downarrow }^{\dagger }-|\Delta
_{i}|^{2}/U$, with $\Delta _{i}=-U\langle c_{i\downarrow }c_{i\uparrow
}\rangle $, $n_{i\sigma }=c_{i\sigma }^{\dagger }c_{i\sigma }$. Here $%
c_{i\sigma }$ denotes the annihilation operator of a fermionic atom with spin $\sigma $ at site $i=(i_{x},i_{y})$. Hereafter, we use $t=1$ as the basic energy unit. For more details, see Methods.
![**Chiral edge states of topological FFLO phases in a 2D strip**. The strip is along the $x$ direction (a); $y$ direction (b). The parameters are $Q_{y}=-0.25$, $\protect\mu =-4t$, $\protect\alpha =2.0t$, $%
\Delta =1.0t$, $h_{z}=-1.2t$, $h_{x}=-0.3t$.[]{data-label="fig-edgestate"}](chiralstate.eps){width="3in"}
In the following, we only present the chiral edge states in the topological gapped FFLO superfluid regime, and assume $\Delta _{i}=\Delta
e^{iQ_{y}i_{y}} $. We consider a 2D strip with width $W=200$, and the results for the strip along $x$ and $y$ directions in the topological FFLO phase are presented in Fig. \[fig-edgestate\]. The linear dispersion of the edge states reads as $$H_{\text{edge}}=\sum_{k}v_{L}\psi _{kL}^{\dagger }k\psi _{kL}-v_{R}\psi
_{kR}^{\dagger }k\psi _{kR},$$where $L$ and $R$ define the left and right edges of the strip, and $v_{L}$ and $v_{R}$ are the corresponding velocities. We have also confirmed that the wavefunctions of the edge states are well localized at two edges. For a strip along the $x$ direction, the particle-hole symmetry as well as the discrete $\mathbb{Z}_{2}$ symmetry for $k_{x}\rightarrow -k_{x}$ ensure the eigenenergies of Eq. \[eq-TB\] always come in pairs ($E_{k}$,$-E_{k}$), thus $v_{R}=v_{L}$. However, when the strip is along the $y$ direction (parallel to the FFLO momentum $\mathbf{Q}$), the eigenenergies no longer come in pairs, therefore $v_{R}\neq v_{L}$. The two chiral edge states with totally different velocities and density of states represent the most remarkable feature of our model. The Chern number $C=1$ in our lattice model, thus only one pair of chiral edge states can be observed.
![**Majorana fermions in a 1D chain.** (a) The BdG quasi-particle excitation energies ($E_{2}$, $E_{1}$, -$E_{1}$, -$E_{2}$) and the order parameter; (b) The spatial profile of the FF type order parameter obtained self-consistently. (c) The wavefunction (WF) of the Majorana zero energy state $\left( U_{\uparrow },V_{\uparrow }\right) $ in the 1D chain. $\left( U_{\downarrow },V_{\downarrow }\right) $ is similar but with different amplitudes. The parameters are $\protect\alpha =2.0t$, $%
h_{x}=-0.5t$, $h_{z}=-1.2t$, $U=4.5t$, $\protect\mu =-2.25t$. []{data-label="fig-mf"}](Majorana1D.eps){width="3in"}
**MFs in 1D Chain**: Topological FFLO superfluid and associated MFs can also be observed in 1D SO coupled Fermi gas when the Hamiltonian (\[eq-TB\]) is restricted to 1D chain. In this case, the system is characterized by a $%
\mathbb{Z}_{2}$ invariant, which can be determined using the similar procedure as discussed above. The only difference is that now not only $k=0$, but also $k=\pi $ needs be taken into account (see Methods). In Fig. [fig-mf]{}a, we see Majorana zero-energy state protected by a large gap ($\sim
0.3t$) emerges in a suitable parameter region. The superfluid order parameter (Fig. \[fig-mf\]b) has the FF form. The local Bogoliubov quasi-particle operator $\gamma (E_{n})=\sum_{i\sigma }u_{i\sigma
}^{n}c_{i\sigma }+v_{i\sigma }^{n}c_{i\sigma }^{\dagger }$, where the zero energy wavefunction $\left( u_{i\uparrow }^{0},u_{i\downarrow
}^{0},v_{i\uparrow }^{0},v_{i\downarrow }^{0}\right) =\left( U_{i\uparrow
}e^{i\phi _{i\uparrow }},U_{i\downarrow }e^{i\phi _{i\downarrow
}},V_{i\uparrow }e^{-i\phi _{i\uparrow }},V_{i\downarrow }e^{-i\phi
_{i\downarrow }}\right) $ satisfies $u_{i\sigma }^{0}=v_{i\sigma }^{0\ast }$ at the left edge and $u_{i\sigma }^{0}=-v_{i\sigma }^{0\ast }$ at the right edge (see Fig. \[fig-mf\]c). This state supports two local MFs at two edges, respectively [@Roman].
[**Discussions**]{}
Our proposed topological FFLO phase may also be realized using semiconductor/superconductor heterostructures. Recently, topological BCS superconductors and the associated MFs have been proposed in such heterostructures [@JSau; @Roman; @Oreg; @Alicea] and some preliminary experimental signatures have been observed [@Mourik; @Deng; @Das; @Rokhinson]. To realize a topological FFLO superconductor, the semiconductor should be in proximity contact with a FFLO superconductor, which introduces finite momentum Cooper pairs. The topological parameter region defined in Eq. (5) still applies except that the order parameter, chemical potential and FFLO vector are external independent parameters. The flexibility of Eq. (5) makes it easier for tuning to the topological region with MFs. Because the FFLO state can sustain in the presence of a large magnetic field, it opens the possibility for the use of many semiconductor nanowires with large spin-orbit coupling but small $g$-factors (e.g, GaSb, hole-doped InSb, etc.).
In summary, we propose that topological FFLO superfluids or superconductors with finite momentum pairings can be realized using SO coupled $s$-wave superfluids subject to Zeeman fields and they support exotic quasi-particle excitations such as chiral edge modes and MFs. The phase transition to the topological phases depends strongly on all physical quantities, including SO coupling, chemical potential, Zeeman field and its orientations, paring strength, FFLO vector $\mathbf{Q}$ and the effective mass of Cooper pairs explicitly, which are very different from topological BCS superfluids/superconductors that are intensively studied recently. These new features not only provide more knobs for tuning topological phase transitions, but also greatly enrich our understanding of topological quantum matters. The topological FFLO phases have not been discussed before, and the phases unveiled in this Letter represent a totally new quantum matter.
[**Methods**]{}
**Momentum space BdG equations**: The partition function at finite temperature $T$ is $Z=\int \mathcal{D}[\psi ,\psi ^{\dagger }]e^{-S[\psi
,\psi ^{\dagger }]}$, where $S[\psi ,\psi ^{\dagger }]=\int d\tau d\mathbf{r}%
\sum_{\sigma =\uparrow ,\downarrow }\psi _{\sigma }(\mathbf{x})^{\dagger
}\partial _{\tau }\psi _{\sigma }(\mathbf{x})+H$, with $H$ defined in Eq. \[eq-H\], and $V_{\text{int}}=g\psi _{\uparrow }^{\dagger }\psi
_{\downarrow }^{\dagger }\psi _{\downarrow }\psi _{\uparrow }$ in real space. The FFLO phase is defined as $g\langle \psi _{\downarrow }(\mathbf{x}%
)\psi _{\uparrow }(\mathbf{x})\rangle =\Delta e^{i\mathbf{Q}\cdot \mathbf{x}%
} $, where $\mathbf{Q}$ is the total momentum of the Cooper pairs and $%
\Delta $ is a spatially independent constant. Here the position dependent phase of $\Delta (\mathbf{x})$ can be gauged out by the transformation $\psi
_{\sigma }\rightarrow \psi _{\sigma }e^{i\mathbf{Q}\cdot \mathbf{x}/2}$. Integrating out the fermion field $\psi $ and $\psi ^{\dagger }$, we obtain $%
Z=\int \mathcal{D}\Delta e^{-S_{\text{eff}}}$, with effective action $S_{%
\text{eff}}=\int d{\tau }d\mathbf{r}{\frac{|\Delta |^{2}}{g}}-{\frac{1}{%
2\beta }}\ln \text{Det}\beta G^{-1}+\text{Tr}(H)$, where $\beta =1/T$, and $%
G^{-1}=\partial _{\tau }+H_{\text{BdG}}$. The order parameter, chemical potential and FFLO vector $\mathbf{Q}$ are determined self-consistently by solving the following equation set $${\frac{\partial S_{\text{eff}}}{\partial \Delta }}=0,\quad {\frac{\partial
S_{\text{eff}}}{\partial \mu }}=-\beta n,\quad {\frac{\partial S_{\text{eff}}%
}{\partial \mathbf{Q}}}=0.$$In our model the deformation of Fermi surface is along the $y$ direction, thus we have $\mathbf{Q}=(0,Q_{y})$, and only three parameters need be determined self-consistently. We determine the different quantum phases using the following criterion. When $E_{g}>0$, $\Delta \neq 0$, we have gapped FFLO phases ($\mathcal{M}=-1$ ($\mathcal{C}=+1$) for topological, and $M=+1$ ($C=0$) for non-topological). When there is a nodal line with $%
E_{g}=0 $ and $\Delta \neq 0$, we have gapless FFLO phases. When $\Delta =0$ (then $\mathbf{Q}=0$ is enforced), we get normal gas phases. It is still possible to observe gapless excitations in the gapless FFLO phase regime, however, we do not distinguish this special condition because gapless excitations are not protected by gaps. In our numerics, the energy and momentum are scaled by Fermi energy $E_{F}$ and its corresponding momentum $%
K_{F}$ in the case without SO coupling and Zeeman fields. The results in Fig. \[fig-Delta\] and Fig. \[fig-Phases\] are determined at $%
n=K_{F}^{2}/2\pi $ and $T=0$.
**Real space BdG equations**: In the tight-binding model of (\[eq-TB\]), the many-body interaction is decoupled in the mean-field approximation. The particle number $n_{i\sigma }=c_{i\sigma }^{\dagger }c_{i\sigma }$ and superfluid pairing $\Delta _{i}=-U\langle c_{i\downarrow }c_{i\uparrow
}\rangle $ are determined self-consistently for a fixed chemical potential. Using the Bogoliubov transformation, we obtain the BdG equation $$\sum_{j}%
\begin{pmatrix}
H_{ij\uparrow } & \alpha _{ij} & 0 & \Delta _{ij} \\
-\alpha _{ij} & H_{ij\downarrow } & -\Delta _{ij} & 0 \\
0 & -\Delta _{ij}^{\ast } & -H_{ij\uparrow } & -\alpha _{ij} \\
\Delta _{ij}^{\ast } & 0 & \alpha _{ij} & -H_{ij\downarrow }%
\end{pmatrix}%
\begin{pmatrix}
u_{j\uparrow }^{n} \\
u_{j\downarrow }^{n} \\
-v_{j\uparrow }^{n} \\
v_{j\downarrow }^{n}%
\end{pmatrix}%
=E_{n}%
\begin{pmatrix}
u_{j\uparrow }^{n} \\
u_{j\downarrow }^{n} \\
-v_{j\uparrow }^{n} \\
v_{j\downarrow }^{n}%
\end{pmatrix}%
, \label{BdG}$$where $H_{ij\uparrow }=-t\delta _{i\pm 1,j}-(\mu +h_z)\delta _{ij}$, $%
H_{ij\downarrow }=-t\delta _{i\pm 1,j}-(\mu -h_z)\delta _{ij}$, $\alpha
_{ij}=\frac{1}{2}(j-i)\alpha \delta _{i\pm 1,j}-h_x \delta_{i,j}$, $%
\left\langle \hat{n}_{i\sigma }\right\rangle =\sum_{n=1}^{2N}[|u_{i\sigma
}|^{2}f(E_{n})+|v_{i\sigma }|^{2}f(-E_{n})]$, $\Delta _{ij}=-U\delta
_{ij}\sum_{n=1}^{2N}[u_{i\uparrow }^{n}v_{i\downarrow }^{n\ast
}f(E_{n})-u_{i\downarrow }^{n}v_{i\uparrow }^{n\ast }f(-E_{n})]$, with $%
f(E)=1/\left( 1+e^{E/T}\right) $. In the tight-binding model, FF phase and LO phase can be determined naturally, which depend crucially on the parameters of the system as well as the position of the chemical potential. The results in Fig. \[fig-edgestate\] and Fig. \[fig-mf\] are obtained at $T=0$.
**Topological boundaries in lattice models:** To determine the topological phase transition conditions, we transform the tight-binding Hamiltonian to the momentum space in Eq. \[eq-bdg\]. Here $\xi _{\mathbf{k}%
}$ is replaced by $-2t\cos (k_{x})-2t\cos (k_{y})-\mu $ for the kinetic energy, and $k_{\alpha }$ by $\sin (k_{\alpha })$ for the SO coupling, where $\alpha =x,y$. The topological boundary conditions can still be determined by the Pfaffian of $\Gamma (\mathbf{K})=H_{\text{BdG}}(\mathbf{K})\Lambda $ at four nonequivalent points, $K_{1}=(0,0)$, $K_{2}=(0,\pi )$, $K_{3}=(\pi
,0)$, $K_{4}=(\pi ,\pi )$ when the system is gapped. At these special points, $\Gamma (\mathbf{k})$ is a skew matrix. The topological phase is determined by $\mathcal{M}=\prod_{i=1}^{4}\text{sign}(\text{Pf}(\Gamma
(K_{i})))=-1$. For uniform BCS superfluids, the Pfaffian at $K_{2}$ and $%
K_{3}$ are identical, thus only $K_{1}$ and $K_{4}$ are essential to determine the topological boundaries. However, in our system, all four points affect the topological boundaries, and the exact expression of $%
\mathcal{M}$ is too complex to present here. In 1D chain, there are only two nonequivalent points at $K_{1}=0$ and $K_{2}=\pi $. We find $\text{Pf}%
(\Gamma (K_{1}))=\Delta ^{2}-(h_{z}-\mu -2t\cos (Q_{y}/2))(h_{z}+\mu +2t\cos
(Q_{y}/2))-(h_{x}+\alpha \sin (Q_{y}/2))^{2}$ , and $\text{Pf}(\Gamma
(K_{2}))=\Delta ^{2}-(h_{z}-\mu +2t\cos (Q_{y}/2))(h_{z}+\mu -2t\cos
(Q_{y}/2))-(h_{x}-\alpha \sin (Q_{y}/2))^{2}$. The topological index in the gapped regime is determined by $\text{sign}(\text{Pf}(\Gamma (K_{1})))\text{%
sign}(\text{Pf}(\Gamma (K_{2})))$.
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**Acknowledgement**
C.Q, Y.X, L.M, C.Z are supported by ARO (W911NF-12-1-0334), AFOSR (FA9550-13-1-0045), and NSF-PHY (1104546). Z.Z., X.Z., and G.G. are supported by the National 973 Fundamental Research Program (Grant No. 2011cba00200), the National Natural Science Foundation of China (Grant No. 11074244 and No. 11274295). M.G is supported in part by Hong Kong RGC/GRF Project 401512, the Hong Kong Scholars Program (Grant No. XJ2011027) and the Hong Kong GRF Project 401113.
**Author contributions** All authors designed and performed the research and wrote the manuscript.
**Competing financial interests**
The authors declare no competing financial interests.
[^1]: These authors contributed equally to this work
[^2]: These authors contributed equally to this work
[^3]: Email: [email protected]
[^4]: Email: [email protected]
[^5]: Email: [email protected]
| ArXiv |
---
abstract: 'The integers $n=\prod_{i=1}^r p_i^{a_i}$ and $m=\prod_{i=1}^r p_i^{b_i}$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. We estimate the number of pairs of exponentially coprime integers $n,m\le x$ having the prime factors $p_1,...,p_r$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$.'
author:
- '[László Tóth]{} (Pécs, Hungary)'
date: 'Pure Math. Appl. (PU.M.A.), 15 (2004), 343-348'
title: '**On exponentially coprime integers**'
---
=6.truein =9.truein =-.5truein =-.8truein
\[section\] \[section\] \[section\]
Mathematics Subject Classification (2000): 11A05, 11A25, 11N37
[**1. Introduction**]{}
Let $n>1$ be an integer of canonical form $n=\prod_{i=1}^r p_i^{a_i}$. The integer $d$ is called an [*exponential divisor*]{} of $n$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i | a_i$ for every $1\le i \le r$, notation: $d|_e n$. By convention $1|_e 1$. This notion was introduced by [M. V. Subbarao]{} [@Su72]. The smallest exponential divisor of $n>1$ is its squarefree kernel $\kappa(n):=\prod_{i=1}^r p_i$.
Let $\tau^{(e)}(n)= \sum_{d|_e n} 1$ and $\sigma^{(e)}(n)=\sum_{d|_e n} d$ denote the number and the sum of exponential divisors of $n$, respectively. Properties of these functions were investigated by several authors, see [@FaSu89], [@KaSu2003], [@PeWu97], [@SmWu97], [@Su72], [@Wu95].
Two integers $n,m >1$ have common exponential divisors iff they have the same prime factors and for $n=\prod_{i=1}^r p_i^{a_i}$, $m=\prod_{i=1}^r p_i^{b_i}$, $a_i,b_i\ge 1$ ($1\le i\le r$), the [*greatest common exponential divisor*]{} of $n$ and $m$ is $$(n,m)_e:=\prod_{i=1}^r p_i^{(a_i,b_i)}.$$
Here $(1,1)_e=1$ by convention and $(1,m)_e$ does not exist for $m>1$.
The integers $n,m >1$ are called [*exponentially coprime*]{}, if they have the same prime factors and $(a_i,b_i)=1$ for every $1\le i\le r$, with the notation of above. In this case $(n,m)_e=\prod_{i=1}^r p_i$. $1$ and $1$ are considered to be exponentially coprime. $1$ and $m>1$ are not exponentially coprime. Exponentially coprime integers were introduced by [J. Sándor]{} [@Sa96].
Let $p_i$ ($1\le i\le r$) be fixed distinct primes and let $P^{(e)}(p_1,...,p_r;x)$ denote the number of pairs $\langle n,m \rangle$ of exponentially coprime integers such that $\kappa(n)=\kappa(m)=\prod_{i=1}^r p_i$ and $n,m\le x$.
In this note we estimate $P^{(e)}(p_1,...,p_r;x)$ and show that the asymptotic density of pairs of exponentially coprime integers having $r$ fixed prime divisors is $(\zeta(2))^{-r}$.
As an open problem we formulate the following: What can be said on the asymptotic density of pairs of exponentially coprime integers if their prime divisors are not fixed ?
For a real $x\ge 1$ and an integer $n\ge 1$ consider the Legendre-type function $L^{(e)}(x,n)$ defined as the number of integers $k\le x$ such that $k$ and $n$ are exponentially coprime.
The following estimate holds:
Let $N(p_1,...,p_r;x)$ denote the number of integers $n\le x$ having the kernel $\kappa(n)=p_1\cdots p_r$. Taking $a_1=\cdots =a_r=1$ we obtain from Theorem 1 the following known estimate, cf. for ex. [@Te95], Ch. III.5 regarding integers free of large prime factors.
The proofs of Theorems 1 and 2 are by induction on $r$, while Corollary 3 follows from Theorem 2 and Corollary 1. First we prove the following lemma.
We will use the well-known estimate: if $s\ge 0$, then $$\phi_s(z,a):=\sum_{n\le z \atop{(n,a)=1}} n^s =\frac{z^{s+1}\phi(a)}{(s+1)a}+O(z^s\theta(a)), \leqno(5)$$ uniformly for $z\ge 1$ and $a\ge 1$.
Induction on $r$. For $r=1$ (4) follows from (5) applied for $s=0$. Suppose formula (4) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{(k_1,a_1)=\cdots =(k_r,a_r)=1 \atop{k_1,...,k_r\ge 1}}} 1 =
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}}
\sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{(k_1,a_1)=\cdots =(k_{r-1},a_{r-1})=1 \atop{k_1,...,k_{r-1}\ge 1}}} 1$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}}
\left(\frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right)
(z-k_rt_r)^{r-1} + O\left( z^{r-2} \sum_{i=1}^{r-1} \theta(a_i) \right) \right)$$ $$= \frac1{(r-1)!} \left( \prod_{i=1}^{r-1} \frac{\phi(a_i)}{a_i t_i}\right)
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} (z-k_rt_r)^{r-1}+
O\left( z^{r-1} \sum_{i=1}^{r-1} \theta(a_i) \right).$$ Using the binomial formula and estimate (5) the sum appearing here is $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{(k_r,a_r)=1 \atop{k_r\ge 1}}} k_r^j$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots -t_{r-1})^{j+1}\phi(a_r)}{(j+1)t_r^{j+1} a_r} +O(z^j \theta(a_r))\right)$$ $$=\frac{\phi(a_r)}{t_ra_r} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O( z^{r-1}\theta(a_r))$$ $$=\frac{\phi(a_r)}{rt_ra_r} z^r + O(z^{r-1}\theta(a_r))$$ and the proof is complete.
Apply Lemma 1 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$.
In order to prove Theorem 2 we need
Induction on $r$, similar to the proof of Lemma 1. We use the well-known estimate: let $s\ge -1$ be a real number, then for $z\ge 3$, $$\sum_{n\le z} \phi(n)n^s =\frac{z^{s+2}}{(s+2)\zeta(2)} + O(z^{s+1}\log z). \leqno(7)$$
For $r=1$ (6) follows from (7) applied for $s=-1$. Suppose formula (6) is valid for $r-1$ and prove it for $r$. $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}} \prod_{i=1}^r \frac{\phi(k_i)}{k_i} = \sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
\sum_{k_1t_1+\cdots +k_{r-1}t_{r-1}\le z-k_rt_r \atop{k_1,...,k_{r-1}\ge 1}} \prod_{i=1}^{r-1} \frac{\phi(k_i)}{k_i}$$ $$=\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
\left(\frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i} \right)
(z-k_rt_r)^{r-1} + O\left( z^{r-2} \log z\right) \right)$$ $$= \frac1{(r-1)! (\zeta(2))^{r-1}} \left( \prod_{i=1}^{r-1} \frac1{t_i}\right)
\sum_{k_rt_r\le z-t_1-\cdots -t_{r-1} \atop{k_r\ge 1}} \frac{\phi(k_r)}{k_r}
(z-k_rt_r)^{r-1}+ O\left( z^{r-1} \log z \right).$$ The sum appearing here is, applying (7), $$\sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \sum_{k_rt_r\le z-t_1-\cdots
-t_{r-1} \atop{k_r\ge 1}} \phi(k_r)k_r^{j-1}$$ $$= \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} z^{r-1-j} t_r^j \left( \frac{(z-t_1-\cdots
-t_{r-1})^{j+1}}{(j+1)t_r^{j+1} \zeta(2)} +O(z^j \log z)\right)$$ $$=\frac1{t_r\zeta(2)} z^r \sum_{j=0}^{r-1} (-1)^j {r-1 \choose j} \frac1{j+1} + O(z^{r-1}\log z)$$ $$=\frac1{r t_r \zeta(2)} z^r + O(z^{r-1}\log z),$$ which completes the proof.
Using estimate (4), $$\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{j_1t_1+\cdots +j_rt_r\le z \atop{(k_1,j_1)=\cdots =(k_r,j_r)=1 \atop{k_1,j_1,...,k_r,j_r\ge 1}}}} 1
=
\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\sum_{j_1t_1+\cdots +j_rt_r\le z \atop{(j_1,k_1)=\cdots =(j_r,k_r)=1 \atop{j_1,...,j_r\ge 1}}} 1$$ $$=\sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\left(\frac1{r!} \left( \prod_{i=1}^r \frac{\phi(k_i)}{k_i t_i} \right) z^r +
O\left(z^{r-1} \sum_{i=1}^r \theta(k_i) \right)\right)$$ $$=\frac{z^r}{r!\prod_{i=1}^r t_i} \sum_{k_1t_1+\cdots +k_rt_r\le z \atop{k_1,...,k_r\ge 1}}
\prod_{i=1}^r \frac{\phi(k_i)}{k_i} + O\left( z^{r-1} \sum_{i=1}^r \sum_{k_1t_1+\cdots +k_rt_r\le z} \theta(k_i) \right)$$ here the $O$-term is $O(z^{r-1}z^{r-1}z\log z)=O(z^{2r-1}\log z)$ and applying Lemma 2 to the main term finishes the proof.
Apply Lemma 3 for $z=\log x$, $t_1=\log p_1,...,t_r=\log p_r$.
This is a direct consequence of Theorem 2 and Corollary 1. The considered asymptotic density is $$\lim_{x\to \infty} P^{(e)}(p_1,...,p_r;x) (N(p_1,...,p_r;x))^{-2}=
%\qquad ( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x \atop{
%p_1^{b_1}\cdots p_r^{b_r}\le x \atop{
%(a_1,b_1)=...=(a_r,b_r)=1}}} 1 ) \cdot
%( \sum_{p_1^{a_1}\cdots p_r^{a_r}\le x} 1 )^{-2} =
(\zeta(2))^{-r}.$$
[99]{}
and [M. V. Subbarao]{}, The maximal order and the average order of multiplicative function $\sigma^{(e)}(n)$, [*Théorie des nombres. Proc. of the Int. Conf. Québec, 1987*]{}, de Gruyter, Berlin – New York, 1989, 201-206.
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[[**László Tóth**]{}\
University of Pécs\
Institute of Mathematics and Informatics\
Ifjúság u. 6\
7624 Pécs, Hungary\
[email protected]]{}
| ArXiv |
---
abstract: 'This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang’s Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.'
author:
- 'Alan Dow[$^1$]{} and Franklin D. Tall[$^2$]{}'
bibliography:
- 'normality.bib'
nocite: '[@*]'
title: Normality versus paracompactness in locally compact spaces
---
[^1]
[^2]
Introduction
============
The space of countable ordinals is locally compact, normal, but not paracompact. The question of what additional conditions make a locally compact normal space paracompact has a long history. At least 45 years ago, it was recognized that subparacompactness plus collectionwise Hausdorffness would do (see e.g. [@T1]), as would perfect normality plus metacompactness [@A]. Z. Balogh proved a variety of results under MA$_{\omega_1}$ [@B1] and **Axiom R** [@B2], and was the first to realize the importance of not including a perfect pre-image of $\omega_1$ (equivalently, the one-point compactification being countably tight [@B1]). However, he assumed collectionwise Hausdorffness in order to obtain paracompactness. A breakthrough came with S. Watson’s proof that:
$V = L$ implies locally compact normal spaces are collectionwise Hausdorff, and hence locally compact normal metacompact spaces are paracompact.
Watson’s proof crucially involved the idea of *character reduction*: if one wants to separate a closed discrete subspace of size $\kappa$, $\kappa$ regular, in a locally compact normal space, it suffices to separate $\kappa$ compact sets, each with an *outer base* of size $\leq \kappa$.
An [**outer base**]{} for a set $K \subseteq X$ is a collection ${\mathcal}{B}$ of open sets including $K$ such that each open set including $K$ includes a member of ${\mathcal}{B}$.
The use of $V = L$ was to get that normal spaces of character $\leq \aleph_1$ are collectionwise Hausdorff [@F], and variations on that theme.
It was known that locally compact normal non-collectionwise Hausdorff spaces could be constructed from MA$_{\omega_1}$, indeed from the existence of a $Q$-set [@T1], so it was a big surprise when G. Gruenhage and P. Koszmider proved that:
MA$_{\omega_1}$ implies locally compact, normal, metacompact spaces are $\aleph_1$-collectionwise Hausdorff and (hence) paracompact.
The next result involving iteration axioms and a positive “normal implies collectionwise Hausdorff" type of result was:
Let $S$ be a coherent Souslin tree (obtainable from $\diamondsuit$ or a Cohen real). Force MA$_{\omega_1}(S)$, i.e. MA$_{\omega_1}$ for countable chain condition posets preserving $S$. Then force with $S$. In the resulting model, there are no first countable $L$-spaces, no compact first countable $S$-spaces, and separable normal first countable spaces are collectionwise Hausdorff.
The first two statements are consequences of MA$_{\omega_1}$ [@Sz]; the last of $V=L$, indeed of $2^{\aleph_0} < 2^{\aleph_1}$. Larson and Todorcevic used this combination to solve *Katětov’s problem*. This idea of combining consequences of a iteration axiom with “normal implies collectionwise Hausdorff" consequences of $V = L$ was exploited in [@LT1] in order to prove the consistency, modulo a supercompact cardinal, of *every locally compact perfectly normal space is paracompact*. The large cardinal was later removed, so that:
If ZFC is consistent, then so is ZFC plus every locally compact perfectly normal space is paracompact.
In the models of [@LT1] and [@DT2], every first countable normal space is collectionwise Hausdorff. This is achieved in two stages. The novel one is:
\[lem15\] Force with a Souslin tree. Then\[LT1\] normal first countable spaces are $\aleph_1$-collectionwise Hausdorff.
This is obtained by showing that if a normal first countable space is not $\aleph_1$-collectionwise Hausdorff, a generic branch of the Souslin tree induces a generic partition of the unseparated closed discrete subspace which cannot be “normalized", i.e. there do not exist disjoint open sets about the two halves of the partition. The argument is a blend of the two usual methods of proving “normal implies $\aleph_1$-collectionwise Hausdorff" results, namely those of adjoining Cohen subsets of $\omega_1$ by countably closed forcing [@T1], [@T2] and using *$\diamondsuit$ for stationary systems on $\omega_1$*, a strengthening of $\diamondsuit$ that holds in $L$ [@F]. It is noteworthy that:
Either force to add $\aleph_2$ Cohen subsets of $\omega_1$, or assume $\diamondsuit$ for stationary subsets of $\omega_1$. Then normal spaces of character $\leq \aleph_1$ are $\aleph_1$-collectionwise Hausdorff.
Once one has normal first countable spaces are $\aleph_1$-collectionwise Hausdorff, it is easy to obtain full collectionwise Hausdorffness by starting with $L$ as the ground model and following [@F]. However, if a supercompact cardinal is involved, instead of $L$ we need to follow the method of [@LT1], based on [@T2]. Namely, first make the supercompact indestructible under countably closed forcing [@L] and then perform an Easton extension, adding $\kappa^+$ Cohen subsets of each regular $\kappa$, before forcing with the Souslin tree.
In order to extend the theorems about locally compact normal spaces being paracompact beyond the realm of first countability, one first needs to get that *locally compact normal spaces are collectionwise Hausdorff*. In [@T3], the second author claimed to have done so, in the model of [@LT1]. The key was to force to expand a closed discrete subspace in a locally compact normal space to a discrete collection of compact sets with countable outer bases and then apply the methods of [@LT1]. Unfortunately the expansion argument was flawed. A corrected argument is presented below, but at the cost of using a stronger iteration axiom (but not a larger large cardinal).
With the conclusion of [@T3] restored, [@T4], [@LT2], and [@T] are re-instated. We shall then proceed to improve the results of the two latter ones.
PFA$(S)[S]$ and the role of $\omega_1$
======================================
*PFA$(S)$* is the Proper Forcing Axiom (PFA) restricted to those posets that preserve the (Souslinity of the) coherent Souslin tree $S$.
*PFA$(S)[S]$ implies $\varphi$* is shorthand for *whenever one forces with a coherent Souslin tree $S$ over a model of PFA$(S)$, $\varphi$ holds.* *$\varphi$ holds in a model of form PFA$(S)[S]$* is shorthand for *there is a coherent Souslin tree $S$ and a model of PFA$(S)$ such that when one forces with $S$ over that model, $\varphi$ holds.*
For discussion of PFA$(S)[S]$, see [@D2], [@To], [@LT1], [@LT2], [@T4], [@T], [@FTT], [@T6].
The following results appear in [@LT2] and [@T], respectively.
\[thm:paracompactcopy\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a perfect pre-image of ${\omega}_1$.
\[thm:paracompactcountablytight\] There is a model of form ${\mathrm}{PFA}(S)[S]$ in which a locally compact normal space is paracompact and countably tight if and only if its separable closed subspaces are Lindelöf and it does not include a perfect pre-image of ${\omega}_1$.
**** is the assertion that every first countable perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.
${\mathrm}{PFA}(S)[S]$ implies ****.
$\mathbf{PPI}$ was originally proved from PFA in [@BDFN]. Using $\mathbf{PPI}$, we are able to weaken “perfect pre-image" to “copy" in the improved version of the first theorem, but provably cannot in the second theorem.
\[thm:paracompactcopyallmodels\] There is a model of form PFA$(S)[S]$ in which a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a copy of $\omega_1$.
There is a locally compact space $X$ (indeed a perfect pre-image of $\omega_1$) which is normal, does not include a copy of $\omega_1$, in which all separable closed subspaces are compact, but $X$ is not paracompact.
It is clear that to establish Theorem \[thm:paracompactcopyallmodels\], it suffices to use \[thm:paracompactcopy\] and apply $\mathbf{PPI}$ after proving:
\[thm34\] ${\mathrm}{PFA}(S)[S]$ implies a hereditarily normal perfect pre-image of ${\omega}_1$ includes a first countable perfect pre-image of ${\omega}_1$.
This follows from:
\[lxb\] Let $X$ be a perfect pre-image of $\omega_1$, and suppose separable subspaces of $X$ are Lindelöf. Then $X$ includes a first countable perfect pre-image of $\omega_1$.
and
\[lxc\] ${\mathrm}{PFA}(S)[S]$ implies compact, separable, hereditarily normal spaces are hereditarily Lindelöf.
Here is the proof of Lemma \[lxb\].
Let $f : X\rightarrow\omega_1$, perfect and onto. Then $X$ is locally compact, countably compact, but not compact. There is a closed $Y \subseteq X$ such that $f' = f|Y$ is perfect, irreducible, and maps $Y$ onto $\omega_1$. So $Y = \bigcup_{\alpha < \omega_1}f'^{-1}(\{\beta : \beta \leq
\alpha\})$. Each $D_\alpha = f'^{-1}(\{\beta : \beta \leq \alpha\})$ is clopen and hence countably compact. It suffices to show $D_\alpha$ is hereditarily Lindelöf, for then points are $G_\delta$ and $D_\alpha$ is first countable. But then $Y$ is first countable, since $D_\alpha$ is open. To show $D_\alpha$ is hereditarily Lindelöf, we need only show it is separable. $f_\alpha = f'|D_\alpha$ is irreducible, for if there were a proper closed subset $A$ of $D_\alpha$ such that $f'(A) = f'(D_\alpha)$, then $f$ would map $A \cup (Y - D_\alpha)$ onto $\omega_1$, contradicting $f$’s irreducibility. But
If $f$ is a closed irreducible map of $X$ onto $Y$ and $E$ is dense in $Y$, then $f^{-1}(E)$ is dense in $X$.
Thus $D_\alpha$ is separable.
Let us construct the example that constrains the hoped-for improvement of Theorem \[thm:paracompactcountablytight\]. Consider a stationary, co-stationary subset $E$ of $\omega_1$ and its Stone-Čech extension $\beta E$. The identity map $\iota$ embeds $E$ into the compact space $\omega_1 + 1$. $\iota$ extends to $\hat{\iota}$ mapping $\beta E$ onto $
\omega_1+1$; we claim that $\hat{\iota}$ maps only one element – call it $z$ – of $\beta E$ to the point $\omega_1$. The reason is that every real-valued continuous function on $E$ is eventually constant. If there were another such point, say $z'$, let $f$ be a continuous real-valued function sending $z$ to $0$ and $z'$ to $1$. Let $U, V$ be open sets about the point $\omega_1$ such that $\hat{\iota}^{-1}(U) \subseteq f^{-1}\left(\left[0, \frac{1}{2}\right]\right)$ and $\hat
{\iota}^{-1}(V) \subseteq f^{-1}\left(\left(\frac{1}{2}, 1\right]\right)$. Then $\hat{\iota}^{-1}(U) \cap \hat{\iota}^{-
1}(V) = \emptyset$, but $U \cap V \cap E$ is cocountable in $E$, contradiction.
Our space $X$ will be $\beta E - {\{z\}}$. $\hat{\iota}|X$ maps $X$ onto $\omega_1$; we claim this map is perfect. By 3.7.16(iii) of Engelking [@E], it suffices to show that $\hat{\iota}[\beta X - X] = \beta \omega_1 - \omega_1$. But $\beta \omega_1 = \omega_1 + 1$ and $\beta X = \beta E$, so this just says $\hat{\iota}(z) = \omega_1$, which we have.
If $H, K$ are disjoint closed subsets of $X$, then their closures in $\beta E$ have at most $z$ in common. Thus their images $\hat{\iota}[H]$ and $\hat{\iota}[K]$ cannot overlap in a subspace with a point of $E$ in its closure. Since $E$ is stationary, their overlap is countable. Then at least one of them is bounded, and hence compact. it is then easy to pull back disjoint open sets to establish normality.
For any perfect pre-image of $\omega_1$, it is easy to see that separable closed subspaces are compact, since they are included in a pre-image of an initial closed segment of $\omega_1$.
It remains to show that $X$ does not include a copy $W$ of $\omega_1$. A standard $\beta \mathbb{N}$ argument shows that no point in $X - E$ is the limit of a convergent sequence, so the set $C$ of all limits of convergent sequences from $W$ is a subset of $E$. But $C$ is homeomorphic to $\omega_1$, so cannot be included in a co-stationary $E$.
There is, however, a satisfactory improvement of Theorem \[thm:paracompactcountablytight\]:
\[thm38\] There is a model of form PFA$(S)[S]$ in which a locally compact, normal, countably tight space is paracompact if and only if its separable closed subspaces are Lindelöf, and it does not include a copy of $\omega_1$.
This follows from:
\[thmConj2\] PFA$(S)[S]$ implies a countably tight, perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.
The proof of Theorem \[thm38\] is essentially the same as the proof in [@T] of our Theorem 2.2.
Countably tight, hereditarily normal perfect pre-images of $\omega_1$ are rather special:
Suppose $\pi : X \to \omega_1$. We say $Y \subseteq X$ is **unbounded** if $\pi(Y)$ is unbounded.
\[thm39\] PFA$(S)[S]$ implies that a countably tight, hereditarily normal, perfect pre-image of $\omega_1$ is the union of a paracompact space with a finite number of disjoint unbounded copies of $\omega_1$.
By \[thmConj2\], the perfect pre-image $X$ includes a copy, $W_1$, of $\omega_1$. If $W_1$ were bounded, then for some $\alpha$, $W_1 \subseteq \pi^{-1}([0, \alpha])$. But $\pi^{-1}([0, \alpha])$ is compact, and $W_1$ – being a countably compact subspace of a countably tight space – is closed in $X$ and hence in $\pi^{-1}([0, \alpha])$. But then $W_1$ is compact, contradiction. Since perfect pre-images of locally compact spaces are locally compact, $X$ is locally compact. Since $W_1$ is closed, $X - W_1$ is open and so is also locally compact. If it is paracompact, we are done; if not, apply \[thmConj2\] to get a copy $W_2$ of $\omega_1$ included in $X - W_1$. Continue. The process must end at some finite stage, since:
Let $X$ be a $T_5$ space, $\pi : X \to \omega_1$ continuous, $\pi^{-1}(\{\alpha\})$ countably compact for all $\alpha \in
S$, a stationary subset of $\omega_1$. Then $X$ cannot include an infinite disjoint family of closed, countably compact subspaces each with unbounded range.
Note that the paracompact subspace is the topological sum of $\leq \aleph_1$ $\sigma$-compact subspaces.
An early version of [@DT1] used the axioms ${\mathbf{\mathop{\pmb{\sum}}}}^-$ (defined in Section 5), ${\mathbf{PPI}}$, and the $\aleph_1$-collectionwise Hausdorffness of first countable normal spaces, as well as \[thm39\] to obtain “countably compact, hereditarily normal manifolds of dimension $> 1$ are metrizable" without the $\mathbf{P}_{22}$ axiom used in [@DT1] to get the stronger assertion in which “countably compact" is omitted.
Both of the conditions for paracompactness in \[thm38\] are necessary:
$\omega_1$ is locally compact, normal, first countable, its separable subspaces are countable, but it is not paracompact.
Van Douwen’s “honest example” [@vD] is locally compact, normal, first countable, separable, does not include a perfect pre-image of $\omega_1$ (because it has a $G_\delta$-diagonal), but is not paracompact.
Strengthenings of [PFA]{}$(S)[S]$
=================================
In addition to “front-loading” a PFA$(S)[S]$ model in order to get full collectionwise Hausdorffness, it has also been useful to employ strengthenings of PFA$(S)$ so as to obtain more reflection. E.g. in [@LT2] and [@T], **** is employed.
$C\subseteq[X]^{<\kappa}$ is **tight** if whenever $\{C_\alpha:\alpha<\delta\}$ is an increasing sequence from $C$ and $\omega<\text{\normalfont cf}(\delta)<\kappa$, $\bigcup\{C_\alpha:\alpha<\beta\}\in C$.
**Axiom R**
If $\mathcal{S}\subseteq[X]^{<\omega_1}$ is stationary and $C\subseteq[X]^{<\omega_2}$ is tight and unbounded, then there is a $Y\in C$ such that $\mathcal{P}(Y)\cap\mathcal{S}$ is stationary in $[Y] ^{<\omega_1}$.
**** (due to Fleissner [@Fle]) was obtained by using what is called *PFA$^{++}(S)$* in [@LT2], before forcing with $S$ [@LT2]. PFA$^{++}(S)$ holds if PFA$(S)$ is forced in the usual Laver-diamond way. Here we shall use a conceptually simple principle, MM$(S)$, which is forced in a more complicated way, but does not require a larger cardinal. The axiom *Martin’s Maximum* was introduced in [@FMS].
Let $\mathcal{P}$ be a partial order such that forcing with $\mathcal{P}$ preserves stationary subsets of $\omega_1$. Let $\mathcal{D}$ be a collection of $\aleph_1$ dense subsets of $\mathcal{P}$. **** asserts that for each such $\mathcal{D}$, there is a $\mathcal{D}$-generic filter included in $\mathcal{P}$.
Assume there is a supercompact cardinal. Then there is a revised countable support iteration establishing [MM]{}.
MM$(S)$ is defined analogously to PFA$(S)$; Miyamoto [@M] proved that there is a “nice” iteration establishing MM$(S)$ but preserving $S$. One can then define MM$(S)[S]$ analogously to PFA$(S)[S]$.
In order to obtain a model of PFA$(S)[S]$ in which Theorem \[thm:paracompactcopyallmodels\] holds, we need to improve the model of [@LT2] so as to not only have **Axiom R** but also:
**LCN($\aleph_1$)**
Every locally compact normal space is $\aleph_1$-collectionwise Hausdorff.
We shall prove that MM$(S)$ implies:
**NSSAT**
NS$_{\omega_1}$ (the non-stationary ideal on $\omega_1$) is $\aleph_2$-saturated.
**SCC**
Strong Chang Conjecture. Let $\lambda>2^{\aleph_2}$ be a regular cardinal. Let $H(\lambda)$ be the collection of hereditarily $<\lambda$ sets. Let $M^*$ be an expansion of $\langle H_\lambda,\in\rangle$. Let $N\prec M^*$ (i.e. $N$ is an elementary submodel of $M^*$) be countable. Then there is an $N'$ such that $N\prec N'\prec M^*$, $N'\cap
\omega_1=N\cap\omega_1$, and $|N\cap\omega_2|=\aleph_1$.
We also note:
$(S)$ implies $2^{\aleph_1}=\aleph_2$.
With these, we can modify the proof in [@LT1] that forcing with a Souslin tree makes *first countable normal spaces $\aleph_1$-collectionwise Hausdorff* to obtain *locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff*, and then, if we wish, front-load the model as in [@LT1] to obtain full collectionwise Hausdorffness, using the character reduction method of [@W]. More precisely, the crucial new step is:
\[thm41\] Suppose there is a model in which there is a Souslin tree $S$ and in which ****, ****, and $2^{\aleph_1}=\aleph_2$ hold. Then $S$ forces that locally compact normal spaces are $\aleph_1$-collectionwise Hausdorff.
It will be convenient to consider the following intermediate proposition, which implies the three things that we want:
**SRP**
Strong Reflection Principle [@To2]. Suppose $\lambda\geq\aleph_2$ and $\mathfrak{Z}\subseteq\mathcal{P}_{\omega_1}(\lambda)$ and that for each stationary $T\subseteq\omega_1$, $$\{\sigma\in\mathfrak{Z}:\sigma\cap\omega_1\in T\}$$ is stationary in $\mathcal{P}_{\omega_1}(\lambda)$. Then for all $X\subseteq\lambda$ of cardinality $\aleph_1$, there exists $Y\subseteq\lambda$ such that:
- $X\subseteq Y$ and $|Y|=\aleph_1$;
- $\mathfrak{Z}\cap\mathcal{P}_{\omega_1}(Y)$ contains a set which is closed unbounded in $\mathcal{P}_{\omega_1}(Y)$.
With regard to **SCC**, Shelah [@S XII.2.2, XII.2.5] proves that:
If there is a semi-proper forcing P changing the cofinality of $\aleph_2$ to $\aleph_0$, then **** holds.
There are various versions of *Namba forcing*, e.g. two in [@S] and one in [@Lar]. All of these change the cofinality of $\aleph_2$ to $\aleph_0$. Larson states in [@Lar p.142] that his version of Namba forcing preserves stationary subsets of $\omega_1$. In [@FMS], it is shown that a principle, **SR**, implies *any forcing that preserves stationary subsets of $\omega_1$ is semi-proper*. **SR** is a consequence of MM [@FMS]. is stronger than **SR** and so:
implies .
$(S)$ implies .
**** implies **** and $2^{\aleph_1}\leq\aleph_2$.
For the proof of \[thm:paracompactcopyallmodels\] we should also remark that:
**** implies ****.
We use an equivalent formulation of **SRP** due to Feng and Jech [@FJ].
**SRP**
For every cardinal $\kappa$ and every $S\subseteq[\kappa]^\omega$, for every regular $\theta>\kappa$, there is a continuous elementary chain $\{N_\alpha:\alpha\in\omega_1\}$ (with $N_0$ containing some given element of $H(\theta)$, e.g. $S$) such that for all $\alpha$, $N_\alpha\cap\kappa\in S$ if and only if there is a countable $M\prec H(\theta)$ such that $N_\alpha\subseteq M$, $M\cap\omega_1=N_\alpha\cap\omega_1$, and $M\cap\kappa\in S$.
Let $\mathcal{S}$ and $\mathcal{C}$ be as in **Axiom R**. Choose $\theta$ sufficiently large so that $\mathcal{S},\mathcal{C}\in H(\theta)$ and so that $\theta^{\aleph_1}=\theta$. Let $\{\mathcal{S},\mathcal{C}\}\in N_0$ and let $\{N_\alpha:\alpha\in\omega_1\}$ be as in **SRP**. By induction on $\alpha\in\omega_1$, choose $Y_\alpha\in C\cap N_{\alpha+1}$ so that $\bigcup (\mathcal{C}\cap N_\alpha)\subseteq Y_\alpha$. Then $\{Y_\alpha:\alpha\in\omega_1\}$ is an increasing chain in $\mathcal{C}$. Therefore $Y=\bigcup_{\alpha\in\omega_1}(N_\alpha\cap\kappa)$ is in $\mathcal{C}$.
$\mathcal{S}^+=\{M\prec H(\theta):M\cap \kappa\in\mathcal{S}\}$ is a stationary subset of $[H(\theta)]^\omega$. This is proved in the same way as 1) of Claim 1.12 on page 196 of [@S]. Since $\{N_\alpha:\alpha\in\omega_1\}$ is an element of $H(\theta)$, there is an $M\in\mathcal{S}^+$ such that $\{N_\alpha:\alpha\in\omega_1\}\in M$. Let ${M\cap\omega_1=\delta}$. Obviously $M\cap\kappa\in\mathcal{S}$, and, by continuity, $N_\delta\subseteq M$ and $M\cap\omega_1= N_\delta\cap\omega_1$. It then follows from **SRP** that $N_\delta\in\mathcal{S}$.
This actually proves that $\{\alpha\in\omega_1:N_\alpha\cap\kappa\in\mathcal{S}\}$ is a stationary subset of $\omega_1$, because we could have put any cub of $\omega_1$ as an element of $M$. Now assume that $\mathfrak{Z}\subseteq[Y]^\omega$ is a cub of $[Y]^\omega$. Choose a strictly increasing $g:\omega_1\to\omega_1$ such that for each $\alpha$, there is a $Z_\alpha\in \mathfrak{Z}$ such that $N_\alpha\cap \kappa\subseteq Z_\alpha\subseteq N_{g(\alpha)}$. If limit $\delta$ satisfies that $g(\alpha)<\delta$ for all $\alpha<\delta$, then we have that $N_\delta\cap\kappa\in\mathfrak{Z}$. This finishes the proof that $\mathcal{S}\cap[Y]^\omega$ is stationary.
Suppose we have a model with a Souslin tree $S$ in which **Axiom R** holds. Then, after forcing with $S$, **** still holds.
This is an improvement over [@LT2], which required a stronger axiom, Axiom R$^{++}$, holding in the model. We will use *t.u.b.* as an abbreviation for *tight unbounded*. We must consider two $S$-names: $\dot{\mathcal{C}}$ and $\dot{\mathcal{X}}$ where $\dot{ \mathcal{C}}$ is forced to be a t.u.b. subset of $[\kappa]^{\omega_1}$ and $\dot{ \mathcal{X}}$ is forced to be a stationary subset of $[\kappa]^\omega$. Let us assume that some $s_0\in S$ forces there is no $Y$ in $\dot{ \mathcal{C}}$ such that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. (It would make the discussion below easier if we just assumed that $s_0$ was the root of $S$ – which one can certainly immediately do if $S$ is a coherent Souslin tree.)
We first show that $\dot C$ contains a t.u.b. $C$ from the ground model. Simply put $Y\in \mathcal{C}$ if every $s\in S$ forces that $Y\in \dot{ \mathcal{C}}$. It is clear that $\mathcal{C}$ is closed under increasing $\omega_1$-chains. Thus we just have to show that it is unbounded. Let us enumerate $S$ as $\{ s_\alpha : \alpha\in \omega_1\}$. Fix any $Y_0\in
[\kappa]^{\omega_1}$. By recursion choose an increasing chain $\{Y_\alpha : \alpha\in \omega_1\}$ so that for each $\alpha$, $\bigcup \{Y_\beta : \beta < \alpha\}\subseteq Y_\alpha$ and there is an extension $s_\beta$ of $s_\alpha $ forcing that $Y_{\alpha+1}\in \dot{ \mathcal{C}}$. This we may do, since $s_\alpha$ forces that $\dot{ \mathcal{C}}$ is unbounded. Now let $Y$ be the union of the chain $\{ Y_\alpha : \alpha\in \omega_1\}$. Note that for each $s\in S$ and each $\beta\in \omega_1$, there is an $\beta<\alpha$ such that $s_\alpha$ is an extension of $s$. It follows that $s$ forces that $\dot{ \mathcal{C}}\,\cap \{ Y_\alpha : \alpha \in
\omega_1\}$ is uncountable, hence $s\Vdash Y\in \dot{ \mathcal{C}}$.
Now we let $\mathcal{X}$ be the set of $x\in [\kappa]^\omega$ such that there is some $s\in S$ extending $s_0$ with $s\Vdash x\in \dot{ \mathcal{X}}$. It is clear that $\mathcal{X}$ is a stationary subset of $[\kappa]^\omega$ because $s_0$ forces that $\mathcal{X}$ meets every cub. Now apply **Axiom R** to choose $Y\in \mathcal{C}$ so that $\mathcal{X}\cap [Y]^\omega$ is a stationary subset of $Y$.
Now we obtain a contradiction (and thus a proof) by showing that there is an extension $s\in S$ of $s_0$ that forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary. Let $\{ y_\alpha :\alpha\in \omega_1\}$ be an enumeration of $Y$. Let $\mathcal{E}$ be the set of $\delta\in \omega_1$ such that $x_\delta = \{y_\alpha :\alpha\in \delta\}\in \mathcal{X}$. Notice that $\{ \{ y_\alpha : \alpha \in \delta \} : \delta\in \omega_1\}$ is a cub in $[Y]^\omega$. Thus it follows that $\mathcal{E}$ is stationary. In fact, if $\mathcal{E}'$ is any stationary subset of $\mathcal{E}$, then $\mathcal{E}'$ is also a stationary subset of $[Y]^\omega$.
For each $\delta\in \mathcal{E}$ choose $s_\delta\in S$ above $s_0$ so that $s_\delta\Vdash x_\delta\in \dot{ \mathcal{X}}$ (as per the definition of $ \mathcal{X}$). Now we have a name $\dot{ \mathcal{E}} = \{ (x_\delta, s_\delta) :
\delta\in \omega_1\}$. We prove that there is some $s\in S$ above $s_0$ that forces that $\dot{ \mathcal{E}}$ is stationary. Thus such an $s$ forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary as required.
Let $s_0$ be on level $\alpha_0$ of $S$. There is a $\gamma>\alpha_0$ so that each member of $S_\gamma$ decides if $\dot{ \mathcal{E}}$ is stationary. Also, for each $\bar s\in S_\gamma$ that forces $\dot{ \mathcal{E}}$ is not stationary, there is a cub $\mathcal{C}_{\bar
s}$ of $\omega_1$ that $\bar s$ forces is disjoint from $\dot{ \mathcal{E}}$. Choose any $\delta $ in the intersection of those countably many cubs that is also in $\mathcal{E}$. Clearly if $\bar s\in S_\gamma$ is compatible with $s_\delta$, then $\mathcal{C}_{\bar s}$ did not exist since $\bar s \cap s_\delta$ would force that $\delta \in
\mathcal{C}_{\bar s}\cap \dot{ \mathcal{E}}$. This completes the proof, since that element $\bar s$ is above $s_0$ and forces that $\dot{ \mathcal{X}}\cap [Y]^\omega$ is stationary.
$(S)[S]$ implies ****.
We next need:
\[P. Larson\]\[larson\] Suppose
- ****, and
- for sufficiently large $\theta$ and stationary $E\subseteq\omega_1$, for any $X\in H(\theta)$, there is a Chang model $M$ with $M\cap\omega_1\in E, X\in M$ and $|M\cap\omega_2|=\aleph_1$.
Then if $\{A_\alpha:\alpha<\omega_2\}$ are stationary subsets of $\omega_1$, $M\cap\omega_1=\delta$ is in uncountably many $A_\alpha,\alpha\in M$.
It is well known that ${\mathrm}{NS}_{\omega_1}$ is $\aleph_1$-complete, since the diagonal union of $\aleph_1$ non-stationary subsets of $\omega_1$ is non-stationary. It follows that $\mathcal{P}(\omega_1)/{\mathrm}{NS}_{\omega_1}$ is a complete Boolean algebra, because (1) says it satisfies the $\aleph_2$-chain condition. Since it is complete, for each $\alpha<\omega_2$ there is a stationary $B_\alpha$ which is the sup of $\{A_\beta:\beta\in(\alpha,\omega_2)\}$. Let $E$ be the inf of the family of $B_\alpha$’s. By saturation, $E$ is really the inf of an $\aleph_1$-sized family, and so is itself stationary. Given any $\alpha\in\omega_2$, we can find an $\eta(\alpha)>\alpha$ such that the diagonal union of $\{A_\beta:\beta\in(\alpha,\eta(\alpha))\}$ includes $E$, mod ${\mathrm}{NS}_{\omega_1}$. It follows that there is a cub $C\subseteq \omega_2$ such that for each $\alpha\in C$, there is a subset of $\{A_\beta:\beta\in(\alpha,\alpha^+)\}$ of cardinality $\aleph_1$ with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$, where $\alpha^+$ denotes the next element of $C$ after $\alpha$.
Now let $M$ be an elementary submodel of a suitable $H(\theta)$, with $\langle A_\alpha:\alpha<\omega_2\rangle$, $E$, and $C\in M$ and $\delta=M\cap\omega_1\in E$, $|M\cap\omega_2|=\aleph_1$. We claim $\delta$ is an element of uncountably many $A_\alpha,\alpha\in M$.
Since the cub $C$ divides $\omega_2$ into $\aleph_2$ disjoint intervals, $C\hspace{.03cm}\cap M$ divides $\omega_2\,\cap M$ into $\aleph_1$ disjoint intervals. Choose any one of these intervals $J$. There is a family $\mathcal{F}_J=\{F_\gamma:\gamma<\omega_1\}$ in $M$ consisting of $A_\alpha$’s indexed in the interval $J$, with diagonal union including $E$, mod ${\mathrm}{NS}_{\omega_1}$. Then there is a cub $D_J$ in $M$ disjoint from $E\setminus\nabla\mathcal{F}_J$. $D_J\cap M$ is unbounded in $M$, so $\delta=M\cap\omega_1\in D_J$, so $\delta\notin E\setminus\nabla\mathcal{F}_J$. Then $\delta\in \nabla\mathcal{F}_J$ so $\delta\in F_\gamma$ for some $\gamma\in M\cap\omega_1$ and therefore $\delta$ is in some $A_\xi$ with $\xi\in J$.
We shall finish the proof that MM$(S)[S]$ implies **LCN**$(\aleph_1)$ in Section 4, but first let us note another advantage of stating MM$(S)[S]$ as a hypothesis is that we can often avoid front-loading to get collectionwise Hausdorffness, since **Axiom R** provides enough reflection. For example,
\[312\] $(S)[S]$ implies a locally compact, hereditarily normal space is hereditarily paracompact if and only if it does not include a copy of $\omega_1$.
As usual, we may assume the space does not include a perfect pre-image of $\omega_1$. The proof for that case in [@T] uses *P-ideal Dichotomy*, ${\mathbf{\mathop{\pmb{\sum}}}}$, $\aleph_1$-collectionwise Hausdorffness, and **Axiom R**. We can get all of these from MM$(S)[S]$. (Todorcevic [@To] proved that PFA$(S)[S]$ implies P-ideal Dichotomy; a proof was published in [@D2].)
Similar considerations enable us to prove:
\[thm312\] $(S)[S]$ implies a locally compact, normal, countably tight space is paracompact if and only if its separable closed subspaces are Lindelöf, and it does not include a copy of $\omega_1$.
We thank Paul Larson for Lemma \[larson\] and several discussions concerning the material in this section. Next, we need to do some topology.
Getting locally compact normal spaces collectionwise Hausdorff
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Let $X$ be a locally compact normal space and suppose $Y$ is a closed discrete subspace of $X$ of size $\aleph_1$. Then there is a locally compact normal space $X'$ with a closed discrete subspace $Y'$ of size $\aleph_1$, such that if $Y'$ is separated in $X'$, then $Y$ is separated in $X$, but each point in $Y'$ has character $\leq\aleph_1$.
By Watson’s character reduction technique [@W], there is a discrete collection of compact subsets of $X$, $\mathcal{K}=\{K_y:y\in Y\}$, such that $y\in K_y$, and each $K_y$ has character $\leq\aleph_1$. Let $X'$ be the quotient of $X$ obtained by collapsing each $K_y$ to a point $y'$. This collapse is a perfect map, so preserves normality and local compactness, and it is clear that $\{y':y'\in Y\}$ is separated if and only if $\{K_y:y\in Y\}$ is separated, and that $Y$ is separated if $\{K_y:y\in Y\}$ is.
\[lem47\] Suppose $X$ is a locally compact normal space of Lindelöf degree $\aleph_1$ with an uncountable closed discrete subspace. Then there is a continuous image of $X$ of weight $\aleph_1$ enjoying the same properties.
Let $\mathcal{U}$ be an open cover of $X$ of size $\aleph_1$ with each member of $\mathcal{U}$ a cozero set with compact closure. Without loss of generality, assume that for each $x\in X$ there is a $U\in\mathcal{U}$ such that $x\in U$ and $U$ meets at most one element of a given closed discrete set $D$ of size $\aleph_1$. Also without loss of generality, assume $\mathcal{U}$ is closed under finite intersections. For each $U\in\mathcal{U}$, let $f_U:X\to[0,1]$ with $U=f^{-1}_U((0,1])$. Define an equivalence relation on $X$ by letting $x_0\!\!\sim\!\! x_1$ if $f_U(x_0)=f_U(x_1)$ for all $U\in\mathcal{U}$. Let $X/\!\!\sim$ be the quotient set, with $\pi:X\to X/\!\!\sim$ the projection. Topologize $X/\!\!\sim$ by taking as base all sets of form $\pi(U)$, $U\in\mathcal{U}$. Then $X/\!\!\sim$ is $\textup{T}_{3 \frac{1}{2}}$ and of weight $\leq\aleph_1$. To see the former, consider $X$ as embedded in $[0,1]^{C^*(X)}$ by $e(x)=(f(x))_{f\in C^*(X)}$. Let $p:[0,1]^{C^*(X)}\to[0,1]^{\{f_U:U\in\mathcal{U}\}}$ be given by $(x_f)_{f\in C^*(X)}\to(x_{f_U})_{U\in\mathcal{U}}$, i.e. $p$ projects onto only those coordinates in $C^*(X)$ which are $f_U$’s. Then $X/\!\!\sim\; =p\circ e(X)$.
The projection map $\pi$ is closed, for let $F\subseteq X$ be closed and suppose $y\in\overline{\pi[F]}$. Claim $y\in\pi[F]$. $y\in\pi[U]$ for some $U\in\mathcal{U}$; note $\pi^{-1}(\pi[U])=U$ for if $\pi(x)\in\pi[U]$, $x\sim x_0$ for some $x_0\in U$. Then $f_V(x)=f_V(x_0)$ for every $V\in\mathcal{U}$. But $U=f_U^{-1}((0,1])$. Thus $f_U(x)=f_U(x_0)\in(0,1]$, which implies $x\in U$. So $\stackrel{ }{\overline{U}=\overline{\pi^{-1}\left(\pi[U]\right)}}$ is compact. Suppose $y\notin\pi[F]$. Then $y\notin\pi[F\cap\overline{U}]$, which is compact. Then $\pi[U]\setminus\pi[F\cap\overline{U}]$ is a neighborhood of $y$ disjoint from $\pi[F]$.
Since $\pi$ is closed and $X$ is normal, $X/\!\!\sim$ is normal. It is clear that $\pi[D]$ is closed discrete. By continuity, $\pi[\overline{U}]\subseteq\overline{\pi[U]}$; $\pi[\overline{U}]$ is a closed set including $\pi[U]$, so including $\stackrel{ }{\overline{\pi[U]}}$, so $\pi[\overline{U}]=\overline{\pi[U]}$, so $X/\!\!\sim$ is covered by open sets with compact closures, so it is locally compact.
\[lem48\] In any model obtained by forcing with a Souslin tree $S$, any locally compact normal space with a dense Lindelöf subspace has countable extent.
Suppose $X_0$ is a locally compact normal space with an uncountable closed discrete subspace, which we may conveniently label as $\omega_1$, and a dense Lindelöf subspace $L$. Via normality, we can find a closed subspace $X_1$ with $\omega_1$ in its interior which is covered by $\aleph_1$-many open sets with compact closures. Without loss of generality, we may assume $X_1=\overline{{\mathrm}{int}\,X_1}$. $L$ is dense in ${\mathrm}{int}\,X_1$, so $L\cap({\mathrm}{int}\,X_1)$ is dense in $X_1$. Then $\stackrel{ }{\overline{L\cap {\mathrm}{int}\,X_1}\cap X_1}$ is a dense Lindelöf subspace of $X_1$.
Thus, without loss of generality, we may as well assume our original space $X_0$ has a cover by $\aleph_1$-many open sets, each with compact closure. Without loss of generality, we may assume each is a cozero set and indeed is $\sigma$-compact. By Lemma \[lem47\], there is a continuous image of $X_0$ — call it $X$ — which is also locally compact, normal, has an uncountable closed discrete subspace, and has weight $\aleph_1$. Since both density and Lindelöfness are preserved by continuous functions, $X$ also has a dense Lindelöf subspace. Thus it suffices to find a contradiction for the special case in which the weight of our space is $\aleph_1$.
For $\delta\in\omega_1$ and a cub $C\subseteq\omega_1$, let $\delta^+(C)$ denote the minimum element of $C$ greater than $\delta$. Without loss of generality, we may assume our cubs only consist of limit ordinals. For a cub $C$, we use ${\mathrm}{Fix}(C)$ to denote the set $\{\delta\in C:\text{order-type}(C\cap\delta)=\delta\}$. Let $S_\delta$ be the $\delta$th level of the Souslin tree.
As usual, we work in the ground model and fix names $\dot{\mathcal{B}} = \{\dot{B}_\alpha:\alpha\in\omega_1\}$ for a base of $X$ consisting of open sets with compact closures. It is convenient to assume that $\{\dot{B}_n:n\in\omega\}$ is forced to have dense union. Again, we let $\omega_1$ label a closed discrete subspace and let $\{\dot{U}(\alpha,\xi):\xi\in\omega_1\}$ be a subset of $\dot{\mathcal{B}}$ forced to be a local base at $\alpha$. Without loss of generality, assume each $B_n$ is disjoint from the closed discrete set $\omega_1$. Fix a cub $C_0$ such that for each $\delta\in C_0$ and each $s\in S_\delta$, $s$ decides all equations of the form $\dot{B}_\alpha\cap\dot{B}_\beta =\emptyset$, for $\alpha,\beta<\delta$. Also assume that for each $s\in S_\delta\;(\delta\in C_0)$ and each $\xi,\beta\in\delta$, there is an $\alpha\in\delta$ such that $s$ forces that $\dot{U}(\xi,\beta)=\dot{B}_\alpha$.
It is convenient to assume that $S$ is $\omega$-branching (specifying any infinite maximal antichain above each element would serve the same purpose). We can use $C_1={\mathrm}{Fix}(C_0)$ to define a partition $\dot{f}$ of $\omega_1$ so that for each $\xi\in\omega_1$ and each $s\in S_{\xi^+(C_1)}$, $s^\smallfrown j$ forces that $\dot{f}(\xi)=j$. Now we choose two (names of) functions $\dot{h}_1$ and $\dot{h}_2$ witnessing normality as follows:
- For each $j\in\omega$ and each $i\in 2$, let $\dot{W}^i_j=\bigcup\{\dot{U}(\xi,h_i(\xi)):\xi\in \dot{f}^{-1}(j)\}$,
- the $\{\dot{W}^1_j:j\in\omega\}$ form a discrete family,
- the closure of $\dot{W}^2_j$ is included in $\dot{W}^1_j$.
Choose any countable elementary submodel $M$ with all the above as members of $M$, such that $\delta = M\cap\omega_1$ is an element of $C_1$. We know that there is a name of an integer $\dot{J}_\delta$ satisfying that it is forced that $\dot{U}(\delta,0)\cap\dot{W}_j$ is empty for all $j\geq\dot{J}_\delta$. Choose any $s\in S$ of height at least $\delta^+(C_1)$ that decides a value $J$ for $\dot{J}_\delta$. Let $\bar{s} = s\upharpoonright\delta^+(C_1)$. Notice that $\bar{s}$ decides the truth value of the equation “$\dot{U}(\delta,0)\cap\dot{B}_\alpha=\emptyset$”, for all $\alpha\in M$. For each $n,j\in\omega$, $s$ and hence $s\upharpoonright\delta$ forces that the closure of $\dot{W}^2_j\cap \dot{B}_n$ is included in $\dot{W}^1_j$. By elementarity and compactness, this implies there is a finite $\dot{F}_{j,n}\subseteq\delta$ such that $s\upharpoonright\delta$ forces that $\dot{W}^2_j\cap \dot{B}_n\subseteq\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\}\subseteq\dot{W}^1_j$. But now $\bar{s}$ forces $\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in \dot{F}_{j,n}\})$ is empty for all $n$ and all $j\geq J$.
On the other hand, fix any $j\geq J$ and consider what $\bar{s}^\smallfrown j$ is forcing. This forces that $\dot{f}(\delta)=j$ and that $\delta\in W^2_j$, and so $\delta$ is in the closure of the union of the sequence $\{\dot{U}(\delta,0)\cap(\bigcup\{\dot{B}_\eta:\eta\in
F_{j,n}\}):n\in\omega\}$. This is a contradiction.
\[cor410\] In any model obtained by forcing with a Souslin tree, if $X$ is locally compact normal, $D$ is a closed discrete subspace of $X$ of size $\aleph_1$ and $\{U_\alpha:\alpha\in\omega_1\}$ are open sets with compact closures, then for any countable $T\subseteq\omega_1$, $\stackrel{}{\overline{\bigcup\{U_\alpha:\alpha\in T\}}}\cap\; D$ is countable.
${\bigcup\{{\overline{U}_\alpha}:\alpha\in T\}}$ is dense in $\overline{\bigcup\{U_\alpha:\alpha\in T\}}$, which is locally compact normal.
Getting back to the proof of \[thm41\], let us assume we are in a model of ${\mathrm}{MM}(S)$ and that we have an $S$-name $\dot{X}$ for a locally compact normal space, with a closed discrete subspace labeled as $\omega_1$, with each of its points having character $\aleph_1$. Let us note that it follows from character reduction and Lemma \[LT1\] that if there is a discrete expansion of $\omega_1$ into compact $G_\delta$’s, then $\omega_1$ will have a separation. In fact, even more, it is shown in [@T3 Theorem 12] that if $\omega_1$ is forced to have an expansion by compact $G_\delta$’s that is $\sigma$-discrete, then $\omega_1$ will be separated. Since our proof is by contradiction, we will henceforth assume that it is forced (by the root of $S$) that there is no expansion of $\omega_1$ into a $\sigma$-discrete family of compact $G_\delta$’s.
For each $\xi,\alpha\in\omega_1$, let $\dot{U}(\xi,\alpha)$ be the name of the $\alpha$th neighbourhood from a local base for $\xi$ with $\dot U(\xi,0)$ forced to have compact closure. Corollary \[cor410\], and the fact that $S$ is ccc, ensure that for each $\delta\in
\omega_1$, every element of $S$ forces that $\omega_1 \cap \overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}$ is bounded by $\gamma$ for some $\gamma\in \omega_1$. Therefore there is a cub $C_0$ such that without loss of generality, we can assume that each of the following is forced by each element of $S$:
1. for each $\delta\in C_0$, $\omega_1\cap
\stackrel{}{\overline{\bigcup\{\dot{U}(\xi,0):\xi<\delta\}}}$ is included in $\delta^+(C_0)$,
2. for all $\beta\neq \xi$ in $\omega_1$, $\beta\notin \dot{U}(\xi,0)$,
3. for all $\xi,\alpha\in \omega_1$ $\dot{U}(\xi,\alpha)\subseteq \dot{U}(\xi,0)$ and has compact closure,
4. for each limit $\delta\in \omega_1$, the sequence $\{\dot{U}(\xi,\alpha):\alpha<\delta\}$ is a *regular filter*, i.e. each finite intersection of these includes the closure of another.
For an $S$-name $\dot{h}$ of a function from $\omega_1$ to $\omega_1$, let $\dot{U}(\xi,\dot{h})$ stand for $\dot{U}(\xi,\dot{h}(\xi))$. For limit $\delta$, let $\dot{Z}(\xi,\delta)$ denote the $S$-name of the compact $G_\delta$ equal to $\bigcap\{\dot{U}(\xi,\alpha):\alpha<\delta\}$. For a cub $C$ and ordinal $\xi$, we also use $\dot{Z}(\xi,C)$ as an abbreviation for $\dot{Z}(\xi,\xi^+(C))$.
Fix an enumeration $\{C_\gamma:\gamma\in\omega_2\}$ for a base for the cubs on $\omega_1$ (each containing only limit ordinals), chosen so that $C_0$ is as above and for $0<\lambda \in \omega_2$, $C_\lambda \subseteq {\mathrm}{Fix}(C_0)$ and $C_\lambda\setminus{\mathrm}{Fix}(C_\gamma)$ is countable for all $0\leq\gamma<\lambda$. We can do this by taking diagonal intersections, since **SRP** implies $2^{\aleph_1}=\aleph_2$.
For each $\delta\in C_0$, let $\beta(\delta) = \delta^+(C_0)$. Since $\dot{Z}(\xi,C_\gamma)\subseteq
\dot{U}(\xi,C_\gamma)$ for all $\xi\in \omega_1$ for all $\delta\in C_\gamma$, $\beta(\delta)<\delta^+(C_\gamma)$, and so it is forced that: $$\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}}
\cap\omega_1\subseteq\beta(\delta).$$
We can also assume that for all cubs $C\subseteq C_0$, there is an $S$-name $\dot A$, that is forced to be a stationary subset of ${\mathrm}{Fix}(C)$ satisfying: $$(\forall s\in S)(\forall \delta)
~~ s\Vdash \left(\delta\in \dot A \ \Rightarrow
(\exists\alpha\in[\delta,\beta(\delta)])\;
\alpha\in\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}}~\right).$$
The reason we can make this assumption is that we are assuming there is no $\sigma$-discrete expansion of $\omega_1$ by compact $G_\delta$’s. If, in the extension, the set $A = \{ \delta :
\overline{\bigcup\{\dot{Z}(\xi,C):\xi<\delta\}} \not\subseteq \delta\}$ were not stationary, then there would be a $\lambda\in \omega_2$ such that $A\cap C_\lambda$ is empty. Since the cub $C_\lambda$ divides $\omega_1$ into countable pieces, we see that we can expand the points in $\omega_1$ into a $\sigma$-discrete collection of compact $G_\delta$’s.
For each $\lambda \in \omega_2$, let $\dot A_\lambda$ denote the name of the stationary set just described. For any $B\subseteq\omega_1$, we will write $$\alpha\in\langle\dot{Z}(\xi,C):\xi<\delta\rangle'$$ to mean that $\alpha$ is a limit point of that sequence of sets.
Fix any function $e:S\to\omega$ with the property that for all $\delta\in\omega_1$, $e\restriction S_\delta$ is one-to-one. For an ordinal $\gamma\in\omega_2$, we use $\dot{f}_\gamma$ for the $S$-name of the function from $\omega_1$ into $\omega$ given by the property that each $s\in S_{\xi^+(C_\gamma)}$ forces that $\dot{f}_\gamma(\xi)=e(s)$. Thus $\dot{f}_\gamma$ partitions $\omega_1$ into a discrete collection of countably many closed subsets. Then let $\{\dot{W}(\gamma,n):n\in\omega\}$ be a discrete collection of open sets separating the $\dot{f}_\gamma^{-1}(n)$’s. Fix $n\in\omega$. By normality, there is an open $\dot{V}_n$ such that $S$ forces $\dot{f}^{-1}_\gamma(n)\subseteq\dot{V}_n\subseteq\dot{\overline{V}}_n\subseteq\dot{W}(\gamma,n)$. For each $\xi\in\dot{f}^{-1}_\gamma(n)$, there is an $\alpha_\xi\in\omega_1$ such that $S$ forces $\dot{U}(\xi,\alpha_\xi)\subseteq\dot{V}_n$. Let $\zeta_n(\gamma)\in\omega_2$ be such that for $\xi\in\dot{f}^{-1}_\gamma(n), \xi<\rho\in C_{\zeta_n(\gamma)}$ implies $\alpha_\xi<\rho$. Then $S$ forces $\{\dot{Z}(\xi,
C_{\zeta_n(\gamma)}):\xi\in\dot{f}_\gamma^{-1}(n)\}\subseteq\dot{V}_n$. We then can find a $C_{\zeta(\gamma)}$ included in each $C_{\zeta_n(\gamma)}$ such that for every $n\in\omega$, $S$ forces $\{\dot{Z}(\xi,C_{\zeta(\gamma)}):\zeta\in\dot{f}^{-1}_\gamma(n)\}\subseteq
\dot{V}_n$. Thus $$\overline{\bigcup\{\dot{Z}(\xi,C_{\zeta(\xi)}):\xi\in f_\gamma^{-1}(n)\}}
\subseteq\dot{W}(\gamma,n).$$ Then we can get a $\zeta(\gamma)$ that works for all $n$.
By recursion on $\gamma\in\omega_2$, we can choose $\zeta(\gamma)\geq\gamma$ as above, so that the sequence $\{\zeta(\gamma):\gamma\in\omega_2\}$ is strictly increasing. For each $\gamma$, we have the $S$-name $\dot A_{\zeta(\gamma)}$ as above. It is immediate that $A_\gamma = \{ \delta :
(\exists s\in S) s\Vdash \delta\in \dot A_{\zeta(\gamma)}\}$ is a stationary set. In other words, $\delta \in A_\gamma$ implies there is some $s\in S$ and $\eta \in [\delta,\beta(\delta)]$ such that $s\Vdash \eta \in
\langle\dot{Z}(\xi,
C_{\zeta(\gamma)}):\xi\in\delta\rangle'$.
By **SCC** and \[larson\] we may assume there is an elementary submodel $M$ of some $\langle H(\theta),\{\langle
\gamma,\zeta(\gamma), A_\gamma\rangle:\gamma\in\omega_2 \}\rangle$, with $M\cap\omega_1=\delta<\omega_1$, $|M\cap\omega_2|=\aleph_1$, and an uncountable $\{\gamma_\alpha:\alpha\in\omega_1\}\subseteq
M\cap\omega_2$, so that $\delta\in A_{\gamma_\alpha}$ for all $\alpha\in\omega_1$.
For each $\alpha\in\omega_1$ choose $s_{\alpha}\in S$, $\eta_\alpha\in[\delta,\beta(\delta)]$ such that $s_\alpha\Vdash
\eta_\alpha\in\langle\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)}):
\xi\in\delta\rangle'$. We may assume $s_\alpha$ is on a level at least as high as $\delta^+(C_{\gamma_\alpha})$. We may also assume that if $\alpha<\beta\in\omega_1$, then $\gamma_\alpha<\gamma_\beta$. We may also assume that the height of $s_\alpha$ is less than the height of $s_\beta$, for $\alpha<\beta$, so that $\{s_\alpha:\alpha\in\omega_1\}$ is an uncountable subset of $S$. Therefore there is an $\eta\in [\delta,\beta(\delta)]$ such that $L = \{ \alpha : \eta_\alpha = \eta\}$ is uncountable. Also, as is well-known for Souslin trees, there is an $\bar{s}\in S$, such that $\{s_\alpha :\alpha\in L\}$ includes a dense subset of $\{s\in S:\bar{s}<s\}$. By passing to an uncountable subset, we may assume that $\bar{s} <s_\alpha$ for all $\alpha\in L$ and that $\bar{s}$ is on a level above $\delta$. Similarly we may assume that for all $\xi,\rho<\delta$, $\bar{s}$ has decided the statement $$\dot{U}(\eta,0)\cap\dot{Z}(\xi,\rho)\neq\emptyset\quad
\text{ for all }\xi,\rho<\delta.$$ Now choose any $\alpha\in L$ (e.g.the least one), and then choose an infinite sequence $\{\beta_l:l\in\omega\}\subseteq
L\setminus(\alpha+1)$ so that $s_{\beta_l}\upharpoonright \delta^+({C_{\gamma_\alpha}})$ are all distinct. For each $l$, let $e(s_{\beta_l}\upharpoonright\delta^+({C_{\gamma_\alpha}})~)=n_l$. **Main Claim:** $\quad\bar{s}\Vdash
(\forall
l\in\omega)\left(\dot{W}(\gamma_\alpha,n_l)\cap\dot{U}(\eta,0)\neq
0\right).$ Once this claim is proven we are done, because we then have that $\bar{s}$ forces that $\dot{U}(\eta,0)$ cannot have compact closure, because it meets infinitely many members of the discrete family $\{\dot{W}(\gamma_\alpha,n):n\in\omega\}$.
To prove the claim, first note that there is a tail of $C_\zeta(\gamma_{\beta_l})\cap\delta$ included in $C_{\zeta(\gamma_{\alpha})}$. To see this, recall $C_{\zeta(\gamma_\alpha)}\setminus{\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is countable, so some tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))$ is included in $C_{\zeta(\gamma_\alpha)}$. By elementarity, since $\gamma_\alpha$ and $\gamma_\beta$ are in $M$, a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap M$ is included in $C_{\zeta(\gamma_\alpha)}\cap M$, so a tail of ${\mathrm}{Fix}(C_\zeta(\gamma_{\beta_l}))\cap\delta$ is included in $C_{\zeta(\gamma_\alpha)}$.
Since there is a tail of $C_{\zeta(\gamma_{\beta_l})}\cap \delta$ included in $C_{\zeta(\gamma_\alpha)}$, $\dot{Z}(\xi,
C_{\zeta(\gamma_{\beta_l})})\subseteq\dot{Z}(\xi,C_{\zeta(\gamma_\alpha)})$ for each $\xi<\delta$ (at least on a tail — which is all that matters for limits above $\delta$). Then $s_{\beta_l}$ forces that $\eta $ is a limit of the sequence $$\langle\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)}):
\xi\in\delta\text{ and }\dot{f}_{\gamma_\alpha}(\xi)= n_l\rangle.$$
Of course this means that $s_{\beta_l}$ forces that $\dot{U}(\eta,0)$ meets $\dot{Z}(\xi, C_{\zeta(\gamma_\alpha)})$ for cofinally many $\xi<\delta$ such that $s_{\beta_l}\upharpoonright\gamma_\alpha\Vdash
\dot{f}_{\gamma_\alpha}(\xi)=n_l$. But $\bar{s}$ has already decided the value of $\dot{f}_{\gamma_\alpha}\upharpoonright\delta$, and $\bar{s}$ already forces $\dot{U}(\eta,0)\cap\dot{Z}(\xi,
C_{\zeta(\gamma_\alpha)})\neq\emptyset$ whenever $s_{\gamma_\beta}$ does. In particular then, $\bar{s}$ forces there is a $\xi$ with $\dot{f}_{\gamma_\alpha}(\xi)=n_l$ (and so $\dot{Z}(\xi,
C_{\zeta(\gamma_\alpha)})\subseteq\dot{W}(\gamma_\alpha,n_l)$) and $\dot{U}(\eta,0)\cap\dot{Z}(\xi,
C_\zeta(\gamma_\alpha))\neq\emptyset$.
For the record, let us state what we have accomplished:
$(S)[S]$ implies **LCN**$(\aleph_1)$.
There is a model of $(S)[S]$ in which **LCN** holds, *i.e. every locally compact normal space is collectionwise Hausdorff.*
Large Cardinals and the MOP
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In [@DT2] we showed that large cardinals are not required to obtain the consistency of every *locally compact perfectly normal space is paracompact*. It is interesting to see which other PFA$(S)[S]$ results can be obtained without large cardinals. The standard method used was pioneered by Todorcevic in [@To3] and given several applications in [@D], all in the context of PFA results. In the context of PFA$(S)[S]$, it is referred to in [@To] and actually carried out in [@DT1] for a version of [P-ideal Dichotomy]{} and for ****. It is routine to get additionally that such models are of form MA$_{\omega_1}(S)[S]$ by interleaving additional forcing. In [@DT2] we pointed out that such methods can give models in which in addition the following holds:
**${\mathbf{\mathop{\pmb{\sum}}}}^{\bm{-}}$(sequential)**
In a compact sequential space, each locally countable subspace of size $\aleph_1$ is $\sigma$-discrete.
A modification of such a proof produces a model in which the following proposition (see [@FTT]) holds:
**${\mathbf{\mathop{\pmb{\sum}}}}$(sequential)**
Let $X$ be a compact sequential space. Let $Y\subseteq X$, $|Y|=\aleph_1$. Suppose $\{W_\alpha\}_{\alpha\in\omega_1}$, $\{V_\alpha\}_{\alpha\in\omega_1}$ are open subsets of $X$ such that:
- $W_\alpha\subseteq\overline{W_\alpha}\subseteq V_\alpha,$
- $|V_\alpha\cap Y|\leq\aleph_0$,
- $Y\subseteq\bigcup\{W_\alpha:\alpha\in\omega_1\}$.
Then $Y$ is $\sigma$-closed discrete in $\bigcup\{W_\alpha:\alpha\in\omega_1\}$.
Without the parenthetical “sequential”, ${\mathbf{\mathop{\pmb{\sum}}}}^-$ and ${\mathbf{\mathop{\pmb{\sum}}}}$ refer to the corresponding propositions obtained by replacing “sequential” by countably tight”, which follow from their sequential versions if one has
**Moore-Mrówka**
Every compact countably tight space is sequential.
It follows easily from **Moore-Mrówka** that *locally compact countably tight spaces are sequential*. A proof of **Moore-Mrówka** from PFA$(S)[S]$ is sketched in [@To] and the author remarks that, by the usual methods, large cardinals are not necessary. Thus, one can obtain a model of MA$_{\omega_1}(S)[S]$ in which, for example, both **PPI** and **${\mathbf{\mathop{\pmb{\sum}}}}$** hold, without the need for large cardinals. Working in such a model, we can establish the following proposition, the conclusion of which was proved from PFA$(S)[S]$ in [@To] and asserted to be obtainable without large cardinals.
If ZFC is consistent, it’s consistent to additionally assume that locally compact, hereditarily normal, separable spaces are hereditarily Lindelöf.
Let $X$ be such a space. By \[lem48\] $X$ has countable spread. So does its one-point compactification $X^*$, which hence is countably tight [@A2]. If $X$ were not hereditarily Lindelöf, it would include a right-separated subspace $\{x_\alpha:\alpha\in\omega_1\}$. Let $\{V_\alpha:\alpha\in\omega_1\}$ be open sets witnessing right-separation. Let $x_\alpha\in W_\alpha\subseteq\overline{W_\alpha}\subseteq V_\alpha$, with $W_\alpha$ open and $\overline{W_\alpha}$ compact. Applying ${\mathbf{\mathop{\pmb{\sum}}}}$ to $X^*$, we see that $\{x_\alpha:\alpha\in\omega_1\}$ is $\sigma$-closed discrete in $W=\bigcup\{W_\alpha:\alpha\in\omega_1\}$. But $W$ is locally compact, separable, and hereditarily normal, so this contradicts \[lem48\].
Also without large cardinals we obtain:
\[thm510\] If ZFC is consistent, it is consistent to additionally assume that each hereditarily normal perfect pre-image of $\omega_1$ includes a copy of $\omega_1$.
Using ${\mathbf{\mathop{\pmb{\sum}}}}$ and **PPI**, we can carry out the proof of Theorem \[thm34\] above.
We also have:
\[thm511\] If ZFC is consistent, it is consistent to assume that every locally compact, first countable, hereditarily normal space with Lindelöf number $\leq\aleph_1$ not including a copy of $\omega_1$ is paracompact.
We use the model of \[thm510\]. In [@T] the second author asserted the following, but under PFA$(S)[S]$ instead of MM$(S)[S]$, which we now see should have been used.
\[lem512\] $(S)[S]$ implies that if $X$ has Lindelöf number $\leq\aleph_1$ and is locally compact, normal, and does not include a perfect pre-image of $\omega_1$, then $X$ is paracompact.
In addition to the topological properties mentioned, the proof used ${\mathbf{\mathop{\pmb{\sum}}}}$ and that the space was $\aleph_1$-collectionwise Hausdorff. For the purposes of \[thm511\], however, we get $\aleph_1$-collectionwise Hausdorff just from the Souslin forcing, since the space is first countable.
MM$(S)[S]$ is also relevant for questions concerning the Baireness of $C_k(X)$, for locally compact $X$ (see [@GM; @MN; @T5]).
A ***moving off** collection* for a space $X$ is a collection ${\mathcal}{K}$ of non-empty compact sets such that for each compact $L$, there is a $K \in {\mathcal}{K}$ disjoint from $L$. A space satisfies the **Moving Off Property** (MOP) if each moving off collection includes an infinite subcollection with a discrete open expansion.
$C_k(X)$, for a space $X$, is the collection of all continuous real-valued functions on $X$, considered as a subspace of the compact-open topology on the Cartesian power $X^{\mathbb{R}}$.
A locally compact space $X$ satisfies the MOP if and only if $C_k(X)$ is Baire, i.e. satisfies the Baire Category Theorem.
\[lem6\] Locally compact, paracompact spaces satisfy the MOP.
\[thm35\] ${\mathrm}{MM}(S)[S]$ implies that normal spaces satisfying the MOP are paracompact if they are:
- locally compact, countably tight, and hereditarily normal, or
- first countable and hereditarily normal, or
- locally compact, countably tight with Lindelöf number $\leq\aleph_1$, or
- first countable, with Lindelöf number $\leq\aleph_1$, or
- locally compact, countably tight, and countable sets have Lindelöf closures.
These all follow easily from \[thmConj2\], \[thm312\], and **Moore-Mrówka**, using:
In a sequential space, countably compact subspaces are closed.
Countably compact spaces satisfying the MOP are compact.
\[lem510\] First countable spaces satisfying the MOP are locally compact.
The one-point compactification of a locally compact space $X$ is countably tight if and only if $X$ does not include a perfect pre-image of $\omega_1$.
If they have the MOP, sequential spaces do not include copies of $\omega_1$, so (1) follows from \[312\]. (2) follows from (1) plus \[lem510\]. (3) follows from \[lem512\] plus \[thmConj2\]. (4) follows from (3) plus \[lem510\]. (5) follows from \[thm312\], \[thmConj2\] and Balogh’s Lemma above.
In the special case of a space with the MOP, we have:
\[thm513\] If ZFC is consistent, then it is consistent to additionally assume that first countable normal spaces satisfying the MOP and with Lindelöf number $\leq\aleph_1$ are paracompact.
Such a space is locally compact and does not include a perfect pre-image of $\omega_1$.
MA$_{\omega_1}$ gives counterexamples for the conclusions of \[thm35\] and \[thm513\]. See e.g. [@T5].
If ZFC is consistent, then it is consistent to assume that first countable hereditarily normal, locally connected spaces satisfying the MOP are paracompact.
The extra ingredient is that the local connectedness will enable us to decompose the space into a sum of pieces with Lindelöf number $\leq\aleph_1$. More precisely,
A space $X$ is of **Type I** if $X=\bigcup\{X_\alpha:\alpha\in\omega_1\}$, where each $X_\alpha$ is open, $\alpha<\beta$ implies $\overline{X}_\alpha\subseteq X_\beta$, and each $X_\alpha$ is Lindelöf.
In [@T], it is shown on page 104 that, assuming ${\mathbf{\mathop{\pmb{\sum}}}}$ and hereditary $\aleph_1$-collectionwise Hausdorffness for a locally compact hereditarily normal space not including a perfect pre-image of $\omega_1$ that the closure of a Lindelöf subspace is Lindelöf. Then we quote:
If $X$ is locally compact, locally connected, and countably tight, then $X$ is a topological sum of Type I spaces if and only if every Lindelöf subspace of $X$ has Lindelöf closure.
Since a topological sum of paracompact spaces is paracompact, this will complete the proof of the Theorem.
It may be of interest that **SRP** implies a weaker version of the conclusion of Theorem \[thm35\].2.
implies every first countable, monotonically normal space satisfying the MOP is paracompact.
Suppose $S$ is a first countable stationary subspace of some regular cardinal. Then each $s\in S$ is an $\omega$-cofinal ordinal.
Each $s\in S$ is either isolated in $S$ or is a limit of some subset of $S$. By first countability, in the latter case, each such $s$ is a limit of a sequence of elements of $S$.
Suppose not. Then by [@BR] the space includes a copy of a stationary subset of some regular cardinal. By [@J 37.18] **SRP** implies that that stationary set includes a copy of a closed unbounded subset of $\omega_1$. That copy is closed, countably compact but not compact, contradicting the MOP.
We conjecture th answer is positive. Large cardinals would be necessary to refute the existence of such a space, since an example can be constructed from the failure of the Covering Lemma for the Core Model K, which entails the consistency of measurable cardinals. We thank Peter Nyikos for referring us to [@G], where that failure is used to construct a locally compact, locally countable, normal, non-paracompact space $X$ on $\kappa^+\times\omega_1$, where $\kappa^+$ is the successor of a singular strong limit cardinal of countable cofinality, such that the spaces $X_\alpha=\alpha\times\omega$ are metrizable for all $\alpha\in\kappa^+$. It follows that closed subspaces of $X$ of size $\leq 2^{\aleph_0}$ are locally compact and metrizable, so satisfy the MOP by \[lem6\]. On the other hand,
If a Hausdorff space $Z$ is locally countable, locally compact, and closed subspaces of $\leq 2^{\aleph_0}$ have the MOP, then $Z$ has the MOP.
It follows that $X$ has the MOP.\
With MM$(S)[S]$ we have:
$(S)[S]$ implies that if $X$ is normal, locally compact, locally countable, and closed subspaces of size $\leq 2^{\aleph_0}$ are metrizable, then $X$ is metrizable.
By the preceding proof, $X$ has the MOP. By \[thm35\], to get that $X$ is paracompact, it suffices to show that countable subspaces of $X$ have Lindelöf closures. But if $Y$ is a countable subset of $X$, $|\overline{Y}|\leq 2^{\aleph_0}$ and hence is separable metrizable and hence Lindelöf. Once we have $X$ paracompact, it follows that $X$ is a topological sum of $\sigma$-compact subspaces. But each of these has size $\leq 2^{\aleph_0}$ and so is metrizable.
**Axiom R** precludes stationary non-reflecting sets of $\omega$-cofinal ordinals in $\omega_s$, and hence the locally compact, $\aleph_1$-collectionwise Hausdorff ladder system space built on such a set; we can therefore ask:
Examples
========
A question left open in [@LT1] is whether, as was shown for adjoining $\aleph_2$ Cohen subsets of $\omega_1$ in [@T1], forcing with a Souslin tree would make normal spaces of character $\aleph_1$ $\aleph_1$-collectionwise Hausdorff. We shall show that the answer is negative by showing:
$_{\omega_1}(S)[S]$ implies that there is a normal non-$\aleph_1$-collectionwise Hausdorff space of character $\aleph_1$.
Let $S\subseteq 2^{<\omega_1}$ be a coherent Souslin tree. Fix a family ${\{a_s:s\in S\}\subseteq[\omega]^\omega}$ so that for $s<t\in S$, $a_t\subseteq^* a_s$ and for each $\gamma\in\omega_1$, $\{a_s:s\in S_\gamma\}$ is pairwise disjoint.
For each limit $\delta\in\omega_1$, let $L_\delta\in\delta^\omega$ be a strictly increasing function with range cofinal in $\delta$ consisting of successor ordinals. For $a\subseteq\omega,$ let $L[a]=\{L_\delta(n):n\in a \}$. The generic $g$ for $S$ will enable us to define the required topology on the set $\omega_1$. We declare each successor ordinal to be isolated. For each limit $\delta$, the neighborhood filter for $\delta$ will be $\{L_\delta[a_s]\cup\{\delta\}:s\in g \}$. The set $C_0$ of limit ordinals is then a closed discrete set. By pressing down, we see that $C_0$ cannot be separated. It remains to show that the space is normal. It suffices to show that if $f$ is an $S$-name of a function from $C_0$ to $2$, then there is a neighborhood assignment $\{\dot{U}_\delta:\delta\in C_0\}$ and a cub $C_1$, such that for each $\alpha<\delta\in C_1$, $S$ forces that if $\dot{f}(\alpha)\neq\dot{f}(\delta)$, then $\dot{U}(\alpha)$ and $\dot{U}(\delta)$ are disjoint.
There is a cub $C_1\subseteq C_0$ so that for all $\delta\in C_1$ and $\alpha<\delta_1$ each $s\in S_\delta$ decides the value of $\dot{f}(\alpha)$. For each $\delta\in C_0$, let $\delta^+$ denote the minimal element of $C_1$ above $\delta$, and choose a function $f_\delta:\omega\to 2$ so that for each $s\in S_{\delta^+}$ and each $n\in a_s$, $s$ forces $\dot{f}(\delta)=f_\delta(n)$. We will define an integer $n_\delta$ such that the value of $\dot{U}_\delta$ is forced by $s\in S_{\delta^+}$ to equal $\{\delta\}\cup L_{\delta}[a_s\setminus n_\delta]$. The sequence of functions $\{f_\delta:\delta\in C_0\}$ will be in the MA$_{\omega_1}(S)$ model.
Let $\mathcal{Q}$ be the poset of partial functions $h$ from $\omega_1$ into $2$ such that ${h=^*\bigcup\{f_\delta\circ L_\delta^{-1}: \delta\in H \}}$, for some $H\in[C_0]^{<\omega}$. $\mathcal{Q}$ is ordered by extension. We claim that in ZFC, $\mathcal{Q}$ is ccc. If so, there will be a generic for $\aleph_1$ dense subsets of $\mathcal{Q}$ in a model of MA$_{\omega_1}(S)$. Let $\mathcal{H}=\{(h_\alpha,H_\alpha):\alpha\in\omega_1 \}$ be a subset of $\mathcal{Q}\times[C_0]^{<\omega}$, where $h_\alpha=\bigcup\{f_\delta\circ L_\delta^{-1}:\delta\in H_\alpha \}$. Choose any countable elementary submodel $M$ with $\mathcal{Q}$ and $\mathcal{H}$ in $M$. Let $\delta=M\cap\omega_1$ and $H_\delta\cap M=H$ and $H_\delta\setminus M=\{\delta_i:i<l\}$. We may assume that $\delta_0=\delta$ and then choose $\alpha_0\in M$ so that $H\subseteq\alpha_0$ and $L_{\delta_i}\cap\delta\subseteq\alpha_0$, for $0<i<l$. Notice that $h_\delta\!\!\upharpoonright\!\!\alpha$ is an element of $M$, for all $\alpha\in M$. In $M$, recursively choose $\alpha_0<\alpha_1<\cdots$ so that $h_{\alpha_{n+1}}\!\!\upharpoonright\!\!\alpha_n = h_\delta\!\!\upharpoonright\!\!\alpha_n$ and dom$(h_{\alpha_{n+1}})\subseteq\alpha_{n+2}$. With $\beta=\sup_n\alpha_n<\delta$, we have that there is an $n\in\omega$ such that $h_\delta\!\!\upharpoonright\!\!\beta = h_\delta\!\!\upharpoonright\!\!\alpha_n$. It follows that $h_\delta\!\!\upharpoonright\!\!\alpha_n\subseteq h_{\alpha_{n+1}}$, and so $h_\delta$ and $h_{\alpha_{n+1}}$ are compatible members of $\mathcal{Q}$.
MA$_{\omega_1}(S)$ implies there is a generic for $\mathcal{Q}$ that adds a function $h$ from $\omega_1$ to $2$ that mod finite extends $f_\delta\circ L_\delta^{-1}$, for all $\delta\in C_0$. Now define $n_\delta$ to be chosen so that $h$ actually extends $f_\delta\circ L_\delta^{-1}[\omega\setminus n_\delta]$. Suppose $\alpha<\delta$, with $\delta\in C_1$, and let $s\in S_{\delta^+}$. Then $s$ forces that $f_\delta\circ L^{-1}_\delta=\dot{f}_\delta$ on $a_s$, and similarly, $s\!\!\upharpoonright\!\alpha^+$ forces that $f_\alpha\circ L^{-1}_\alpha=\dot{f}_\alpha$ on $a_{s\upharpoonright\alpha^+}$. Also, $h$ agrees with $f_\delta\circ L_\delta^{-1}$ on $a_s\setminus n_\delta$ and with $f_\alpha\circ L_\alpha^{-1}$ on $a_{s\upharpoonright\alpha^+}\setminus n_\alpha$. Thus if $\beta\in L_\delta[a_s\setminus n_\delta]\cap L_{\alpha}[a_{s\upharpoonright\alpha^+}\setminus n_\alpha]$, then $h(\beta)=\dot{f}(\alpha)=\dot{f}(\delta)$. This completes the proof that the space is normal.
The strategy attempted in [@T3] was to expand a closed discrete subspace of a locally compact normal space to a discrete collection of compact $G_\delta$’s. There are limitations on such an approach, given by the following example.
MA$_{\omega_1}(S)[S]$ implies there is a locally compact space of character $\aleph_1$ which includes a normalized closed discrete set which does not have a normalized discrete expansion by compact $G_\delta$’s.
We modify the previous example. Let $\mathcal{A}_s$ denote the Boolean subalgebra of $\mathcal{P}(\omega)$ generated by $[\omega]^{<\omega}\cup\{a_s:s\in S\}$. In the forcing extension by $S$, let $x_g$ denote the member of the Stone space $\mathcal{S}(\mathcal{A}_s/\text{FIN})$ containing $\{a_s:s\in g\}$.
In the forcing extension, our space has the base set $(\omega_1\setminus C_0)\cup(C_0\times\mathcal{S}(\mathcal{A}_s))$. The points of $\omega_1\setminus C_0$ are isolated. For each $\delta\in C_0$ and $x\in\mathcal{S}(\mathcal{A}_s/\text{FIN})$, a neighborhood of $(\delta,x)$ must include $U_\delta(a)=L_\delta[a]\cup(\{\delta\}\times a^*)$ for some $a\in x$, where $a^* = \{p\in\mathcal{S}(\mathcal{A}_\mathcal{S}):a\in p\}$. Notice that $U_\delta(a)$ is disjoint from $\{\gamma\}\times\mathcal{S}(\mathcal{A}_s/\text{FIN})$, for all $\gamma\neq\delta$. It follows immediately that the sequence $D=\{(\gamma,x_g):\delta\in C_0\}$ is a closed discrete subset. It also follows from the proof of the normality of the previous example that $D$ is normalized.
Now we show that $D$ does not have a normalized discrete expansion by compact $G_\delta$’s, indeed by any $G_\delta$’s. Assume that $\{\dot{Z}_\delta:\delta\in C_0\}$ is a sequence of $S$-names so that $\dot{Z}_\delta$ is forced to be a $G_\delta$ containing $(\delta,x_g)$. There is a cub $C_1$ such that for each $\alpha\in C_0$ and each $s\in S_{\alpha^+}$ (again, $\alpha^+$ is the minimal element of $C_1$ above $\alpha$), $s$ forces that $\dot{Z}_\alpha$ contains $\{\alpha\}\times a_s^*$. Since $S$ is ccc, the cub $C_1$ can be chosen to be a member of the PFA$(S)$ model.
We use $C_1$ to define a partition of $C_0$: for each $\alpha\in C_0$, we define $\dot{f}(\alpha)$ to equal the value $g(\alpha^+)$ (i.e. the element of $S_{\alpha^+}$ that $g$ picks). Thus if $\delta$ is a limit of $C_1$ and $s\in S_\delta$, then $s$ forces a value for $\dot{f}\!\!\upharpoonright\!\!\delta$. Then a potential normalizing expansion would consist of a sequence $\{\dot{n}_\alpha:\alpha\in C_0\}$ of $S$-names of integers for which $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cup(\{\alpha\}\times a_{g\upharpoonright\alpha^+}^*)$ is an open neighborhood of $\dot{Z}_\alpha$. There is a cub $C_2\subseteq C_1$ so that for each $\delta\in C_2$ and each $s\in S_\delta$, $s$ forces a value on $\dot{n}_\alpha$ for all $\alpha<\delta$. We may choose any $s_0\in g$ so that $s_0$ forces that $L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]\cap L_\delta[a_{g\upharpoonright\delta^+}\setminus\dot{n}_\delta]$ is empty whenever $\dot{f}(\alpha)\neq\dot{f}(\delta)$. Working in $V[g]$, we prove there is a stationary $E$ satisfying that $L_\delta[a_{g\upharpoonright\delta^+}]\cap\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$ is infinite, for all $\delta\in E$. If not, then there would be an assignment $\langle m_\delta:\delta\in C\rangle$ (for some cub $C$) so that $L_\delta[a_{g\upharpoonright\delta^+}\setminus m_\delta]$ would be disjoint from $\bigcup\{L_\alpha[a_{g\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha\in\delta\}$, for all $\delta\in C$. Pressing down, we would arrive at a contradiction.
Let $\dot{E}$ denote the $S$-name of the stationary set whose existence was shown in the previous paragraph. Choose any $s$ above $s_0$ and any $\delta\in C_2$ such that $s$ forces that $\delta\in\dot{E}$. Without loss of generality, the height of $s$ is $\geq\delta^+$, but note that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{n}_\alpha$, for all $\alpha<\delta$. This means that $s\!\!\upharpoonright\!\!\delta^+$ forces that $\delta\in \dot{E}$, since it will also decide the value of $L_\delta[a_{g\upharpoonright\delta^+}]$. We also have that $s\!\!\upharpoonright\!\!\delta$ forces a value on $\dot{f}\!\!\upharpoonright\!\!\delta$ and so we can choose a value $e\in\{0,1\}$ so that $s\!\!\upharpoonright\!\!\delta$ forces that $L_\delta[a_{s\upharpoonright\delta^+}]$ intersected with $\{L_\alpha[a_{s\upharpoonright\alpha^+}\setminus\dot{n}_\alpha]:\alpha<\delta\text{ and }\dot{f}(\alpha)=e\}$ is infinite. We now have a contradiction, since $s\!\!\upharpoonright\!\!\delta^+\cup\{(\delta^+,1-e)\}$ forces that the assigned neighborhood of $\delta$ must meet the assigned neighborhood of $\alpha$, for some $\alpha<\delta$ with $\dot{f}(\alpha)=e\neq\dot{f}(\delta)$.
Point-countable type
====================
There is another normal-implies-collectionwise-Hausdorff result holding in $L$ for which we don’t know whether it holds in our MM$(S)[S]$ model:
A space is of **point-countable type** if each point is a member of a compact subspace which has a countable outer neighbourhood base.
Spaces of point-countable type simultaneously generalize locally compact and first countable spaces, and V$=$L implies normal spaces of point-countable type are collectionwise Hausdorff [@W].
Does MM$(S)[S]$ imply normal spaces of point-countable type are $\aleph_1$-collectionwise Hausdorff?
The usual arguments would show that if so, in our front-loaded model of MM$(S)[S]$, normal spaces of point-countable type would be collectionwise Hausdorff.
**Acknowledgement.** We thank Peter Nyikos for catching errors in an earlier version of this manuscript.
[Alan Dow, Department of Mathematics and Statistics, University of North Carolina, Charlotte, North Carolina 28223]{}
[*e-mail address:*]{} [[email protected]]{}
[Franklin D. Tall, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, CANADA]{}
[*e-mail address:*]{} [[email protected]]{}
[^1]:
[^2]:
| ArXiv |
---
abstract: 'We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring, and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin-Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.'
address:
- |
Petter Andreas Bergh\
Institutt for matematiske fag\
NTNU\
N-7491 Trondheim\
Norway
- |
David A. Jorgensen\
Department of mathematics\
University of Texas at Arlington\
Arlington\
TX 76019\
USA
author:
- 'Petter Andreas Bergh & David A. Jorgensen'
title: 'Realizability and the Avrunin-Scott theorem for higher-order support varieties'
---
[^1]
Introduction {#sec:intro}
============
Support varieties for modules over commutative local complete intersections were introduced in [@Avramov] and [@AvramovBuchweitz], inspired by the cohomological varieties of modules over group algebras of finite groups. These geometric invariants encode several homological properties of the modules. For example, the dimension of the variety of a module equals its complexity. In particular, a module has finite projective dimension if and only if its support variety is trivial.
In this paper, we define higher-order support varieties for pairs of modules over complete intersections. These varieties are defined in terms of Grassmann varieties of subspaces of the canonical vector space associated to the defining regular sequence of the complete intersection. Thus, for a fixed dimension $d$, the support varieties of order $d$ are subsets of the Grassmann variety of $d$-dimensional subspaces of the canonical vector space, under a Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$. For $d=1$, we recover the classical support varieties: the varieties of order $1$ are precisely the projectivizations of the support varieties defined in [@AvramovBuchweitz].
We show that several of the results that hold for classical support varieties also hold for the higher-order varieties. Among these is the realizability result: we give a complete description of the closed subsets of the Grassmann variety that occur as higher-order support varieties. We also prove a higher-order Avrunin-Scott result for group algebras of finite elementary abelian groups. Namely, we extend the notion of $r$-rank varieties from [@CarlsonFriedlanderPevtsova] to higher-order rank varieties of pairs of modules and show that these varieties are isomorphic to the higher-order support varieties.
In Section 2 we give our definition of higher-order support varieties, and prove some of their elementary properties. In particular, we show that they are well-defined, independent of the choice of corresponding intermediate complete intersection, and are in fact closed subsets of the Grassmann variety. In Section 3 we discuss the realizability question, and in Section 4 we prove the higher-order Avrunin-Scott result.
Higher-order support varieties {#sec:hdsv}
==============================
In this section and the next, we fix a regular local ring $(Q, {\operatorname{\mathfrak{n}}\nolimits}, k)$ and an ideal $I$ generated by a regular sequence of length $c$ contained in ${\operatorname{\mathfrak{n}}\nolimits}^2$. We denote by $R$ the complete intersection ring $$R = Q/I,$$ and by $V$ the $k$-vector space $$V=I/{\operatorname{\mathfrak{n}}\nolimits}I.$$ For an element $f\in I$, we let $\overline f$ denote its image in $V$.
If the codimension of the complete intersection $R=Q/I$ is at least 2, then $V$ has dimension at least 2, and it makes sense to consider subspaces $W$ of $V$. Each such subspace has many corresponding complete intersections, in the following sense: if $W$ is a subspace of $V$, then choosing preimages in $I$ of a basis of $W$ we obtain another regular sequence [@BrunsHerzog Theorem 2.1.2(c,d)], and the ideal $J\subseteq I$ it generates. We thus get natural projections of complete intersections $Q\to Q/J\to R$. We call $Q/J$ a *complete intersection intermediate to $Q$ and $R$*, or when the context is clear, simply an *intermediate complete intersection*.
We now give our definition of higher-order support variety. We fix a basis of $V$, and let ${\operatorname{G}\nolimits}_d(V)$ denote the Grassmann variety of $d$th order subspaces of $V$ under the Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$ with respect to the chosen basis of $V$.
We set $$V_R^d(M,N)=\{p_W\in{\operatorname{G}\nolimits}_d(V)\mid {\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)\ne 0 \text{ for infinitely many $i$}\},$$ where $W$ is a $d$th order subspace of $V$, $p_W$ is the corresponding point in the Grassmann variety ${\operatorname{G}\nolimits}_d(V)$, and $Q/J$ is an intermediate complete intersection corresponding to $W$. We also define ${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d(M,k)$.
We note that ${\operatorname{V}\nolimits}_R^1(M,N)$ is the projectivization of the affine support variety ${\operatorname{V}\nolimits}_R(M,N)$ defined in [@AvramovBuchweitz].
There are two aspects of the definition which warrant further discussion.
1. \[independent\] The definition is independent of the chosen intermediate complete intersection $Q/J$ corresponding to $W$, and
2. \[closed\] ${\operatorname{V}\nolimits}_R^d(M,N)$ is a closed set in $G_d(V)$.
We next give proofs of these two statements.
Let $Q/J$ and $Q/J'$ be two complete intersections intermediate to $Q$ and $R$. The condition that $$(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$$ in $V$ defines an equivalence relation on the set of such intermediate complete intersections. The following result addresses (\[independent\]) above.
Suppose that $Q/J$ and $Q/J'$ are equivalent complete intersections intermediate to $Q$ and $R$, that is, $(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ in $V$. Then for all finitely generated $R$-modules $M$ and $N$ one has ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{Ext}\nolimits}_{Q/J'}^i(M,N)=0$ for all $i\gg 0$.
Let $W=(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ and consider the natural map of $k$-vector spaces $\varphi_{J}:J/{\operatorname{\mathfrak{n}}\nolimits}J\to W\subseteq V$ defined by $f+{\operatorname{\mathfrak{n}}\nolimits}J \mapsto f+{\operatorname{\mathfrak{n}}\nolimits}I$. This is an isomorphism: it is onto by construction, and one-to-one since $J\cap {\operatorname{\mathfrak{n}}\nolimits}I={\operatorname{\mathfrak{n}}\nolimits}J$. The condition that $(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ is equivalent to $\varphi_J(J/{\operatorname{\mathfrak{n}}\nolimits}J)=\varphi_{J'}(J'/{\operatorname{\mathfrak{n}}\nolimits}J')$. By [@BerghJorgensen Proposition 3.2], one has the equality $\varphi_J({\operatorname{V}\nolimits}_{Q/J}(M,N))=\varphi_{J'}({\operatorname{V}\nolimits}_{Q/J'}(M,N))$, where ${\operatorname{V}\nolimits}_{Q/J}(M,N)$ denotes the affine support variety of $M$ and $N$ over the complete intersection $Q/J$. By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5] one has that ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_{Q/J}(M,N)=\{0\}$. The same holds over $Q/J'$, and thus the result follows by the injectivity of $\varphi_{J}$.
Next, we address the second point in the remark.
\[proposition:closed\] For all finitely generated $R$-modules $M$ and $N$ one has that ${\operatorname{V}\nolimits}_R^d(M,N)$ is a closed set in $G_d(V)$.
This result follows from an incidence correspondence (see, for example, [@Harris Example 6.14]), as we now describe. Set $$\Gamma =\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \mid x\in W\cap{\operatorname{V}\nolimits}_R^1(M,N)\}.$$ Since $\Gamma$ is an incidence correspondence, it is a closed subset of the product space ${\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V)$. We have the two natural projections $$\xymatrixrowsep{2pc}
\xymatrixcolsep{0pc}
\xymatrix
{
& {\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \ar[dl]_\pi \ar[dr]^{\pi'}& \\
{\operatorname{G}\nolimits}_d(V) & & {\operatorname{G}\nolimits}_1(V)
}$$ Now by classical results from elimination theory (see, for example, [@Eisenbud Theorem 14.1]), the image of $\Gamma$ under $\pi$ is closed in ${\operatorname{G}\nolimits}_d(V)$. It suffices now to show that $\pi(\Gamma)={\operatorname{V}\nolimits}_R^d(M,N)$.
We have $p_W\in\pi(\Gamma)$ if and only if $x\in{\operatorname{V}\nolimits}_R^1(M,N)$ for some $x\in W$. This is equivalent to ${\operatorname{Ext}\nolimits}_{Q/(f)}^i(M,N)\ne 0$ for infinitely many $i$ and for some $f\in I$ with $\overline f=x\in W$. By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5], this condition is the same as ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)\ne 0$ for infinitely many $i$. By definition, this happens if and only if $p_W\in{\operatorname{V}\nolimits}_R^d(M,N)$.
\(1) Let ${\operatorname{\mathcal{T}}\nolimits}=\{(p_W,x)\in {\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \mid x\in W\}$. Then the map $\tau:{\operatorname{\mathcal{T}}\nolimits}\to G_d(V)$ given by $\tau(p_W,x)=p_W$ is the tautological bundle over the Grassmann variety ${\operatorname{G}\nolimits}_d(V)$. For $\Gamma$ as in the proof of Proposition \[proposition:closed\], we have $\Gamma\subseteq{\operatorname{\mathcal{T}}\nolimits}$, and $\tau(\Gamma)={\operatorname{V}\nolimits}_R^d(M,N)$. Thus ${\operatorname{V}\nolimits}_R^d(M,N)$ may be interpreted as the image under the tautological bundle of the fiber of ${\operatorname{V}\nolimits}_R^1(M,N)$ in ${\operatorname{\mathcal{T}}\nolimits}$.
\(2) In the definition of ${\operatorname{V}\nolimits}_R^d(M,N)$, a specific basis of $V$ was chosen. We remark that the definition is independent of the choice of basis, in the sense that if another basis of $V$ is chosen, then the two higher-order support varieties are isomorphic. Indeed, this is true for the first order affine varieties ${\operatorname{V}\nolimits}_R(M,N)$ by [@AvramovBuchweitz Remark 2.3]. It then follows that the same is true for the projectivizations ${\operatorname{V}\nolimits}_R^1(M,N)$, namely, there is an automorphism $\xi:{\operatorname{G}\nolimits}_1(V) \to {\operatorname{G}\nolimits}_1(V)$ such that if ${\operatorname{V}\nolimits}_R^1(M,N)$ is the support variety with respect to the first basis, and ${\operatorname{V}\nolimits}_R^1(M,N)'$ is the support variety with respect to the second, then $\xi({\operatorname{V}\nolimits}_R^1(M,N))={\operatorname{V}\nolimits}_R^1(M,N)'$. The general result for the higher-order support varieties follows from the incidence correspondence from the proof above.
We now give basic properties of higher-order support varieties, akin to those of the one-dimensional affine support varieties.
\[thm:props\] The following hold for finitely generated $R$-modules $M$ and $N$.
1. ${\operatorname{V}\nolimits}_R^d(k)={\operatorname{G}\nolimits}_d(V)$.
2. ${\operatorname{V}\nolimits}_R^d(M,N)={\operatorname{V}\nolimits}_R^d(N,M)$. For $d=1$, we moreover have ${\operatorname{V}\nolimits}_R^1(M,N)={\operatorname{V}\nolimits}_R^1(M)\cap{\operatorname{V}\nolimits}_R^1(N)$.
3. ${\operatorname{V}\nolimits}_R^d(M,M)={\operatorname{V}\nolimits}_R^d(k,M)={\operatorname{V}\nolimits}_R^d(M)$.
4. If $M'$ is a syzygy of $M$ and $N'$ is a syzygy of $N$, then ${\operatorname{V}\nolimits}_R^d(M,N)={\operatorname{V}\nolimits}_R^d(M',N')$. \[syzygy\]
5. If $0\to M_1\to M_2\to M_3\to 0$ and $0\to N_1\to N_2\to N_3\to 0$ are short exact sequences of finitely generated $R$-modules, then for $\{h,i,j\}=\{1,2,3\}$ there are inclusions $${\operatorname{V}\nolimits}_R^d(M_h,N)\subseteq{\operatorname{V}\nolimits}_R^d(M_i,N)\cup{\operatorname{V}\nolimits}_R^d(M_j,N);$$ $${\operatorname{V}\nolimits}_R^d(M,N_h)\subseteq{\operatorname{V}\nolimits}_R^d(M,N_i)\cup{\operatorname{V}\nolimits}_R^d(M,N_j).$$
6. If $M$ is Cohen-Macaulay of codimension $m$, then $${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d({\operatorname{Ext}\nolimits}_R^m(M,R)).$$ In particular, if $M$ is a maximal Cohen-Macaulay $R$-module, then ${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d({\operatorname{Hom}\nolimits}_R(M,R))$.
7. \[regseq\] If $x_1,\dots,x_d$ is an $M$-regular sequence, then $${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d(M/(x_1,\dots,x_d)M).$$
The proof of properties (1)–(7) for the affine one-dimensional support varieties ${\operatorname{V}\nolimits}_R(M,N)$ are given in [@AvramovBuchweitz] (see also [@BerghJorgensen].) Since ${\operatorname{V}\nolimits}_R^1(M,N)$ is simply the projectivization of ${\operatorname{V}\nolimits}_R(M,N)$, the same properties also hold for these varieties. Finally, properties (1)–(7) for $d>1$ follow from the $d=1$ case, as we now indicate.
For a subset $X$ of ${\operatorname{G}\nolimits}_1(V)$, we let $$\Gamma(X)=\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V)\mid x\in W\cap X\}.$$ The proofs make repeated use of the fact that ${\operatorname{V}\nolimits}_R^d(M,N)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))$, where $\pi$ is as in the proof of Proposition \[proposition:closed\]. For example, for (1) we have ${\operatorname{V}\nolimits}_R^d(k)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(k)))=\pi({\operatorname{G}\nolimits}_1(V))={\operatorname{G}\nolimits}_d(V)$.
For (2), we use the fact that ${\operatorname{V}\nolimits}_R^1(M,N)={\operatorname{V}\nolimits}_R^1(M)\cap{\operatorname{V}\nolimits}_R^1(N)={\operatorname{V}\nolimits}_R^1(N,M)$. Therefore ${\operatorname{V}\nolimits}_R^d(M,N)=\pi\left(\Gamma\left({\operatorname{V}\nolimits}_R^1(M,N)\right)\right)=\pi\left(\Gamma\left({\operatorname{V}\nolimits}_R^1(N,M)\right)\right)
={\operatorname{V}\nolimits}_R^d(N,M)$
To prove (3), we use the equalities ${\operatorname{V}\nolimits}_R^d(M,M)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,M)))=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(k,M)))={\operatorname{V}\nolimits}_R^d(k,M)$. The remaining equality and (4) are proved similarly.
To prove (5), we use the fact that for subsets $X$ and $Y$ of ${\operatorname{G}\nolimits}_1(V)$ one has $\Gamma(X\cup Y)=\Gamma(X)\cup\Gamma(Y)$. (We also use the fact that $\pi$ preserves unions, and both $\pi$ and $\Gamma$ preserve containment.) Therefore $$\begin{aligned}
{\operatorname{V}\nolimits}_R^d(M_h,N)&=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_h,N)))\\
&\subseteq\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_i,N)\cup{\operatorname{V}\nolimits}_R^1(M_j,N)))\\
&=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_i,N)))\cup\pi(\Gamma(V_R^1(M_j,N)))\\
&={\operatorname{V}\nolimits}_R^d(M_i,N)\cup{\operatorname{V}\nolimits}_R^d(M_j,N).\end{aligned}$$
The proofs of (6) and (7) are analogous to the proofs of [@AvramovBuchweitz Theorem 5.6(10)] and [@AvramovIyengar 7.4] (see also [@BerghJorgensen Theorem 2.2(7) and (8)].)
We can extend Proposition 2.4(1) of [@AvramovBuchweitz], to a sort of generalized Dade’s Lemma, in the projective context.
Fix $1\le d\le c$. Then ${\operatorname{Ext}\nolimits}_R^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_R^d(M,N)=\emptyset$.
By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5], ${\operatorname{Ext}\nolimits}_R^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_R^1(M,N)=\emptyset$. The latter holds if and only if $\Gamma=\Gamma({\operatorname{V}\nolimits}_R^1(M,N))=\emptyset$, which in turn holds if and only if ${\operatorname{V}\nolimits}_R^d(M,N)=\pi(\Gamma)=\emptyset$, where $\Gamma$ and $\pi$ are from the proof of Proposition \[proposition:closed\].
Realizability
=============
In this section we give a complete description of which closed subsets of ${\operatorname{G}\nolimits}_d(V)$ can possibly occur as the $d$th order support variety ${\operatorname{V}\nolimits}_R^d(M,N)$ of a pair of finitely generated $R$-modules $(M,N)$. The basis of the description is the following result in the first order case.
\[dimensionone\] Every closed subset of ${\operatorname{G}\nolimits}_1(V)$ is the support variety of some finitely generated $R$-module. Specifically, if $Z$ is a closed subset of ${\operatorname{G}\nolimits}_1(V)$, then there exists a finitely generated $R$-module $M$ such that $Z={\operatorname{V}\nolimits}_R^1(M,k)$.
This is well-known in the affine case, see, for example, [@Bergh]. Since every closed set in ${\operatorname{G}\nolimits}_1(V)$ is the projectivization of a cone in $V$, and ${\operatorname{V}\nolimits}_R^1(M,N)$ is the projectivization of ${\operatorname{V}\nolimits}_R(M,N)$, the result follows.
The framework of the proof of Proposition \[proposition:closed\] allows us to complete the description of realizable higher-order varieties. Recall that $\pi$ denotes the projection map ${\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V)\to{\operatorname{G}\nolimits}_d(V)$.
\[dimensiond\] For a closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$, set $$\Gamma(Z)=\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V) \mid x\in W\cap Z\}.$$ Let $Y$ be a closed subset of ${\operatorname{G}\nolimits}_d(V)$. Then $Y={\operatorname{V}\nolimits}_R^d(M,N)$ for a pair of finitely generated $R$-modules $(M,N)$ if and only if $Y=\pi(\Gamma(Z))$ for some closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$.
Suppose that $Y={\operatorname{V}\nolimits}_R^d(M,N)$ for a pair of finitely generated $R$-modules $(M,N)$. Then the proof of Proposition \[proposition:closed\] shows that $Y=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))$.
Conversely, suppose that $Y=\pi(\Gamma(Z))$ for some closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$. Then Theorem \[dimensionone\] shows that $Z={\operatorname{V}\nolimits}_R^1(M,N)$ for some pair of finitely generated $R$-modules $(M,N)$. Thus $Y=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))={\operatorname{V}\nolimits}_R^d(M,N)$, again from the proof of Proposition \[proposition:closed\].
Theorem \[dimensiond\] shows that, in contrast to first order support varieties, the realizability of varieties in ${\operatorname{G}\nolimits}_d(V)$ for $d>1$ as $d$th order support varieties of a pair of finitely generated $R$-modules is more restrictive. Indeed, consider a smallest nontrivial first order support variety ${\operatorname{V}\nolimits}_R^1(M,N)$, namely, one consisting of a single point $x$. Then ${\operatorname{V}\nolimits}_R^d(M,N)$ consists of all $d$-dimensional planes in $V$ containing $x$. Changing the basis of $V$ if necessary, we can assume that $x=(1,0,\dots,0)\in{\operatorname{G}\nolimits}_1(V)$. Then there is an obvious bijective correspondence between $d$-dimensional subspaces of $V$ containing $x$, and $(d-1)$-dimensional subspaces of a $(c-1)$-dimensional $k$-vector space. Thus $\dim{\operatorname{V}\nolimits}_R^d(M,N)=\dim {\operatorname{G}\nolimits}_{d-1}(k^{c-1})=(d-1)(c-d)$. In particular, we have $\dim{\operatorname{V}\nolimits}_R^{c-1}(M,N)=c-2$, which is of codimension one in ${\operatorname{G}\nolimits}_{c-1}(V)$, and this is when ${\operatorname{V}\nolimits}_R^1(M,N)$ is nontrivially as small as possible.
The following example illustrates the previous discussion.
Let $k$ be a field (of arbitrary characteristic), and $Q=k[[x_1,\dots,x_c]]$. Then $Q$ is a regular local ring with maximal ideal ${\operatorname{\mathfrak{n}}\nolimits}=(x_1,\dots,x_c)$. For $I=(x_1^2,\dots,x_c^2)$, the quotient ring $R=Q/I$ is a codimension $c$ complete intersection. Let $M=R/(x_1)$. Then it is not hard to show that relative to the basis $\overline {x_1^2},\dots,\overline {x_c^2}$ of $V=I/{\operatorname{\mathfrak{n}}\nolimits}I$, the one-dimensional support variety of $M$ is ${\operatorname{V}\nolimits}^1_R(M,k)=\{(1,0,\dots,0)\}$. Thus we have $\dim{\operatorname{V}\nolimits}_R^d(M,k)=(d-1)(c-d)$, for $1\le d\le c-1$.
Higher-order rank varieties and a higher-order Avrunin-Scott Theorem
====================================================================
In this final section we consider complete intersections of a special form, namely, those which arise as the group algebra $kE$ of a finite elementary abelian $p$-group $E$, where $k$ has characteristic $p$. In this case one has $$kE\cong k[x_1,\dots,x_c]/(x_1^p,\dots,x_c^p).$$ Note that by assigning $\deg x_i=1$ for $1\le i \le c$, the $k$-algebra $kE$ is standard-graded. We let $kE_1$ denote the degree-one component of $kE$; this is a $k$-vector space of dimension $c$. For any linear form $u$ of $kE_1$ one has $u^p=0$, and thus the subalgebra $k[u]$ of $kE$ generated by $u$ is isomorphic to $k[x]/(x^p)$ (for $x$ an indeterminate). Since $k[u]$ is a principal ideal ring, every finitely generated $k[u]$-module is a direct sum of a free module and a torsion module. Recall from [@Carlson] that the *rank variety* ${\operatorname{W}\nolimits}_E(M)$ of a $kE$-module $M$ is the set of those linear forms $u\in kE_1$ such that the torsion part of $M$ as a $k[u]$-module is nonzero. It was conjectured by Carlson [@Carlson] and subsequently proven by Avrunin and Scott [@AvruninScott] that the rank variety and the group cohomological support variety ${\operatorname{V}\nolimits}_{kE}(M)$ of a $kE$-module agree.
Recall that $I$ denotes the ideal $(x_1^p,\dots,x_c^p)$, and $V$ the $k$-vector space $I/{\operatorname{\mathfrak{n}}\nolimits}I$, where ${\operatorname{\mathfrak{n}}\nolimits}$ is the maximal ideal $(x_1,\dots,x_c)$. We now want to show that the classical Avrunin-Scott theorem mentioned above is a special case of a more general result involving the higher-order varieties. We generalize the definition of $d$th order rank varieties from [@CarlsonFriedlanderPevtsova] (which they call $d$-rank varieties) to $d$th order rank varieties ${\operatorname{W}\nolimits}^d_E(M,N)$ of pairs of modules $(M,N)$. Fix a basis of $kE_1$, and consider the Grassmann variety ${\operatorname{G}\nolimits}_d(kE_1)$ of $d$-dimensional subspaces of $kE_1$ under the Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$ with respect to the chosen basis.
We set $${\operatorname{W}\nolimits}^d_E(M,N)=\{p_W\in {\operatorname{G}\nolimits}_d(kE_1) \mid {\operatorname{Ext}\nolimits}^i_{k[W]}(M,N)\ne 0 \text{ for infinitely many $i$}\}$$ where ${\operatorname{G}\nolimits}_d(kE_1)$ is the Grassmann variety of $d$-dimensional subspaces of $kE_1$, $p_W$ is the point in ${\operatorname{G}\nolimits}_d(kE_1)$ corresponding to the $d$-dimensional subspace $W$, and $k[W]$ is the subalgebra of $kE$ generated by $W$.
Consider the Frobenius isomorphism $\Phi:k \to k$ given by $\Phi(a)=a^p$. We have a $\Phi$-equivariant isomorphism of $k$-vector spaces $$\alpha: kE_1 \to V$$ defined as follows. For $u\in kE_1$, we choose a preimage $\widetilde u$ in $Q$, and then we set $\alpha(u)=\widetilde u^p+{\operatorname{\mathfrak{n}}\nolimits}I\in V$. It is clear that $\alpha$ is a $\Phi$-equivariant homomorphism of $k$-vector spaces, which is defined independent of the choice of preimage. Since $k$ is algebraically closed, it contains $p$th roots, and therefore $\alpha$ is onto. Since $\dim kE_1=\dim V$, $\alpha$ is also one-to-one.
Taking as a basis for $V$ the image under $\alpha$ of the chosen basis of $kE_1$, we obtain an induced $\Phi$-equivariant isomorphism of Grassmann varieties $$\beta:{\operatorname{G}\nolimits}_d(kE_1) \to {\operatorname{G}\nolimits}_d(V)$$ with respect to these bases. Specifically, let $p_W$ be a point in ${\operatorname{G}\nolimits}_d(kE_1)$, and $W$ the associated $d$-dimensional subspace of $kE_1$. Let $\widetilde W^p$ denote the ideal of $Q$ generated by the $p$th powers of linear preimages in $Q$ of a basis of $W$. Then $\beta(p_W)$ is the point in ${\operatorname{G}\nolimits}_d(V)$ (with respect to the chosen basis of $V$) corresponding to the subspace $\widetilde W^p+{\operatorname{\mathfrak{n}}\nolimits}I/{\operatorname{\mathfrak{n}}\nolimits}I$.
\[ASthm\] Given finitely generated $kE$-modules $M$ and $N$, one has $$\beta({\operatorname{W}\nolimits}_E^d(M,N))={\operatorname{V}\nolimits}^d_{kE}(M,N).$$
The proof relies on the following lemma, which is a statement extracted from the proof of [@Avramov Theorem (7.5)]. For completeness we include the proof here.
For any non-zero linear form $u\in kE_1$ we choose a preimage $\widetilde u$ in $Q=k[x_1,\dots,x_c]$, which is also a linear form, and define a homomorphism from $\mu :k[u]\to Q/(\widetilde u^p)$ by sending $u$ to $\widetilde u+(\widetilde u^p)$. Note that $Q/(\widetilde u^p)$ is free when regarded as module over $k[u]$ via $\mu$. We have the commutative diagram of ring homomorphisms $$\xymatrixrowsep{2pc}
\xymatrixcolsep{1.9pc}
\xymatrix{
& Q/(\widetilde u^p) \ar[d]\\
k[u] \ar@{^{(}->}[r] \ar[ur]^\mu & kE
}$$ where the vertical map is the natural projection. In particular, the action of $k[u]$ on a $kE$-module $M$ factors through $\mu$.
Let $M$ be a finitely generated $kE$-module. Then $M$ has finite projective dimension over $k[u]$ if and only if it has finite projective dimension over $Q/(\widetilde u^p)$.
The proof follows part of that of [@Avramov Theorem (7.5)]. Suppose that $M$ has finite projective dimension over $Q/(\widetilde u^p)$. Since $Q/(\widetilde u^p)$ is free over $k[u]$ any free resolution of $M$ over $Q/(\widetilde u^p)$ is also one of $M$ over $k[u]$. Thus $M$ has a finite free resolution over $k[u]$.
Conversely, suppose $M$ is free as a $k[u]$-module. Let $F$ be a minimal free resolution of $M$ over $Q/(\widetilde u^p)$. Since $F$ is also a free resolution of $M$ over $k[u]$ and ${\operatorname{Tor}\nolimits}_i^{k[u]}(M,k)=0$ for all $i>0$, we see that $F\otimes_{k[u]}k$ is a minimal free resolution of $M \otimes_{k[u]}k$ over $Q/(\widetilde u^p)\otimes_{k[u]}k\cong Q/(\widetilde u)$. Since $Q/(\widetilde u)$ is regular and $F\otimes_{k[u]}k$ is a minimal, we must have that $F_c\otimes_{k[u]}k=0$, and this implies $F_c=0$. Thus $F$ is a finite free resolution, and so $M$ has finite projective dimension over $Q/(\widetilde u)$.
We now give a proof of Theorem \[ASthm\].
Suppose that $p_W\in{\operatorname{W}\nolimits}_E^d(M,N)$. Then by definition there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{k[W]}(M,N)$. Therefore, by Dade’s Lemma, there exist infinitely nonzero ${\operatorname{Ext}\nolimits}_{k[u]}^i(M,N)$ for some linear form $u\in W$. Thus both $M$ and $N$ have infinite projective dimension over $k[u]$. Therefore, by the lemma, both $M$ and $N$ have infinite projective dimension over $Q/(\widetilde u^p)$, and so it follows from [@AvramovBuchweitz Proposition 5.12] that there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{Q/(\widetilde u^p)}(M,N)$. This implies that there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{Q/(\widetilde W^p)}(M,N)$, where $\widetilde W^p$ represents the ideal generated by the $p$th powers of linear preimages in $Q$ of a basis of $W$. This gives $\beta(p_W)\in{\operatorname{V}\nolimits}^d_{kE}(M,N)$.
For the reverse containment we just retrace our steps, noting that any $f\in I$ is equivalent mod ${\operatorname{\mathfrak{n}}\nolimits}I$ to an element of the form $a_1x_1^p+\cdots+a_cx_c^p=(\sqrt[p]{a_1}x_1+\cdots+\sqrt[p]{a_c}x_c)^p$, $a_i\in k$, and hence it is clear how to employ the previous lemma.
[CFP]{} L.L. Avramov, *Modules of finite virtual projective dimension*, Invent. Math. 96 (1989), no. 1, 71–101. L.L. Avramov, R.-O. Buchweitz, *Support varieties and cohomology over complete intersections*, Invent. Math. 142 (2000), no. 2, 285–318. L.L. Avramov, S. B. Iyengar, *Constructing modules with prescribed cohomological support*, Illinois J. Math. 51 (2007), no. 1, 1–20. G.S. Avrunin, L.L. Scott *Quillen stratification for modules*, Invent. Math. 66 (1982), no. 2, 277–286. P.A. Bergh *On support varieties for modules over complete intersections*, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3795–3803. P.A. Bergh, D.A. Jorgensen *Support varieties over complete intersections made easy*, preprint. W. Bruns, J. Herzog *Cohen-Macaulay Rings*, Cambridge studies in advanced mathematics 39, Cambridge University Press, Cambridge, 1993. J.F. Carlson, *The varieties and the cohomology ring of a module*, J. Algebra 85 (1983), no. 1, 104–143. J.F. Carlson, E.M. Friedlander, J. Pevtsova *Representations of elementary abelian $p$-groups and bundles on Grassmannians*, Adv. Math. 229 (2012), no. 5, 2985–3051. D. Eisenbud, *Commutative algebra, with a view toward algebraic geometry*, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995. J. Harris, *Algebraic Geometry, a First Course* Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992.
[^1]: Part of this work was done while we were visiting the Mittag-Leffler Institute in February and March 2015. We would like to thank the organizers of the Representation Theory program.
| ArXiv |
---
abstract: |
We show that a strong P-Cygni feature seen in the far-UV spectra of some very hot (${\mbox{\,$T_{eff}$}}\gtrsim 85$ kK) central stars of planetary nebulae (CSPN), which has been previously identified as [$\lambda$]{}977, actually originates from [$\lambda$]{}973. Using stellar atmospheres models, we reproduce this feature seen in the spectra of two \[WR\]-PG 1159 type CSPN, Abell 78 and NGC 2371, and in one PG 1159 CSPN, K 1-16. In the latter case, our analysis suggests an enhanced neon abundance. Strong neon features in CSPN spectra are important because an overabundance of this element is indicative of processed material that has been dredged up to the surface from the inter-shell region in the “born-again” scenario, an explanation of hydrogen-deficient CSPN. Our modeling indicates the [$\lambda$]{}973 wind feature may be used to discern enhanced neon abundances for stars showing an unsaturated P-Cygni profile, such as some PG 1159 stars. We explore the potential of this strong feature as a wind diagnostic in stellar atmospheres analyses for evolved objects. For the \[WR\]-PG 1159 objects, the line is present as a P-Cygni line for ${\mbox{\,$T_{eff}$}}\gtrsim
85$ kK, and becomes strong for $100 \lesssim {\mbox{\,$T_{eff}$}}\lesssim 155$ kK when the neon abundance is solar, and can be significantly strong beyond this range for higher neon abundances. When unsaturated, [*i.e.*]{}, for very high and/or very low mass-loss rates, it is sensitive to ${\mbox{\,$\dot{M}$}}$ and very sensitive to the neon abundance. The classification is consistent with recent identification of this line seen in absorption in many PG 1159 spectra.
author:
- 'J.E. Herald, L. Bianchi'
- 'D.J. Hillier'
title: 'DISCOVERY OF NeVII IN THE WINDS OF HOT EVOLVED STARS[^1]'
---
INTRODUCTION {#sec:intro}
============
Central stars of planetary nebulae (CSPN) represent an evolutionary phase the majority of low/intermediate mass stars will experience. A small subset of CSPN have been termed “PG 1159-\[WR\]” stars, as they represent objects transitioning from the \[WR\] to the PG 1159 class. The former are objects moving along the constant-luminosity branch of the H-R diagram which have optical spectra rich in strong emission line features, similar to those of Wolf-Rayet (WR) stars, which represent a late evolutionary stage of massive stars (the “\[WR\]” designation is meant to distinguish the two). The majority of \[WR\] CSPN show prominent carbon features and are termed “\[WC\]”, while a handful show strong oxygen lines (“\[WO\]”). This difference is believed to reflect a difference in the ionization of the winds rather than in the elemental abundances [@crowther:02]. Unlike massive WR stars, CSPN of the nitrogen-rich \[WN\] subtype are very rare, the two candidates being LMC-N66 in the LMC [@pena:04] and PM5 in the Galaxy [@morgan:03]. The PG 1159 class marks the entry point onto the white-dwarf cooling sequence, and these stars display mainly absorption line profiles in the optical, as their stellar wind has almost all but faded. Both classes are examples of hydrogen-deficient CSPN, which presumably make up 10-20% of the CSPN population ([@demarco:02; @koesterke:98b] and references therein), and are believed to represent subsequent evolutionary stages based on their similar parameters and abundances. An explanation for the origin of such objects is the “born-again” scenario (see [@iben:95; @herwig:99] and references therein). In this scenario, helium shell flashes produce processed material between the H- and He-burning shells (the intershell region). This material is enriched in He from CNO hydrogen burning, but through 3$\alpha$ process burning it also becomes enriched in C, O, Ne, and deficient in Fe (see, [*e.g.*]{}, [@werner:04], and references therein). After the star initially moves off the asymptotic giant-branch (AGB), it experiences a late helium-shell flash, causing the star to enter a second (or “born-again”) AGB phase. Flash induced mixing dredges the processed intershell material to the surface, resulting in H-deficient surface abundances. When the star enters its second post-AGB phase, its spectrum can develop strong wind features causing it to resemble that of (massive) WR stars, perhaps because the chemically enriched surface material increases the efficiency of radiative momentum transfer to the wind. Eventually, as the wind fades, the object moves onto the white dwarf cooling sequence. As this happens, observable wind spectral features may only be present in the far-UV and UV regions.
@herald:04b (hereafter, HB04) modeled the far-UV and UV spectra of four Galactic CSPN, including Abell 78 (A78 hereafter), considered the proto-typical transition star and candidate for the born-again scenario. @crowther:98 classified it as a PG 1159-\[WO1\] star based on its high / ratio. HB04 also presented an analysis of NGC 2371, which the authors argue is of similar nature, although possessing a wind of even higher ionization. In those analyses, HB04 were unable to reproduce the prominent P-Cygni feature seen in the spectra of both stars at $\sim 975$ Å, identified in the spectra of A78 as [$\lambda$]{}977 in the past [@koesterke:98b]. Prominent [$\lambda$]{}977 P-Cygni profiles do occur in the far-UV spectra of CSPN of cooler temperatures (${\mbox{\,$T_{eff}$}}\lesssim 80$ kK, see, [*e.g.*]{}, [@herald:04a]), as well as in massive hot stars. However, effective temperatures of the transition objects (such as A78) are found to be very high ([*i.e.*]{}, $\gtrsim 90$ kK) from both stellar atmospheres codes ([*e.g.*]{}, [@werner:03], HB04) and nebular line analyses ([*e.g.*]{}, [@kaler:93; @grewing:90]). HB04 noted that although this feature has been assumed to be from , a strong transition from an ion of relatively low ionization potential (48 eV) in such highly-ionized winds was strongly questionable. HB04 investigated and excluded the possibility that the line originated from a highly ionized iron species.
@werner:04 reported the identification of a narrow absorption feature at [$\lambda$]{}973 in the spectra of several PG 1159 stars as . In some cases, this line is superimposed on a broad P-Cygni feature, which they identified as [$\lambda$]{}977. Given that the high ionization potential of (207 eV) is more consistent with the conditions expected in objects of such high temperatures, we were motivated to include neon in the model atmospheres of A78 and NGC 2371 presented in HB04. Neon had not been included in any previous modeling. We present the results of this analysis, and report that neon can adequately account for this hitherto unexplained wind feature. Additionally, we show that is also responsible for the broad P-Cygni feature seen in the spectra of the PG 1159 star K 1-16. We also investigate, with a grid of models, the usefulness of this line as a wind diagnostic for very hot CSPN. As the neon abundance is also of interest with respect to massive stars with winds, this work may have application to the study of the hottest Wolf-Rayet stars as well. This paper is arranged as follows: the observations are described in § \[sec:obs\]. The models are described in § \[sec:modeling\]. Our results are discussed in § \[sec:discussion\] and our conclusions in § \[sec:conclusions\].
OBSERVATIONS AND REDUCTION {#sec:obs}
==========================
The data sets utilized in this paper are summarized in Table \[tab:obs\]. For NGC 2371 and K 1-16, we have used far-UV data from *Far-Ultraviolet Spectroscopic Explorer* (FUSE), and for A78, from the *Berkeley Extreme and Far-UV Spectrometer* (BEFS). For NGC 2371, we have also made use of a UV *International Ultraviolet Explorer* (IUE) spectrum. The data characteristics, and the reduction of the FUSE data, are described in HB04. The data were acquired from the MAST archive.
For K 1-16, the FUSE data were reduced in a similar manner as described in HB04, except with the latest version of the FUSE pipeline (CALFUSE v2.4). The count-rate plots show that the star was apparently out of the aperture during part of the observation, and data taken during that period was omitted from the reduction process.
The radial velocities of NGC 2371 and A78 are $+20.6$ and $+17$ , respectively [@acker:92]. All observed spectra presented in this paper have been velocity-shifted to the rest-frame of the star based on these values.
The far-UV spectra of our sample are shown in Fig. \[fig:fuv\_all\], along with our models (described in § \[sec:modeling\]). They are mainly dominated by two strong P-Cygni features - [$\lambda\lambda$]{}1032,38 and [$\lambda$]{}973. Both features are saturated in the spectra of NGC 2371 and A78, while they are unsaturated in that of K 1-16. The numerous absorption lines seen are due to the Lyman and Werner bands of molecular hydrogen (), which resides in both the interstellar and circumstellar medium (discussed in HB04).
Close inspection of the P-Cygni profile reveals that there does appear to be some absorption due to [$\lambda$]{}977 in each case, as well as emission in the case of NGC 2371. This apparently arises from absorption by cooler carbon material in the circumstellar environment, perhaps similar to the “carbon curtain” @bianchi:87b invoked to explain the similar [$\lambda\lambda$]{}1334.5,1335.7 features seen in the spectra of NGC 40 (that CSPN has a temperature of ${\mbox{\,$T_{eff}$}}= 90$ kK, as estimated from the UV spectrum, too hot for to be present in the stellar atmosphere).
MODELING {#sec:modeling}
========
To analyze the spectra of our sample, we have computed non-LTE line-blanketed models which solve the radiative transfer equation in an extended, spherically-symmetric expanding atmosphere. The models are identical to those described in HB04, except neon is now included in the model atmospheres. The reader is referred to that work for a more detailed description of the models, here we give only a summary.
The intense radiation fields and low wind densities of CSPN invalidate the assumptions of local thermodynamic equilibrium, and their extended atmospheres necessitate a spherical geometry for solving the radiative transfer equation. To model these winds, we have used the CMFGEN code [@hillier:98; @hillier:99b; @hillier:03]. The detailed workings of the code are explained in the references therein. To summarize, the code solves for the non-LTE populations in the co-moving frame of reference. The fundamental photospheric/wind parameters include , , , the elemental abundances and the velocity law (including ). The *stellar radius* () is taken to be the inner boundary of the model atmosphere (corresponding to a Rosseland optical depth of $\sim20$). The temperature at different depths is determined by the *stellar temperature* , related to the luminosity and radius by $L = 4\pi{\mbox{\,$R_{*}$}}^2\sigma{\mbox{\,$T_{*}$}}^4$, whereas the *effective temperature* () is similarly defined but at a radius corresponding to a Rosseland optical depth of 2/3. The luminosity is conserved at all depths, so $L =
4\pi{\mbox{\,$R_{2/3}$}}^2\sigma{\mbox{\,$T_{eff}$}}^4$. We assume what is essentially a standard velocity law $v(r) = {\mbox{\,v$_{\infty}$}}(1-r_0/r)^\beta$ where $r_0$ is roughly equal to , and $\beta =1$.
For the model ions, CMFGEN utilizes the concept of “superlevels”, whereby levels of similar energies are grouped together and treated as a single level in the rate equations [@hillier:98]. Ions and the number of levels and superlevels included in the model calculations are listed in Table \[tab:ion\_tab\]. The atomic data references are given in HB04, except for neon (discussed in § \[sec:neon\]). The parameters of the models presented here are given in Table \[tab:mod\_param\_dist\].
Abundances {#sec:abund}
----------
Throughout this work, the nomenclature $X_i$ represents the mass fraction of element $i$, “” denotes the solar abundance, with the values for “solar” taken from @grevesse:98 (their solar abundance of neon is 1.74 by mass). As explained in HB04, an abundance pattern of ,,= 0.54, 0.36, 0.08 was adopted to model these hydrogen deficient objects. The nitrogen abundance was taken to be = 0.01, and solar values were adopted for the other elements, except for iron. HB04 and @werner:03 found a sub-solar iron abundance was required to match observations of A78, and our models of that star have ${\mbox{\,$X_{Fe}$}}= 0.03{\mbox{\,$\rm{X_{\odot}}$}}$.
Neon {#sec:neon}
----
The prominent far-UV feature arises from the $2p^1P^o - 2p^2
\:^1D$ transition. As discussed by @werner:04, there is some uncertainty in the corresponding wavelength. We adopt 973.33 Å, the value found in the Chianti database [@young:03] and which was was measured by @lindeberg:72. The corresponding lower and upper level energies are 214952.0 and 317692.0 $\rm{cm^{-1}}$, respectively.
The neon atomic data was primarily taken from the Opacity Project [@seaton:87; @opacity:95; @opacity:97] and the Atomic Spectra Database at the NIST Physical Laboratory. For , energy level data have been taken from NIST with the exception of the $2p^2 \:^1D$ level, for which we have used the value from the Chianti database. Individual sources of atomic data (photo-ionization and cross-sections) include the following: @luo:89a (), @tully:90 (), and @peach:88 ().
RESULTS {#sec:discussion}
=======
The goal of this work was to test whether the inclusion of neon in the models of HB04 could account for the strong P-Cygni feature appearing at $\sim975$ Å in the spectra of two transition stars (NGC 2371 and A78), which previously has lacked a plausible explanation (HB04). The previous common identification with [$\lambda$]{}977 was questioned by HB04, as the presence of this ion would imply a much lower , inconsistent with other spectral diagnostics. @koesterke:98b speculated that for A78, neglected iron lines might sufficiently cool the outer layers of the (otherwise hot) wind to allow for the formation of . However, HB04 computed models which included highly ionized iron, and excluded this explanation for the observed feature. As we discuss below, we find that this P-Cygni line originates from in both stars. We also show that this is the case for a PG 1159 star, K 1-16. Additionally, we explore the usefulness of this feature as a diagnostic for stellar parameters, which is very important given the scarcity of diagnostic lines at high effective temperatures (discussed by HB04).
NGC 2371 & A78 {#sec:transition}
--------------
We initially re-calculated the NGC 2371 and A78 best-fit models of HB04 (Table \[tab:mod\_param\_dist\]) including neon at solar abundance. The resulting models reproduced the [$\lambda$]{}973 P-Cygni feature at a strength comparable to the observations, (although a bit weak in both cases), showing that is indeed responsible for this line. We also computed models with higher neon abundances, which is a predicted consequence of the “born-again” scenario (see § \[sec:diag\]). In Fig. \[fig:fuv\_all\], we show the ${\mbox{\,$X_{Ne}$}}=
10{\mbox{\,$\rm{X_{\odot}}$}}$ models, which are nearly indistinguishable from the solar abundance models (see § \[sec:diag\]). We have applied the effects of and absorption to the model spectra as described in HB04. The feature is weaker in the observations of A78 than in those of NGC 2371, and its blue P-Cygni edge is more severely affected by absorption from , making assessments of the quality of its model fit more uncertain. We note here that the parameters derived by HB04 were determined from a variety of diagnostic lines, and the listed uncertainties take into account all the different adjustments needed to fit them all, not just the ones shown here. In addition, the inclusion of neon in the calculation does not change the ionization significantly for other abundant ions in the wind (as can be seen in Fig. \[fig:fuv\_all\], where the HB04 models are also plotted), thus there was no need for a revision of the stellar parameters.
K 1-16 {#sec:k116}
------
Based on the similar parameters ( and abundances) of NGC 2371 and A78 to those of PG 1159 stars discussed by @werner:04, we suspected that the broad P-Cygni profile identified therein as [$\lambda$]{}977 in the spectra of a few of their objects originated from as well. @werner:04 do classify a line from their (static) models, but only attribute this identification to a narrow absorption feature, while identifying the broad P-Cygni wind feature as .
We decided to test this hypothesis for the case of the PG 1159 star K 1-16. @koesterke:98b determined the following parameters for K 1-16 from a hydrostatic analysis: ${\mbox{\,$T_{*}$}}= 140$ kK, $\log{L/{\mbox{\,$\rm{L_{\odot}}$}}}
= 3.6$ (which imply ${\mbox{\,$R_{*}$}}= 0.11$ ), ${\mbox{\,$\log{g}$}}=6.1$, , , = 0.38, 0.56, 0.06, and the following parameters from a wind-line analysis: ${\mbox{\,v$_{\infty}$}}= 4000$ and $\log{{\mbox{\,$\dot{M}$}}} = -8.1$ from the resonance lines. Although the mass-loss rate is lower, the other parameters are close to those NGC 2371, so we first took the parameters of our NGC 2371 model shown in Fig. \[fig:fuv\_all\], and scaled them to K 1-16’s radius of ${\mbox{\,$R_{*}$}}= 0.11$ as determined by @koesterke:98b. Further scaling of the model flux is needed to match the observed flux levels of K 1-16, and this scaling is equivalent to the star lying at a distance of 2.05 kpc. The only distance estimates to this star are statistical (based on nebular relations), ranging from 1.0 to 2.5 kpc ([@cahn:92; @maciel:84], respectively). The FUSE spectrum of K 1-16 (Fig. \[fig:fuv\_all\]) shows a unsaturated P-Cygni profile of comparable strength to the feature (which is also unsaturated), in contrast to that of NGC 2371, where the profiles are saturated and that of is stronger than that of . We therefore decreased the mass-loss rate of the scaled model with solar neon abundance until the feature was fit adequately (the resulting parameters of the best-fit model for K 1-16 are listed in Table \[tab:mod\_param\_dist\]). The resulting model’s feature is unsaturated and is weak compared to the observations, as shown in Fig. \[fig:k116\_XNe\]. We also computed an enriched model with ${\mbox{\,$X_{Ne}$}}=10$ (also shown). As the figure illustrates, the feature (unsaturated in this case) is now very sensitive to the abundance (unlike the cases of NGC 2371 and A78), and the profile of the Ne-enriched model is now too strong. Fig. \[fig:fuv\_all\] shows both the and lines of the Ne-enriched K 1-16 model.
Our models undoubtedly show that , not , accounts for the broad P-Cygni profile seen in the PG 1159 stars as well. Furthermore, our models demonstrate how this feature can be used to detect a supersolar neon abundance in the case of an unsaturated [$\lambda$]{}973 profile. Although these results suggest a supersolar abundance for K 1-16, a more complete photospheric/wind-line analysis should be performed, given that other parameters influence the strength of this line (§ \[sec:diag\]).
To account for the interstellar absorption, we have used the parameters utilized by @kruk:98 ($\log{N({\mbox{\rm{\ion{H}{1}}}})} =
20.48$ cm$^{-2}$, $b=20$ ). We modeled the absorption using $\log{N({\mbox{\rm{H}$_2$}})} = 16.0$ cm$^{-2}$, which produces fits adequate for our purpose (Figs. \[fig:fuv\_all\] and \[fig:k116\_XNe\]). We have assumed and gas temperatures of 80 K, and used the same methods described in HB04 to calculate the absorption profiles. We found a slightly higher reddening value than that of @kruk:98 to produce better results (${\mbox{\,$E_{\rm{B-V}}$}}= 0.025$ vs. 0.02 mag). We note that the far-UV spectrum of K 1-16 shows very little absorption from compared to other CSPN ([*e.g.*]{}, [@herald:02; @herald:04a; @herald:04b]), presenting a very “clean” example of a far-UV CSPN spectrum.
[$\lambda$]{}973 as a DIAGNOSTIC {#sec:diag}
---------------------------------
To investigate the potential of this feature as a diagnostic of stellar parameters, we computed exploratory models varying either the neon abundance, the mass-loss rate or the temperature of the A78 and NGC 2371 models (while keeping the other parameters the same) to study the sensitivity of this line to each parameter.
Evolutionary calculations of stars experiencing the “born-again” scenario ([*e.g.*]{}, see [@herwig:01]) predict a neon abundance of $\sim2$% by mass in the intershell region, produced via $\:^{14}\rm{N}(\alpha,\gamma)\:^{18}\rm{F}(\rm{e}^{+}\nu)\:^{18}\rm{O}(\alpha,\gamma)\:^{22}\rm{Ne}$. This material later gets “dredged up” to the surface, resulting in a surface abundance enhancement of up to 20 times the solar value. We have thus calculated models with super-solar neon abundances (with ${\mbox{\,$X_{Ne}$}}= 10$ , and 50 ) to gauge the effects on the [$\lambda$]{}973 feature (shown in Fig. \[fig:diag\]). As expected, because the feature is nearly saturated in both cases, the line shows only a weak dependence on . The enriched models do result in a better fit than the solar abundance models. However, given the sensitivity of this line to ${\mbox{\,$T_{eff}$}}$ (discussed below) and the uncertainty in this parameter (see HB04), we cannot make a definitive statement about the neon abundance of these transition objects based solely on this wind line. On the other hand, the strength of the line depends dramatically on the neon abundance in parameter regimes where it is not saturated, [*e.g.*]{}, for very high or very low mass-loss rates, as discussed below and shown in Figs. \[fig:k116\_XNe\]-\[fig:k116\_mdot\].
To test the sensitivity of the [$\lambda$]{}973 feature to , we have computed a range of models varying the mass-loss rates of our models (with solar neon abundance) while keeping the other parameters fixed for each. We find virtually no change while the [$\lambda$]{}973 profile remains saturated, until the change in induces a significant change in the ionization of the wind. This is shown for the model parameters of K 1-16 in Fig. \[fig:k116\_mdot\], where the profile is essentially unchanged for $5{\mbox{\,$\rm x 10^{-8}$}} <
{\mbox{\,$\dot{M}$}}< 1{\mbox{\,$\rm x 10^{-7}$}}$ , and then weakens dramatically as lowered to $1{\mbox{\,$\rm x 10^{-8}$}}$ . However, if the atmosphere is Ne-enriched, this limit could be significantly lower, as illustrated in the $5{\mbox{\,$\rm x 10^{-9}$}}$ (${\mbox{\,$T_{eff}$}}=135$ kK) case.
The ionization structures of neon for the A78 and NGC 2371 models (with ${\mbox{\,$X_{Ne}$}}=1{\mbox{\,$\rm{X_{\odot}}$}}$) are shown in Fig. \[fig:ion\]. In the cooler A78 model, is only dominant deep in the wind, with being dominant in the outer layers. We have explored the temperature sensitivity of the feature by adjusting the luminosity of our default models while keeping the other parameters ([*i.e.*]{}, and ) the same. The results are shown in Fig. \[fig:diag\]. For the A78 model parameters, the [$\lambda$]{}973 wind feature weakens as the temperature is decreased, becoming insignificant for ${\mbox{\,$T_{eff}$}}\lesssim
85$ kK. For the NGC 2371 model parameters, it weakens significantly as the temperature is lowered from $\simeq 130$ kK to $\simeq 110$ kK. The profile is fairly constant for $130 \lesssim {\mbox{\,$T_{eff}$}}\lesssim
145$ kK, and then starts to weaken as the temperature is increased, as ceases to be dominant in the outer wind (around ${\mbox{\,$T_{eff}$}}\simeq
150$ kK), and becomes a pure absorption line for ${\mbox{\,$T_{eff}$}}\gtrsim
170$ kK. However, these thresholds are dependent on the neon abundance, as illustrated by the ${\mbox{\,$T_{eff}$}}= 165$ kK models. For that temperature, the solar neon abundance model shows only a weak, unsaturated P-Cygni profile, but in the ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$ model, the line increases dramatically, becoming saturated again. This shows the potential of this line to exist in strength over a wide range of effective temperatures, as well as being an Ne-abundance diagnostic. The presence of as a P-Cygni profile sets a lower limit to ($\sim 80$ kK), quite independent of the neon abundance.
Although our modeling indicates the [$\lambda$]{}973 feature may not be useful in diagnosing a super-solar neon abundance in the case of NGC 2371 (because it is saturated) and A78 (because its blue edge is obscured by absorption), it does show other neon transitions in the far-UV and UV which do not produce significant spectral features for a solar neon abundance, but do for enriched neon abundances (see examples in Fig. \[fig:neon\_lines\]). The strongest examples are the [$\lambda$]{}2213.13 and [$\lambda$]{}2229.05 transitions from the $3d \:^2D - 3p \:^2P^o$ triplet which become evident in models of both objects when ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$. For this abundance, $3p \:^2P^o - 3s \:^2S$ transitions are also seen at [$\lambda$]{}2042.38 and [$\lambda$]{}2055.94 in the A78 models, and the $3s^1S
- 3p^1 P^o$ transition ([$\lambda$]{}3643.6) in the models of NGC 2371 (this line has been used by [@werner:94] to deduce enhanced neon abundances in a few PG 1159 stars, including K 1-16). Also seen in the ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$ models of NGC 2371 is the $3p^3P^o -
3d \:^3D$ multiplet, which @werner:04 observed at positions shifted about 6 Å blueward of the wavelengths listed in the NIST database (the strongest observed component occurs at [$\lambda$]{}3894). For ${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$, in the model of NGC 2371, $2p^2 \:^4P - 2p^2
\:^2P^o$ features are seen from 993 Å to 1011 Å, as well as blend of - transitions at $\sim2300$ Å. Although the resolution and/or quality of the available IUE data in this range are not sufficient to rigorously analyze these lines, we note that the observations seem to favor a higher neon abundance, based on a significantly strong feature at 2230 Å in the IUE observations of NGC 2371 that is only matched by the ${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$ model.
We note that introducing neon at solar abundance in the model atmospheres of these objects does not have a significant impact on the ionization structure of other relevant ions in the wind. For A78, at ${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$, the ionization structures of other elements changes slightly, but not enough to result in spectral differences. Very high neon abundances (${\mbox{\,$X_{Ne}$}}= 50{\mbox{\,$\rm{X_{\odot}}$}}$) result in a less ionized wind, with, for example, the [$\lambda$]{}1371 feature strengthening as this ion becomes more dominant in the outer parts of the wind. The neon ionization structure is most dramatically affected - with becoming the dominant ion in the outer parts of the wind. For NGC 2371, introducing neon at solar abundance reduces the ionization of the wind slightly, with the effect becoming more significant at ${\mbox{\,$X_{Ne}$}}\ge 10{\mbox{\,$\rm{X_{\odot}}$}}$ when it leads to stronger and features at UV wavelengths. Thus, high neon abundances can significantly influence the atmospheric structure, and fitting a spectrum using the same set of non-neon diagnostics with a model with an enriched neon abundance generally seems to require a higher luminosity.
CONCLUSIONS {#sec:conclusions}
===========
We have shown that the strong P-Cygni wind feature seen around 975 Å (hitherto unidentified or mistakenly identified as [$\lambda$]{}977) in the far-UV spectra of very hot (${\mbox{\,$T_{eff}$}}\gtrsim
100$ kK) CSPN can be reproduced by models which include neon in the stellar atmosphere calculations. We have demonstrated this identification in the case of A78, a transitional \[WO\]-PG 1159 star, and in a similar object with winds of even higher ionization, NGC 2371. Through a comparison of our models with the far-UV spectrum of the PG 1159-type CSPN K 1-16, we have also demonstrated that the broad wind feature seen at this wavelength in some PG 1159 objects originates not from (as indicated in [@werner:04]), but from as well.
Our grid of models show that [$\lambda$]{}973.33 is a very strong wind feature detectable at solar abundance levels, in contrast to photospheric optical neon features [@werner:04], in CSPN of high stellar temperatures (${\mbox{\,$T_{eff}$}}\gtrsim 85$ kK). For the parameters of NGC 2371 ($\log{{\mbox{\,$\dot{M}$}}} = -7.1$ ) and A78 ($\log{{\mbox{\,$\dot{M}$}}} =
-7.3$ ), the strength of the feature peaks for $130 \lesssim
{\mbox{\,$T_{eff}$}}\lesssim 145$ kK, and weakens dramatically for ${\mbox{\,$T_{eff}$}}\gtrsim
160$ (these cutoffs depend on the value of and the neon abundance). For an enhanced neon abundance (${\mbox{\,$X_{Ne}$}}= 10{\mbox{\,$\rm{X_{\odot}}$}}$), the feature remains strong even in models of very high temperatures (${\mbox{\,$T_{eff}$}}\gtrsim 165$ kK) or very low mass-loss rates (${\mbox{\,$\dot{M}$}}\simeq
1{\mbox{\,$\rm x 10^{-8}$}}$ ), while the lower limits remains approximately the same. We note here that the far-UV spectra of these objects show the feature being weaker that the line, while in some PG 1159 stars ([*e.g.*]{}, K 1-16 and Longmore 4), they are of comparable strength. Since PG 1159 stars represent a more advanced evolutionary stage when the star is getting hotter and the wind is fading, [$\lambda$]{}973 may be the last wind feature to disappear if the atmosphere is enriched in neon.
In hydrogen-deficient objects, an enhanced neon abundance lends credence to evolutionary models which have the star experiencing a late helium shell flash, and predict a neon enrichment of about 20 times the solar value. When saturated ([*e.g.*]{}, in the case in NGC 2371), the feature is insensitive to , and only weakly sensitive to the neon abundance. Although models of these objects with enriched neon abundances do result in better fits for our transition objects, the sensitivity of the feature to ${\mbox{\,$T_{eff}$}}$ prevents us making a quantitative statement regarding abundances based on this feature alone. Other far-UV/UV lines from (at 2042, 2056, 2213, and 2229 Å) and (at 3644 Å) which only appear in Ne-enriched models, could in principle be used for this purpose, but we lack observations in this range of sufficient quality/resolution to make a quantitative assessment. In the case of K 1-16, [$\lambda$]{}973 is unsaturated, and our models require an enhanced neon abundance to fit it simultaneously with the [$\lambda\lambda$]{}1032,38 profile. This result is in line with those of @werner:94, who derived a neon abundance of 20 times the solar value for this object from analysis of the [$\lambda$]{}3644 line. The neon overabundance is further evidence that this PG 1159 object has experienced the “born-again” scenario.
[$\lambda$]{}973 has diagnostic applications not only to late post-AGB objects, but also for evolved massive stars. Evolutionary models predict the surface neon abundance to vary dramatically as Wolf-Rayet stars evolve (see, [*e.g.*]{}, [@meynet:05]). For cooler WR stars (such as those of the WN-type), the neon abundance can be estimated from low-ionization features in the infrared and ultraviolet ([*e.g.*]{}, \[\] 12.8$\mu$m, \[\] 15.5 $\mu$m, [$\lambda$]{}2553). The [$\lambda$]{}973 feature may provide a strong neon diagnostic for hotter, more evolved WR stars. For example, the WO star Sanduleak 2 has ${\mbox{\,$T_{*}$}}\simeq 150$ kK [@crowther:00], and appears to have a feature at that wavelength in a FUSE archive spectrum. Neon enhancements produced in massive stars may explain the discrepancy in the $\:^{22}$Ne/$\:^{20}$Ne ratio between the solar system and Galactic cosmic ray sources (see, [*e.g.*]{}, [@meynet:01]).
We are grateful to the anonymous referee for a careful reading of the manuscript and their constructive comments. We are indebted to the members of the Opacity Project and Iron Project and to Bob Kurucz for their continuing efforts to compute accurate atomic data, without which this project would not have been feasible. The SIMBAD database was used for literature searches. This work has been funded by NASA grants NAG 5-9219 (NRA-99-01-LTSA-029) and NAG-13679. The BEFS and IUE data were obtained from the Multimission Archive (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
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[^1]: Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer and data from the MAST archive. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985.
| ArXiv |
---
abstract: 'Solar flares involve the sudden release of magnetic energy in the solar corona. Accelerated nonthermal electrons have often been invoked as the primary means for transporting the bulk of the released energy to the lower solar atmosphere. However, significant challenges remain for this scenario, especially in accounting for the large number of accelerated electrons inferred from observations. Propagating magnetohydrodynamics (MHD) waves, particularly those with subsecond/second-scale periods, have been proposed as an alternative means for transporting the released flare energy, 1[likely alongside the electron beams]{}, while observational evidence remains elusive. Here we report a possible detection of such waves in the 1[late]{} impulsive phase of a two-ribbon flare. This is based on ultrahigh cadence dynamic imaging spectroscopic observations of a peculiar type of decimetric radio bursts obtained by the Karl G. Jansky Very Large Array. Radio imaging at each time and frequency pixel allows us to trace the spatiotemporal motion of the source, which agrees with the implications of the frequency drift pattern in the dynamic spectrum. The radio source1[, propagating at 1000–2000 km s$^{-1}$ in projection, shows close spatial and temporal association with transient brightenings on the flare ribbon]{}. In addition, multitudes of subsecond-period oscillations are present in the radio emission. We interpret the observed radio bursts as 1[short-period]{} MHD wave packets propagating along newly reconnected magnetic flux tubes linking to the flare ribbon. The estimated energy flux carried by the waves is comparable to that needed to account for the plasma heating 1[during the late impulsive phase of this flare]{}.'
author:
- 'Sijie Yu ()'
- 'Bin Chen ()'
bibliography:
- 'Yu2018.bib'
title: 'Possible Detection of Subsecond-period Propagating Magnetohydrodynamics Waves in Post-reconnection Magnetic Loops during a Two-ribbon Solar Flare'
---
Introduction {#sec:intro}
============
An outstanding question in solar flare studies is how a large amount of magnetic energy released in a flare (up to 10$^{33}$ erg) is converted into other forms of energy in accelerated particles, heated plasma, waves/turbulence, and bulk motions, and transported throughout the flare region. The collisional thick-target model (CTTM; @1971SoPh...18..489B), along with the framework of the standard CSHKP flare scenario [@1964NASSP..50..451C; @1966Natur.211..695S; @1974SoPh...34..323H; @1976SoPh...50...85K], assumes that 1[a considerable fraction]{} of the magnetic energy released via reconnection goes into acceleration of charged electrons and ions to nonthermal energies in the solar corona 1[[@2004JGRA..10910104E; @2005JGRA..11011103E; @2012ApJ...759...71E]]{}. The downward-propagating electrons along the reconnected, close field lines slam into the dense chromosphere and lose most of their energy through Coulomb collisions. This sudden energy loss results in the 1[intense]{} heating of the chromospheric material within a confined region at the footpoints of the closed arcades, driving hot and dense material upward and filling the arcades — a process known as “chromospheric evaporation.” The arcades, in turn, accumulate a large emission measure at high temperatures, thereby appearing particularly bright in extreme ultraviolet (EUV) and soft X-ray (SXR) wavelengths (see, e.g., a recent review by @2017LRSP...14....2B).
The CTTM model has been successful in accounting for a variety of flare phenomena, most notably the “Neupert effect”: The high-energy, hard X-ray (HXR) emission tends to coincide temporally with the rate of the rising lower-energy, SXR emission during the primary phase of energy release (also known as the “impulsive phase”) of a flare . Other outstanding examples include the decreasing height and area of HXR footpoint sources with increasing energy. However, significant challenges remain for the CTTM model (see, e.g., and references therein). One challenge is the so-called “number problem”: the total number of nonthermal electrons required to account for the observed HXR, (E)UV, or white light (WL) footpoint sources or flare ribbons can be very large compared to that available in the corona [e.g., @2007ApJ...656.1187F; @2011ApJ...739...96K]. Similar implications have been argued based on observations of coronal HXR sources — the inferred number density of nonthermal electrons is a large fraction of, or in some cases, nearly equal to, the total electron density available in the corona [@2007ApJ...669L..49K; @2008ApJ...678L..63K; @2010ApJ...714.1108K]. 1[This requires electrons to replenished the corona at the same rate as nonthermal electrons precipitate from it, otherwise the coronal acceleration region would be quickly evacuated. A scenario that invokes return currents, which involve electrons flowing up from the chromosphere into the corona to neutralize the depletion of the coronal electrons, has been suggested to alleviate the difficulty ]{}. 1[Nevertheless, these considerations have led various authors to suggest alternative scenarios that invoke electron (re)acceleration in the lower, denser solar atmosphere .]{} Other mechanisms have also been proposed for heating the chromospheric plasma, such as thermal conduction or magnetohydrodynamics (MHD) waves . In all cases, alternative means, 1[possibly operating alongside accelerated electrons]{} as in the CTTM model, are postulated to transport a sizable portion of the released flare energy from the reconnection region, presumably located in the corona, downward to spatially confined regions in the lower solar atmosphere.
One excellent way to provide such focused energy transport other than electron beams is via propagating plasma waves within reconnected flare arcades . A variety of plasma waves, including Alfvén waves and fast-mode and slow-mode magnetosonic waves, can arise as a natural consequence of the flare energy being released in an impulsive fashion (see, e.g., recent studies by @2017ApJ...847....1T and @2018ApJ...860..138P). As argued by @2008ApJ...675.1645F and @2013ApJ...765...81R, plasma waves are capable of carrying a significant amount of flare energy, which may be comparable to that needed to power the radiative emissions of a flare. 1[An intriguing recent numerical study by @2016ApJ...818L..20R demonstrated that the waves can drive chromospheric evaporation in a strikingly similar fashion to the way electron beams do. Their results were then confirmed by @2016ApJ...827..101K, who further showed that the detailed shapes of certain chromospheric lines could be used as a potential observational test to distinguish between the wave- and electron-beam heating scenarios.]{}
Observationally, flare-associated quasi-periodic pulsations (QPPs) with different periods ranging from $<$1 s to tens of minutes have been detected at virtually all wavelengths. One of the main origins for the QPPs is thought to be MHD oscillations or waves (see, e.g., @2009SSRv..149..119N for a review). Observational evidence for large-scale wave-like phenomena associated with flares has also frequently been reported using spatially-resolved imaging data (see reviews by e.g., @2012SoPh..281..187P [@2014SoPh..289.3233L; @2015LRSP...12....3W; @2017SoPh..292....7L], a study of a large sample of such events in @2013ApJ...776...58N, and a most recent observation of the 2017 September 10 X8.2 flare in @2018ApJ...864L..24L). Observational evidence that links the response in the lower solar atmosphere to downward-propagating MHD waves, however, is rather rare. One outstanding example was from @2016NatCo...713104L, who found a sudden sunspot rotation during the impulsive phase of a flare based on observations from the Goode Solar Telescope of the Big Bear Solar Observatory (GST/BBSO), possibly triggered by downward-propagating waves generated by the release of flare energy. Another interesting study by @2015ApJ...810....4B reported long-period ($\sim$140 s), slow ($\sim$20 km s$^{-1}$) oscillating flare ribbons based on observations by the *Interface Region Imaging Spectrograph*, although the authors interpreted the oscillating phenomenon in terms of instabilities in the reconnection current sheet rather than MHD waves. It is worthwhile to point out that, in the Earth’s magnetosphere, direct evidence for Alfvén waves propagating along the outer boundary of the “plasma sheet” (which is analogous to newly reconnected flare loops) has been reported based on *in situ* measurements. These waves have been argued to be responsible in transporting a significant amount of energy flux (in the form of Poynting flux) from the site of energy release in the magnetotail toward the Earth, which, in turn, powers the auroral emission that is analogous to flare ribbons on the Sun [@2000JGR...10518675W; @2002JGRA..107.1201W; @2000GeoRL..27.3169K].
Recently, numerical and analytical models have been developed to investigate 1[energy transport and deposition]{} from the corona to the low solar atmosphere by MHD waves [@2013ApJ...765...81R; @2016ApJ...827..101K; @2016ApJ...818L..20R; @2017ApJ...847....1T; @2018ApJ...853..101R]. An important finding is that short-period MHD waves, especially those having periods of about one second or less, carry a significant amount of energy [@2017ApJ...847....1T], suffer much less energy loss when propagating out from the corona to the lower solar atmosphere [@2013ApJ...765...81R; @2018ApJ...860..138P], and are much more efficient in dissipating the energy in the upper chromosphere than their long-period counterparts [@2008ApJ...675.1645F; @2013ApJ...765...81R; @2016ApJ...818L..20R]. Therefore, these short-period MHD waves are thought to be a potential candidate for an alternative carrier for energy released in flares. Subsecond-period ($P<1$ s) QPPs have frequently been reported in radio and X-ray light curves and/or dynamic spectra (e.g., ). 1[However, most of the large-scale wave-like phenomena detected on the basis of imaging data fall into the long-period regime ($>$10 s, e.g., @2009SSRv..149..119N), with some rare exceptions from eclipse observations [e.g., @2002SoPh..207..241P].]{} This is mainly due to the limitation on temporal cadence of current WL/EUV imaging instrumentation, or the lack of radio/X-ray imaging capability at high temporal cadence with sufficient dynamic range or counting statistics.
Here we report ultrahigh cadence (0.05 s) spectroscopic imaging of a peculiar type of radio bursts in the decimetric wavelength range (“[dm-$\lambda$]{}” hereafter) that is likely associated with subsecond-period MHD waves propagating along flaring arcades. The bursts were recorded by the Karl G. Jansky Very Large Array (VLA) in a *GOES*-class C7.2 flare that is associated with a failed filament eruption and large-scale coronal EUV waves. We further show that these MHD waves may carry a significant amount of energy flux that is comparable to the average energy flux needed for driving the plasma heating at the flare ribbons. In Section \[sec-obs\], we present VLA dynamic imaging spectroscopic observations of the radio bursts, supported by complementary magnetic, EUV, and X-ray data. In Section \[sec-discussion\], we interpret the observations within a physical scenario that involves propagating short-period MHD wave packets and discuss their energetics. We briefly summarize our findings in Section \[sec-conclusion\].
Observations {#sec-obs}
============
Event overview {#sec-overview}
--------------
The VLA is a general-purpose radio interferometer operating at $<$1–50 GHz. It has completed a major upgrade [@2011ApJ...739L...1P] and was partially commissioned for solar observation in late 2011 [@2013ApJ...763L..21C]. It is capable of making broadband radio imaging spectroscopic observations in more than one thousand spectral channels with ultrahigh time resolution of tens of millisecond-scale. Recent studies with the VLA have demonstrated its unique power in using coherent solar radio bursts to diagnose the production and transport of energetic electrons in solar flares by utilizing its imaging capabilities with spectrometer-like time and spectral resolution [@2013ApJ...763L..21C; @2014ApJ...794..149C; @2015Sci...350.1238C; @2018ApJ...866...62C; @2017ApJ...848...77W].
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The event under investigation occurred on 2014 November 11 in NOAA active region (AR) $12201$, located at $44\degr$ east from the central meridian. It is a *GOES*-class C7.2 solar flare (flare identifier “SOL2014-11-01T16:39:00L085C095” following the IAU convention suggested by @2010SoPh..263....1L). This event was well observed by the Atmospheric Imaging Assembly (AIA; @2012SoPh..275...17L) and the Helioseismic and Magnetic Imager (HMI; @2012SoPh..275..207S) aboard the *Solar Dynamics Observatory* (*SDO*). The impulsive phase of the flare started from $\sim$16:39 UT and was partially covered by *RHESSI* [@2002SoPh..210....3L] until 16:42 UT, when the spacecraft entered the South Atlantic Anomaly (SAA). The VLA was used to observe the Sun from 16:30:10 UT to 20:40:09 UT and captured the entire flare. The observations were made in frequency bands between 1 and 2 GHz with 50 ms cadence and 2 MHz spectral resolution in dual circular polarizations. The 27-antenna array was in the C configuration (maximum baseline length 3 km), yielding an intrinsic angular resolution of $35''.7\times 16''.3$ at $\nu=1$ GHz at the time of the observation (and this scales linearly with $1/\nu$). The deconvolved synthesis images are restored with a $30''$ circular beam.
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Figures \[fig-overview\] and \[fig-euv\] show an overview of the time history and general context of the flare event. The *GOES* 1–8 Å SXR flux starts to rise at 16:39 UT and peaks at around 16:46 UT, during which time a filament is seen to erupt (green arrows in Figure \[fig-euv\]) but it does not fully detach from the surface and forms a coronal mass ejection—a phenomenon known as a “failed eruption.” During this period, both the HXR light curve (blue curve in Figure \[fig-overview\](D)) and the SXR derivative (red curve in Figure \[fig-overview\](D)) display multiple bursty features, which is characteristic of the flare’s impulsive phase, during which the primary energy release occurs. Precipitating nonthermal electrons lose most of their energy in the dense chromosphere, resulting in HXR sources at the footpoints of the reconnected flare arcades via bremsstrahlung radiation (contours in Figure \[fig-euv\](B)). Bright flare ribbons, visible in UV/EUV passbands (shown in Figure \[fig-euv\] in purple, which is mostly contributed by AIA 304 Å), are formed due to heating of the chromospheric/photospheric material by precipitated nonthermal electrons or by other means. The evaporated chromospheric material fills the flare arcades and forms bright coronal loops, best seen in EUV passbands that are sensitive to relatively high coronal temperatures (green and blue colors in Figure \[fig-euv\], which show AIA 211 and 94 Å bands that correspond to plasma temperatures of 2 MK and 7 MK, respectively). Many of the impulsive peaks in the SXR derivative have counterparts in the light curves from the Radio Solar Telescope Network (RSTN) (Figure \[fig-overview\](B) and (C)), which are also visible in the VLA 1–2 GHz dynamic spectrum as short-duration radio bursts (Figure \[fig-overview\](A)), suggesting that they are both closely associated with accelerated nonthermal electrons. The [dm-$\lambda$]{} bursts have complex fine spectrotemporal structures, especially in the lower-frequency portion of the radio dynamic spectrum.
The radio bursts under study appear during the late impulsive phase (shaded area in Figure \[fig-overview\](A–C) demarcated with vertical dashed lines). Two main episodes can be distinguished in the dynamic spectrum, each of which lasts for $\sim$10–20 seconds (referred to as “Burst 1” and “Burst 2” hereafter). An enlarged view of these bursts is available in Figures \[fig-ribbonbrightenings\](A) and \[fig-radio-spec-imaging\](A). From the imaging data, the bursts have a peak brightness temperature $T_B$ of $\sim1.1 \times 10^7$ K. 1[The total flux density is $\sim1$ sfu (solar flux unit; 1 sfu $= 10^4$ Jy).]{} In addition, the bursts are nearly 100% polarized with left-hand circular polarization (LCP). These properties are consistent with radio emission associated with a coherent radiation mechanism. In the dynamic spectrum, the bursts appear as arch-shaped emission lanes, which display a low-high-low frequency drift pattern. The frequency drift rate $d\nu/dt$ is between 60 and 200 MHz/s (or a relative drift rate of $\dot{\nu}/\nu\approx0.04$–0.2), which is about one order of magnitude lower than type III radio bursts emitted by beams of fast electrons, but similar to fiber bursts and lace bursts in the same frequency range . Such bursts with an intermediate frequency drift rate are sometimes referred to as the “intermediate drift bursts” . The multiple episodes of positive- and negative-drifting features resemble to some extent the “lace bursts” in the literature . However, the emission lanes of these bursts appear to be much smoother, while the lace bursts, at least from the few cases reported in the literature, have a much more fragmentary and chaotic appearance.
Radio imaging of the bursts places the burst source (red contours in Figures \[fig-euv\](E) and (F)) near the northern flare ribbon. The location of the radio bursts is also very close to the *RHESSI* 12–25 keV HXR footpoint source, shown in Figure \[fig-euv\](B) as white contours, albeit the latter is obtained several minutes earlier (at 16:40 UT) before the spacecraft enters the SAA. 1[A more detailed investigation reveals a close temporal and spatial association between the radio bursts and the transient (E)UV brightenings at the northern flare ribbon. Figure \[fig-ribbonbrightenings\](B) shows an AIA 304 Å background-detrended image sequence during the time interval of the radio dynamic spectrum shown in Figure \[fig-ribbonbrightenings\](A). During this period, the northern ribbon features the appearance of two transient EUV brightenings during radio Bursts 1 and 2, and the location of the brightenings is very close to the radio source (red).]{} The appearance of the radio source during the flare’s impulsive phase, as well as its close spatial and temporal association with the ribbon brightenings, suggests that the radio source is intimately related to the release and transport of the flare energy. More detailed discussions of the spectral, temporal, and spatial features of the bursts based on radio dynamic imaging spectroscopy will be presented in the next subsection.
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Another interesting feature of this event is that it is accompanied by large-scale, 1[fast-propagating]{} disturbances 1[(“PDs” hereafter), observed in EUV,]{} during the impulsive and gradual phase of the flare; they are usually interpreted as propagating MHD waves in the corona [@2012SoPh..281..187P; @2013ApJ...776...58N; @2014SoPh..289.3233L; @2015LRSP...12....3W; @2017SoPh..292....7L; @2018ApJ...864L..24L]. Using AIA 171, 193, and 211 Å running-ratio images (ratio of current frame to the previous frame), a large-scale 1[PD feature]{} (denoted as 1[“PD1”]{} in Figure \[fig-LSWave\]) is present in the area between AR 12201 and AR 12200. In addition, another large-scale 1[PD]{} appears to move outward above the limb (denoted 1[“PD2”]{} in Figure \[fig-LSWave\]). The temporal evolution of the two 1[PDs]{} is displayed in the time-distance plots in Figure \[fig-LSWave\](G) and (H), made along two slices labeled “S1” and “S2” in Figure \[fig-LSWave\](A), respectively. The initialization of the large-scale 1[PDs]{} coincides with the onset of the flare, demonstrating their close association with the flare energy release. The large-scale 1[PDs]{} propagate at a speed of 400–500 km s$^{-1}$, with 1[PD1]{} clearly experiencing multiple deflections by magnetic structures of the ARs. We note that the radio bursts are observed during the period when 1[PD1]{} remains in the flaring region (Figure \[fig-LSWave\](C)). This is a strong indication of the presence of ubiquitous MHD disturbances in and around the flaring region during the time of the radio bursts.
Radio Dynamic Spectroscopic Imaging {#sec-spec-imaging}
-----------------------------------
The capability of simultaneous imaging and dynamic spectroscopy offered by the VLA allows each pixel in the dynamic spectrum to form a radio image. As an example, Figure \[fig-radio-spec-imaging\](B) shows a three-dimensional (3D) rendering of a VLA spectral image cube taken for Burst 2 within a 100 ms integration (at 16:46:18.2 UT; the timing is shown as the vertical dotted line in panel (A)). The two horizontal slices in Figure \[fig-radio-spec-imaging\] (B) indicate the radio images at the peak frequencies of the two emission lanes at that time (circles in panel (A)). The same two radio images are shown in Figures \[fig-radio-spec-imaging\](C) and (D) as green and blue contours overlaid on the AIA EUV 304 Å image and the HMI photospheric line-of-sight (LOS) magnetogram respectively. As discussed in the previous subsection, the radio sources are located near the northern flare ribbon. In the magnetogram, this flare ribbon corresponds to a region with a positive magnetic polarity. As the bursts are 100% LCP, they are likely polarized in the sense of $o$ mode.
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We produce an independent 3D spectral image cube for each time pixel when the radio burst of interest is present in the radio dynamic spectrum, thereby creating a four-dimensional (4D) spectrotemporal image cube. From the 4D cube we are able to derive the spectrotemporal variation intrinsic to this radio source of interest by isolating its flux from all other sources present on the solar disk in the spatial domain, resulting in a spatial resolved, or “vector” radio dynamic spectrum of the source (Figure \[fig-radio-spec-imaging\](E) and (F)). This technique was first introduced by @2015Sci...350.1238C based on VLA dynamic spectroscopic imaging data, and was subsequently applied in a number of recent studies with VLA data [@2017ApJ...848...77W; @2018ApJ...866...62C]. A similar approach is discussed in a recent study by @2017SoPh..292..168M based on data from the Murchison Widefield Array. The resulting vector dynamic spectra show clearer features of the radio bursts than the cross-power dynamic spectra obtained at short baselines (which are a proxy for the total-power dynamic spectra; Figure \[fig-radio-spec-imaging\](A)). The improvement, however, is not substantial, which is consistent with the imaging results in which this burst source is shown as the dominant emission on the solar disk. To highlight the fine structure of the bursts, we further enhance the vector dynamic spectrum by using the contrast-limited adaptive histogram equalization technique [@Pizer1987], shown in Figures \[fig-radio-spec-imaging\](G) and (H).
Bursts 1 and 2 share similar spectrotemporal features. They contain at least one emission lane that starts with a positive drift rate toward higher frequency ($d\nu/dt>0$, sometimes referred to in the literature as “reverse drift” because “normal drift” bursts show negative frequency drifts). It then turns over rather smoothly at the highest frequency point and drifts toward lower frequency with a negative frequency drift rate ($d\nu/dt<0$). The total frequency variation $\Delta \nu_{\rm tot}/\nu$ can be up to 30%. Burst 2 undergoes two repeated cycles of positive-to-negative frequency drift. At least three distinct emission lanes are clearly visible (denoted as “L1”, “L2”, and “L3” in Figure \[fig-radio-spec-imaging\](F)) with two additional faint lanes that can only been distinguished in the enhanced dynamic spectrum (arrows in Figure \[fig-radio-spec-imaging\](H)). Although the three bright emission lanes of Burst 2 occur close together in time, they differ in their intensity, peak emission frequency, frequency drift rate, and frequency turnover time $t_{\rm o}$ (defined as the time when the emission frequency reaches the highest value and the frequency drift rate $\dot{\nu}$ goes to nearly zero; it is indicated by orange arrows in Figure \[fig-radio-spec-imaging\](F) and (H)). The average instantaneous frequency bandwidth $\Delta \nu$ of the emission lanes is about 60–100 MHz, corresponding to a relative frequency bandwidth $\Delta \nu/\nu\approx 6\%$.
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More detailed inspection of the dynamic spectral features of the stronger burst (Burst 2; based on the full 50 ms cadence data) reveals multitudes of very short, subsecond-scale fine structures on each emission lane (Figure \[fig-wavelet\](A)). Figures \[fig-wavelet\](B–E) provide an enlarged view of four segments of the emission lanes for Burst 2 (labelled as “S1”, “S2”, “S3”, “S4” in Figure \[fig-wavelet\](A)), which have been detrended to remove their overall frequency drift pattern. The bursts appear to oscillate quasi-periodically in their emission frequency around the central “ridge” of the emission lane. 1[We use a damped oscillation profile $$\delta\nu(t) = \delta\nu_0\exp(-t/\tau_A)\sin\bigg[\frac{2\pi t}{P/(1-t/\tau_P)^3}\bigg]$$ to fit the four segments (Figure \[fig-wavelet\](B–E)). The oscillations have an amplitude of $\delta\nu_0\approx10$–$30\ \rm{MHz}$ (or a relative amplitude of $\delta\nu/\nu$ of $\sim$1–2$\%$), period of $P\approx0.3$–1.0 s, and damping times of $\tau_A\approx0.5$–5 s in amplitude and $\tau_P \gtrsim$ 30 s in period.]{} Wavelet analysis of such oscillation patterns in emission frequency confirms that the oscillations display very short, subsecond-scale periods ranging from $\sim$0.3–1.0 s (Figure \[fig-wavelet\](F–I)).
Radio imaging of each pixel in the dynamic spectrum where the bursts are found provides key information on the spatial variation of the radio source as a function of time and frequency. For each image at a given frequency $\nu$ and time $t$, we fit the source with a 2D Gaussian function and determine the source centroid $I_{\rm pk}(\theta, \phi, \nu, t$), where $I_{\rm pk}$ is the peak intensity, and $\theta$ and $\phi$ are the centroid position in helioprojective longitude and latitude. As shown in several previous studies, the uncertainty of the centroid location for unresolved, point-like sources (which is likely the case for the coherent bursts under study) is determined by $\sigma\approx\theta_{\rm FWHM}/\mathrm{S/N}\sqrt{8\ln 2}$, where $\theta_{\rm FWHM}$ is the FWHM beamwidth and S/N is the ratio of the peak flux to the root-mean-square noise of the image [@1988ApJ...330..809R; @1997PASP..109..166C; @2018ApJ...866...62C]. In our data, typical values are $\theta_{\rm FWHM}\approx 30''$ and $\mathrm{S/N}\gtrsim20$, which give $\sigma\lesssim0.6''$. However, as discussed later in Section \[sec-discussion\], the bursts are likely associated with fundamental plasma radiation, which is known to be prone to scattering effects as the radiation propagates through the inhomogeneous corona toward the observer [@1994ApJ...426..774B; @2017NatCo...8.1515K; @2018ApJ...856...73C; @2018SoPh..293..132M]. Therefore, the estimate of uncertainty given above should only be considered as a lower limit. In fact, by obtaining the centroid locations of all frequency-time pixels on the emission lane within a small time period ($\sim$0.5 s) and frequency range ($\sim$50 MHz), we find that they are distributed rather randomly within an area of a FWHM size of $\sim 2''\times2''$. Hence we estimate the actual position uncertainty of the centroids as $\sigma\approx1''$.
We focus on Burst 2 for detailed investigations of the spatial, temporal, and spectral variation of the source centroid since it has the best S/N. For each emission lane, we first extract all time and frequency pixels where the intensity exceeds 50% of its peak intensity. An example for such a selection for lane L1 of Burst 2 is shown in Figure \[fig-L1-3d\](A) enclosed by the white contour. Figure \[fig-L1-3d\](B) shows the resulting centroid positions as a function of frequency (colored dots from blue to red in increasing frequency) for emission lane L1. To further improve positional accuracy and reduce cluttering in the figure, each dot in the plot represents the average position for centroids at all frequency pixels across the emission lane (that have an intensity above 50% of the peak) for a given time $t$, with the color representing their mean frequency. The background of Figure \[fig-L1-3d\](B) is the HMI photospheric magnetogram shown in grayscale, overlaid with the AIA 1600 Å image. The latter clearly shows the double flare ribbons in red.
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Figure \[fig-L1\_time\](A) shows the same distribution of radio centroids derived from emission lane L1 as in Figure \[fig-L1-3d\](B), but colored in time instead. It displays an evident motion in projection: the radio source first moves toward the flare ribbon as frequency increases (blue to red color in Figure \[fig-L1-3d\](B) and blue to white color in Figure \[fig-L1\_time\](A)) until it reaches the maximum frequency at the lowest height, and then bounces back in the opposite direction away from the ribbon as frequency decreases (red to blue color in Figure \[fig-L1-3d\](B) and white to red color in Figure \[fig-L1\_time\](A)). The average speed in projection is $\sim$1000–2000 km s$^{-1}$, which is typical for propagating Alfvén or fast-mode magnetosonic waves in the low corona [e.g., @2013ApJ...776...58N; @2018ApJ...864L..24L].This is a strong indication that the radio emission is associated with a propagating Alfvén or fast-mode MHD disturbance in a magnetic tube in the close vicinity of the flare ribbon. As discussed in Section \[sec-overview\], the presence of ubiquitous MHD disturbances in the flaring region is strongly implicated by the observation of large-scale, fast 1[PDs]{} observed by *SDO*/AIA at about the same time.
Motion of Radio source motion in 3D {#sec-3d}
-----------------------------------
In order to place the location of the radio centroids into the physical context of the flare, we perform potential field extrapolation based on the *SDO*/HMI LOS photospheric data right after the flare peak at 17:00 UT [@2014SoPh..289.3549B; @2014SoPh..289.3483H] to derive the coronal magnetic field. Selected magnetic field lines from the extrapolation results are shown in Figure \[fig-L1-3d\](B) for regions around the location of the radio burst centroids and the postflare arcades. It is shown that the spatial distribution of the positions of the radio centroids at different frequencies tends to follow the magnetic field lines (yellow) rooted around the northern flare ribbon, with its higher-frequency end located closer to the ribbon. This is consistent with the expectation for plasma radiation, in which emission occurs at a higher emission frequency in regions with higher plasma density, which are typically located at lower coronal heights ($\nu\approx s\nu_{pe}\approx 8980s\sqrt{n_e}$ Hz, where $s$ is the harmonic number, $\nu_{\rm pe}$ is the electron plasma frequency, and $n_e$ is the local electron density in cm$^{-3}$).
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Since the emission is highly polarized, it is reasonable to assume fundamental plasma radiation as the emission mechanism responsible (i.e., harmonic number $s$=1 and $\nu \approx \nu_{\rm pe}$). In this case, the emission frequency $\nu$ of the radio source centroid $I_{\rm pk}(\theta, \phi, \nu, t$) can be directly translated into the plasma density of the source $n_e$. By further assuming a coronal density model $n_e(h)$ where $h$ is the coronal height, we can thus map the measured centroid locations in 2D projection to three dimensional (3D) locations in the corona, i.e., from $I_{\rm pk}(\theta, \phi, \nu, t$) to $I_{\rm pk}(\theta, \phi, h, t$). A similar practice has been used in , and more recently @2017ApJ...848...77W, for deriving 3D trajectories of [dm-$\lambda$]{} fiber bursts in the corona. Here we adopt a barometric density model with an exponential form $$n_\mathrm{e}(h)=n_{e0}\,\mathrm{exp}\left(-\frac{h-h_0}{L_n}\right)
\label{equ-density-model}$$ where $h$ is the height above the solar surface, $L_n$ is the density scale height, and $n_{e0}$ is the density at a reference height $h_0$. Such a density model describes the variation in density for an isothermal, plane-parallel atmosphere under hydrostatic equilibrium (e.g., @2005psci.book.....A), and has been widely used in the literature as a zero-order approximation for estimating the coronal heights of various solar coherent radio bursts . For simplicity, we fix the parameters $n_{e0}$ and $h_0$ to be $\sim 3\times 10^{10}\ \mathrm{cm^{-3}}$ and $\sim$2000 km at the top of the chromosphere according to the VAL model [@1981ApJS...45..635V], and investigate the effect of different choices of $L_n$ on the resulting 3D distribution of the radio source centroids.
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Figures \[fig-L1-3d\](C) and (D) demonstrate the inferred 3D distributions of the radio source centroids with different choices of $L_n$ from 10 Mm to 70 Mm, viewed from the eastern and northern sides of the AR respectively. Each set of 3D centroid positions at a given $L_n$ is shown as dots of the same color (from red to blue in increasing $L_n$). It is obvious from the figure that the choice of a greater value of $L_n$ yields a more stretched distribution of the radio centroids in height, and vice versa. Such a proportionality between the vertical extent $h_{\rm tot}$ of the radio sources and $L_n$ is straightforward to find if we combine the barometric density model (Eq. \[equ-density-model\]) with the frequency–density relation for plasma radiation $\nu \propto \sqrt{n_e}$, which gives $h_{\rm tot} \approx 2L_n\Delta\nu_{\rm tot}/\nu$, where $\Delta\nu_{\rm tot}$ is the total frequency width of the radio burst determined from the dynamic spectrum. More importantly, different choices of $L_n$ affect how the radio source centroids are distributed with regard to the extrapolated magnetic field lines in 3D. For small $L_n$ values, the centroids tend to be distributed across the field lines within a small range of vertical heights, while for $L_n$ values in the intermediate range ($\sim$35–50 Mm), the spatial extent of the centroids tends to agree with the direction of the extrapolated field lines. As discussed earlier, the temporal evolution of a radio source (1000–2000 km s$^{-1}$ in projection) is consistent with a physical motion of the emission source at Alfvénic or fast-mode magnetosonic speed. Because the radio source appears to be closely associated with the flare ribbon both spatially and temporally (see Section \[sec-overview\]), we assume that the radio source moves along (or within a small angle with respect to) the reconnected magnetic loops that link to the flare ribbon. In this case, the corresponding $L_n$ values fall into the $\sim$35–50 Mm range. For subsequent discussions, we will adopt $L_n=40$ Mm, with the understanding that this parameter is not very well determined due to the inherent limitations of magnetic field extrapolation the uncertainty on the exact direction of propagation of the radio source in 3D, and it may vary from burst to burst.
Figure \[fig-L1\_time\](B)–(D) shows the inferred 3D spatial and temporal evolution of the radio centroids of emission lane L1 after adopting the coronal density model with $L_n=40$ Mm, viewing from the east side of the AR. It is clear that the radio source first moves downward along a converging magnetic field tube (panels B and C) and then bounces backward after it reaches the lowest altitude (or highest frequency). We also extend the same analysis to emission lanes L2 and L3 of Burst 2. The results show a similar spatiotemporal evolution of the radio source centroids as lane L1 (Figures \[fig-L2\] and \[fig-L3\]). 1[We caution that the absolute height of the radio source as well as the point of reflection, however, depends strongly on the selection of parameters in the coronal density and magnetic field model adopted here]{}, which may very well be different for radio bursts propagating along different flare loops. Therefore, the 3D source evolution shown in Figures \[fig-L1\_time\]–\[fig-L3\] should only be considered as a qualitative representation.
DISCUSSIONS {#sec-discussion}
===========
We briefly summarize the observational results in the previous section as follows.
1. The radio bursts of interest appeared during the late impulsive phase of a C7.2 two-ribbon solar flare that was associated with a failed filament eruption, when large-scale, fast-propagating EUV disturbances were observed throughout the flare region.
2. The location of the radio source coincides with the northern flare ribbon and HXR footpoints. 1[In addition, the radio source appears to show close spatial and temporal association with transient (E)UV brightenings on the ribbon.]{}
3. The bursts have a high brightness temperature of $>10^7$ K and are completely polarized in the sense of $o$ mode.
4. The bursts consist of multiple emission lanes that exhibit a low–high–low frequency drift pattern in the radio dynamic spectrum with a moderate relative frequency drift rate of $\dot{\nu}/\nu \lesssim 0.2 \rm{s}^{-1}$, which is typical for intermediate drift bursts in the decimetric wavelength range.
5. Imaging at all time and frequency pixels where the bursts are present shows that the radio source propagates at a speed of 1–2 Mm s$^{-1}$ in projection. The low–high–low frequency drift pattern corresponds to the source firstly moving downward along the flaring loop before it reaching the lowest point and bouncing back upward.
6. Some of the emission lanes consist of multitudes of subsecond-period oscillations in emission frequency with an amplitude of 1[$\delta\nu/\nu\approx$1–2$\%$]{}.
What is the nature of the propagating radio source that is reflected at or near the flare ribbon? First, it is most likely associated with fundamental plasma radiation, which is due to the nonlinear conversion from plasma Langmuir waves induced by the presence of nonthermal electrons. This is because that the bursts have a narrow frequency bandwidth ($\delta \nu/\nu \approx 6\%$) and fast temporally varying features, and are nearly 100% polarized. Second, the propagation speed of the emission source (1–2 Mm s$^{-1}$ in projection) is too slow for electron beams emitting type III bursts (which usually propagate at 0.1$c$–0.5$c$, see, e.g., @2013ApJ...763L..21C [@2018ApJ...866...62C; @2017ApJ...851..151M]), but 1[likely]{} too fast for slow-mode magnetosonic waves, 1[unless the temperature in the source reaches over $\sim$50 MK]{}. The most probable candidate for the radio-emission-carrying disturbance is thus Alfvénic or fast-mode magnetosonic waves, which propagate at $\sim$1–4 Mm s$^{-1}$ under typical coronal conditions. The Alfvénic or fast-mode waves can be excited by a broadband driver, such as the impulsive flare energy release, and propagate outward from the site of energy release. For fast-mode waves to achieve focused, field-aligned energy transport, an overdense magnetic tube would be required to act as a waveguide , which, in our case, can be the freshly reconnected flaring loops that connect to the flare ribbons. The observed reflection of the waves at or near the flare ribbon may be due to sharp gradients at and/or below the transition region . However, this is less clear from our observations regarding the physical connection between the nonthermal electrons (responsible for the production of Langmuir waves) and the MHD waves: the energetic electrons could be accelerated locally within the waves by a variety of means, including acceleration by parallel electric field, turbulence, or a first-order Fermi process with the wavefront acting as a moving mirror (e.g., @2008ApJ...675.1645F), or they could originate from an acceleration site elsewhere (e.g., at the reconnection site or the top of the flare loops) but be trapped with the propagating MHD waves.
It is particularly intriguing that some of the emission lanes show fast, subsecond-scale quasi-periodic oscillations in the emission frequency with an amplitude of $\delta\nu/\nu\approx$1[1–2$\%$]{}. Under the plasma radiation scenario, $\delta\nu/\nu$ can be directly translated into small density perturbations of $\delta n_e/n_e\approx 2\delta\nu/\nu\approx $1[2–4$\%$]{}. 1[If these small-amplitude oscillations in frequency can be interpreted as weak density perturbations associated with the propagating waves, the scenario of fast-mode magnetosonic mode scenario would be more probable, because pure Alfven modes are incompressible.]{} We note that such small density disturbances are hardly detectable by current EUV or SXR imaging instrumentation, mainly because the resulting small fluctuation level in the EUV/SXR intensity $\delta I/I\lesssim 2\delta n_e/n_e\approx 4\%$ is very difficult to detect against the background. In addition, the subsecond periodicity of the density perturbations is at least an order of magnitude below the time cadence of the current EUV/SXR imaging instrumentation (e.g., 12 s for *SDO*/AIA). We note, however, that subsecond-scale oscillations in the solar corona have been reported in the literature based on non-imaging radio or X-ray light curves or dynamic spectra during flares . @1987SoPh..111..113A summarized the possible mechanisms into three categories: (1) quasi-periodic injections of nonthermal electrons, (2) fast cyclic self-organizing systems of plasma instabilities associated with the wave–particle or wave–wave interaction processes, and (3) MHD oscillations. While we cannot completely rule out the other possibilities, the observed oscillations in radio emission frequency (or plasma density), combined with the fast-moving radio source with a speed characteristic of Alfvénic or fast-mode waves, are a strong indication of a weakly compressible, propagating MHD wave packets in the flaring loops that cause localized quasi-periodic modulations of the plasma density along their way.
The spatial scale of the radio-emitting fast-wave packages can be inferred from the instantaneous frequency bandwidth $\Delta \nu/\nu$ of individual emission lanes based on the plasma radiation scenario: $\Delta L = 2L_n(\Delta \nu/\nu)$, where $L_n=n_e/(dn_e/dl)$ is the density scale height. For a magnetic loop under hydrostatic equilibrium, the density gradient is along the vertical direction $z$, and the density scale height is $L_n=n_e/(dn_e/dl)=2k_BT/(\mu m_Hg)\approx46 T_{\rm MK}$ Mm, where $g$ is the gravitational acceleration near the solar surface, $m_H$ is the mass of the hydrogen atom, $T_{\rm MK}$ is the coronal temperature in megakelvin, and $\mu\approx 1.27$ is the mean molecular weight for typical coronal conditions [@2005psci.book.....A]. In this case, a frequency bandwidth of $\Delta \nu/\nu \approx 6\%$ implies a vertical extent of the source of $\Delta L_z \approx 5.5T_{\rm MK}$ Mm. Such an estimate of the source size is not inconsistent with the distribution on a small scale of a few megameters of the radio source centroids across all frequencies on the emission lane at a given time in the plane of the sky $\Delta L_{\parallel}$, although the latter is complicated by the scattering of the radio waves due to coronal inhomogeneities (see discussions in Section \[sec-spec-imaging\]). It is interesting to note that this size estimation is about an order of magnitude smaller than the apparent size of each radio image (with a half-power-full-maximum size of $\sim$30–50 Mm; see Figure \[fig-radio-spec-imaging\](D)). Such an extended radio image can likely be attributed to the angular broadening of the radio source caused by random scattering of the radio waves traversing the inhomogeneous corona [@1994ApJ...426..774B]. Indeed, @1994ApJ...426..774B estimated an angular broadening of a few tens of arcseconds at our observing frequency and source longitude, which is of the same order of magnitude as our apparent source size.
The wavelength associated with the subsecond-period oscillations can be estimated via $\lambda\approx v_{\rm p}P$, where $v_{\rm p}$ is the phase speed of the waves, taken to be of the same order of magnitude as the observed wave speed $\sim$3 Mm s$^{-1}$ (after assuming an inclination angle of 60$^\circ$ inferred from the extrapolation of the magnetic field, see Section \[sec-3d\]) that presumably represents the group speed of the wave packet $v_{\rm g}$ (see, e.g., @1984ApJ...279..857R for discussions regarding the relation between $v_{\rm p}$ and $v_{\rm g}$), and $P$ is the wave period, taken to be $\sim$0.5 s from the observed periods of the density fluctuations (see Figure \[fig-wavelet\]). Therefore, the wavelength of the oscillations is estimated as $\lambda\approx 1.5$ Mm, much smaller than the size of the propagating radio source ($\Delta L > \Delta L_z \approx 5.5T_{\rm MK} $ Mm). We therefore argue that each radio source is likely a propagating MHD wave packet that consists of multiple short-period oscillations.
During each burst period, multiple emission lanes are present in the radio dynamic spectrum with almost synchronous frequency drift behavior (which is particularly clear for Burst 2; see Figure \[fig-radio-spec-imaging\](H)). Imaging results of the different emission lanes suggest that they are all located at the same site and share very similar spatiotemporal behavior in projection, but show subtle differences (see Figures \[fig-L1\_time\](A), \[fig-L2\](C), and \[fig-L3\](C)). Their different emission frequencies, however, imply that the corresponding propagating disturbances have different plasma densities. Some other types of solar [dm-$\lambda$]{} bursts, in particular, zebra-pattern bursts (ZBs), also display multiple emission lanes. One leading theory for ZBs attributes the observed multiple lanes to radio emission at the plasma upper-hybrid frequency $\nu_{\rm uh}$ that coincides with harmonics of the electron gyrofrequency $\nu_{\mathrm{ce}}$, i.e., $\nu\approx\nu_{\rm uh}\approx (\nu_{\mathrm{pe}}^2+\nu_{\mathrm{ce}}^2)^{1/2}\approx s\nu_{\mathrm{ce}}$ . However, unlike the ZBs, the frequency spacing between different emission lanes in this burst is irregular and varies in time. Moreover, although the frequency turnover times of different emission lanes $t_{\rm o}$ are very close to each other, they differ by $\sim$0.5–0.8 s (orange arrows in Figure \[fig-radio-spec-imaging\](F) and (H)) and does not show a systematic lag in frequency as is usually present in ZBs [@2007SoPh..246..431C; @2007SoPh..241..127K; @2013ApJ...777..159Y]. Therefore, we argue that the different emission lanes are not due to harmonics of a particular plasma wave mode. Instead, they are related to different wave packets, which are triggered by the same impulsive energy release event, propagating in magnetic flux tubes with different plasma properties.
{width="\textwidth"}
The schematic in Figure \[fig-cartoon\] summarizes our interpretation of the observed radio bursts in terms of propagating MHD wave packets that contain multiple subsecond-period oscillations within the context of the filament eruption and two-ribbon flare. As introduced in Section \[sec:intro\], subsecond-period MHD waves may 1[be a viable mechanism]{} responsible for transporting a substantial amount of the magnetic energy released in the corona downward to the lower atmosphere, resulting in intense plasma heating and/or particle acceleration. Let us consider the scenario of fast-mode MHD waves guided by dense magnetic flux tubes as an example . The kinetic energy flux associated with propagating MHD waves can be estimated as $F_K\approx\frac{1}{2}\rho\delta v^2 v_g$ [@2007Sci...317.1192T; @2014ApJ...795...18V], where $\rho\approx m_Hn_e$ is the mass density, $\delta v$ is the amplitude of the velocity perturbation, and $v_g$ is the group speed of the propagating MHD wave. Estimates for both $\rho$ and $v_g$ can be conveniently obtained from our observations of the radio emission frequency and the radio source motion. Although the velocity perturbation $\delta v$ is not directly measured by our observations, it is intimately related to the observed density perturbation amplitude $\delta\rho \approx m_H \delta n_e$ through the continuity equation in the regime of small perturbation: $$\frac{d(\delta\rho)}{dt}=-\rho_0 \nabla \cdot \delta v,$$ It is beyond the scope of the current study to examine the detailed relation for all possible wave modes propagating in coronal loops with different density profiles. Nevertheless, under typical coronal conditions, it has been shown by previous studies that, under typical coronal conditions, $\delta v/v_g$ is of the same order of magnitude as $\delta n_e/n_e$ for fast-mode MHD waves propagating along dense coronal loops . The latter is found to be $\delta n_e/n_e\approx2\delta\nu/\nu\approx$1[2–4$\%$]{}. Following these assumptions, we estimate the energy flux as 1[(2–8)$\times 10^8$]{} erg s$^{-1}$ cm$^{-2}$, with $n_e\approx 2\times 10^{10}$ cm$^{-3}$, $\delta v/v_g \approx $1[2–4$\%$]{}, and $v_g\approx 3$ Mm s$^{-1}$.
{width="\textwidth"}
Is the estimated energy flux carried by the MHD disturbances energetically important in this flare? The energy flux required to power flares can be inferred using a variety of observational diagnostic methods including broadband imaging of flare ribbons in WL and UV [@2007ApJ...656.1187F; @2012ApJ...752..124Q; @2013ApJ...770..111L], as well as HXR spectroscopic and imaging observations of flare footpoints [@2007ApJ...656.1187F]. Here we adopt the method developed by @2012ApJ...752..124Q to estimate the energy flux 1[needed to account for flare heating]{} based on *SDO*/AIA 1600 Å UV observations of the flare ribbons. The energy flux $F_i(t)$ of flare heating is related to UV 1600 Å ribbon brightening at pixel $i$ as $$F_i(t)=\lambda I_i^{\rm pk} \exp\left[-\frac{(t-t_i^{\rm pk})^2}{2\tau_i^2}\right] \mathrm{erg\ s^{-1}\ cm^{-2}},$$\[eq-energy\]
where the exponential term is the Gaussian function used to fit the rise phase of the light curve of the UV count rate that has a characteristic rise time $\tau_i$ and peaks at $t_i^{\rm pk}$, and $\lambda$ is the scaling factor that converts the observed peak UV count rate $I_i^{\rm pk}$ at pixel $i$ (in DN s$^{-1}$ pixel$^{-1}$, where DN is data numbers) to 1[the estimated energy flux responsible for the flare heating (erg s$^{-1}$ cm$^{-2}$), which depends not only on the mechanism of UV radiation upon heating in the lower atmosphere, but also on the instrument response. @2012ApJ...752..124Q and @2013ApJ...770..111L performed detailed modeling studies of loop heating of two flares, and found that $\lambda$ generally lies in the range (2-3)$\times10^5\ \mathrm{erg\ DN^{-1}\ pixel/cm^{-2}}$ to best match the model-computed *GOES* SXR light curves with the observations. Here we take $\lambda \approx 2.7\times10^5\ \mathrm{erg\ DN^{-1}\ pixel/cm^{-2}}$ quoted in @2012ApJ...752..124Q for our order-of-magnitude estimate]{}. We have traced all pixels in AIA 1600 Å UV images that show flare ribbon brightenings, which are shown in Figure \[fig-ribbon\](A) colored by their peak time $t_i^{\rm pk}$ from purple to red. The flare ribbons show an evident separating motion during the impulsive phase of the flare, which is characteristic of two-ribbon flares and has been considered as one of the primary evidence for magnetic-reconnection-driven flare energy release [@2002ApJ...565.1335Q]. The corresponding light curves of the UV count rate for all ribbon pixels are shown in Figure \[fig-ribbon\](B), again colored by their peak time (only the rising portion of the light curve is shown). The UV ribbon brightenings agree very well in time with the *GOES* SXR derivative (thick red curve in Figure \[fig-ribbon\](C)), suggesting that heating of the flare loops is mainly driven by the “evaporation” of the heated chromospheric plasma. The estimated energy flux averaged over all ribbon pixels $\overline{F}(t)$ based on Eq. \[eq-energy\] is shown as the blue curve in Figure \[fig-ribbon\](C). 1[Also shown is the average $\overline{F}(t)$ estimated using only pixels of the northern ribbon (dashed blue curve), with which the radio bursts appear to be associated temporally and spatially (see Figure \[fig-ribbonbrightenings\]).]{} The values are in the range of $10^8$–$10^9$ $\mathrm{erg\ s^{-1}\ cm^{-2}}$, which is typical for *GOES* C-class flares. At the time of the radio burst, the average $\overline{F}(t)$ 1[at the northern ribbon]{} is about 1[$4\times 10^8\ \mathrm{erg\ s^{-1}\ cm^{-2}}$]{}, which is comparable to the estimated energy flux carried by the observed subsecond-period MHD wave packets.
We note, however, taht such coherent-burst-emitting waves can only be observed when the following conditions are met. 1[(1) Flare-accelerated electrons are present in the vicinity of the MHD waves. (2) Conditions are satisfied for inducing nonlinear growth of Langmuir waves and the subsequent conversion to transverse radio waves. (3) The radio waves are emitted within the bandwidth of the instrument (1–2 GHz in our case). (4) The instrument is sensitive enough to distinguish the radio bursts from the background active region and flare emission—the flux density of the bursts is only $\sim 1$ sfu (1 sfu = 10$^4$ Jansky) in our case, which is barely above the noise level of most non-imaging solar radio spectrometers.]{} For these reasons, the radio bursts appear relatively rare, and thus their volume filling factor in the entire flaring region is essentially unknown. 1[Moreover, although possible signatures of wave damping seem to be present in some bursts that we observe (see Figure \[fig-wavelet\](B–E)), which may be due to energy loss during their propagation, the fraction of total energy deposited to the lower solar atmosphere from the waves remains undetermined in this study.]{} However, considering the presence of ubiquitous large-scale fast EUV waves throughout the active region around the same time, it is reasonable to postulate that these short-period waves are also ubiquitously present in the flaring region. If this is the case, these waves may 1[play a role in transporting the released flare energy during the late impulsive phase of this flare, likely alongside the accelerated electrons, and the subsequent heating of the flare ribbons and arcades]{}.
Conclusion {#sec-conclusion}
==========
Here we report radio imaging of propagating MHD waves along post-reconnection flare loops during the late impulsive phase of a two-ribbon flare. This is based on observations of a peculiar type of [dm-$\lambda$]{} radio bursts recorded by the VLA. In the radio dynamic spectrum, the bursts show a low–high–low frequency drift pattern with a moderate frequency drift rate of $\dot{\nu}/\nu \lesssim 0.2$. VLA’s unique capability of imaging with spectrometer-like temporal and spectral resolution (50 ms and 2 MHz) allows us to image the radio source at every pixel in the dynamic spectrum where the burst is present. In accordance with its low–high–low frequency drift behavior, we find that the radio source firstly moves downward toward a flare ribbon before it reaches the lowest height and turns upward. The measured speed in projection is $\sim$1–2 Mm/s, which is characteristic of Alfvénic or fast-mode MHD waves in the low corona. Furthermore, we find that the bursts consist of many subsecond, quasi-periodic oscillations in emission frequency, interpreted as fast oscillations within propagating MHD wave packets. As illustrated in Figure \[fig-cartoon\], these wave packets are likely triggered by the impulsive flare energy release, and subsequently propagate downward along the newly reconnected field lines down to the flare ribbons. From the observed density oscillations and the source motion, we estimate that these wave packets carry an energy flux of 1[(2–8)$\times 10^8$ erg s$^{-1}$ cm$^{-2}$]{}, which is comparable to the average energy flux required for driving the flare heating 1[during the late impulsive phase of the flare estimated from the UV ribbon brightenings. In addition, the radio source seems to show a close spatial and temporal association with the transient brightenings on the flare ribbon]{}. As introduced in Section \[sec:intro\], such subsecond-period MHD waves have long been postulated as an alternative or complementary means for transporting the bulk of energy released in flares alongside electron beams, resulting in strong plasma heating and/or particle acceleration. Here we provide, to the best of our knowledge, the first possible observational evidence for these subsecond-period MHD waves propagating in post-reconnection magnetic loops derived from imaging and spectroscopy data, and demonstrate 1[their possible role in driving plasma heating during the late impulsive phase of this flare event. Future studies are required to, first of all, investigate their presence in other flare events, and moreover, establish whether or not they are energetically important in transporting the released flare energy during different flare phases]{}.
We thank Sophie Musset for her help in producing the *RHESSI* X-ray image. We also thank Tim Bastian, John Wygant, Lindsay Glesener, Kathy Reeves, and Dale Gary for helpful discussions, 1[as well as an anonymous referee who provided constructive comments to improve the paper]{}. The National Radio Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc. This work made use of open-source software packages including CASA [@2007ASPC..376..127M], SunPy , and Astropy . B.C. and S.Y. are supported by NASA grant NNX17AB82G and NSF grant AGS-1654382 to the New Jersey Institute of Technology.
| ArXiv |
---
abstract: 'We consider stability properties of spherically symmetric spacetimes of stars in metric $f(R)$ gravity. We stress that these not only depend on the particular model, but also on the specific physical configuration. Typically configurations giving the desired $\gamma_{\rm PPN} \approx 1$ are strongly constrained, while those corresponding to $\gamma_{\rm PPN} \approx 1/2$ are less affected. Furthermore, even when the former are found strictly stable in time, the domain of acceptable static spherical solutions typically shrinks to a point in the phase space. Unless a physical reason to prefer such a particular configuration can be found, this poses a naturalness problem for the currently known metric $f(R)$ models for accelerating expansion of the Universe.'
author:
- Kimmo Kainulainen
- Daniel Sunhede
title: 'On the stability of spherically symmetric spacetimes in metric $f(R)$ gravity'
---
Introduction {#sec:Intro}
============
The observation that the expansion of the Universe appears to be accelerating [@astier; @spergel] has provoked discussion of a number of models for extended gravity involving nonlinear interactions in the Ricci scalar $R$: $$S = \frac{1}{2\kappa} \int {\rm d}^{4}x \sqrt{-g} [R + f(R)]
+ S_{\rm m} \,.
\label{eq:action}$$ Here $\kappa \equiv 8\pi G$, $S_{\rm m}$ is the usual matter action and $f(R)$ describes the new physics in the gravity sector; setting $f(R) = -2\Lambda$ corresponds to the canonical Einstein-Hilbert action in General Relativity (GR) with a cosmological constant $\Lambda$. The idea is that if cosmological data could be fitted by the use of some nontrivial function $f(R)$, one might avoid the theoretical difficulties and fine-tuning issues related to a pure cosmological constant. However, it has been shown that when understood as a [*metric*]{} theory, the action (\[eq:action\]) can lead to predictions that are not consistent with Solar System measurements [@chiba; @erickcek; @Kainulainen:2007bt]. While observations require a parameter $|\gamma_{\rm PPN}-1|\lesssim 10^{-4}$ [@obsongamma] in the Parametrized Post-Newtonian (PPN) formalism, the value predicted in metric $f(R)$ theories is typically $\gamma_{\rm PPN} \approx 1/2$. This is certainly the case [@chiba; @erickcek; @Kainulainen:2007bt] for the first simple $f(R)$ models suggested in the literature [@vollick; @carroll]. It is however difficult to make a completely generic prediction of this result and there have been many arguments both for [@chiba; @erickcek; @Kainulainen:2007bt; @metricFail] and against [@metricPass; @Zhang:2007ne; @Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq; @Clifton:2008jq] metric $f(R)$ gravity failing Solar System tests. In particular, more complicated $f(R)$ functions have since been suggested which claim to yield $\gamma_{\rm PPN} \approx 1$ [@Zhang:2007ne; @Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq].
In this paper we set up the conditions which the function $f(R)$ must fulfill, so that a solution to the field equations which is compatible with Solar System observations exists, in particular with $\gamma_{\rm PPN} \approx 1$. However, we will also argue that the mere existence of such a solution does not imply that a model is consistent with observations. Since metric $f(R)$ gravity is a fourth-order theory, spacetime geometry and matter are not in as strict a correspondence as in General Relativity; depending on the boundary conditions on the metric, a given matter distribution can be consistent with different static spacetimes and with different values of $\gamma_{\rm PPN}$. Moreover, no physical principle tells us that only the boundary conditions corresponding to $\gamma_{\rm PPN} \approx 1$ solutions should be acceptable. The question is then, which solutions are the most natural ones? How plausible is it that the collapse of a protostellar dust cloud leads to the formation of the spacetime observed in the Solar System? To answer these questions one would ideally like to study the full dynamical collapse, and given a domain of reasonable initial conditions, determine the attractor in the configuration space of possible solutions. This computation is beyond the scope of this paper however. We will instead approach the problem by studying how the time stability argument constrains the phase space of configurations with the desired properties.
The conditions that a generic metric $f(R)$ model should satisfy in order to yield acceptable solutions are: first, the Ricci scalar should closely follow the trace of the energy-momentum tensor inside a changing matter distribution, where at the same time the dimensionless quantities $f/R$ and $F \equiv \partial f/\partial R$ should remain much smaller than 1 at regions of high density. Second, the effective mass term $m^2_R$ for a perturbation in the Ricci scalar should be positive in order to assure that the GR-like $\gamma_{\rm PPN} \approx 1$ configurations are stable in time. Third, the mass $m_R^2$ should remain small so that a finite domain of static, GR-like configurations exist. This is guaranteed if $m_R^2 \lesssim 1/r_{\odot}^2$, where $r_{\odot}$ is the radius of the Sun. If this last condition is not fulfilled, the domain of GR-like configurations shrinks to essentially a point in the phase space, while a continuum of equally good, but observationally excluded, solutions still exists. In such a case the credibility of the theory requires an argument as to why the particular GR-like configuration should be preferred. None of the models so far proposed in the literature, including Refs. [@Hu:2007nk; @Nojiri:2007as; @Nojiri:2007cq], satisfy all of these constraints, and we also failed to construct a model that would. Largely this failure comes from the difficulty to keep both the function $F$ and $m_R^2 \sim 1/(3F_{,R})$ small simultaneously when the Ricci scalar follows the matter distribution, $R \approx \kappa\rho$.
It should be noted that the above considerations only apply for the [*desired*]{} GR-like $\gamma_{\rm PPN} \approx 1$ configurations when a model is tuned to mimic a (very small) cosmological constant. Metric models with a true cosmological constant plus some additional sufficiently small $f(R)$ correction can be perfectly fine. Thus the above arguments do not exclude generic $f(R)$ modifications, such as might arise from quantum corrections, to the Einstein-Hilbert action. One should also note that it is in general easy to construct stable attractor solutions yielding $\gamma_{\rm PPN} \approx 1/2$ in metric $f(R)$ theory. It is the precision data from the Solar System which makes these solutions unacceptable.
The paper is organized as follows. We start by reviewing the Solar System constraints in Sec. \[sec:ppn\]. We consider the Dolgov-Kawasaki time instability [@Dolgov:2003px] in Sec. \[sec:timestab\] and discuss the corresponding stability criterion for spherically symmetric configurations. Section \[sec:paltrack\] considers static configurations and the possibility for metric $f(R)$ gravity to follow stable, GR-like solutions that are compatible with Solar System constraints. We find that the condition for finding a finite *domain* of boundary conditions giving rise to a GR-like metric is nearly orthogonal to the time stability condition. Finally, Sec. \[sec:summary\] contains our conclusions and discussion.
Solar System constraints and the Palatini track {#sec:ppn}
===============================================
Let us begin by reviewing the main constraints from the Solar System observations on static solutions in metric $f(R)$ gravity. Varying the action (\[eq:action\]) with respect to the metric gives the equation of motion: $$\begin{aligned}
(1+F) R_{\mu \nu} - \frac{1}{2} (R+f) g_{\mu \nu} && \nonumber \\
- \nabla_\mu\nabla_\nu F+ g_{\mu \nu}\Box F & = & \kappa T_{\mu \nu} \,,
\label{eq:eom}\end{aligned}$$ where $F \equiv f_{,R} = \partial f/\partial R$ and $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$. Taking the trace of this equation one finds: $$\Box F - \frac{1}{3}(R(1-F) + 2f) = \frac{1}{3}\kappa T \,.
\label{eq:trace}$$ If $F\rightarrow 0$ and $f\rightarrow R$ this equation reduces to the standard algebraic GR relation between the Ricci scalar and the trace of the energy-momentum tensor $T$. In a generic metric $f(R)$ theory $R$ is a dynamical variable however, and the theory may exhibit an instability which we will discuss in the next section. Assuming a static, spherically symmetric metric $g_{\mu \nu}$, $$ds^2 \equiv g_{\mu \nu} x^{\mu} x^{\nu} =
-e^{A(r)}{\rm d}t^2 + e^{B(r)}{\rm d}r^2 + r^2{\rm d}\Omega^2 \,,
\label{eq:metric}$$ the full field equations (\[eq:eom\]) reduce to the following source equations for the metric functions $A$ and $B$ in the weak field limit (to first order in small quantities): $$\begin{aligned}
(rB)' & \approx & \kappa \rho r^2 \bigg( 1
- \frac{1}{3}\bigg[ \frac{1+3F}{1+F}
- \frac{R}{\kappa\rho}\frac{1 + \frac{F}{2}+\frac{f}{2R}}{1+F}
\bigg] \bigg) \nonumber \\
& & - \gamma rA' \,,
\label{eq:sourceB} \\
A' & \approx & \frac{1}{1+\gamma} \left( \frac{B}{r}
- \frac{r}{2(1+F)}\bigg[ FR - f + \frac{4}{r}F' \bigg] \right) \,,\qquad
\label{eq:sourceA}\end{aligned}$$ where $\gamma \equiv rF'/2(1+F)$, a prime refers to a derivative with respect to $r$, and we have neglected pressure so that $T \approx -\rho$. The parameter $\gamma$ and the terms in the square brackets highlight the deviation from General Relativity. The value of $\gamma_{\rm PPN} \approx - B/A$ far away from a gravitational source depends on the continuous evolution of $A$ and $B$ throughout the Sun. It is particularly sensitive to the evolution through the core where the density is the highest. Hence, to obtain $\gamma_{\rm PPN} \approx 1$ and the correct gravitational strength in the Solar System, the only solution, not obviously dependent on an enormous amount of fine-tuning [^1], is that the extra terms in Eqns. (\[eq:sourceB\]-\[eq:sourceA\]) must remain small throughout the interior of the Sun. Now, if the extra terms can be neglected in the $B'$ equation, one finds that $B \lesssim 10^{-6}$ throughout the interior of the Sun [@Kainulainen:2007bt]. It then becomes clear from the $A'$ equation that $f/R$, $F$, and $rF'$ need to be very small compared to 1. However, to make the correction vanish in the $B'$ equation one in addition needs to require that the Ricci scalar traces the matter density $R/\kappa \rho \approx 1$. So, barring perhaps some fantastic fine-tunings, the only possibility is that one finds a configuration for which: $$F \ll 1 \,, \quad
f/R \ll 1 \,, \quad {\rm and} \quad
R \approx \kappa \rho \,.
\label{eq:conditions}$$ Note that this limit was also discussed in Ref. [@Hu:2007nk]. Here we see that the above conditions are a necessary requirement for fulfilling the local gravity constraints.
In GR the equation $R = -\kappa T$ is of course exact (remember that we are neglecting pressure throughout so that $T\approx -\rho$), but this is in general very difficult to arrange in a metric $f(R)$ model. The problem lies in the dynamical nature of the Ricci scalar in metric $f(R)$ gravity. To see this, consider the static trace equation (\[eq:trace\]) in the weak field limit: $$\begin{aligned}
F'' + \frac{2}{r}F' & = & \frac{1}{3} \big(R - \kappa \rho - FR + 2f \big)
\nonumber \\
& \equiv & \frac{1}{3} \big(\Sigma(F) - \kappa \rho \big) \,,
\label{eq:traceWeak}\end{aligned}$$ where we have again assumed that pressure is negligible. Assume now that $F=F_0$ at the center of the Sun. If the nonlinear term $\Sigma(F)$ is small compared to $\kappa \rho$, then the solution for $F$ becomes: $$F(r) = - \int_0^r {\rm d}r' \frac{2G m(r')}{3r'^2} + F_0 \,,
\label{eq:Fsol}$$ where $m(r) \equiv \int_0^r {\rm d}r' 4\pi r'^2 \rho$. In the case of the Sun this implies that $F$ evolves only little: $|F(r)-F_0| \lesssim 10^{-6}$ [@Kainulainen:2007bt]. The solution (\[eq:Fsol\]) is in general not compatible with $R \approx \kappa \rho$ as required by the conditions (\[eq:conditions\]).
Let us now set $\Sigma(F_0) = \kappa \rho_0$ at $r = 0$. If the gradients somehow remain small throughout the evolution, then the solution follows the Palatini trace equation: $$\Sigma(F) = R(1-F) + 2f = 8\pi G \rho \,.
\label{eq:tracePal}$$ For small $F$ and $f/R$ this evolution [*would*]{} be consistent with the condition $R\approx \kappa \rho$ and, since we are following the Palatini track, give $\gamma_{\rm PPN} \approx 1$ (see [*e.g. *]{}[@solarPal]). Whether such a solution actually exists is more difficult to prove. However, one can study under which conditions such a solution, if it exists, would be an attractor and whether it would also be sufficiently stable in time.
Finally, let us note that if $\Sigma(F) \ll \kappa \rho$ so that $F$ is given by Eqn. (\[eq:Fsol\]), small $F$ and $f/R$ result in a different class of solutions with $R/\kappa \rho \ll 1$. In this case the field Eqns. (\[eq:sourceB\]-\[eq:sourceA\]) reduce to $$\begin{aligned}
(rB)' & \approx & \frac {2}{3}\kappa \rho r^2 \,,
\label{eq:sourceB2} \\
A' & \approx & \frac{B}{r} - 2F' \,.
\label{eq:sourceA2}\end{aligned}$$ It is easy to show that together with Eqn. (\[eq:Fsol\]) these give $\gamma_{\rm PPN} \approx 1/2$.
Time stability {#sec:timestab}
==============
Let us first consider the time stability of spherically symmetric configurations in generic $f(R)$ models. Perturbing around some arbitrary configuration, $R(r) \rightarrow \widetilde{R}(r,t) = R(r) + \delta R(r,t)$, and expanding to first order in $\delta R$, $\delta R'$ and $\dot{\delta R}$, where the prime refers to a derivative with respect to $r$ and the dot to a derivative with respect to $t$, one can write the trace equation (\[eq:trace\]) in the following form in the weak field limit: $$\begin{aligned}
(\partial_t^2 - \vec{\nabla}^2) \delta R & = &
- m_R^2\delta R
+ 2\frac{F_{,RR}}{F_{,R}}R'\delta R'
\nonumber \\
& & + \vec{\nabla}^2 R + \frac{1}{3F_{,R}}\Delta
+ \frac{F_{,RR}}{F_{,R}} (R')^2 \,, \quad
\label{eq:tracePert}\end{aligned}$$ where $\Delta \equiv -\kappa T - R(1-F) - 2f$ and $$m_R^2 \equiv \frac{1}{3F_{,R}}(1-F-\varepsilon) \,,
\label{eq:mR}$$ with $$\begin{aligned}
\varepsilon \phantom{.} \delta R & \equiv &
R(\widetilde{F}-F) - 2(\widetilde{f} - f)
+ \bigg(1-\frac{\widetilde{F}_{,R}}{F_{,R}} \bigg)\Delta
\nonumber \\
&& {}+ 3 F_{,RR}\bigg(\frac{\widetilde{F}_{,RR}}{F_{,RR}}
- \frac{\widetilde{F}_{,R}}{F_{,R}}\bigg)(R')^2 \,.
\label{eq:epsilon}\end{aligned}$$ Here a tilde is used to denote that a quantity is perturbed, [*i.e. *]{}it is a function of $\widetilde{R}(r,t)$ as opposed to the background value $R(r)$. Assuming that the configuration $R(r)$ we are perturbing around is a solution to the static equation, the second line in Eqn. (\[eq:tracePert\]) drops out. Moreover, for most cases the gradient term proportional to $\delta R'$ is completely negligible inside a stellar object and can be dropped as well. See Fig. \[fig:gradient\] for some examples. The behavior of the perturbation around a static, spherically symmetric solution is thus governed by the equation
![The gradient term proportional to $\delta R'$ in Eqn. (\[eq:delR\]), normalized to $m_R^2/r_{\odot}$, for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black). The actual density profile used in all figures corresponds to the known density profile of the Sun with a central density of $150$ g/cm$^3$ and with a roughly exponential dependence on $r$. We have also superimposed a constant dark matter distribution on the profile of the Sun with $\rho_{\rm DM} = 0.3$ GeV/cm$^3$.[]{data-label="fig:gradient"}](Figs/figGrad.eps){width="8cm"}
![The parameter $F$ given as a function of the radius for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black).[]{data-label="fig:F"}](Figs/figF.eps){width="8cm"}
$$\begin{aligned}
(\partial_t^2 - \vec{\nabla}^2) \delta R & = &
- m_R^2 \delta R \,.
\label{eq:delR}\end{aligned}$$
Note that the mass $m_R^2$ only depends on the background value of the Ricci scalar $R(r)$.
Table \[table1\] and \[table2\] show the components of $m_R^2$ in some particular models and the corresponding parameter values used in all figures are displayed in Table \[table3\]. As discussed in the previous section, $F$ needs to be small compared to one for GR-like configurations (see Fig. \[fig:F\], we will discuss this constraint further below). When this is the case one typically finds that also $\varepsilon \ll 1$, so that $m_R^2 \approx 1/3F_{,R}$. It then follows that if $F_{,R} < 0$, then $m_R^2 < 0$ and the coefficient of $\delta R$ is negative for the configuration in question, so that system exhibits an instability. This is the instability first found by Dolgov and Kawasaki in the context of an $f(R) = -\mu^4/R$ model [@Dolgov:2003px] (see Ref. [@Faraoni:2006sy] for a more general case). It is important to note that the instability depends not only on the model, but also on the particular configuration. Certain configurations in a given model are more stable than others and the instability may even vanish in some cases.
The nature of the instability is most transparent in the special case with constant curvature. Then $m_R$ is a constant and one can obtain an exact solution for $\delta R(r,t)$. Expanding $\delta R$ in Fourier modes, one finds that a mode with wave vector $\vec{k}$ has the time dependence $$\begin{aligned}
\delta R_k(\vec{k},t) \sim e^{\pm i\sqrt{k^2 + m_R^2}t} \,,
\label{eq:delRsol}\end{aligned}$$ so that for negative $m_R^2 \sim 1/3F_{,R}$, all modes with $k < |m_R|$ are unstable. This does not necessarily rule out a model however. If for example $|m_R| \sim H_0$, then the instability time is much longer than the lifetime of the Solar System and the model is safe. Moreover, whenever $|m_R|^{-1}$ is much larger than the size of the physical system under consideration, only modes corresponding to scales much larger than the system are unstable and this can not alter its local geometry.
Now, assume that we have a GR-like solution, such that $R \sim -\kappa T \approx 8\pi G \rho$. One then has $$\frac{R}{\mu^2} \sim 10^{29} \left(\frac{\Lambda}{\mu^2} \right)
\left(\frac{\rho}{{\rm g}/{\rm cm}^3}\right) \,,
\label{eq:Rmagn}$$ where we have used $\Lambda \approx 0.73 \kappa \rho_{\rm crit}$. Hence, for a pure $-\mu^4/R$ model the mass squared is on the order of $$m_R^2 \sim -(10^{-26} \textrm{ s})^{-2}\left(\frac{\Lambda}{\mu^2} \right)^2
\left(\frac{\rho}{{\rm g}/{\rm cm}^3}\right)^3 \,.$$ This system is violently unstable at normal densities for
all scales larger than $\sim 10^{-18}{\rm~m}$, if $\mu$ is fixed to account for the present accelerating expansion of the Universe.
As was pointed out by Dick [@Dick:2003dw] and later discussed by Nojiri & Odintsov [@Nojiri:2003ft], adding a conformal term $\alpha R^2/\mu^2$ can stabilize this system; for $f(R) = -\mu^4/R + \alpha R^2/\mu^2$ the previous approximation for the mass reads: $$m_R^2 \sim -\frac{R^3}{6\mu^4}\left(\frac{1}{1-\alpha R^3/\mu^6}\right)
\sim \frac{\mu^2}{\alpha} \,,
\label{eq:mRmagn}$$ where the last step assumes that $\alpha R^3/\mu^6 > 1$. If $\alpha \sim 1$ this may be true even for $R \sim \mu^2$ so that one always finds a very small positive mass $m_R^2 \sim \mu^2/\alpha \sim (10^{18}\textrm{ s})^{-2}$.
However, the above stabilization mechanism runs into problems with the conditions in Eqn. (\[eq:conditions\]). Indeed, for $\alpha \sim 1$ and $R \approx \kappa \rho \gg \mu^2$ one has $$F = \frac{\mu^4}{R^2} + \alpha \frac{R}{\mu^2} \gg 1\,,$$ so that the configuration would clearly not be GR-like [^2]. The problem is that changing $F$ modifies the effective strength of the gravitational constant $G_{\rm eff} = G/(1+F)$, which controls the buildup of the gravitational potential inside the star. In fact, for $\alpha \sim 1$ the effect is so strong that it would weaken the gravitational force so much as to prohibit the growth of any density contrasts much above the critical density. This argument can be turned around to a constraint: in order for the function $F$ to remain small inside the densest objects we have reasonably accurate information on, the neutron stars, one has to have $\alpha \kappa \rho_{\rm nucl}/\mu^2 \ll 1$, where $\rho_{\rm nucl}$ is the nuclear density [^3]. Since $\mu^2 \sim \kappa \rho_{\rm crit}$ we find that $$\alpha \lesssim \frac{\rho_{\rm crit}}{\rho_{\rm nucl}} \sim 10^{-45} \,.$$ This is quite a stringent constraint, but it does not rule out the model based on the required time stability. Indeed, for example with $\alpha = 10^{-47}$ one has $\alpha (\kappa \rho )^3/\mu^6 = 1$ when $\rho \sim 10^{16}\rho_{\rm crit}$. For any density higher than this value, the system is stable in time with a very [*large*]{} positive mass squared given by the formula in Eqn. (\[eq:mRmagn\]): $m_R^2 \sim \mu^2/\alpha \sim (10^{-6} {\rm~s})^{-2}$. There is a caveat to this argument however, since for these parameters the gradient term proportional to $\delta R'$ becomes very large inside the Sun (see Fig. (\[fig:gradient\])) and the simplified equation (\[eq:delR\]) can no longer be trusted.
The complete mass squared function for the model with a fine-tuned conformal $\alpha R^2$ term, using the exact expression (\[eq:mR\]), is shown in Fig. \[fig:mR2\] (dashed green curve). The lower panel displays the absolute value $|m_R^2|$ and the upper panel the sign of $m_R^2$. In the model at hand the mass would remain large and *positive* throughout the entire interior of the Sun, which is the necessary condition for time stability. The GR-like configuration does become unstable at low densities, but this would not necessarily change the value of $\gamma_{\rm PPN}$ in the Solar System.
![The mass squared $m_R^2$ in units $1/r_{\odot}^2$ for various $f(R)$ models ($R = \kappa \rho$): $-\mu^4/R$ (solid blue), $-\mu^4/R + \alpha R^2/\mu^2$ (dashed green), Hu & Sawicki (dot-dashed red) [@Hu:2007nk], and $\alpha R \log{(R/\mu^2)}$ (solid black). The horizontal solid gray line corresponds to the limit $1/r_{\odot}^2$. For the $-\mu^4/R$ and $\alpha R\log{(R/\mu^2)}$ models we have also plotted $1/3F_{,R}$ with dotted blue and dotted black lines, respectively. The upper panel displays the corresponding sign of the mass squared where we have excluded the $-\mu^4/R$ model for which $m_R^2$ is strictly negative.[]{data-label="fig:mR2"}](Figs/figM2.eps){width="8cm"}
$\displaystyle f(R)$ Parameter values
------------------------------------------------------------------ -- -- -------------------------------------------------------------------------------------------
$\displaystyle -\frac{\mu^4}{R}$ $\displaystyle \mu^2 = 4\Lambda/\sqrt{3}$
$\displaystyle -\frac{\mu^4}{R} + \alpha \frac{R^2}{\mu^2}$ $\displaystyle \mu^2 = 4\Lambda/\sqrt{3} \,, \quad \alpha = 10^{-47}$
$\displaystyle -\mu^2 \frac{c_1(R/\mu^2)^n}{c_2(R/\mu^2)^n + 1}$ $\displaystyle \mu^2 = (8315 {\rm~Mpc})^{-1} \,, \quad n = 1\,,$
$\displaystyle c_1/c_2 = 6 \times 0.76/0.24 \,,$
$\displaystyle \phantom{\Bigg|}c_1/c_2^2 = 10^{-6}\times41^{n+1}/n $
$\displaystyle \alpha R \log{\frac{R}{\mu^2}}$ $\displaystyle \mu^2 = 4\Lambda e^{(1-\alpha)/\alpha} \,, \quad \alpha = 1/\log{10^{32}}$
: Chosen parameter values for the different $f(R)$ models in Figs. \[fig:gradient\]-\[fig:mR2\], where $\Lambda = 0.73 \kappa \rho_{\rm crit}$. For the third model, originally suggested in Ref. [@Hu:2007nk], we have used values given in the original publication. Note that a value $n=4$, which was also discussed in [@Hu:2007nk], would result in an even larger value of $m_R^2$ in this scenario.[]{data-label="table3"}
Fig. \[fig:mR2\] also displays $m_R^2$ for several other models (for parameter values used in each model see Table \[table3\]): the solid blue line represents the simple $-\mu^4/R$ model, which has a very large negative mass inside the Sun, and the dash-dotted red curve shows the mass function in a model by Hu & Sawicki (HS) [@Hu:2007nk]. The HS model fulfills the conditions (\[eq:conditions\]) by construction, and its very large positive mass guarantees time stability. The fact that $m_R^2$ becomes negative around $r \sim 6r_{\odot}$ in the HS model is caused by the $\varepsilon$ term in the complete expression (\[eq:mR\]), but this does not necessarily have any effect on $\gamma_{\rm PPN}$. Moreover, this feature is sensitive to the particular form of the exterior density profile (where we have neglected for example the Solar wind) and it is not important for our main results. Overall one sees that the expression $m_R^2 \approx 1/3F_{,R}$ is a very good approximation for the first three models described in Table \[table1\], except for a small region around $r \sim 6r_{\odot}$ where the $\varepsilon$ term may come into play. For the logarithmic model this approximation is only good for very low densities and we will discuss this in more detail in section \[sec:paltrack\].
We can summarize this section as follows: the stability of a static, spherically symmetric GR-like configuration with $R \approx \kappa \rho$ is predominantly governed by the mass term $m_R^2 \sim 1/3F_{,R}$. If $F_{,R} < 0$, all perturbations with wavelengths larger than $1/|m_R|$ will be unstable so that for a large mass $|m_R|$, the curvature inside a stellar object will evolve rapidly before some nonperturbative effect stabilizes the system. Hence, in order for a model to be stable, $m_R^2$ should be positive throughout the Sun for GR-like configurations [^4]. Both the model with a fine-tuned conformal $\alpha R^2$ term (apart from the caveat mentioned above) and the HS scenario do satisfy all constraints discussed so far.
An upper bound on $m_R^2\,$? {#sec:paltrack}
============================
As mentioned in the introduction, a given matter distribution can be consistent with many different static geometries, depending on how the boundary conditions are defined at the center of the star. One always requires that the exterior metric is asymptotically flat and so different solutions are characterized by different values of $\gamma_{\rm PPN}$. There appears to be no [*a priori*]{} preference of one solution to another and indeed the question is: what is the most probable configuration to arise through gravitational collapse? Lacking a dynamical calculation we are restricted here to study how special the GR-like solutions are in the phase space.
Consider a static solution $R(r) = R_T(r) + \delta(r)$ where $R_T$ is the solution to the Palatini trace equation and $\delta/R_T \ll 1$ so that $R$ remains very close to the Palatini track. Note that the function $\delta(r)$ is not a true perturbation since $R_T(r)$ is not a solution to the complete metric trace equation (\[eq:trace\]). However, one can easily obtain the equation governing $\delta$ via Eqn. (\[eq:tracePert\]), giving $$\begin{aligned}
\vec{\nabla}^2 \delta & = &
\frac{1}{3F_{,R}}(1-F-\varepsilon)\delta
- 2\frac{F_{,RR}}{F_{,R}}R'_T\delta' \nonumber \\
& & - \vec{\nabla}^2 R_T - \frac{F_{,RR}}{F_{,R}} (R'_T)^2 \,,
\label{eq:del}\end{aligned}$$ where $F$ and its derivatives are functions of the “background” value $R_T(r)$. Similarly, the “perturbed” quantities in the definition for $\varepsilon$ are functions of $R(r) = R_T(r) + \delta(r)$.
In analogy with the above analysis for $\delta R(r,t)$, let us consider a constant density object so that $R_T = {\rm const.}$, giving $$\begin{aligned}
\vec{\nabla}^2 \delta & = & m_R^2 \delta \,.
\label{eq:delConst}\end{aligned}$$ The mass term $m_R^2$ is of course the same mass that appears in the equation for $\delta R(r,t)$. Now, for $m_R^2 < 0$, the solution for $\delta(r)$ is decaying so that the Palatini track acts as an attractor for the solution $R(r)$. This is exactly the behavior that was demonstrated by solving the full field equations in Ref. [@Kainulainen:2007bt] for a $f(R) = -\mu^4/R$ model. However, if $m_R^2 > 0$, the solution for $\delta(r)$ will also contain a growing component: $$\delta(r) = \frac{C_1}{r}e^{+m_R r} + \frac{C_2}{r}e^{-m_R r}\,.
\label{eq:delSol}$$ The fine-tuning problem we have to face is manifest from this equation: the time stability argument of the preceding section requires that $m_R^2>0$, so that Eqn. (\[eq:delSol\]) with its growing mode is what describes the deviation away from GR-like solutions.
From Sec. \[sec:timestab\] we already know that setting up a GR-like configuration requires fine-tuning at the center of the star. However, what Eqn. (\[eq:delSol\]) implies is worse: starting from $r=0$ we could always choose a boundary condition $(F_0,F'_0)$ that kills the growing mode, but as any perturbation around this solution would be exponentially enhanced, the boundary condition must be set with an incredible precision when $m_R^2$ is large. Numerically such a solution can always be found by use of a differentiation method that kills the growing mode as was done in Ref. [@Hu:2007nk]. However, different choices of boundary conditions would lead to other physically, but not observationally, acceptable spacetimes. For example, if one starts from a point a little off from the GR track, $\delta$ initially grows exponentially pulling the solution away from $R \approx \kappa \rho$. Then the nonlinear terms typically become negligible in Eqn. (\[eq:traceWeak\]) and $R(F)$ starts to approach the solution of Eqn. (\[eq:Fsol\]). For $R/\kappa\rho \ll 1$ this limit corresponds to the evolution of $A$ and $B$ given by Eqns. (\[eq:sourceB2\]-\[eq:sourceA2\]), which leads to $\gamma_{\rm PPN} \approx 1/2$. Thus, for a large $m_R^2$ the nearly singular static GR-like solution is surrounded by a continuum of equally acceptable configurations, however with observationally excluded values for $\gamma_{\rm PPN}$.
Hence, given no physical reason to prefer a given set of boundary conditions, it would appear more natural to expect that the metric around a generic star would correspond to $\gamma_{\rm PPN}$ different from one. As stated above, to make a definitive statement would require solving the dynamical problem of collapse, but this is beyond the scope of the present work. Nevertheless we believe that we have identified a potential problem for metric $f(R)$ gravity models: for an $f(R)$ model to be credible, it is not sufficient to provide a mere proof of existence of a GR-like solution, but one should also give an argument as to why this particular solution is preferred.
The situation would be ameliorated if the growing mode is not excluded, but the length scale dictating the growth of the perturbations, $1/m_R$, is small enough. Roughly one should have $$m_R^2 \lesssim \frac{1}{r_{\odot}^2} \,,
\label{eq:constraint}$$ throughout the Sun. However, both the HS scenario and the fine-tunded $f(R) = -\mu^4/R + \alpha R^2/\mu^2$ model fail this constraint by a large margin, as can be seen from Fig. \[fig:mR2\]. This is also the case for the model in Ref. [@Zhang:2007ne] where a stabilizing conformal term creates a behavior very similar to the $\alpha R^2$ model discussed here. The same argument also applies to more recent models introduced in Refs. [@Nojiri:2007as; @Nojiri:2007cq]. These scenarios behave very similar to the HS model at late times, but were designed to also account for inflation at very high energies. For example, for the model suggested in Ref. [@Nojiri:2007cq], $$f(R) = \frac{\alpha R^{m+l} - \beta R^n}{1 + \gamma R^l} \,,
\label{eq:NojOdmodel}$$ where the authors chose $m = l = n$ for simplicity, and $n \ge 2$, one can show that the mass squared is given by [@Nojiri:2007cq] $$m_{R}^2 \sim + \frac{R_I}{3n(n-1)}\left( \frac{R_I}{R} \right)^{n-1} \,.
\label{eq:NojOdmass}$$ Here $R_I \sim (10^{15} {\rm~GeV})^2$ is set to the scale of inflation, and so this mass is enormous in comparison with the bound (\[eq:constraint\]) inside the Sun.
The generic problem is that a small value of $m_R^2 \sim 1/3F_{,R}$ requires a large value for $F_{,R}$. However, at the same time one also needs $F \ll 1$ in order to obtain a reasonable gravitational potential. This tension is what makes it difficult to find a suitable function $f(R)$. Let us illustrate the problem further by trying to construct an explicit model by Þrst making sure that the toughest requirement is satisfied. At the center of Sun $R \approx \kappa\rho \sim 10^{31} \Lambda \sim 10^{-3}/r_{\odot}^2$, which is much smaller than the upper limit on $m_R^2$. Thus we can take $$m_R^2 \sim +R \,.$$ Using this together with the formula $F_{,R} \sim 1/m_R^2$, we can construct a candidate model: $$f(R) = \alpha R\log{\frac{R}{\mu^2}} \,,
\label{eq:logmodel}$$ where $\mu^2 = 4\Lambda e^{(1-\alpha)/\alpha}$ in order to obtain the desired accelerating expansion of the Universe at present. Furthermore, demanding that $F \ll 1$ in the interior of the Sun yields $\alpha \lesssim 0.01$ so that $\mu^2 \gtrsim e^{100} \Lambda$ [^5]. So, curiously enough the Solar System constraints would force this model to create the desired accelerating expansion without an extremely small energy scale $\sim \sqrt{\Lambda}$.
Unfortunately, there is a flaw in the above argumentation, since we implicitly assumed that $\varepsilon \ll 1$ by assuming $m_R^2 \sim 1/3F_{,R}$. This assumption was fine for the discussion in the previous sections, but it fails here. Indeed, when $F_{,R}$ is large, the gradient term proportional to $[3F_{,R}](R'/R)^2$ in $\varepsilon$ may also become large (see Table \[table2\]). We can estimate the size of this term using an exponential density profile for the Sun $\rho \sim \rho_0/(1 + e^{\xi(r-r_{\odot})})$, where $\xi \sim 10 r_{\odot}^{-1}$: $$\Big(\frac{R'}{R} \Big)^{\! 2} \approx \Big(\frac{\rho'}{\rho} \Big)^{\! 2}
\sim \xi^2 \sim 100 \frac{1}{r_{\odot}^2} \,.$$ Now, since $3F_{,R} \gtrsim r_{\odot}^2$ one gets $[3F_{,R}](R'/R)^2 \gg 1$ and our simple estimate for the mass fails. A more careful estimate in the model (\[eq:logmodel\]) finds that the mass is dominated by the gradient term and one has $$m_R^2 \approx m_R^2\big|_{\rm grad} = -\Big(\frac{R'}{R} \Big)^{\! 2}
\sim -100 \frac{1}{r_{\odot}^2} \,.
\label{eq:m2RlogR}$$ This mass squared is actually negative, so the fine-tuning problem is no longer an issue. However, we have recreated a time instability corresponding to the characteristic length scale $\xi^{-1}$ of the system. This behavior is clearly visible in Fig. \[fig:mR2\] where we have plotted both the full mass squared $m_R^2$ (solid black) and the bare function $1/3F_{,R}$ (dotted black) for the $\alpha R\log{(R/\mu^2)}$ model.
Summary and Discussion {#sec:summary}
======================
We have shown in this paper that attempts to find stable static solutions with $\gamma_{\rm PPN} \approx 1$ in metric $f(R)$ models, designed to also account for the accelerating expansion of the Universe, lead to a string of constraints on the model parameters. One must find a configuration for which simultaneously $F \equiv \partial f/\partial R$ and $f/R$ remain small compared to one in the interior of the star, where the strength of the gravitational field is built up, while the Ricci scalar traces the matter distribution: $R\approx \kappa \rho$. (See also Ref. [@Hu:2007nk]). In addition, for the configuration to be stable in time, the effective mass term $m_R^2$ for a perturbation in the Ricci scalar needs to be either positive [@Dolgov:2003px], or if negative, $|m_R|^{-1}$ must be much larger than the size of the physical system under consideration. Furthermore, we showed that unless $m_R^2 \lesssim 1/r_{\odot}^2$, the domain of GR-like static configurations shrinks to essentially a point in the phase space, while for example a continuum of solutions corresponding to $\gamma_{\rm PPN} \approx 1/2$ exists.
Hence, in particular for large positive $m_R^2$, it would appear more natural to expect that the metric around a generic star would correspond to $\gamma_{\rm PPN}$ different from one. To make a more definitive statement one should solve the dynamical gravitational collapse of a protostellar dust cloud, which is beyond the scope of this paper however. Moreover, to a degree the problem with the boundary conditions would merely be translated to setting the initial conditions for the collapse. Nevertheless, we believe that we have identified a potential problem in that to make a given metric $f(R)$ model credible, one should give an argument as to why the GR-like configurations should be preferred. Otherwise, if the PPN and stability constraints are supplemented by our fine-tuning argument, it seems unlikely that any $f(R)$ model can pass the test – unless one gives up the hope that the theory is also responsible for the accelerating expansion of the Universe. This is because the condition $m_R^2 \lesssim 1/r_{\odot}^2$, combined with $m_R^2 \sim 1/3F_{,R}$, implies that $F_{,R}$ needs to be large. However, since at the same time one needs $F \ll 1$, a tension is created that makes finding a suitable function $f(R)$ very difficult.
Let us finally note that while completely GR-like configurations are hard to construct in $f(R)$ models, it does not mean that such theories would be somehow fundamentally ill. In fact many metric $f(R)$ theories could describe gravitational physics quite well in most situations; it is the very precise information on $\gamma_{\rm PPN}$ from Solar System experiments which eventually forces one to set $R \approx \kappa \rho$. Indeed, if one looks even at the simplest model with $f(R) = -\mu^4/R$, one finds that setting $R$ essentially to any other value than $\kappa \rho$ gives $\gamma_{\rm PPN} \approx 1/2$ [@Kainulainen:2007bt]. Moreover, whenever one has $R \sim \mu^2$, one finds $m_R^2 \sim \mu^2$, so that these configurations are effectively free of any stability problems. Thus, one sees that the Dolgov-Kawasaki instability and fine-tuning problems depend not only on the theory, but also on the particular given *configuration* within a given model. Even in the simplest $-\mu^4/R$ model one can construct sufficiently stable spacetimes for stellar objects that are consistent with the accelerating expansion of the Universe; these are only excluded from describing reality by the very precise PPN constraints.
This work was partly (DS) supported by a grant from the Finnish Cultural Foundation.
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[^1]: By this we mean that it is in principle possible that an evolution of $B/A$ quite different from the one in GR, could actually lead to the same value for the exterior of a star. However, away from the GR track the outcome becomes sensitive to the form of the density profile, which leads to even more uncertainties as to how the actual dynamical gravitational collapse would proceed.
[^2]: In general, $F>1$ implies that the leading approximation in Eqn. (\[eq:mRmagn\]) is no longer valid. However, it still holds for $f(R) = -\mu^4/R + \alpha R^2/\mu^2$, since the contribution of the conformal term in $F$ will cancel with the leading term in $\varepsilon$ where additional terms remain small compared to one.
[^3]: The loophole to this argument is if one only follows $R \approx \kappa \rho$ inside the Sun, but that the Ricci scalar is allowed to significantly deviate from this relation inside neutron stars. In such a case one could maintain $F \ll 1$ for much larger values of $\alpha$, resulting in a smaller $m_R^2$ for the Sun. This scenario seems very contrived however.
[^4]: Configurations with a small enough negative $m_R^2$ can be accepted as well. The minimum requirement is then that at least all perturbations with wavelengths smaller than the size of the physical system under consideration should remain stable.
[^5]: However, note that since $F = \alpha(\log{(R/\mu^2)} + 1) = \alpha\log{(R/4\Lambda) - 1 + 2\alpha}$, there is some tension between getting $F \ll 1$ inside *both* the Sun and a neutron star. Although this may indeed prove relevant in further considerations, it is not of importance for the discussion at hand.
| ArXiv |
---
abstract: 'The transport and magnetic properties of correlated La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ ultrathin films, grown epitaxially on SrTiO$_{3}$, show a sharp cusp at the structural transition temperature of the substrate. Using a combination of experiment and theory we show that the cusp is a result of resonant coupling between the charge carriers in the film and a soft phonon mode in the SrTiO$_{3}$, mediated through oxygen octahedra in the film. The amplitude of the mode diverges towards the transition temperature, and phonons are launched into the first few atomic layers of the film affecting its electronic state.'
author:
- 'Y. Segal'
- 'K. F. Garrity'
- 'C. A. F. Vaz'
- 'J. D. Hoffman'
- 'F. J. Walker'
- 'S. Ismail-Beigi'
- 'C. H. Ahn'
bibliography:
- 'cusp.bib'
title: 'Resonant phonon coupling across the La$_{1-x}$Sr$_{x}$MnO$_{3}$/SrTiO$_{3}$ interface'
---
The coupling of phonons to charge carriers is a process of key importance for a broad set of phenomena, ranging from carrier mobility in semiconductors to Cooper pairing. In recent times, phonon effects at interfaces emerged as a topic of great importance in the understanding and design of nano-structured materials [@interfacialphonons]. Coupling between charge, structure and magnetic ordering is particularly strong in the Mn oxides [@RefWorks:462], which are used as a component in heterostructure multiferroics [@CarlosPRLPaper]. In these materials, localized spins and mobile carriers reside on the Mn sites, each surrounded by an oxygen octahedra. Intersite hopping occurs through orbital overlap of the Mn with neighbouring oxygens, making it highly sensitive to the static orientation of the octahedra and to phonons that alter the octahedra’s orientation [@RefWorks:463]. This interplay between structure and properties has been exploited to control the electronic phase of CMR films via strain, and also via coherent photoexcitation of a specific octahedra vibration mode [@RefWorks:454].\
In this Letter, we use a specially designed thin film device to isolate and characterize phonon-carrier coupling within a few atomic layers of an interface between the perovskite SrTiO$_{3}$ (STO) and the CMR oxide La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ (LSMO). A soft octahedral rotation phonon with a divergent amplitude in the STO couples to the corresponding mode of the film. This coupling results in a marked change in the electronic and magnetic properties, including a sharp cusp in the resistivity and a dip in the magnetic moment. The sensitivity of LSMO to octahedra orientation allows us to experimentally probe the microscopic character of this interfacial phonon coupling, and compare it to theory. The thin film devices consist of La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ films grown by molecular beam epitaxy on TiO$_{2}$-terminated STO (001) substrates and overlaid by Pb(Zr$_{0.2}$Ti$_{0.8}$)O$_{3}$ (PZT), which is used to provide ferroelectric field effect modulation of the number and distribution of carriers in the film. Details concerning fabrication and structural characterization are described elsewhere [@CarlosGrowthPaper]. In the bulk LSMO phase diagram, the $x=0.5$ composition separates the ferromagnetic metallic phase from an insulating antiferromagnetic phase [@RefWorks:476]. When grown commensurate to the STO, the substrate induces tensile strain in the film, which is known to stabilize an A-type antiferromagnetic metallic phase (AF-M) [@RefWorks:414]. Using X-ray diffraction, we verified that our films are under tensile strain, with $c/a=0.975$, in agreement with previous studies [@RefWorks:414].
Transport measurements of an 11unit cell (uc) LSMO film are shown in Fig.\[fig:Transport\]a. The broad peak in resistivity at 250K corresponds to a metal-insulator transition, typical of this material. In addition, a unique feature is observed in our films: a large and sharp resistance peak centered around 108K, which corresponds to the temperature of the STO soft phonon peak. We observe further that the magnitude of the resistivity cusp decreases when the thickness of the film increases by a few unit cells. Indeed, in previous studies of films $\approx$80uc thick, only a trace of this feature was observed [@RefWorks:477]. This film thickness dependence implies that the strength of the mechanism creating the cusp decays quickly away from the STO/LSMO interface. We can verify this by switching the polarization state of the PZT. When the PZT is switched to the “depletion” state, holes are removed from the top layer of the LSMO, pushing the conducting region closer to the substrate. The opposite occurs in the “accumulation” state [@CarlosPRLPaper]. We find that the PZT has a pronounced effect on the cusp (Fig.\[fig:Transport\]b), making it much larger in the depletion state, in agreement with the notion of a rapid decay into the film. We note, however, that presence of PZT is not required to observe the effect: the same features are observed on uncapped LSMO films. We also observe a striking dip in the magnetic moment centered around the STO transition temperature (Fig.\[fig:Transport\]a). While the majority of the LSMO is in an antiferromagnetic-metallic state, a small ferromagnetic component remains [@RefWorks:477]. The dip in magnetic moment corresponds to a decrease in magnetic order within the ferromagnetic phase.\
![\[fig:Transport\]Enhanced carrier-phonon scattering. (a) Left axes: Resistivity of an 11uc La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ film showing a strong cusp at 108K. The PZT overlayer is in the depletion state. Right axis: Magnetic moment of a 15uc La$_{0.55}$Sr$_{0.45}$MnO$_{3}$ film. The moment is measured along the [\[]{}100[\]]{} direction under an applied magnetic field of 1kOe. A dip in the moment is observed, overlapping the temperature range of the resistivity cusp (emphasized by grey box). The dashed line is a linear interpolation between the edges of the dip region. b) The resistivity of the 11uc film for the two polarization states of the PZT. c) Energy of the $\Gamma_{25}$ phonon mode in STO, showing the softening around the STO transition temperature (after Ref.[@RefWorks:458]). Lines are a guide to the eye. Below the structural phase transition the mode splits due to the breaking of cubic symmetry.](fig1){width="8.5cm"}
We attribute the transport and magnetism anomaly to a coupling between the LSMO and the phonon softening that occurs in STO around the 108K structural transition. The $\Gamma_{25}$ $(111)$ zone edge phonon [@RefWorks:460; @RefWorks:458] becomes lower in energy as the transition is approached from both temperature directions. Fig.\[fig:Transport\]c, reproduced from Ref.[@RefWorks:458], shows the $\Gamma_{25}$ phonon energy as a function of temperature. The softening leads to a divergent increase in mode occupation or amplitude. The motion associated with this mode is a rotation of the TiO$_{6}$ octahedra. Below the transition temperature, the octahedra stabilize into a rotated antiferrodistortive (AFD) configuration accompanied by a tetragonal distortion of the unit cell. Since the film is mechanically constrained to the substrate at the atomic level, motions of the TiO$_{6}$ octahedra couple to the MnO$_{6}$ ones, inducing both static and dynamic changes in their configuration.
![Side view of STO-LSMO interface geometry. The plot shows calculated ground-state atomic positions. Away from from the interface, the STO is fixed to have bulk-like octahedral rotations around the $x$ axis (into the page). The LSMO geometry at the interface is modified by the STO; however, the LSMO relaxes to its bulk-like octahedral rotations around both in-plane axes within 2-3 unit cells. \[fig:interface\]](fig2)
We examine two mechanisms whereby the resistance of the LSMO layer might increase: $(i)$ static changes of the LSMO structure causing a change of electronic band parameters; $(ii)$ decreased carrier relaxation times due to enhanced phonon scattering, i.e. a dynamic effect. The static and dynamic contributions are reflected in the expression for the conductivity in the relaxation time approximation $\sigma_{ij}\propto\tau m_{ij}^{-1}$, where $\tau$ is the relaxation time and $m_{ij}^{-1}$ is the reciprocal effective mass tensor [@ashcroft].
To treat the temperature-dependent character of the coupling phenomena, we perform finite temperature simulations by building a classical model of the energetics of the system as a function of oxygen displacements. Our model includes harmonic coupling between oxygens, 4$^{th}$ order on-site anharmonic terms to stabilize the symmetry breaking, and lowest order coupling between oxygen displacements and stress, thus capturing the STO phase transition [@sto1]. Model parameters are obtained via density functional theory calculations using the spin-polarized PBE GGA functional [@GGA] and ultrasoft pseudopotentials [@ultrasoft]. Ground states for both bulk strained LSMO (using the virtual crystal approximation [@vca_vand]) and the LSMO/STO system are calculated, reproducing the experimental A-type ordering. In addition, the ground state calculation shows how the octahedral orientation is continuous going from substrate to film (see Fig.\[fig:interface\]), as was recently shown in a similar theoretical study [@rondinellioctahedra]. The harmonic interatomic force constants are calculated with DFT perturbation theory (DFPT) [@ModelH2], and the remaining parameters are fit to strained bulk calculations. We then perform classical Monte Carlo sampling on this model in a periodic box. The box contains $10\times10\times100$ perovskite unit cells composed of 60 STO and 40 LSMO unit cells in the $z$ direction.
To evaluate the role of static structural changes, we compute the conductivity tensor of bulk strained LSMO for the static octahedra rotation angles obtained from the Monte-Carlo model. The conductivity is calculated from direct first principles evaluation of the reciprocal effective mass tensor, by summing over all bands at the Fermi energy [@ashcroft]. The upper bound of conductivity change is estimated by using the angles at the LSMO/STO interface, which change the most due to the substrate-film coupling. We find that the static coupling effect appears only below the phase transition temperature. The phase transition causes the octahedra angles in the LSMO to increase somewhat; however, the magnitude of resulting change in conductivity is too small to account the experimental findings. The computed Mn-O-Mn hopping elements change only by 1-2% due to static structural changes, while the experimental conductivity changed by more than 10%.
Because the static structural change manifests only below 108K and does not yield a large enough change in conductivity, we examine whether the $\Gamma_{25}$ phonon might extend into the LSMO and cause dynamic carrier scattering. Hence, we compute the oxygen-oxygen correlation matrix $c_{ij}=\langle x_{i}x_{j}\rangle-\langle x_{i}\rangle\langle x_{j}\rangle$ where $x_{i}$ are the oxygen displacements from their equilibrium pseudocubic positions. We find that near the STO phase transition, the correlation length in STO diverges, as expected. Furthermore, oxygen motions in the interfacial LSMO layers become correlated with those deep in the STO (Fig.\[fig:correlation\]a) demonstrating that STO soft phonons extend into the LSMO. We quantify this relation more precisely by extracting the dominant eigenvectors of the correlation matrix, which are the softest phonon modes in the finite-temperature harmonic system. Fig. \[fig:correlation\]b shows the lowest frequency eigenvector: it decays exponentially into the LSMO with a decay length of 2.3 unit cells.
![\[fig:correlation\]DFT/Monte Carlo results of phonon transfer from STO to LSMO. a) Absolute magnitude of the correlation function of oxygen displacements with those deep in STO versus layer number at $T=T_{C}$. $x$ and $z$ indicate displacements stemming from octahedra rotation around the $x$ and $z$ axis respectively. (The $y$ component is equal to $x$ by symmetry). b) $x$ and $z$ components of the lowest frequency eigenvector at $T=T_{C}$. Layer-to-layer sign changes reflect the AFD nature of the oxygen displacments. ](fig3)
Building upon our theoretical results, we use the following simple model to fit the experimental resistivity data: the scattering due to the soft mode is $cne^{-2z/\lambda}$, where $n$ is the $\Gamma_{25}$ occupation number in STO (given by the Bose-Einstein function using the energies in Fig.\[fig:Transport\]c). $c$ is a conversion factor from $n$ to resistivity (linearly related in phonon scattering theory [@Ziman]); and $\lambda$ is the decay length of the induced octahedra motion amplitude ($\lambda/2$ is the decay length for the phonon number). The total conductivity is obtained by summing over the film layers: $\sigma=\sum\limits _{layers}(\rho_{\mathrm{base}}+cne^{-2z/\lambda})^{-1}$. $\rho_{\mathrm{base}}$ is the unperturbed LSMO resistivity, which we find by passing a smooth line under the cusp, following the method used in an electron paramagnetic resonance study of the softening-induced disorder in STO [@PhysRevB.7.1052]. This model describes the experimental data well, as shown in Fig.\[fig:fits\], and yields a decay length of 1.8uc, in good agreement with theory. This verifies our attribution of the resistivity cusp to resonant coupling of the divergent $\Gamma_{25}$ mode into the LSMO, where it strongly scatters the carriers by disturbing the Mn-O-Mn hopping path.
The phonon coupling picture also explains the dip in magnetic moment, when phonon-magnon interactions are taken into account. For the manganites, it has been shown theoretically [@RefWorks:484] that when phonons involving Mn or O distortions are added to the Heisenberg Hamiltonian, the magnon spectrum is softened. The phonons injected from the substrate cause a softening of the magnon spectrum in the LSMO, with maximal softening occuring at the STO transition. Magnon occupation increases with spectrum softening, leading to the reduction in the magnetic moment. The overlap between the temperature range of the transport cusp and the moment dip is striking (Fig.\[fig:Transport\]a), confirming that they are both driven by $\Gamma_{25}$ phonon softening.\
![\[fig:fits\]Model fitting: resistivity of an 11uc film of La$_{0.53}$Sr$_{0.47}$MnO$_{3}$ as a function of temperature, for several out-of-plane magnetic field values. The PZT overlayer is in the depletion state. Black dashed lines are interpolations excluding the cusp and red dashed lines are fits to $\Gamma_{25}$ phonon coupling.](fig4)
Previous work on the influence of the STO transition on manganite films dealt with effects attributed to the appearance of $a$ and $c$ domains in the STO and the resulting change in the strain state of the film below the transition temperature [@RefWorks:432; @RefWorks:434; @RefWorks:431]. In the current work, we find that effects appear both above and below the transition temperature and correspond to the temperature range of phonon softening. In addition, the short decay length that we find does not agree with a strain-mediated phenomenon. The LSMO remains strained to the substrate up to a thickness of at least 80 uc [@RefWorks:414], so that changes related to strain should be evident at these thicknesses as well. These two facts preclude the strain configuration of the STO below the transition from being the source of the cusp. To further verify that the cusp feature is independent of the $a/c$ domain structure formed in the STO, we applied an electric field of $2\times10^{5}$V/m using a back gate on the substrate. This field should break the symmetry between $a$ and $c$ domains and lead to a different domain structure compared to a zero field case. No difference in resistivity was observed with and without the electric field.
Our observation of resonant phonon-carrier coupling illuminates a key feature of conduction in LSMO. The coupling manifests strongly near the $x=0.5$ composition, while films of similar thickness at the $x=0.2$ composition [@CarlosPRLPaper] showed a resistivity cusp much smaller in magnitude. We relate this effect to an increased carrier coupling to the $\Gamma_{25}$ phonon in the AF-M phase of the LSMO. In this phase, the Mn $d_{x^{2}-y^{2}}$ $e_{g}$ orbital is occupied while the $d_{3z^{2}-r^{2}}$ orbital is depopulated [@RefWorks:489]. This causes the carriers’ wavefunctions to be concentrated on the $xy$ MnO$_{2}$ planes, which underpins the 2D character of metallicity and ferromagnetism in this phase, in contrast to the 3D character of the $x=0.2$ to $0.4$ composition range. The transport measurements of our thin films probe carrier hopping in the $x$ and $y$ directions. Perturbation of the bridging oxygen positions due to the $\Gamma_{25}$ phonon will have a larger scattering effect on carriers in the AF-M phase compared to the 3D FM phase. This is because in the AF-M case, the electron density case is concentrated closer to the perturbed oxygens in the $xy$ plane through which the conduction occurs. This configuration also explains why an out-of-plane magnetic field reduces the effect of the phonon coupling, as can be seen in Fig. \[fig:fits\]. The magnetic field causes the Mn spins to cant so that they are partially aligned out-of-plane. This allows for some inter-plane hopping and reduces the confinement of carriers to the $xy$ MnO$_{2}$ planes, similarly to the “spin valve” effect in A-type Nd$_{0.45}$Sr$_{0.55}$MnO$_{3}$ [@spinvalve].
In conclusion, we show how a single phonon mode originating in the substrate extends resonantly across an epitaxial interface and into the film. The effects of this coupling are amplified by the properties of both materials: phonon softening in the substrate causes the phonon amplitude to diverge, while the LSMO’s electronic phase and charge distribution are tuned using strain and a ferroelectric gate.
This work was supported by the National Science Foundation under Contract MRSEC No. DMR-0520495, DMR-1006265, and FENA. Computational resources were provided by Yale High Performance Computing, partially funded by grant CNS 08-21132 and by TeraGrid/NCSA under grant number TG-MCA08X007.
| ArXiv |
---
abstract: |
Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\operatorname{\textup{PU}}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with an essentially unique boundary map.
The Toledo invariant is a multiplicative constant, hence it remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive, the centralizer of $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$ is compact, $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations.
We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.
address: 'Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy'
author:
- 'A. Savini'
bibliography:
- 'biblionote.bib'
date: '. ©[ A. Savini 2020]{}'
title: Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
---
[^1]
.
Introduction
============
Given a torsion-free lattice $\Gamma \leq G$ in a semisimple Lie group $G$, any representation $\rho:\Gamma \rightarrow H$ into a locally compact group $H$ induces a well-defined map at the level of continuous bounded cohomology groups. Hence fixed a preferred bounded class in the cohomology of $H$, one can pullback it and compare the resulting class with the fundamental class determined by $\Gamma$ via Kronecker pairing. This is a standard way to obtain *numerical invariants* for representations, whose importance has become evident in the study of rigidity and superrigidity properties. Indeed a numerical invariant has bounded absolute value and the maximum is attained if and only if the representation can be extended to a representation $G \rightarrow H$ of the ambient group.
Several examples of these phenomena are given by the work of Bucher, Burger, Iozzi [@iozzi02:articolo; @bucher2:articolo; @BBIborel] in the case of representations of real hyperbolic lattices, by Burger and Iozzi [@BIcartan] and by Duchesne and Pozzetti [@Pozzetti; @duchesne:pozzetti] for complex hyperbolic lattices and by the work of Burger, Iozzi and Wienhard [@BIW07; @BIW09; @BIW1] when the target group is of Hermitian type. In the latter case, of remarkable interest is the analysis of the representation space $\textup{Hom}(\Gamma,G)$ when $G$ is a group of Hermitian type and $\Gamma$ is a lattice in a finite connected covering of $\operatorname{\textup{PU}}(1,1)$, that is a hyperbolic surface group. Burger, Iozzi and Wienhard [@BIW1] exploited the existence of a natural Kähler structure on the Hermitian symmetric space associated to $G$ in order to define the notion of *Toledo invariant* of a representation $\rho:\Gamma \rightarrow G$. That invariant has bounded absolute value and its maximality has important consequences on the Zariski closure $\mathbf{H}=\overline{\rho(\Gamma)}^Z$ of the image of the representation. Indeed the authors show that in the case of maximality $\mathbf{H}$ is reductive, $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$ has compact centralizer and it is of tube type and the representation $\rho$ is injective with discrete image and it preserves a unique maximal tube-type subdomain [@BIW1 Theorem 5]. A domain is of *tube-type* if it can be written in the form $V+i\Omega$, where $V$ is a real vector space and $\Omega \subset V$ is an open convex cone. Maximal tube-type subdomains in a Hermitian symmetric space $\operatorname{\mathcal{X}}$ generalize the notion of complex geodesic in $\operatorname{\mathbb{H}}^n_{\operatorname{\mathbb{C}}}$ and they are all $G$-conjugated.
Partial results in the direction of [@BIW1 Theorem 5] were obtained by several authors. For instance when $G=\operatorname{\textup{PU}}(n,1)$ with $n \geq 2$, Toledo [@Toledo89] proved that maximal representations must preserve a complex geodesic. It is worth mentioning also the papers by Hernández [@Her91], by Koziarz and Maubon [@koziarz:maubon] and by Bradlow, García-Prada and Gothen [@garcia:geom; @garcia:dedicata]. In the latter case those results were obtained using different techniques based on the notion of Higgs bundle.
It is worth noticing that in the particular case of split real groups and surfaces without boundary, the set of maximal representations contains the Hitchin component [@hitchin]. The Hitchin component has been sistematically studied by serveral mathematicians. For instance Labourie [@labourie] focused his attention on the Asonov property, whereas Fock and Goncharov [@Fock:adv; @fock:hautes] related the Hitchin component with the notion of Lusztig’s positivity.
A crucial point in the proof of [@BIW1 Theorem 5] is that maximal representations are *tight*, that is the seminorm of the pullback class is equal to the norm of the bounded Kähler class. The tightness property has an analytic counterpart in terms of maps between symmetric spaces and Burger, Iozzi and Wienhard [@BIW09] give a complete characterization of tight subgroups of a Lie group of Hermitian type.
Recently the author [@savini3:articolo] together with Moraschini [@moraschini:savini; @moraschini:savini:2] and Sarti [@savini:sarti] has applied bounded cohomology techniques to the study measurable cocycles with an essentially unique boundary map. The existence of a boundary map allows to define a pullback in bounded cohomology as in [@burger:articolo] and hence to develop a theory of numerical invariants, called *multiplicative constants*, also in the context of measurable cocycles.
The main goal of this paper is the study of measurable cocycles of surface groups. Let $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$. Consider a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$ and let $\mathbf{G}$ be a semisimple real algebraic group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is of Hermitian type. If a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ admits an essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow G$, then we can apply the theoretical background developed in [@moraschini:savini; @moraschini:savini:2] to defined the *Toledo invariant of $\sigma$*. In an analogous way to what happens for representations, the Toledo invariant is constant along $G$-cohomology classes and has absolute value bounded by $\operatorname{rk}(\operatorname{\mathcal{X}})$, the rank of the symmetric space $\operatorname{\mathcal{X}}$ associated to $G$. Thus it makes sense to speak about *maximal measurable cocycles*. This will be a particular example of *tight cocycles* (see Definition \[def:tight:cocycle\]).
Maximality allows to give a characterization of the *algebraic hull* of a measurable cocycle, as stated in the following
\[teor:maximal:alghull\] Let $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$ and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Let $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is maximal, then
1. the algebraic hull $\mathbf{H}$ is reductive
2. the centralizer $Z_G(H)$ is compact;
3. the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is Hermitian of tube-type;
4. it holds $\mathbf{H}(\operatorname{\mathbb{R}}) \subset \textup{Isom}(\operatorname{\mathcal{T}})$ for some maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$. Equivalently there exists a cocycle cohomologous to $\sigma$ which preserves $\operatorname{\mathcal{T}}$.
The above theorem should be interpreted as a suitable adaptation of [@BIW1 Theorem 5] to the context of maximal measurable cocycles. The first two properties are immediate consequences of the tightness of maximal cocycles, as shown in Theorem \[teor:alg:hull:tight\]. The tube-type condition is more involving and it is proved in Theorem \[teor:symmetric:tube\].
We conclude with some remarks about boundary maps of maximal Zariski dense cocycles. For representations, the relation between maximality and boundary maps preserving positivity of triples were studied by Guichard [@Guichard], Labourie [@labourie] and Fock and Goncharov [@fock:hautes]
Here we attempt to extend [@BIW1 Theorem 5.2] to the context of measurable cocycles. Given a maximal Zariski dense cocycle, we can construct a boundary map which has left-continuous (respectively right-continuous) slices. Moreover each slice preserves *transversality* and it is *monotone*, as proved in Theorem \[teor:boundary:map\]. Unfortunately, to get the statement, we need to make an additional assumption on the measurable map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. More precisely we need to assume that the essential image of almost every slice intersects nicely all closed algebraic subset of $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ (Assumption \[ass:zariski:zero:measure\]). This assumption is clearly verified by cocycles cohomologous to maximal Zariski dense representations, but we do not know more generally under which conditions of both $\sigma$ or $\phi$ this is true and it would be interesting to know it. The proof of Theorem \[teor:boundary:map\] follows the line of [@BIWL Section 8] and of [@BIW1 Theorem 5.2].
Plan of the paper {#plan-of-the-paper .unnumbered}
-----------------
In Section \[sec:preliminary\] we recall the preliminary definitions and results that we need in the paper. In Section \[sec:measurable:cocycles\] we remind the notion of measurable cocycle and of cohomology class determined by a cocycle. Of particular importance for our purpose will be the definition of algebraic hull. Then we conclude the section with some elements of boundary theory. Section \[sec:burger:monod\] is devoted to continuous and continuous bounded cohomology. We remind the functorial approach by Burger and Monod and we recall the definition of pullback induced by a boundary map. The last part is devoted to Hermitian symmetric spaces (Section \[sec:hermitian:groups\]).
The main theorem of paper is proved in Section \[sec:maximal:cocycles\]. We first introduce the notion of Toledo invariant of a measurable cocycle in Section \[sec:toledo:invariant\]. In Section \[sec:maximal:cocycle:thm\] it appears the definition of maximal cocycle. Maximal cocycles are tight by Proposition \[prop:maximal:tight\] and this result togheter with Theorem \[teor:symmetric:tube\] allows to prove Theorem \[teor:maximal:alghull\]. We conclude with Section \[sec:boundary:map\], where we prove Theorem \[teor:boundary:map\].
Preliminary definitions and results {#sec:preliminary}
===================================
Measurable cocycles {#sec:measurable:cocycles}
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The following section is devoted to a quick review about measurable cocycles theory. We are going to recall the definitions of both measurable cocycle and cohomology class. Then we will introduce the notion of algebraic hull and we will conclude the section with some elements of boundary theory. For a more detailed discussion about those topics we refer the reader to the work of both Furstenberg [@furst:articolo73; @furst:articolo] and Zimmer [@zimmer:preprint; @zimmer:annals; @zimmer:libro].
Consider two locally compact second countable groups $G,H$ endowed with their Haar measurable structure. Given a standard Borel measure space $(\Omega,\mu_\Omega)$ we say that it is a *$G$-space* if $G$ acts on $\Omega$ by measure-preserving transformations. Additionally if $\mu_\Omega$ is a probability measure, we are going to call $(\Omega,\mu_\Omega)$ a *standard Borel probability $G$-space*. Given another measure space $(\Theta,\nu)$, we are going to denote by $\textup{Meas}(X,Y)$ the space of measurable functions with the topology of the convergence in measure.
\[def:measurable:cocycle\] Let $G,H$ two locally compact second countable groups and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $G$-space. A measurable function $\sigma:G \times \Omega \rightarrow H$ is a *measurable cocycle* if it holds $$\label{eq:measurable:cocycle}
\sigma(g_1 g_2,s)=\sigma(g_1,g_2 s)\sigma(g_2,s) \ ,$$ for almost every $g_1,g_2 \in G$ and almost every $s \in \Omega$.
Measurable cocycles are quite ubiquitous in mathematics and Equation can be suitably interpreted as a naive generalization to the measurable context of the chain rule for differentiation of smooth functions. By writing a measurable cocycle $\sigma$ as an element $\sigma \in \textup{Meas}(G,\textup{Meas}(\Omega,H))$, Equation boils down the cocycle condition. Indeed $\sigma$ may be interpreted as a Borel $1$-cocycle in the sense of Eilenberg-MacLane (see [@feldman:moore; @zimmer:preprint] for more details about this interpretation). Following this line, one could naturally ask when two different cocycles are cohomologous.
\[def:cohomologous:cocycles\] Let $\sigma:G \times \Omega \rightarrow H$ be a measurable cocycle and let $f:\Omega \rightarrow H$ be a measurable function. The *$f$-twisted cocycle of $\sigma$* is defined as $$\sigma^f:G \times \Omega \rightarrow H, \ \ \sigma^f(g,s):=f(gs)^{-1}\sigma(g,s)f(s) \ .$$ We say that two cocycles $\sigma_1,\sigma_2:G \times \Omega \rightarrow H$ are *cohomologous* if there exists a measurable function $f:\Omega \rightarrow H$ such that $$\sigma_2^f=\sigma_1 \ .$$
Choosing a measurable function $f:\Omega \rightarrow H$ is a typical way to construct cocycles starting from representations. Indeed, given a continuous representation $\rho:G \rightarrow H$, one can verifiy that the measurable function $$\sigma_\rho:G \times \Omega \rightarrow H \ , \ \ \sigma_\rho(g,s):=\rho(g) \ ,$$ is a measurable cocycle as a consequence of the morphism condition. This allows to see representation theory into the wider world of measurable cocycles theory. Additionally this offers us the possibility to interpret the notion of cohomologous cocycles as a generalization of conjugated representations.
Given a representation $\rho:G \rightarrow H$, if the image is not closed, it is quite natural to consider its closure, which it is still a subgroup of $H$. Unfortunately the image of a cocycle has no structure a priori. Nevertheless, if $H$ corresponds to the real points of a real algebraic group, then there is a notion which is in some sense similar to take the closure of the image of a representation.
Suppose that $\mathbf{H}$ is a real algebraic group. Let $\sigma:G \times \Omega \rightarrow \mathbf{H}(\operatorname{\mathbb{R}})$ be a measurable cocycle. The *algebraic hull associated to $\sigma$* is (the conjugacy class of) the smallest algebraic subgroup $\mathbf{L}$ of $\mathbf{H}$ such that $\mathbf{L}(\operatorname{\mathbb{R}})$ contains the image of a cocycle cohomologous to $\sigma$.
As proved in [@zimmer:libro Proposition 9.2] this notion is well-defined by the descending chain condition on algebraic subgroups and it depends only the cohomology class of the cocycle.
We conclude this brief discussion about measurable cocycle introducing some elements of boundary theory. In order to do this, we are going to assume that $G$ is a semisimple Lie group of non-compact type. Let $Q$ be any parabolic subgroup of $G$ and let $Y$ be a measurable $H$-space.
\[def:boundary:map\] Let $\sigma:G \times \Omega \rightarrow H$ be a measurable cocycle. A *(generalized) boundary map* is a measurable map $\phi:G/Q \times \Omega \rightarrow Y$ which is $\sigma$-equivariant, that is $$\phi(g \xi,g s)=\sigma(g,s)\phi(\xi,s) \ ,$$ for every $g \in G$ and almost every $\xi \in G/Q, s \in \Omega$.
It is easy to check that, if $\phi:G/Q \times \Omega \rightarrow Y$ is a boundary map for $\sigma$, then $\phi^f:G/Q \times \Omega \rightarrow Y, \ \phi^f(\xi,s):=f(s)^{-1}\phi(\xi,s)$ is a boundary map for $\sigma^f$ for any measurable function $f:\Omega \rightarrow H$.
The existence and the uniqueness of a boundary map associated to a cocycle $\sigma$ rely on the dynamical properties of $\sigma$. For a more detailed discussion about it we refer the reader to [@furst:articolo]. Boundary maps for measurable cocycles will be crucial to define a pullback map in bounded cohomology imitating the work done by Burger and Iozzi [@burger:articolo] in the case of representations.
Continuous bounded cohomology and functorial approach {#sec:burger:monod}
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Given a locally compact group $G$ we are going to remind the notion of continuous and continuous bounded cohomology groups of $G$. A remarkable aspect of continuous bounded cohomology is that it can be computed using any strong resolution by relatively injective modules. We are going to give few details about this functorial apprach and we will conclude the section by introducing the notion of pullback along a boundary map associated to a measurable cocycle. For more details about continuous bounded cohomology and its functorial approach we refer to the work of Burger and Monod [@burger2:articolo; @monod:libro], whereas we refer to the papers of the author together with Moraschini [@moraschini:savini; @moraschini:savini:2] for more details about pullback along boundary maps.
Consider a *Banach $G$-module* $E$, that is $E$ is a Banach space with an isometric action $\pi:G \rightarrow \textup{Isom}(E)$. We are going to assume that $E$ is the dual of some separable Banach space, so that it makes sense to speak about the weak-${}^\ast$ topology on $E$.
We consider the set $$\begin{aligned}
\operatorname{\textup{C}}^\bullet_{cb}(G;E):=\{ f : G^{\bullet+1} \rightarrow E \ | & \ f \ \textup{is continuous and} \\
&\lVert f \rVert_\infty:=\sup_{g_0,\ldots,g_{\bullet}}\lVert f(g_0,\ldots,g_{\bullet}) \rVert_E < \infty \} \ , \end{aligned}$$ where $\lVert \ \cdot \ \rVert_E$ denotes the norm on the space $E$. Each $\operatorname{\textup{C}}^\bullet_{cb}(G;E)$ is a normed via the supremum norm and it can be endowed with an isometric action by $G$ defined by $$(gf)(g_0,\ldots,g_\bullet):=\pi(g)f(g^{-1}g_0,\ldots,g^{-1}g_\bullet) \ ,$$ where $f \in \operatorname{\textup{C}}^\bullet_{cb}(G;\operatorname{\mathbb{R}})$ and $g,g_0,\ldots,g_\bullet \in G$. Defining the *standard homogeneous coboundary operator* by $$\delta^\bullet:\operatorname{\textup{C}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{C}}^{\bullet+1}_{cb}(G;E) \ ,$$ $$\delta^\bullet(f)(g_0,\ldots,g_{\bullet+1}):=\sum_{i=0}^{\bullet+1}(-1)^i f(g_0,\ldots,\hat g_i,\ldots,g_{\bullet+1}) \ ,$$ we get a cochain complex $(\operatorname{\textup{C}}^\bullet_{cb}(G;E),\delta^\bullet)$.
\[def:bounded:cohomology\] Let $G$ be a locally compact group and let $E$ be a Banach $G$-module. The *$k$-th continuous bounded cohomology group* of $G$ with coefficients in $E$ is the $k$-th cohomology group of the $G$-invariant subcomplex $(\operatorname{\textup{C}}^\bullet_{cb}(G;E)^G,\delta^\bullet)$, that is $$\operatorname{\textup{H}}^k_{cb}(G;E):=\operatorname{\textup{H}}^k(\operatorname{\textup{C}}^\bullet_{cb}(G;E)^G) \ ,$$ for every $k \geq 0$.
It is worth noticing that each cohomology group $\operatorname{\textup{H}}^\bullet_{cb}(G;E)$ has a natural seminormed structure inherited by the normed structure on the continuous bounded cochains.
By dropping the assumption of boundedness one can define similarly the complex of continuous cochains $(\operatorname{\textup{C}}^\bullet_c(G;E),\delta^\bullet)$ and the standard inclusion $i:\operatorname{\textup{C}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{C}}^\bullet_c(G;E)$ induces a map at a cohomological level $$\textup{comp}^\bullet:\operatorname{\textup{H}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{H}}^\bullet_c(G;E) \ ,$$ called *comparison map*.
Computing continuous bounded cohomology of a locally compact group $G$ using only the definition given above may reveal quite difficult. For this reason Burger and Monod [@burger2:articolo; @monod:libro] introduced a way to compute continuous bounded cohomology groups based on the notion of resolutions. More precisely the authors showed [@burger2:articolo Corollary 1.5.3] that given any Banach $G$-module $E$ and any strong resolution $(E^\bullet,d^\bullet)$ of $E$ by relatively injective Banach $G$-modules, it holds $$\operatorname{\textup{H}}^k_{cb}(G;E) \cong \operatorname{\textup{H}}^k((E^\bullet)^G) \ ,$$ for every $k \geq 0$. Since we will not need the notion of strong resolution and of relatively injective Banach $G$-module, we omit them and we refer to the book of Monod [@monod:libro] for more details.
Unfortunately the isomorphism given above it is not isometric a priori, that is it may not preserve the seminormed structure. Nevertheless there are specific resolutions for which the isomorphism it is actually isometric. This is the case for instance when we consider the resolution of essentially bounded weak-$^\ast$ measurable functions $(\operatorname{\textup{L}}^\infty_{\textup{w}^\ast}((G/Q)^{\bullet+1};E),\delta^\bullet)$ on the quotient $G/Q$ [@burger2:articolo Theorem 1], where $G$ is a semisimple Lie group of non-compact type and $Q$ is any parabolic subgroup. We are going to exploit this result for the Shilov boundary of a Hermitian symmetric space.
We conclude this brief section by recalling the pullback determined by a boundary map associated to a measurable cocycle. Suppose that $G$ is a semisimple Lie group of non-compact type and consider a parabolic subgroup $Q \leq G$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $G$-space and let $Y$ be any measurable $H$-space, where $H$ is another locally compact group. Denote by $(\operatorname{\mathcal{B}}^\infty_{\textup{w}^\ast}(Y^{\bullet+1};E),\delta^\bullet)$ the complex of weak-$^\ast$ measurable bounded functions on $Y$ (with the injection of coefficients, the latter is a strong resolution of $E$ by [@burger:articolo Proposition 2.1]). Given a boundary map $\phi:G/Q \times \Omega \rightarrow Y$ associated to a measurable cocycle $\sigma:G \times \Omega \rightarrow H$, there exists a natural map defined at the level of cochains as $$\operatorname{\textup{C}}^\bullet(\Phi^\Omega):\operatorname{\mathcal{B}}^\infty(Y^{\bullet+1};E)^H \rightarrow \operatorname{\textup{L}}^\infty((G/Q)^{\bullet+1};E)^G \ ,$$ $$\operatorname{\textup{C}}^\bullet(\Phi^\Omega)(\psi)(\xi_0,\ldots,\xi_\bullet):=\int_{\Omega} \psi(\phi(\xi_0,s),\ldots,\phi(\xi_\bullet,s))d\mu_\Omega(s) \ .$$
As shown by the author and Moraschini [@moraschini:savini; @moraschini:savini:2], the above map is a chain map which does not increase the norm and it induces a well define map in cohomology $$\operatorname{\textup{H}}^\bullet(\Phi^\Omega):\operatorname{\textup{H}}^\bullet(\operatorname{\mathcal{B}}(Y^{\bullet+1};E)^H) \rightarrow \operatorname{\textup{H}}^\bullet_{cb}(G;E) \ , \ \operatorname{\textup{H}}^\bullet(\Phi^\Omega)([\psi]):=[\operatorname{\textup{C}}^\bullet(\Phi^\Omega)(\psi)] \ .$$ The map $\operatorname{\textup{H}}^\bullet(\Phi^\Omega)$ is the *pullback induced by the boundary map $\phi$*. We are going to use the pullback map in order to define properly the Toledo invariant of a measurable cocycle of a surface group.
Lie groups of Hermitian type {#sec:hermitian:groups}
----------------------------
In this section we are going to recall the main definitions and result about Lie groups of Hermitian type. We are going to remind the notion of Shilov boundary for a Hermitian symmetric space and we are going to define a suitable cocycle on it, called Bergmann cocycle, which will enable us to define the notion of maximality for measurable cocycles of surface groups. For a more detailed discussion about these notions, we refer mainly to the work of Burger, Iozzi and Wienhard [@BIW07; @BIW09; @BIW1].
\[def:hermitian:symmetric:space\] Let $\operatorname{\mathcal{X}}$ be a Riemannian symmetric space and denote by $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$ the connected component of the identity of the isometry group associated to $\operatorname{\mathcal{X}}$. We say that $\operatorname{\mathcal{X}}$ is *Hermitian* if there exists a $G$-invariant complex structure $\operatorname{\mathcal{J}}$ on $\operatorname{\mathcal{X}}$. Given a semisimple real algebraic group $\mathbf{G}$, we say that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is *of Hermitian type* if its symmetric space $\operatorname{\mathcal{X}}$ is Hermitian.
Among all the possible ones, a family of examples of particular interest in this paper will be the one of Hermitian symmetric spaces of tube-type. We say that a Hermitian symmetric space $\operatorname{\mathcal{X}}$ is *of tube-type* if it is biholomorphic to a complex subset of the form $V+i\Omega$, where $V$ is a real vector space and $\Omega \subset V$ is a proper convex cone. A typical example is given by the hyperbolic space $\operatorname{\mathbb{H}}^2$ associated to the group $\operatorname{\textup{PU}}(1,1)$, or more generally to the symmetric space associated to $\operatorname{\textup{PU}}(p,p)$ when $p \geq 2$.
A Hermitian symmetric space $\operatorname{\mathcal{X}}$ can be bihomolorphically realized as bounded convex domain $\operatorname{\mathcal{D}}_{\operatorname{\mathcal{X}}}$ in $\operatorname{\mathbb{C}}^n$. For such a realization, the group $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$ acts via biholomorphisms and its action can be extended in a continuous way to the boundary $\partial \operatorname{\mathcal{D}}_{\operatorname{\mathcal{X}}}$. Unfortunately the latter is not a homogeneous $G$-space, but it admits a unique closed $G$-orbit. The latter will be identified with the Shilov boundary.
More precisely we give first the following
\[def:shilov:boundary\] Let $\operatorname{\mathcal{D}}\subset \operatorname{\mathbb{C}}^n$ be a bounded domain. The *Shilov boundary* $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{D}}}$ of $\operatorname{\mathcal{D}}$ is the unique closed subset of $\partial \operatorname{\mathcal{D}}$ such that, given a function $f$ continuous on $\overline{\operatorname{\mathcal{D}}}$ and holomorphic on $\operatorname{\mathcal{D}}$, then $$\max_{\overline{D}}|f|=\max_{\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{D}}}} |f| \ .$$ Given a Hermitian symmetric space $\operatorname{\mathcal{X}}$, we denote by $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ the Shilov boundary associated to the bounded realization of $\operatorname{\mathcal{X}}$ and we call it *the Shilov boundary of $\operatorname{\mathcal{X}}$*.
As already anticipated the Shilov boundary associated to a Hermitian symmetric space $\operatorname{\mathcal{X}}$ is a homogeneous $G$-space. Indeed if we denote by $\mathbf{G}$ the algebraic group associated to the complexified Lie algebra of $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$, then there exists a maximal parabolic subgroup $\mathbf{Q} \subset \mathbf{G}$ such that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ can be identified with $\mathbf{G}/\mathbf{Q}(\operatorname{\mathbb{R}})$. In particular $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is an amenable $G$-space in the sense of Section \[sec:burger:monod\] and hence the resolution of essentially bounded functions on it computes isometrically the continuous bounded cohomology of $G$.
In order to describe more accurately the second bounded cohomology group of $G$, recall that if $\operatorname{\mathcal{X}}$ is Hermitian, then there exists a $G$-invariant complex structure $\operatorname{\mathcal{J}}$ on it. If we denote by $g$ the $G$-invariant Riemannian metric on $\operatorname{\mathcal{X}}$, we can define the *Kähler form* $$(\omega_{\operatorname{\mathcal{X}}})_a(X,Y):=g_a(X,\operatorname{\mathcal{J}}_a Y) \ ,$$ for any $X,Y \in T_a \operatorname{\mathcal{X}}$. Being $G$-invariant, the form $\omega_{\operatorname{\mathcal{X}}}$ is automatically closed by Cartan’s Lemma. Define now $$\label{eq:cocycle:symmetric:space}
\beta_{\operatorname{\mathcal{X}}}: (\operatorname{\mathcal{X}})^{(3)} \rightarrow \operatorname{\mathbb{R}}, \ \ \beta_{\operatorname{\mathcal{X}}}(a_1,a_2,a_3):=\int_{\Delta(a_1,a_2,a_3)} \omega_{\operatorname{\mathcal{X}}} \ ,$$ where $\Delta(a_1,a_2,a_3)$ is any triangle with geodesic sides determined by $a_1,a_2,a_3 \in \operatorname{\mathcal{X}}$. Since $\omega_{\operatorname{\mathcal{X}}}$ is closed, by Stokes Theorem the function $\beta_{\operatorname{\mathcal{X}}}$ is an alternating $G$-invariant cocycle on $\operatorname{\mathcal{X}}$. Remarkably the cocycle $\beta_{\operatorname{\mathcal{X}}}$ can be extended to a strict measurable $G$-invariant cocycle on the Shilov boundary $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ [@BIW07 Corollary 3.8] and its absolute value is bounded $\operatorname{rk}(\operatorname{\mathcal{X}})$. We are going to denote such an extension with $\beta_{\operatorname{\mathcal{X}}}$ with an abuse of notation. As previously said in Section \[sec:burger:monod\] the cocycle $\beta_{\operatorname{\mathcal{X}}} \in \operatorname{\textup{L}}^\infty((\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)};\operatorname{\mathbb{R}})^G$ determines canonically a class in $\operatorname{\textup{H}}^2_{cb}(G;\operatorname{\mathbb{R}})$.
We call *Bergmann cocycle* the measurable extension $\beta_{\operatorname{\mathcal{X}}}: \check{\operatorname{\mathcal{S}}}^{(3)} \rightarrow \operatorname{\mathbb{R}}$ to the Shilov boundary of the cocycle defined by Equation .
We denote by $\kappa^b_G \in \operatorname{\textup{H}}^2_{cb}(G;\operatorname{\mathbb{R}})$ the class determined by $\beta_{\operatorname{\mathcal{X}}}$ and we call it *bounded Kähler class*.
Recall that two points $\xi,\eta \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ are *transverse* if they lie in the unique open $G$-orbit in $(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^2$. We conclude the section by recalling some properties of the Bergmann cocycle when $\operatorname{\mathcal{X}}$ is a Hermitian symmetric space of tube-type. As stated in [@BIW1 Lemma 5.5], if $\operatorname{\mathcal{X}}$ is of tube-type then
1. the cocycle $\beta_{\operatorname{\mathcal{X}}}$ takes values in the discrete set $\{-\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}, - \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}+1 , \ldots , \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-1, \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \}$;
2. if it holds $|\beta_{\operatorname{\mathcal{X}}}(\xi,\eta,\omega)|=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$, then $\xi,\eta,\omega$ are pairwise transverse;
3. we can decompose $$(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}= \sqcup_{i=0}^{\operatorname{rk}(\operatorname{\mathcal{X}})} \operatorname{\mathcal{O}}_{-\operatorname{rk}(\operatorname{\mathcal{X}})+2i} \ ,$$ where $\operatorname{\mathcal{O}}_{-\operatorname{rk}(\operatorname{\mathcal{X}})+2i}$ is the open subset of $(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$ where $\beta_{\operatorname{\mathcal{X}}}$ is identically equal to $-\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}+i$;
4. given $\xi, (\xi_n)_{n \in \operatorname{\mathbb{N}}}, (\xi'_n)_{n \in \operatorname{\mathbb{N}}}$ where $\xi,\xi_n,\xi'_n \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, if $\lim_{n \to \infty} \xi_n =\xi$ and $\beta_{\operatorname{\mathcal{X}}}(\xi,\xi_n,\xi'_n)=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$ then $\lim_{n \to \infty} \xi'_n=\xi$.
Maximal measurable cocycles of surface groups {#sec:maximal:cocycles}
=============================================
Let $L$ be a finite connected covering of the group $\operatorname{\textup{PU}}(1,1)$ and consider a torsion-free lattice $\Gamma \leq L$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Given an irreducible Hermitian symmetric space $\operatorname{\mathcal{X}}$, we are going to denote by $G=\text{Isom}^\circ(\operatorname{\mathcal{X}})$ the connected component of the identity of the isometry group of $\operatorname{\mathcal{X}}$. In this section we are going to introduce the notion of *Toledo invariant* associated to a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with an essentially unique boundary map. Since this numerical invariant will have absolute value bounded from above by the rank of $\operatorname{\mathcal{X}}$, it will make sense to talk about *maximal cocycles*. Maximal cocycles will be particular examples of *tight cocycles*. We are going to introduce the notion of *tightness* which will have important consequences on the algebraic hull.
We are going to show that if a maximal cocycle is Zariski dense then the Hermitian symmetric space $\operatorname{\mathcal{X}}$ must be of tube-type. Hence there are no maximal Zariski dense cocycle in Hermitian Lie group which are not of tube-type. Moreover, maximality affects also the regularity property of the slices of boundary maps.
The Toledo invariant of a measurable cocycle {#sec:toledo:invariant}
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Let $L$ be a finite connected covering of the group $\operatorname{\textup{PU}}(1,1)$ and consider a torsion-free lattice $\Gamma \leq L$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Denote by $G=\text{Isom}^\circ(\operatorname{\mathcal{X}})$ the connected component of the identity of the isometry group of an irreducible Hermitian symmetric space $\operatorname{\mathcal{X}}$. Let $\mathbf{G}$ be the connected Lie group associated to the complexified Lie algebra of $G$, so that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$. Let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. Here $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is the Shilov boundary associated to the symmetric space $\operatorname{\mathcal{X}}$.
Recall by Section \[sec:burger:monod\] that the existence of the boundary map $\phi$ induces a pullback map in cohomology $$\operatorname{\textup{H}}^\bullet(\Phi^\Omega): \operatorname{\textup{H}}^\bullet(\operatorname{\mathcal{B}}^\infty((\check{S}_{\operatorname{\mathcal{X}}})^{\bullet+1};\operatorname{\mathbb{R}})^G) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \ .$$ In particular we are allowed to consider the pullback of the Bergmann cocycle $\beta_{\operatorname{\mathcal{X}}}$. Since $\Gamma$ is a lattice of $L$, we have a well-defined *transfer map*, which is given at the level of cochains by $$\hat{\operatorname{\textup{T}}}^\bullet_b:\operatorname{\textup{L}}^\infty((\operatorname{\mathbb{S}}^1)^{\bullet+1};\operatorname{\mathbb{R}})^\Gamma \rightarrow \operatorname{\textup{L}}^\infty((\operatorname{\mathbb{S}}^1)^{\bullet+1};\operatorname{\mathbb{R}})^L \ ,$$ $$\hat{\operatorname{\textup{T}}}^\bullet_b(\psi)(\xi_0,\ldots,\xi_\bullet):=\int_{\Gamma \backslash L} \psi(\overline{g}\xi_0,\ldots,\overline{g}\xi_\bullet)d\mu_{\Gamma \backslash L}(\overline{g}) \ ,$$ where $\overline{g}$ denotes the equivalence class of $g$ in $\Gamma \backslash L$ and $\mu_{\Gamma \backslash L}$ is the normalized $L$-invariant measure on the quotient. Being a chain map, $\hat \operatorname{\textup{T}}^\bullet_b$ induces a well-defined map in cohomology called *transfer map* $$\operatorname{\textup{T}}^\bullet_b:\operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_{cb}(L;\operatorname{\mathbb{R}}), \hspace{5pt} \operatorname{\textup{T}}^\bullet_b([\psi]):=[\hat \operatorname{\textup{T}}^\bullet_b(\psi)] \ .$$
It is worth recalling that the bounded Kähler class $\kappa^b_L$ is a generator of the group $\operatorname{\textup{H}}^2_{cb}(L;\operatorname{\mathbb{R}})$ and it is represented by the Bergmann cocycle on the circle $\beta_{\operatorname{\mathbb{S}}^1}$ (which is nothing else than the orientation cocycle). In this particular setting, we are allowed to give the following
\[def:toledo:inv\] Let $\Gamma \leq L$ be a torsion-free lattice and $(\Omega,\mu_\Omega)$ a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. The *Toledo invariant* $\textup{t}_b(\sigma)$ associated to $\sigma$ is defined as $$\label{eq:toledo}
\operatorname{\textup{T}}^2_b \circ \operatorname{\textup{H}}^2(\Phi^\Omega)([ \beta_{\operatorname{\mathcal{X}}}])=\textup{t}_b(\sigma)[\beta_{\operatorname{\mathbb{S}}^1}]=\textup{t}_b(\sigma)\kappa^b_L \ .$$
Since the $\Gamma$-action on the circle is doubly ergodic and the cocycles are alternating, Equation holds actually at the level of cochains, that is $$\begin{aligned}
\label{eq:formula}
\int_{\Gamma \backslash L} \int_\Omega & \beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))d\mu_\Omega(s)d\mu_{\Gamma \backslash L}(\overline{g}) =\\
=& \text{t}_b(\sigma)\beta_{\operatorname{\mathbb{S}}^1}(\xi,\eta,\omega) \nonumber \ ,\end{aligned}$$ and the equation holds for *every* $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$, as a consequence of either Burger and Iozzi [@BIcartan] or Pozzetti [@Pozzetti], for instance. Notice that Equation is simply a suitable adaptation of [@BIW1 Corollary 4.4] to the context of measurable cocycles.
It is immediate to verify that the Toledo invariant is a *multiplicative constant* in the sense of [@moraschini:savini:2 Definition 3.16]. Indeed following the notation of that paper, the setting required by [@moraschini:savini:2 Definition 3.16] is satisfied and one has $$\textup{t}_b(\sigma)=\lambda_{\beta_{\operatorname{\mathcal{X}}},\beta_{\operatorname{\mathbb{S}}^1}}(\sigma) \ .$$ Thanks to this analogy, one can immediately argue that $\textup{t}_b(\sigma)$ is invariant along the $G$-cohomology class of $\sigma$ and its absolute value can be bounded from above as follows $$|\text{t}_b(\sigma)| \leq \operatorname{rk}(\operatorname{\mathcal{X}}) \ .$$
\[oss:alternative:definition\] We could have defined the Toledo invariant in a different way. Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Denote by $\Sigma$ the finite-area surface obtained as the quotient of $\operatorname{\mathbb{H}}^2_{\operatorname{\mathbb{R}}}$ by $\Gamma$, that is $\Sigma=\Gamma \backslash \operatorname{\mathbb{H}}^2_{\operatorname{\mathbb{R}}}$. If $\Gamma$ is *uniform* we know that $\Sigma$ is closed, whereas when $\Gamma$ is *non-uniform* then the surface $\Sigma$ has finitely many cusps. In the latter case we are going to denote by $S$ a *compact core* of $\Sigma$, otherwise we set $S=\Sigma$.
Following [@moraschini:savini Section 3.4] we can define the following composition of functions $$\label{eq:j:composition}
\operatorname{\textup{J}}^\bullet_{S, \partial S}: \operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Sigma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Sigma,\Sigma \setminus S) \rightarrow \operatorname{\textup{H}}^\bullet_b(S,\partial S) \ ,$$ where the first map is the isomorphism given by the Gromov’s Mapping Theorem [@Grom82; @Ivanov; @FM:grom], the second map is obtained by the long exact sequence in bounded cohomology [@BBFIPP] and the last map is induced by the homotopy equivalence $(\Sigma,\Sigma \setminus S) \simeq (S, \partial S)$.
Given a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, we could have defined the *Toledo number* of the cocycle $\sigma$ as $$\operatorname{\textup{T}}_b(\sigma):= \langle \textup{comp}^2_{S, \partial S} \circ \operatorname{\textup{J}}^2_{S, \partial S} \circ \operatorname{\textup{H}}^2(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]),[S,\partial S] \rangle$$
To compare the two different definitions of the Toledo invariant, one can follows the same strategy of the proofs of either [@moraschini:savini Proposition 1.2, Proposition 1.6] or [@moraschini:savini:2 Proposition 5.5]. In this way it is possible to show that $$\label{eq:alternative:toledo}
\textup{t}_b(\sigma)=\frac{\textup{T}_b(\sigma)}{|\chi(\Sigma)|} \ ,$$ where $\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$. Notice that Equation is analogous to the one obtained by Burger, Iozzi and Wienhard [@BIW1 Theorem 3.3]. In particular $\textup{T}_b(\sigma)$ is an invariant of the $G$-cohomology class of $\sigma$ and it holds the following estimate $$|\textup{T}_b(\sigma)| \leq \operatorname{rk}(\operatorname{\mathcal{X}}) |\chi(\Sigma)| \ .$$
Maximal measurable cocycles of surface groups {#sec:maximal:cocycle:thm}
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In this section we are going to introduce the notion of maximality. Maximal measurable cocycles represent the first example of tight cocycles and this has important consequences on their algebraic hull. Additionally, if they are Zariski dense then the target must be a Hermitian Lie group of tube-type.
We start by giving the definition of maximality.
\[def:maximal:cocycle\] Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. We say that $\sigma$ is *maximal* if it holds $\text{t}_b(\sigma)=\operatorname{rk}(\operatorname{\mathcal{X}})$.
In order to show that maximal cocycles are tight, we need first to introduce the notion of tightness for measurable cocycles of surface groups. Inspired by the notion for representations studied by Burger, Iozzi and Wienhard [@BIW09], we can give the following
\[def:tight:cocycle\] Let $\Gamma \leq L$ be a torsion-free lattice and $(\Omega,\mu_\Omega)$ a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. We say that $\sigma$ is *tight* if it holds $$\lVert \operatorname{\textup{H}}^2(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty=\frac{\operatorname{rk}\operatorname{\mathcal{X}}}{2} \ .$$
Clearly the definition above mimic the one given for representations. Indeed it is immediate to check that if the cocycle is cohomologous to the one induced by a representation, Definition \[def:tight:cocycle\] boils down to the standard one. Another important aspect is that the tightness property is invariant along the $G$-cohomology class of a given cocycle [@moraschini:savini:2 Proposition 3.12]. Notice that we could have introduced the notion of tightness in a much more general setting, but this would be not so useful for our purposes.
The deep study of tight representations done by Burger, Iozzi and Wienhard [@BIW09] enables us to state the following theorem which characterizes the algebraic hull of a tight cocycle and which is a direct consequence of [@BIW09 Theorem 3], where a full characterization of tight subgroups is given.
\[teor:alg:hull:tight\] Let $\Gamma$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$ and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider $\mathbf{G}$ a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Given a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$, assume that there exists an essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is tight then
1. $\mathbf{H}$ is a reductive group;
2. the centralizer $Z_G(H)$ is compact;
3. if $\operatorname{\mathcal{Y}}$ is the symmetric space associated to $H$, there exists a unique $H$-invariant complex structure on $\operatorname{\mathcal{Y}}$ such that the inclusion $H \rightarrow G$ is tight and positive.
Since the cocycle is tight and this condition is invariant along the $G$-cohomology class of $\sigma$, the inclusion $i:H \rightarrow G$ is tight. The conclusion follows by direct application of [@BIW09 Theorem 7.1] which characterize tight subgroups of $G$.
The next step is to prove that maximal cocycles are tight in the sense of Definition \[def:tight:cocycle\], similarly for what happens in the case of representations [@BIW1 Lemma 6.1]. This result will have important consequence for the algebraic hull of a maximal cocycle as a direct application of Theorem \[teor:alg:hull:tight\].
\[prop:maximal:tight\] Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. If $\sigma$ is maximal then it is tight.
Suppose that $\sigma:\Gamma \times \Omega \rightarrow G$ is maximal. Then it holds $\textup{t}_b(\sigma)=\operatorname{rk}\operatorname{\mathcal{X}}$. By definition we have that $$\textup{T}^2_b \circ \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}])=\operatorname{rk}(\operatorname{\mathcal{X}}) \kappa^b_L \ ,$$ and hence it follows $$\operatorname{rk}(\operatorname{\mathcal{X}}) = \lVert \operatorname{rk}(\operatorname{\mathcal{X}}) \kappa^b_L \rVert_\infty=\lVert \textup{T}^2_b \circ \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert \leq \rVert \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty \ .$$ Since the pullback is norm non-increasing, we have also that $\lVert \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty \leq \operatorname{rk}(\operatorname{\mathcal{X}})$, whence we must have equality and the cocycle $\sigma$ is tight.
Before moving in our discussion, we need to remind briefly some notation regarding the triple products studied by Burger, Iozzi and Wienhard [@BIW07]. If we denote by $(\check{S}_{\operatorname{\mathcal{X}}})^{(3)}$ the set of triples of distinct points in $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, the *Hermitian triple product* is defined as
$$\langle \langle \cdot , \cdot , \cdot \rangle \rangle: (\check{S}_ {\operatorname{\mathcal{X}}})^{(3)} \rightarrow \operatorname{\mathbb{R}}^\times \backslash \operatorname{\mathbb{C}}^\times \ ,$$ $$\langle\langle \xi,\eta,\omega \rangle\rangle=e^{i \pi p_{\operatorname{\mathcal{X}}} \beta_{\operatorname{\mathcal{X}}}(\xi,\eta,\omega)} \ \ \text{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for every $(\xi,\eta,\omega) \in \check{S}^{(3)}_{\operatorname{\mathcal{X}}}$. The number $p_{\operatorname{\mathcal{X}}}$ is an integer defined in terms of the root system associated to $G$.
Recall that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is a homogeneous $G$-space, which can be realized as the quotient $G/Q$, where $Q=\mathbf{Q}(\operatorname{\mathbb{R}})$ and $\mathbf{Q}$ is a maximal parabolic subgroup of $\mathbf{G}$. Burger, Iozzi and Wienhard were able to extend the Hermitian triple product to a *complex Hermitian triple product* $\langle\langle \cdot, \cdot, \cdot \rangle\rangle_{\operatorname{\mathbb{C}}}$ defined on $(\mathbf{G}/\mathbf{Q})^3$ with values into $\Delta^\times \backslash A^\times$. Here $A^\times$ is the group $\operatorname{\mathbb{C}}^\times \times \operatorname{\mathbb{C}}^\times$ endowed with real structure $(\lambda,\mu) \mapsto (\overline{\mu},\overline{\lambda})$ and $\Delta^\times$ is the image of the diagonal embedding of $\operatorname{\mathbb{C}}^\times$. More precisely, the authors [@BIW07 Section 2.4] showed that the following diagram commutes $$\xymatrix{
(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)} \ar[rr]^{\langle \langle \cdot, \cdot, \cdot \rangle\rangle} \ar[d]^{(\imath)^3} && \operatorname{\mathbb{R}}^\times \backslash \operatorname{\mathbb{C}}^\times \ar[d]^\Delta\\
(\mathbf{G}/\mathbf{Q})^3 \ar[rr]^{\langle \langle \cdot, \cdot, \cdot \rangle\rangle_{\operatorname{\mathbb{C}}}} && \Delta^\times \backslash A^\times \ .
}$$ where $\imath:\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}} \rightarrow \mathbf{G}/\mathbf{Q}$ is the map given by the $G$-equivariant identification of $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ with $(\mathbf{G}/\mathbf{Q})(\operatorname{\mathbb{R}})$ and $\Delta$ is the diagonal embedding.
Given any pair of distinct points $(\xi,\eta) \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(2)}$, following [@BIW07 Section 5.1], we denote by $\operatorname{\mathcal{O}}_{\xi,\eta}$ the open Zariski subset of $\mathbf{G}/\mathbf{Q}$ on which the map $$p_{\xi,\eta}:\operatorname{\mathcal{O}}_{\xi,\eta} \rightarrow \Delta^\times \backslash A^\times, \hspace{5pt} p_{\xi,\eta}(\omega):=\langle \langle \xi, \eta, \omega \rangle \rangle_{\operatorname{\mathbb{C}}} \ ,$$ is well-defined. Burger, Iozzi and Wienhard [@BIW07 Lemma 5.1] proved that if there exists an integer $m \in \operatorname{\mathbb{Z}}\setminus \{ 0 \}$ such that $\omega \mapsto p_{\xi,\eta}(\omega)^m$ is constant, then $\operatorname{\mathcal{X}}$ must be of tube-type.
Now we can proceed proving the following theorem, which should be thought of as a generalization of [@BIW1 Theorem 4.1(1)].
\[teor:symmetric:tube\] Let $L$ be a finite connected covering of $\operatorname{\textup{PU}}(1,1)$ and let $\Gamma \leq L$ be a torsion-free lattice. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space and let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle with essentially unique boundary map. If $\sigma$ is maximal and Zariski dense, then $\operatorname{\mathcal{X}}$ must be of tube-type.
Consider a positively oriented triple of distinct points $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$. By the maximality assumption we have that $\textup{t}_b(\sigma)=\operatorname{rk}(\operatorname{\mathcal{X}})$ and by substituting this value in Equation we obtain $$\label{eq:formula:maximal}
\int_{\Gamma \backslash L}\int_\Omega \beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))d\mu_\Omega(s)d\mu_{\Gamma \backslash L}(\overline{g})=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ .$$ Hence for almost every $\overline{g} \in \Gamma \backslash L$ and almost every $s \in \Omega$ it holds $$\beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ and by the equivariance of the map $\phi$ it follows $$\label{eq:almost:every:maximal}
\beta_{\operatorname{\mathcal{X}}}(\phi(g\xi,s),\phi(g\eta,s),\phi(g\omega,s))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every $g \in L$ and almost every $s \in \Omega$.
For almost every $s \in \Omega$, we know that the $s$-slice $\phi_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is measurable and, by Equation \[eq:almost:every:maximal\] it satisfies $$\label{eq:maximal:slice}
\beta_{\operatorname{\mathcal{X}}}(\phi_s(g\xi),\phi_s(g\eta),\phi_s(g\omega))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every $g \in L$. Since the same reasoning applies to a negatively oriented triple, we must have $$\label{eq:maximal:slice:triples}
\beta_{\operatorname{\mathcal{X}}}(\phi_s(\xi),\phi_s(\eta),\phi_s(\omega))=\pm \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every triple $\xi,\eta,\omega$ such that $\beta_{\operatorname{\mathbb{S}}^1}(\xi,\eta,\omega)=\pm 1/2$. The equation above implies that $$\label{eq:hermitian:product}
\langle \langle \phi_s(\xi),\phi_s(\eta),\phi_s(\omega) \rangle\rangle^2 = 1 \ \ \textup{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for almost every $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$ distinct.
Fix now a pair $(\xi,\eta) \in (\operatorname{\mathbb{S}}^1)^{2}$ such that Equation holds for almost every $\omega \in \operatorname{\mathbb{S}}^1$. Since $G$ acts transitively on the set of maximal triples on $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ by [@BIW09 Theorem 3.8(3)], for a fixed pair of distinct points $\xi_0,\eta_0 \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, there exists a measurable function $f:\Omega \rightarrow G$ such that $$\phi_s(\xi)=f(s)\xi_0 \ , \hspace{10pt} \phi_s(\eta)=f(s)\eta_0 \ , \ (\xi_0,\eta_0,f(s)^{-1}\phi_s(\omega)) \ \text{is maximal}$$ for almost every $\omega \in \operatorname{\mathbb{S}}^1, s \in \Omega$. For such a measurable function $f$, we consider $\sigma^f$ and the map $\phi^f$ as the ones defined in Section \[sec:measurable:cocycles\]. For the ease of notation we are going to write $\alpha=\sigma^f$ and $\psi=\phi^f$. By the choice of the map $f$, Equation can be rewritten as $$\langle \langle \xi_0,\eta_0,\psi_s(\omega) \rangle\rangle^2 = 1 \ \ \textup{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for almost every $\omega \in \operatorname{\mathbb{S}}^1$. The previous equation implies that $\psi_s(\omega) \in \operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ for almost every $\omega \in \operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$. We denote by $E$ the subset of full measure in $\operatorname{\mathbb{S}}^1 \times \Omega$ such that $\psi_s(\omega) \in \operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ for all $E$. Define $$E^\Gamma:=\bigcap_{\gamma \in \Gamma} \gamma E \ ,$$ which has full measure being a countable intersection of full measure sets (notice that $\Gamma$ preserves the measure class on $\operatorname{\mathbb{S}}^1 \times \Omega$). Since $\sigma$ is Zariski dense, the cocycle $\alpha$ is Zariski dense too. Since the Zariski closure of $\psi(E^\Gamma)$ is preserved by the algebraic hull of $\alpha$ which coincides with $\mathbf{G}$, the set $\psi(E^\Gamma)$ is Zariski dense in $\mathbf{G}/\mathbf{Q}$, whence is $\psi(E^\Gamma)$ Zariski dense in $\operatorname{\mathcal{O}}_{\xi_0,\eta_0}$. Thus the map $\omega \rightarrow p_{\xi_0,\eta_0}(\omega)^2$ is constant on $\operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ and $\operatorname{\mathcal{X}}$ is of tube-type, as claimed.
An important consequence of the previous theorem is the following
Let $L$ be a finite connected covering of $\operatorname{\textup{PU}}(1,1)$ and let $\Gamma \leq L$ be a torsion-free lattice. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. There is no maximal Zariski dense cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ when $G$ is not of tube-type.
As a consequence of Theorem \[teor:symmetric:tube\], if $\mathbf{H}$ is the algebraic hull of a maximal cocycle $\sigma$ and $H=\mathbf{H}(\operatorname{\mathbb{R}})$, then $H^\circ$ must be a Hermitian group of tube-type.
The following theorem collects all the properties we discovered about the algebraic hull of a maximal cocycle and it should be thought of as a statement equivalent to [@BIW1 Theorem 5] in the context of measurable cocycles.
[teor:maximal:alghull]{} Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Let $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is maximal, then
1. the algebraic hull $\mathbf{H}$ is reductive
2. the centralizer $Z_G(H)$ is compact;
3. the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is Hermitian of tube-type;
4. it holds $\mathbf{H}(\operatorname{\mathbb{R}}) \subset \textup{Isom}(\operatorname{\mathcal{T}})$ for some maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$. Equivalently there exists a cocycle cohomologous to $\sigma$ which preserves $\operatorname{\mathcal{T}}$.
Being maximal, the cocycle $\sigma$ is tight by Proposition \[prop:maximal:tight\]. Hence we can apply Theorem \[teor:alg:hull:tight\] to get properties $1)$ and $2)$. Additionally by Theorem \[teor:symmetric:tube\] the symmetric space $\operatorname{\mathcal{Y}}$ must be of tube-type, whence point $3)$.
The inclusion $i: H \rightarrow G$ is tight because the cocycle $\sigma$ is tight. Since the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is of tube-type and the inclusion is tight, by [@BIW09 Theorem 9(1)] there exists a unique maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$ preserved by $H$. By uniqueness, $\operatorname{\mathcal{T}}$ must be preserved by the whole $\mathbf{H}(\operatorname{\mathbb{R}})$ and we are done.
Regularity properties of boundary maps {#sec:boundary:map}
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Imitating what happens in the context of representations, we are going to study the regularity properties of boundaries map associated to maximal measurable cocycles. Given a maximal Zariski dense measurable cocycle, under suitable hypothesis on the push-forward measure with respect to the slices of the boundary map, we are going to show that there exists an essentially unique equivariant measurable map with left-continuous (respectively right-continuous) slices which preserve transversality and maximality. We are going to follow the line of [@BIW1 Section 5].
Before introducing the setup of the section, we say that a measurable map $\phi:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is *maximal* if it satisfies Equation . Notice that almost every slice of a boundary map associated to a maximal cocycle is maximal.
\[setup:boundary:map\] From now until the end of the section we are going to assume the following
- $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$;
- $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space;
- $\sigma:\Gamma \times \Omega \rightarrow G$ is a maximal Zariski dense cocycle with boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$;
- denote by $\{E_s\}_{s \in \Omega}$ the family of essential graphs $E_s=\textup{EssGr}(\phi_s)$ associated to the slices, that is the support of the push-forward of the Lebesgue measure on $\operatorname{\mathbb{S}}^1$ with respect to the map $\xi \mapsto (\xi,\phi_s(\xi)) \in \operatorname{\mathbb{S}}^1 \times \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$.
Having introduced the setup we needed, we can now move on proving the following
\[lemma:maximality:triples\] In the situation of Setup \[setup:boundary:map\], suppose that $E_s$ is maximal. Let $(\xi_i,\eta_i) \in E_s$ for $i=1,2,3$ be points such that $\xi_1,\xi_2,\xi_3$ are pairwise distinct and $\eta_1,\eta_2,\eta_3$ are pairwise transverse. Then it holds $$\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\eta_3)=\operatorname{rk}(\operatorname{\mathcal{X}}) \beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$
Denote by $I_i$ for $i=1,2,3$ open paiwise non-intersecting intervals such that $\xi_i \in I_i$ and for any $\omega_i \in I_i$ it holds $$\beta_{\operatorname{\mathbb{S}}^1}(\omega_1,\omega_2,\omega_3)=\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$ Consider a open neighborhood $U_i$ of $\eta_i$, for $i=1,2,3$, such that $U_1 \times U_2 \times U_3 \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$. Then the measurable set $$A_i=\{ \omega \in I_i \ | \ \phi_s(\omega_i) \in U_i \} \ ,$$ is a set of positive measure, since $\eta_1,\eta_2,\eta_3$ are in the essential image of $\phi_s$. Since we assumed the slice $E_s$ is maximal, for almost every $(\omega_1,\omega_2,\omega_3) \in A_1 \times A_2 \times A_3$ we have that $$\beta_{\operatorname{\mathcal{X}}}(\phi_s(\omega_1),\phi_s(\omega_2),\phi_s(\omega_3))=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\omega_1,\omega_2,\omega_3)=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$
By setting $\varepsilon=2\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3)$, we have that $|\varepsilon|=1$ and for almost every $(\omega_1,\omega_2,\omega_3) \in A_1 \times A_2 \times A_3$ we have that $$(\phi_s(\omega_1),\phi_s(\omega_2),\phi_s(\omega_3)) \in U_1 \times U_2 \times U_3 \cap \operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}{\operatorname{\mathcal{X}}}} \ ,$$ where $\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}{\operatorname{\mathcal{X}}}}$ is the open set in $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}^3$ on which $\beta_{\operatorname{\mathcal{X}}}$ is identically equal to $\operatorname{rk}(\operatorname{\mathcal{X}})/2$. By the arbitrary choice of the neighborhood $U_i$, must have $(\eta_1,\eta_2,\eta_3) \in \overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}}$.
Since we have that $$\overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}} \cap (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}=\overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}} \cap ( \sqcup_{i=0}^{\operatorname{rk}(\operatorname{\mathcal{X}})} \operatorname{\mathcal{O}}_{-\operatorname{rk}\operatorname{\mathcal{X}}+ 2 i}) = \operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}$$ and $(\eta_1,\eta_2,\eta_3) \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$, the triple is maximal and the claim follows.
In order to proceed we have now to discuss a condition we have to impose on the slices of the boundary map. Recall that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ can be identified with $G/Q$, where $Q$ is a maximal parabolic subgroup. We denote by $\textbf{V}_\xi \subset \mathbf{G}/\mathbf{Q}$ the Zariski closed set of points transverse to $\xi$ and set $V_\xi:=\textbf{V}_\xi(\operatorname{\mathbb{R}})$, the set of points transvers to $\xi$ in the Shilov boundary.
Burger, Iozzi and Wienhard [@BIW1 Proposition 5.2] proved that the boundary map associated to a Zariski dense representation has very strong properties, since its essential image intersects any proper Zariski closed set of the Shilov boundary in a set of measure zero. The author wonder under which hypothesis the same property should hold for almost every slice of a boundary map associated to a cocycle. Here we are going to assume it. More precisely
\[ass:zariski:zero:measure\] In the situation of Setup \[setup:boundary:map\], we suppose that for every propery Zariski closed set $\mathbf{V} \subset \mathbf{G}/\mathbf{Q}$ it holds $$\nu(\phi_s^{-1}(\mathbf{V}(\operatorname{\mathbb{R}})))=0 \ ,$$ for almost every $s \in \Omega$. Here $\nu$ is the round measure on $\operatorname{\mathbb{S}}^1$.
Assumption \[ass:zariski:zero:measure\] is clearly satisfied by cocycles which are cohomologous to a Zariski dense representation $\rho:\Gamma \rightarrow G$. We are not aware if this property can be extended to a wider class of cocycles.
\[lemma:transverse\] Let $E_s$ be a maximal slice satisfying Assumption \[ass:zariski:zero:measure\] and let $(\xi_1,\eta_1), (\xi_2,\eta_2) \in E_s$ with $\xi_1 \neq \xi_2$. Then $\eta_1$ and $\eta_2$ are transverse.
For any distinct $\xi,\omega \in \operatorname{\mathbb{S}}^1$ we denote by $$((\xi,\omega)):=\{ \eta \in \operatorname{\mathbb{S}}^1 \ | \ \beta_{\operatorname{\mathbb{S}}^1}(\xi,\zeta,\omega)=\frac{1}{2} \} \ .$$ Thanks to Assumption \[ass:zariski:zero:measure\], we can suppose that the essential image of the slice $\phi_s$ meets any Zariski closed set in a measure zero set. Hence we can find $\alpha_1 \in ((\xi_1,\xi_2))$ such that $\phi_s(\alpha_1)$ is transverse to both $\eta_1$ and $\eta_2$. In the same way there will exist a point $\alpha_2 \in ((\xi_2,\xi_1))$ such that $\phi_s(\alpha_2)$ is transverse to $\eta_1$ and $\eta_2$.
Using now jointly Lemma \[lemma:maximality:triples\] and the cocycle condition on $\beta_{\operatorname{\mathcal{X}}}$ we get $$\begin{aligned}
0&=\beta_{\operatorname{\mathcal{X}}}(\phi_s(\alpha_1),\eta_1,\phi_s(\alpha_2))-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))+\\
&+\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\phi_s(\alpha_2))-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\eta_2))=\\
&=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))+\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\eta_2)) \ . \\\end{aligned}$$ The previous line implies that $\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$ and hence $\eta_1$ and $\eta_2$ are transverse.
Given now any subset $A \subset \operatorname{\mathbb{S}}^1$ we put $$F_A^s:=\{ \eta \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}} \ | \ \exists \ \xi \in A \ : \ (\xi,\eta) \in E_s \} \ .$$ We define also $$((\xi,\omega]]:=((\xi,\omega)) \cup \{ \omega \} \ .$$
\[lemma:one:point\] Let $s \in \Omega$ be a point such that $E_s$ is a maximal slice satisfying Assumption \[ass:zariski:zero:measure\]. Let $\xi \neq \omega$ be two points in $\operatorname{\mathbb{S}}^1$. Then $\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ and $\overline{F^s_{[[\omega,\xi))}} \cap F^s_{\xi}$ consist each of one point.
We prove that $\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ consists of exactly one point. The same strategy can be applied to $\overline{F^s_{[[\omega,\xi))}} \cap F^s_{\xi}$ to prove the same statement.
Let $\eta,\eta' \in \overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ and consider $(\xi_n,\eta_n) \in E_s$ a sequence such that $$\xi_n \in ((\xi,\omega]], \ \ \ \lim_{n \to \infty} \xi_n=\xi, \ \ \ \lim_{n \to \infty} \eta_n=\eta \ .$$
Given any $\zeta \in ((\xi,\omega))$, we can apply the same reasoning of [@BIW1 Lemma 5.8], to say that $$\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi = \overline{F^s_{((\xi,\zeta]]}} \cap F^s_\xi \ .$$ Thanks to the previous equation, consider a sequence $(\omega_n,\eta'_n) \in E_s$ so that $$\omega_n \in ((\xi,\xi_n)), \ \ \ \lim_{n \to \infty} \omega_n=\xi, \ \ \ \lim_{n \to \infty} \eta'_n=\eta' \ .$$
Applying Lemma \[lemma:transverse\] we have that $\eta,\eta'_n,\eta_n$ are pairwise transverse. Hence we can apply Lemma \[lemma:maximality:triples\] to the triples $(\xi,\omega_n,\xi_n)$ and $(\eta,\eta_n',\eta_n)$ to get $$\beta_{\operatorname{\mathcal{X}}}(\eta,\eta_n',\eta_n)=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\xi,\omega_n,\xi_n)=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ .$$ Since $\lim_{n \to \infty} \eta_n=\eta$, Property $4)$ of Section \[sec:hermitian:groups\] of the Bergmann cocycles $\beta_{\operatorname{\mathcal{X}}}$ forces $\lim_{n \to \infty} \eta'_n=\eta$ and hence $\eta=\eta'$.
In this way wet get immediately the following
\[cor:two:points\] Let $s \in \Omega$ be a point such that $E_s$ is a maximal slice satisfying Assumption \[ass:zariski:zero:measure\]. For every $\xi \in \operatorname{\mathbb{S}}^1$ the set $F^s_\xi$ contains either one or two points.
Consider $\omega_-, \xi, \omega_+ \in \operatorname{\mathbb{S}}^1$ and let $\eta \in F^s_\xi$. Since it holds $$F^s_\xi=\left( \overline{F^s_{[[\omega_-,\xi))}} \cap F^s_{\xi} \right) \cup \left( \overline{F^s_{((\xi,\omega_+]]}} \cap F^s_\xi \right)\ ,$$ the claim follows by Lemma \[lemma:one:point\].
We are know ready to prove the main theorem of the section which extends in some sense [@BIW1 Theorem 5.1] to the context of measurable cocycles.
\[teor:boundary:map\] In the situation of Assumption \[ass:zariski:zero:measure\], there exist two measurable maps $$\phi^\pm:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$$ such that
1. the slice $\phi^+_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is right continuous for almost every $s \in \Omega$;
2. the slice $\phi^-_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is left continuous for almost every $s \in \Omega$;
3. the maps $\phi^\pm$ are measurable and $\sigma$-equivariant;
4. for every $\xi \neq \omega $ in $\operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$, $\phi^\varepsilon_s(\xi)$ is transverse to $\phi^\delta_s(\omega)$, where $\varepsilon, \delta \in \{ \pm \}$;
5. almost every slice is monotone, that is for every $\xi,\omega,\zeta \in \operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$ it holds $$\beta_{\operatorname{\mathcal{X}}}(\phi_s^\varepsilon(\xi),\phi_s^\delta(\omega),\phi_s^\theta(\zeta))=\operatorname{rk}(\operatorname{\mathcal{X}}) \beta_{\operatorname{\mathbb{S}}^1}(\xi,\omega,\zeta) \ ,$$ where $\varepsilon,\delta,\theta \in \{ \pm \}$.
Additionally $\phi^\pm$ are essentially unique.
By assumption we know that for almost every $s \in \Omega$, the slice $\phi_s$ is maximal and it satisfies Assumption \[ass:zariski:zero:measure\]. For any such $s$, we define for every $\xi \in \operatorname{\mathbb{S}}^1$ the following maps $$\phi_s^+(\xi)=\overline{F^s_{[[\omega_-,\xi))}} \cap F^s_{\xi}\ ,
\phi_s^-(\xi)=\overline{F^s_{((\xi,\omega_+]]}} \cap F^s_\xi \ ,$$ where $\omega_- ,\xi,\omega_+$ is a positively oriented triple in $\operatorname{\mathbb{S}}^1$ and $\omega_\pm$ are arbitrary. The right continuity of $\phi^+_s$ and the left continuity of $\phi^-_s$ are clear by their definitions. We can define $$\phi^\pm:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}, \ \phi_s^\pm(\xi,s):=\phi_s^\pm(\xi) \ .$$ The measurability of the functions $\phi^\pm_s$ comes from the fact the slice $\phi_s^\pm$ are measurable and varies measurably with respect to $s$ by the measurability of $\phi$. The $\sigma$-equivariance of the latter implies that $\phi^\pm$ are $\sigma$-equivariant.
Finally property $4)$ follows by Lemma \[lemma:transverse\] and property $5)$ follows by Lemma \[lemma:maximality:triples\]. The essential uniqueness is a consequences of the assumption on the essential uniqueness of the boundary map.
[^1]:
| ArXiv |
---
abstract: 'In this paper, the exact dynamics of open quantum systems in the presence of initial system-reservoir correlations is investigated for a photonic cavity system coupled to a general non-Markovian reservoir. The exact time-convolutionless master equation incorporating with initial system-reservoir correlations is obtained. The non-Markovian dynamics of the reservoir and the effects of the initial correlations are embedded into the time-dependent coefficients in the master equation. We show that the effects induced by the initial correlations play an important role in the non-Markovian dynamics of the cavity but they are washed out in the steady-state limit in the Markovian regime. Moreover, the initial two-photon correlation between the cavity and the reservoir can induce nontrivial squeezing dynamics to the cavity field.'
author:
- 'Hua-Tang Tan'
- 'Wei-Min Zhang'
title: 'Dynamics of open quantum systems with initial system-reservoir correlations'
---
= 10000
Introduction
============
The study of dynamics of open quantum systems continuously receives attentions because of its fundamental importance in quantum physics and also because of the rapid development of quantum technologies. Previous studies on the dynamics of open quantum systems mainly lie on the Lindblad-type master equation [@bm1; @bm2; @bm3], where the characteristic time of the environment is sufficiently shorter than that of the system such that the non-Markovian memory effect is negligible, and so does for initial system-reservoir correlations. However, the new development in ultrafast photonics, ultracold atomic physics, nanoscience and technology as well as quantum information science strongly suggests that the non-Markovian dynamics in ultrafast and ultrasmall open systems should play an important role, and the associated effects (including the initial system-reservoir correlations) should be fully taken into account. To this end, the more rigorous approach is demanded for the study of non-Markovian dynamics of open quantum systems incorporating with the initial system-reservoir correlations.
The exact description of open quantum systems has indeed been explored extensively in the literature, mainly focusing on quantum Brown motion based on Feynman-Vernon influence functional [@Fey63118; @Cal83587; @Haa852462; @Hu922843; @Hal962012; @Food01105020] and stochastic diffusion Schrödinger equation [@Str981699; @Str994909; @Yu04062107]. Extending the Feynman-Vernon influence functional to other open quantum systems has also achieved a great success recently, including the exact master equation for electron systems and the nonequilibrium quantum transport theory in various nanostructures [@Tu08235311; @Tu09631; @Jin10083013] and the exact master equation for micro- or nanocavities in photonic crystals and the quantum transport theory for photonic crystals [@Xio10012105; @Wu1018407; @Lei104570]. However, in most of these investigations, the system and the reservoir are often assumed to be initially uncorrelated with each other [@Leg871]. Realistically, it is possible and often unavoidable in experiments that the system and its environment are correlated closely at the beginning, especially for the cases of the system strongly coupled to the reservoir [@ee]. Various initial-correlation induced effects have been investigated in different open quantum systems [@src0; @src1; @src2; @src3; @src4; @src5; @src7; @src8; @src6; @src9; @src10]. For example, it has been recently shown that the initial correlations between a qubit and its environment can lead to the distance growth of two quantum states over its initial value [@src7; @src8]. It has also been demonstrated that the initial correlations have nontrivial differences in quantum tomography process [@src6]. Besides, it has been found that the initial system-reservoir correlations have significant effects on the entanglement in a two-qubit system [@src9; @src10].
In this paper, the dynamics of open quantum systems in the presence of initial system-reservoir correlations is investigated with a photonic cavity system coupled to a non-Markovian reservoir as a specific example. By solving the exact dynamics of the cavity system, the effects of the initial correlations are explicitly built into the equations of motion for the intensity and the two-photon correlation function of the cavity field. We then obtain the exact master equation incorporating with the initial correlations which induce new terms and also modify the time-dependent dissipation and fluctuation coefficients in the master equation. Taking a nanocavity coupled to a coupled resonator optical waveguide (serving as a structured reservoir) as an experimentally realizable system, we find that the effects of the initial correlations are fragile for a Markovian reservoir but play an important role in the non-Markovian regime. In fact, in the strong non-Markovian regime, the initial two-photon correlation between the cavity and the reservoir can induce oscillating squeezing dynamics in the cavity. But in the Markovian regime, the initial correlations will be washed out in the steady-state limit.
The rest of the paper is organized as follows. In Sec. II, the dynamics of open quantum systems with initial system-reservoir correlations is formulated for a photonic cavity system coupled to a general non-Markovian reservoir. In Sec. III, we construct the exact time-convolutionless master equation incorporating with the initial correlations, where the effects from the initial correlations are explicitly embedded into the time-dependent coefficients in the master equation. In Sec. IV, an experimentally realizable example is considered to analytically and numerically examine the influence of the initial correlations on the dynamics of open quantum systems. At last, a summary is given in Sec. V.
Non-Markovian dynamics with initial system-reservoir correlations
=================================================================
To be specific, we consider here a single-mode photonic cavity system coupled to a general non-Markovian reservoir, where the single-mode cavity system could be a nanocavity in nanostructures or photonic crystals, and the non-Markovian environment may be a structured photonic reservoir [@stru-reservoir]. The Hamiltonian of the system can be expressed as a Fano-type model of a localized state coupled with a continuum [@Fano611866]: $$\begin{aligned}
H=\omega_c a^\dag a+\sum_{k}\omega_k b_k^\dag b_k\ +\sum_kV_k (a
b_k^\dag +b_k a^\dag),\label{H1}\end{aligned}$$ where the first term is the Hamiltonian of the cavity field with frequency $\omega_c$, and $a^\dag$ and $a$ are the creation and annihilation operators of the cavity field; the second term describes a general non-Markovian reservoir which is modeled as a collection of infinite photonic modes, where $b_k^\dag$ and $b_k$ are the corresponding creation and annihilation operators of the $k$-th photonic mode with frequency $\omega_k$. The third term characterizes the system-reservoir coupling with the coupling strength $V_k$ between the cavity field and the $k$-th photonic mode. For convenience, we take $\hbar=1$ throughout the paper.
We shall use the equation of motion approach to solve the dynamics of the cavity system and the reservoir, from which the general initial correlations between the cavity and the reservoir can be fully taken into account. The time evolution of the cavity field operator $a(t)=e^{iHt}ae^{-iHt}$ and the reservoir field operators $b_k(t)=e^{iHt}b_ke^{-iHt}$ in the Heisenberg picture obey the equations of motion
$$\begin{aligned}
&\frac{d}{dt}a(t)=-i[a(t), H]=-i\omega_c a(t)-i\sum_k V_k b_k(t),\\
&\frac{d}{dt}b_k(t)=-i[b_k(t),H]=-i\omega_k b_k(t)-iV_k a(t).
\label{bk}\end{aligned}$$
Solving Eq. (\[bk\]) for $b_k(t)$ $$\begin{aligned}
b_k(t)=b_k(0)e^{-i\omega_k t}-iV_k\int_0^t d\tau
a(\tau)e^{-i\omega_k (t-\tau)},\end{aligned}$$ we obtain $$\begin{aligned}
\frac{d}{dt}a(t)=-i\omega_c a(t) -\int_0^t d\tau g(t-\tau)a(\tau)
\nonumber\\-i\sum_k V_k b_k(0)e^{-i\omega_k t}. \label{lat}\end{aligned}$$ Here, the memory kernel $g(\tau)=\sum_{k}|V_k|^2e^{-i\omega_k\tau}$ characterizes the non-Markovian dynamics of the reservoir. For a continuous reservoir spectrum, we have $g(\tau)=\int_{0}^\infty
\frac{d\omega}{2\pi}J(\omega)e^{-i\omega\tau}$, where $J(\omega)=2\pi
\varrho(\omega)|V(\omega)|^2$ is the spectral density of the reservoir, with $\varrho(\omega)$ being the density of states and $V(\omega)$ the coupling between the cavity and the reservoir in the frequency domain.
Because of the linearity of Eq.(\[lat\]), the cavity field operator $a(t)$ can be expressed, in terms of the initial field operators $a(0)$ and $b_k(0)$ of the cavity and the reservoir, as $$\begin{aligned}
a(t)=u(t)a(0)+f(t) , \label{at}\end{aligned}$$ where the time-dependent coefficient $u(t)$ and $f(t)$ are determined from Eq.(\[lat\]) and given by
$$\begin{aligned}
\frac{d}{dt}u(t)=-i\omega_c u(t)-\int_0^t d\tau
g(t-\tau)u(\tau),\label{ut}
\\
\frac{d}{dt}f(t)= -i\omega_c f(t)-\int_0^t d\tau g(t-\tau)f(\tau) \label{ft} \nonumber\\
-i\sum_kV_k b_k(0)e^{-i\omega_k \tau},\end{aligned}$$
subjected to the initial conditions $u(0)=1$ and $f(0)=0$. The integrodifferential equation (\[ut\]) shows that $u(t)$ is just the propagating function of the cavity field (the retarded Green function in nonequilibrium Green function theory [@green]). In addition, $f(t)$ is in fact an operator coefficient and its solution can be obtained analytically from the inhomogeneous equation of Eq. (\[ft\]): $$\begin{aligned}
f(t)=-i\sum_kV_k b_k(0)\int_0^t d\tau e^{-i\omega_k \tau}u(t-\tau).
\label{ress}\end{aligned}$$
From Eqs. (\[at\])-(\[ress\]) we can determine the exact non-Markovian dynamics of the cavity field coupled to a general reservoir with arbitrary initial system-reservoir correlations, upon a given initial state $\rho_{\rm tot}(0)$ of the whole system. In the Heisenberg picture, quantum states are time-independent. Once $\rho_{\rm
tot}(0)=\rho_{\rm tot}$ is given, the time evolution of any physical observable can be obtained directly from Eqs. (\[at\])-(\[ress\]) through the relation $$\begin{aligned}
\langle f(a^\dag(t),a(t))\rangle = {\rm
tr}[f(a^\dag(t),a(t))\rho_{\rm tot}].\end{aligned}$$ For example, the time evolution of the expectation values $\langle
a(t) \rangle$, $n(t) \equiv \langle a^\dag(t) a(t) \rangle$, and $s(t) \equiv \langle a(t) a(t)\rangle$, which respectively describe the cavity amplitude, the cavity intensity, and the two-photon correlation of the cavity field, can be expressed explicitly by the following solution
\[e\] $$\begin{aligned}
&\langle a (t) \rangle=u(t)\langle a (0)\rangle + \upsilon_0(t),\label{e1}\\
& n(t)=|u(t)|^2 n(0)
+2{\rm Re}[u^*(t)\nu_1(t)]+\upsilon_1(t),\label{e2}\\
& s(t)=u^2(t) s (0) +2u(t)\nu_2(t)+\upsilon_2(t) , \label{e3}\end{aligned}$$
where $\langle a (0)\rangle, n(0)$ and $ s (0)$ are the corresponding initial conditions. Other time-dependent functions in Eq. (\[e\]) are given by
\[core\]
$$\begin{aligned}
\nu_1(t)&=\langle a^\dag(0)f(t)\rangle
=-i\int_0^t\sum_k V_k \langle a^\dag(0)b_k(0)\rangle e^{-i\omega_k \tau}u(t-\tau)d\tau,\label{u1}\\
\nu_2(t)&=\langle a(0)f(t)\rangle
=-i\int_0^t\sum_k V_k \langle a(0) b_k(0)\rangle e^{-i\omega_k \tau}u(t-\tau)d\tau,\label{u2}\\
\upsilon_0(t)&=-i\int_0^t\sum_k V_k \langle b_k(0)\rangle
e^{-i\omega_k\tau}u(t-\tau)d\tau,\label{u0}\\
\upsilon_1(t)&=\langle f^\dag(t)f(t)\rangle =\int_0^t d\tau\int_0^t
d\tau'\sum_{kk'} V^*_k V_{k'}\langle b^\dag_k(0)
b_{k'}(0)\rangle e^{-i(\omega_{k'}\tau'-\omega_k\tau)}u^*(t-\tau)u(t-\tau'),\label{v1}\\
\upsilon_2(t)&=\langle f(t)f(t)\rangle =-\int_0^t d\tau\int_0^t
d\tau'\sum_{kk'} V_k V_{k'}\langle b_k(0) b_{k'}(0)\rangle
e^{-i(\omega_k\tau+\omega_{k'}\tau')}u(t-\tau)u(t-\tau')\label{v2}.\end{aligned}$$
In these solutions, $\upsilon_j(t)$ ($j=0,1,2$) characterize respectively the contributions from the initial field amplitudes $\langle b_k(0)\rangle$, the initial photon scattering amplitudes $\langle b^\dag_k(0) b_{k'}(0)\rangle$ and the initial two-photon correlations $\langle b_k(0) b_{k'}(0)\rangle$ of all the photonic modes in the reservoir. While $\nu_1(t)$ and $\nu_2(t)$ correspond to the contributions from the different initial system-reservoir correlations $\langle a(0)b_k^\dag(0)\rangle$ and $\langle
a(0)b_k(0)\rangle$, respectively.
If the initial state of the total system is uncorrelated, and the reservoir is in a thermal equilibrium state, i.e., $$\begin{aligned}
\rho_{\rm tot}(0)=\rho(0) \times \rho_R(0), ~~
\rho_R(0)=\frac{e^{-\beta H_R}}{tr e^{-\beta H_R}},\end{aligned}$$ with $H_R=\sum_k\omega_kb_k^\dag b_k$ and $\beta=1/k_BT$ being the initial temperature of the reservoir, it is easy to check that $\nu_i(t)=0, \upsilon_i(t)=0$ except for $v_1(t)$ which is given by $$\begin{aligned}
\upsilon_1(t) =\int_0^t d\tau\int_0^t d\tau'
u(t-\tau')\widetilde{g}(\tau'-\tau) u^*(t-\tau). \label{vts}\end{aligned}$$ Here, $\widetilde{g}(\tau)=\sum_k |V_k|^2 \bar{n}_k e^{-i\omega_k
\tau}$ and $\bar{n}_k=\langle b^\dag_k(0) b_{k}(0)\rangle=
1/(e^{\beta \omega_k} -1)$ is the initial photonic distribution function of the reservoir. Then Eq. (\[e\]) reproduces the same solution solved from the exact master equation without initial system-reservoir correlations [@Xio10012105]. However, as we see, the exact non-Markovian dynamics in Eq.(\[e\]) derived via the equation of motion approach has explicitly included the effects induced by the initial correlations between the system and the reservoir.
Exact master equation with initial system-reservoir correlations
================================================================
To further understand the effects of the initial system-reservoir correlations on the dynamics of open quantum systems, we shall attempt to derive the exact master equation for the reduced density matrix of the cavity system $\rho(t)$. In the literature, exact master equations for open systems are mostly derived without initial correlations, such as the systems associated with quantum Brown motions [@Hu922843; @Hal962012; @Food01105020], quantum dot systems in various nanostructures [@Tu08235311; @Tu09631] and cavity systems coupled to structured reservoirs as well as general non-Markovian reservoirs [@An07042127; @Xio10012105; @Wu1018407]. Here, we concentrate the exact master equation for the photonic system in the presence of initial Gaussian correlated states. Based on the bilinear operator structure of the system as well as the techniques in deriving exact master equation for the cavity system described by Eq. (\[H1\]) [@Xio10012105; @Wu1018407], the master equation with the initial system-reservoir correlations would have a general time-convolutionless form as follows: $$\begin{aligned}
\dot{\rho}(t)=& -i\Delta(t)[a^\dag a, \rho] \notag \\
& +\gamma_1(t)(2a\rho a^\dag -a^\dag a\rho-\rho_a a^\dag a)\nonumber\\
&+\gamma_2(t)(a\rho a^\dag+a^\dag\rho a -a^\dag a\rho-\rho a a^\dag)\nonumber\\
&+\gamma_3^*(t)(2a\rho a-aa\rho-\rho aa)\nonumber\\
&+\gamma_3(t)(2a^\dag \rho a^\dag -a^\dag a^\dag \rho-\rho a^\dag
a^\dag),\label{me}\end{aligned}$$ where the coefficient $\Delta(t)$ is the renormalized cavity frequency, $\gamma_1(t)$ and $\gamma_2(t)$ usually denote respectively the dissipation (damping) and fluctuation (noise) due to the back-reactions between the system and the reservoir, and $\gamma_3(t)$ is related to a two-photon decoherence process. As we see, the first three terms have the standard form as the exact master equation for the Hamiltonian in Eq. (\[H1\]) without the initial correlations [@Xio10012105; @Wu1018407], but with the coefficients modified by the initial correlation $\langle
a(0)b^\dag_k(0) \rangle$. The last two terms are contributed from the two-photon correlation $\langle b_k(0)b_{k'}(0) \rangle$ in the reservoir as well as by the initial system-reservoir two-photon correlation $\langle a(0)b_k(0) \rangle$.
To figure out the time-convolutionless coefficients in Eq. (\[me\]), we compute the physical observables in Eq. (\[e\]) from the above master equation. From Eq. (\[me\]), it is easy to find that
\[opem\] $$\begin{aligned}
&\frac{d}{dt}\langle a(t)\rangle=-[\gamma_1(t)+i\Delta(t)]\langle a(t) \rangle, \\
&\frac{d}{dt}n(t)=-2\gamma_1(t)n(t)+2\gamma_2(t), \\
&\frac{d}{dt}s(t)=-2[\gamma_1(t)+i\Delta(t)]s(t)-2\gamma_3(t).\label{meq}\end{aligned}$$
On the other hand, with Eq.(\[at\]) we obtain $$\begin{aligned}
\frac{d}{dt}a(t)=\frac{\dot{u}(t)}{u(t)}a(t)-\frac{\dot{u}(t)}{u(t)}f(t)
+\dot{f}(t).\label{at3}\end{aligned}$$ Note that the photonic modes in the reservoir usually cannot be a coherent state so that $\langle b_k(0)\rangle=0$. Then using Eq. (\[at3\]), we find that
\[opem1\] $$\begin{aligned}
&\frac{d}{dt}\langle a(t)\rangle=\frac{\dot{u}(t)}{u(t)}\langle a(t)\rangle, \\
&\frac{d}{dt}n(t)=2{\rm
Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]n(t) +\dot{\upsilon}_1(t) -2{\rm Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]\upsilon_1(t)\nonumber\\
&~~~~~~~~~~ ~~~~~~ +2{\rm Re}\Big[u^*(t)\dot{\nu}_1(t)-\frac{\dot{u}(t)u^*(t)}{u(t)}\nu_1(t)\Big] ,\\
&\frac{d}{dt}s(t)=2\frac{\dot{u}(t)}{u(t)}s(t) +\dot{\upsilon}_2(t) -2\frac{\dot{u}(t)}{u(t)}\upsilon_2(t)
\nonumber\\
&~~~~~~~~~~ ~~~~~~~~~~~~~~~ +2u(t)\dot{\nu}_2(t)
-2\dot{u}(t)\nu_2(t).\label{ap1}\end{aligned}$$
By comparing Eq. (\[opem\]) with Eq. (\[opem1\]), the coefficients $\Delta(t)$ and $\gamma_j(t)$ in the master equation can be uniquely determined and given by
\[ecoff\] $$\begin{aligned}
&\Delta(t)=-{\rm Im}\Big[\frac{\dot{u}(t)}{u(t)}\Big],~~
\gamma_1(t)=-{\rm Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big],\\
&\gamma_2(t)=\dot{\upsilon}_1(t)+2{\rm Re}\Big[ u(t)\dot{\nu}_1^*(t)
-\frac{\dot{u}(t)}{u(t)}[ \upsilon_1(t)+u^*(t)\nu_1(t)]\Big],\\
&\gamma_3(t)=-\frac{1}{2}\dot{\upsilon}_2(t)
+\frac{\dot{u}(t)}{u(t)}\upsilon_2(t)-
u(t)\dot{\nu}_2(t)+\dot{u}(t)\nu_2(t),\end{aligned}$$
which shows that the coefficients $\gamma_2(t)$ and $\gamma_3(t)$ in the master equation depend explicitly on the initial correlations $\langle a(0)b_k^\dag(0)\rangle$ and $\langle a(0)b_k(0)\rangle$ in the presence of the initial Guassian correlated states of the whole system.
If the reservoir is initially in a thermal state uncorrelated to the system, we have $\langle a(0)b^\dag_k(0) \rangle =\langle a(0)b_k(0)
\rangle=\langle b_k(0)b_{k'}(0) \rangle = 0$ except for $\langle
b^\dag_k(0)b_{k'}(0)=\bar{n}_k$. Accordingly, from Eq.(\[core\]) we have $\upsilon_2(t)=0=\nu_1(t)=\nu_2(t)$ so that $\gamma_3(t)=0$ and $$\begin{aligned}
\gamma_2(t)=\dot{\upsilon}_1(t)-2{\rm
Re}\Big[\frac{\dot{u}(t)}{u(t)}\Big]\upsilon_1(t),\end{aligned}$$ where $\upsilon_1(t)$ is then given by Eq. (\[vts\]). Consequently, the master equation (\[me\]) in this situation is reduced to the exact master equation for the cavity system coupled with a general non-Markovian reservoir presented recently in Ref. [@Xio10012105; @Wu1018407], which is obtained originally using the Feynman-Vernon influence functional. In addition, if there are no initial correlations but the reservoir involves initially two-photon correlation, namely, $\langle a(0)b^\dag_k(0) \rangle
=\langle a(0)b_k(0) \rangle=0$ but $\langle b_k(0)b_{k'}(0) \rangle
\neq 0$, then we have $\nu_1(t)=0=\nu_2(t)$ but $\upsilon_2(t)\neq0$. As a result, the coefficient $\gamma_3(t) \neq
0$, which induces a two-photon decoherence process in the cavity [@TMSS1]. However, if the initial states of the whole system only contains the two-photon correlation $\langle a(0)b_k(0)\rangle$ but the reservoir itself stays in an initial thermal state, then we have $\nu_1(t)=0=\upsilon_2(t)$ but $\nu_2(t) \neq 0$. This situation also leads to a non-zero $\gamma_3(t)$ which is essentially equivalent to the situation in which the reservoir involves initially two-photon correlation but without the initial system-reservoir correlations.
Therefore, the master equation, Eq. (\[me\]) with the time-dependent coefficients in Eq. (\[ecoff\]), describes the exact non-Markovian dynamics of a cavity system coupled with a general reservoir involving two-photon correlation in the presence of the quadratic-type initial correlations between the system and reservoir. It shows explicitly that the initial correlation $\langle
a(0)b_k^\dag(0)\rangle$ modifies the fluctuation coefficient $\gamma_2(t)$ but without altering the damping (dissipation) rate $\gamma_1(t)$, which in turn changes the cavity field intensity given by Eq. (\[e2\]) without changing the cavity field amplitude of Eq. (\[e1\]). The initial correlation $\langle
a(0)b_k(0)\rangle$ affects on the two-photon decoherence process which leads to a two-photon process $s(t)=\langle a(t)a(t) \rangle$ of the cavity field. It should be pointed out that if the system and the reservoir are initially in non-Gaussian correlated states, the form of Eq. (\[me\]) may need to be modified further. Nevertheless, the master equation of Eq. (\[me\]) is exact for the initial Gaussian correlated states of the whole system, and it remains in a time-convolutionless form in which the non-Markovain memory dynamics is fully embedded into the time-dependent coefficients. As we see, all these time-dependent coefficients are determined by a unique function, the cavity field propagating function $u(t)$, through the relations given by Eqs. (\[ecoff\]) and (\[core\]). While the propagating function $u(t)$ is determined by Eq. (\[ut\]) in which the integral kernel contains all the non-Markovian memory effects characterizing the back-reactions between the system and the reservoir.
Examples with initial system-reservoir correlations
===================================================
In this section, we shall take two different initial correlated states to examine the effects of the initial correlation on the non-Markovian dynamics in such an open system. To be more specific, we consider an experimentally realizable nanocavity system. Fig. 1 is a schematic plot for a single-mode nanocavity coupled to a coupled resonator optical waveguide (CROW) structure. The nanocavity could be a point defect created in photonic crystals and the waveguide consists of a linear defects in which light propagates due to the coupling of the adjacent defects. The CROW is called as a structured reservoir which possess strong non-Markovian effects [@FanoAnderson-2; @Wu1018407]. The Hamiltonian of the whole system is given by $$\begin{aligned}
H&=&\omega_ca^\dag a+\sum_{n}\omega_0b_n^\dag b_n
+\lambda (ab_1^\dag+b_1a^\dag)\nonumber\\
&&-\sum_{n}\lambda_0(b_nb_{n+1}^\dag+b_{n+1}b_n^\dag),\label{H2}\end{aligned}$$ where $a$ and $a^\dag$ are the annihilation and creation operators of the nanocavity field with frequency $\omega_c$, and the annihilation and creation operators $b_n$ and $b_n^\dag$ describe the resonators at site $n$ in the waveguide with identical frequency $\omega_0$. The frequencies $\omega_c$ and $\omega_0$ are tunable by changing the size of the relevant defects. The third terms describes the coupling of the nanocavity field to the resonator at the first site in the waveguide with the coupling strength $\lambda$ which is also controllable experimentally by adjusting the distance between defects. The last term characterizes the photon hopping between two consecutive resonators in the waveguide structure with the controllable hopping amplitude $\lambda_0$.
Consider the waveguide is semi-infinite long, then performing the following Fourier transformation $b_k=\sqrt{2/\pi}\sum_{n=1}^\infty\sin(nk)b_n$, where the operators $b_k$ and $b_k^\dag$ correspond to the Bloch modes of the waveguide, the Hamiltonian of Eq. (\[H2\]) becomes $$\begin{aligned}
H&=&\omega_ca^\dag a+\sum_{k}\omega_kb_k^\dag b_k
+\sum_{k}g_k(ab_k^\dag+b_ka^\dag),\label{H3}\end{aligned}$$ where $$\begin{aligned}
\omega_k=\omega_0-2\lambda_0\cos k~,
~~g_k=\sqrt{\frac{2}{\pi}}\lambda\sin k. \label{strength}\end{aligned}$$ with $0\le k\le \pi$. As we see, Eq. (\[H3\]) is reduced to the same form of Eq. (\[H1\]) for the system considered in Secs. II-III.
Initially system-reservoir correlated squeezed state
----------------------------------------------------
For the above specific physical system, we shall first consider an initial system-reservoir correlated state with two-photon correlation $\langle a(0)b_{k}(0)\rangle \neq 0$. We assume that the cavity field is correlated initially with the first resonator mode of the CROW in terms of a two-mode entangled squeezed vacuum state [@TMSS1] as $$\begin{aligned}
|\psi_{ab_1}(0)\rangle=\exp(-r_se^{-i\theta_s}ab_1+r_se^{i\theta_s}a^\dag
b_1^\dag)|0_a0_{b_1}\rangle,\label{cstate1}
%&=&\sec r_s\sum_{n=0}^\infty[-\exp(i\theta_s)\tanh %r_s]^n|n_an_{b_1}\rangle.\nonumber\\\end{aligned}$$ and the other resonators in the CROW are in vacuum, with $r_s$ and $\theta_s$ being the squeezing parameter and the reference phase, respectively. The strength of the nonclassical correlations (entanglement) contained in the above state increases with the increasing of the squeezing parameter $r_s$ [@TMSS2]. The reduced density matrices of the cavity field and the resonator mode from Eq. (\[cstate1\]) is a mixed state which can be expressed as $$\begin{aligned}
\rho_{a/b_1} (0)=\sum_{n=0}^\infty\frac{\sinh^{2n}r_s}
{(\sinh^2r_s+1)^{n+1}}|n_{a/b_1}\rangle\langle
n_{a/b_1}|,\label{rstate}\end{aligned}$$ which is indeed of a single-mode thermal state with average thermal photon number $n_{a/b_1}(0)=\sinh^2r_s$. Based on the same Fourier transformation, it follows that the initial system-reservoir correlations are then given by
\[tpc\] $$\begin{aligned}
&&\langle a(0)b_k(0)\rangle=\sqrt{\frac{1}{2\pi}}\sinh2r_se^{i\theta_s}\sin k,\\
&&\langle a(0)b_k^\dag(0)\rangle=0,\end{aligned}$$
namely, the initial Gaussian state only has initial two-photon correlation between the system and the reservoir.
With the initial system-reservoir correlations in Eq. (\[tpc\]), we obtain from Eq. (\[core\]) that $\nu_1(t)=0=\upsilon_0(t)=\upsilon_2(t)$ and
\[core1s\] $$\begin{aligned}
&\nu_2(t)=-i\frac{\sinh2r_se^{i\theta_s}}{\sqrt{2\pi}}\mathcal{F}(t),
\\ &\upsilon_1(t)=\frac{2}{\pi}\sinh^2r_s|\mathcal{F}(t)|^2,\end{aligned}$$
where $$\begin{aligned}
\mathcal{F}(t)&=\int_0^td\tau\sum_kg_k\sin(k)e^{-i\omega_k\tau}u(t-\tau)\notag
\\
&=\frac{\eta}{\sqrt{2\pi}}\int_0^td\tau\int_0^\infty d\omega
\sin[k(\omega)]e^{-i\omega \tau}u(t-\tau).\end{aligned}$$ The last line of the above equation has been applied to the waveguide band structure given in Eq. (\[strength\]), so that $\eta=\frac{\lambda}{\lambda_0}$ and $\sin[k(\omega)]=\frac{1}{2\lambda_0}\sqrt{4\lambda_0^2-(\omega-\omega_0)^2}$ with $\omega_0-2\lambda_0\leq\omega\leq \omega_0+2\lambda_0$.
After obtaining the time-dependent functions $\nu_j(t)$ and $\upsilon_j(t)$ given above, Eq. (\[e\]) becomes
$$\begin{aligned}
&\langle a(t)\rangle =0 , \\
&n(t)=|u(t)|^2n_a(0)+\upsilon_1(t),\\
&s(t)=2u(t)\nu_2(t).\label{pjgz}\end{aligned}$$
This solution indicates that for the given initial thermal state $\rho_a(0)$ in Eq.(\[rstate\]), the cavity field at time $t$ is in a squeezed thermal state [@marian], which can be expressed as $$\begin{aligned}
\rho(t)=S_a[r(t)]\rho_{\rm th}(t) S_a^\dag[r(t)],\end{aligned}$$ where the single-mode squeezing operator $$\begin{aligned}
S_a[r(t)]=\exp[-\frac{r(t)}{2}e^{-i\theta(t)}a^2+\frac{r(t)}{2}e^{i\theta(t)}a^{\dag2}],\end{aligned}$$ with the squeezing parameters $$\begin{aligned}
r(t)=\frac{1}{4}\ln\frac{n(t)+|s(t)|+1/2}{n(t)-|s(t)|+1/2},\end{aligned}$$ and $\theta(t)=\arg[s(t)]$. The thermal state $$\begin{aligned}
\rho_{\rm th}(t)=\sum_{k}\frac{[\bar{n}(t)]^n}{[\bar{n}(t)+1]^{k+1}}|n_a\rangle\langle
n_a|,\end{aligned}$$ where the average thermal photon number $\bar{n}(t)=\sqrt{(n(t)+1/2)^2-|s(t)|^2}-1/2$. By defining the quadrature operators $X=(a + a^\dag )/\sqrt{2}$ and $Y=(a - a^\dag
)/\sqrt{2i}$, the covariance matrix are given by $$\begin{aligned}
\begin{pmatrix} \Delta X^2 & \Delta\{XY\} \\
\Delta \{YX\}& \Delta Y^2 \end{pmatrix}
=\Big[\bar{n}(t) + \frac{1}{2}\Big]\Big[\frac{\cosh 2r(t)}{2} I ~~\notag \\
+ \frac{\sinh 2r(t)}{2} \begin{pmatrix} \cos\theta(t) & \sin\theta(t) \\
\sin\theta(t) & -\cos\theta(t) \end{pmatrix} \Big].\end{aligned}$$ If $\bar{n}(t)=0$, the above covariance matrix is reduced to the standard form for a pure squeezed state [@Zhang90867]. Obviously, the squeezed thermal state squeezes the thermal-state fluctuation $\bar{n}(t) +1/2$. Thus, the squeezing in the squeezed thermal state can be described by the squeezing parameter $r(t)$. If there is no initial system-reservoir correlation, then we have $\nu_2(t)=0$ so that $s(t)=0$ which leads to the squeezing parameter $r(t)=0$.
In Fig. \[fig2\], the time evolution of the cavity intensity $n(t)$ and the squeezing parameter $r(t)$ are plotted for the different coupling strengths $\eta=\lambda/\lambda_0$. As shown in Fig. \[fig2\] (a), for a weak coupling ($\eta=0.4$), the cavity intensity decays monotonically and eventually approaches to zero, as a result of the Markovian damping dynamics at zero temperature. Also, the small but non-zero squeezing parameter $r(t)$ indicates that the initial two-photon correlation $\langle a(0)b_k(0)\rangle$ between the system and reservoir induces a small squeezing effect to the cavity field in the beginning. However, the long-time behavior of the squeezing parameter shows that the effect of the initial system-reservoir correlation is washed out in the long-time limit, which is also consistent with the Markovian dynamics. In contrast, by increasing the coupling strength, as depicted in Fig. \[fig2\] (b), the cavity intensity decays rather fast in the beginning and then it revives and damps with oscillation in which some non-Markovian memory effect appears. Interestingly, the squeezing parameter $r(t)$ shows a similar behavior of the non-Markiovan effect, except for the beginning where the initial two-photon correlation $\langle a(0)b_1(0)\rangle$ generates a stronger squeezing effect to the cavity field, in comparison with the weak coupling case. When the coupling strength continues increasing to $\eta=2.0$ (the strong non-Markovian regime [@Wu1018407]), the cavity intensity decays faster in the very beginning and then revives and keeps oscillating without damping from then on, see Fig. \[fig2\] (c). In this situation, we find that the initial-correlation-induced squeezing dynamics also oscillates over all the time. Therefore, we can conclude that the initial two-photon correlations $\langle a(0)b_k(0)\rangle$ can lead to a nontrivial squeezing dynamics of the cavity field, as a consequence of strong non-Markovian memory dynamics, but it is negligible in the steady-state limit in the Markovian regime.
Initially system-reservoir correlated mixed thermal states
----------------------------------------------------------
Next, we investigate the effect of the initially system-reservoir correlated state with the correlation $\langle
a(0)b_k^\dag(0)\rangle \neq 0$. To this end, we consider an initially mixed state $$\begin{aligned}
\rho_{ab_1}(0)=B(\vartheta)\rho_a\otimes\rho_{b_1}B^\dag(\vartheta),
\label{ics}\end{aligned}$$ where the operator $B(\vartheta)=\exp[\frac{\vartheta}{2}(ab_1^\dag-a^\dag b_1)]$ and the density operators $\rho_{a/b_1}$ represent the thermal states with average thermal photon numbers $\bar{n}_{a/b_1}$. This initially correlated state can be formed via the bilinear coupling between the cavity field $a$ and the resonator mode $b_1$ in the thermal states, and note that nonclassical entanglement are not present in this initially correlated state [@kim]. A direct calculation shows that the initial system-reservoir correlations
$$\begin{aligned}
&\langle a(0)b_k(0)\rangle=0, \\
& \langle
a(0)b_k^\dag(0)\rangle=\frac{\sin\vartheta}{\sqrt{2\pi}}(\bar{n}_a-\bar{n}_{b_1})\sin
k.\end{aligned}$$
For the initially correlated state of Eq.(\[ics\]), it is easy to find that the reduced density matrices $\rho_a(0)$ and $\rho_{b_1}(0)$ of the cavity field and the resonator mode $b_1$ are also the thermal states with the average thermal photon numbers $n_{a}(0)=\frac{1}{2}[\bar{n}_a+\bar{n}_{b_1}+(\bar{n}_a-\bar{n}_{b_1})\cos\vartheta]$ and $n_{b_1}(0)=\frac{1}{2}[\bar{n}_a+\bar{n}_{b_1}-(\bar{n}_a-\bar{n}_{b_1})\cos\vartheta]$, respectively. From Eq.(\[core\]), we obtain $$\begin{aligned}
\nu_1(t)&=&-i
\frac{(\bar{n}_a-\bar{n}_{b_1})\sin \vartheta}{\sqrt{2\pi}}\mathcal{F}(t),\\
\upsilon_1(t)&=&\frac{2n_{b_1}(0)}{\pi}|\mathcal{F}(t)|^2,\end{aligned}$$ and $\nu_2(t)=0$, and $\upsilon_0(t)=0$, $\upsilon_2(t)=0$. Thus, the corresponding physical observables of the cavity system for the above initially correlated state of Eq.(\[ics\]) are given by $$\begin{aligned}
n(t)
=|u(t)|^2n_a(0)+2Re[u^*(t)\nu_1(t)]+\upsilon_1(t),\end{aligned}$$ and $\langle a(t)\rangle=0$, $ s(t) =0$. It indicates that the cavity state remains in a thermal state over all the time with the cavity field intensity $\sim n(t)$.
In Fig. \[fig3\], the time evolution of the cavity field intensity $n(t)$ is plotted for different coupling strengths $\eta$ between the nanocavity and the waveguide. Fig. \[fig3\] (a) shows the the average photon number for a weak coupling $\eta=0.4$. It reveals that the intensity of the cavity field decays monotonically to a steady-state value, as a character of the Markovian dynamics. The initial system-reservoir correlation $\langle
a(0)b_k^\dag(0)\rangle$ leads to the intensity oscillating around the decay line of the case of the initially uncorrelated state. The amplitude of the local oscillations increases in the beginning and then decreases to a unnoticeable effect as time develops. In other words, the effect of the initial correlation $\langle
a(0)b_k^\dag(0)\rangle$ will be washed out in the steady limit. With the increasing of the coupling strength, the intensity no longer monotonically decays and some revival phenomena occur as a character of the non-Markovian memory dynamics [@Wu1018407], as depicted in Fig. \[fig3\] (b). When the coupling is increased to $\eta=2.0$ as a strong coupling value, we see from Fig. \[fig3\] (c) that the intensity and the initial system-reservoir induced oscillation keep oscillating in the whole time regime. In other words, the effect resulted from the initial system-reservoir correlation in the non-Markovian regime will not be washed out by the interaction between the system and the reservoir.
Summary
=======
In summary, we investigate the dynamics of open quantum systems in the presence of initial system-reservoir correlations. We take the photonic cavity system coupled to a non-Markovian reservoir as a specific open quantum system. By solving the exact dynamics of the cavity system, the effects of the initial correlations are explicitly built into the solution of the cavity field intensity and the two-photon correlation function. We also derive the time-convolutionless exact master equation which incorporates with the initial system-reservoir correlations. The non-Markovian memory effects are fully embedded into the time-dependent coefficients in the master equation. The fluctuation coefficient $\gamma_2(t)$ is modified by the initial system-reservoir photonic scattering correlation but the frequency shift $\Delta (t)$ of the cavity and the dissipation coefficient $\gamma_1(t)$ remain unchanged. However, the initial two-photon correlation between the system and the reservoir induces two-photon decoherence terms in the master equation, which can lead to photon squeezing in the cavity. We also take a nanocavity coupled to a coupled resonator optical waveguide (serving as a structured reservoir) as an experimentally realizable system, from which we find that the effects of the initial correlations are fragile for a Markovian reservoir but play an important role in the non-Markovian regime. In fact, in the strong non-Markovian regime, the initial two-photon correlation between the cavity and the reservoir can induce oscillating squeezing dynamics in the cavity. But in Markovian regime, the effects of the initial system-reservoir correlations will be washed out in the steady-state limit.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported by the National Science Council of ROC under Contract No. NSC-99-2112-M-006-008-MY3, the National Center for Theoretical Science of Taiwan, National Natural Science Foundation of China (Grant No.10804035), and SDRF of CCNU (Grant No. CCNU 09A01023).
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| ArXiv |
---
abstract: |
It is shown that two gravitating scalar fields may form a thick brane in 5D spacetime. The necessary condition for the existence of such a regular solution is that the scalar fields potential must have local and global minima.
Key words: thick brane, scalar fields
author:
- 'Vladimir Dzhunushaliev [^1]'
title: Thick brane solution in the presence of two interacting scalar fields
---
Introduction
============
In recent years there has been a revived interest in theories having a greater number of spatial dimensions than the three that are observed. In contrast to the original Kaluza-Klein theories of extra dimensions, the recent incarnations of extra dimensional theories allow the extra dimensions to be large and even infinite in size (in the original Kaluza-Klein theories the extra dimensions were curled up or compactified to the experimentally unobservable small size of the Planck length: $10^{-33}$cm). These new extra dimensional theories have opened up new avenues to explaining some of the open questions in particle physics (the hierarchy problem, nature of the electro-weak symmetry breaking, explanation of the family structure) and astrophysics (the nature of dark matter, the nature of dark energy) [@arkani] - [@gogberashvili]. In addition they predict new experimentally measurable phenomenon in high precision gravity experiments, particle accelerators, and in astronomical observations.
Most of the brane world models use infinitely thin branes with delta-like localization of matter. However these models are generally regarded as an approximation since any fundamental underlying theory, such as quantum gravity or string theory, must contain a fundamental length beyond which a classical space-time description is impossible. It is therefore necessary to justify the infinitely thin brane approximation as a well-defined limit of a smooth structure – a thick brane – obtainable as a solution to coupled gravitational and matter field equations. One early example of a thick brane comes from the 5 dimensional model considered in [@rubakov] where one had a topologically non-trivial field configuration for the scalar field.
In Ref. [@akama], the picture is presented that our universe is a dynamically localized 3-brane in a higher dimensional space (”brane world“ ). As an example, the dynamics of the Nielsen-Olesen vortex type in six dimensional spacetime is adopted to localize our space-time within a 3-brane. At low energies, everything is trapped in the 3-brane, and the Einstein gravity is induced through the fluctuations of the 3-brane.
It is therefore of great interest to formulate some general requirements on the brane world design leading to the appearance of stable, thick branes having a well-defined zero thickness limit and able to trap ordinary matter. Taking a physically reasonable stress-energy tensor it was shown that in 6 dimensions [@gogberashvili2] and also higher extra dimensions [@singleton] one can trap all the Standard Model fields using gravity alone.
In Ref’s [@bronnikov] thick brane world models are studied as $\mathbb Z_2$-symmetric domain walls supported by a scalar field with an arbitrary potential $V(\phi)$ in 5D general relativity and it was shown that in the framework of 5D gravity, a globally regular thick brane always has an anti-de Sitter asymptotic and is only possible if the scalar field potential $V(\phi)$ has an alternating sign.
In Ref’s [@Gremm1] - [@barcelo] some properties of brane models was investigated: localization of gravity, graviton ground state, stability and so on.
In Ref. [@Barbosa-Cendejas:2005kn] a comparative analysis of localization of 4D gravity on a non $Z_2$-symmetric scalar thick brane in both 5-dimensional Riemannian space time and pure geometric Weyl integrable manifold is presented.
Multidimensional space-times with large extra dimensions turned out to be very useful when addressing several problems of the recent non–supersymmetric string model realization of the Standard Model at low energy with no extra massless matter fields [@kokorelis].
In Ref. [@Dzhunushaliev:2003sq] it is shown that two interacting non-gravitating scalar fields with a non-trivial potential may have a regular spherically symmetric solution. This solution shows that one can avoid the Derrick’s theorem [@derrick] forbidding the existence of regular static solution in the spacetime with the dimension greater 2 for scalar fields if the potential has a local minimum besides global one. This result allows us to assume that the inclusion of gravitation may not destroy the regularity of similar solutions in 5D spacetime. In Ref. [@Bronnikovc] one example of spherically symmetric solution with a gravitating scalar field is given but in contrast with the solution that will be presented here the potential of the scalar field in Ref. [@Bronnikovc] is negative.
The goal of this investigation is to show that there exists a new kind of thick brane solutions that is different with thick brane solutions found in Ref’s [@DeWolfe:1999cp] [@Bronnikov:2005bg]. We will show that the asymptotical behavior of one scalar field allow us to offer trapping of Maxwell electrodynamics and spinor fields on the brane. Especially it is necessary to note that the consideration of two scalar fields allow us to obtain the regular thick brane solution with the potential bounded from below.
Initial equations
=================
We consider 5D gravity + two interacting fields. The key for the existence of a regular solution here is that the scalar fields potential have to have *local* and *global* minima, and at the infinity the scalar fields tend to a local but *not* to global minimum.
The 5D metric is $$ds^2 = a(y) \eta_{\mu \nu} dx^\mu dx^\nu - dy^2,
\label{sec2-10}$$ where $\mu ,\nu = 0,1,2,3$; $y$ is the $5^{th}$ coordinate; $\eta_{\mu \nu} = \left\{ +1, -1, -1, -1 \right\}$ is the 4D Minkowski metric. The Lagrangian for scalar fields $\phi$ and $\chi$ is $$\mathcal L = \frac{1}{2} \nabla_A \phi \nabla^A \phi +
\frac{1}{2} \nabla_A \chi \nabla^A \chi - V(\phi, \chi) ,
\label{sec2-20}$$ where $A= 0,1,2,3,5$. The potential $V(\phi, \chi)$ is $$V(\phi, \chi) = \frac{\lambda_1}{4} \left(
\phi^2 - m_1^2
\right)^2 +
\frac{\lambda_2}{4} \left(
\chi^2 - m_2^2
\right)^2 + \phi^2 \chi^2 - V_0 ,
\label{sec2-30}$$ where $V_0$ is a constant which can be considered as a 5D cosmological constant $\Lambda$. We consider the case when the functions $\phi, \chi$ are $\phi(y), \chi(y)$. The 5D Einstein and scalar field equations are $$\begin{aligned}
R^A_B - \frac{1}{2} \delta^A_B R &=& \varkappa T^A_B ,
\label{sec2-40}\\
\frac{1}{\sqrt{G}} \nabla_A \left(
\sqrt{G} G^{AB} \nabla_B \phi
\right) &=& - \frac{\partial V\left( \phi, \chi \right)}{\partial \phi} ,
\label{sec2-50}\\
\frac{1}{\sqrt{G}} \nabla_A \left(
\sqrt{G} G^{AB} \nabla_B \chi
\right) &=& - \frac{\partial V\left( \phi, \chi \right)}{\partial \chi} ,
\label{sec2-60}\end{aligned}$$ where $\varkappa$ is the 5D gravitational constant; $G_{AB}$ is the 5D metric and $G$ is the corresponding determinant. After substituting metric into Eq’s - we have the following equations $$\begin{aligned}
-3 \frac{a''}{a} - 3 \frac{a'^2}{a^2} &=& \frac{\varkappa}{4} \left[
\phi'^2 + \chi'^2 + \frac{\lambda_1}{2} \left(
\phi^2 - m_1^2
\right)^2 + \frac{\lambda_2}{2} \left(
\chi^2 - m_2^2
\right)^2 + 2 \phi^2 \chi^2 - 2 V_0
\right] ,
\label{sec2-70}\\
- 6 \frac{a'^2}{a^2} &=& \frac{\varkappa}{4} \left[
- \phi'^2 - \chi'^2 + \frac{\lambda_1}{2} \left(
\phi^2 - m_1^2
\right)^2 + \frac{\lambda_2}{2} \left(
\chi^2 - m_2^2
\right)^2 + 2 \phi^2 \chi^2 - 2 V_0
\right] ,
\label{sec2-80}\\
\phi'' + 4 \frac{a'}{a} \phi' &=& \phi \left[
2 \chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right)
\right] ,
\label{sec2-90}\\
\chi'' + 4 \frac{a'}{a} \chi' &=& \chi \left[
2 \phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right)
\right] ,
\label{sec2-100}\end{aligned}$$ where $\frac{d (\cdots)}{ dy} = (\cdots)'$. Let us introduce the following dimensionless functions $a/\sqrt{\varkappa/6} \rightarrow a$, $\phi \sqrt{\varkappa/3} \rightarrow \phi$, $ \chi\sqrt{\varkappa/3} \rightarrow \chi$, $2\left( \varkappa/6 \right)^2 V_0 \rightarrow V_0$, $m_{1,2}\sqrt{\varkappa/3} \rightarrow m_{1,2}$, $\lambda_{1,2}/2 \rightarrow \lambda_{1,2}$ and the dimensionless variable $y/\sqrt{\varkappa/6} \rightarrow y$.
After algebraical transformations Eq’s - have the following form $$\begin{aligned}
\frac{a''}{a} - \frac{a'^2}{a^2} &=&
- \frac{1}{2}\left( \phi'^2 + \chi'^2 \right),
\label{sec2-75}\\
\frac{a'^2}{a^2} &=& \frac{1}{8} \left[
\phi'^2 + \chi'^2 - \frac{\lambda_1}{2} \left(
\phi^2 - m_1^2
\right)^2 - \frac{\lambda_2}{2} \left(
\chi^2 - m_2^2
\right)^2 - \phi^2 \chi^2 + 2 V_0
\right] ,
\label{sec2-85}\\
\phi'' + 4 \frac{a'}{a} \phi' &=& \phi \left[
\chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right)
\right] ,
\label{sec2-95}\\
\chi'' + 4 \frac{a'}{a} \chi' &=& \chi \left[
\phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right)
\right] .
\label{sec2-105}\end{aligned}$$ It is easy to see that Eq. is the consequence of Eq. : if we take a derivative from the LHS and RHS of Eq. then we shall receive Eq. . The boundary conditions are $$\begin{aligned}
a(0) &=& a_0 ,
\label{sec2-110}\\
a'(0) &=& 0 ,
\label{sec2-120}\\
\phi(0) &=& \phi_0 , \quad \phi'(0) = 0 ,
\label{sec2-130}\\
\chi(0) &=& \chi_0 , \quad \chi'(0) = 0 .
\label{sec2-140}\end{aligned}$$ The boundary condition - and Eq. give us the following constraint $$V_0 = \frac{\lambda_1}{4} \left(
\phi^2_0 - m_1^2
\right)^2 + \frac{\lambda_2}{4} \left(
\chi^2_0 - m_2^2
\right)^2 + \frac{1}{2} \phi^2_0 \chi^2_0,
\label{sec2-150}$$
Numerical investigation {#num}
=======================
For the numerical calculations we choose the following parameters values $$a_0 = \phi_0 = 1, \quad
\chi_0 = \sqrt{0.6}, \quad
\lambda_1 = 0.1, \quad
\lambda_2 = 1.0 .
\label{sec3-10}$$ We apply the methods of step by step approximation for finding of numerical solutions using the MATHEMATICA package (the details of similar calculations can be found in Ref. [@Dzhunushaliev:2003sq], the corresponding MATHEMATICA program can be found in \*.tar.gz file of the archived version of this paper [@Dzhunushaliev:2006vv]).
**Step 1**. On the first step we solve Eq. (having zero approximations $a_0(y) = a_0, \chi_0(y) = m_1 \tanh y$). The regular solution exists for a special value $m^*_{1,i}$ only. For $m_1 < m^*_{1,i}$ the function $\chi_i(y) \rightarrow +\infty$ and for $m_1 > m^*_{1,i}$ the function $\chi_i(y) \rightarrow -\infty$ (here the index $i$ is the approximation number). One can say that in this case we solve *a non-linear eiqenvalue problem*: $\chi_i^*(y)$ is the eigenstate and $m_{1,i}^*$ is the eigenvalue on this Step.
**Step 2**. On the second step we solve Eq. using zero approximation $a_0(y)$ for the function $a(y)$ and the first approximation $\chi_1^*(y)$ for the function $\chi(y)$ from the Step 1. For $m_2 < m^*_{2,1}$ the function $\phi_1(y) \rightarrow +\infty$ and for $m_2 > m^*_{2,1}$ the function $\phi_1(y) \rightarrow -\infty$. Again we have *a non-linear eiqenvalue problem* for the function $\phi_1(y)$ and $m^*_{2,1}$.
**Step 3**. On the third step we repeat the first two steps that to have the good convergent sequence $\phi_i^*(y), \chi_i^*(y)$. Practically we have made three approximations.
**Step 4**. On the next step we solve Eq. which gives us the function $a_1(y)$.
**Step 5**. On this step we repeat Steps 1-4 necessary number of times that to have the necessary accuracy of definition of the functions $a^*(y), \phi^*(y), \chi^*(y)$.
After Step 5 we have the solution presented on Fig. \[fig1\]. These numerical calculations give us the eigenvalues $m_1^* \approx 2.122645756$, $m_2^* \approx 1.3721439906788$ and eigenstates $a^*(y), \phi^*(y), \chi^*(y)$. The derived solution was verified by using the standard numerical method of solving the differential equations in the MATHEMATICA package (the corresponding MATHEMATICA program can be found in \*.tar.gz file of the archived version of this paper [@Dzhunushaliev:2006vv]).
It easy to see that the asymptotical behavior of the solution is $$\begin{aligned}
a(y) &\approx& a_\infty e^{-k_a y} , \quad
k_a^2 = \frac{1}{4} \left( V_0 - \frac{\lambda_2}{4} m_2^4 \right) ,
\label{sec3-20}\\
\phi(y) &\approx& m_1 + \phi_\infty e^{-k_\phi y} ,\quad
k_\phi = 2k_a + \sqrt{4 k_a^2 + 2 \lambda_1 m_1^2} ,
\label{sec3-30}\\
\chi(y) &\approx& \chi_\infty e^{-k_\chi y} ,\quad
k_\chi = 2k_a + \sqrt{4 k_a^2 + m_1^2 - \lambda_2 m_2^2} ,
\label{sec3-40}\end{aligned}$$ where $a_\infty, \phi_\infty, \chi_\infty$ are constants. The dimensionless energy density is $$e(y) = 2 \left( \frac{\varkappa}{3} \right)^2 \varepsilon(y) =
\frac{1}{4} \left[
\phi'^2 + \chi'^2 + \frac{\lambda_1}{2} \left( \phi^2 - m_1^2 \right)^2 +
\frac{\lambda_2}{2} \left( \chi^2 - m_2^2 \right)^2 +
\phi^2 \chi^2 - 2 V_0
\right]
\label{sec3-50}$$ and it is presented in Fig. \[fig2\].
Taking into account that the quantity $V\left( \phi(\infty), \chi(\infty) \right)$ is absolutely similar to a 5D cosmological constant, we can introduce a dimensionless brane tension $$\sigma = 2 \int \limits_0^\infty \biggl[
e(y) - V\Bigl( \phi(\infty), \chi(\infty) \Bigl)
\biggl] dy \approx 0.74 .
\label{sec3-55}$$
According to Eq. one can define the thickness $\Delta$ of the presented thick brane as $$\Delta \approx \max \left\{ k_\phi, k_\chi \right\}.
\label{sec3-60}$$ The key role for understanding why such regular solution may exist belongs to the fact that the potential has the local and global minima. The profile of the potential $V(\phi, \chi)$ is presented in Fig. \[fig3\].
![The profile of the potential $V(\phi, \chi)$.[]{data-label="fig3"}](potential){height="9cm" width="9cm"}
Trapping of the matter
======================
Now we would like to consider trapping of the electromagnetic and spinor fields on the above derived thick brane. The Lagrangian of interacting electromagnetic and scalar fields is taken from : $$\label{Lagr_int}
L_{eff}=-\frac{1}{4} \tilde{F}_{BC} \tilde{F}^{BC} +
\alpha \phi^2 \tilde{A}_B \tilde{A}^B - m^2 \tilde{A}_B \tilde{A}^B ,$$ where $\tilde{F}_{BC} = \tilde{A}_{B,C} - \tilde{A}_{C,B}$ is the 5D electromagnetic tensor with 5-dimensional vector potential $\tilde{A}_B(x^{D})$ and scalar field $\varphi(y)$ depending only on the extra coordinate $y$; $\alpha$ - an arbitrary constant, $m$ is the mass of vector field $\tilde{A}_B$.
The 5D Maxwell equations will be: $$\label{max}
D_C \tilde{F}^{BC}=\tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right].$$ Let us rewrite Eq. (\[max\]) as follows: $$\label{max1}
D_{\nu} \tilde{F}^{B \nu}+D_5 \tilde{F}^{B 5}=
\tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right].$$ We will use the gauge $\tilde{A}_5=0$ and search for a solution of (\[max1\]) in the form: $$\begin{aligned}
D_\nu \tilde{F}^{\mu \nu} &=& 0,
\label{max2}\\
D_5 \tilde{F}^{B 5} &=& \tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right]
\label{max3}\end{aligned}$$ For the solution we will use the following ansatz $$\tilde{A}^B(x^{D}) = A^B(x^{\mu}) f(y),$$ where $A^B(x^{\mu})$ is the 4D electromagnatic potential function only on 4D coordinates. Then from Eq’s we will have $$\begin{aligned}
D_\nu F^{\mu \nu} &=& 0,
\label{sec4-10}\\
f^{\prime \prime} + 4\frac{a^{\prime}}{a} f^{\prime} &=&
\frac{1}{a^4} \frac{d}{dy} \left( a^4 \frac{df}{dy} \right) =
f \left( \alpha \phi^2 - m^2 \right)
\label{sec4-20}\end{aligned}$$ where the first equation is the usual 4D Maxwell equations on the brane. The solution of the second equation on the background of the thick brane is presented in Fig.\[EM\]. Here it is necessary to note that again the regular solution $f(y)$ exists for an exceptional value of the parameter $m$ only. It is easy to see from Eq. : this equation is exactly Schrodinger equation with the potential $\alpha \phi^2$ (which is a hole). Eq. has a regular solution describing a particle in a hole for an exceptional value of $m$ that is an eigenvalue of the Schrodinger equation .
As one can see, the EM field is trapped on the 4D brane. In this case the electromagnetic fields in the bulk are $$\tilde{A}(x^B) = A(x^\mu) f(y)$$ where $f(y)$ is the exponentially decreasing function.
Let us consider further the question about trapping of fermion fields on the brane. In the simplest case such a possibility was pointed out in Ref. [@Rub] at consideration of the brane model as the model of domain wall. In this work the model of one real scalar field $\phi$ with two degenerated minima was introduced for description of the domain wall in 5D spacetime. In this case existing kink solution has its asymptotes in these minima with constant values of the field $\phi$. In our case similar situation occurs: two scalar fields $\phi, \chi$ create the system with two local minima, and the solutions tend asymptotically to one of these minima where the field $\chi$ tends to zero and $\phi$ to the constant values as in the case from Ref. [@Rub].
It allows us to investigate trapping of fermions on the brane for our case by analogy with Ref. [@Rub]. The curved space 5D gamma matrices are $$\Gamma^{\mu} = \frac{1}{\sqrt{a}} \gamma^{\mu},\qquad \Gamma^{r}=-i \gamma^{5},$$ where $\gamma^{\mu}$ and $\gamma^{5}$ are the usual Dirac matrices in 4D theory $$\gamma^\mu = \left\{ \left(
\begin{array}{cc}
0 & I \\
I & 0
\end{array}
\right),
\left(
\begin{array}{cc}
0 & \vec \sigma \\
-\vec \sigma & 0
\end{array}
\right)
\right\}, \quad
\gamma^5 = \left(
\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}
\right)
\label{1.3}$$ where $\sigma^i$ are usual Pauli matrices in flat spacetime. Then using the action for interacting scalar $\varphi$ and fermion $\Psi$ fields we have $$\label{action_int}
S_{\Psi}=\int d^4x\, dr \left( i \overline{\Psi} \Gamma^A D_A \Psi -
h \phi \overline{\Psi} \Psi \right)$$ here $\phi$ is the scalar field from the Lagrangian and $h$ is a constant. The Dirac equation can be written in the form $$\label{Dirac_eq}
i \Gamma^A D_A \Psi-h \phi(y) \Psi =0.$$ Here $D_A=\partial_A+\Upsilon_A$, where pseudo-connection $\Upsilon_A$ can be defined as follows [@Ying] $$\Upsilon_A=\frac{1}{2} e_{\bar{M}}^N \left(\partial_A e_N^{\bar{M}}-\partial_N e_A^{\bar{M}}\right),$$ where the vielbein $e_A^{\bar{M}}$ is defined via $g_{A B}= e_A^{\bar{M}} e_B^{\bar{N}} \eta_{\bar{M} \bar{N}}$, and the inverse vielbein $e_{\bar{M}}^A$ via $g^{A B}= e_{\bar{M}}^A e_{\bar{N}}^B \eta^{\bar{M} \bar{N}}$. For our case $\Upsilon_A=\left(0, 0, 0, 0, a^{\prime}/a\right)$.
Let us consider ansatz $$\label{ansatz1}
\Psi (x^B) = \psi(x^\mu) \Psi_0(y)$$ If we are interested in localization of zero modes, then, as it was shown in Ref. [@Jack], there are the solutions of Eq. (\[Dirac\_eq\]) with 4D mass $m=0$. For the zero mode $\gamma^{\mu}D_{\mu}\psi=0$, and the Dirac equation (\[Dirac\_eq\]) turns out in the equation: $$\label{Dirac_eq_1}
\gamma^5 \left( \partial_r +\frac{a^{\prime}}{a}\right) \Psi_0 =
h \phi(y) \Psi_0,$$ where $^\prime$ means the derivative with respect to $r$. Eq. with account of has the following solution: $$\label{Dirac_sol}
\Psi = \exp{\left[ -\int_{0}^{r} dr^{\prime}\left(\frac{a^{\prime}}{a}+ h \phi(r^{\prime})\right) \right]}
\psi (x^\mu),$$ where $\psi (x^\mu)$ is the usual solution of 4D Weyl equation, and the condition $\gamma_5 \Psi_0=-\Psi_0$ is taken into account. As it was shown in Section \[num\], the sum $\left(a^{\prime}/a+ h \phi\right)$ tends asymptotically to some constant. So the zero mode (\[Dirac\_sol\]) is localized near $r=0$, i.e. on the brane, and decreases exponentially at large $r$: $\Psi_0 \propto \exp{(-m_5 \left| y \right|)}$.
Let us note that one can include the function $\chi$ in Eq’s and by the following way:\
$\alpha \phi^2 \tilde{A}_B \tilde{A}^B \stackrel{\text{change}}{\longrightarrow}
\alpha \left( \phi^2 + \chi^2 \right) \tilde{A}_B \tilde{A}^B$ and $h \phi \overline{\Psi} \Psi \stackrel{\text{change}}{\longrightarrow}
h \left( \phi + \chi \right) \overline{\Psi} \Psi$ but it does not matter because the asymptotical behavior of the function $\phi \stackrel{r \rightarrow \infty}{\longrightarrow} m_1$ is important only (as $\chi \stackrel{r \rightarrow \infty}{\longrightarrow} 0$).
Let us note that we do not consider trapping of scalar fields on this brane. The reason is very simple: we have shown *exactly* that two scalar fields with Lagrangian are confined on the brane. The situation is even better: these scalar fields create the brane ! It is necessary note that in the process of numerical calculation we have obtained a domain wall solution without gravity, i.e. two scalar fields can create the solution with the planar symmetry and switching on the gravity does not destroy this solution.
Discussion and conclusions
==========================
Now we would like to list the essential specialities of the presented solution:
1. The existence of the solution crucially depends on the number of interacting scalar fields $(n > 1)$ and the presence of the non-trivial potential $V(\phi, \chi)$ which has local and global minima. At the infinity the scalar fields tend to local minimum and the potential has alternating sign when $r \in [0, \infty]$ that to the existence of the presented solution. The numerical investigation shows that in the presence of one scalar field the similar solution does not exist.
2. The advantage of the presented solution is that the asymptotical behavior of the scalar field $\phi$ allow us to obtain trapping of electromagnetic and spinor fields on the brane.
3. Let us note that the thick brane solution presented here differs from the thick brane solutions presented in Ref’s [@DeWolfe:1999cp] and [@Bronnikov:2005bg] that:
1. thick brane solution from Ref. [@DeWolfe:1999cp] is obtained for the scalar field with the potential unbounded from below that in contrast with our potential which is bounded from below.
2. in Ref. [@Bronnikov:2005bg] the thick brane solution is obtained for scalar fields having non-trivial asymptotical topological structure in contrast with our solution.
4. The solution is topologically trivial. It means that at the infinity two scalar fields do not form a hedgehog configuration in contrast with the thick brane solutions presented in Ref. [@Bronnikov:2005bg].
5. The quantity $V(\phi(\infty), \chi(\infty))$ can be considered as a 5D cosmological constant $\Lambda$.
6. \[4\] In Ref. [@Dzhunushaliev:2003sq] it is shown that after some simplification and assumtions the SU(3) gauge Lagrangian can be reduced to the Lagrangian describing interacting scalar fields $\phi$ and $\chi$. This remark allows us to assume that a real thick brane can be formed by a 5D gauge condensate which is described by interacting scalar fields.
7. According to the previous item (\[4\]) the 5D mechanism of trapping the matter on a thick brane may be similar to the confinement mechanism in 4D quantum chromodynamics. In this case trapping of the corresponding quantum gauge fields on the thick brane is non-perturbative and can not be investigated using Feynman diagram technique.
8. If the thick brane is formed with the help of a gauge condensate then the problem of the stability of the thick brane becomes very non-trivial. It occurs because a non-static condensate has to be described in much more complicated manner than static condensate in Ref. [@Dzhunushaliev:2003sq]. It is connected to that fact that the change in time of quantum object is connected not only to the change of this quantity but also to the change of a wave function as well.
9. From the mathematical point of view the presented solution is an eigenstate for a nonlinear eigenvalue problem.
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[^1]: Senior Associate of the Abdus Salam ICTP
| ArXiv |
---
abstract: 'Attitude control systems naturally evolve on nonlinear configuration spaces, such as ${\ensuremath{\mathsf{S}}}^2$ and ${\ensuremath{\mathsf{SO(3)}}}$. The nontrivial topological properties of these configuration spaces result in interesting and complicated nonlinear dynamics when studying the corresponding closed loop attitude control systems. In this paper, we review some global analysis and simulation techniques that allow us to describe the global nonlinear stable manifolds of the hyperbolic equilibria of these closed loop systems. A deeper understanding of these invariant manifold structures are critical to understanding the global stabilization properties of closed loop control systems on nonlinear spaces, and these global analysis techniques are applicable to a broad range of problems on nonlinear configuration manifolds.'
author:
- 'Taeyoung Lee, Melvin Leok, and N. Harris McClamroch[^1][^2][^3][^4] [^5]'
bibliography:
- 'CDC11.bib'
title: |
**Stable Manifolds of Saddle Points\
for Pendulum Dynamics on ${\ensuremath{\mathsf{S}}}^2 $ and ${\ensuremath{\mathsf{SO(3)}}}$**
---
Introduction
============
Global nonlinear dynamics of various classes of closed loop attitude control systems have been studied in recent years. An overview of results on attitude control of a rotating rigid body is given in [@ChSaMcCSM11]. Closely related results on attitude control of a spherical pendulum (with attitude an element of the two-sphere ${\ensuremath{\mathsf{S}}}^2$) and of a 3D pendulum (with attitude an element of the special orthogonal group ${\ensuremath{\mathsf{SO(3)}}}$) are given in [@ChMcIJRNC07; @ChMcBeAut08; @ChaMcCITAC09]. These and other similar publications address the global closed dynamics of smooth vector fields. Assuming that the closed loop vector field has an asymptotically stable equilibrium, as desired in attitude stabilization problems, additional hyperbolic equilibria necessarily exist. The domain of attraction of the asymptotically stable equilibrium is contained in the complement of the union of the stable manifolds of the hyperbolic equilibria. These geometric factors motivate the current paper, in which new analytical and computational results on the stable manifolds of the hyperbolic equilibria are obtained.
To make the development concrete, the presentation is built around two specific closed loop vector fields: one for the attitude dynamics of a spherical pendulum and one for the attitude dynamics of a 3D pendulum. In analyzing these two cases, we introduce new analytical and computational tools that are broadly applicable to studying the geometry of more general attitude control systems.
Spherical Pendulum
==================
A spherical pendulum is composed of a mass $m$ connected to a frictionless pivot by a massless link of length $l$. It is acts under uniform gravity, and it is subject to a control moment $u$. The configuration of a spherical pendulum is described by a unit-vector $q\in{\ensuremath{\mathbb{R}}}^3$, representing the direction of the link with respect to a reference frame.
Therefore, the configuration space is the two-sphere ${\ensuremath{\mathsf{S}}}^2=\{q\in{\ensuremath{\mathbb{R}}}^3\,|\, q\cdot q =1\}$. The tangent space of the two-sphere at $q$, namely ${\ensuremath{\mathsf{T}}}_q{\ensuremath{\mathsf{S}}}^2$, is the two-dimensional plane tangent to the unit sphere at $q$, and it is identified with ${\ensuremath{\mathsf{T}}}_q{\ensuremath{\mathsf{S}}}^2\simeq\{\omega\in{\ensuremath{\mathbb{R}}}^3\,|\, q\cdot \omega =0\}$, using the following kinematics equation: $$\begin{aligned}
\dot q= \omega\times q,\end{aligned}$$ where the vector $\omega\in{\ensuremath{\mathbb{R}}}^3$ represents the angular velocity of the link. The equation of motion is given by $$\begin{gathered}
\dot\omega = \frac{g}{l}q\times e_3 + \frac{1}{ml^2} u,\end{gathered}$$ where the constant $g$ is the gravitational acceleration, and the vector $e_3=[0,0,1]\in{\ensuremath{\mathbb{R}}}^3$ denotes the unit vector along the direction of gravity. The control moment at the pivot is denoted by $u\in{\ensuremath{\mathbb{R}}}^3$.
Control System
--------------
Several proportional-derivative (PD) type control systems have been developed on ${\ensuremath{\mathsf{S}}}^2$ in a coordinate-free fashion [@BulMurN95; @BulLew05]. Here, we summarize a control system that stabilizes a spherical pendulum to a fixed desired direction $q_d\in{\ensuremath{\mathsf{S}}}^2$.
Consider an error function on ${\ensuremath{\mathsf{S}}}^2$, representing the projected distance from the direction $q$ to the desired direction $q_d$, given by $$\begin{aligned}
\Psi(q,q_d)=1-q\cdot q_d.\end{aligned}$$ The derivative of $\Psi$ with respect to $q$ along the direction $\delta q =\xi\times q$, where $\xi\in{\ensuremath{\mathbb{R}}}^3$ and $\xi\cdot q=0$, is given by $$\begin{aligned}
{\ensuremath{\mathbf{D}}}_q \Psi(q,q_d)\cdot\delta q = -(\xi\times q)\cdot q_d = (q_d\times q)\cdot \xi.\end{aligned}$$ For positive constants $k_q,k_\omega$, the control input is chosen as: $$\begin{aligned}
u = ml^2(- k_\omega \omega - k_q q_d\times q -\frac{g}{l}q\times e_3).\end{aligned}$$ The corresponding closed loop dynamics are given by $$\begin{gathered}
\dot \omega = -k_\omega\omega - k_q q_d\times q,\label{eqn:dotw}\\
\dot q= \omega\times q.\label{eqn:dotq}\end{gathered}$$
This yields two equilibrium solutions: (i) the desired equilibrium $(q,\omega)=(q_d,0)$; (ii) additionally, there exists another equilibrium $(-q_d,0)$ at the antipodal point on the two-sphere.
It can be shown that the desired equilibrium is asymptotically stable by using the following Lyapunov function: $$\begin{aligned}
\mathcal{V} = \frac{1}{2} \omega\cdot\omega + k_q \Psi(q,q_d).\end{aligned}$$ In this paper, we analyze the local stability of each equilibrium by linearizing the closed loop dynamics to study the equilibrium structures more explicitly. In particular, we develop a coordinate-free form of the linearized dynamics of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}, in the following section.
Linearization
-------------
A variation of a curve $q(t)$ on ${\ensuremath{\mathsf{S}}}^2$ is a family of curves $q^\epsilon(t)$ parameterized by $\epsilon\in{\ensuremath{\mathbb{R}}}$, satisfying several properties [@BulLew05]. It cannot be simply written as $q^\epsilon(t)=q(t)+\epsilon\delta q(t)$ for $\delta q(t)$ in ${\ensuremath{\mathbb{R}}}^3$, since in general, this does not guarantee that $q^\epsilon(t)$ lies in ${\ensuremath{\mathsf{S}}}^2$. In [@LeeLeoIJNME08], an expression for a variation on ${\ensuremath{\mathsf{S}}}^2$ is given in terms of the exponential map as follows: $$\begin{aligned}
q^\epsilon(t) = \exp(\epsilon \hat\xi(t))q(t),\label{eqn:qe}\end{aligned}$$ for a curve $\xi(t)$ in ${\ensuremath{\mathbb{R}}}^3$ satisfying $\xi(t)\cdot q(t)=0$ for all $t$. The *hat map* $\hat\cdot:{\ensuremath{\mathbb{R}}}^3\rightarrow{\ensuremath{\mathfrak{so}(3)}}$ is defined by the condition that $\hat x y =x\times y$ for any $x,y\in{\ensuremath{\mathbb{R}}}^3$. The resulting infinitesimal variation is given by $$\begin{aligned}
\delta q (t) = \frac{d}{d\epsilon}\bigg|_{\epsilon=0} q^\epsilon(t) = \xi(t)\times q(t). \label{eqn:delq}\end{aligned}$$ The variation of the angular velocity can be written as $$\begin{aligned}
\omega^\epsilon(t) = \omega(t) + \epsilon \delta \omega(t),\label{eqn:we}\end{aligned}$$ for a curve $\delta w(t)$ in ${\ensuremath{\mathbb{R}}}^3$ satisfying $q(t)\cdot w(t)=0$ for all $t$. Hereafter, we do not write the dependency on time $t$ explicitly.
The time-derivative of $\delta q$ can be obtained either from [(\[eqn:delq\])]{} or by substituting [(\[eqn:qe\])]{}, [(\[eqn:we\])]{} into [(\[eqn:dotq\])]{}, and considering the first order terms of $\epsilon$. In either case, we have $$\begin{aligned}
\delta\dot q= \dot\xi\times q + \xi\times(\omega\times q) = \delta\omega\times q + \omega\times(\xi\times q).\end{aligned}$$ Using the vector cross product identity $a\times (b\times c)=(a\cdot c)b-(a\cdot b) c$ for any $a,b,c\in{\ensuremath{\mathbb{R}}}^3$, this can be written as $$\begin{gathered}
\dot\xi\times q + (\xi\cdot q)w -(\xi\cdot\omega) q = \delta\omega\times q +(\omega\cdot q)\xi -(\omega\cdot\xi)q.\end{gathered}$$ Since $\xi\cdot q=0$, $\omega\cdot q=0$, this reduces to $$\begin{gathered}
\dot\xi\times q = \delta\omega\times q.\end{gathered}$$ Since both sides of the above equation are perpendicular to $q$, this is equivalent to $q\times(\dot\xi\times q) = q\times(\delta\omega\times q)$, which yields $$\begin{gathered}
\dot \xi - (q\cdot\dot\xi) q = q\times(\delta\omega\times q).\end{gathered}$$ Since $\xi\cdot q =0$, we have $\dot\xi\cdot q +\xi\cdot\dot q=0$. Using this, the above equation can be rewritten as $$\begin{aligned}
\dot \xi & = -(\xi\cdot(\omega\times q))q+ q\times(\delta\omega\times q)\nonumber\\
& = (qq^T\hat\omega) \xi +(I-qq^T)\delta\omega.\label{eqn:dotxi}\end{aligned}$$ This corresponds to the linearized equation of motion for [(\[eqn:dotq\])]{}. Similarly, by substituting [(\[eqn:delq\])]{}, [(\[eqn:we\])]{} into [(\[eqn:dotw\])]{}, we obtain $$\begin{aligned}
\delta\dot\omega &= -k_\omega\delta\omega -k_qq_d\times(\xi\times q)\nonumber\\
&=-k_\omega\omega +k_q\hat q_d \hat q\,\xi,\label{eqn:dotdelw}\end{aligned}$$ which is the linearized equation for [(\[eqn:dotw\])]{}.
Equations [(\[eqn:dotxi\])]{}, [(\[eqn:dotdelw\])]{} can be written in a matrix form as $$\begin{aligned}
\dot x =
\begin{bmatrix}\dot\xi \\ \delta\dot\omega \end{bmatrix}
=\begin{bmatrix} qq^T\hat\omega & I-qq^T\\k_q\hat q_d\hat q & -k_wI\end{bmatrix}
\begin{bmatrix}\xi \\ \delta\omega \end{bmatrix}=Ax,\label{eqn:xdot}\end{aligned}$$ where the state vector of the linearized controlled system is $x=[\xi;\delta\omega]\in{\ensuremath{\mathbb{R}}}^6$. A spherical pendulum has two degrees of freedom, but this linearized equation of motion evolves in ${\ensuremath{\mathbb{R}}}^6$ instead of ${\ensuremath{\mathbb{R}}}^4$. Since $q\cdot\omega=0$ and $q\cdot\xi=0$, we have the following two additional constraints on $\xi,\delta\omega$: $$\begin{aligned}
Cx=
\begin{bmatrix} q^T & 0 \\ -\omega^T\hat q & q^T\end{bmatrix}
\begin{bmatrix}\xi \\ \delta\omega \end{bmatrix}
=\begin{bmatrix} 0 \\ 0 \end{bmatrix}.\label{eqn:con}\end{aligned}$$ Therefore, the state vector $x$ should lie in the null space of the matrix $C\in{\ensuremath{\mathbb{R}}}^{2\times 4}$. However, this is not an extra constraint that should be imposed when solving [(\[eqn:xdot\])]{}. As long as the initial condition $x(0)$ satisfies [(\[eqn:con\])]{}, the structure of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}, and [(\[eqn:xdot\])]{}, guarantees that the state vector $x(t)$ satisfies [(\[eqn:con\])]{} for all $t$, i.e. $\frac{d}{dt}C(t)x(t) =0$ for all $t\geq 0$ when $C(0)x(0)=0$. This means that the null space of $C$ is a flow-invariant subspace.
Equilibrium Solutions
---------------------
We choose the desired direction as $q_d=e_3$. The equilibrium solution $(q_d,0)=(e_3,0)$ is referred to as the hanging equilibrium, and the additional equilibrium solution $(-q_d,0)=(-e_3,0)$ is referred to as the inverted equilibrium. We study the eigen-structure of each equilibrium using the linearized equation [(\[eqn:xdot\])]{}. To illustrate the ideas, the controller gains are selected as $k_q=k_\omega=1$.
### Hanging Equilibrium
The eigenvalues $\lambda_i$, and the eigenvectors $v_i$ of the matrix $A$ at the hanging equilibrium $(e_3,0)$ are given by $$\begin{gathered}
\lambda_{1,2}=(-1\pm\sqrt{3}i)/2,\;
\lambda_{3,4}=\lambda_{1,2},\;\lambda_5=0,\;\lambda_6=-1,\\
v_{1,2}= e_1 + (-1\pm\sqrt{3}i)e_4/2,\;
v_{3,4}= e_2 + (-1\pm\sqrt{3}i)e_5/2,\\
v_5=e_3,\quad v_6=e_6,\end{gathered}$$ where $e_i\in{\ensuremath{\mathbb{R}}}^6$ denotes the unit-vector whose $i$-th element is one, and other elements are zeros. Note that there are repeated eigenvalues, but we obtain six linearly independent eigenvectors, i.e., the geometric multiplicities are equal to the algebraic multiplicities.
The basis of the null space of the matrix $C$, namely $\mathcal{N}(C)$ is $\{e_1,e_2,e_4,e_5\}$. The solution of the linearized equation can be written as $x(t)=\sum_{i=1}^6 c_i \exp(\lambda_it) v_i$ for constants $c_i$ that are determined by the initial condition: $x(0)=\sum_{i=1}^6 c_i v_i$. But, the eigenvectors $v_5,v_6$ do not satisfy the constraint given by [(\[eqn:con\])]{}, since they do not lie in $\mathcal{N}(C)$. Therefore, the constants $c_5,c_6$ are zero for initial conditions that are compatible with [(\[eqn:con\])]{}. We have $\mathrm{Re}[\lambda_i]<0$ for $1\leq i\leq 4$. Therefore, the equilibrium $(q,\omega)=(e_3,0)$ is asymptotically stable.
### Inverted Equilibrium
The eigenvalues $\lambda_i$, and the eigenvectors $v_i$ of the matrix $A$ at the inverted equilibrium $(-e_3,0)$ are given by $$\begin{gathered}
\lambda_{1,2}=-(\sqrt{5}+1)/2,\lambda_{3,4}=(\sqrt{5}-1)/2,\lambda_5=0,\lambda_6=-1,\nonumber\\
v_{1}= e_1 -(\sqrt{5}+1)e_4/2,\, v_2=e_2-(\sqrt{5}+1)e_5/2,\label{eqn:v1v2}\\
v_3=(\sqrt{5}+1)e_1/2 +e_4,\, v_4=(\sqrt{5}+1)e_2/2+e_5,\nonumber\\
v_5=e_3,\, v_6=e_6.\nonumber\end{gathered}$$ The basis of $\mathcal{N}(C)$ is $\{e_1,e_2,e_4,e_5\}$. Hence, the eigenvectors $v_5,v_6$ do not lie in $\mathcal{N}(C)$. Therefore, the solution can be written as $x(t)=\sum_{i=1}^4 c_i \exp(\lambda_it) v_i$ for constants $c_i$ that are determined by the initial condition.
We have $\mathrm{Re}[\lambda_{1,2}]<0$, and $\mathrm{Re}[\lambda_{3,4}]>0$. Therefore, the inverted equilibrium $(q,\omega)=(-e_3,0)$ is a hyperbolic equilibrium, and in particular, a saddle point.
Stable Manifold for the Inverted Equilibrium {#sec:SM}
--------------------------------------------
### Stable Manifold
The saddle point $(-e_3,0)$ has a stable manifold $W^s$, which is defined to be $$\begin{aligned}
W^s(-e_3,0) &= \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\, \lim_{t\rightarrow\infty} \mathcal{F}^t(q,\omega) = (-e_3,0)\},\end{aligned}$$ where $\mathcal{F}^t:(q(0),\omega(0))\rightarrow(q(t),\omega(t))$ denotes the flow map along the solution of [(\[eqn:dotw\])]{}, [(\[eqn:dotq\])]{}. The existence of $W^s(-e_3,0)$ has nontrivial effects on the overall dynamics of the controlled system. Trajectories in $W^s(-e_3,0)$ converge to the antipodal point of the desired equilibrium $(e_3,0)$, and it takes a long time period for any trajectory near $W^s(-e_3,0)$ to asymptotically converge to the desired equilibrium $(e_3,0)$.
According to the stable and unstable manifold theorem [@Kuz98], a local stable manifold $W^s_{loc}(-e_3,0)$ exists in the neighborhood of $(-e_3,0)$, and it is tangent to the stable eigenspace $E^s(-e_3,0)$ spanned by the eigenvectors $v_1$ and $v_2$ of the stable eigenvalues $\lambda_{1,2}$. The (global) stable manifold can be written as $$\begin{aligned}
W^s(-e_3,0) & = \bigcup_{t>0} \mathcal{F}^{-t} ( W^s_{loc}(-e_3,0)),\label{eqn:Ws}\end{aligned}$$ which states that the stable manifold $W^s$ can be obtained by globalizing the local stable manifold $W^s_{loc}$ by the backward flow map.
This yields a method to compute $W^s(-e_3,0)$ [@KraOsiIJBC05]. We choose a small ball $B_\delta\subset W^s_{loc}(-e_3,0)$ with a radius $\delta$ around $(-e_3,0)$, and we grow the manifold $W^s(-e_3,0)$ by evolving $B_\delta$ under the flow $\mathcal{F}^{-t}$. More explicitly, the stable manifold can be parameterized by $t$ as follows: $$\begin{aligned}
W^s(-e_3,0) =\{ \mathcal{F}^{-t} (B_\delta)\}_{t>0}.\label{eqn:Wc}\end{aligned}$$
We construct a ball in the stable eigenspace of $(-e_3,0)$ with sufficiently small radius $\delta$, i.e. $B_\delta\subset E^s_{loc}(-e_3,0)$. From the stable eigenvectors $v_1,v_2$ at [(\[eqn:v1v2\])]{}, $E^s_{loc}(-e_3,0)$ can be written as $$\begin{aligned}
E^s_{loc}& (-e_3,0) = \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\,
q=\exp(\alpha_1\hat e_1+\alpha_2\hat e_2)(-e_3),\nonumber\\
& \omega=-\hat q^2(-(\sqrt{5}+1)/2)(\alpha_1 e_1+\alpha_2e_2)\text{ for $\alpha_1,\alpha_2\in{\ensuremath{\mathbb{R}}}$}\},\label{eqn:Esloc}\end{aligned}$$ where $-\hat q^2$ in the expression for $\omega$ corresponds to the orthogonal projection onto the plane normal to $q$, as required due to the constraint $q\cdot\omega=0$.
We define a distance on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$ as follows: $$\begin{aligned}
d_{{\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2} ((q_1,\omega_1),(q_2,\omega_2)) = \sqrt{\Psi(q_1,q_2)} + \|\omega_1-\omega_2\|.\label{eqn:dis}\end{aligned}$$ For $\delta>0$, the subset $B_\delta$ of $E^s_{loc}(-e_3,0)$ is parameterized by $\theta\in{\ensuremath{\mathsf{S}}}^1$ as $$\begin{aligned}
B_\delta & = \{ (q,\omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2\,|\,
q=\exp(\alpha_1\hat e_1+\alpha_2\hat e_2)(-e_3),\nonumber\\
& \omega=-\hat q^2(-(\sqrt{5}+1)/2)(\alpha_1 e_1+\alpha_2e_2),\text{ where}\nonumber\\
& \text{$\alpha_1=\frac{\delta}{1/\sqrt{2}+(\sqrt{5}+1)/2}\cos\theta,$\;}\nonumber\\
& \text{$\alpha_2=\frac{\delta}{1/\sqrt{2}+(\sqrt{5}+1)/2}\sin\theta$,} \text{ for $\theta\in{\ensuremath{\mathsf{S}}}^1$} \}.\label{eqn:Bdelta}\end{aligned}$$ The given choice of the constants $\alpha_1,\alpha_2$ guarantees that any point in $B_\delta$ has a distance $\delta$ to $(-e_3,0)$ according to the distance metric [(\[eqn:dis\])]{}.
### Variational Integrators
The parameterization of the stable manifold $W_s$ in [(\[eqn:Wc\])]{} requires the computation of the backward flow map $\mathcal{F}^{-t}$. However, general purpose numerical integrators may not preserve the structure of the two-sphere or the underlying dynamic characteristics, such as energy dissipation rate, accurately, and they may yield qualitatively incorrect numerical results in simulating a complex trajectory over a long-time period [@HaiLub00].
Geometric numerical integration is concerned with developing numerical integrators that preserve geometric features of a system, such as invariants, symmetry, and reversibility. In particular, variational integrators are geometric numerical integrators for Lagrangian or Hamiltonian systems, constructed according to Hamilton’s principle. They have desirable computational properties of preserving symplecticity and momentum maps, and they exhibit good energy behavior [@MarWesAN01]. A variational integrator is developed for Lagrangian or Hamiltonian systems evolving on the two-sphere in [@LeeLeoIJNME08]. It preserves both the underlying symplectic properties and the structures of the two-sphere concurrently.
A variational integrator on ${\ensuremath{\mathsf{S}}}^2$ for the controlled dynamics of a spherical pendulum can be written in a backward-time integration form as follows: $$\begin{aligned}
q_{{k}} & = -{\ensuremath{\left( h\omega_{k+1} - \frac{h^2}{2ml^2} M_{k+1} \right)}}\times q_{k+1}\nonumber\\
&\quad + {\ensuremath{\left( 1-{\ensuremath{\left\| h\omega_{k+1} - \frac{h^2}{2ml^2} M_{k+1} \right\|}}^2 \right)}}^{1/2} q_{k+1},\label{eqn:qk}\\
\omega_{{k}} & = \omega_{k+1} - \frac{h}{2ml^2} M_k - \frac{h}{2ml^2} M_{k+1},\label{eqn:wk}\end{aligned}$$ where the constant $h>0$ is time step, the subscript $k$ denotes the value of a variable at the time $t_k=kh$, and $M_k = ml^2(-k_\omega \omega_k - k_q q_d\times q_k)$. For given $(q_{k+1},\omega_{k+1})$, we first compute $M_{k+1}$. Then, $q_k$ is obtained by [(\[eqn:qk\])]{}, followed by $M_k$, and $\omega_k$ is computed by [(\[eqn:wk\])]{}. This yields an explicit, discrete inverse flow map $\mathcal{F}_d^{-h}((q_{k+1},\omega_{k+1}))=(q_{k},\omega_{k})$.
### Visualization
We choose 100 points on the surface of $B_\delta$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:Ws\]]{} for several values of $t$. Each colored curve on the sphere represents a trajectory on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$, since at any point $q$ on the curve, the direction of $\dot q=\omega\times q$ is tangent to the curve at $q$, and the magnitude of $\dot q$ is indirectly represented by color shading.
We observe the following characteristics of the stable manifold $W_s(-e_3,0)$ of the inverted equilibrium:
- The boundary of the stable manifold $W_s(-e_3,0)\subset {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{S}}}^2$ parameterized by $t$ is circular when projected onto ${\ensuremath{\mathsf{S}}}^2$.
- Each trajectory in $W_s(-e_3,0)$ is on a great circle, when projected onto ${\ensuremath{\mathsf{S}}}^2$. According to the closed loop dynamics [(\[eqn:dotw\])]{}, and the given initial condition at the surface of $B_\delta$, the direction of $\dot\omega$ is always parallel to $\omega$. Therefore, the direction of $\omega$ is fixed, and the resulting trajectory of $q$ is on a great circle. This also corresponds to the fact that the eigenvalue $\lambda_1$ for the first mode representing the rotations about the first axis is equal to the eigenvalue $\lambda_2$ for the second mode representing the rotations about the second axis at [(\[eqn:v1v2\])]{}, i.e. the convergence rates of these two rotations are identical.
- The angular velocity decreases to zero as the direction of the pendulum $q$ converges to $-e_3$.
- The stable manifold $W_s(-e_3,0)$ may cover ${\ensuremath{\mathsf{S}}}^2$ multiple times if $t$ is sufficiently large, as illustrated at [Fig. \[fig:Ws9\]]{}. Therefore, at any point $q\in{\ensuremath{\mathsf{S}}}^2$, we can choose $\omega$ such that $(q,\omega)$ lies in the stable manifold $W^s(-e_3,0)$ (the corresponding value of $\omega$ is not unique, since if it is sufficiently large, $q$ can traverse the sphere several times before converging to $-e_3$). This is similar to *kicking* a damped spherical pendulum carefully such that it converges to the inverted equilibrium.
3D Pendulum
===========
A 3D pendulum is a rigid body supported by a frictionless pivot acting under a gravitational potential. This is a generalization of a planar pendulum or a spherical pendulum, as it has three rotational degrees of freedom. It has been shown that a 3D pendulum may exhibit irregular maneuvers [@ChaLeeJNS11].
We choose a reference frame, and a body-fixed frame. The origin of the body-fixed frame is located at the pivot point. The attitude of a 3D pendulum is the orientation of the body-fixed frame with respect to the reference frame, and it is described by a rotation matrix representing the linear transformation from the body-fixed frame to the reference frame. The configuration manifold of a 3D pendulum is the special orthogonal group, ${\ensuremath{\mathsf{SO(3)}}}=\{R\in{\ensuremath{\mathbb{R}}}^{3\times 3}\,|\, R^T R=I,\mathrm{det}[R]=1\}$.
The equations of motion for a 3D pendulum are given by $$\begin{gathered}
J\dot\Omega + \Omega\times J\Omega = mg \rho\times R^T e_3 + u,\label{eqn:Wdot2}\\
\dot R = R\hat\Omega,\label{eqn:Rdot}\end{gathered}$$ where the matrix $J\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ is the inertia matrix of the pendulum about the pivot, and $\rho\in{\ensuremath{\mathbb{R}}}^3$ is the vector from the pivot to the center of mass of the pendulum represented in the body-fixed frame. The control moment at the pivot is denoted by $u\in{\ensuremath{\mathbb{R}}}^3$.
Control System
--------------
Several control systems have been developed on ${\ensuremath{\mathsf{SO(3)}}}$ [@BulLew05; @ChaMcCITAC09; @LeePACC11]. Here, we summarize a control system to stabilize a 3D pendulum to a fixed desired attitude $R_d\in{\ensuremath{\mathsf{SO(3)}}}$. Consider an attitude error function given by $$\begin{aligned}
\Psi(R,R_d)=\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ (I-R_d^TR)G \right]}}}},\end{aligned}$$ for a diagonal matrix $G=\mathrm{diag}[g_1,g_2,g_3]\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ with $g_1,g_2,g_3>0$. The derivative of this attitude error function with respect to $R$ along the direction of $\delta R= R\hat\eta$ for $\eta\in{\ensuremath{\mathbb{R}}}^3$ is given by $$\begin{aligned}
{\ensuremath{\mathbf{D}}}_R & \Psi(R,R_d)\cdot\delta R = -\frac{1}{2}{\mbox{tr}\ensuremath{\negthickspace{\ensuremath{\left[ R_d^TR\hat\eta G \right]}}}} \\
& = \frac{1}{2}(GR_d^TR -R^TR_d G)^\vee \cdot\eta\equiv e_R\cdot \eta,\end{aligned}$$ where we use the property that $\mathrm{tr}[\hat x A]=-x\cdot(A-A^T)^\vee$ for any $x\in{\ensuremath{\mathbb{R}}}^3,A\in{\ensuremath{\mathbb{R}}}^{3\times 3}$. The *vee map*, $\vee:{\ensuremath{\mathfrak{so}(3)}}\rightarrow{\ensuremath{\mathbb{R}}}^3$, denotes the inverse of the hat map. An attitude error vector is defined as $e_R = \frac{1}{2}(GR_d^TR -R^TR_d G)\in{\ensuremath{\mathbb{R}}}^3$. For positive constants $k_\Omega,k_R$, we choose the following control input: $$\begin{aligned}
u = -k_R e_R -k_\Omega \Omega -mg \rho\times R^T e_3.\end{aligned}$$ The corresponding closed loop dynamics are given by $$\begin{gathered}
J\dot\Omega =- \Omega\times J\Omega -k_R e_R -k_\Omega \Omega,\label{eqn:Wdot2}\\
\dot R = R\hat\Omega.\label{eqn:Rdot}\end{gathered}$$ This system has four equilibria: in addition to the desired equilibrium $(R_d,0)$, there exist three other equilibria at $(R_d\exp (\pi\hat e_i,0),0)$ for $i\in\{1,2,3\}$, which correspond to the rotation of the desired attitude by $180^\circ$ about each body-fixed axis.
The existence of additional, undesirable equilibria is due to the nonlinear topological structure of ${\ensuremath{\mathsf{SO(3)}}}$, and it cannot be avoided by constructing a different control system (as long as it is continuous). It has been shown that it is not possible to design a continuous feedback control stabilizing an attitude globally on [$\mathsf{SO(3)}$]{} [@BhaBerSCL00; @KodPICDC98].
The stability of the desired equilibrium can be studied by using the following Lyapunov function, $$\begin{aligned}
\mathcal{V} =\frac{1}{2}\Omega\cdot J\Omega + k_R \Psi(R,R_d).\end{aligned}$$ In this paper, we analyze the stability of each equilibrium by linearizing the closed loop dynamics to study the equilibrium structures more explicitly.
Linearization
-------------
A variation in ${\ensuremath{\mathsf{SO(3)}}}$ can be expressed as [@LeeLeoPICCA05]: $$\begin{aligned}
R^\epsilon=R\exp(\epsilon\hat\eta),\quad \Omega^\epsilon=\Omega +\epsilon\delta\Omega,\label{eqn:delRdelW}\end{aligned}$$ for $\eta,\delta\Omega\in{\ensuremath{\mathbb{R}}}^3$. The corresponding infinitesimal variation of $R$ is given by $\delta R = R\hat\eta$. Substituting this into [(\[eqn:Rdot\])]{}, $$\begin{aligned}
R\hat\Omega\hat\eta + R\hat{\dot\eta}= R\hat\eta\hat\Omega + R\delta\hat\Omega.\end{aligned}$$ Using the property $\hat x \hat y -\hat y\hat x=\widehat{x\times y}$ for any $x,y,\in{\ensuremath{\mathbb{R}}}^3$, this can be rewritten as $$\begin{aligned}
\dot\eta = \delta\Omega -\hat\Omega\eta.\label{eqn:etadot}\end{aligned}$$ Similarly, by substituting [(\[eqn:delRdelW\])]{} into [(\[eqn:Wdot2\])]{}, we obtain $$\begin{aligned}
J\delta\dot\Omega & = -\delta\Omega\times J\Omega -\Omega\times J\delta\Omega\nonumber\\
&\quad -\frac{1}{2}k_R (GR_d^T R\hat\eta +\hat\eta R^T R_d G) -k_\Omega\delta\Omega,\nonumber\\
& = (\widehat{J\Omega}-\hat\Omega J -k_\Omega I) \delta\Omega -\frac{1}{2}k_RH\eta,\label{eqn:delWdot}\end{aligned}$$ where $H=\mathrm{tr}[R^T R_d G]I-R^T R_d G\in{\ensuremath{\mathbb{R}}}^{3\times 3}$, and we used the property, $\hat x A + A^T\hat x = \mathrm{tr}[A]I-A$ for any $x\in{\ensuremath{\mathbb{R}}}^3, A\in{\ensuremath{\mathbb{R}}}^{3\times 3}$. Equations [(\[eqn:etadot\])]{},[(\[eqn:delWdot\])]{} can be written in matrix form as $$\begin{aligned}
\dot x & = \begin{bmatrix}\dot\eta \\ \delta\dot\Omega \end{bmatrix}
=\begin{bmatrix}
-\hat\Omega & I\\
-\frac{1}{2}k_R J^{-1}H & J^{-1}(\widehat{J\Omega}-\hat\Omega J -k_\Omega I)
\end{bmatrix}
\begin{bmatrix}\eta \\ \delta\Omega \end{bmatrix}\nonumber\\
& = Ax.\label{eqn:xdotSO3}\end{aligned}$$ This corresponds to the linearized equation of motion of [(\[eqn:Wdot2\])]{}, [(\[eqn:Rdot\])]{}.
Equilibrium Solutions
---------------------
We choose the desired attitude as $R_d=I$. In addition to the desired equilibrium $(I,0)$, there are three additional equilibria, namely $(\exp(\pi\hat e_1),0)$, $(\exp(\pi\hat e_2),0)$, $(\exp(\pi\hat e_3),0)$. We study the eigen-structure of each equilibrium using the linearized equation [(\[eqn:xdotSO3\])]{}. We assume that $$\begin{aligned}
J=\mathrm{diag}[3,2,1]\,\mathrm{kgm^2},\; G=\mathrm{diag}[0.9,1,1.1],\; k_R=k_\Omega=1.\end{aligned}$$
### Equilibrium $(I,0)$
The eigenvalues of the matrix $A$ at the desired equilibrium $(I,0)$ are given by $$\begin{gathered}
\lambda_{1,2}=-0.1667\pm0.5676i,\\
\lambda_{3,4}=-0.25\pm 0.6614i,\\
\lambda_{5,6}=-0.5\pm 0.8367i.\end{gathered}$$ This equilibrium is an asymptotically stable focus.
### Equilibrium $(\exp(\pi\hat e_1),0)$
At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered}
\lambda_1=-0.7813,\quad v_1= e_1-0.7813 e_4,\nonumber\\
\lambda_2=-0.5854,\quad v_2=e_2-0.5854e_5,\nonumber\\
\lambda_3=-1.0477,\quad v_3=e_3-1.0477e_6,\label{eqn:v3_SO31}\\
\lambda_4=0.4480,\quad v_4=e_1+0.4480e_4,\nonumber\\
\lambda_5=0.0854,\quad v_5=e_2+0.0854e_5,\nonumber\\
\lambda_6=0.0477,\quad v_6=e_3+0.0477e_6.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where three modes are stable, and three modes are unstable.
### Equilibrium $(\exp(\pi\hat e_2),0)$
At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered}
\lambda_1=-0.3775,\quad v_1= e_1-0.3775 e_4,\nonumber\\
\lambda_2=-1,\quad v_2=e_2-e_5,\label{eqn:v2_SO32}\\
\lambda_3=-0.9472,\quad v_3=e_3-0.9472e_6,\nonumber\\
\lambda_4=-0.0528,\quad v_4=e_3-0.0528e_6,\nonumber\\
\lambda_5=0.0442,\quad v_5=e_1+0.0442e_4,\nonumber\\
\lambda_6=0.5,\quad v_6=e_2+5e_5.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where four modes are stable, and two modes are unstable.
### Equilibrium $(\exp(\pi\hat e_3),0)$
At this equilibrium, the eigenvalues and the eigenvectors of $A$ are given by $$\begin{gathered}
\lambda_1=-0.0613,\quad v_1= e_1-0.0613 e_4,\nonumber\\
\lambda_2=-0.2721,\quad v_2= e_1-0.2721 e_4,\nonumber\\
\lambda_3=-0.1382,\quad v_3= e_2-0.1382 e_5,\nonumber\\
\lambda_4=-0.3618,\quad v_4= e_2-0.3618 e_5,\nonumber\\
\lambda_5=-1.5954,\quad v_5= e_3-1.5954 e_6,\label{eqn:v5_SO33}\\
\lambda_6= 0.5954,\quad v_6= e_2+0.5954 e_6.\nonumber\end{gathered}$$ Therefore, this equilibrium is a saddle point, where five modes are stable, and one mode is unstable.
Stable Manifolds for the Saddle Points
--------------------------------------
The eigen-structure analysis shows that there exist multi-dimensional stable manifolds for each saddle point. They have zero measure as the dimension of stable manifold is less than the dimension of ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}$. But, the existence of these stable manifolds may have nontrivial effects on the attitude dynamics.
We numerically characterize these stable manifolds using backward time integration, as discussed in Section \[sec:SM\].
The stable eigenspace for each saddle point can be written as $$\begin{aligned}
&E^s_{loc} (\exp(\pi\hat e_1),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\
&\; R=\exp(\pi\hat e_1)\exp(\alpha_1\hat e_1+\alpha_2\hat e_2+\alpha_3\hat e_3),\\
&\; \Omega=-0.7813\alpha_1e_1-0.5854\alpha_2e_2-1.0477\alpha_3e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\\
&E^s_{loc} (\exp(\pi\hat e_2),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\
&\; R=\exp(\pi\hat e_2)\exp(\alpha_1\hat e_1+\alpha_2\hat e_2+(\alpha_3+\alpha_4)\hat e_3),\\
&\; \Omega=-0.37\alpha_1e_1-\alpha_2e_2-(0.94\alpha_3+0.05\alpha_4)e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\\
&E^s_{loc} (\exp(\pi\hat e_3),0) = \{ (R,\Omega)\in {\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}\,|\,\nonumber\\
&\; R=\exp(\pi\hat e_3)\exp((\alpha_1+\alpha_2)\hat e_1+(\alpha_3+\alpha_4)\hat e_2+\alpha_5\hat e_3),\\
&\; \Omega=-(0.06\alpha_1+0.27\alpha_2)e_1-(0.13\alpha_3+0.36\alpha_4)e_2\\
&\;\quad -1.59\alpha_5e_3\text{ for $\alpha_i\in{\ensuremath{\mathbb{R}}}$}\},\end{aligned}$$
We define a distance on ${\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}$ as follows: $$\begin{aligned}
d_{{\ensuremath{\mathsf{T}}}{\ensuremath{\mathsf{SO(3)}}}} ((R_1,\Omega_1),(R_2,\Omega_2)) = \sqrt{\Psi(R_1,R_2)} + \|\Omega_1-\Omega_2\|.\end{aligned}$$
A variational integrator for the attitude dynamics of a rigid body on ${\ensuremath{\mathsf{SO(3)}}}$ is developed in [@LeeLeoPICCA05; @LeeLeoCMDA07]. It can be rewritten in a backward integration form as follows: $$\begin{gathered}
h (\Pi_{k+1}-\frac{h}{2}M_{k+1})^\wedge = J_dF_k - F_k^TJ_d,\label{eqn:Fk}\\
R_k = R_{k+1}F_k^T,\label{eqn:Rk}\\
\Pi_k = F_k\Pi_{k+1}-\frac{h}{2}F_kM_{k+1}-\frac{h}{2} M_k,\label{eqn:Pik}\end{gathered}$$ where $M_{k}=u_k+mg\rho\times R^T e_3\in{\ensuremath{\mathbb{R}}}^3$ is the external moment, $\Pi_k=J\Omega_k\in{\ensuremath{\mathbb{R}}}^3$ is the angular momentum. The matrix $J_d\in{\ensuremath{\mathbb{R}}}^{3\times 3}$ denotes a non-standard inertia matrix given by $J_d = \frac{1}{2}\mathrm{tr}[J]I-J$, and the rotation matrix $F_k\in{\ensuremath{\mathsf{SO(3)}}}$ represent the relative attitude update between two integration time steps. For given $(R_{k+1},\Pi_{k+1})$, we first compute $M_{k+1}$, and solve [(\[eqn:Fk\])]{} for $F_k$. Then, $R_k$ is obtained by [(\[eqn:Rk\])]{}, and $\Pi_k$ is computed by [(\[eqn:Pik\])]{}. This yields a discrete inverse flow map, $\mathcal{F}^{-h}_d(R_{k+1},\Pi_{k+1})\rightarrow(R_{k},\Pi_{k})$.
### Visualization of $W_s(\exp(\pi\hat e_1),0)$
In [@LeeLeoPICDC08], a method to visualize a function or a trajectory on ${\ensuremath{\mathsf{SO(3)}}}$ is proposed. Each column of a rotation matrix represents the direction of a body-fixed axis, and it evolves on ${\ensuremath{\mathsf{S}}}^2$. Therefore, a trajectory on ${\ensuremath{\mathsf{SO(3)}}}$ can be visualized by three curves on a sphere, representing the trajectory of three columns of a rotation matrix. The direction of the angular velocity should be chosen such that the corresponding time-derivative of the rotation matrix is tangent to the curve, and the magnitude of angular velocity can be illustrated by color shading. An example of visualizing a rotation about a single axis is illustrated in [Fig. \[fig:visSO3\_demo\]]{}.
We choose 112 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_1),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM1\]]{} for several values of $t$.
In each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_1)=[e_1,-e_2,-e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_1),0)$:
- When $t\leq 15$, the trajectories in $W_s(\exp(\pi\hat e_1),0)$ are close to rotations about the third body-fixed axis $e_3$ to $\exp(\pi\hat e_1)$. This is consistent with the linearized dynamics, where the eigenvalue of the third mode, corresponding to the rotations about $e_3$, has the fastest convergence rate, as seen in [(\[eqn:v3\_SO31\])]{}.
- When $t\geq 15$, the first mode representing the rotations about $e_1$ starts to appear, followed by the second mode representing the rotation about $e_2$. This corresponds to the fact that the first mode has a faster convergence rate than the second mode, i.e. $|\lambda_1|>|\lambda_2|$.
- As $t$ is increased further, the third body-fixed axis leaves the neighborhood of $-e_3$, and it exhibit the following pattern:
{width="0.32\columnwidth"}
- The stable manifold $W_s(\exp(\pi\hat e_1),0)$ covers a certain part of ${\ensuremath{\mathsf{SO(3)}}}$, when projected on to it. So, when an initial attitude is chosen such that its third body-fixed axis is sufficiently close to $-e_3$, there possibly exist multiple initial angular velocities such that the corresponding solution converges to $\exp(\pi\hat e_1)$ instead of the desired attitude $R_d=I$.
### Visualization of $W_s(\exp(\pi\hat e_2),0)$
We choose 544 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_2),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM2\]]{} for several values of $t$.
In each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_2)=[-e_1,e_2,-e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_2),0)$:
- When $t\leq 12$, the trajectories in $W_s(\exp(\pi\hat e_2),0)$ is close to the rotations about the second body-fixed axis $e_2$. As $t$ increases, rotations about $e_3$ starts to appear. This corresponds to the linearized dynamics where the second mode representing rotations about $e_2$ has the fastest convergence rate, followed by the third mode at [(\[eqn:v2\_SO32\])]{}.
- As $t$ is increased further, nonlinear modes become dominant. The trajectories in $W_s(\exp(\pi\hat e_2),0)$ almost cover [$\mathsf{SO(3)}$]{}. This suggests that for any initial attitude, we can choose several initial angular velocities such that the corresponding solutions converges to $\exp(\pi\hat e_2)$.
### Visualization of $W_s(\exp(\pi\hat e_3),0)$
Similarly, we choose 976 points on the surface of $B_\delta\subset E^s_{loc} (\exp(\pi\hat e_3),0)$ with $\delta=10^{-6}$, and each point is integrated backward using [(\[eqn:qk\])]{}, [(\[eqn:wk\])]{} with timestep $h=0.002$. The resulting trajectories are illustrated in [Fig. \[fig:SM3\]]{} for several values of $t$.
At each figure, three body-fixed axes of the desired attitude $R_d=[e_1,e_2,e_3]$, and three body-fixed axes of the additional equilibrium attitude $\exp(\pi\hat e_3)=[-e_1,-e_2,e_3]$ are shown. From these computational results, we observe the following characteristics on the stable manifold $W_s(\exp(\pi\hat e_3),0)$:
- When $t\leq 8$, the trajectories in $W_s(\exp(\pi\hat e_3),0)$ are close to the rotations about the third body-fixed axis $e_3$. This corresponds to the linearized dynamics where the fifth mode representing rotations about $e_3$ has the fastest convergence rate given in [(\[eqn:v5\_SO33\])]{}.
- The rotations about $e_3$ are still dominant, even as $t$ is increased further. For the given simulation times, all trajectories in $W_s(\exp(\pi\hat e_3),0)$ are close to rotations about $e_3$.
Conclusions
===========
Stable manifolds of saddle points that arise in the closed-loop dynamics of two pendulum models are characterized numerically, and several properties are observed. Although the analytical and computational results have been presented for a spherical pendulum and a 3D pendulum, the methods presented naturally extend to any closed loop attitude control system with configurations in either ${\ensuremath{\mathsf{S}}}^2$ or ${\ensuremath{\mathsf{SO(3)}}}$.
[^1]: Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 39201 [[email protected]]{}
[^2]: Melvin Leok, Mathematics, University of California at San Diego, La Jolla, CA 92093 [[email protected]]{}
[^3]: N. Harris McClamroch, Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109 [[email protected]]{}
[^4]: ^$*$^This research has been supported in part by NSF under grants CMMI-1029551.
[^5]: ^$\dagger$^This research has been supported in part by NSF under grants DMS-0726263, DMS-1001521, DMS-1010687, and CMMI-1029445.
| ArXiv |
---
abstract: 'TIMMI2 diffraction–limited mid-infrared images of a multipolar proto-planetary nebula IRAS 16594$-$4656 and a young \[WC\] elliptical planetary nebula IRAS 07027$-$7934 are presented. Their dust shells are for the first time resolved (only marginally in the case of IRAS 07027$-$7934) by applying the Lucy-Richardson deconvolution algorithm to the data, taken under exceptionally good seeing conditions ($\leq$0.5). IRAS 16594$-$4656 exhibits a two-peaked morphology at 8.6, 11.5 and 11.7 $\mu$m which is mainly attributed to emission from PAHs. Our observations suggest that the central star is surrounded by a toroidal structure observed edge-on with a radius of 0.4$\arcsec$ ($\sim$640 AU at an assumed distance of 1.6 kpc) with its polar axis at P.A.$\sim$80, coincident with the orientation defined by only one of the bipolar outflows identified in the HST optical images. We suggest that the material expelled from the central source is currently being collimated in this direction and that the multiple outflow formation has not been coeval. IRAS 07027$-$7934 shows a bright, marginally extended emission (FWHM=0.3$\arcsec$) in the mid-infrared with a slightly elongated shape along the N-S direction, consistent with the morphology detected by HST in the near-infrared. The mid-infrared emission is interpreted as the result of the combined contribution of small, highly ionized PAHs and relatively hot dust continuum. We propose that IRAS 07027$-$7934 may have recently experienced a thermal pulse (likely at the end of the AGB) which has produced a radical change in the chemistry of its central star.'
author:
- 'D. A. García-Hernández, A. Manchado,P. García-Lario, A. Benítez Cañete, J. A. Acosta-Pulido, and A. M. Pérez García'
title: |
Revealing the mid-infrared emission structure of\
IRAS 16594$-$4656 and IRAS 07027$-$7934[^1]
---
Introduction
============
Planetary nebulae (PNe) are the result of the evolution of low- to intermediate-mass stars (0.8–8 M$_{\odot}$). These stars experience a phase of extreme mass loss during the previous asymptotic giant branch (AGB) that causes the ejection of the stellar envelope. When this mass loss ceases the AGB phase ends and the star evolves into a short-lived evolutionary stage called the ‘post-AGB’ or ‘proto-PN’ (PPN) phase just before the star becomes a PN. At present, the formation of axisymmetric structures in PNe (ranging from elliptical to bipolar) is believed to be completed by the end of the AGB phase (Balick & Frank 2002; Van Winckel 2003). But unlike for PNe, the study of PPNe is more difficult since their central stars (CSs) are usually too cool to photoionize the gas. Therefore, we cannot study the formation of axisymmetric morphologies in PPNe by mapping the ionized gas. We must use alternative techniques based on the analysis of: (i) the light scattered by the surrounding dust at optical wavelenghts; (ii) the neutral molecular gas in the envelope in the near-infrared (H$_2$), submilimeter (e.g. CO) or radio domain (e.g OH, SiO, H$_{2}$O, CO); and (iii) the dust emission emerging at mid- to far-infrared wavelengths.
PPNe generally show a double-peaked spectral energy distribution (SED) (Kwok 1993; Volk & Kwok 1989; van der Veen, Habing & Geballe 1989) with the photospheric emission coming from the central star dominating in the optical range and a strong infrared excess indicating the presence of a cool detached envelope (T$_{d}$$\sim$150–300 K). This strong infrared excess is produced by the thermal emission of the dust present in their circumstellar shells previously expelled during the AGB phase. The processes that lead to a wide variety of different morphologies observed in PNe (e.g. Manchado et al. 2000) are, however, still unknown. Several mechanisms have been proposed: the interaction of stellar winds (e.g. Mellema 1993), binary systems as central stars (e.g. Bond & Livio 1990; Morris 1987), non-radial pulsations (e.g. Soker & Harpar 1992) or the influence of magnetic fields (e.g. Pascoli 1992; Soker & Harpar 1992; García-Segura et al. 1999). To establish which one(s) of the above is the dominant process, it is essential to study these morphologies as early as possible after the departure from the spherical symmetry takes place, that is, in the PPN phase. Only recently, with the help of high spatial resolution observations it has been possible to study the intrinsic axisymmetric nature of the dust shells around a few compact PPNe at subarcsec level (Meixner et al. 1997; Meixner et al. 1999; Ueta et al. 2001).
In this paper, we present for the first time mid-infrared images (8–13 $\mu$m) at subarcsec level of a PPN IRAS 16594$-$4656 (hereafter I16594) and of a very young \[WC\] PN IRAS 07027$-$7934 (hereafter I07027) with the aim of mapping the dust emission originated in the innermost regions of their circumstellar dust shells. The observations made in the mid-infrared are presented in Sect. 2 while the data reduction process is described in Sect. 3. We show the results obtained in Sect. 4, which are later discussed in Sect. 5. The main conclusions derived from our analysis are given in Sect. 6.
Mid-infrared Observations
=========================
The observations were carried out on 2001 October 9 and 10, using the imaging mode of TIMMI2 (Reimann et al. 2000; Käufl et al. 2003) attached to the ESO 3.6m telescope (La Silla, Chile). TIMMI2 has an array of 320 $\times$ 240 pixels with a pixel scale of 0.2$\arcsec$ $\times$ 0.2$\arcsec$ resulting in a field of view of 64$\arcsec$ $\times$ 48$\arcsec$. The observational conditions were very good (photometric and with a stable seeing of around 0.5$\arcsec$) and, thus, we could obtain mid-infrared images (at 8.6 $\mu$m \[N1-filter\], 11.5 $\mu$m \[N11.9-filter\] and 11.7 $\mu$m \[SiC-filter\]) of I16594 and I07027 at the diffraction limit of the telescope. The standard nodding/chopping observational technique was used in order to cancel the thermal emission from the atmosphere and from the telescope. An on-chip nodding/chopping throw of 15" along the north-south direction was selected. Due to the short integration times required to avoid saturation in the mid-infrared, each image is a combination of a large number of individual sub-images $\sim$50–100, each one with an integration time between 18 and 40 ms and a chopping frequency around 6 Hz. Total on-source integration times were typically of $\sim$2 minutes. The mid-infrared photometric standard stars HD 29291, HD 156277, HD 196171 and HD 6805 (Doublier et al. 2004) were also observed at different air masses every night to determine the photometric flux conversion from ADUs (Analog to Digital Units) to Jy and to measure the telescope point spread function (PSF).
Data Reduction
==============
The data reduction process includes bad pixel correction and the combination of all the images into one single image per filter using standard tasks in IRAF[^2]. The flux calibration was made using the conversion factors derived from the observation of standard stars at different air masses. The variation of these conversion factors with air mass was slightly different during the two nights of observation due to the different atmospheric conditions. On October 9 this variation was very small ($<$10%), so a single averaged conversion factor was used for all the observations performed during the night. However, on October 10 we found larger variations ($\sim$30%). Thus, for I07027 we took the conversion factors derived from the observations of the standard star HD 156277, observed closer in time and at a similar air mass. For each filter the size and morphology of the target stars in our programme was compared with a mean PSF derived from the observation of the standard stars used for the flux calibration. The object name, observing date, filters, central wavelength and width of the different filters used, total on-source integration time, object size, PSF size, integrated and peak fluxes, are listed in Table 1. From the internal consistency of the measurements made on the standard stars we estimate that the photometric uncertainty of our observations is of the order of $\sim$10%.
The observed PSFs are dominated by diffraction effects. Thus, the Lucy-Richardson deconvolution algorithm as implemented in IRAF (task [lucy]{}) was used in order to recover the actual emission structure of the targets in each filter and remove the effects induced by the telescope PSF, which was found to be very stable. In order to study the goodness of the deconvolution process the standard stars observed for flux calibration purposes were also deconvolved with the same PSF used for the target stars. This exercise is useful to confirm that the deconvolution process does not introduce undesired artifacts. We found that in all cases the deconvolved images of the standard stars spread over only one or two pixels, showing a quasi-point-like brightness distribution. As an example, one of the deconvolved standard stars is shown together with its corresponding raw image in Figure 1.
Results
=======
Mid-infrared Morphology of IRAS 16594$-$4656
--------------------------------------------
Figure 2 shows the raw images of I16594 taken with TIMMI2 (left panel) together with the images obtained after applying the Lucy-Richardson deconvolution algorithm (right panel) above mentioned. The PSFs used for the deconvolution are also shown for comparison. I16594 shows already in the raw images an extended elongated morphology surrounding a complex inner core emission (with a FWHM=1.5" or 3 times the PSF FWHM) which is resolved in more detail after deconvolution. The source center has been determined by averaging the central coordinates of the elliptical isophotes within 20–40% of the peak intensity (this way we avoid any contamination from the core structure). The images displayed in Figure 2 have been centered at this position.
The deconvolved 8.6, 11.5 and 11.7 $\mu$m images are shown in the right panel of Figure 2. The overall elliptical shape of the nebula is clear in all three filters with its major axis oriented along the east-west direction (P.A.$\sim$80) and extends out to at least 3.5$\arcsec$ $\times$ 2.1$\arcsec$ at 5% of the peak intensity (10-$\sigma$ level above the sky background) at 8.6 $\mu$m. In addition, a conspicuous double-peaked morphology in the innermost region of the nebulosity is also recovered, suggesting the presence of an equatorial density enhancement (e.g. a dust torus). The two detected peaks are oriented approximately along the north-south direction (P.A.$\sim$$-$10) perpendicular to the axis of symmetry defined by the outer elliptical emission. The measured separation between the two peaks is always 0.8$\arcsec$, independent of the filter considered (see Figure 3). In addition, the north-peak (P.A.$\sim$$-$10) is about a factor 2 brighter than the south peak (see Figure 3). This finding is not unique to I16594. Indeed, Ueta et al.(2001) found a similar asymmetric profile in IRAS 22272$+$5435. The origin of this asymmetric appearance of the dust torus is still unclear. Ueta et al. (2001) argued that this can be attributed to asymmetric mass loss and/or an inhomogeneity in the dust distribution. We are confident that the deconvolved structure is real because a very similar emission structure is observed in all three filters. Note that a negligible contribution to the observed flux at 10 $\mu$m is expected from the central star of I16594 if this has a B7 spectral type as suggested by Van de Steene, Wood & van Hoof (2000). Hrivnak, Kwok & Su (1999) found that the central star contributes only 3% to the total flux detected in the mid-infrared. We can therefore safely assume that we are just observing the emission structure of the dust in the shell alone. The mid-infrared morphology seen in the deconvolved images is thus interpreted as the evidence of the presence of a dusty toroidal structure with a 0.4$\arcsec$ radius size seen nearly edge-on. Recently, Ueta et al. (2005) based on polarization data also suggest an orientation of the dust torus close to edge-on and they indicate that an inclination angle of roughly 75 ${\hbox{$^{\circ}$}}$ with respect to the line of sight is derived from a 2-D dust emission model. This adds I16594 to the short list of PPN where a similar dust torus has been resolved at subarcsec scale.
Mid-infrared Morphology of IRAS 07027$-$7934
--------------------------------------------
The mid-infrared morphology of I07027 is clearly less complex than the one observed in I16594 as deduced already from the raw images shown in Figure 4 (left panel). In this case, only a slightly extended and asymmetric source is detected in the two images available, which correspond to the filters N1 and N11.9 (centered at 8.6 and 11.5 $\mu$m, respectively). This can just be due to the larger distance to this source (see Section 5.2.2). Note, that the low level extension to the west of the peak emission seen in both filters seems to be a PSF effect, as it is also observed in the standard stars, although not so prominent.
The deconvolved images of I07027 in the 8.6 and 11.5 $\mu$m filters are shown in Figure 4 (right panel). After the deconvolution, a very bright and slightly elongated, marginally extended (FWHM=0.3$\arcsec$) emission core is recovered in both filters, oriented along the north-south direction. A similar orientation is observed in recent HST-NICMOS images of I07027 taken in the near-infrared (see Section 5.2). Unfortunately, our TIMMI2 data cannot confirm precisely whether the mid-IR emission peak is exactly coincident with the near-IR emission peak. We propose that the emission core which is detected in the mid-infrared must be coincident with the location of the central star, as the peak emission is observed exactly at the centre of the slight extended emission.
Discussion
==========
IRAS 16594$-$4656
-----------------
### IRAS 16594$-$4656 in the literature
I16594 (=GLMP 507) was first identified as a PPN candidate on the basis of its IRAS colors by Volk & Kwok (1989) and van der Veen, Habing & Geballe (1989). It shows a double-peaked spectral energy distribution dominated by a strong mid- to far- infrared dust emission component which is much brighter than the peak in the near-infrared (Van de Steene, van Hoof & Wood 2000). The first indication of the C-rich chemistry of I16594 was the detection of CO molecular emission in its envelope (with V$_{exp}$$\sim$16 km s$^{-1}$) by Loup et al. (1990), and the non-detection of OH maser emission (te Lintel Hekkert et al. 1991). More recently, García-Lario et al. (1999) studied the ISO spectrum of this source and confirmed this classification based on the detection of the characteristic IR emission features generally attributed to PAHs (at 3.3, 6.2, 7.7, 8.6 and 11.3 $\mu$m) together with relatively strong features at 12.6 and 13.4 $\mu$m which indicates a high degree of hydrogenation in these PAHs. The ISO spectrum also reveals the presence of strong 21, 26 and 30 $\mu$m dust emission features (see Figure 5), adding I16594 to a short list of known PPNe displaying this set of still unidentified features.
The optical spectrum of I16594 shows only the hydrogen Balmer emission lines over an extremely reddened stellar continuum (E$_{B-V}$=1.8, Van de Steene & van Hoof 2003) consistent with a B7 spectral type if dereddened. HST optical images show the presence of a bright central star surrounded by a multiple-axis bipolar nebulosity (seen in scattered light) with a complex morphology at some intermediate viewing angle (see Figure 6). The size of this optical nebulosity is 6.3$\arcsec$ $\times$ 3.3$\arcsec$ at 3$\sigma_{sky}$ level (Hrivnak, Kwok & Su 1999).
In the literature there are several indications of the presence of a circumstellar disc or a torus (an equatorial density enhancement) around I16594. The highly collimated structure seen in the HST optical images and the non-detected radio-continuum emission ($<$10 $mJy$) by Van de Steene & Pottasch (1993) suggest that the emission lines observed in the optical spectrum are the result of shock excitation produced by a fast bipolar wind from the central source in interaction with the slow AGB wind. In agreement with this hypothesis García-Hernández et al. (2002) reported the detection of H$_2$ shock-excited emission in I16594, later confirmed by Van de Steene & van Hoof (2003) through a more detailed analysis of the H$_2$ spectrum. They postulate that the H$_2$ emission originates mainly where the stellar wind is funnelled through a circumstellar disc or torus. More recently, Hrivnak, Kelly & Su (2004) presented HST-NICMOS near-infrared images of I16594 which show that this emission is originated in regions where shocks must be taking place. Polarization measurements originally taken by Su et al. (2003) and later analyzed by Ueta et al. (2005), who presented a PSF subtracted map of the polarized light, suggest the presence of an equatorial enhancement in I16594 as well. However, Van de Steene, van Hoof & Wood (2000) failed to detect any extended emission in their N-band TIMMI images of I16594 in a previous attempt to search for mid-infrared emission coming from this torus, but they observed the source with a lower spatial resolution (pixel scale of 0.66$\arcsec$), and under poor weather conditions.
### A Dusty Toroidal Structure around IRAS 16594$-$4656
There exists more than a dozen PPN shells that have been resolved in the mid-infrared so far. However, only a few of them show some structure at mid-infrared wavelengths. Meixner et al. (1999) found two different classes of mid-infrared morphologies. They distinguish those sources with a mid-infrared core/elliptical structure from those with a toroidal one and they argue that this morphological dichotomy is due to a difference in optical depth. In their sample there are only 4 out of 6 toroidal PPN/PNe in which the central dust torus is well resolved in two emission peaks. This work adds I16594 to this short list. In Table 2 we list the few known toroidal PPN/PNe sample together with some of their main observational characteristics, such as the spectral type of the central star, C/O ratio, evolutionary classification, optical morphology, and the list of mid-infrared dust emission features detected.
An inspection of Table 2 clearly indicates that I16594 is now a toroidal-PPN with the earliest spectral type known. The other PPNe with mid-infrared toroidal structures have all F-G spectral types, while IRAS 21282$+$5050 is already a young PN with an O9-type central star. It seems that many of these sources show a C-rich chemistry (indicated by the presence of PAH emission features) but the number of objects considered is still small and the statistics are very poor. It is interesting to remark the fact that all mid-infrared toroidal-PPN/PNe have bipolar/multipolar optical morphologies where the central star is clearly seen. In contrast, the central star is rarely seen in the mid-infrared core/elliptical class sources described by Meixner et al. (1999) and almost all of them display bipolar morphologies in the optical. In addition, the mid-infrared core/elliptical sources are typically O-rich and show deep silicate absorption features at 9.8 $\mu$m in their mid-infrared spectra, indicating that they may be optically thick at mid-infrared wavelengths (Meixner et al. 1999). The different optical morphology (with or without a visible central star) and the apparent differences in dust properties (optical thickness in the mid-infrared) suggest that mid-infrared toroidal PPNe might be surrounded by a dust torus which is optically thin at mid-infrared wavelengths and, thus, not able to obscure the central star in the optical domain, while, in contrast, mid-infrared core/elliptical PPNe would be surrounded by an optically thick dust torus/disk which would completely obscure the central star in the optical (Meixner et al. 1999, 2002; Ueta et al. 2000, 2003).
Our deconvolved mid-infrared images of I16594, of much better quality than those previously reported by Van de Steene, van Hoof & Wood (2000), reveal directly for the first time the presence of an optically thin dusty toroidal structure with a radius of 0.4$\arcsec$. Unfortunately, the distance determinations to I16594 are quite uncertain and, thus, a direct transformation of this observed size into an absolute physical value is not straightforward. Estimations based on the observed reddening are hampered by the fact that the overall extinction is always a combination of interstellar and circumstellar reddening. And in the case of I16594 there seems to be a considerable contribution from the circumstellar component. Calculations made by Van de Steene & van Hoof (2003) based on the intrinsic colors expected for a B7 central star in the optical and in the near-infrared suggest a total extinction of A$_{V}$=7.5 mag with R$_{V}$=4.2. With this value for the extinction and the flux calibration from the Kurucz model a distance of (2.2$\pm$0.4) L$_4$$^{1/2}$ kpc is obtained, where L$_4$ is in units of 10$^4$ L$_{\odot}$. We have tried to derive our own distance estimate to I16594 based on the analysis of the overall SED, from the optical to the far-infrared. For this we put together the IRAS fluxes at 12, 25, 60 and 100 $\mu$m, the near-infrared JHKL magnitudes from @gl97 and the BVRI magnitudes from @hr99. The observed BVRI and JHKL fluxes were corrected for extinction using the total extinction of A$_{V}$=7.5 mag determined by @vv03 and the extinction law from @ca89. Then, a distance-dependent luminosity was obtained by integrating the observed flux at all wavelengths and extrapolating the IRAS fluxes to the infinite following Myers et al. (1987). This way, a distance of 2.1 L$_4$$^{1/2}$ kpc is obtained, in very good agreement with the previous determination by Van de Steene & van Hoof (2003). Assuming a luminosity of 6,000 L$_{\odot}$, which is the theoretical luminosity expected for a post-AGB star with a core mass of 0.60 M$_{\odot}$ (Schönberner 1987), a distance of 1.6 kpc to I16594 is derived, value that will be adopted in the following discussion. The value of 0.60 M$_{\odot}$ is chosen for the mass of the core because the mass distribution of planetary nebulae central stars is strongly peaked at this value (Stasynska, Gorny & Tylenda 1997).
At a distance of 1.6 kpc, the extended emission detected in our deconvolved mid-infrared images of I16594 would correspond to a dusty toroidal structure with a radius of $\sim$640 AU. Assuming that the CO emission detected towards I16594 is a good tracer of the dusty torus structure and considering the CO expansion velocity of 16 km s$^{-1}$ measured by Loup et al. (1990), a dynamical age of the dusty torus structure of $\sim$190 yr can be estimated. This dynamical age is quite consistent with a source which has left the AGB very recently.
### Dust Temperature
The radiation transfer equation in the interior of a dust cloud adopts a simple form when the energy source is a single exciting star under the optically thin approximation and assuming thermal equilibrium. Under these conditions, the mean color temperature of the dust can be obtained from a simple equation (see e.g. Evans 1980) which relates the measured fluxes S$_{\nu1,2}$ at two different wavelenghts $\lambda_{1,2}$ and the dust emissivity index (which depends on the assumed dust model), assuming a homogeneous dust distribution throughout the cloud. It should be noted that the assumption of the central star as the only source of energy for dust heating is appropiate for I16594 because direct stellar ratiation is the dominant heating source for the circumstellar dust grains in the shell. In particular, we have investigated whether dust heating due to line emission could also contribute to the observed emission and found that this effect is negligible, as the value of the Infrared Excess(IRE) for I16594, defined as the ratio between the observed total far infrared flux and the expected far infrared flux due to absorption by dust of Ly $\alpha$ photons (see e.g. Zijlstra et al.1989), is $\sim$ 400.
In principle, one could construct color temperature maps for the dust from the analysis of the 8.6 and 11.5 $\mu$m TIMMI2 images of any given source, as long as these bands are representative of the dust continuum emission. Unfortunately, in the case of I16594, the 8.6 and 11.5 $\mu$m emission is strongly affected by the PAH emission features which are clearly visible in the ISO spectrum (see Figure 5). Thus, the temperature values derived this way are not expected to represent realistic estimations of any physical temperature in the shell. The same problem is found if we try to derive the dust temperature from the IRAS photometry at 12 and 25 $\mu$m, since both filters are also strongly affected by the presence of dust features, as ISO spectroscopy reveals. This is confirmed by the strongly different mean dust temperatures T$_{8.6/11.5}$ of 227 K and T$_{12/25}$ of 129 K, derived (assuming a dust emissivity index of 1) from our mid-infrared data and from the IRAS photometry at 12 and 25 $\mu$m, respectively.
A more reliable dust temperature can be directly estimated from the observed size of 0.4$\arcsec$ for the inner radius of the dusty torus assuming that this is the equilibrium radius for the bulk of the dust emitting at mid-infrared wavelengths. Based on the formula worked out by Scoville & Kwan (1976), this can be calculated using the equation: $$T_{d} =
1.64f^{-1/5}r_{eq}^{-2/5}L_{*}^{1/5}$$ where *T$_{d}$* is the dust temperature in K, *f* is the emissivity of the dust, r$_{eq}$ is the equilibrium radius in $pc$, and *L$_{*}$* is the source luminosity in L$_{\odot}$.
We decided to use for our calculations a basic dust model composed by hydrogenated amorphous carbon grains (HACs; type BE of Colangeli et al. 1995), whose emissivity index is $\sim$1. The selection of this dust model to reproduce the dust continuum emission observed in I16594 is justified by the presence of highly hydrogenated PAHs in the ISO spectrum (García-Lario et al. 1999). In order to calculate the dust emissivity, the mass extinction coefficient value for hydrogenated amorphous carbon was taken from appendix A of Colangeli et al. (1995) at the central wavelength between the two filters. Then, a typical grain density of 1.81 g cm$^{-3}$ (Koike, Hasegawa & Manabe 1980) was assumed. Finally, this quantity was multiplied by a dust grain size in the range 0.001-0.1 $\mu$m obtaining a dust emissivity *f*. Note that a dust grain size of 0.01 $\mu$m is a reasonable mid-range size for circumstellar carbon dust (e.g. Jura, Balm & Kahane 1995). The dust temperature in thermal equilibrium at 0.4$\arcsec$ (or $\sim$640 AU at the assumed distance of 1.6 kpc) can then be derived using the dust emissivity *f* and the assumed luminosity of the source (6,000 L$_{\odot}$). This way, a dust temperature T$_{d}$=237 K is found for a mid-range dust grain size of 0.01 $\mu$m. A smaller or a larger dust grain size of 0.001 and 0.1 $\mu$m would imply dust temperatures of 376 and 150 K, respectively.
We are conscious that the assumption of spherical geometry may not be valid for the circumstellar envelope of I16594 where the emission is clearly asymmetric and the geometry assumes a toroidal shape, according to our mid-IR images. Note that adopting a more complex, axysimmetric geometry would esentially translate into grains being more effectively heated in the biconical opening angle defined by the dust torus because of the different local optical depth. In spite of this, our simple model can be used irrespective of the shell geometry when applied to dust grains at the inner radius of the shell. The use of more detailed axysimmetric, multiple grain size models is beyond the scope of this paper. In addition, the presence of a dust torus close to the star with respect the spherical case mainly influences the optical and near-infrared radiation. A large effect on the mid- to far-IR emission is not expected (see e.g. Ueta & Meixner 2003). In this sense, spatially unresolved SEDs do not provide any spatial information necessary to constrain the geometry and inclination angle of the PPN dusty shells.
### Comparison with ISO data
Another dust temperature estimate can be derived by fitting one (or more) blackbodies to the available ISO data by considering fluxes representative of the underlying continuum at carefully selected wavelengths not affected by any dust feature. We did this by selecting the ISO fluxes at 6.0, 9.4, 14.3, 18.0, and 45.0 $\mu$m plus the IRAS fluxes at 60 and 100 $\mu$m. The best fit to the overall SED is obtained with a combination of two blackbodies (with an emissivity index of 1) with temperatures of 273 K and 130 K, respectively, as we can see in Figure 5, where we display the SED of I16594 from 1 to 100 microns together with the two blackbodies. We find that actually the warm component (at 273 K) dominates in the wavelength range of the N1-filter (at 8.6 $\mu$m) while the cool component (at 130 K) dominates in the N11.9-filter range (at 11.5 $\mu$m). Overimposed on the continuum emission, strong PAH features are also clearly contributing to the observed emission. Note that the PAH emission features observed at the ISO short wavelengths as well as the dust features at 21, 26 and 30 $\mu$m, the latter extending from 20 to 40 $\mu$m, are intentionally excluded from the fitting because they are not representative of the dust continuum emission. In particular, the 30 $\mu$m feature overlaps with the 26 $\mu$m feature, and even with the 21 $\mu$m feature and the continuum level is well below the flux detected by ISO at 22–24 microns (see e.g. the analysis of the similar sources IRAS 20000$+$3239 and HD 56126 shown in Fig. 10 of Hony, Waters & Tielens 2002). At present, most of these features remain still unidentified, although several possible carriers have been proposed in the literature, e.g. fullerenes, TiC, SiC for the 21 $\mu$m feature (García-Lario et al. 1999; von Helden et al. 2000; Speck & Hofmeister 2003); MgS for the broad 30 $\mu$m feature (Hony, Waters, & Tielens 2002 and references therein).
Using the above two dust temperatures we can estimate the size of the dust grains which are expected to emit in equilibrium at the distance of 0.4$\arcsec$ from the central star which is derived from our mid-IR images. This is found to correspond to small dust grains with a size of 0.005 $\mu$m in the case of the warm dust component emitting at 273 K which dominates at 8.6 $\mu$m (comparable to the typical size of small PAH clusters). Dust grains with the same size emitting at 130 K (note that this cold dust emission dominates at 11.5 and 11.7 $\mu$m) would need to be located at $\sim$4075 AU from the central star, which corresponds to a projected $\sim$2.5$\arcsec$ on the sky at the assumed distance. This is considerably beyond the observed extension of the inner shell in the mid-infrared. A surface brightness of $\sim$350 mJy/pixel can be roughly estimated for the continuum emission expected under these conditions, well above ($\sim$50-$\sigma$) our detection limit. The fact that we do not detect this extended emission in our images suggests that the angular size of the region giving rise to the bulk of the hot dust emission is much smaller than that of the region emitting at 130 K. Note that, assuming e.g. that the cold dust emission extends homogeneously over the larger aperture used by ISO, we find that the surface brightness would be just below the 3-$\sigma$ level of the sky background and, as such, undetectable in our TIMMI2 images. The similar extension and morphology of the mid-infrared emission observed at 8.6, 11.5 and 11.7 $\mu$m suggests that the contribution from PAHs observed in the ISO spectrum must be dominant in our TIMMI2 images, and that these PAHs may be well mixed with the small, hot dust grains responsible for the underlying continuum, being mainly distributed along the torus.
Considering the information available and the limited spectral coverage, an alternative scenario which cannot be ruled out completely might be that both small, hot dust grains and large, cold dust grains could be co-located in the dust torus. This would be possible if a larger grain size ($\geq$0.1 $\mu$m) is assumed for the cold dust. Note that, in a non-spherical (torus) distribution of the dust, the shielding can become very efficient and the density very high in the outer equatorial regions, where the dust can grow and get colder, protected both from the radiation from central star and from the ISM UV radiation field. This would explain the larger size of the cold dust grains in the torus. In contrast, small, hot dust grains are expected to dominate in the inner boundary of the torus. Unfortunately, the spatial resolution of our images is not enough to resolve the grain size distribution within the torus.
### Collimated outflows in IRAS 16594$-$4656
I16594 has also been observed by the HST in the optical, through the broad F606W continuum filter with the Wide Field Planetary Camera (WFPC2) under proposal 6565 (P.I.: Sun Kwok), and in the near-infrared, through the narrow F212N (H$_2$) and F215N (H$_2$-continuum) filters with the Near Infrared Camera and Multi Object Spectrometer (NICMOS) under proposal 9366 (P.I.: Bruce Hrivnak). In the optical, I16594 shows a flower-shaped morphology where several petals (or bipolar lobes) can be identified at the opposite sides of the central star with different orientations, which has been suggested to be a result of episodic mass ejection (Hrivnak, Kwok, & Su 1999). Similar structures have also been detected in other PPNe (e.g. Hen 3-1475; Riera et al. 2003) and in more evolved PNe (e.g. NGC 6881; Guerrero & Manchado 1998) and they have been interpreted as the result of episodic mass loss from a precessing central source (e.g. García-Segura & López 2000). From the HST optical images (taken from the HST Data Archive) we identify pairs of elongated structures with at least four different bipolar axes at P.A.$\sim$34${\hbox{$^{\circ}$}}$, $\sim$54${\hbox{$^{\circ}$}}$, $\sim$84${\hbox{$^{\circ}$}}$ and $\sim$124${\hbox{$^{\circ}$}}$.
In Figure 6 we have displayed the contour map of the deconvolved mid-infrared images of I16594 obtained with TIMMI2 in the N1 and N11.9 filters overlaid on the optical HST-WFPC2 image taken in the F606W filter. Remarkably, we can see that the axis of symmetry defined by the mid-infrared emission nicely coincides with only one of the bipolar axes that can be identified in the optical images, in particular with that oriented at P.A.$\sim$84${\hbox{$^{\circ}$}}$. If this emission is a good tracer of the hot dust in the envelope and we accept that this hot dust must have been recently ejected from the central star we can interpret the observed spatial distribution in the mid-infrared as the result of the preferential collimation of the outflow material along this direction in the most recent past.
Remarkably, the H$_2$ shocked emission detected with HST-NICMOS in the near-infrared is also found mainly distributed following the same bipolar axis (Hrivnak, Kelly & Su 2004) and nicely coincides with the mid-IR emission seen in our TIMMI2 images. This is shown in Figure 7, where the H$_2$ continuum-subtracted HST-NICMOS image is shown together with a contour map of the deconvolved mid-infrared image taken in the N11.9 filter. Note that the H$_2$ image (at 2.122 $\mu$m) showed in Figure 7 was continuum-subtracted using the HST-NICMOS image taken in the adjacent continuum at 2.15 $\mu$m (both images were also taken from the HST Data Archive). Interestingly, we found that the H$_2$ emission is mainly coming from the walls of the bipolar lobe oriented at P.A.$\sim$84${\hbox{$^{\circ}$}}$ identified in the HST optical images. In addition, four additional clumps of much weaker H$_2$ emission are detected at the end of each of the other two point-symmetric outflows associated to I16594 (Hrivnak, Kelly & Su 2004). The stronger emission detected along the walls of this bipolar lobe suggests that the interaction of the fast wind from the central star with the slowly moving AGB wind is currently taking place preferentially also along this axis of symmetry. This suggests that the formation of the multiple outflows observed in I16594 has not been simultaneous. The rest of bipolar outflows observed at other orientations in the optical images taken with HST must then be interpreted as the result of past episodic mass loss ejections. As such, they must contain much cooler dust grains which are then only detectable in the optical because of their scattering properties.
IRAS 07027$-$7934
-----------------
### IRAS 07027$-$7934 in the literature
I07027 (=GLMP 170) is a very peculiar young PN. It has a central star that was classified by Menzies & Wolstencroft (1990) as of \[WC11\]-type. At present, there are only about half a dozen PNe with a central star classified as \[WC11\]. They all have stellar temperatures between $\sim$28,000 and 35,000 K (Leuenhagen & Hamann 1998) and are supposed to be in the earliest observable phase of its PN evolution, soon after the onset of the ionization in their circumstellar envelopes. I07027 is also among the brightest IRAS PNe and it has IRAS colors similar to other young PNe (Zijlstra 2001). The youth of I07027 as a PN is also evidenced by the detection of OH maser emission at 1612 MHz (Zijlstra et al. 1991), which is usually observed in their precursors, the OH/IR stars, but very rarely in PNe. The OH emission is single-peaked, which is interpreted as being detected only coming from the blue side of the shell, as the consequence of the ionized inner region being optically thick at 1612 MHz. This is supported by the shift in velocity with respect to the CO emission, which has also been detected toward this source, and from which an expansion velocity of 14.5 km s$^{-1}$ is derived (Zijlstra et al. 1991).
The detection of strong PAH features and crystalline silicates in the ISO spectrum (Cohen et al. 2002; Peeters et al. 2002) indicates the simultaneous presence of oxygen and carbon-rich dust in the envelope. Remarkably, all other \[WC\] CSPNe observed with ISO show a mixed chemistry as well (Cohen et al. 2002) but I07027 is the only known \[WC\] star belonging to the rare group of PNe with OH maser emission, and therefore it links OH/IR stars with carbon-rich PNe.
Zijlstra et al. (1991) published an H$\alpha$ image of I07027 taken with the ESO 3.5m NTT telescope. This image shows a stellar core with non-gaussian wings extending to a maximum diameter of about 15$\arcsec$, which may be mostly due to light scattered by neutral material and dust grains in the envelope. García-Hernández et al. (2002) detected H$_2$ fluorescence-excited emission from this source, in agreement with the round/elliptical H$\alpha$ morphology of the nebula and the temperature of the central star.
I07027 had never been imaged in the mid-infrared before. Thus, our observations are the first attempt to reveal the spatial distribution of the warm dust in this peculiar object.
### The Marginally Extended Mid-infrared Core of IRAS 07027$-$7934
The deconvolved mid-infrared images of I07027 displayed in Figure 4 show a slightly extended emission at 8.6 and 11.5 $\mu$m. This mid-infrared emission is only marginally resolved (with a FWHM=0.3$\arcsec$ as compared to the typical PSF size of FWHM$\leq$0.2$\arcsec$ measured in the deconvolved standard stars) and is elongated along the north-south direction. Zijlstra et al. (1991) predicted for this source a radio flux density of 10 $mJy$ assuming E$_{B-V}$=1.1 and T$_{e}$=10$^4$ K. In addition, by using a plausible radio brightness temperature of 10$^3$ K they predicted an angular diameter of $\sim$0.3$\arcsec$ for the ionized region. This size for the ionized region is consistent with the measured size of the bright mid-infrared core seen in our deconvolved images of I07027.
Unfortunately, there are no HST images of I07027 available in the optical but it has very recently been observed in the near-infrared through the broad F110W (J-band) and F160W (H-band) continuum filters with NICMOS under proposal 9861 (P.I.: Raghvendra Sahai). In Figure 8 we have displayed the still unpublished near-infrared HST images of I07027 (taken from the HST Data Archive) together with the contour levels of the deconvolved mid-infrared image taken by us in the N11.9 filter. In the F160W filter, I07027 shows a bright extended core (with FWHM=0.25$\arcsec$), which is slightly elongated along the north-south direction, in agreement with the mid-infrared structure seen in our deconvolved TIMMI2 images. This core is surrounded by a fainter elliptical nebulosity extended along the NW-SE direction with a total size of $\sim$1.6$\arcsec$ $\times$ 2.1$\arcsec$ at 1% of the peak intensity. The HST image in the F110W filter shows a slightly less extended emission of $\sim$1.1$\arcsec$ $\times$ 1.5$\arcsec$ (at 1% of the peak intensity) and shows a very similar morphology. In this case, a central point source is clearly detected which corresponds very probably to the central star, which is barely detected in the F160W image.
Similarly to what we did for I16594 we have also estimated the distance to I07027. In this case we constructed the SED of I07027 by combining the available IRAS fluxes at 12, 25, 60 and 100 $\mu$m with the JHKL and BVRI photometry taken from García-Lario et al. (1997) and Zijlstra et al. (1991), respectively. The observed fluxes were also corrected for reddening adopting the extinction law from Cardelli, Clayton & Mathis (1989) and the value of E$_{B-V}$=1.1 derived by Zijlstra et al. (1991) through the measurement of nearby stars, with R$_{V}$=3.1. Then, a distance of 4.1 L$_4$$^{1/2}$ kpc is obtained. Note that I07027 is located at a much higher galactic latitude (b=$-$26) than I16594 (b=$-$3) and at such high galactic latitudes so much interestellar reddening is unexpected. Thus, we interpret that the observed reddening E$_{B-V}$=1.1 is mainly circumstellar in origin. On the other hand, most of the flux is emitted in the infrared where the effect of the interstellar/circumstellar extinction is mild. This is probably the reason why a very similar luminosity of 4.2 L$_4$$^{1/2}$ kpc was obtained by Surendiranath (2002), who derived this value by integrating the photometric fluxes from 0.36 $\mu$m to 100 $\mu$m, but without introducing any correction for extinction. Assuming a standard luminosity of 6,000 L$_{\odot}$ for I07027, a distance of 3.2 kpc is derived, in agreement with the distance of 3–5 kpc suggested by Zilstra et al. (1991). At this distance, the core size would correspond to $\sim$960 AU.
### Dust Temperature
As for I16594, stellar light must be the dominant heating source for the circumstellar dust grains in I07027. This is confirmed by Zijlstra et al. (1991), who derived an Infrared Excess (IRE) of 93 for this source, indicating that dust heating by line emission can also be neglected in I07027. Again, we cannot interpret our mid-infrared observations of I07027 in terms of dust temperatures in the shell because the ISO spectrum of I07027 (Cohen et al. 2002) shows that the 8.6 and 11.5 $\mu$m filters are also heavily affected by strong PAH emission features. In particular, the PAH emission features around 8 $\mu$m (at $\sim$7.7 and 8.6 $\mu$m) are much stronger in this case than the feature located at 11.3 $\mu$m. Thus, the dust temperature values derived would be unrealistically high. Using IRAS data and assuming a dust emissivity index of 1, a mean dust temperature T$_{12/25}$ of 148 K is derived, while the TIMMI2 data gives a T$_{8.6/11.5}$ of $\sim$363 K. The strong differences in the derived temperatures confirm that the PAH emission is dominating the emission observed in the mid-infrared.
In contrast to I16594, the ISO spectrum of I07027 shows much weaker emission features at 12.6 and 13.4 $\mu$m, which are the signatures of the CH out-of-plane bending vibrations for hydrogens in positions duo and trio, respectively (Pauzat, Talbit & Ellinger 1997) and indicate that the PAH population in I07027 is largely dehydrogenated. Then, for the modelling of the dust emitting at mid-IR wavelengths we made the same assumptions as in the case of I16594 (see Section 5.1.3) but this time we adopted a composition dominated by dehydrogenated amorphous carbon grains (type ACAR of Colangeli et al. 1995). Under these assumptions and taking into account the dust equilibrium radius to be consistent with the 0.3$\arcsec$ (or $\sim$960 AU at the assumed distance of 3.2 kpc) of the shell (which is the radius at $\sim$90% of the peak intensity) seen in our mid-infrared deconvolved images we obtain a dust temperature of 219 K for a mid-range dust grain size of 0.01 $\mu$m. For a larger dust grain size of 0.1 $\mu$m a smaller dust temperature of 138 K is derived while a smaller dust grain size of 0.001 $\mu$m yields a dust temperature of 347 K. Note that if the dust grains were located closer to the central star than the 0.3$\arcsec$ derived from our mid-infrared images, the dust temperatures above derived should then be considered as lower limits.
### Comparison with ISO data
The validity of the range of possible dust temperatures derived from our TIMMI2 observations can be further explored by looking at the ISO spectrum originally published by Cohen et al. (2002). In a similar way as we did for I16594, the SED can be fitted by a two-component dust continuum with temperatures of T$_{BB1}$=430 K and T$_{BB2}$=110 K, respectively. The warm component in this case completely dominates in the wavelength range of our TIMMI2 observations (where also strong PAH features are found) while the cool component dominates at longer wavelengths, where crystalline silicate dust features are also detected on top of the continuum emission.
By forcing the dust equilibrium radius to be consistent with the 0.3$\arcsec$ seen in our mid-IR deconvolved images, we need to assume in this case a very small grain size of $<$0.001 $\mu$m in order to reproduce the dust temperature of 430 K derived from the ISO spectrum. This suggests that the mid-infrared emission at $\sim$430 K must be the result of the combined contribution of small PAH molecules, located very close to the central star, and relatively hot dust continuum. In this case, the PAH population must be subject to a relatively strong UV field, consistent with the narrow features detected by ISO (in contrast to the broader features observed in I16594). Actually, the ISO spectrum of I07027 shows that the PAH emission features at 3.3 and 11.3 $\mu$m are weak compared with the emission features located at 6.2, 7.7, and 8.6 $\mu$m, which is also indicating a high degree of ionization in the population of PAHs (see Figure 2 in Allamandola, Hudgins & Sandford 1999). Note, however, that if the UV radiation field becomes too strong the PAH molecules can be destroyed, especially the small ones with a size $\sim$20$-$30 carbon atoms (see e.g. Allain, Leach & Sedlmayr 1996). This means that the C-rich dust seen in the ISO spectrum subject to the UV irradiation coming from the central star must be shielded from the stronger ISM UV radiation field by the outer layers of the circumstellar shell, where the OH maser emission is originated.
For the cool dust emitting at 110 K, a different dust model was assumed, composed mainly of astronomical silicates. This choice takes into account the O-rich nature of the crystalline silicates detected in the ISO spectrum at wavelengths longer than 25 $\mu$m. The crystalline silicates are expected to be formed in the circumstellar dust shells of evolved stars at temperatures in the range 60$-$160 K (Molster et al. 2002b). The mean emissivity value adopted between 25 and 60 $\mu$m was taken from Figure 5 in Draine & Lee (1984). If we try to confine this cool O-rich dust to the observed extension of 0.3 $\arcsec$ we would need to adopt a very large dust grain size of $>$0.1 $\mu$m. Note that for this O-rich cool component we derive dust equilibrium radii of $\sim$0.02, $\sim$0.05 and 0.17 pc (or 1.1$\arcsec$, 3.4$\arcsec$ and 10.4$\arcsec$ at 3.2 kpc) for dust grain sizes of 0.1, 0.01 and 0.001 $\mu$m, respectively, which in all cases are inconsistent with our mid-infrared observations. These calculations indicate that independent of the dust grain size considered, the O-rich cool dust must be located much farther away from the central source than the C-rich warm dust emission (at 430 K) detected in the mid-infrared This different relative distribution of O-rich and C-rich dust would also be consistent with the detection of OH maser emission from the outer shell and suggests that the material expelled by the central star during the previous AGB phase was predominantly O-rich.
### Evolutionary Status of IRAS 07027$-$7934
At present, the evolutionary status of I07027 is not well understood. The hydrogen-deficiency of the central star together with the mixed dust chemistry (C-rich and O-rich) is a common finding among the limited sample of known \[WC\] PNe (De Marco & Soker 2002; Cohen et al. 2002). The most promising scenarios to explain the current observational properties of this rare class of PNe are: (i) the so-called ‘disk-storage’ scenario (Jura, Chen & Plavchan 2002; Yamamura et al. 2000); (ii) a final thermal pulse while the star was still in the AGB; or (iii) a late thermal pulse during the post-AGB evolution (Herwig et al. 1997, 1999; Herwig 2000, 2001; Blöcker 2001).
The disk-storage scenario invokes the presence of a binary system in which the O-rich silicates are trapped in a disk formed by a past mass transfer event, with the C-rich particles being more widely distributed in the nebula as a result of recent ejections of C-rich material. This type of dusty disk structures have been detected in some PPN/PNe with binary \[WC\] central stars like CPD$-$56${\hbox{$^{\circ}$}}$8032 (De Marco, Barlow & Cohen 2002) or in the Red Rectangle (HD 44179) (Waters et al. 1998), but no firm evidence of the presence of any disk-like structure nor of the binarity of I07027 exists yet.
Both a final thermal pulse in the AGB and a late thermal pulse during the post-AGB phase can eventually produce a sudden switch to a C-rich chemistry and a strong stellar wind, which is also characteristic of these \[WC\] CSPNe. However, because of the short lifetime of stars in the post-AGB phase, the latter is expected to be a rare phenomenon. Models predict that post-AGB stars which experience a late thermal pulse evolve back into the AGB (the so-called “*born-again*” scenario, e.g., Herwig 2001; Blöcker 2001). As a result of this, they show a fast spectroscopic evolution in the H-R diagram as well as peculiar spectroscopic features (e.g., Asplund et al. 1999; Lechner & Kimeswenger 2004; Hajduk et al. 2005) which are not observed in I07027, nor in any other known \[WC\] CSPNe.
A final thermal pulse in the AGB phase seems to be a more plausible explanation since it does not require the assumption of exotic scenarios. As we have discussed in Section 5.2.4, the emission detected in our mid-infrared images can be mainly attributed to ionized PAHs plus thermal emission from relatively warm dust ($\sim$430 K) located very close to the central source. The OH maser emission detected by Zijlstra et al. (1991) supports the idea that the envelope of I07027 was until very recently O-rich. It is very difficult to explain how a low-mass disk around a binary system, which could act as an oxygen-rich reservoir, may be able to sustain such a luminous maser emission. Attending to geometry considerations, Zijlstra et al. (1991) suggests that the star must have changed its chemistry within the last 500 yrs. I07027 may have experienced a final thermal pulse in the AGB which has produced the recent switch to a C-rich chemistry. All C-rich material would then be warm as a consequence of its very recent formation and, thus, located very close to the central source (as it is actually observed) while the cooler O-rich material ejected during the previous AGB phase is then found now only farther away from the central source.
In contrast, the typical disk sources with dual chemistry which are known to be binary systems show a completely different relative distribution of O-rich and C-rich dust. Waters et al. (1998) found that the PAH emission at 11.3 $\mu$m has a clumpy nature and comes from the extended nebula around HD 44179, while the O-rich material is located in a circumbinary disk. More recently, the bipolar post-AGB star IRAS 16279$-$4757 has been studied in the mid-infrared by Matsuura et al. (2004). They found that the PAH emission is enhanced at the outflow, while the continuum emission is located towards the center. Thus, they suggest the presence of a dense O-rich torus around an inner, low density C-rich region and a C-rich bipolar outflow resembling the morphology attributed to HD 44179. The observational characteristics of I07027 indicate a totally different formation mechanism, which are only consistent with a very recent change of chemistry from O-rich to C-rich.
Conclusions
===========
We have presented diffraction limited mid-infrared images of the PPN I16594 and the \[WC\] PN I07027 at 8.6, 11.5 and 11.7 $\mu$m taken under exceptionally good seeing conditions ($\leq$0.5). By applying the Lucy-Richardson deconvolution algorithm, we have resolved, for the first time, the subarcsecond dust shell structures around both objects.
I16594 displays two emission peaks in the innermost region of the circumstellar dust shell at the three wavelengths observed. This two-peaked mid-infrared morphology is interpreted as an equatorial density enhancement revealing the presence of a dusty toroidal structure with a 0.4$\arcsec$ radius size (or $\sim$640 AU corresponding to a dynamical age of $\sim$190 yr at the assumed distance of 1.6 kpc). The observed size is used to derive the dust temperature at the inner radius of the shell. This result has been combined with the information derived from the ISO observations of I16594 to conclude that the mid-infrared emission detected in our TIMMI2 images must be dominated by PAH molecules or clusters which must be mainly distributed along the torus, as suggested by the similar size and morphology observed in all filters. We have also found that the axis of symmetry observed in the mid-infrared is well aligned with only one of the bipolar outflows (at P.A.$\sim$84${\hbox{$^{\circ}$}}$) seen as optical reflection nebulae in the optical HST images. We suggest that the multiple outflow formation has not been coeval and that, at present, the outflow material is being ejected in this direction. Consistently, the H$_2$ shocked-emission seen in the HST NICMOS image is mainly distributed along the same bipolar axis where the fast post-AGB wind is interacting with the slow moving material ejected during the previous AGB phase. The presence of several other bipolar outflows at a variety of position angles may be the result of past episodic mass loss events.
I07027 exhibits a slightly asymmetric mid-IR emission core which is only marginally extended along the north-south direction with FWHM=0.3$\arcsec$ at 8.6 and 11.5 $\mu$m. This is the same orientation observed in recent HST images of the source taken in the near-infrared. The mid-infrared emission is attributed to a combination of emission from highly ionized, small PAH molecules plus relatively warm dust continuum located very close to the central star. The characteristics of the PAH emission observed in the ISO spectrum are also consistent with this interpretation. Taking into account the spatial distribution of the C-rich material deduced from our observations and because the OH maser emission from I07027 is expected to be located in the external and cooler regions, we propose that the dual chemistry observed in I07027 must be interpreted as the consequence of a recent thermal pulse (probably at the end of the previous AGB phase) which has switched the chemistry of the central star from the original O-rich composition to a C-rich one within the last 500 yrs. This might be the commom mechanism which originates the dual chemistry and strong stellar winds usually observed in other \[WC\]-type CSPNe.
DAGH is grateful to Eva Villaver for her useful comments. AM and PGL acknowledge support from grants AYA 2001$-$1658 and AYA 2003$-$9499, respectively, from the Spanish Ministerio de Ciencia y Tecnología (MCYT).
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[ccccccccc]{} IRAS 16594$-$4656&2001 Oct 9&N1&8.6(1.67)&120.4&$\sim$1.7 x 1.6&0.57&14&5\
$\dots$&$\dots$&N11.9&11.5(1.89)&107.5&$\sim$1.8 $\times$ 1.6&0.73&40&13\
$\dots$&$\dots$&SiC&11.7(3.21)&161.3&$\sim$1.8 $\times$ 1.6&0.75&35&11\
IRAS 07027$-$7934&2001 Oct 10 &N1&8.6(1.67)&120.4&0.65&0.57&19&23\
$\dots$&$\dots$&N11.9&11.5(1.89)&107.5&0.83&0.73&19&17\
[cccccccccc]{} 07134$+$1005 & R & 1& F5 Iab & C & PPN & S+B&y&y&1\
16594$-$4656 & R & 2& B7 & C & PPN & S+M&y&y&2\
17436$+$5003 & R & 3& F3 Ib & O & PPN & S+B&n&n&3\
19114$+$0002 & R$^{*}$ & 4& G5 Ia & O & PPN/SG & S+M&n&n&3\
21282$+$5050 & R & 1& O9& C & Young PN & S+B&y&n&4\
22223$+$4327 & U & 4& G0 Ia & C & PPN & S+M &y&y&5\
22272$+$5435 & R & 5& G5 & C & PPN & S+M&y&y&6\
[^1]: Based on observations collected at the European Southern Observatory (La Silla, Chile), on observations made with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) with the participation of ISAS and NASA, and on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555
[^2]: The Image Reduction and Analysis Facility software package (IRAF) is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation
| ArXiv |
---
abstract: |
By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant $I_{\rm q}(t)$ is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear $I_{\rm l}(t)$ and quadratic $I_{\rm
q}(t)$ (where the latter $I_{\rm q}(t)$ cannot be written as the squared of the former $I_{\rm l}(t)$, [*i.e.*]{}, the relation $I_{\rm q}(t)= cI_{\rm l}^{2}(t)$ does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar [*et al.*]{} more recently is the one corresponding to the linear $I_{\rm l}(t)$, one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment \[Phys. Rev. A [**68**]{}, 016101 (2003)\] of Bekkar [*et al.*]{} on Guedes’s work \[Phys. Rev. A [**63**]{}, 034102 (2001)\] and Guedes’s corresponding reply \[Phys. Rev. A [**68**]{}, 016102 (2003)\].\
\
[*PACS:*]{} 03.65.Fd, 03.65.Ge\
[*Keywords:*]{} exact solutions, Lewis-Riesenfeld invariant formulation, unitary transformation
address: |
$^{1}$ Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation,\
Zhejiang University, Hangzhou SpringJade 310027, P.R. China\
$^{2}$ Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China
author:
- 'Jian-Qi Shen $^{1,}$$^{2}$ [^1]'
title: 'Solutions of the Schrödinger equation for the time-dependent linear potential[^2]'
---
Introduction
============
Recently, Guedes used the Lewis-Riesenfeld invariant formulation[@Lewis] and solved the one-dimensional Schrödinger equation with a time-dependent linear potential[@Guedes]. More recently, Bekkar [*et al.*]{} pointed out that[@Bekkar] the result obtained by Guedes is merely the particular solution (that corresponds to the null eigenvalue of the linear Lewis-Riesenfeld invariant) rather than a general one. In the comment[@Bekkar], Bekkar [*et al.*]{} stated that they correctly used the invariant method[@Lewis] and gave the general solutions of the time-dependent Schrödinger equation with a time-dependent linear potential[@Bekkar]. However, in the present paper, I will show that although the solutions of Bekkar [*et al.*]{} is more general than that of Guedes[@Guedes], what they finally achieved in their comment[@Bekkar] is still [*not*]{} the [*general*]{} solutions, either. On the contrary, I think that their result [@Bekkar] also belongs to the particular one. The reason for this may be as follows: according to the Lewis-Riesenfeld invariant method[@Lewis], the solutions of the time-dependent Schrödinger equation can be constructed in terms of the eigenstates of the Lewis-Riesenfeld (L-R) invariants. It is known that both the squared of a L-R invariant (denoted by $I(t)$) and the product of two L-R invariants are also the invariants, which agree with the Liouville-Von Neumann equation $\frac{\partial
}{\partial t}I(t)+\frac{1}{i}\left[ I(t),H(t)\right] =0$, and that if $I_{a}$ and $I_{b}$ are the two L-R invariants of a certain time-dependent quantum system and $|\psi(t)\rangle$ is the solution of the time-dependent Schrödinger equation (corresponding to one of the invariants, say, $I_{a}$), then $I_{b}|\psi(t)\rangle$ is another solution of this quantum system. So, in an attempt to obtain the [*general*]{} solutions of a time-dependent system, one should first analyze the complete set of all L-R invariants of the system under consideration. Historically, in order to obtain the complete set of invariants, Gao [*et al.*]{} suggested the concept of basic invariants which can generate the complete set of invariants[@Gao], as stated in Ref.[@Gao], the basic invariants can be called invariant generators. As far as Bekkar [*et al.*]{}’s result[@Bekkar] is concerned, the obtained solutions are the ones corresponding only to the linear invariant ([*i.e.*]{}, $I_{\rm l
}(t)=A(t)p+B(t)q+C(t)$) that is simply one of the L-R invariants, which form a complete set. It is apparently seen that the quadratic form, $I_{\rm
q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A'(t)p+B'(t)q+C'(t)$, is also the one that can satisfy the Liouville-Von Neumann equation, since it is readily verified that the Lie algebraic generators of $I_{\rm q}(t)$ form a Lie algebra, which possesses the following commutators $$\begin{aligned}
\left[q^{2}, p^{2}\right]=2i(pq+qp), \quad \left[pq+qp,
q^{2}\right]=-4iq^{2}, \quad
\left[pq+qp, p^{2}\right]=4ip^{2}, \nonumber \\
\left[q, p^{2}\right]=2ip, \quad \left[p, q^{2}\right]=-2iq,
\quad \left[q, pq+qp\right]=2iq, \quad \left[p,
pq+qp\right]=-2ip. \label{algebra}\end{aligned}$$ However, for the cubic-form invariant, it is easily seen that there exists no such closed Lie algebra. This point holds true also for the algebraic generators in any high-order L-R invariants $I_{\rm l}^{n}$. So, it is concluded that for the driven oscillator, only the linear $I_{\rm l}(t)$ and quadratic $I_{\rm
q}(t)$ will form a complete set of L-R invariants. Note that here $I_{\rm q}(t)$ should not be the squared of $I_{\rm l}(t)$, [*i.e.*]{}, $I_{\rm q}(t)\neq cI_{\rm l}^{2}(t)$, where $c$ is an arbitrary c-number. It is emphasized here that Bekkar [*et al.*]{}’s solution is the one constructed only in terms of the eigenstates of the linear invariant $I_{\rm l}(t)$. Even though only for the linear invariant $I_{\rm l}(t)$ Bekkar [*et al.*]{}’s result[@Bekkar] can truly be viewed as the complete set of solutions, it still cannot be considered general one of the Schrödinger equation, since the latter should contain those corresponding to the quadratic invariant $I_{\rm q}(t)$. In brief, Bekkar [*et al.*]{}’s solution and my solution, which will be found in what follows, together constitute the complete set of solutions of the Schrödinger equation involving a time-dependent linear potential.
On the complete set of L-R invariants
=====================================
According to the L-R invariant theory[@Lewis], if the eigenstate of the linear invariant $I_{\rm l}(t)$ corresponding to $\lambda_{n}$, [*i.e.*]{}, one of the eigenvalues of $I_{\rm
l}(t)$, is $|\lambda_{n}, t\rangle$, then the solution of the Schrödinger equation can be written in the form $$|\Psi(t)\rangle_{\rm Schr}=\sum_{n}c_{n}\exp
\left[\frac{1}{i}\phi_{n}(t)\right]|\lambda_{n}, t\rangle
\label{2}$$ with $c_{n}$’s and $\phi_{n}(t)$’s being the time-independent coefficients and time-dependent phases[@Lewis; @Gao], respectively. This, therefore, means that the solutions of the time-dependent Schrödinger equation can be constructed in terms of the complete set of eigenvector basis, $\{|\lambda_{n},
t\rangle\}$, of $I_{\rm l}(t)$. Moreover, one can readily verify that the squared, $I_{\rm l}^{2}$, of the linear invariant is the one satisfying the Liouville-Von Neumann equation, and that $I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}$ is also a solution (but not another new general one) of the same time-dependent Schrödinger equation, since it is readily verified that $I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}$ can also be the linear combination of the eigenstate basis set $\{|\lambda_{n},
t\rangle\}$ of $I_{\rm l}(t)$, [*i.e.*]{}, $$I_{\rm l}(t)|\Psi(t)\rangle_{\rm Schr}=\sum_{n}b_{n}\exp
\left[\frac{1}{i}\phi_{n}(t)\right]|\lambda_{n}, t\rangle,
\label{1}$$ where the time-independent coefficients $b_{n}$’s are taken $b_{n}=\lambda_{n}c_{n}$, which is obtained via the comparison of the expression (\[1\]) with (\[2\]).
Thus, the above discussion shows that the linear invariant $I_{\rm
l}$ and its squared $I_{\rm l}^{2}$ have the same eigenstate basis set and therefore $I_{\rm l}$ and $I_{\rm l}^{2}$ cannot form a complete set of L-R invariants. In contrast, if for any c-number $c$, the quadratic $I_{\rm q}$ cannot be written as the squared of linear $I_{\rm l}$ with various integral constants $A_{0}$, $B_{0}$ and $C_{0}$ (for the definition of $A_{0}$, $B_{0}$ and $C_{0}$, see, for example, in Ref.[@Bekkar]), namely, the relation $I_{\rm q}=cI_{\rm l}^{2}$ is always not true, then $\{I_{\rm l}, I_{\rm q}\}$ is the complete set of L-R invariants, which enables us to obtain the general solutions (complete set of solutions) of the time-dependent Schrödinger equation.
Perhaps someone will ask such question as, “Does there really exist such quadratic $I_{\rm q}$ that can always not be written in the form $cI_{\rm l}^{2}$?” or “Maybe any $I_{\rm q}$ that satisfies the Liouville-Von Neumann equation can surely be written as the squared of certain $I_{\rm l}$. Really?” Now I will discuss these questions. Consider a given quadratic invariant $I_{\rm q}$ that is written $I_{\rm
q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A'(t)p+B'(t)q+C'(t)$ whose time-dependent parameters are determined by the Liouville-Von Neumann equation, and a certain linear invariant $I_{\rm l
}(t)=A(t)p+B(t)q+C(t)$, the squared of which is $I_{\rm
l}^{2}=A^{2}p^{2}+AB(pq+qp)+B^{2}q^{2}+2C(Ap+Bq+\frac{C}{2})$. Since the functions $A$, $B$ and $C$ can also be determined by the Liouville-Von Neumann equation, the only retained parts left to us to determine is the integral constants $A_{0}$, $B_{0}$ and $C_{0}$. Choose the appropriate integral constants in $A$, $B$ and $C$, and let $I_{\rm q}$ be the squared of $I_{\rm l}$ (should such case exist), and then we have $$\begin{aligned}
D=cA^{2}, \quad E=cAB, \quad F=cB^{2},
\nonumber \\
A'=2cAC, \quad B'=2cBC, \quad
C'=cC^{2}. \label{eqq6}\end{aligned}$$ If a given $I_{\rm q}$ can really be written as the squared of $I_{\rm l}$, the above six equations are just used to determine the c-number $c$ and the suitable integral constants $A_{0}$, $B_{0}$ and $C_{0}$ in the functions $A$, $B$ and $C$. It is seen that there are only four numbers expected to be determined, and that, in contrast, we have six equations. So, it is possible that there exist potential parameters $c$ and $A_{0}$, $B_{0}$, $C_{0}$ which will not agree with Eqs.(\[eqq6\]) always for a given parameter set $\{D, E, F, A', B', C'\}$, or, for a given parameter set $\{D, E, F, A', B', C'\}$ there are always no such parameters $c$ and $A_{0}$, $B_{0}$, $C_{0}$ which satisfy Eqs.(\[eqq6\]). The existence of $I_{\rm q}$ that cannot be written as the squared of any $I_{\rm l}$ is thus demonstrated.
So, in the above we indicate that such two invariants $I_{\rm l}$ and $I_{\rm q}$ (which are independent) form a complete set of L-R invariants.
Unitary transformation associated with L-R invariants
=====================================================
Now I will solve the time-dependent Schrödinger equation, of which the time-dependent Hamiltonian[@Guedes] is given $$H(t)=\frac{p^{2}}{2m}+f(t)q,$$ by making use of the Lewis-Riesenfeld invariant theory[@Lewis]. The time-dependent L-R invariant used here takes the form $$I_{\rm q}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}+A(t)p+B(t)q+C(t).
\label{eq11}$$ With the help of the Liouville-Von Neumann equation, one can arrive at $$\begin{aligned}
\dot{D}+\frac{2E}{m}=0, \quad \dot{E}+\frac{F}{m}=0, \quad
\dot{F}=0,
\nonumber \\
\dot{A}+\frac{B}{m}-2Df=0, \quad \dot{B}-2Ef=0, \quad
\dot{C}-fA=0\end{aligned}$$ with dot denoting the derivative with respect to time $t$. The above six equations (referred to as the auxiliary equations[@Gao]) can be used to determine all the time-dependent parameters $A(t)$, $B(t)$, $C(t)$ and $D(t)$, $E(t)$, $F(t)$.
In accordance with the L-R theory, solving the eigenstates of the invariant (\[eq11\]) will enable us to obtain the solutions of the time-dependent Schrödinger equation. But, unfortunately, it is not easy for us to immediately solve the eigenvalue equation of the time-dependent invariant (\[eq11\]), for the invariant (\[eq11\]) involves the time-dependent parameters. So, in the following we will use the invariant-related unitary transformation formulation[@Gao], under which the [*time-dependent*]{} invariant in (\[eq11\]) can be transformed into a [*time-independent*]{} one $I_{V}$, and if the eigenstates of $I_{V}$ can be obtained conveniently, the eigenstates of $I_{\rm q}(t)$ can then be easily achieved.
Here we will employ two time-dependent unitary transformation operators $$V_{1}(t)=\exp [\eta(t)q+\beta(t)p], \quad
V_{2}(t)=\exp [\alpha(t)p^{2}+\rho(t)q^{2}] \label{eq21}$$ to get a [*time-independent*]{} $I_{V}$. The time-dependent parameters $\eta$, $\beta$, $\alpha$ and $\rho$ in (\[eq21\]) are purely imaginary functions, which will be determined in the following subsections. Since the canonical variables (operators) $q$ and $p$ form a non-semisimple Lie algebra, here the first step is to transform $I_{\rm q}(t)$ into $I_{1}(t)$, [*i.e.*]{}, $I_{1}(t)=V_{1}^{\dagger}(t)I_{\rm q}(t)V_{1}(t)$, which no longer involves the canonical variables $q$ and $p$, and the retained Lie algebraic generators in $I_{1}(t)$ are only $p^{2}$, $pq+qp$, $q^{2}$. Note that these three generators also form a Lie algebra (see the commutators (\[algebra\])) . The second step is to obtain the time-independent $I_{V}$, which will be gained via the calculation of $I_{V}=V_{2}^{\dagger}(t)I_{1}(t)V_{2}(t)$. In this step, the obtained $I_{V}$ has no other generators (and time-dependent c-numbers) than $p^{2}$ and $q^{2}$, namely, $I_{V}$ may be written in the form $I_{V}=\varsigma(p^{2}+q^{2})$ with $\varsigma$ being a certain parameter independent of time. It is well known that the eigenvalue equation of $I_{V}$ is of the form $I_{V}|n, q\rangle=(2n+1)\varsigma|n, q\rangle$, where $|n,
q\rangle$ stands for the familiar harmonic-oscillator wavefunction. Hence, the eigenstates of the time-dependent L-R invariant $I_{\rm q}(t)$ in (\[eq11\]) can be achieved and the final result is $V_{1}(t)V_{2}(t)|n, q\rangle$ with the eigenvalue being $(2n+1)\varsigma$.
The calculation of $I_{1}(t)=V_{1}^{\dagger}(t)I_{\rm q}(t)V_{1}(t)$
--------------------------------------------------------------------
By the aid of the Glauber formula, one can arrive at $$\begin{aligned}
I_{1}(t)&=&Dp^{2}+E(pq+qp)+Fq^{2}+[A+2i(E\beta-D\eta)]p+[B+2i(F\beta-E\eta)]q
\nonumber \\
&+&C-[-i(B\beta-A\eta)+D\eta^{2}+F\beta^{2}-2E\beta\eta].\end{aligned}$$ If the two relations $$A+2i(E\beta-D\eta)=0, \quad B+2i(F\beta-E\eta)=0
\label{eq22}$$ are satisfied, then we can obtain[^3] $$I_{1}(t)=D(t)p^{2}+E(t)(pq+qp)+F(t)q^{2}.$$ It follows from (\[eq22\]) that the time-dependent parameters in the unitary transformation $V_{1}(t)$ are expressed by $$\eta=\frac{EB-FA}{2i(E^{2}-DF)}, \quad
\beta=\frac{DB-EA}{2i(E^{2}-DF)}.$$
The calculation of $I_{V}=V_{2}^{\dagger}(t)I_{1}(t)V_{2}(t)$
-------------------------------------------------------------
By using the Glauber formula, one can arrive at $$I_{V}\equiv V^{\dagger}_{2}(t)I_{1}(t)V_{2}(t)={\mathcal
D}p^{2}+{\mathcal E}(pq+qp)+{\mathcal F}q^{2},$$ where ${\mathcal D}$, ${\mathcal E}$ and ${\mathcal F}$ are of the form $$\begin{aligned}
{\mathcal
D}&=&D+\frac{4iE\alpha}{(16\rho\alpha)^{\frac{1}{2}}}\sinh
(16\rho\alpha)^{\frac{1}{2}}
+\frac{-8(F\alpha-D\rho)\alpha}{16\rho\alpha}\left[\cosh (16\rho\alpha)^{\frac{1}{2}}-1\right], \nonumber \\
{\mathcal
E}&=&\frac{2i(F\alpha-D\rho)}{(16\rho\alpha)^{\frac{1}{2}}}\sinh
(16\rho\alpha)^{\frac{1}{2}}+E\cosh (16\rho\alpha)^{\frac{1}{2}},
\nonumber \\
{\mathcal
F}&=&F+\frac{-4iE\rho}{(16\rho\alpha)^{\frac{1}{2}}}\sinh
(16\rho\alpha)^{\frac{1}{2}}
+\frac{8(F\alpha-D\rho)\rho}{16\rho\alpha}\left[\cosh
(16\rho\alpha)^{\frac{1}{2}}-1\right], \label{eqqq}\end{aligned}$$ respectively. It follows that if the following two equations are satisfied, $$E=\zeta\sinh (16\rho\alpha)^{\frac{1}{2}}, \quad
\frac{2i(F\alpha-D\rho)}{(16\rho\alpha)^{\frac{1}{2}}}=-\zeta\cosh
(16\rho\alpha)^{\frac{1}{2}}, \label{eq23}$$ then the coefficients of $pq+qp$ in $I_{V}$ is vanishing. In order that we can analyze the above equations (\[eq23\]) conveniently, the time-dependent parameters $\alpha$, $\rho$ (which are expected to be determined) and $F$, $D$ are respectively parameterized to be $$\alpha=\frac{u\theta}{4}, \quad \rho=\frac{v\theta}{4}, \quad
F=h\cosh (\sqrt{uv}\theta), \quad D=g\cosh
(\sqrt{uv}\theta). \label{eq24}$$ Substitution of the expressions (\[eq24\]) into (\[eq23\]) yields $$E=\zeta \sinh (\sqrt{uv}\theta), \quad
\frac{i(hu-gv)}{2\sqrt{uv}}=-\zeta, \label{eq25}$$ which can determine $\zeta$ and $\theta$ (expressed in terms of $E$, $h$, $g$ and $u$, $v$). It is noted that if the functions $u$ and $v$ are finally determined, then the time-dependent parameters $\alpha$ and $\rho$ in the unitary transformation operator $V_{2}(t)$ (\[eq21\]) can be obtained.
In what follows we will determine $u$ and $v$ via setting ${\mathcal D}={\mathcal F}=\varsigma$ with $\varsigma$ being constant ([*i.e.*]{}, time-independent). Insertion of (\[eq24\]) into (\[eqqq\]) will yield $$D+\frac{hu-gv}{2v}[\cosh (\sqrt{uv}\theta)-1]=\varsigma, \quad
F-\frac{hu-gv}{2u}[\cosh (\sqrt{uv}\theta)-1]=\varsigma.
\label{eq26}$$ Eq.(\[eq26\]) can determine the functions $u$ and $v$, although the problem is very complicated. Here it should be noted that $\theta$ which has been determined by (\[eq25\]) is also the function of $u$ and $v$. Thus, in principle, we can obtain the time-dependent functions $\alpha$ and $\rho$ in the second unitary transformation operator $V_{2}(t)=\exp
[\alpha(t)p^{2}+\rho(t)q^{2}]$.
Now under the unitary transformation $V_{1}(t)V_{2}(t)$ the time-dependent invariant $I_{\rm q}(t)$ is changed into a time-independent one, [*i.e.*]{}, $$I_{V}\equiv\left[V_{1}(t)V_{2}(t)\right]^{\dagger}I_{\rm
q}(t)\left[V_{1}(t)V_{2}(t)\right]=\varsigma(p^{2}+q^{2})$$ whose eigenvalue is $(2n+1)\varsigma$ and the corresponding eigenstate is $|n, q\rangle$ that is the familiar stationary harmonic-oscillator wavefunction, and the eigenvalue equation of the time-dependent invariant $I_{\rm q}(t)$ is thus given as follows $$I_{\rm q}(t)V_{1}(t)V_{2}(t)|n, q\rangle=(2n+1)\varsigma V_{1}(t)V_{2}(t)|n, q\rangle.$$
The solutions of the time-dependent Schrödinger equation
--------------------------------------------------------
According to the L-R invariant theory, the particular solution $\left| n, t\right\rangle _{\rm Schr}$ of the time-dependent Schrödinger equation is different from the eigenfunction of the invariant $I_{\rm q}(t)$ only by a phase factor $\exp \left[\frac{1}{i}\phi _{n}(t)\right]$, the time-dependent phase of which is written as $$\phi _{n}(t)=\int_{0}^{t}\langle n, q|\left [
V_{1}(t')V_{2}(t')\right]^{\dagger}\left[H(t')-i\partial/\partial
t' \right]\left[V_{1}(t')V_{2}(t')\right]|n, q\rangle {\rm d}t'.$$ This phase $\phi _{n}(t)$ can be calculated with the help of the Glauber formula and the Baker-Campbell-Hausdorff formula[@Wei; @EPJD].
The particular solution $\left| n, t\right\rangle _{\rm Schr}$ of the time-dependent Schrödinger equation corresponding to the invariant eigenvalue $(2n+1)\varsigma$ is thus of the form $$\left| n, t\right\rangle _{\rm Schr}=\exp \left[\frac{1}{i}\phi
_{n}(t)\right]V_{1}(t)V_{2}(t)|n, q\rangle.$$ Hence the general solution of the Schrödinger equation can be written in the form $$|\Psi(q, t)\rangle_{\rm Schr}=\sum_{n}c_{n}\left| n,
t\right\rangle _{\rm Schr},$$ where the time-independent c-number $c_{n}$’s are determined by the initial conditions, [*i.e.*]{}, $c_{n}= _{\rm Schr}\langle n,
t=0|\Psi(q, t=0)\rangle_{\rm Schr}$.\
\
In the above we thus found the general solutions of the Schrödinger equation for the time-dependent linear potential, which corresponds only to the quadratic-form invariant (\[eq11\]). It is concluded here that the solutions obtained above does not form a complete set of solutions of this time-dependent Schrödinger equation, and that Bekkar [*et al.*]{}’s solution and my solution presented here will constitute together such complete set of solutions of the Schrödinger equation.
Discussions and Conclusions
===========================
\(i) In the present paper we show that since Bekkar [*et al.*]{}’s solution[@Bekkar] has not yet contain those corresponding to the quadratic invariant, it is not the true general solution of the Schrödinger equation for the time-dependent linear potential. Instead, it is the solution corresponding only to the linear L-R invariant. The obtained solution here is the one that corresponds to the invariant (\[eq11\]), which is of the quadratic form. Since the linear and quadratic invariants form a complete set of L-R invariants, Bekkar [*et al.*]{}’s solution[@Bekkar] and my solution presented here constitute such complete set of solutions of the Schrödinger equation involving a time-dependent linear potential.\
\
(ii) It is well known that in quantum optics there are three kinds of photonic quantum states, [*i.e.*]{}, Fock state, coherent state and squeezed state. From my point of view, the calculation of the variations of creation and annihilation operators ($a^{\dagger}$, $a$) of photons under the translation ([*e.g.*]{}, $V_{1}$ of (\[eq21\])) and squeezing transformation ([*e.g.*]{}, $V_{2}$ of (\[eq21\])) operators shows that the variations of $a^{\dagger}$ and $a$ are exactly analogous to that of space-time coordinate variations under the translation, Lorentz rotation (boosts) and dilatation (scale) transformation[@Fulton] and thus these three quantum states (coherent, squeezed and Fock states) correspond to the above three conformal transformations, respectively. I think that this connection between them is of physical interest and deserves further consideration.\
\
(iii) Guedes recently stated that in order to obtain the general solutions of the Schrödinger equation one must follow the L-R invariant theory [*step by step*]{}[@Guedes2]. I don’t approve of this point of view, however. Personally speaking, in fact, the L-R method has only one step, namely, the particular solution of the time-dependent Schrödinger equation is different from the eigenfunction of the invariant only by a time-dependent phase factor. In Ref.[@Bekkar] and [@Shenarxiv], although we follow the L-R method step by step, what we obtained still cannot be viewed as the general solutions of the Schrödinger equation. For this reason, I think that “step by step” is not the essence of getting the general solutions of Schrödinger equation. Instead, the key point for the present subject is that one should first find the complete set of all L-R invariants of the time-dependent quantum systems under consideration. For some systems in the Hamiltonian there may exist no such closed Lie algebra as (\[algebra\]), the complete set of exact solutions can be found by working in a sub-Hilbert-space corresponding to a particular eigenvalue of one of the invariants, namely, only in the sub-algebra (quasi-algebra) corresponding to a particular eigenvalue of this invariant will such time-dependent quantum systems (which have no closed Lie algebra) be solvable[@japan]. For the time-dependent quantum systems, there are no other eigenvalue equations of Hamiltonian than that of the L-R invariants with time-dependent eigenvalues. The complete set of invariants, instead of the time-dependent Hamiltonian, can describe completely the time-dependent quantum systems. For this reason, it is essential to find the complete set of invariants for the time-dependent Hamiltonian of a given quantum system.\
\
(iv) In the Ref.[@Guedes], the author says that to the best of his knowledge there was no publication reporting the solution of the Schrödinger equation for the system described by $H(t)=\frac{p^{2}}{2m}+f(t)q $ without considering approximate and/or numerical calculations. I think that this is, however, not the true case. In the literature, at least in the early of 1990’s, Gao [*et al.*]{} had reported their investigation of the driven generalized time-dependent harmonic oscillator which is described by the following Hamiltonian $H(t)=\frac{1}{2}[X(t)q^{2}+Y(t)(pq+qp)+Z(t)p^{2}]+F(t)q$[@Gao]. It is believed that my solution presented here is only the special case of what they obtained[@Gao].\
\
**Acknowledgements** This project was supported partially by the National Natural Science Foundation of China under the project No. $90101024$.
H.R. Lewis, Jr. and W.B. Riesenfeld, J. Math. Phys.(N.Y) [**10**]{}, 1458 (1969).
I. Guedes, Phys. Rev. A [**63**]{}, 034102 (2001). H. Bekkar, F. Benamira, and M. Maamache, Phys. Rev. A [**68**]{}, 016101 (2003) X.C. Gao, J.B. Xu, and T.Z. Qian, Phys. Rev. A [**44**]{}, 7016 (1991). J.Q. Shen, arXiv: math-ph/0301026 (2003). J. Wei and E. Norman, J. Math. Phys.(N.Y) [**4**]{}, 575 (1963). J.Q. Shen, H.Y. Zhu, and P. Chen, Euro. Phys. J. D [**23**]{}, 305 (2003). I. Guedes, Phys. Rev. A [**68**]{}, 016102 (2003). T. Fulton, F. Rohrlich, L. Witten, Rev. Mod. Phys. [**34**]{}, 442 (1962). J.Q. Shen, H.Y. Zhu, and H. Mao, J. Phys. Soc. Jpn. [**[71]{}**]{}, 1440 (2002).
[^1]: Electronic address: [email protected]
[^2]: I think that this paper will be a supplement to the recent comment \[Phys. Rev. A [**68**]{}, 016101 (2003)\] of Bekkar [*et al.*]{} on Guedes’s work \[Phys. Rev. A [**63**]{}, 034102 (2001)\] and Guedes’s reply to Bekkar [*et al.*]{}’s comment. It will be submitted nowhere else for publication, just uploaded at the e-print archives.
[^3]: In general, for the case of three-generator Hamiltonian (the generators of which form a non-semisimple algebra), the time-dependent c-number $C(t)-[-i(B\beta-A\eta)+D\eta^{2}+F\beta^{2}-2E\beta\eta]$ in $I_{1}(t)$ are vanishing. See, for example, in Ref.[@Shenarxiv], which is a special case of the present problem.
| ArXiv |
---
abstract: 'Establishing the security of continuous-variable quantum key distribution against general attacks in a *realistic* finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. Indeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on de Finetti reductions only provide security for unrealistically large block lengths. We address this problem here by considering a new type of *Gaussian* de Finetti reduction, that exploits the invariance of some continuous-variable protocols under the action of the unitary group $U(n)$ (instead of the symmetric group $S_n$ as in usual de Finetti theorems), and by introducing generalized $SU(2,2)$ coherent states. Crucially, combined with an energy test, this allows us to truncate the Hilbert space globally instead as at the single-mode level as in previous approaches that failed to provide security in realistic conditions. Our reduction shows that it is sufficient to prove the security of these protocols against *Gaussian* collective attacks in order to obtain security against general attacks, thereby confirming rigorously the widely held belief that Gaussian attacks are indeed optimal against such protocols.'
author:
- Anthony Leverrier
title: 'Security of continuous-variable quantum key distribution via a Gaussian de Finetti reduction'
---
Quantum key distribution (QKD) is a cryptographic primitive aiming at distributing large secret keys to two distant parties, Alice and Bob, who have access to an authenticated classical channel. Mathematically, a QKD protocol ${\mathcal{E}}$ is described by a quantum channel, that is a completely positive trace-preserving (CPTP) map transforming an input state, typically a large bipartite entangled state shared by Alice and Bob, into two keys, ideally two identical bit strings unknown to any third party. Establishing the security of the protocol against arbitrary attacks means proving that the map ${\mathcal{E}}$ is approximately equal to an ideal protocol ${\mathcal{F}}$. An operational way of quantifying the security is by bounding the completely positive trace distance, or diamond distance between the two maps [@PR14]: the protocol is said to be ${\varepsilon}$-secure if $\|{\mathcal{E}}- {\mathcal{F}}\|_{\diamond} \leq {\varepsilon}$. If ${\mathcal{E}}$ and ${\mathcal{F}}$ act on some Hilbert space ${\mathcal{H}}$ and $\Delta = {\mathcal{E}}- {\mathcal{F}}$, then the diamond norm is defined as $$\begin{aligned}
\|\Delta\|_\diamond = \sup_{\rho \in {\mathfrak{S}}({\mathcal{H}}\otimes {\mathcal{H}}') } \|(\Delta \otimes {\mathbbm{1}}_{{\mathcal{H}}'}) (\rho) \|_1 \label{eqn:diamond1}\end{aligned}$$ where $\|\cdot\|_1$ is the trace norm and ${\mathfrak{S}}({\mathcal{H}}\otimes {\mathcal{H}}')$ is the set of normalized density matrices (positive operators of trace 1) on ${\mathcal{H}}\otimes {\mathcal{H}}'$ with ${\mathcal{H}}' \cong {\mathcal{H}}$ (see *e.g.* [@wat16]). Computing an upper bound of Eq. is very challenging in general because the Hilbert space ${\mathcal{H}}= {\mathcal{H}}_1^{\otimes n}$ has a dimension scaling exponentially with the number $n$ of quantum systems shared by Alice and Bob. Typical values of $n$ range in the millions or billions.
In order to estimate the diamond norm, it is natural to exploit all the symmetries displayed by $\Delta$. For instance, if ${\mathcal{E}}$ is a QKD protocol involving many 2-qubit pairs, such as BB84 for instance [@BB84], then $\Delta$ might be covariant under any permutation of these pairs. For such maps, Christandl, König and Renner [@CKR09] showed that the optimization of Eq. can be dramatically simplified provided that one is only interested in a polynomial approximation of $\|\Delta\|_\diamond$: indeed, it is then sufficient to consider a *single* state, called a “de Finetti state”, instead of optimizing over ${\mathcal{H}}\otimes {\mathcal{H}}' \cong ({\mathbb{C}}^4)^n \otimes ({\mathbb{C}}^4)^n$. More precisely, this de Finetti state is a purification of $ \tau_{{\mathcal{H}}} = \int \sigma_{{\mathcal{H}}_1}^{\otimes n} \mu(\sigma_{{\mathcal{H}}_1})$, where ${\mathcal{H}}_1 \cong {\mathbb{C}}^{4}$ is the single-system Hilbert space and $\mu(\cdot)$ is the measure on the space of density operators on a single system induced by the Hilbert-Schmidt metric.
This approach, called a *de Finetti reduction*, has been applied successfully to analyze the security of QKD protocol such as BB84 [@SLS10] or qudit protocols [@SS10]. Indeed, computing the value of $\|(\Delta \otimes {\mathbbm{1}}_{{\mathcal{H}}'}) \tau_{{\mathcal{H}}{\mathcal{H}}'}\|_1$ for some purification $\tau_{{\mathcal{H}}{\mathcal{H}}'}$ of $\tau_{{\mathcal{H}}}$ is usually tractable and is closely related to the task of establishing the security of the QKD protocol against *collective attacks*, corresponding to restricting the inputs of ${\mathcal{E}}$ to i.i.d. states of the form $\sigma_{{\mathcal{H}}_1}^{\otimes n}$. A full security proof then consists of two steps: proving the security against these restricted collective attacks, and applying the de Finetti reduction to obtain security (with a polynomially larger security parameter) against general attacks.
An outstanding problem in the theory of QKD is to address the security of protocols with continuous variables, that is protocols encoding the information in the continuous degrees of freedom of the quantified electro-magnetic field [@WPG12; @DL15]. From a practical point of view, the essential difference between continuous-variable (CV) protocols and discrete-variables ones lies in the detection method: CV protocols rely on coherent detection, either homodyne or heterodyne depending on whether one or two quadratures are measured for each mode, while discrete-variable protocols use photon counting. The main theoretical difference is the Hilbert space ${\mathcal{H}}$, which is *infinite-dimensional* for CV QKD, corresponding to a $2n$-mode Fock space: ${\mathcal{H}}= F({\mathbb{C}}^n \otimes {\mathbb{C}}^n)=\bigoplus_{k=0}^\infty \mathrm{Sym}^k({\mathbb{C}}^{n} \otimes {\mathbb{C}}^{n})$, where $\mathrm{Sym}^k(H)$ stands for the symmetric part of $H^{\otimes k}$. Note that the definition of Eq. is formally restricted to finite-dimensional spaces, but we will ignore this issue here because one can always truncate ${\mathcal{H}}$ to make its dimension finite (arbitrary large) and will therefore assume that the supremum can still be taken on ${\mathcal{H}}\otimes {\mathcal{H}}'$ for ${\mathcal{H}}' \cong {\mathcal{H}}$. For later convenience, let us denote ${\mathcal{H}}$ by $F_{1,1,n}$ and ${\mathcal{H}}\otimes {\mathcal{H}}'$ by $F_{2,2,n} :=F({\mathbb{C}}^{2n} \otimes {\mathbb{C}}^{2n}) \cong F_{1,1,n}\otimes F_{1,1,n}$.
A possible strategy to prove the security of such CV protocols is to follow the same steps as for BB84: first establish the security against collective attacks, then prove that this implies security against general attacks (with a reasonable loss). For protocols involving a Gaussian modulation of coherent states and heterodyne detection [@WLB04], composable security against collective attacks was recently demonstrated in [@Lev15]. The second step is to apply the de Finetti reduction outlined above. The difficulty here comes from the infinite dimensionality of the Fock space ${\mathcal{H}}$. In order to apply the technique of [@CKR09], it is therefore needed to truncate the Fock space in a suitable manner. This can be achieved with the help of an energy test, but unfortunately, the local dimension of $\overline{{\mathcal{H}}}_1$, the truncated single-mode space, needs to grow like the logarithm of $n$, for the technique to apply [@LGRC13]. Indeed, the technique of [@CKR09] was developed for finite-dimensional systems, and the energy test needs to enforce that with high probability, *each the unmeasured modes* contains a number of photons below some given threshold. Such a guarantee can only be obtained for a threshold increasing logarithmically with $n$. The dimension of the total truncated Hilbert space is then super-exponential in $n$, on the order of $(\log n)^{Cn}$, for some constant $C>1$. Since the loss in the security parameter obtained with [@CKR09] is superpolynomial in the dimension of the total Hilbert space, this means that if the protocol is ${\varepsilon}$-secure against collective attacks, this approach only shows that the protocol is also ${\varepsilon}'$-secure against general attacks with ${\varepsilon}' = {\varepsilon}\times 2^{\mathrm{polylog}(n)}$. While this gives a proof that the protocol is asymptotically secure in the limit of infinitely large block lengths, it fails to provide any useful bound in practical regimes where $n \sim 10^6 - 10^9$. We note that a related strategy relies on the exponential de Finetti theorem but fails similarly to provide practical security bounds in the finite-size regime [@Ren08; @RC09].
Let us also mention that there exists a CV QKD protocol with proven security where Alice sends squeezed states to Bob instead of coherent states [@CLV01]. This protocol can be analyzed thanks to an entropic uncertainty relation [@FBB12], but this technique requires the exchange of squeezed states, which makes the protocol much less practical. Moreover, this approach does not recover the secret key rate corresponding to Gaussian attacks in the asymptotic limit of large $n$, even though these attacks are expected to be optimal. Here, in contrast, we are interested in the security of CV protocols based on the exchange of coherent states.
The idea that we exploit in this paper is that CV QKD protocols not only display the permutation invariance common to most QKD protocols, but also a specific symmetry with a continuous-variable flavor [@LKG09]. This new symmetry is linked to the unitary group $U(n)$ instead of the symmetric group $S_n$. More precisely, the protocols are covariant if Alice and Bob process their $n$ respective modes with linear-optical networks acting like the unitary $u \in U(n)$ on Alice’s annihilation operators and its complex conjugate $\overline{u}$ on Bob’s annihilation operators.
Our main technical result is an upper bound on $\|\Delta\|_{\diamond}$ for maps $\Delta$ covariant under a specific representation of the unitary group. For such maps, we show that is it sufficient to consider again a single state, which is the purification of a specific mixture of *Gaussian* i.i.d. states. This in turn will imply that it is sufficient to establish the security of the protocol against *Gaussian* collective attacks in order to prove the security of the protocol against general attacks. An important technicality is that we still need to truncate the total Hilbert space to replace it by a finite-dimensional one. Crucially, this truncation can now be done globally and not for single-mode Fock spaces as in [@LGRC13] and this is this very point that makes our approach so effective. Indeed, in our security proof, we argue that it is sufficient to consider states that are invariant under the action of $U(n)$ and such states live in a very small subspace of the ambient Fock space. More precisely, the dimension of the restriction of this subspace to states containing $K$ photons grows polynomially in $K$, instead of exponentially in the case of the total Fock space. This phenomenon is reminiscent of the fact that the dimension of the symmetric subspace of $({\mathbb{C}}^{\otimes d})^{\otimes n}$ only grows polynomially in $n$ if the local dimension $d$ is constant.
The consequence is that the security loss due to the reduction from general to collective attacks will not scale like $2^{\mathrm{polylog}(n))}$ anymore, but rather like $O(n^4)$, which behaves *much more nicely* for typical values of $n$, and yields the first practical security proof of a CV QKD protocol with coherent states against general attacks. Indeed, our security reduction performs even better than the original de Finetti reduction developed for BB84, where the security loss scales like $O(n^{15})$ [@CKR09].
Ideally, truncating the Fock space could be done by projecting the quantum state given as an input to $\Delta$ onto a finite dimensional subspace with say, less than $K$ photons (where the value of $K$ scales linearly with the total number of modes). Of course, such a projection ${\mathcal{P}}$ is unrealistic, and one will instead apply an energy test ${\mathcal{T}}$ that passes if the energy measured on a small number $k \ll n$ of modes is below some threshold and will abort the protocol otherwise. Such an idea was already considered in previous works dealing with the security of CV QKD [@RC09; @LGRC13; @fur14]. An application of the triangle inequality (see Lemma \[lem:sec-red\] in the appendix) yields: $$\begin{aligned}
\|\Delta \circ {\mathcal{T}}\|_\diamond \leq \|\Delta \circ {\mathcal{P}}\|_\diamond + 2 \|({\mathbbm{1}}-{\mathcal{P}})\circ {\mathcal{T}}\|_\diamond. \label{eqn:triangle}\end{aligned}$$ In other words, it is sufficient for our purposes to show the security of the protocol restricted to input states subject to a maximum photon number constraint, provided that we can bound the value of $\|({\mathbbm{1}}-{\mathcal{P}})\circ {\mathcal{T}}\|_\diamond$, which corresponds to the probability that the energy test passes but that the number of photons in the remaining modes is large.
[**Analysis of the energy test**]{}.—We show that $\|({\mathbbm{1}}-{\mathcal{P}})\circ {\mathcal{T}}\|_\diamond$ is indeed small for a maximal number of photons $K$ scaling linearly with $n$ (see Appendix \[sec:test\]). The energy test ${\mathcal{T}}(k, d_A, d_B)$ depends on 3 parameters: the number $k$ of additional modes that will be measured for the test and maximum allowed average energies $d_A$ and $d_B$ for Alice and Bob’s modes. The input of the state is a $2(n+k)$-mode state. Alice and Bob should symmetrize this state by processing them with random conjugate linear-optical networks and measure the last $k$ modes with heterodyne detection, corresponding to a projection of standard (Glauber) coherent states. If the average energy per mode is below $d_A$ for Alice and $d_B$ for Bob, the test passes and Alice and Bob apply the protocol ${\mathcal{E}}_0$ to their remaining modes. Otherwise the protocol simply aborts. These thresholds $d_A$, $d_B$ should be chosen large enough to ensure that the energy test passes with large probability. Note that the symmetrization of the state can be done on the classical data for the protocols of Refs [@WLB04; @POS15] since these protocols require both parties to measure all the modes with heterodyne detection, which itself commutes with the action of the linear-optical networks. For this, Alice and Bob need to multiply their measurement results (gathered as vectors for $\mathbbm{R}^{2n}$) by an identical random orthogonal matrix. There is also hope that this symmetrization can be further simplified, but we do not address this issue here.
[**An upper bound on $\|\Delta \circ {\mathcal{P}}\|_\diamond$ via de Finetti reduction**]{}.— This requires two main ingredients: first, a proof that any mixed state on $F_{1,1,n}$ that is invariant under the action of the unitary group admits a purification in the *symmetric subspace* $F_{2,2,n}^{U(n)}$, and second, that Gaussian states resolve the identity on the symmetric subspace. The symmetric subspace $F_{2,2,n}^{U(n)}$ was introduced and studied in Ref. [@lev16] and is defined as follows: $$\begin{aligned}
F_{2,2,n}^{U(n)} = \left\{|\psi\rangle \in F_{2,2,n} \: :\: W_u |\psi\rangle = |\psi\rangle, \forall u \in U(n) \right\},\end{aligned}$$ where $u \mapsto W_u$ is a representation of the unitary group $U(n)$ on the Fock space $F_{2,2,n}$ corresponding to mapping the $4n$ annihilation operators $\vec{a} = (a_1, \ldots, a_n), \vec{b} =(b_1, \ldots, b_n), \vec{a}' = (a'_1, \ldots, a'_n), \vec{b}'=(b'_1, \ldots, b'_n)$ of each of the $n$ modes of ${\mathcal{H}}_A, {\mathcal{H}}_B, {\mathcal{H}}_{A'}, {\mathcal{H}}_{B'}$ to $u \vec{a}, \overline{u} \vec{b}, \overline{u} \vec{a}', u \vec{b}'$. Here $\overline{u}$ denotes the complex conjugate of $u$ and $F_{2,2,n} = {\mathcal{H}}_A\otimes {\mathcal{H}}_B \otimes {\mathcal{H}}_{A'}\otimes {\mathcal{H}}_{B'}$.
In Ref. [@lev16], a full characterization of the symmetric subspace $F_{2,2,n}^{U(n)}$ is given. Let us introduce the four operators $Z_{11}, Z_{12}, Z_{21}, Z_{22}$ defined by: $$\begin{aligned}
Z_{11} &= \sum_{i=1}^n a_i^\dagger b_i^\dagger, \quad Z_{12} =\sum_{i=1}^n a_i^\dagger a'^\dagger_i,\\
Z_{21} &= \sum_{i=1}^n b_i^\dagger b'^\dagger_i, \quad Z_{22} =\sum_{i=1}^n a'^\dagger_i b'^\dagger_i.\end{aligned}$$ We now define the so-called $SU(2,2)$ *generalized coherent states* [@per72; @per86]: to any $2\times 2$ complex matrix $\Lambda = \left(\begin{smallmatrix} \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \end{smallmatrix} \right)$ such that $\Lambda \Lambda^\dagger \prec {\mathbbm{1}}_2$ (that is, with a spectral norm strictly less than 1), we associate the $4n$-mode Gaussian state $|\Lambda, n\rangle = |\Lambda,1\rangle^{\otimes n}$ given by $$\begin{aligned}
|\Lambda,n\rangle = \mathrm{det} (1-\Lambda\Lambda^\dagger)^{n/2} \exp\left( \sum_{i,j=1}^2 \lambda_{ij} Z_{ij}\right) |\mathrm{vacuum}\rangle.\end{aligned}$$ Since the polynomial $\sum_{i,j=1}^2 \lambda_{ij} Z_{ij}$ is quadratic in the creation operators, the generalized coherent state is a Gaussian state. More specifically, it corresponds to $n$ copies of a centered 4-mode pure Gaussian state whose covariance matrix is a function of $\Lambda$ (see the discussion in Section 3 of Ref. [@lev16] for details).
These generalized coherent states span the symmetric subspace [@lev16], and moreover, for $n\geq 4$, they resolve the identity on the symmetric subspace [@lev16]: $$\begin{aligned}
\int_{{\mathcal{D}}} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda) = {\mathbbm{1}}_{F_{2,2,n}^{U(n)}} \label{eqn:maintext-id}\end{aligned}$$ where ${\mathcal{D}}$ is the set of $2\times 2$ matrices $\Lambda$ such that $\Lambda\Lambda^\dagger \prec {\mathbbm{1}}_2$ and $\mathrm{d}\mu_n(\Lambda)$ is the invariant measure on ${\mathcal{D}}$ given by $$\begin{aligned}
\mathrm{d}\mu_n (\Lambda) = \frac{(n-1)(n-2)^2(n-3)}{\pi^{4}\det(\mathbbm{1}_2 - \Lambda \Lambda^\dagger)^4 } \d \lambda_{11} \d \lambda_{12}\d \lambda_{21} \d \lambda_{22}. \end{aligned}$$
Since the space $F_{2,2,n}^{U(n)}$ is infinite-dimensional, the integral of Eq. is not normalizable. In order to obtain an operator with finite norm, we consider the finite-dimensional subspace $F_{2,2,n}^{U(n), \leq K}$ of $F_{2,2,n}^{U(n)}$ spanned by states with less than $K$ “excitations”: $$\begin{aligned}
\mathrm{Span}\left\{ (Z_{11})^i (Z_{12})^j (Z_{21})^k (Z_{22})^\ell |\mathrm{vac}\rangle \: : \: i+j+k+\ell \leq K \right\}.\end{aligned}$$ We show in Appendix \[sec:finite\] that an approximate resolution of the identity still holds for this space when restricting the coherent states $|\Lambda,n\rangle$ to $\Lambda \in {\mathcal{D}}_\eta$ for ${\mathcal{D}}_\eta = \left\{ \Lambda \in {\mathcal{D}}\: : \: \eta {\mathbbm{1}}_2 -\Lambda\Lambda^\dagger \succeq 0Ê\right\}$ for $\eta \in [0,1[$. Let us denote by $\Pi_{\leq K}$ the identity onto the subspace $F_{2,2,n}^{U(n), \leq K}$ and introduce the relative entropy $D(x||y) = x \log \frac{x}{y} + (1-x) \log \frac{1-x}{1-y}$.
\[thm:finite-version\] For $n\geq 5$ and $\eta \in [0,1[$, if $K \leq\frac{\eta N}{1-\eta}$ for $N=n-5$, then the operator inequality $$\begin{aligned}
\int_{\mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n | \d \mu_n(\Lambda)\geq (1-{\varepsilon}) \Pi_{\leq K} \label{eqn:approximate-main}\end{aligned}$$ holds with ${\varepsilon}= 2 N^4 (1+K/N)^7 \exp\left(-N D\left(\frac{K}{K+N} \big\| \eta \right) \right)$.
This approximate resolution of the identity allows us to bound the diamond norm of maps which are covariant under the action $W_u$ of the unitary group $U(n)$, provided that the total photon number of the input state is upper bounded by some known value $K$. Let us define $\tau_{{\mathcal{H}}}^\eta$ to be the normalized state corresponding to the left-hand side of Eq. , and $\tau_{{\mathcal{H}}{\mathcal{N}}}^{\eta}$ a purification of $\tau_{{\mathcal{H}}}^\eta$.
\[thm:postselection\] Let $\Delta: \mathrm{End}(F_{1,1,n}^{\leq K}) \to \mathrm{End}(\mathcal{H}')$ such that for all $u \in U(n)$, there exists a CPTP map $\mathcal{K}_u: \mathrm{End}(\mathcal{H}') \to \mathrm{End}(\mathcal{H}')$ such that $\Delta \circ W_u = \mathcal{K}_u \circ \Delta$, then $$\begin{aligned}
\|\Delta\|_\diamond \leq \frac{K^4}{50} \|(\Delta \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1,\end{aligned}$$ for $\eta = \frac{K-n+5}{K+n-5}$, provided that $n \geq N^*(K/(n-5))$.
The function $N^*$ is defined in Eq. and its argument is an upper bound on the average number of photons per mode. One has for instance $N^*(21) \approx 10^4$, $N^*(60) \approx 10^5$.
Similarly as in [@CKR09] for the case of permutation invariance, Theorem \[thm:postselection\] shows that one can obtain a polynomial approximation of degree 4 (if the average number of photons per mode is constant) of the diamond norm by simply evaluating the trace norm of the map on a very simple state, namely a purification of a mixture of Gaussian i.i.d. states. We note that we restricted the analysis to $SU(2,2)$ coherent states here because they are the relevant ones for cryptographic applications, but our results can be extended to $SU(p,q)$ coherent states for arbitrary integers $p, q$. In that case, the prefactor of the diamond norm approximation would be a polynomial of degree $pq$.
[**Security reduction to Gaussian collective attacks**]{}.—We now explain how to obtain a bound on $\|(\Delta \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1$, if we already know that the initial protocol (without the energy test) is ${\varepsilon}$-secure against collective attacks. Let us therefore assume that we are given such a CV QKD protocol ${\mathcal{E}}_0$ acting on $2n$-mode states shared by Alice and Bob which is, in addition, covariant under the action of the unitary group (*i.e.* there exists $\mathcal{K}_u$ such that ${\mathcal{E}}_0 \circ W_u = {\mathcal{K}}_u \circ {\mathcal{E}}_0$). Examples of such protocols are the no-switching protocol [@WLB04] and the measurement-device-independent protocol of Ref. [@POS15], provided that they are suitably symmetrized. We define ${\mathcal{E}}:= {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{T}}$, where ${\mathcal{R}}$ is an additional privacy amplification step that reduces the key by $\lceil 2 \log_2 \tbinom{K+4}{4} \rceil$ bits.
Recall that by definition, the QKD protocol ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks if $$\begin{aligned}
\| (({\mathcal{E}}_0-{\mathcal{F}}_0)\otimes \mathrm{id})(|\Lambda,n\rangle \langle \Lambda,n|)\|_1\leq {\varepsilon}\end{aligned}$$ for all $\Lambda \in {\mathcal{D}}$. It means that the protocol is shown to be secure for input states of the form ${\mathrm{tr}}_{{\mathcal{H}}_{A'}{\mathcal{H}}_{B'}} (|\Lambda, n\rangle\langle \Lambda,n|)$, which are nothing but i.i.d. bipartite Gaussian states. By linearity, we immediately obtain that $\|(\Delta \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}} \|_1 \leq {\varepsilon}$. To finish the proof, we need to take into account the extra system ${\mathcal{N}}$ given to Eve. This system can be chosen of dimension $\tbinom{K+4}{4}$ and the leftover hashing lemma of Renner [@Ren08] says that by shortening the final key of the protocol by $2 \log_2 (\mathrm{dim} \, {\mathcal{N}})$, one ensures that the protocol remains ${\varepsilon}$-secure. This is the role of the map ${\mathcal{R}}$. Overall, we find that $\|(({\mathcal{E}}-{\mathcal{F}}) \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1 \leq {\varepsilon}$.
[**Results**]{}.—Putting everything together, we show that if ${\mathcal{E}}_0$ is covariant under the action of the unitary group and ${\varepsilon}$-secure against Gaussian collective attacks, then the protocol ${\mathcal{E}}= {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{T}}$ is ${\varepsilon}'$-secure against general attacks, with $$\begin{aligned}
{\varepsilon}' =\frac{K^4}{50} {\varepsilon}\label{eqn:final-result-main}\end{aligned}$$ for $K = \max \Big\{1, n(d_A + d_B)\Big(1 + 2 \sqrt{\frac{\ln (8/{\varepsilon})}{2n}} + \frac{\ln (8/{\varepsilon})}{n}\Big)\Big(1-2{\sqrt{\frac{\ln (8/{\varepsilon})}{2k}}}\Big)^{-1}\Big\}$. The full proof is presented in Appendix \[sec:general-proof\]. The advantage of our approach compared to the previous results of [@LGRC13] is two-fold: first the improvement of the prefactor in Eq. from $2^{\mathrm{polylog}(n)}$ to $O(n^4)$ yields security for practical settings; second, it is only required to establish the security of the protocol against Gaussian collective attacks in order to apply our security reduction, a task arguably much simpler than addressing the security against collective attacks in the case of CV QKD.
[**Discussion**]{}.—Despite their wide range of application, there is a regime where “standard” de Finetti theorems fail, namely when the local dimension is not negligible compared to the number $n$ of subsystems [@CKMR07]. In particular, these techniques do not apply directly to CV protocols where the local spaces are infinite-dimensional Fock spaces. In this work, we considered a natural symmetry displayed by some important CV QKD protocols, which are covariant under the action of beamsplitters and phase-shifts on their $n$ modes [@LKG09]. For such protocols, one legitimately expects that stronger versions of de Finetti theorems should hold. In particular, a widely held belief that it is enough to consider *Gaussian* i.i.d. input states instead of all i.i.d. states in order to analyze the security of the corresponding protocol.
We proved this statement rigorously here. Our main tool is a family of $SU(2,2)$ generalized coherent states that resolve the identity of the subspace spanned by states invariant under the action of $U(n)$. This implies that in some applications such as QKD, it is sufficient to consider the behaviour of the protocol on these states in order to obtain guarantees that hold for arbitrary input states.
Let us conclude by discussing the issue of active symmetrization. For the proof above to go through, it is required that the protocols are covariant under the action of the unitary group. Such an invariance can be enforced by symmetrizing the classical data held by Alice and Bob. However, this step is computationally costly and it would be beneficial to bypass it. We believe that this should be possible. Indeed, it is often argued that a similar step is unnecessary when proving the security of BB84 for instance, and there is no fundamental reason to think that the situation is different here. Moreover, we already know of security proofs based on the uncertainty principle [@TR11; @TLG12; @DFR16] where such a symmetrization is not required.
I gladly acknowledge inspiring discussions with Matthias Christandl and Tobias Fritz.
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In Section \[sec:symm-sub\], we recall the main results from Ref. [@lev16] about the symmetric subspace $F_{2,2,n}^{U(n)}$ and the generalized $SU(2,2)$ coherent states. In Section \[sec:lemmas\], we present a series of technical lemmas and prove in Section \[sec:finite\] that bounded-energy generalized coherent states approximately resolve the identity on $F_{2,2,n}^{U(n), \leq K}$. In Section \[sec:general-proof\], we explain how to perform the security proof of the protocol and show that bounding the norm of $\Delta = {\mathcal{E}}- {\mathcal{F}}$ decomposes into separate tasks. In Section \[sec:generalization\], we derive our generalization of the de Finetti reduction of [@CKR09] to maps that are covariant under the action of the unitary group $U(n)$. In Section \[sec:collective\], we show how to reduce the security analysis against general attacks to a security analysis against Gaussian collective attacks, if the photon number of the input states is bounded. Finally, in Section \[sec:test\], we analyze the energy test and show how it provides the restriction on the input states required for the proof of Section \[sec:collective\] to go through.
The symmetric subspace $F_{2,2,n}^{U(n)}$ and generalized $SU(2,2)$ coherent states {#sec:symm-sub}
===================================================================================
In this section, we recall some results from Ref. [@lev16] where the symmetric subspace $F_{p,q,n}^{U(n)}$ is considered for arbitrary integers $p,q$ and specialize them to the case where $p=q=2$.
The symmetric subspace $F_{2,2,n}^{U(n)}$
-----------------------------------------
Let $H_A \cong H_B \cong H_{A'} \cong H_{B'} \cong {\mathbb{C}}^n$ and define the Fock space $F_{2,2,n}$ as $$\begin{aligned}
F_{2,2,n} := \bigoplus_{k=0}^\infty \mathrm{Sym}^k(H_A \otimes H_B \otimes H_{A'} \otimes H_{B'}),\end{aligned}$$ where $\mathrm{Sym}^k(H)$ is the symmetric part of $H^{\otimes k}$.
In this paper, we will use both the standard Hilbert representation and the Segal-Bargmann representation of $F_{2,2,n}$. Using the Segal-Bargmann representation, the Hilbert space $F_{2,2,n}$ is realized as a functional space of complex holomorphic functions square-integrable with respect to a Gaussian measure, $F_{2,2,n} \cong L^2_{\mathrm{hol}}({\mathbb{C}}^{4n}, \| \cdot\|)$, with a state $\psi \in F_{2,2,n}$ represented by a holomorphic function $\psi(z,z')$ with $z \in {\mathbb{C}}^{2n}, z' \in {\mathbb{C}}^{2n}$ satisfying $$\begin{aligned}
\label{eqn:norm}
\|\psi\|^2 := \langle \psi, \psi\rangle = \frac{1}{\pi^{4n}}\int \exp(-|z|^2 -|z'|^2) |\psi(z,z')|^2 \d z \d z'< \infty\end{aligned}$$ where $\d z := \prod_{k=1}^n \prod_{i=1}^2 \mathrm{d}z_{k,i}$ and $\d z' := \prod_{k=1}^n \prod_{j=1}^2 \mathrm{d}z_{k,j}'$ denote the Lebesgue measures on ${\mathbb{C}}^{2n}$ and ${\mathbb{C}}^{2n}$, respectively, and $|z|^2 := \sum_{k=1}^n\sum_{i=1}^2 |z_{k,i}|^2, |z'|^2 := \sum_{k=1}^n \sum_{j=1}^2 |z_{k,j}'|^2$. A state $\psi$ is therefore described as a holomorphic function of $4n$ complex variables $(z_{1,1}, z_{n,1}; z_{1,2}, \ldots, z_{n,2}; z_{1,1}', \ldots, z_{n,1}'; z_{1,2}', \ldots, z_{n,2}')$. In the following, we denote by $z_i$ and $z_j'$ the vectors $(z_{1,i}, \ldots, z_{n,i})$ and $(z_{1,j}', \ldots, z_{n,j}')$, respectively, for $i,j \in \{1,2\}$. With these notations, the vector $z_1$ is associated to the space $H_A$, the vector $z_1'$ to $H_B$, the vector $z_2$ to $H_B'$ and the vector $z_2'$ to $H_A'$. These notations are chosen so that the unitary $u \in U(n)$ acts as $u$ on $z_1, z_2$, and $\overline{u}$ on $z'_1, z_2'$.
Let $\mathfrak{B}(F_{2,2,n})$ denote the set of bounded linear operators from $F_{2,2,n}$ to itself and let $\mathfrak{S}(F_{2,2,n})$ be the set of quantum states on $F_{2,2,n}$: positive semi-definite operators with unit trace.
Formally, one can switch from the Segal-Bargmann representation to the representation in terms of annihihation and creation operators by replacing the variables $z_{k,1}$ by $a_k^\dagger$, $z_{k,2}$ by $b'^\dagger_k$, $z'_{k,1}$ by $b_k^\dagger$ and $z'_{k,2}$ by $a'^\dagger_k$. The function $f(z,z')$ is therefore replaced by an operator $f(a^\dagger, b^\dagger, a'^\dagger, b'^\dagger)$ and the corresponding state in the Fock basis is obtained by applying this operator to the vacuum state.
The metaplectic representation of the unitary group $U(n) \subset Sp(2n,{\mathbb{R}})$ on $ F_{2,2,n}$ associates to $u \in U(n)$ the operator $W_u$ performing the change of variables $z \to uz$, $z' \to \overline{u} z'$: $$\begin{aligned}
U(n) & \to \mathfrak{B}(F_{2,2,n})\\
u & \mapsto W_u = \big[ \psi(z_1, z_2, z_1', z_2') \mapsto \psi(u z_1, u z_2, \overline{u} z_1', \overline{u} z_2')\big]\end{aligned}$$ where $\overline{u}$ denotes the complex conjugate of the unitary matrix $u$. In other words, the unitary $u$ is applied to the modes of $F_A \otimes F_{B'}$ and its complex conjugate is applied to those of $F_B \otimes F_{A'}$.
The states that are left invariant under the action of the unitary group $U(n)$ are relevant for instance in the context of continuous-variable quantum key distribution, and we define the symmetric subspace as the space spanned by such invariant states.
For integer $n \geq 1$, the *symmetric subspace* $F_{2,2,n}^{U(n)}$ is the subspace of functions $\psi \in F_{2,2,n}$ such that $$\begin{aligned}
W_u \psi = \psi \quad \forall u \in U(n).\end{aligned}$$
The name *symmetric subspace* is inspired by the name given to the subspace $\mathrm{Sym}^n(\mathbbm{C}^d)$ of $(\mathbbm{C}^d)^{\otimes n}$ of states invariant under permutation of the subsystems: $$\begin{aligned}
\mathrm{Sym}^n(\mathbbm{C}^d) := \left\{|\psi\rangle \in(\mathbbm{C}^d)^{\otimes n} \: : \: P(\pi) |\psi\rangle = |\psi\rangle, \forall \pi \in S_n \right\}\end{aligned}$$ where $\pi \mapsto P(\pi)$ is a representation of the permutation group $S_n$ on $(\mathbbm{C}^d)^{\otimes n}$ and $P(\pi)$ is the operator that permutes the $n$ factors of the state according to $\pi \in S_n$. See for instance [@har13] for a recent exposition of the symmetric subspace from a quantum information perspective.
In [@lev16], a full characterization of the symmetric subspace $F_{2,2,n}^{U(n)}$ is given. It is helpful to introduce the four operators $Z_{11}, Z_{12}, Z_{21}, Z_{22}$ defined by: $$\begin{aligned}
Z_{11} = \sum_{i=1}^n z_{i,1} z'_{i,1} \quad & \leftrightarrow \quad \sum_{i=1}^n a_i^\dagger b_i^\dagger\\
Z_{12} =\sum_{i=1}^n z_{i,1} z'_{i,2} \quad & \leftrightarrow \quad \sum_{i=1}^n a_i^\dagger a'^\dagger_i,\\
Z_{21} = \sum_{i=1}^n z_{i,2} z'_{i,1} \quad & \leftrightarrow \quad \sum_{i=1}^n b_i^\dagger b'^\dagger_i, \quad \\
Z_{22} =\sum_{i=1}^n z_{i,2} z'_{i,2} \quad & \leftrightarrow \quad \sum_{i=1}^n a'^\dagger_i b'^\dagger_i.\end{aligned}$$
For integer $n\geq 1$, let $E_{2,2,n}$ be the space of analytic functions $\psi$ of the $4$ variables $Z_{1,1}, \ldots, Z_{2,2}$, satisfying $\|\psi\|_E^2 < \infty$, that is $E_{2,2,n} = L^2_{\mathrm{hol}}({\mathbb{C}}^{pq}, \|\cdot\|_E)$.
In [@lev16], is was proven that $E_{2,2,n}$ coincides with the symmetric subspace $F_{2,2,n}^{U(n)}$.
\[thm:charact-symm\] For $n \geq 2$, the symmetric subspace $F_{2,2,n}^{U(n)}$ is isomorphic to $E_{2,2,n}$.
In other words, any state in the symmetric subspace can be written as $$\begin{aligned}
|\psi\rangle = f\big(\sum_{i=1}^n a_i^\dagger b_i^\dagger, \sum_{i=1}^n a_i^\dagger a'^\dagger_i, \sum_{i=1}^n b_i^\dagger b'^\dagger_i, \sum_{i=1}^n a'^\dagger_i b'^\dagger_i \big) |\mathrm{vacuum}\rangle\end{aligned}$$ for some function $f$. Said otherwise, such a state is characterized by only 4 parameters instead of $4n$ for an arbitrary state in $F_{2,2,n}$; or else, the symmetric subspace is isomorphic to a 4-mode Fock space (with “creation” operators corresponding to $Z_{11}, Z_{12}, Z_{21}, Z_{22}$, instead of the ambient $4n$-mode Fock space.
Coherent states for $SU(2,2)/SU(2)\times SU(2) \times U(1)$ {#sec:CS}
-----------------------------------------------------------
In this section, we first review a construction due to Perelomov that associates a family of generalized coherent states to general Lie groups [@per72], [@per86]. In this language, the standard Glauber coherent states are associated with the Heisenberg-Weyl group, while the atomic spin coherent states are associated with $SU(2)$. The symmetric subspace $F_{2,2,n}^{U(n)}$ is spanned by $SU(2,2)$ coherent states, where $SU(2,2)$ is the special unitary group of signature $(2,2)$ over ${\mathbb{C}}$: $$\begin{aligned}
SU(2,2) := \left\{ A \in M_{4}({\mathbb{C}}) \: : \: A {\mathbbm{1}}_{2,2} A^\dagger ={\mathbbm{1}}_{2,2}Ê\right\}\end{aligned}$$ where $M_{4}({\mathbb{C}})$ is the set of $4\times 4$-complex matrices and ${\mathbbm{1}}_{2,2} = {\mathbbm{1}}_{2} \oplus (-{\mathbbm{1}}_2)$.
In Perelomov’s construction, a *system of coherent states of type* $(T, |\psi_0\rangle)$ where $T$ is the representation of some group $G$ acting on some Hilbert space $\mathcal{H} \ni |\psi_0\rangle$, is the set of states $\left\{|\psi_g\rangle \: : \: |\psi_g\rangle = T_g |\psi_0\rangle\right\}$ where $g$ runs over all the group $G$. One defines $H$, the *stationary subgroup* of $|\psi_0\rangle$ as $$\begin{aligned}
H := \left\{g \in G \: :Ê\: T_g |\psi_0\rangle = \alpha |\psi_0\rangle \, \text{for} \, |\alpha|=1Ê\right\},\end{aligned}$$ that is the group of $h \in G$ such that $|\psi_h\rangle $ and $|\psi_0\rangle$ differ only by a phase factor. When $G$ is a connected noncompact simple Lie group, $H$ is the maximal subgroup of $G$. In particular, for $G = SU(2,2)$, one has $H= SU(2,2) \cap U(4) = SU(2) \times SU(2)\times U(1)$ and the factor space $G/H$ corresponds to a Hermitian symmetric space of classical type (see *e.g.* Chapter X of [@hel79]). The generalized coherent states are parameterized by points in $G/H$. For $G/H = SU(2,2)/SU(2)\times SU(2) \times U(1)$, the factor space is the set ${\mathcal{D}}$ of $2\times 2$ matrices $\Lambda$ such that $\Lambda \Lambda^\dagger < {\mathbbm{1}}_{p}$, i.e. the singular values of $\Lambda$ are strictly less than 1. $$\begin{aligned}
{\mathcal{D}}= \left\{ \Lambda \in M_{2}({\mathbb{C}}) \: : \:\mathbbm{1}_2 - \Lambda\Lambda^\dagger >0 \right\},\end{aligned}$$ where $A>0$ for a Hermitian matrix $A$ means that $A$ is positive definite.
We are now ready to define our coherent states for the noncompact Lie group $SU(2,2)$.
\[defn:CS\] For $n \geq 1$, the coherent state $\psi_{\Lambda,n}$ associated with $\Lambda \in {\mathcal{D}}$ is given by $$\begin{aligned}
\psi_{\Lambda,n}(Z_{1,1}, \ldots, Z_{2,2}) = \det (1-\Lambda \Lambda^\dagger)^{n/2} \det \exp (\Lambda^T Z)
\end{aligned}$$ where $Z$ is the $2\times 2$ matrix $\left[ Z_{i,j}\right]_{i,j \in \{1,2\}}$.
In the following, we will sometimes abuse notation and write $\psi_{\Lambda}$ instead of $\psi_{\Lambda,n}$, when the parameter $n$ is clear from context.
We note that the coherent states have a tensor product form in the sense that $$\begin{aligned}
\psi_{\Lambda,n}=\psi_{\Lambda,1}^{\otimes n}.\end{aligned}$$ We will also write $|\Lambda,n\rangle = |\Lambda,1\rangle^{\otimes n}$ for $\psi_{\Lambda,n}$. Such a state is called *identically and independently distributed* (i.i.d.) in the quantum information literature.
The main feature of a family of coherent states is that they resolve the identity. This is the case with the $SU(2,2)$ coherent states introduced above: see Ref. [@lev16].
\[thm:resol\] For $n \geq 4$, the coherent states resolve the identity over the symmetric subspace $F_{2,2,n}^{U(n)}$: $$\begin{aligned}
\int_{{\mathcal{D}}} |\Lambda,n\rangle \langle \Lambda,n| \mathrm{d}\mu_n(\Lambda) = \mathbbm{1}_{F_{2,2,n}^{U(n)}}, \end{aligned}$$ where $\mathrm{d}\mu_n(\Lambda)$ is the invariant measure on ${\mathcal{D}}$ given by $$\begin{aligned}
\label{eqn:mu}
\mathrm{d}\mu_n (\Lambda) = \frac{(n-1)(n-2)^2(n-3)}{\pi^{4}\det(\mathbbm{1}_2 - \Lambda \Lambda^\dagger)^4 } \prod_{i=1}^2 \prod_{j=1}^{2} \mathrm{d} \mathfrak{R}(\Lambda_{i,j}) \mathrm{d} \mathfrak{I}(\Lambda_{i,j}), \end{aligned}$$ where $\mathfrak{R}(\Lambda_{i,j})$ and $\mathfrak{I}(\Lambda_{i,j})$ refer respectively to the real and imaginary parts of $\Lambda_{i,j}$. This operator equality is to be understood for the weak operator topology.
Technical lemmas {#sec:lemmas}
================
In this section, we prove or recall a number of technical results that will be useful for analyzing the finite energy version of the de Finetti theorem in \[sec:finite\].
Tail bounds
-----------
For positive integers $k,n >0$, the Beta and regularized (incomplete) Beta functions are given respectively by $$\begin{aligned}
B(k,n) = \int_0^1 t^{k-1} (1-t)^{n-1} \d t = \frac{(k-1)!(n-1)!}{(n+k-1)!}, \quad B(x;k,n) = \int_0^x t^{k-1}(1-t)^{n-1} \d t,\end{aligned}$$ for $x >0$. Finally, the regularized Beta function is defined as $$\begin{aligned}
I_{x}(k,n) = \frac{B(x; k,n)}{B(k,n)}.\end{aligned}$$
Let us recall the Chernoff bound for a sum of independent Bernoulli variables.
\[thm:chernoff\] Let $X_1, \ldots, X_n$ be independent random variables on $\{0,1\}$ with $\mathrm{Pr}[X_i=1]=p$, for $i=1, \ldots, n$. Set $X = \sum_{i=1}^n X_i$. Then for any $t \in [0,1-p]$, we have $$\begin{aligned}
\mathrm{Pr}[X \geq (p+t)n ] \leq \exp \left(-n D(p+t||p) \right),\end{aligned}$$ where the relative entropy is defined as $D(x||y) = x \log \frac{x}{y} + (1-x) \log \frac{1-x}{1-y}$.
Pinsker’s inequality gives a lower bound on $D(x||y)$ as a function of the total variation distance between the two probability distributions.
\[lem:pinsker\] For $0 < y < x <1$, it holds that $$\begin{aligned}
D(x \| y) \geq \frac{2}{\ln 2} (x-y)^2.\end{aligned}$$
We now prove a tail bound for the regularized Beta function.
\[lem:tail-bound\] For integers $k,n >0$, it holds that $$\begin{aligned}
1-I_{\eta}(k,n) \leq \exp\left(-(n+k-1) D\left(\frac{k-2}{n+k-1} \| \eta \right) \right),\end{aligned}$$ provided that $\eta \geq (k-2)/(n+k-1)$.
The incomplete Beta function can be related to the tail of the binomial distribution as follows: $$\begin{aligned}
1- I_{\eta}(k,n) &= F(k-1,n+k-1,\eta) \label{eqn:injected}\end{aligned}$$ where $F(K,N,p)$ is the probability that there are at most $K$ successes when drawing $N$ times from a Bernoulli distribution with success probability $p$. Equivalently, if $X_i$ are $\{0,1\}$-random variables such that $\mathrm{Pr}[X_i=1] = 1-p$ for $i = 1, \ldots, n$, then $$\begin{aligned}
F(K,N,p) = \mathrm{Pr}[ X \geq N-K+1],\end{aligned}$$ where $X = \sum_{i=1}^{n} X_i$. The Chernoff bound of Theorem \[thm:chernoff\] yields $$\begin{aligned}
F(K,N,p) \leq \exp \left(-N D(1-p+t||1-p) \right)\end{aligned}$$ for $t = p - \frac{K-1}{N}$, provided that $N-K+1 \geq (1-p)N$, *i.e.* $p \geq (K-1)/N$ or $\eta \geq (K-1)/N$. Taking $K = k-1, N = n+k-1$ and $p=\eta$, and injecting into Eq. , gives $$\begin{aligned}
1- I_{\eta}(k,n) & \leq \exp \left(-(n+k-1) D\left(1-\frac{k-2}{n+k-1}||1- \eta \right) \right)\\
& \leq \exp \left(-(n+k-1) D\left(\frac{k-2}{n+k-1}||\eta \right) \right),\end{aligned}$$ which holds provided that $\eta \geq (k-2)/(n+k-1)$. This proves the claim.
Energy cutoff
-------------
The resolution of the identity of Theorem \[thm:resol\] involves operators which are not trace-class, as well as coherent states with arbitrary large energy. The natural solution to get operators with finite norm is to replace the domain $\mathcal{D}$ by a cut-off versions $\mathcal{D}_\eta$ defined by $$\begin{aligned}
\mathcal{D}_\eta := \left\{ \Lambda \in M_{p,q} ({\mathbb{C}}) \: : \: \eta {\mathbbm{1}}_p - \Lambda \Lambda^\dagger \geq 0\right\},\end{aligned}$$ for $\eta \in [0,1[$. Note that $$\begin{aligned}
\lim_{\eta \to 1} \mathcal{D}_\eta = \mathcal{D}.\end{aligned}$$ The integration over $\mathcal{D}_\eta$ can then be performed by first integrating the measure $\d \mu_n(\Lambda)$ on the “polar variables”, and only later on the “radial” variables corresponding to the singular values of $\Lambda$.
For a fixed pair of squared singular values $(x,y)$, let $V_{x,y}$ be the set of matrices $\Lambda \in \mathcal{D}$ with squared singular values $(x,y)$, *i.e.*, $$\begin{aligned}
V_{x,y} := \left\{u \big[\begin{smallmatrix} \sqrt{x} & 0 \\ 0 & \sqrt{y}\end{smallmatrix}\big] v^\dagger \: : \: u, v \in U(2)Ê\right\}.\end{aligned}$$ We further define the operator $P_{x,y}$ corresponding to the integral of $|\Lambda,n\rangle \langle \Lambda, n|$ over $V_{x,y}$: $$\begin{aligned}
P_{x,y} := \int_{V_{x,y}} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_{x,y}(\Lambda) \geq 0 \label{eqn:Pxy}\end{aligned}$$ where $\mathrm{d}\mu_{x,y} (\Lambda)$ is the Haar measure on $V_{x,y}$ and the normalization is chosen so that ${\mathrm{tr}}\, P_{x,y} = 1$.
We have the following equivalent version of the resolution of the identity of Theorem \[thm:resol\].
\[thm:resol2\] For $n \geq 4$, it holds that: $$\begin{aligned}
\int_{0}^1 \int_0^1 q(x,y) P_{x,y} \d x \d y = \mathbbm{1}_{F_{2,2,n}^{U(n)}},\end{aligned}$$ where the distribution $q(x,y)$ is given by $$\begin{aligned}
q(x,y) := \frac{(n-1)(n-2)^2(n-3) (x-y)^2}{2(1-x)^4 (1-y)^4}. \label{eqn:Qxy}\end{aligned}$$
We wish to integrate $|\Lambda, n \rangle \langle \Lambda, n| \d\mu_n(\Lambda)$ over the “polar” variables. For this, we perform the singular value decomposition of $\Lambda$, which reads $\Lambda = u \Sigma v^\dagger$, where $u, v \in U(2)$ and $\Sigma = \mathrm{diag}(\sigma_1, \sigma_2)$, with $\sigma_1, \sigma_2 \in [0,1[$.
The Jacobian for the singular value decomposition is [@mui82]: $$\begin{aligned}
\mathrm{d}\Lambda = (\sigma_1^2-\sigma_1^2)^2 \sigma_1 \sigma_2 (u^\dagger \mathrm{d} u) \mathrm{d} \Sigma (v^\dagger \mathrm{d} v).\end{aligned}$$
Exploiting this Jacobian and performing the change of variables $x = \sigma_1^2$, $y=\sigma_2^2$, one obtains that the resolution of the identity of Theorem \[thm:resol\] can be written: $$\begin{aligned}
C \int_0^1 \d x \int_0^1 \d y \frac{(x-y)^2}{(1-x)^4(1-y)^4} P_{x,y} = \mathbbm{1}_{F_{2,2,n}^{U(n)}},\end{aligned}$$ for the appropriate constant $C$. Here, we have used that $\det(\mathbbm{1}_2 - \Lambda \Lambda^\dagger)^4 = (1-x)^4 (1-y)^4$ for any $\Lambda \in V_{x,y}$.
The constant $C$ can be determined by considering the overlap between ${\mathbbm{1}}_{F_{2,2,n}^{U(n)}}$ and the vacuum state: $$\begin{aligned}
1 & = \langle 0 | {\mathbbm{1}}_{F_{2,2,n}^{U(n)}}| 0\rangle\\
&= C \int_0^1 \d x \int_0^1 \d y \int \frac{(x-y)^2}{(1-x)^4(1-y)^4} \big \langle 0 \big|u \big[\begin{smallmatrix} \sqrt{x} & 0 \\ 0 & \sqrt{y}\end{smallmatrix}\big] v^\dagger, n\big\rangle \big\langle u \big[\begin{smallmatrix} \sqrt{x} & 0 \\ 0 & \sqrt{y}\end{smallmatrix}\big] v^\dagger, n\big|0 \big\rangle \d u \d v\\
&= C \int_0^1 \d x \int_0^1 \d y\frac{(x-y)^2}{(1-x)^4(1-y)^4}(1-x)^n (1-y)^n \int \d u \d v\\
&= C \int_0^1 \d x \int_0^1 \d y\frac{(x-y)^2}{(1-x)^4(1-y)^4}(1-x)^n (1-y)^n\\
&= C \frac{2}{(n-1)(n-2)^2(n-3)},
\end{aligned}$$ where we used that $ \big \langle 0 \big|u \big[\begin{smallmatrix} \sqrt{x} & 0 \\ 0 & \sqrt{y}\end{smallmatrix}\big] v^\dagger, n\big\rangle = (1-x)^{n/2} (1-y)^{n/2}$ for any $u, v\in U(2)$ and that the measures $\d u$ and $\d v$ are normalized.
Let $K \geq 0$ be an integer. We define $V_{=K}$ as the subspace of $F_{2,2,n}^{U(n)}$ spanned by vectors with $K$ pairs of excitations: $$\begin{aligned}
V_K := \mathrm{Span}\{Z_{1,1}^i Z_{1,2}^j Z_{2,1}^k Z_{2,2}^\ell |0\rangle \: : \: i + j + k+\ell = K; i, j,k, \ell \in {\mathbb{N}}\},\end{aligned}$$ and the projector $\Pi_{=K}$ to be the orthogonal projector onto $V_{=K}$. Physically, this is the subspace of the Fock space restricted to states containing $2K$ photons in total in the $4n$ optical modes.
Moreover, let us denote by $a_k^n := \tbinom{n+k-1}{k}$ the number of configurations of $k$ particles in $n$ modes.
\[lem:Piq\] For $K \in {\mathbb{N}}$ and $x, y \in [0,1[$, we have $$\begin{aligned}
{\mathrm{tr}}\left[\Pi_{=K} P_{x,y} \right] = \sum_{k_1+k_2=K} a_{k_1}^n a_{k_2}^n (1-x)^n (1-y)^n x^{k_1} y^{k_2}.\end{aligned}$$
The total photon number distribution of a state $|\Lambda, n\rangle$ is invariant under local unitaries $u, v \in U(2)$ applied on the creation operators of $F_A$ or $F_B$. This means that this distribution only depends on the squared singular values of the matrix $\Lambda$. In particular, denoting by $|(\sqrt{x},\sqrt{y}),n\rangle$ the coherent state corresponding to the matrix $\mathrm{diag}(\sqrt{x}, \sqrt{y})$, we obtain: $$\begin{aligned}
{\mathrm{tr}}\left[\Pi_{=K} P_{x,y} \right] = \langle (\sqrt{x},\sqrt{y}),n | \Pi_{=K} | (\sqrt{x},\sqrt{y}),n\rangle.\end{aligned}$$ Since this coherent state is given by $$\begin{aligned}
| (\sqrt{x},\sqrt{y}),n\rangle := (1-x)^{n/2} (1-y)^{n/2} \exp(\sqrt{x} Z_{11}) \exp(\sqrt{y} Z_{22}),\end{aligned}$$ it implies that $$\begin{aligned}
{\mathrm{tr}}\left[\Pi_{=K} P_{x,y} \right] =\sum_{k_1+k_2=K} a_{k_1}^n a_{k_2}^n (1-x)^n (1-y)^n x^{k_1} y^{k_2}.\end{aligned}$$
Let us define the operator $\overline{P}_{\eta}$ as $$\begin{aligned}
\overline{P}_{\eta} := \int_{\mathcal{D} \setminus \mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda).\end{aligned}$$
\[eqn:crucial-step\] For $n \geq 38$, $K \in {\mathbb{N}}$ and $\eta \in [0,1[$ such that $K \leq \frac{\eta}{1-\eta}(n-5) $, it holds that $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) \leq 2 N^4 (1+\alpha)^7 \exp\left(-N D\left(\frac{\alpha}{\alpha+1} \big\|\eta \right) \right),\end{aligned}$$ where $N = n-5$ and $\alpha := K/N$.
For any non negative distribution $f(x,y) \geq 0$ symmetric in $x$ and $y$, *i.e.* such that $f(x,y) = f(y,x)$, it holds that $$\begin{aligned}
\int_{\overline{\mathcal{E}}_\eta} f(x,y) \d x \d y &\leq 2\int_{\eta}^1 \d x \int_0^1 \d y f(x,y).\end{aligned}$$ Since $q(x,y) {\mathrm{tr}}\left[P_{=K} \Pi_{x,y} \right]$ is such a distribution, it holds that $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta)\leq 2\int_{\eta}^1 \d x \int_0^1 \d y q(x,y) \, {\mathrm{tr}}\left[P_{=K} \Pi_{x,y} \right].\end{aligned}$$ Lemma \[lem:Piq\] then yields $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) &\leq 2 \sum_{k_1+k_2=K} a_{k_1}^n a_{k_2}^n \int_{\eta}^1 \d x (1-x)^n x^{k_1} \int_0^1 \d y q(x,y) (1-y)^n y^{k_2}\\
&\leq (n-1)(n-2)^2(n-3) \sum_{k_1+k_2=K} a_{k_1}^n a_{k_2}^n \int_{\eta}^1 \d x (1-x)^{n-4} x^{k_1} \int_0^1 \d y (x-y)^2 (1-y)^{n-4} y^{k_2}\\
&\leq (n-1)(n-2)^2(n-3) \sum_{k_1+k_2=K} a_{k_1}^n a_{k_2}^n \int_{\eta}^1 \d x (1-x)^{n-4} x^{k_1} \int_0^1 \d y (1-y)^{n-4} y^{k_2}\end{aligned}$$ where we used the trivial bound $ (x-y)^2 \leq 1$ for $0 \leq x,y \leq 1$ in the last inequality.
The normalization of the Beta function reads $$\begin{aligned}
\int_0^1 (1-y)^{n} y^k \d y =\frac{k! n!}{(n+k+1)!},\end{aligned}$$ which gives $$\begin{aligned}
\label{eqn:interm}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) &\leq (n-2) \sum_{k_1+k_2=K} (n+k_2-1)(n+k_2-2) \int_{\eta}^1 a_{k_1}^n (1-x)^{n-4} x^{k_1} \d x\end{aligned}$$ Lemma \[lem:tail-bound\] allows us to bound the integral: $$\begin{aligned}
\int_{\eta}^1 a_{k_1}^{n-4} (1-x)^{n-4} x^{k_1} \d x \leq\exp\left(-(n+k_1-5) D\left(\frac{n-3}{n+k_1-5} \big\| 1-\eta \right) \right),\end{aligned}$$ provided that $1-\eta \leq (n-3)/(n+k_1-5)$.
If $\frac{n-5}{n+K-5} \geq 1-\eta$, this term can be bounded uniformly as $$\begin{aligned}
\int_{\eta}^1 a_{k_1}^{n-4} (1-x)^{n-4} x^{k_1} \d x \leq\exp\left(-N D\left(\frac{N}{N+K} \big\| 1-\eta \right) \right),\end{aligned}$$ where we defined $N := n-5$. Injecting this in Eq. , we obtain $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) &\leq (N+3) \sum_{k_1+k_2=K} (N+k_2+4)(N+k_2+3) \frac{(N+k_1+4)! N!}{(N+k_1)! (N+4)!} \exp\left(-N D\left(\frac{N}{N+K} \big\| 1-\eta\right) \right) \nonumber\\
&\leq \frac{(K+1) (N+K+4)^6}{(N+1)^3} \exp\left(-N D\left(\frac{N}{N+K} \big\| 1-\eta \right) \right) \label{eqn:bound33}\end{aligned}$$ Imposing in addition that $N\geq 4$, *i.e.* $n\geq 9$, so that $N+K+4 \leq 2(N+K)$, one finally obtains the bound: $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) \leq 64 \frac{(N+K)^7}{N^3} \exp\left(-N D\left(\frac{N}{N+K} \big\| 1-\eta \right) \right).\end{aligned}$$ One can get a better bound by choosing $N \geq 33$, *i.e.* $n\geq 38$: in that case, one can check that for any $K \geq 0$, it holds that $$\begin{aligned}
\left( 1 + \frac{4}{N+K}\right)^6 \leq 2,\end{aligned}$$ which gives $(N+K+4)^6 \leq 2(N+K)^6$. Injecting this into Eq. yields $$\begin{aligned}
{\mathrm{tr}}(\Pi_{=K} \overline{P}_\eta) \leq 2 \frac{(N+K)^7}{N^3} \exp\left(-N D\left(\frac{N}{N+K} \big\| 1-\eta \right) \right).\end{aligned}$$
\[lem:proj\] For any nonnnegative operator $A \geq 0$ and projector $\Pi$ with $\mathrm{rank}(\Pi) < \infty$, it holds that: $$\Pi A \Pi \leq {\mathrm{tr}}[\Pi A ] \Pi.$$
The support of $\Pi A \Pi$ is contained in that of ${\mathrm{tr}}[\Pi A ] \Pi$. Since both operators are positive semi-definite, the only thing we need to prove is that for any $\lambda \in \mathrm{spec}(\Pi A \Pi)$, it holds that $$\begin{aligned}
\lambda \leq {\mathrm{tr}}[\Pi A]\end{aligned}$$ since all the nonzero eigenvalues of ${\mathrm{tr}}[\Pi A ] \Pi$ are equal to ${\mathrm{tr}}[\Pi A ]$. The sum of the eingenvalues of an operator is equal to its trace, which gives $$\begin{aligned}
\sum_{\lambda \in \mathrm{spec}(\Pi A \Pi)}Ê\lambda = {\mathrm{tr}}(\Pi A \Pi).\end{aligned}$$ Moreover, since all these eigenvalues are nonnegative, we have that $\lambda_{\max} (\Pi A \Pi) \leq \sum_{\lambda \in \mathrm{spec}(\Pi A \Pi)}Ê\lambda$, which concludes the proof.
Finite energy version of de Finetti theorem {#sec:finite}
===========================================
In this section, we establish a *de Finetti reduction*, similar to the one obtained in [@CKR09] in the case of permutation invariance. Such a reduction uses as a main tool as statement analogous to the resolution of the identity $$\begin{aligned}
\mathbbm{1}_{\mathrm{Sym}} \leq C_{n,d} \int \left(|\phi\rangle \langle \phi|\right)^{\otimes n} \d \mu(\phi)\end{aligned}$$ where $C(n,d)$ is a polynomial in $n$ provided the local dimension $d$ is finite.
In the case of continuous-variable protocols, the local dimension is infinite and we need to find a better reduction. This is indeed possible provided we have bounds on the maximum energy (or total number of photons) of the states under consideration.
For $\eta \in [0,1[$, define the sets $\mathcal{E}_\eta = [0, \eta] \times [0,\eta]$ and $\overline{\mathcal{E}}_\eta = [0,1[^2 \setminus \mathcal{E}_\eta$.
We introduce the following positive operators $$\begin{aligned}
P_\eta &:= \int_{\mathcal{E}_\eta} q(x,y) P_{x,y} \d x \d y = \int_{\mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda), \label{eqn:Peta} \\
\overline{P}_{\eta} &:= \int_{\overline{\mathcal{E}}_\eta} q(x,y) P_{x,y} \d x \d y = \int_{\mathcal{D} \setminus \mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda),\end{aligned}$$ where the equalities follow from the fact that one can integrate over $\mathcal{D}_\eta$ by first integrating over $V_{x,y}$ and then over $\mathcal{E}_\eta$. We recall that the operator $P_{xy}$ is defined in Eq. and that the distribution $q(x,y)$ is defined in Eq. . The resolution of the identity over $F_{2,2,n}^{U(n)}$ (Theorem \[thm:resol\]) immediately implies that $$\begin{aligned}
P_\eta + \overline{P}_\eta = {\mathbbm{1}}_{F_{2,2,n}^{U(n)}}.\end{aligned}$$
Let $K \geq 0$ be an integer. We recall that $V_{=K}$ is the subspace of $F_{2,2,n}^{U(n)}$ spanned by vectors with $K$ pairs of excitations: $$\begin{aligned}
V_K := \mathrm{Span}\{Z_{1,1}^i Z_{1,2}^j Z_{2,1}^k Z_{2,2}^\ell |0\rangle \: : \: i + j + k+\ell = K; i, j,k, \ell \in {\mathbb{N}}\}.\end{aligned}$$ The subspace $V_{\leq K}$ is defined as $V_{\leq K} := \bigoplus_{k=0}^K V_{=k}$. The projector $\Pi_{=K}$ is the orthogonal projector onto $V_{=K}$ and the projector $\Pi_{\leq K}$ is defined as $$\begin{aligned}
\Pi_{\leq K} : = \sum_{k=0}^K \Pi_{=k}.\end{aligned}$$
\[thm:finite-version\] For $n\geq 5$ and $\eta \in [0,1[$, if $K \leq\frac{\eta}{1-\eta} (n-5) $, then the following operator inequality holds $$\begin{aligned}
\int_{\mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n | \d \mu_n(\Lambda)\geq (1-{\varepsilon}) \Pi_{\leq K}\end{aligned}$$ with $$\begin{aligned}
{\varepsilon}:= 2 N^4 (1+\alpha)^7 \exp\left(-N D\left(\frac{\alpha}{\alpha+1} \big\| \eta \right) \right).\end{aligned}$$ for $\alpha = K/N$ and $N=n-5$.
In particular, choosing $K$ such that $\alpha = \frac{1+\eta}{1-\eta} = \frac{K}{N}$ and using Pinsker’s inequality (Lemma \[lem:pinsker\]) yields $$\begin{aligned}
{\varepsilon}\leq \frac{2 (N+K)^7}{N^3}\exp\left(- \frac{2N^3}{(N+K)^2 \ln 2} \right).\end{aligned}$$
The resolution of the identity reads $$\begin{aligned}
\int_{\overline{\mathcal{E}}_\eta} P_{x,y}q(x,y) \d x \d y+\int_{{\mathcal{E}}_\eta} P_{x,y}q(x,y) \d x \d y = {\mathbbm{1}}_{F_{2,2,n}^{U(n)}} = \sum_{k=0}^\infty\Pi_{=k} .\end{aligned}$$ For all $k \leq K$, the projector $\Pi_{=k}$ can be written as: $$\begin{aligned}
\Pi_{=k} = \int_{\overline{\mathcal{E}}_\eta} \Pi_{=k}P_{x,y}\Pi_{=k} q(x,y) \d x \d y+\int_{{\mathcal{E}}_\eta} \Pi_{=k} P_{x,y} \Pi_{=k} q(x,y) \d x \d y.\end{aligned}$$ In particular, since $k \leq K \leq \frac{\eta}{1-\eta} (n-5)$, we have $$\begin{aligned}
\int_{{\mathcal{E}}_\eta} P_{x,y}q(x,y) \d x \d y &\geq \int_{{\mathcal{E}}_\eta} \Pi_{=k}P_{x,y} \Pi_{=k} q(x,y) \d x \d y \nonumber\\
&\geq \Pi_{=k} - \int_{\overline{\mathcal{E}}_\eta} \Pi_{=k}P_{x,y} \Pi_{=k} q(x,y) \d x \d y \nonumber \\
&\geq \Pi_{=k} - \int_{\overline{\mathcal{E}}_\eta} {\mathrm{tr}}[ \Pi_{=k}P_{x,y}] \Pi_{=k} q(x,y) \d x \d y \label{eqn640}\\
&\geq (1-{\varepsilon})\Pi_{=k} \label{eqn641}\end{aligned}$$ where we used Lemma \[lem:proj\] in Eq. and the upper bound resulting from Lemma \[eqn:crucial-step\]: $$\begin{aligned}
\int_{\overline{\mathcal{E}}_\eta} {\mathrm{tr}}\left[ \Pi_{=k}P_{x,y}\right] q(x,y) \d x \d y \leq {\varepsilon}\end{aligned}$$ in Eq. . It follows that: $$\begin{aligned}
\int_{{\mathcal{E}}_\eta} P_{x,y}q(x,y) \d x \d y \geq (1-{\varepsilon}) \Pi_{=k} \end{aligned}$$ for all $k \leq K$. This finally implies that $$\begin{aligned}
\int_{{\mathcal{E}}_\eta} P_{x,y}q(x,y) \d x \d y \geq(1-{\varepsilon}) \sum_{k \leq K} \Pi_{=k} = (1-{\varepsilon}) \Pi_{\leq K}.\end{aligned}$$
The crucial property of Theorem \[thm:finite-version\] that will be important for application is that the volume of $\mathcal{D}_\eta$ is finite, and scales as a low degree polynomial in $n$ and $K$.
\[thm:volume\] For $n \geq 38$, $K \geq n-5$ and $\eta = \frac{K-n+5}{K+n-5}$, it holds that $$\begin{aligned}
T(n,\eta) := {\mathrm{tr}}\int_{\mathcal{D}_\eta} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_n(\Lambda) \leq \frac{K^4}{100}.\end{aligned}$$
The volume of $\mathcal{D}_\eta$ is given by $$\begin{aligned}
{\mathrm{tr}}\int_{\mathcal{D}_\eta} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_n(\Lambda) &= \int_0^{\eta} \int_0^{\eta} q(x,y)\d x \d y \\
&= \frac{(n-1)(n-2)^2(n-3)\eta^4}{12(1-\eta)^4} \\
&\leq \frac{n^4 \eta^4}{12 (1-\eta)^4}\\
& \leq \frac{n^4(1+\eta)^4}{192(1-\eta)^4}\end{aligned}$$ where we used that $\eta \leq (1+\eta)/2$ in the last equation.
In particular, choosing $K$ such that $\eta = \frac{K-n+5}{K+n-5}$ gives $\frac{1+\eta}{1-\eta} = \frac{K}{n-5}$, and therefore $$\begin{aligned}
{\mathrm{tr}}\int_{\mathcal{D}_\eta} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_n(\Lambda) \leq \frac{n^4 K^4}{192(n-5)^4}.\end{aligned}$$
For $n \geq 38$, it holds that $\frac{1}{192} \left(\frac{n}{n-5}\right)^4 \leq \frac{1}{100}$, which finally gives $$\begin{aligned}
{\mathrm{tr}}\int_{\mathcal{D}_\eta} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_n(\Lambda) \leq \frac{K^4}{100}.\end{aligned}$$
For future reference, let us not that that under the same assumptions as in the theorem, the following inequality also holds: $$\begin{aligned}
T(n,\eta) +1 \leq \frac{K^4}{100}. \label{eqn:tighter}\end{aligned}$$ This inequality will later be useful to analyze Eq. at the end of Section \[sec:general-proof\] and obtain Eq. (5) in the main text.
In other words, the volume $T(n,\eta)$ of ${\mathcal{D}}_\eta$ is upper bounded by a polynomial of degree 4 in the number of modes (or equivalently in the total energy).
Let us define the function $N^* : [1,\infty[ \to {\mathbb{N}}$ such that $$\begin{aligned}
N^*(\alpha) = \max\left\{ 38, \min \left\{N \in {\mathbb{N}}\: : \: 2(1+\alpha)^7 N^4 \exp\left(- \frac{2N}{(1+\alpha)^2 \ln 2} \right) \leq \frac{1}{2} \right\}\right\}. \label{eqn:N*}\end{aligned}$$ For instance, $N^*(21) \approx 10^4$, $N^*(60) \approx 10^5$.
We obtain:
\[corol16\] For $K \geq n-5$, if $n \geq N^*\big( \frac{K}{n-5} \big)-5$ then, for $\eta^* = \frac{K-n+5}{K+n-5}$, it holds that $$\begin{aligned}
&\int_{\mathcal{D}_{\eta^*}} |\Lambda,n\rangle \langle \Lambda,n | \d \mu_n(\Lambda)\geq \frac{1}{2} \Pi_{\leq K} \\
&{\mathrm{tr}}\int_{\mathcal{D}_{\eta^*}} |\Lambda,n\rangle\langle \Lambda,n| \d \mu_n(\Lambda) \leq \frac{K^4}{100}.\end{aligned}$$
Security proof for a modified CV QKD protocol {#sec:general-proof}
=============================================
In this section, we recall some facts about security proofs for QKD protocols and explain how to obtain a secure protocol from an initial protocol ${\mathcal{E}}_0$ known to be secure against Gaussian collective attacks, by prepending an energy test and adding an additional privacy amplification test. These various steps will then be detailed in the subsequent sections.
[**QKD protocols and their security**]{}.— A QKD protocol is a CP map from the infinite-dimensional Hilbert space $(\mathcal{H}_A\otimes \mathcal{H}_B)^{\otimes n}$, corresponding to the initially distributed entanglement, to the set of pairs $(S_A,S_B)$ of $\ell$-bit strings (Alice and Bob’s final keys, respectively) and $C$, a transcript of the classical communication. In order to assess the security of a given QKD protocol $\mathcal{E}$ in a composable framework, one compares it with an ideal protocol [@MKR09; @PR14]. The action of an ideal protocol $\mathcal{F}$ is defined by concatenating the protocol $\mathcal{E}$ with a map $\mathcal{S}$ taking $(S_A,S_B,C)$ as input and outputting the triplet $(S,S,C)$ where the string $S$ is a perfect secret key (uniformly distributed and unknown to Eve) with the same length as $S_A$, that is $\mathcal{F} = \mathcal{S}\circ\mathcal{E}$. Then, a protocol will be called *$\epsilon$-secure* if the advantage in distinguishing it from an ideal version is not larger than $\epsilon$. This advantage is quantified by (one half of) the diamond norm defined by $$||\mathcal{E} - \mathcal{F}||_\diamond := \sup_{\rho_{ABE} } \left\|(\mathcal{E}-\mathcal{F})\otimes \mathrm{id}_\mathcal{K} (\rho_{ABE})\right\|_1,$$ where the supremum is taken over density operators on $(\mathcal{H}_A\otimes \mathcal{H}_B)^{\otimes n} \otimes \mathcal{K}$ for any auxiliary system $\mathcal{K}$. The diamond norm is also known as the *completely bounded trace norm* and quantifies a notion of distinguishability for quantum maps [@wat16].
Our main technical result is a reduction of the security against general attacks to that against Gaussian collective attacks, for which security has already been proved in earlier work, for instance in [@Lev15]. Let us therefore suppose that our CV QKD protocol of interest, $\mathcal{E}_0$, is secure against Gaussian collective attacks. We will slightly modify it by prepending an initial test $\mathcal{T}$. More precisely, $\mathcal{T}$ is a CP map taking a state in a slightly larger Hilbert space, $(\mathcal{H}_A\otimes \mathcal{H}_B)^{\otimes (n+k)}$, applying a random unitary $u \in U(n+k)$ to it (corresponding to a network of beamsplitters and phaseshifters), measuring the last $k$ modes and comparing the measurement outcome to a threshold fixed in advance. The test succeeds if the measurement outcome (related to the energy) is small, meaning that the global state is compatible with a state containing only a low number of photons per mode. Such a state is well-described in a low dimensional Hilbert space, as we will discuss in Section \[sec:test\]. Depending on the outcome of the test, either the protocol aborts, or one applies the original protocol $\mathcal{E}_0$ on the $n$ remaining modes.
For the test to be practical, it is important that the legitimate parties do not have to physically implement the transformation $u \in U(n+k)$. Rather, they can both measure their $n+k$ modes with heterodyne detection, perform a random rotation of their respective classical vector in ${\mathbb{R}}^{2(n+k)}$ according to $u \in U(n+k) \cong O(2(n+k)) \cap Sp(2(n+k))$.
In this paper, we assume that this symmetrization step is performed, as it is anyway required for the security proof of the protocol against collective attacks [@Lev15]. We believe, however, that this step might not be required for establishing the security of the protocol and leave it as an important open question for future work. In particular, recent proof techniques in discrete-variable QKD have shown that the permutation need not be applied in practice [@TLG12].
In section \[sec:generalization\], we will prove a de Finetti reduction that allows to upper bound the diamond distance between two quantum channels, provided that they display the right invariance under the action of the unitary group $U(n)$ and that the input states have a maximum number of photons. We address this second issue by introducing another CP map $\mathcal{P}$ which projects a state acting on $F_{1,1,n} = (\mathcal{H}_A\otimes \mathcal{H}_B)^{\otimes n}$ onto a low-dimensional Hilbert space $F_{1,1,n}^{\leq K} $ with less than $K$ photons overall in the $2n$ modes shared by Alice and Bob. Here, the value of $K$ scales linearly with $n$.
Let us denote by ${\mathcal{E}}_0$ a CV QKD proven ${{\varepsilon}}$-secure against Gaussian collective attacks, for instance as in [@Lev15]. This means that (see Section \[sec:collective\] for details) $$\begin{aligned}
\|(({\mathcal{E}}_0 - {\mathcal{F}}_0)\otimes {\mathbbm{1}}) (|\Lambda,n\rangle \langle \Lambda, n|)\|_1 \leq {{\varepsilon}},\end{aligned}$$ for any generalized coherent state $|\Lambda,n\rangle$. Here ${\mathcal{F}}_0 := {\mathcal{S}}\circ {\mathcal{E}}_0$ and ${\mathcal{S}}$ is a map that replaces the output key of ${\mathcal{E}}_0$ by an independent and uniformly distributed string of length $\ell$ when ${\mathcal{E}}_0$ did not abort, and does nothing otherwise.
Here ${\mathcal{E}}_0$ maps an arbitrary density operator $\rho_{AB} \in \mathfrak{S}(F_{1,1,n})$ to a state $\rho_{S_A, S_B, C}$ where the registers are all classical and store respectively Alice’s final key, Bob’s final key and a transcript of the classical communication.
Let us define the following maps: $$\begin{aligned}
{\mathcal{T}}&: {\mathcal{B}}(F_{1,1,n+k}) \to {\mathcal{B}}(F_{1,1,n}) \otimes \{\mathrm{passes} / \mathrm{aborts}\},\\
{\mathcal{P}}&: {\mathcal{B}}(F_{1,1,n}) \to {\mathcal{B}}(F_{1,1,n}^{\leq K}),\\
{\mathcal{R}}&: \{0,1\}^{\ell} \times \{0,1\}^{\ell} \to \{0,1\}^{\ell'} \times \{0,1\}^{\ell'},\end{aligned}$$ where
- ${\mathcal{T}}(k, d_A, d_B)$ takes as input an arbitrary state $\rho_{AB}$ on $F_{1,1,n+k}$, maps it to $V_u \rho_{AB} V_u^{\dagger}$ where the unitary $u$ is chosen from the Haar measure on $U(n+k)$, measures the last $k$ modes for $A$ and $B$ with heterodyne detection and check whether the measurement outputs pass the test if the $k$ outcomes $\alpha_1, \cdots, \alpha_k$ of Alice and $\beta_1, \cdots, \beta_k$ of Bob satisfy $$\begin{aligned}
\sum_{i=1}^k |\alpha_i|^2 \leq k d_A \quad \text{and} \quad \sum_{i=1}^k |\beta_i|^2 \leq k d_B.\end{aligned}$$ If they pass the test, the map returns the state on the first $n$ modes (that were not measured) as well as the flag “passes”. Otherwise, it returns the vacuum state and the flag “aborts”.
- ${\mathcal{P}}$ is the projector onto the finite-dimensional subspace $F_{1,1,n}^{\leq K}$ (corresponding to states with at most $K$ photons in the $2n$ modes): it maps any state $\rho \in {\mathcal{B}}(F_{1,1,n})$ to $\Pi_{\leq K} \rho \Pi_{\leq K} \in {\mathcal{B}}(F_{1,1,n}^{\leq K})$. This trace non-increasing map is introduced as a technical tool for the security analysis but need not be implemented in practice. It simply ensures that the states that are fed to the original QKD protocol ${\mathcal{E}}_0$ live in a finite-dimensional subspace. In the text, we will alternatively denote this projection by ${\mathcal{P}}^{\leq K}$ or ${\mathcal{P}}(n,K)$, depending on which parameters we wish to make explicit.
- ${\mathcal{R}}$ takes two $\ell$-bit strings as input and returns $\ell'$-bit strings (for $\ell' < \ell$).
We finally define our CV QKD protocol ${\mathcal{E}}$ as $$\begin{aligned}
{\mathcal{E}}= {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{T}}\end{aligned}$$ and the ideal protocol as ${\mathcal{F}}= {\mathcal{S}}\circ {\mathcal{E}}$. Abusing notation slightly, the map ${\mathcal{S}}$ now acts on strings of length $\ell'$ instead of $\ell$.
\[lem:sec-red\] Let $\overline{{\mathcal{E}}}$ be the protocol ${\mathcal{R}}\circ {\mathcal{E}}_0$ where the inputs are restricted to the finite-dimensional subspace ${\mathcal{B}}(F_{1,1,n}^{\leq K})$, and $\overline{{\mathcal{F}}} = {\mathcal{S}}\circ \overline{{\mathcal{E}}}$. Then the security of $\overline{{\mathcal{E}}}$ implies the security of ${\mathcal{E}}$: $$\begin{aligned}
\label{eqn:sec-red}
||\mathcal{E} - \mathcal{F}||_\diamond &\leq ||\overline{\mathcal{E}} - \overline{\mathcal{F}}||_\diamond + 2 || ({\mathbbm{1}}- \mathcal{P}) \circ \mathcal{T}||_\diamond,\end{aligned}$$ provided that the quantity $|| ({\mathbbm{1}}- \mathcal{P}) \circ \mathcal{T}||_\diamond$ can be made arbitrarily small.
We define (virtual) protocols $\tilde{\mathcal{E}}:= {\mathcal{R}}\circ \mathcal{E}_0 \circ \mathcal{P} $ and $\tilde{\mathcal{F}}:= \mathcal{S} \circ \tilde{\mathcal{E}}$. The security of the protocol $\mathcal{E}$ is then a consequence of the following derivation: $$\begin{aligned}
||\mathcal{E} - \mathcal{F}||_\diamond &\leq ||\tilde{\mathcal{E}}\circ {\mathcal{T}}- \tilde{\mathcal{F}} \circ {\mathcal{T}}||_\diamond + ||\mathcal{E} - \tilde{\mathcal{E}} \circ {\mathcal{T}}||_\diamond+ ||\mathcal{F} - \tilde{\mathcal{F}} \circ {\mathcal{T}}||_\diamond \nonumber \\
&\leq ||(\tilde{\mathcal{E}} - \tilde{\mathcal{F}}) \circ {\mathcal{T}}||_\diamond + ||{\mathcal{R}}\circ \mathcal{E}_0 \circ (\mathrm{id}- \mathcal{P}) \circ \mathcal{T}||_\diamond +||{\mathcal{S}}\circ {\mathcal{R}}\circ \mathcal{E}_0 \circ (\mathrm{id}- \mathcal{P}) \circ \mathcal{T}||_\diamond \nonumber\\
&\leq ||\tilde{\mathcal{E}} - \tilde{\mathcal{F}}||_\diamond + 2 || ({\mathbbm{1}}- \mathcal{P}) \circ \mathcal{T}||_\diamond ,\end{aligned}$$ where we used the triangle inequality and the fact that the CP maps ${\mathcal{T}}$, ${\mathcal{R}}\circ \mathcal{E}_0$ and $\mathcal{S}$ cannot increase the diamond norm.
Since $\overline{E} \circ {\mathcal{P}}= \tilde{E}$ and ${\mathcal{P}}$ is trace non-increasing, we finally obtain that $$\begin{aligned}
||\mathcal{E} - \mathcal{F}||_\diamond &\leq ||\overline{\mathcal{E}} - \overline{\mathcal{F}}||_\diamond + 2 || ({\mathbbm{1}}- \mathcal{P}) \circ \mathcal{T}||_\diamond.\end{aligned}$$
Bounding the two terms in the right hand side of Eq. is done with the two following theorems, which will be proven in Sections \[sec:collective\] and \[sec:test\], respectively.
\[thm:diamond-protocol\] With the previous notations, if ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks, then $$\begin{aligned}
||\overline{\mathcal{E}} - \overline{\mathcal{F}}||_\diamond \leq 2 T(n,\eta) {\varepsilon}\end{aligned}$$ where $T(n,\eta) =(n-1)(n-2)^2(n-3) \frac{\eta^4}{12(1-\eta)^4}$ and $\overline{{\mathcal{E}}} = {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{P}}^{\leq K}$.
\[thm:test\] For integers $n,k \geq 1$, and $d_A, d_B >0$, define $K = n(d'_A + d'_B)$ for $d'_{A/B} = d_{A/B} g(n,k,{\varepsilon}/4)$ for the function $g$ defined in Eq. . Then $$\begin{aligned}
\big\| \big({\mathbbm{1}}- {\mathcal{P}}(n,K)\big) \circ {\mathcal{T}}(k, d_A, d_B)\big\|_{\diamond} \leq {\varepsilon}.\end{aligned}$$
Putting everything together yields our main result.
\[thm:main\] If the protocol ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks, then the protocol ${\mathcal{E}}= {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{P}}$ is ${\varepsilon}'$-secure against general attacks with $$\begin{aligned}
{\varepsilon}' \leq 2 T(n,\eta) {\varepsilon}+ {\varepsilon}.\end{aligned}$$
Putting everything together, we show that if ${\mathcal{E}}_0$ is covariant under the action of the unitary group and ${\varepsilon}$-secure against Gaussian collective attacks, then the protocol ${\mathcal{E}}= {\mathcal{R}}\circ {\mathcal{E}}_0 \circ {\mathcal{T}}$ is ${\varepsilon}'$-secure against general attacks, with $$\begin{aligned}
{\varepsilon}' = 2{\varepsilon}( T(n, \eta)+1) \label{eqn:final-result}\end{aligned}$$ for $T(n,\eta) \leq \frac{1}{12} \left(\frac{\eta n}{1-\eta}\right)^4$, $\eta = \frac{K-n+5}{K+n-5}$ and $K = n(d_A + d_B)\left(1 + 2 \sqrt{\frac{\ln (8/{\varepsilon})}{2n}} + \frac{\ln (8/{\varepsilon})}{n}\right)\left(1-2{\sqrt{\frac{\ln (8/{\varepsilon})}{2k}}}\right)^{-1}$. The first term in Eq. results from the de Finetti reduction and the second term results for the energy test failure probability.
In particular, for $n \geq 38$ and $K \geq n-5$, we obtain the bound $T(n,\eta) +1 \leq \frac{K^4}{100}$ from Eq. . This yields ${\varepsilon}' \leq \frac{K^4}{50} {\varepsilon}$, which corresponds to Eq. (5) in the main text.
Generalization of the postselection technique of Ref. [@CKR09] {#sec:generalization}
==============================================================
The goal of this section is to prove the following theorem (Theorem 2 in the main text).
\[thm:postselection\] Let $\Delta: \mathrm{End}(F_{1,1,n}^{\leq K}) \to \mathrm{End}(\mathcal{H}')$ such that for all $u \in U(n)$, there exists a CPTP map $\mathcal{K}_u: \mathrm{End}(\mathcal{H}') \to \mathrm{End}(\mathcal{H}')$ such that $\Delta \circ u = \mathcal{K}_u: \circ \Delta$, then $$\begin{aligned}
\|\Delta\|_\diamond \leq 2 T(n,\eta) \|(\Delta \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1,\end{aligned}$$ for $\eta = \frac{K-n+5}{K+n-5}$, provided that $n \geq N^*(K/(n-5))$.
One way to make sure that the input of the map is indeed restricted to states with less than $K$ photons is to replace $\Delta$ by $\Delta \circ {\mathcal{P}}^{\leq K}$.
In the following, for conciseness, we will denote by $\mathcal{H}$ the symmetric subspace: $$\begin{aligned}
{\mathcal{H}}:= F_{2,2,n}^{U(n)}.\end{aligned}$$
Let $\tau^\eta_{\mathcal{H}}$ be the normalized state corresponding to the projector $P_\eta$ defined in Eq. : $$\begin{aligned}
\tau^\eta_{\mathcal{H}} = T(n,\eta)^{-1} \int_{ \mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda)\end{aligned}$$ where $$\begin{aligned}
T(n,\eta) := {\mathrm{tr}}(P_\eta) = \frac{(n-1)(n-2)^2(n-3)\eta^4}{12(1-\eta)^4}. \label{eqn:defT}\end{aligned}$$
Consider an orthonormal basis $\left\{Ê|\nu_i\rangle \right\}$ of $F_{2,2,n}^{U(n)}$ and define the non normalizable operator $$\begin{aligned}
|\Phi\rangle_{{\mathcal{H}}{\mathcal{N}}} := \sum_{i} |\nu_i\rangle_{{\mathcal{H}}} |\nu_i\rangle_{{\mathcal{N}}}.\end{aligned}$$
A conjecture for an explicit such orthonormal basis was given in [@lev16], but we do not need to have such an explicit basis for our present purpose.
Let us further define the state $|\Phi^\eta\rangle \in F_{2,2,n}^{U(n)} \otimes F_{2,2,n}^{U(n)}$: $$\begin{aligned}
|\Phi^\eta\rangle = \left(\sqrt{ \tau^\eta} \otimes {\mathbbm{1}}\right) |\Phi\rangle.\end{aligned}$$ It is well-known that $|\Phi^\eta\rangle$ is a purification of $\tau_{\mathcal{H}}^{\eta}$: $$\begin{aligned}
{\mathrm{tr}}_{\mathcal{N}} \left(|\Phi^\eta\rangle\langle \Phi^\eta|_{\mathcal{H}\mathcal{N}} \right)=\tau_{\mathcal{H}}^{\eta}.\end{aligned}$$
Recall that $F_{2,2,n}^{U(n), \leq K}$ denotes the finite-dimensional subspace of $F_{2,2,n}^{U(n)}$ with less than $K$ excitations.
\[lem:measurement\] Let $\rho$ be an arbitrary density operator on $F_{2,2,n}^{U(n), \leq K}$. Then there exists a binary measurement $\mathcal{M} = \{M_{\mathcal{N}}, {\mathbbm{1}}_{\mathcal{N}}-M_{\mathcal{N}}\}$ on ${\mathcal{N}}$ applied to $|\Phi^\eta\rangle \in {\mathcal{H}}\otimes {\mathcal{N}}$ that successfully prepares the state $\rho$ with probability at least $\frac{1}{2T(n,\eta)}$.
To avoid cluttering up the notations, let us write $\tau$ instead of $\tau^\eta_{{\mathcal{H}}}$. Recall that $\tau \geq p {\mathbbm{1}}_{F_{2,2,n}^{U(n), \leq K}}$ with $p = \frac{1}{2T(n,\eta)}$, as a consequence of Corollary \[corol16\].
Let us define the non negative operator $M := p \tau^{-1/2} \rho \tau^{-1/2}$. Since $p^{-1} \tau \geq {\mathbbm{1}}$ on the support of $\rho \leq {\mathbbm{1}}$, the operator $M$ satisfies $$\begin{aligned}
0 \leq M \leq {\mathbbm{1}}.\end{aligned}$$ Let us define the measurement $\mathcal{M} = \{ M, {\mathbbm{1}}-M\}$. Performing this measurement on state $|\Phi^\eta\rangle$ prepares the state $$\begin{aligned}
{\mathrm{tr}}_{{\mathcal{N}}} \left( (1 \otimes M^{1/2}) |\Phi^\eta\rangle \langle \Phi^\eta | (1 \otimes M^{1/2}) \right)\end{aligned}$$ with probability $ \langle \Phi^\eta | (1 \otimes M) |\Phi^\eta\rangle$. This state can be written: $$\begin{aligned}
{\mathrm{tr}}_{{\mathcal{N}}} \left( (1 \otimes M^{1/2}) |\Phi^\eta\rangle \langle \Phi^\eta | (1 \otimes M^{1/2})\right) &= {\mathrm{tr}}_{{\mathcal{N}}} \left( (1 \otimes M^{1/2})\left(\sqrt{ \tau} \otimes {\mathbbm{1}}\right) |\Phi\rangle \langle \Phi | \left(\sqrt{ \tau} \otimes {\mathbbm{1}}\right) (1 \otimes M^{1/2}) \right) \nonumber \\
&= {\mathrm{tr}}_{{\mathcal{N}}} \left( (\tau^{1/2} \otimes M^{1/2}) \sum_{i,j} |\nu_i \rangle \langle \nu_j| \otimes |\nu_i \rangle \langle \nu_j| (\tau^{1/2} \otimes M^{1/2}) \right) \nonumber \\
&= \sum_{i,j} \tau^{1/2} |\nu_i \rangle \langle \nu_j| \tau^{1/2} \langle \nu_j| M^{1/2} M^{1/2} |\nu_i \rangle \nonumber\\
&= \sum_{i,j} \tau^{1/2} |\nu_i \rangle \langle \nu_j| \tau^{1/2} \langle \nu_i| M^{1/2} M^{1/2} |\nu_j \rangle \label{eqn:inv} \\
&= \sum_{i,j} \tau^{1/2} |\nu_i \rangle \langle \nu_i| M^{1/2} M^{1/2} |\nu_j \rangle \langle \nu_j| \tau^{1/2} \nonumber \\
&= \tau^{1/2} M^{1/2} M^{1/2} \tau^{1/2} \nonumber \\
&= \tau^{1/2} p \tau^{-1/2} \rho \tau^{-1/2} \tau^{1/2} \nonumber \\
&= p \rho, \nonumber\end{aligned}$$ and it is obtained with probability $p$. In Eq. , we used that $M$ is symmetric, that is $\langle \lambda_i |M |\lambda_j\rangle = \langle \lambda_j |M |\lambda_i\rangle$.
\[lem:dim\] For $k\geq 0$ and $n\geq 4$, the dimensions of $V_{=K}$ and $V_{\leq K} = F_{2,2,n}^{U(n), \leq K}$ are given by $$\begin{aligned}
\mathrm{dim} \, V_{=K} = \tbinom{K+3}{3} \quad \text{and} \quad \mathrm{dim} \, V_{\leq K} = \tbinom{K+4}{4}.\end{aligned}$$
It was proven in [@lev16] that the vectors $(Z_{1,1})^i (Z_{1,2})^j (Z_{2,1})^k (Z_{2,2})^{\ell}$ are independent (provided than $n\geq 4$), which means that the dimension of $V_{=K}$ is the cardinality of the sets of quadruples $\{(i,j,k,\ell) \in {\mathbb{N}}^4 \: : \: i+j+k+\ell =K\}$. This number is $\tbinom{K+3}{3}$. More generally, the number of $t$-uples of nonnegative integers that sum to $K$ is $\tbinom{n+K-1}{n-1}$. Since the subspaces $V_{=K}$ are orthogonal, it follows that $\mathrm{dim} \, V_{\leq K} = \sum_{k=0}^K \mathrm{dim} \, V_{=k}$, which can be computed explicitly. Alternatively, one can see that the space $V_{\leq K}$ of quadruples $(i,j,k,\ell)$ summing to $K-m$ for some integer $m \leq K$ corresponds to the space of $5$-uples $(i,j,k,\ell, m)$ that sum to $K$.
\[lem:design\] For any $K$ and $n$ integers, there exists a finite subset $\mathcal{U} \subset U(n)$, such that for any state $\rho$ with support on $F_{1,1,n}^{\leq K}$, the subspace of $F_{1,1,n}$ restricted to states with less than $K$ photons, the following holds: $$\begin{aligned}
\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} V_u \rho V_u^\dagger = \int V_u \rho V_u^\dagger \d u,\end{aligned}$$ where $\d u$ is the normalized Haar measure on $U(n)$.
Note that by definition of the Haar measure, the state $ \int V_u \rho V_u^\dagger \d u$ is invariant under the application of any unitary $u' \in U(n)$: $V_{u'} \int V_u \rho V_u^\dagger \d u V_{u'}^\dagger = \int V_u \rho V_u^\dagger \d u$, which means that it has support on $F_{1,1,n}^{U(n), \leq K}$.
By linearity, it is sufficient to establish the lemma for pure states $|\psi\rangle \in F_{1,1,n}^{\leq K}$. Such a state can be written as $$\begin{aligned}
|\psi\rangle = \sum_{\substack{k_1, \ldots, k_n, \ell_1, \ldots \ell_n\\ \sum k_i + \ell_i \leq K}} \lambda_{k_1 \ldots k_n, \ell_1 \ldots \ell_n} \prod_{i=1}^n \left(a_i^\dagger\right)^{k_i} \left(b_i^\dagger\right)^{\ell_i} |0\rangle.\end{aligned}$$
Applying $V_u$ maps $a_i^\dagger$ to $\sum_{j=1}^n u_{i,j} a_j^\dagger$ and $b_i^\dagger$ to $\sum_{j=1}^n \overline{u}_{i,j} b_j^\dagger$. In other words, the function $f: u \mapsto V_u |\psi\rangle \langle \psi | V_u^\dagger$ is a polynomial of degree at most $K$ in $u$ and $\overline{u}$. Taking $\mathcal{U}$ to be a $K$-design of $U(n)$, we obtain that $$\begin{aligned}
\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} f(u) = \int f(u) \d u,\end{aligned}$$ which proves the result.
We recall the following theorem that was established in [@lev16].
\[theo:purification\] Any density operator $\rho \in \mathfrak{S}(F_{1,1,n})$ invariant under $U(n)$ admits a purification in $F_{2,2,n}^{U(n)}$.
\[lem:symmetrization\] It is sufficient to consider states $\rho_{{\mathcal{H}}\mathcal{N}} $ with support on $F_{2,2,n}^{U(n),\leq K}$ when computing the diamond norm of Theorem \[thm:postselection\].
Consider a state $\rho_{{\mathcal{H}}{\mathcal{N}}}$ with support on $F_{2,2,n}^{\leq K}$. Let $\mathcal{U}$ be a finite set of unitaries as promised by Lemma \[lem:design\]. Let $\{ |u\rangle_{\mathcal{C}}\}_{u \in \mathcal{U}}$ be an orthogonal basis for some classical register $\mathcal{C}$. The following sequence of equalities holds: $$\begin{aligned}
\|(\Delta \otimes {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}}\|_1 &= \frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} \|( \Delta \otimes {\mathbbm{1}}) (\rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}}) \|_1 \nonumber \\
&= \left\|\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ( \Delta \otimes {\mathbbm{1}}) (\rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}}) \right\|_1 \label{eqn00} \\
&= \left\|\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ( \mathcal{K}_u \circ \Delta \otimes {\mathbbm{1}}) (\rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}}) \right\|_1 \label{eqn01} \\
&= \left\|\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ( \Delta \circ u \otimes {\mathbbm{1}}) (\rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}}) \right\|_1 \label{eqn02} \\
&= \left\| ( \Delta \otimes {\mathbbm{1}}) \left(\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ((u \circ {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}})\right) \right\|_1 \nonumber \end{aligned}$$ where we used that the classical states $|u\rangle$ are all pairwise orthogonal in Eq. , that ${\mathcal{K}}_u$ is trace preserving in Eq. , that ${\mathcal{K}}_u \circ \Delta = \Delta \circ u$ in Eq. . Consider now the reduced state $\tilde{\rho}_{{\mathcal{H}}}$: $$\begin{aligned}
\tilde{\rho}_{{\mathcal{H}}} = {\mathrm{tr}}_{{\mathcal{N}}{\mathcal{C}}}\left(\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ((u \circ {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}})\right) =\frac{1}{|{\mathcal{U}}|} \sum_{u \in \mathcal{U}} V_u \rho_{{\mathcal{H}}} V_u^\dagger = \int V_u \rho_{{\mathcal{H}}} V_u^{\dagger} \d u\end{aligned}$$ where the last equality follows from Lemma \[lem:design\]. Theorem \[theo:purification\] now assures the existence of some purification $\tilde{\rho}_{{\mathcal{H}}{\mathcal{N}}}$ of $\tilde{\rho}_{{\mathcal{H}}}$ in $F_{2,2,n}^{U(n), \leq K}\cong {\mathcal{H}}\otimes {\mathcal{N}}$. In particular, there exists a CPTP map $g: \mathrm{End}({\mathcal{N}}) \to \mathrm{End}({\mathcal{N}}\otimes {\mathcal{C}})$ such that $$\begin{aligned}
\frac{1}{|\mathcal{U}|} \sum_{u \in \mathcal{U}} ((u \circ {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}} \otimes |u\rangle \langle u|_{\mathcal{C}}) = ({\mathbbm{1}}_{\mathcal{H}}\otimes g) \tilde{\rho}_{{\mathcal{H}}{\mathcal{N}}}.\end{aligned}$$ Since $g$ is trace preserving, it further implies that $$\begin{aligned}
\|(\Delta \otimes {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}}\|_1 =\|(\Delta \otimes {\mathbbm{1}}) ({\mathbbm{1}}_{\mathcal{H}}\otimes g) \overline{\rho}_{{\mathcal{H}}{\mathcal{N}}}\|_1 =\|(\Delta \otimes {\mathbbm{1}}) \tilde{\rho}_{{\mathcal{H}}{\mathcal{N}}}\|_1,\end{aligned}$$ which concludes the proof
We are now in position to prove Theorem \[thm:postselection\].
[thm:postselection]{} Let $\Delta: \mathrm{End}(F_{1,1,n}^{\leq K}) \to \mathrm{End}(\mathcal{H}')$ such that for all $u \in U(n)$, there exists a CPTP map $\mathcal{K}_u: \mathrm{End}(\mathcal{H}') \to \mathrm{End}(\mathcal{H}')$ such that $\Delta \circ u = \mathcal{K}_u: \circ \Delta$, then $$\begin{aligned}
\|\Delta\|_\diamond \leq 2 T(n,\eta) \|(\Delta \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1,\end{aligned}$$ for $\eta = \frac{K-n+5}{K+n-5}$, provided that $n \geq N^*(K/(n-5))$.
According to Lemma \[lem:symmetrization\], it is sufficient to prove the theorem for a state $\rho_{{\mathcal{H}}{\mathcal{N}}}$ on $F_{2,2,n}^{U(n), \leq K}$. Lemma \[lem:measurement\] guarantees the existence of a trace-non-increasing map $\mathcal{T}$ from a copy of $F_{2,2,n}^{U(n), \leq K}$ to ${\mathbb{C}}$ such that $$\begin{aligned}
\rho_{{\mathcal{H}}{\mathcal{N}}} = {2T(n,\eta)}({\mathbbm{1}}\otimes \mathcal{T}) (|\Phi^\eta\rangle \langle \Phi^\eta |).\end{aligned}$$ This gives $$\begin{aligned}
(\Delta \otimes {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}} = {2T(n,\eta)}(\Delta \otimes \mathcal{T}) (|\Phi^\eta\rangle \langle \Phi^\eta |)\end{aligned}$$ and finally that $$\begin{aligned}
\|(\Delta \otimes {\mathbbm{1}}) \rho_{{\mathcal{H}}{\mathcal{N}}}\|_1 ={2T(n,\eta)} \|(\Delta \otimes \mathcal{T}) (|\Phi^\eta\rangle \langle \Phi^\eta |)\|_1.\end{aligned}$$
Security against collective attacks provides a bound on $\| {\mathcal{R}}\circ \Delta \circ {\mathcal{P}}\|_{\diamond}$ {#sec:collective}
=======================================================================================================================
In order to exploit Theorem \[thm:postselection\], one needs an upper bound on $\|(({\mathcal{R}}\circ \Delta \circ {\mathcal{P}}^{\leq K})\otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1$. We will see that such a bound can be obtained if the protocol is known to be secure against Gaussian collective attacks. For this, we follow the same strategy as in [@CKR09]. Let us first recall the definition of being secure against Gaussian collective attacks.
The QKD protocol ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks if $$\begin{aligned}
\| (({\mathcal{E}}_0-{\mathcal{F}}_0)\otimes \mathrm{id})(|\Lambda,n\rangle \langle \Lambda,n|)\|_1\leq {\varepsilon}\label{eqn:sec-coll}\end{aligned}$$ for all $\Lambda \in {\mathcal{D}}$.
We show the following result.
With the previous notations, if ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks, then $$\begin{aligned}
\|({\mathcal{R}}\circ \Delta \circ {\mathcal{P}}^{\leq K} \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1 \leq {\varepsilon},\end{aligned}$$ where $\tau^\eta_{\mathcal{H}\mathcal{N}}$ is a purification of $\tau^\eta_{\mathcal{H}}$. Here ${\mathcal{R}}$ is an additional privacy amplification step that reduces the key by $\lceil 2 \log_2 \tbinom{K+4}{4} \rceil$ bits and ${\mathcal{P}}^{\leq K}$ is the projection onto $F_{1,1,n}^{\leq K}$.
Recall that $$\begin{aligned}
\tau^\eta_{\mathcal{H}} = T(n,\eta)^{-1} \int_{\mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda)\end{aligned}$$ where $$\begin{aligned}
T(n,\eta) := {\mathrm{tr}}(P_\eta) = \frac{(n-1)(n-2)^2(n-3)\eta^4}{12(1-\eta)^4}.\end{aligned}$$ By linearity, it holds that $$\begin{aligned}
\| (({\mathcal{E}}-{\mathcal{F}})\otimes \mathrm{id})( \tau^\eta_{\mathcal{H}})\|_1 &=
\| (({\mathcal{E}}-{\mathcal{F}})\otimes \mathrm{id})( T(n,\eta)^{-1} \int_{\Lambda \in \mathcal{D}_\eta} |\Lambda,n\rangle \langle \Lambda,n| \d \mu_n(\Lambda) )\|_1 \\
&\leq {\varepsilon}T(n,\eta)^{-1}\left\| \int_{\Lambda \in \mathcal{D}_\eta} \d \mu_n(\Lambda) \right\|_1 \\ &= {\varepsilon}\end{aligned}$$ In order to obtain the theorem, we need to consider a purification of $\tau_{{\mathcal{H}}{\mathcal{N}}}^{\eta}$. Since ${\mathcal{P}}^{\leq K}$ restricts the states to live in a space of dimension at most $\mathrm{dim} \, F_{2,2,n}^{U(n), \leq K} = \tbinom{K+4}{4}$ (according to Lemma \[lem:dim\]), it implies that the purifying system ${\mathcal{N}}$ can be chosen of this dimension. Giving this extra system to Eve can at most provide her with a limited amount of information. Applying an additional privacy amplification step ${\mathcal{R}}$ ensures that the protocol remains ${\varepsilon}$-secure for the state $\tau_{{\mathcal{H}}{\mathcal{N}}}^\eta$ thanks to the leftover hashing lemma (Theorem 5.1.1 of [@Ren08]): $$\begin{aligned}
\|({\mathcal{R}}\circ \Delta \circ {\mathcal{P}}^{\leq K} \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}\mathcal{N}} \|_1 \leq \|( \Delta \circ {\mathcal{P}}^{\leq K} \otimes \mathrm{id}) \tau^\eta_{\mathcal{H}} \|_1.\end{aligned}$$
Combining this result with Theorem \[thm:postselection\] yields Theorem \[thm:diamond-protocol\].
[thm:diamond-protocol]{} With the previous notations, if ${\mathcal{E}}_0$ is ${\varepsilon}$-secure against Gaussian collective attacks, then $$\begin{aligned}
\|{\mathcal{R}}\circ ({\mathcal{E}}_0-{\mathcal{F}}_0) \circ {\mathcal{P}}^{\leq K} \|_{\diamond} \leq 2 T(n,\eta) {\varepsilon}.\end{aligned}$$
Energy test {#sec:test}
===========
The goal of this section is to prove the following result.
[thm:test]{} For integers $n,k \geq 1$, and $d_A, d_B >0$, define $K = n(d'_A + d'_B)$ for $d'_{A/B} = d_{A/B} g(n,k,{\varepsilon}/4)$ for the function $g$ defined in Eq. . Then $$\begin{aligned}
\big\| \big({\mathbbm{1}}- {\mathcal{P}}(n,K)\big) \circ {\mathcal{T}}(k, d_A, d_B)\big\|_{\diamond} \leq {\varepsilon}.\end{aligned}$$
For $d >0$, let us introduce the following operators on ${\mathcal{H}}^{\otimes n}$ for a single-mode Fock space ${\mathcal{H}}$: $$\begin{aligned}
T_n^{d} &:= \frac{1}{\pi^n} \int_{\sum_{i=1}^n |\alpha_i|^2 \geq n d} |\alpha_1\rangle\langle \alpha_1| \otimes \ldots \otimes |\alpha_n\rangle \langle \alpha_n| \d \alpha_1 \ldots \alpha_n\\
U_n^{d} &:= \sum_{m = n d+1}^\infty \Pi_{m}^n,\end{aligned}$$ where $\Pi_m^n$ is the projector onto the subspace of ${\mathcal{H}}^{\otimes n}$ spanned by Fock states containing $m$ photons: $$\begin{aligned}
\Pi_m^n = \sum_{m_1+\ldots+m_n=m} |m_1, \ldots, m_n\rangle \langle m_1, \ldots, m_n|.\end{aligned}$$ In words, $T_n^{d}$ is the sum of the projectors onto products of coherent states such that the total squared amplitude is greater than $n d$ and $U_n^{d}$ is the projector onto Fock states containing more that $n d$ photons. Intuitively, both operators should be “close” to each other. This is formalized with the following lemma that was proven in [@LGRC13].
\[lem:LGRC\] For any integer $n$ and any $d\geq0$, it holds that $$\begin{aligned}
U_n^{d} \leq 2 T_n^{d}.\end{aligned}$$
The following lemma results from the definitions of $U_n^d$ and ${\mathcal{P}}^{\leq K}$, the projector onto $F_{1,1,n}^{\leq K}$.
\[lem:obs\] For any $d_A, d_B \geq 0$ and integer $K$ such that $K \leq n(d_A+d_B)$, it holds that $$\begin{aligned}
{\mathbbm{1}}_{{\mathcal{H}}_A^{\otimes n} \otimes {\mathcal{H}}_B^{\otimes n}} - {\mathcal{P}}^{\leq K} \leq U_n^{d_A}\otimes {\mathbbm{1}}_{{\mathcal{H}}_B^{\otimes n}} + {\mathbbm{1}}_{{\mathcal{H}}_A^{\otimes n}} \otimes U_n^{d_B}.\end{aligned}$$
The left hand side is the projector onto the states of ${\mathcal{H}}_A^{\otimes n} \otimes {\mathcal{H}}_B^{\otimes n}$ containing strictly more than $K$ photons. Any such state must contain either at least $n d_A$ photons in ${\mathcal{H}}_A^{\otimes n}$ or at least $K - n d_A$ photons in ${\mathcal{H}}_B^{\otimes n}$, for any possible value of $d_A$. This proves the claim.
Combining Lemmas \[lem:LGRC\] and \[lem:obs\], we obtain the immediate corollary.
\[cor:proj\] For any $d_A, d_B \geq 0$ and integer $K$ such that $K \leq n(d_A+d_B)$, it holds that $$\begin{aligned}
{\mathbbm{1}}_{{\mathcal{H}}_A^{\otimes n} \otimes {\mathcal{H}}_B^{\otimes n}} - {\mathcal{P}}^{\leq K} \leq 2 T_n^{d_A}\otimes {\mathbbm{1}}_{{\mathcal{H}}_B^{\otimes n}} + 2{\mathbbm{1}}_{{\mathcal{H}}_A^{\otimes n}} \otimes T_n^{d_B}.\end{aligned}$$
Recall that the heterodyne measurement corresponds to a projection onto (Glauber) coherent states, and is described by the resolution of the identity: $$\begin{aligned}
{\mathbbm{1}}_{{\mathcal{H}}^{\otimes k}} = \frac{1}{\pi^k} \int_{{\mathbb{C}}^k} |\alpha_1\rangle \langle \alpha_1| \otimes \ldots \otimes |\alpha_k\rangle \langle \alpha_k| \d\alpha_1 \ldots \d \alpha_k.\end{aligned}$$ In other words, measuring a state $\rho$ on ${\mathcal{H}}^{\otimes k}$ with heterodyne detection outputs the result $(\alpha_1, \ldots, \alpha_k) \in {\mathbb{C}}^k$ with probability $$\begin{aligned}
\mathrm{Pr}_\rho(\alpha_1, \ldots, \alpha_k) =\frac{1}{\pi^k} {\mathrm{tr}}( \rho |\alpha_1\rangle \langle \alpha_1| \otimes \ldots \otimes |\alpha_k\rangle \langle \alpha_k|).\end{aligned}$$
Laurent and Massart [@LM00] established the following tail bounds for $\chi^2(D)$ distributions.
\[lem:LM\] Let $U$ be a $\chi^2$ statistic with $D$ degrees of freedom. For any $x >0$, $$\begin{aligned}
\mathrm{Pr}[U-D \geq 2\sqrt{D x} + 2 Dx] \leq \exp(-x) \quad \text{and} \quad \mathrm{Pr}[D-U \geq 2\sqrt{Dx}] \leq \exp(-x). \end{aligned}$$
A state $\rho$ on ${\mathcal{H}}^{\otimes n} = F({\mathbb{C}}^n)$ is said *rotationally invariant* is $V_u \rho V_u^\dagger = \rho$ for all $u \in U(n)$.
In particular, the state $\int V_u \rho V_U^\dagger \d u$ is invariant if $\d u$ is the Haar measure on $U(n)$.
\[lem:36\] Let $\rho$ be an rotationally invariant state on ${\mathcal{H}}^{\otimes (n+k)}$. Then, for any $d >0$, $$\begin{aligned}
{\mathrm{tr}}\left[ ( T_n^{d'} \otimes ({\mathbbm{1}}-T_k^d)) \rho \right] \leq {\varepsilon},\end{aligned}$$ for $d' = g(n,k,{\varepsilon}) d$ and $$\begin{aligned}
g(n,k,{\varepsilon}) = \frac{1 + 2 \sqrt{\frac{\ln (2/{\varepsilon})}{2n}} + \frac{\ln (2/{\varepsilon})}{n}}{1-2{\sqrt{\frac{\ln (2/{\varepsilon})}{2k}}}}. \label{eqn:g}\end{aligned}$$
By definition, ${\mathrm{tr}}[T_k^d \rho]$ is the probability that the outcome $(\alpha_1, \ldots, \alpha_k)\in {\mathbb{C}}^k$ obtained by measuring the last $k$ modes of the state $\rho$ with heterodyne detection satisfies $\sum_{i=1}^k |\alpha_i|^2 \geq k d$. Similarly, ${\mathrm{tr}}\left[ ((T_n^{d'} \otimes ({\mathbbm{1}}-T_k^d)) \rho \right]$ is the probability that the outcome of measuring the $n+k$ modes of $\rho$ with heterodyne detection yields a vector $(\alpha_1, \ldots, \alpha_{n+k})$ such that $$\begin{aligned}
Y_n := \sum_{i=1}^n |\alpha_i|^2 \geq n d' \quad \text{and} \quad Y_k :=\sum_{i=1}^k |\alpha_{n+i}|^2 \leq k d.\end{aligned}$$ Since the state is rotationally invariant, it means that the random vector $(\alpha_1, \ldots, \alpha_{n+k})$ is uniformly distributed on the sphere of radius $M$ in ${\mathbb{C}}^{n+k}$, conditioned on the fact that the modulus is $\sqrt{\sum_{i=1}^{n+k} |\alpha_i|^2}=M$. Equivalently, one can consider the $2(n+k)$-dimensional real vector $({\mathfrak{R}}(\alpha_1), {\mathfrak{I}}(\alpha_1), \ldots, {\mathfrak{R}}(\alpha_1), {\mathfrak{I}}(\alpha_1))$ which is uniformly distributed over the sphere in ${\mathbb{R}}^{2(n+k)}$. Here ${\mathfrak{R}}(\alpha_1)$ and ${\mathfrak{I}}(\alpha)$ refer respectively to the real and imaginary part of $\alpha$. We obtain $$\begin{aligned}
{\mathrm{tr}}\left[ ( T_n^{d'} \otimes ({\mathbbm{1}}-T_k^d)) \rho \right] &= \mathrm{Pr}[(Y_n \geq nd' ) \wedge (Y_k \leq kd)]\\
& \leq \mathrm{Pr}[kd Y_n \geq nd' Y_k]\end{aligned}$$ where the inequality is a simple consequence of the fact that the rectangle $[nd',\infty] \times [0, kd]$ is a subset of the triangle $\{ (x,y) \in [0,\infty]^2 \: : \: kd x \geq nd' y\}$.
It is well-known that the uniform distribution over the unit sphere of ${\mathbb{R}}^{2(n+k)}$ can be generated by sampling $2(n+k)$ normal variables with 0 mean and unit variance. In that case, the squared norm $\sum_{i=1}^n |\alpha_i|^2$ is simply a $\chi^2$ variable with $2n$ degrees of freedom while $\sum_{i=1}^k |\alpha_{n+i}|^2$ corresponds to an independent $\chi^2$ variable with $2k$ degrees of freedom. Let us denote by $Z_n$ and $Z_k$ the corresponding random variables: $Z_n \sim \chi^2(2n)$, $Z_k \sim \chi^2(2k)$. Since $(Y_n, Y_k)$ and $(Z_n, Z_k)$ follow the same distribution, up to rescaling, we obtain that $$\begin{aligned}
\mathrm{Pr}[kd Y_n \geq nd' Y_k] = \mathrm{Pr}[kd Z_n \geq nd' Z_k].\end{aligned}$$ This is particularly useful because it means that there is no need to enforce normalization explicitly. Finally, using now that the triangle $\{ (x,y) \in [0,\infty]^2 \: : \: kd x \geq dd' y\}$ is a subset of the union of the rectangles $[\alpha nd',\infty]\times [0,\infty]$ and $[0,\infty] \times [0,\alpha kd]$ for any $\alpha >0$, it follows that $$\begin{aligned}
\mathrm{Pr}[kd Z_n \geq nd' Z_k] \leq \mathrm{Pr}[ Z_n \geq \alpha n d'] + \mathrm{Pr}[ Z_k \leq \alpha k d].\end{aligned}$$ Choosing $\alpha$ such that $$\begin{aligned}
\alpha k d = 2k \left(1 - 2 \sqrt{\frac{\ln({\varepsilon}/2)}{2k}}\right)\end{aligned}$$ and applying the lower bounds on the tails of the $\chi^2$ distribution given in Lemma \[lem:LM\] gives $$\begin{aligned}
\mathrm{Pr}[ Z_n \geq \alpha n d'] \leq \frac{{\varepsilon}}{2}, \quad \mathrm{Pr}[ Z_k \leq \alpha k d] \leq \frac{{\varepsilon}}{2}.\end{aligned}$$ This establishes that $$\begin{aligned}
{\mathrm{tr}}\left[ ( T_n^{d'} \otimes ({\mathbbm{1}}-T_k^d)) \rho \right] \leq {\varepsilon}, \end{aligned}$$ which concludes the proof.
We are now ready to define and analyze the energy test. Alice and Bob perform a random rotation of their data according to a unitary $u \in U(n)$ chosen from the Haar measure on $U(n)$, and measure the last $k$ modes of their respective state with heterodyne detection. They compute the squared norm of their respective vectors and obtain two values $Y_A$ for Alice and $Y_B$ for Bob. The test depends on three parameters: the number $k$ of modes which are measured, a maximum value for Alice $d_A$ and a maximum value for Bob, $d_B$. The test $\mathcal{T}(k, d_A, d_B)$ passes if $$\begin{aligned}
Y_A \leq k d_A \quad \text{and} \quad Y_B \leq k d_B.\end{aligned}$$
We are interested in the probability of passing the test and failing for the remaining modes to contain less than $K$ photons, more precisely in the quantity $$\begin{aligned}
\|({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}}\|_{\diamond}.\end{aligned}$$
Let us denote by $\mathrm{Inv}( {\mathfrak{S}}({\mathcal{H}}^{\otimes (n+k)}))$ the set of density matrices which are invariant under the action of $U(n+k)$.
[thm:test]{} For integers $n,k \geq 1$, and $d_A, d_B >0$, define $K = n(d'_A + d'_B)$ for $d'_{A/B} = d_{A/B} g(n,k,{\varepsilon}/4)$ for the function $g$ defined in Eq. . Then $$\begin{aligned}
\big\| \big({\mathbbm{1}}- {\mathcal{P}}(n,K)\big) \circ {\mathcal{T}}(k, d_A, d_B)\big\|_{\diamond} \leq {\varepsilon}.\end{aligned}$$
Writing ${\mathcal{P}}$ and ${\mathcal{T}}$ for conciseness, the definition of the diamond norm yields: $$\begin{aligned}
\|({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}}\|_{\diamond} &= \max_{\rho \in {\mathcal{H}}_{AB}^{\otimes (n+k)} \otimes {\mathcal{H}}_{AB}^{\otimes (n+k)}} \big\| \big(({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}})\otimes {\mathbbm{1}}_{{\mathcal{H}}_{AB}^{\otimes (n+k)}}\big) (\rho)\big\|_{1} \nonumber\\
&= \max_{\rho \in {\mathfrak{S}}({\mathcal{H}}_{AB}^{\otimes (n+k)})} \|({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}}(\rho)\|_{1} \label{eqn:nonneg}\\
& \leq \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{AB}^{\otimes (n+k)}\right)} \|({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}}(\rho)\|_{1} \label{eqn:invar}\\
& \leq \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{AB}^{\otimes (n+k)}\right)} \| (U_n^{d'_A} \otimes {\mathbbm{1}}+{\mathbbm{1}}\otimes U_n^{d'_B}) \circ \big( ({\mathbbm{1}}- T_k^{d_A} )\otimes ({\mathbbm{1}}-T_k^{d_B}) \big)(\rho)\|_{1} \label{eqn:sum} \\
& = \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{AB}^{\otimes (n+k)}\right)} \| (U_n^{d'_A} \circ ({\mathbbm{1}}-T_k^{d_A}) + U_n^{d'_B} \circ ({\mathbbm{1}}- T_k^{d_B})) (\rho)\|_{1} \nonumber \\
& \leq \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{A}^{\otimes (n+k)}\right)} \| (U_n^{d'_A} \circ ({\mathbbm{1}}-T_k^{d_A} )) (\rho)\|_{1} + \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{B}^{\otimes (n+k)}\right)} \| ( U_n^{d'_B} \circ ({\mathbbm{1}}-T_k^{d_B})) (\rho)\|_{1} \label{eqn:triang}\\
& \leq 2 \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{A}^{\otimes (n+k)}\right)} \| (T_n^{d'_A} \circ ({\mathbbm{1}}-T_k^{d_A}) ) (\rho)\|_{1} +2 \max_{\rho \in \mathrm{Inv}\left( {\mathfrak{S}}({\mathcal{H}}_{B}^{\otimes (n+k)}\right)} \| ( T_n^{d'_B} \circ ({\mathbbm{1}}-T_k^{d_B})) (\rho)\|_{1}\\
&\leq {\varepsilon}\label{eqn:final}\end{aligned}$$ where we used that $\big(({\mathbbm{1}}- {\mathcal{P}}) \circ {\mathcal{T}})\otimes {\mathbbm{1}}_{{\mathcal{H}}_{AB}^{\otimes (n+k)}}\big) (\rho)$ is a nonnegative operator in Eq. , the fact that both ${\mathcal{P}}$ and ${\mathcal{T}}$ are rotationally invariant in Eq. , Lemma \[lem:obs\] in Eq. , the triangle inequality in Eq. , Lemma \[lem:36\] in Eq. .
| ArXiv |
---
abstract: 'Bulk and decay properties, including deformation energy curves, charge mean square radii, Gamow-Teller (GT) strength distributions, and $\beta$-decay half-lives, are studied in neutron-deficient even-even and odd-$A$ Hg and Pt isotopes. The nuclear structure is described microscopically from deformed quasiparticle random-phase approximation calculations with residual interactions in both particle-hole and particle-particle channels, performed on top of a self-consistent deformed quasiparticle Skyrme Hartree-Fock basis. The observed sensitivity of the, not yet measured, GT strength distributions to deformation is proposed as an additional complementary signature of the nuclear shape. The $\beta$-decay half-lives resulting from these distributions are compared to experiment to demonstrate the ability of the method.'
author:
- 'J. M. Boillos'
- 'P. Sarriguren'
title: 'Effects of deformation on the beta-decay patterns of light even-even and odd-mass Hg and Pt isotopes'
---
Introduction
============
Neutron-deficient isotopes in the lead region are nowadays well established examples of the shape coexistence phenomenon in nuclei [@heyde11; @julin01]. They have been subject of much experimental and theoretical interest in the last years. The first direct evidence of the shape coexistence in the region $Z\approx 82$ was obtained in neutron-deficient Hg isotopes from isotope shift measurements [@bonn72]. Those measurements showed a sharp transition in the nuclear size between the ground states of $^{187}$Hg and $^{185}$Hg that was interpreted [@frauendorf75] as a change from a weak oblate shape in the heavier isotopes to a more deformed prolate shape in the lighter ones from calculations based on Strutinsky’s shell correction method. Later, new isotope shift measurements [@ulm86] revealed a weakly oblate deformed character of the ground states of the even-mass Hg isotopes down to $A=182$, with an odd-even staggering persisting down to $^{181}$Hg. The radius of the oblate isomeric state in $^{185}$Hg follows the trend of the even-even ground-state radii.
Shape evolution and shape coexistence in the region of $\beta$-unstable nuclei with $Z\approx 82$ were subsequently studied experimentally by $\gamma$-ray spectroscopy in the $\alpha$-decay of the products created in fusion-evaporation reactions (see Ref. [@julin01] and references therein). Maybe, the most singular case corresponds to $^{186}$Pb, where two excited $0^+$ states below 700 keV [@andreyev00] have been found. Furthermore, low-lying excited $0^+$ states have been experimentally observed at excitation energies below 1 MeV [@julin01; @andreyev00] in all even Pb isotopes between $A=184$ and $A=194$. Similarly, $0^+_2$ excited states below 1 MeV have been found in neutron-deficient Hg isotopes from $A=180$ up to $A=190$ [@julin01].
The spectroscopy of the Hg isotopes [@julin01; @hannachi; @lane95] shows a nearly constant behavior of the energy of the yrast states in the range $A=190-198$, which are interpreted as members of a rotational band on top of a weakly deformed oblate ground state. For lighter isotopes, $0^+_2$ excited states appear at low energies, decreasing in excitation energy up to $A=182$. They are interpreted as the band-heads of prolate configurations. Their excited states become yrast above $4^+$ for $A<186$, whereas the $2^+$ levels become close enough in energy to the weakly deformed states, opening the possibility of mixing strongly with them. Nevertheless, to determine the magnitude and type of deformation of the bands and their mixing, spectroscopy studies are not enough and the electromagnetic properties (E2 transition strengths) of the low-lying states have to be determined. Lifetime measurements in neutron-deficient Hg isotopes have been performed in the last years [@grahn09; @scheck10; @gaffney14]. More recently [@bree14], Coulomb-excitation experiments have been performed to study the electromagnetic properties of light Hg isotopes $^{182-188}$Hg. In these experiments, the deformation of the ground state and low-lying excited states were deduced, confirming the presence of two different coexisting structures in the light even-even Hg isotopes that are pure at higher spin values and mix at low excitation energy. The ground states of Hg isotopes in the mass range $A=182-188$ are found to be weakly deformed and of predominantly oblate nature, while the excited $0^+_2$ states in $^{182,184}$Hg exhibit a larger deformation. Similarly, low-lying states in light Pt isotopes have been studied experimentally with $\gamma$-ray spectroscopy [@cederwall90; @dracoulis91; @davidson99], showing that shape coexistence of states with different deformation is still present in neutron-deficient Pt isotopes with $Z=78$. Moderate odd-even staggering was also found in very light Pt isotopes from laser spectroscopy [@leblanc99].
From the theoretical point of view different types of models have been used to explain the coexistence of several $0^+$ states at low energies [@heyde11]. In a shell model picture, the excited $0^+$ states are interpreted as multi particle-hole excitations. Protons and neutrons outside the inert core interact through pairing and quadrupole interactions to generate deformed structures. Within a mean-field description of the nuclear structure, the presence of several minima at low energies in the energy surface, corresponding to different $0^+$ states, is understood as due to the coexistence of various collective nuclear shapes. In the mean-field approach, the energy of the different shape configurations can be evaluated with constrained calculations, minimizing the Hartree-Fock energy under the constraint of keeping fixed the nuclear deformation. The resulting total energy plots versus deformation are called in what follows deformation-energy curves (DEC). These calculations have become more and more refined with time, resulting in accurate descriptions of the nuclear shapes and the configurations involved. Calculations based on phenomenological mean fields and Strutinsky method [@bengtsson], are already able to predict the existence of several competing minima in the deformation-energy surface of neutron-deficient Pt, Hg, and Pb isotopes. Self-consistent mean-field calculations with non-relativistic Skyrme [@bender04; @yao13] and Gogny [@delaroche; @libert; @egido; @rayner10], as well as relativistic [@niksic02] energy density functionals have been carried out. Inclusion of correlations beyond mean field [@bender04; @yao13; @delaroche; @libert; @egido; @rayner10] are needed to obtain a detailed description of the spectroscopy. They involve symmetry restoration by means of angular momentum and particle number projection and configuration mixing within a generator coordinate method. It is shown that the underlying mean field picture of coexisting shapes is in general supported, except in those cases where the deformed mean-field structures appear at close energies. In this case mixing can be important, affecting B(E2) strengths and their corresponding $\beta$ deformation parameters. The basic picture is also confirmed from recent calculations within the interacting boson model with configuration mixing carried out for Hg [@nomura13; @gramos14hg] and Pt [@morales08; @gramos09; @gramos11; @gramos14pt] isotopes.
Triaxiality in this mass region has also been explored systematically [@yao13; @rayner10; @nomura13; @gramos14pt; @nomura11], showing that although the axial deformations seem to be the basic ingredients, triaxiality may play a role in some cases. A systematic survey of energy surfaces in the $(\beta ,\gamma )$ plane with the Gogny D1S interaction can be found in the Bruyères-le-Châtel database [@web_Gogny].
On the other hand, it has been shown [@frisk95; @sarri98; @sarri99] that the decay properties of $\beta$-unstable nuclei may depend on the nuclear shape of the decaying nucleus. In particular, the Gamow-Teller (GT) strength distributions corresponding to $\beta^+$/EC-decay of proton-rich nuclei in the mass region $A\approx 70$ have been studied systematically [@sarri01prc; @sarri01npa; @sarri05epja; @sarri09] as a function of the deformation, using a deformed quasiparticle random-phase approximation (QRPA) approach built on a self-consistent Hartree-Fock (HF) mean field with Skyrme forces and pairing correlations. The study has also been extended to stable $pf$-shell nuclei [@sarri03; @sarri13] and to neutron-rich nuclei in the mass region $A\approx 100$ [@sarri_pere]. This sensitivity of the GT strength distributions to deformation has been exploited to determine the nuclear shape in neutron-deficient Kr and Sr isotopes by comparing theoretical results with $\beta$-decay measurements using the total absorption spectroscopy technique (TAS) [@isolde].
Similar studies for the decay properties of even-even neutron-deficient Pb, Po, and Hg isotopes were initiated in Refs. [@sarri05prc; @moreno06] to predict the extent to which GT strength distributions may be used as fingerprints of the nuclear shapes in this mass region. In those works, it was shown that the existence of shape isomers, as well as the location of their equilibrium deformations, are rather stable and independent on the Skyrme and pairing forces. It was also found that the GT strength distributions calculated at the various equilibrium deformations exhibit specific features that can be used as signatures of the shape isomers and, what is important, these features remain basically unaltered against changes in the Skyrme and pairing forces.
In this paper we extend those calculations by studying the DECs and the GT strength distributions of neutron-deficient $^{174-204}$Hg and $^{170-192}$Pt isotopes, focusing on their dependence on deformation. In addition, we also include as a novelty the decay properties of the odd-$A$ isotopes and discuss the sensitivity of the decay patterns to the spin-parity of the decaying nucleus. The aim here is to identify possible signatures of the shape of the nucleus in the decay patterns. This study is timely because the possibility to carry out these measurements in odd-$A$ nuclei is being considered at present at ISOLDE/CERN [@algora]. A program aiming to measure the Gamow-Teller strength distributions in neutron-deficient isotopes in the lead region with TAS techniques started with $^{188,190,192}$Pb isotopes. These data have been already analyzed and submitted for publication [@submitted; @thesis]. Similar measurements have been carried out in $^{182,183,184,186}$Hg and are being presently analyzed [@algora].
The paper is organized as follows. In Sec. II we present briefly the main features of our theoretical framework. Section III contains our results for the energy deformation curves and GT strength distributions in the neutron-deficient Hg and Pt isotopes relevant for $\beta^+$/EC-decay. We also compare the experimental half-lives with our results and discuss the GT strength distributions and their sums in various ranges of excitation energies. Section IV contains the main conclusions.
Theoretical Formalism {#sec2}
=====================
In this section we present a summary of the theoretical formalism used in this paper to describe the $\beta$-decay properties in Hg and Pt neutron-deficient isotopes. More details of the microscopic calculations can be found in Refs. [@sarri98; @sarri99; @sarri01prc; @sarri01npa]. The method starts with a self-consistent calculation based on a deformed Hartree-Fock mean field obtained with effective two-body density-dependent Skyrme interactions including pairing correlations in BCS approximation. From these calculations we obtain energies, occupation probabilities and wave functions of the single-particle states. Most of the calculations in this work have been performed with the interaction SLy4 [@sly4], being among the most successful and extensively studied Skyrme force in the last years [@bender08; @bender09; @stoitsov]. Furthermore, comparison with other widely used Skyrme forces like the simpler Sk3 [@sk3] and SGII [@sg2] that has been shown to provide good spin-isospin properties, will be shown in some instances.
The solution of the HF equation is found by using the formalism developed in Ref. [@vautherin], assuming time reversal and axial symmetry. The single-particle wave functions are expanded in terms of the eigenstates of an axially symmetric harmonic oscillator in cylindrical coordinates, using twelve major shells. The method also includes pairing between like nucleons in BCS approximation with fixed gap parameters for protons and neutrons, which are determined phenomenologically from the odd-even mass differences through a symmetric five-term formula involving the experimental binding energies [@audi12]. In those cases where experimental information for masses is still not available, same pairing gaps as the closer isotope measured are used.
The DECs are analyzed as a function of the quadrupole deformation parameter $\beta$ from constrained HF calculations. Calculations for GT strengths are performed subsequently at the equilibrium shapes of each nucleus, that is, for the solutions (in general deformed) for which minima are obtained in the energy curves.
It is worth mentioning some existing works in this mass region based on mean-field approaches other than the present Skyrme HF+BCS calculations. In particular, mean-field studies of structural changes with the Gogny interaction can be found in Ref. [@nomura13] for Hg isotopes and in Refs. [@rayner10; @gramos14pt; @nomura11] for Pt isotopes. The clear advantage of the finite-range Gogny force over the contact Skyrme force is that pairing correlations can be treated self-consistently using the same interaction through a Hartree-Fock-Bogoliubov (HFB) calculation. Triaxial landscapes were studied in those references, showing that the (axial) prolate and oblate minima, which are well separated by high-energy barriers in the $\beta$ degree of freedom, are in many cases softly linked along the $\gamma$ direction. Indeed, some axial minima become saddle points when the $\gamma$ degree of freedom is included in the analysis. The differences found with the present HF+BCS approach for the axial equilibrium values are not significant, but the topology of the surfaces are somewhat different. Similarities and differences of the various topologies are discussed in the next section.
In the case of odd-$A$ nuclei, the ground state is expressed as a one-quasiparticle (1qp) state, which is determined by finding the blocked state that minimizes the total energy. In the present study we use the equal filling approximation (EFA), a prescription widely used in mean-field calculations to treat the dynamics of odd nuclei preserving time-reversal invariance [@rayner2]. In this approximation the unpaired nucleon is treated on equal footing with its time-reversed state by putting half a nucleon in a given orbital and the other half in the time-reversed partner. This approximation has been found to be equivalent to the exact blocking when the time-odd fields of the energy density functional are neglected and then, it is sufficiently precise for most practical applications [@schunck10]. Effects of time-odd terms in HFB calculations have also been studied in Ref. [@hellemans12]. An extension of beyond-mean-field calculations, where the generator coordinate method is built from self-consistently blocked 1qp HFB states for odd-mass nuclei has recently been presented in Ref. [@bally14].
The deformation in the decaying nuclei in both even-even and odd-$A$ cases, is self-consistently determined. In the odd-$A$ case, the core polarization induced by the odd particle is then taken into account. The effect found is however very small and we get very similar axial deformations in the even-even and neighbor odd-A nuclei. The small effect can be also observed in the Gogny database [@web_Gogny], comparing the DECs of the even-even and nearest odd-$A$ isotopes.
Since the GT operator of the allowed transitions is a pure spin-isospin operator without any radial dependence, one expects the spatial functions of the parent and daughter wave functions to be as close as possible in order to overlap maximally. Then, transitions connecting different radial structures in the parent and daughter nuclei will be suppressed. Thus, we assume similar shapes for the decaying parent nucleus and for all populated states in the daughter nucleus, neglecting core polarization effects in the daughter nuclei. This is a common assumption to deformed QRPA calculations [@moller1]. That core polarization effects are small in both odd-odd case in relation to even-even parent and odd-even (even-odd) case in relation to the even-odd (odd-even) parent can be seen in the Gogny database [@web_Gogny], where potential energy surfaces obtained from Gogny HFB calculations are shown all along the nuclear chart. By comparing the surfaces of parent (Hg, Pt) and daughter (Au, Ir) isotopes considered in this work, one realizes that the profiles are very similar with practically no effect from core polarization due to the odd particles.
The reduction in the transitions connecting different shapes have been quantified in the case of double $\beta$ decay [@alvarez04]. It has been shown that the overlaps between the wave functions in the intermediate nucleus reached from different shapes of the parent and daughter nuclei are dramatically reduced when the deformations differ from each other. Only with similar deformations the overlap is significant. Consequently, given the small polarization effects and the suppression of the overlaps with different deformations, we consider in this work only GT transitions between parent and daughter partners with like deformations.
To describe GT transitions, a spin-isospin residual interaction is added to the mean field and treated in a deformed proton-neutron QRPA [@moller1; @moller2; @homma; @moller3; @moller08; @hir1; @hir2; @frisk95; @sarri01npa]. This interaction contains two parts, particle-hole (ph) and particle-particle (pp). The interaction in the ph channel is responsible for the position and structure of the GT resonance [@homma; @sarri01npa] and it can be derived consistently from the same Skyrme interaction used to generate the mean field, through the second derivatives of the energy density functional with respect to the one-body densities. The ph residual interaction is finally expressed in a separable form by averaging the Landau-Migdal resulting force over the nuclear volume, as explained in Ref. [@sarri98]. The pp component is a neutron-proton pairing force in the $J^\pi=1^+$ coupling channel, which is also introduced as a separable force [@hir1; @hir2; @sarri01npa]. Its strength is usually fitted to reproduce globally the experimental half-lives. Various attempts have been made in the past to fix this strength [@homma], arriving to expressions that depend on the model used to describe the mean field, Nilsson model in the above reference. In previous works we studied the sensitivity of the GT strength distributions to the various ingredients contributing to the deformed QRPA calculations, namely to the nucleon-nucleon effective interaction, to pairing correlations, and to residual interactions. We found different sensitivities to them. In this work, all of these ingredients have been fixed to the most reasonable choices found previously [@sarri05prc; @moreno06]. In particular we use the coupling strengths $\chi ^{ph}_{GT}=0.08$ MeV and $\kappa ^{pp}_{GT} = 0.02$ MeV for the ph and pp channels, respectively. The technical details to solve the QRPA equations have been described in Refs. [@hir1; @hir2; @sarri98]. Here we only mention that, because of the use of separable residual forces, the solutions of the QRPA equations are found by solving first a dispersion relation, which is an algebraic equation of fourth order in the excitation energy $\omega$. Then, for each value of the energy, the GT transition amplitudes in the intrinsic frame connecting the ground state $| 0^+\rangle $ of an even-even nucleus to one phonon states in the daughter nucleus $|\omega_K \rangle \, (K=0,1) $ are found to be
$$\left\langle \omega _K | \sigma _K t^{\pm} | 0 \right\rangle =
\mp M^{\omega _K}_\pm \, ,
\label{intrinsic}$$
where $t^+ |\pi \rangle =|\nu \rangle,\, t^- |\nu \rangle =|\pi \rangle$ and $$\begin{aligned}
M_{-}^{\omega _{K}}&=&\sum_{\pi\nu}\left( q_{\pi\nu}X_{\pi
\nu}^{\omega _{K}}+ \tilde{q}_{\pi\nu}Y_{\pi\nu}^{\omega _{K}}
\right) , \\
M_{+}^{\omega _{K}}&=&\sum_{\pi\nu}\left(
\tilde{q}_{\pi\nu} X_{\pi\nu}^{\omega _{K}}+
q_{\pi\nu}Y_{\pi\nu}^{\omega _{K}}\right) \, ,\end{aligned}$$ with $$\tilde{q}_{\pi\nu}=u_{\nu}v_{\pi}\Sigma _{K}^{\nu\pi },\ \ \
q_{\pi\nu}=v_{\nu}u_{\pi}\Sigma _{K}^{\nu\pi},
\label{qs}$$ in terms of the occupation amplitudes for neutrons and protons $v_{\nu,\pi}$ ($u^2_{\nu,\pi}=1-v^2_{\nu,\pi}$) and the matrix elements of the spin operator, $\Sigma _{K}^{\nu\pi}=\left\langle \nu\left| \sigma _{K}\right|
\pi\right\rangle $, connecting proton and neutron single-particle states, as they come out from the HF+BCS calculation. $X_{\pi\nu}^{\omega _{K}}$ and $Y_{\pi\nu}^{\omega _{K}}$ are the forward and backward amplitudes of the QRPA phonon operator, respectively.
Once the intrinsic amplitudes in Eq. (\[intrinsic\]) are calculated, the GT strength $B_{\omega}(GT^\pm)$ in the laboratory system for a transition $I_iK_i (0^+0) \rightarrow I_fK_f (1^+K)$ can be obtained as $$\begin{aligned}
B_{\omega}(GT^\pm )& =& \sum_{\omega_{K}} \left[ \left\langle \omega_{K=0}
\left| \sigma_0t^\pm \right| 0 \right\rangle ^2 \delta (\omega_{K=0}-
\omega ) \right. \nonumber \\
&& \left. + 2 \left\langle \omega_{K=1} \left| \sigma_1t^\pm \right|
0 \right\rangle ^2 \delta (\omega_{K=1}-\omega ) \right] \, ,
\label{bgt}\end{aligned}$$ in $[g_A^2/4\pi]$ units. To obtain this expression, the initial and final states in the laboratory frame have been expressed in terms of the intrinsic states using the Bohr-Mottelson factorization [@bm].
When the parent nucleus has an odd nucleon, the ground state can be expressed as a one-quasiparticle (1qp) state in which the odd nucleon occupies the single-particle orbit of lowest energy. Then two types of transitions are possible. One type is due to phonon excitations in which the odd nucleon acts only as a spectator. These are three-quasiparticle (3qp) states and the GT transition amplitudes in the intrinsic frame are basically the same as in the even-even case in Eq. (\[intrinsic\]), but with the blocked spectator excluded from the calculation. The other type of transitions are those involving the odd nucleon state (1qp), which are treated by taking into account phonon correlations in the quasiparticle transitions in first-order perturbation. The transition amplitudes for the correlated states can be found in Refs. [@hir2; @sarri01prc].
In this work we refer the GT strength distributions to the excitation energy in the daughter nucleus. In the case of even-even decaying nuclei, the excitation energy of the 2qp states in the odd-odd daughter nuclei is simply given by
$$E_{\mbox{\scriptsize{ex}}\, [(Z,N)\rightarrow (Z-1,N+1)]}=\omega -E_{\pi_0} -
E_{\nu_0} \, ,
\label{eexeven}$$
where $E_{\pi_0}$ and $E_{\nu_0}$ are the lowest quasiparticle energies for protons and neutrons, respectively. In the case of an odd-$A$ nucleus we have to deal with 1qp and 3qp transitions. For Hg and Pt isotopes we have odd-neutron parents decaying into odd-proton daughters. The excitation energies for 1qp transitions are
$$E_{\mbox{\scriptsize{ex,1qp}}\, [(Z,N-1)\rightarrow (Z-1,N)]}=E_\pi-E_{\pi_0} \, .
\label{eex1qp}$$
The excitation energy with respect to the ground state of the daughter nucleus for 3qp transitions is
$$E_{\mbox{\scriptsize{ex,3qp}}\, [(Z,N-1)\rightarrow (Z-1,N)]}=
\omega +E_{\nu,\mbox{\scriptsize{spect}}}-E_{\pi_0} \, .
\label{eex3qp}$$
Therefore, the lowest excitation energy of 3qp type is of the order of twice the neutron pairing gap and then, the strength contained below typically 2-3 MeV in the odd-$A$ nuclei corresponds to 1qp transitions.
The $\beta$-decay half-life is obtained by summing all the allowed transition strengths to states in the daughter nucleus with excitation energies lying below the corresponding $Q_{EC}$ energy, i.e., $Q_{EC}=Q_{\beta^+} +2m_e= M(A,Z)-M(A,Z+1)+2m_e $, written in terms of the nuclear masses $M(A,Z)$ and the electron mass ($m_e$), and weighted with the phase-space factors $f(Z,Q_{EC}-E_{ex})$,
$$T_{1/2}^{-1}=\frac{\left( g_{A}/g_{V}\right) _{\rm eff} ^{2}}{D}
\sum_{0 < E_{ex} < Q_{EC}}f\left( Z,Q_{EC}-E_{ex} \right) B(GT,E_{ex}) \, ,
\label{t12}$$
with $D=6200$ s and $(g_A/g_V)_{\rm eff}=0.77(g_A/g_V)_{\rm free}$, where 0.77 is a standard quenching factor. In this work we use experimental $Q_{EC}$ values [@audi12]. In $\beta^+$/EC decay, $f( Z,Q_{EC}-E_{ex})$ contains two parts, positron emission and electron capture. The former, $f^{\beta^\pm}$, is computed numerically for each value of the energy including screening and finite size effects, as explained in Ref. [@gove],
$$f^{\beta^\pm} (Z, W_0) = \int^{W_0}_1 p W (W_0 - W)^2 \lambda^\pm(Z,W)
{\rm d}W\, ,
\label{phase}$$
with
$$\lambda^\pm(Z,W) = 2(1+\gamma) (2pR)^{-2(1-\gamma)} e^{\mp\pi y}
\frac{|\Gamma (\gamma+iy)|^2}{[\Gamma (2\gamma+1)]^2}\, ,$$
where $\gamma=\sqrt{1-(\alpha Z)^2}$ ; $y=\alpha ZW/p$ ; $\alpha$ is the fine structure constant and $R$ the nuclear radius. $W$ is the total energy of the $\beta$ particle, $W_0$ is the total energy available in $m_e c^2$ units, and $p=\sqrt{W^2 -1}$ is the momentum in $m_e c$ units.
The electron capture phase factors, $f^{EC}$, have also been included following Ref. [@gove]:
$$f^{EC}=\frac{\pi}{2} \sum_{x} q_x^2 g_x^2B_x \, ,$$
where $x$ denotes the atomic sub-shell from which the electron is captured, $q$ is the neutrino energy, $g$ is the radial component of the bound state electron wave function at the nucleus, and $B$ stands for other exchange and overlap corrections [@gove].
![(Color online) Deformation energy curves for even-even $^{174-196}$Hg isotopes obtained from constrained HF+BCS calculations with the Skyrme forces Sk3, SGII, and SLy4.[]{data-label="fig_e_beta_hg"}](fig1_hg_beta){width="80mm"}
![(Color online) Same as in Fig. \[fig\_e\_beta\_hg\], but for $^{170-192}$Pt isotopes.[]{data-label="fig_e_beta_pt"}](fig2_pt_beta){width="80mm"}
![(Color online) Isotopic evolution of the quadrupole deformation parameter $\beta$ of the various energy minima for Hg (a) and Pt (b) isotopes. The dashed lines join the deformations corresponding to the lowest HF+BCS minimum in the DECs obtained with SLy4.[]{data-label="fig_beta_A"}](fig3_beta_A){width="80mm"}
![(Color online) Calculated $\delta \langle r_c^2 \rangle $ in Hg isotopes with various deformations compared to experimental data from Refs. [@bonn72; @ulm86; @angeli04; @lee78]. []{data-label="fig_hg_dr2"}](fig4_hg_rc){width="80mm"}
![(Color online) Same as in Fig. \[fig\_hg\_dr2\], but for Pt isotopes. Experimental data are from Refs. [@leblanc99; @angeli04; @lee88; @sauvage00]. []{data-label="fig_pt_dr2"}](fig5_pt_rc){width="80mm"}
![(Color online) Folded GT strength distributions in $^{182,184,186}$Hg as a function of the excitation energy in the daughter nucleus for oblate and prolate shapes obtained with the Skyrme forces SGII and SLy4.[]{data-label="fig_hg_force"}](fig6_hg_force){width="85mm"}
![(Color online) Accumulated GT strengths in $^{184}$Hg calculated with the Skyrme interaction SLy4 for various values of the coupling strength of the ph residual interaction for a fixed value of the pp residual interaction. []{data-label="fig_hg184_ph"}](fig7_hg184_ph){width="85mm"}
![(Color online) Accumulated GT strengths in $^{184}$Hg calculated with the Skyrme interaction SLy4 for various values of the coupling strength of the pp residual interaction for a fixed value of the ph residual interaction. []{data-label="fig_hg184_pp"}](fig8_hg184_pp){width="85mm"}
![(Color online) (Left) Folded GT strength distributions in even Hg isotopes for prolate and oblate shapes using SLy4. (Right) Accumulated GT strength for the various shapes in the energy range below 7 MeV. The vertical lines correspond to the $Q_{EC}$ energies. No quenching factors are included.[]{data-label="fig_bgt_hg"}](fig9_bgt_hg){width="80mm"}
![(Color online) Same as in Fig. \[fig\_bgt\_hg\], but for even Pt isotopes.[]{data-label="fig_bgt_pt"}](fig10_bgt_pt){width="80mm"}
![(Color online). Same as in Fig. \[fig\_bgt\_hg\], but for odd-$A$ Hg isotopes.[]{data-label="fig_bgt_hg_odd"}](fig11_bgt_hg_odd){width="80mm"}
![(Color online) Same as in Fig. \[fig\_bgt\_pt\], but for odd-$A$ Pt isotopes.[]{data-label="fig_bgt_pt_odd"}](fig12_bgt_pt_odd){width="80mm"}
![(Color online) GT strength distribution in the odd isotope $^{181,183,185,187}$Hg for various $K^{\pi}$ values and deformations (see text).[]{data-label="fig_odd"}](fig13_hg_odd){width="85mm"}
![(Color online) Same as in Fig. \[fig\_odd\], but for Pt isotopes.[]{data-label="fig_odd_pt"}](fig14_pt_odd){width="85mm"}
![(Color online) Experimental $\beta^+$/EC-decay half-lives in Hg isotopes compared with the results of QRPA calculations with SLy4. The results obtained with the ground state shapes are connected with a dashed line.[]{data-label="fig_t12_hg"}](fig15_t12_hg){width="80mm"}
![(Color online) Same as in Fig. \[fig\_t12\_hg\], but for Pt isotopes.[]{data-label="fig_t12_pt"}](fig16_t12_pt){width="80mm"}
Results and discussion {#results}
======================
In this section we first discuss the energy curves and shape coexistence expected, discussing the shape evolution in Hg and Pt isotopic chains. Then, we present the results obtained for the Gamow-Teller strength distributions in the neutron-deficient $^{176-192}$Hg and $^{172-186}$Pt isotopes with special attention to their dependence on the nuclear shape and discuss their relevance as signatures of deformation to be explored experimentally. Finally, we discuss the half-lives and compare them with the experimental values.
Equilibrium deformations
------------------------
We show in Figs. \[fig\_e\_beta\_hg\] and \[fig\_e\_beta\_pt\] the DECs calculated with three Skyrme forces, Sk3, SGII, and SLy4, for Hg and Pt isotopes, respectively. The energies are shown as a function of the quadrupole deformation parameter calculated microscopically as $\beta=\sqrt{\pi/5}\ Q_p/(Z\langle r_c^2 \rangle)$, defined in terms of the proton quadrupole moment, $Q_p$, and charge m.s. radius, $\langle r_c^2 \rangle $. We get similar qualitative results with the three Skyrme forces considered. More specifically, we obtain the same patterns of shape coexistence with minima located at practically the same deformations although the relative energies may change from one force to another. Thus, we focus the discussion on the SLy4 interaction.
In the case of Hg isotopes (Fig. \[fig\_e\_beta\_hg\]) we get prolate and oblate minima in all the isotopes from $A=174$ up to $A=196$. We can see that the ground state is predicted to be prolate for $^{174-182}$Hg and oblate for $^{184-196}$Hg isotopes. The transition occurs smoothly around $^{184}$Hg for SLy4, where we obtain two coexisting shapes at the same energy and it takes place around $^{186}$Hg ($^{188}$Hg) with SGII (Sk3). Similarly, in the case of Pt isotopes (Fig. \[fig\_e\_beta\_pt\]) we get prolate and oblate minima in all the isotopes from $A=170$ up to $A=192$, but in this case the ground state is always prolate except in the heavier isotopes, $^{190,192}$Pt, where the oblate shape becomes ground state with the three forces. The transition is very smooth and the two shapes are practically degenerate between $^{184}$Pt and $^{190}$Pt for SLy4. Except for the very light isotopes, we observe in both isotopic chains the existence of rather sharp oblate and prolate energy minima, close in energy and separated by very high energy barriers, giving raise to shape coexistence. These findings are in qualitative agreement with recent calculations [@yao13; @nomura13; @gramos14hg; @rayner10; @gramos14pt; @nomura11]. Looking in more detail the results from different calculations, one observes differences and similarities within the various approaches. There are robust features common to all calculations, such as the existence of oblate and prolate minima located at similar deformations and separated by spherical barriers, or the isotopic evolution from oblate shapes in the heavier isotopes to prolate shapes in the lighter ones. But there are also particular features that change according to the different calculations, such as the height of the barriers or the relative energies between the minima that finally determines the exact isotope where the shape transition takes place. Obviously, the exact location of the shape transition is very sensitive to small details of the calculation because the shape transition occurs precisely around the region where the energies of the competing shapes are practically degenerate. Thus, it is not surprising that the shape transition in Pt isotopes predicted in Ref. [@yao13] within a beyond mean field approach with the Skyrme SLy6 occurs at $A=186-188$ instead of $A=182-184$ in our calculation. In the same line triaxial D1M-Gogny calculations predict a smooth shape transition at $A=184-186$ [@nomura13]
Similarly, the shape transition in Pt isotopes in our calculations takes place at $A=188-190$. This agrees with triaxial calculations with the Gogny force that exhibit a smooth transition at $A=186-190$, passing through a soft triaxial solution [@rayner10; @gramos14pt; @nomura11], as well as with the calculations in [@sarri08; @robledo09]. In particular, the DECs in Pt isotopes were studied in Ref. [@sarri08], comparing the effects of different interactions (SLy4, SLy6, Gogny) and pairing treatments (constant strength, constant pairing gaps, density-dependent zero-range forces). Little changes in the energy profiles were found within those treatments, but still enough to change the absolute minimum from one deformation to another in the transitional region around $^{188}$Pt, where the energies are practically degenerate. Nevertheless, for the purpose of this work, the exact location at which the shape transitions occur is not of relevance. The important aspect in this work is that a shape competition is taken place and whether the sensitivity of the B(GT) profiles to deformation can be used as a fingerprint of the nuclear shape. Then, we choose in this work a reasonable mean-field based on the Skyrme SLy4 with constant pairing gaps. to be used as a starting point for a QRPA calculation of the decay properties.
To illustrate better the role of deformation in the isotopic evolution, we show in Fig. \[fig\_beta\_A\] the quadrupole deformation parameter $\beta$ of the various energy minima as a function of the mass number $A$, for Hg (a) and Pt (b) isotopic chains. The dashed lines join the deformations corresponding to the lowest HF+BCS minimum in the DECs obtained with SLy4. Starting from the heaviest isotopes in Fig. \[fig\_beta\_A\], we get spherical shapes, as they correspond to the $N=126$ neutron shell closure. Moving into the neutron-deficient region, we observe the appearance of both oblate and prolate shapes with increasing quadrupole moments. The shape of the minimum energy changes from oblate in the heavier isotopes to prolate in the lighter ones at $^{182-184}$Hg and $^{188-190}$Pt for SLy4. The shapes reach maximum quadrupole deformations of about $\beta=0.3$ in the prolate sector and about $\beta=-0.2$ in the oblate one.
Charge radii and their differences have been shown [@rayner1; @rayner2] to be suitable quantities to study the evolution of the nuclear-shape changes as they can be measured with remarkable precision using laser spectroscopic techniques [@cheal]. They are calculated by folding the proton distribution of the nucleus with the finite size of the protons and the neutrons. The m.s. radius of the charge distribution in a nucleus can be expressed as [@negele] $$\langle r^2_c \rangle = \langle r^2_p \rangle _Z+
\langle r^2_c \rangle _p +(N/Z)
\langle r^2_c \rangle _n + r^2_{CM}
\, , \label{rch}$$ where $ \langle r^2_p \rangle _Z$ is the m.s. radius of the point proton distribution in the nucleus
$$\langle r_p^2 \rangle _Z = \frac{ \int r^2\rho_p({\vec r})d{\vec r} }
{\int \rho_p({\vec r})d{\vec r}} \, , \label{r2pn}$$
$ \langle r^2_c \rangle _p=0.80$ fm$^2$ [@sick03] and $ \langle r^2_c \rangle _n=-0.12$ fm$^2$ [@gentile11] are the m.s. radii of the charge distributions in a proton and a neutron, respectively. $r^2_{CM}$ is a small correction due to the center of mass motion. It is worth noticing that the most important correction to the point proton m.s. nuclear radius, coming from the proton charge distribution $ \langle r^2_c \rangle _p$, vanishes when isotopic differences are considered, since it does not involve any dependence on $N$.
The variations of the charge radii trends in isotopic chains are related to deformation effects and can be used as signatures of shape transitions. For an axially symmetric static quadrupole deformation $\beta$ the increase of the charge radius with respect to the spherical value is given to first order by
$$\langle r^2 \rangle = \langle r^2 \rangle _{\rm sph} \left(
1+\frac{5}{4\pi} \beta^2 \right) \, ,$$
where usually $\langle r^2 \rangle _{\rm sph}$ is taken from the droplet model. In this work we analyze the effect of the quadrupole deformation on the charge radii from a microscopic self-consistent approach.
One should notice that our calculations at the mean-field level correspond to the oblate and prolate mean-field solutions and, consequently, they don’t correspond to the actual ground state to which the experimental radii are referred.
In Figs. \[fig\_hg\_dr2\]-\[fig\_pt\_dr2\] we show the differences $\delta \langle r^2_c \rangle ^{A,{\rm ref}}= \langle r^2_c
\rangle ^A - \langle r^2_c \rangle ^{{\rm ref}}$, where the reference isotope is $A=198$ ($A=194$) for the Hg (Pt) isotopic chain. Our calculations are compared with experimental data measured by laser spectroscopy and compiled in Ref. [@angeli04]. For Hg isotopes, the experiment [@bonn72; @ulm86; @lee78] shows an even-odd staggering in the lighter isotopes ($A=181-186$), with larger radii in the odd-$A$ isotopes. When we compare the data for light Hg isotopes with our calculations we see that the even-even isotopes are well described with an oblate shape, whereas the odd-$A$ isotopes are rather associated with a prolate shape. We also observe in our calculations a bump in the oblate radii around $A=190$ and a more pronounced one in the prolate radii around $A=188$ that are related to the shape variation of the energy minima. In the case of Pt isotopes, the experimental radii [@leblanc99; @lee88; @sauvage00] in the interval $A=178-188$ are in between the oblate and prolate radii of reference, pointing out that strong mixing between these two structures is necessary to describe the $0^+$ ground state. The agreement with experiment is reasonable in the heavier Hg and Pt isotopes for both oblate and prolate radii, indicating that these nuclei are approaching a spherical shape.
Gamow-Teller strength distributions
-----------------------------------
In this subsection we study the energy distribution of the Gamow-Teller strengths calculated at the equilibrium shapes that minimize the energy of the nucleus. But before showing the results of our calculations it is worth discussing briefly the expected sensitivity of these calculations to the choice of the nucleon-nucleon effective Skyrme interaction, as well as to the coupling strengths of the residual forces.
Figure \[fig\_hg\_force\] illustrates the sensitivity of the GT strength distributions to the Skyrme interaction. We show in this figure continuous distributions obtained by folding the strength at each excitation energy with 1 MeV width Breit-Wigner functions. The results correspond to the Skyrme interactions SLy4 and SGII, and for three Hg isotopes, $^{182,184,186}$Hg. For a given type of deformation (oblate or prolate), we observe very similar decay patterns for both interactions, with slightly lower strength in the case of SGII. On the other hand, for a given Skyrme force the dependence on the deformation is manifest. This example demonstrates that the profiles of the GT strength distributions are characteristic of the nuclear shape and depend little on the details of the two-body force. This marked sensitivity to deformation can then be used to get information about the nuclear shape of the decaying nucleus, something that has been exploited in the past in other mass regions [@isolde].
In the next two figures we discuss the effect of the residual force on the GT strength distributions, using $^{184}$Hg as an example. In this case, for a better comparison, we plot the summed strengths that give us the total strength contained below a given energy. In Fig. \[fig\_hg184\_ph\] we can see the effect of the ph residual force. For that purpose we show the results obtained with a fixed value of the pp interaction ($\kappa ^{pp}_{GT}=0.02$ MeV) for $\chi ^{ph}_{GT}=0.08$ MeV (a), $\chi ^{ph}_{GT}=0.15$ MeV (b), and $\chi ^{ph}_{GT}=0.20$ MeV (c). As $\chi ^{ph}_{GT}$ increases, the strength is reduced, especially in the low-energy region, but the profiles of both prolate and oblate shapes remain basically the same. This reduction has immediate consequences on the half-lives that increase with increasing values of $\chi ^{ph}_{GT}$. Similarly, we show in Fig. \[fig\_hg184\_pp\] the effect of the pp residual force by taking fixed the ph residual interaction ($\chi ^{ph}_{GT}=0.08$ MeV) and varying the value of the pp interaction from $\kappa ^{pp}_{GT}=0$ (a) to $\kappa ^{pp}_{GT}=0.02$ MeV (b), and finally to $\kappa ^{pp}_{GT}=0.04$ MeV (c). As $\kappa ^{pp}_{GT}$ increases the strength is reduced and slightly shifted to lower energies, but again the prolate and oblate profiles persist.
In the next figures, Figs. \[fig\_bgt\_hg\]-\[fig\_bgt\_pt\_odd\], we show the GT strength distributions for oblate and prolate shapes as a function of the excitation energy in the daughter nucleus obtained with SLy4 and with the residual forces written in Sec. \[sec2\]. Although a similar figure to Fig. \[fig\_bgt\_pt\] was already presented in Ref. [@moreno06], for the sake of completeness and to facilitate the comparison, we also show here those results for the Pt isotopes. In the first two figures we show the results for the even Hg and Pt isotopes, whereas in the last two figures we present the results for the odd-$A$ isotopes. In the left panels we can see continuous GT strength distributions resulting from a folding procedure using 1 MeV width Breit-Wigner functions on the discrete spectrum. On the other hand, in the right panels we plot the accumulated GT strength up to a reduced energy range that covers the $Q_{EC}$ energies represented by the vertical arrows. Thus, we can see in more detail both the strength distribution and the total GT strength contained in the energy window relevant to the $\beta$-decay and to the half-lives. In particular, the crossing of the curves with the $Q_{EC}$ vertical arrows shows us the total GT strength available by $\beta$-decay and eventually measurable. It should be noted that no quenching factor is included in these distributions and therefore one should consider a reduction of this strength prior to comparison with future experiments.
The left panels in Figs. \[fig\_bgt\_hg\] and \[fig\_bgt\_pt\] show the GT strength distributions for the even-even Hg and Pt isotopes, respectively. The strength increases as we move away from the valley of stability to more and more neutron-deficient (lighter) isotopes (note the different scales). On a global scale the strength distribution from different shapes differ mainly in the low energy region. With minor exemptions, oblate shapes produce more strength at low energies and therefore smaller half-lives. In all cases we observe a strong peak (or double peak) at low excitation energy (below 5 MeV) and little strength above this energy, except in the heavier isotopes where a bump at high energy is developed. The differences between oblate and prolate shapes can be better appreciated in the accumulated plots displayed in the right-hand sides. In general we observe that the results from oblate shapes are more fragmented and the strength in the accumulated plots increases steadily. Conversely, prolate shapes produce in most cases a strong peak at low excitation energy an very little strength above. The location of the $Q_{EC}$ energies determines the GT strength distribution available in the decay and thus, contributing to the half-lives. Clearly, $Q_{EC}$ energies increase when moving away from stability.
The next two figures, Figs. \[fig\_bgt\_hg\_odd\] and \[fig\_bgt\_pt\_odd\], contain the GT strength distributions for the odd-even Hg and Pt isotopes, respectively. In the case of odd nuclei the spin and parity of the nucleus are determined by those of the odd nucleon. In principle one would sit the odd nucleon in the single-particle orbit that minimizes the energy. However, it turns out that for deformed nuclei in this mass region several states with different spin projections and parities are very close to the Fermi surface at practically the same energy, and tiny details in the interaction can change the ground state from one to another. Given that the spin ($J$) and parity ($\pi $) of these Hg and Pt isotopes are known experimentally, we have chosen these assignments for our odd nucleons that correspond in all cases to states close to the Fermi energy. Namely, the experimental $J^\pi $ assignments of the odd-$A$ Hg isotopes are given by $J^\pi =7/2^-$ for $^{177,179}$Hg; $J^\pi =1/2^-$ for $^{181,183,185}$Hg; and $J^\pi =3/2^-$ for $^{187,189,191}$Hg. Similarly, for odd-$A$ Pt isotopes they are given by $J^\pi =5/2^-$ for $^{173}$Pt; $J^\pi =7/2^-$ for $^{175}$Pt; $J^\pi =5/2^-$ for $^{177}$Pt; $J^\pi =1/2^-$ for $^{179,181,183}$Pt; $J^\pi =9/2^+$ for $^{185}$Pt; and $J^\pi =3/2^-$ for $^{187}$Pt. Besides these values, for each nucleus, we also consider in our calculations the spin and parity corresponding to the energy minimum of the other nuclear shape. All of these values appear as labels in each isotope, where solid (dashed) lines stand for prolate (oblate) shapes.
In the odd-$A$ isotopes we observe a displacement of the GT strength to higher excitation energies with respect to the even neighbor isotopes. This is due to the character of the excitation mentioned in the previous section, where we discussed that 3qp transitions, similar to those of the even isotopes but with the odd orbital blocked, appear only at energies above twice the pairing gap, typically 2-3 MeV. Similarly, the $Q_{EC}$ values are displaced in an equivalent amount since the mass differences involved in the $Q_{EC}$ definitions are sensitive to the pairing energy in a similar way.
To show further the sensitivity of GT strength distributions to the spin and parity of the odd-A parent nucleus, we show in Fig. \[fig\_odd\] and Fig. \[fig\_odd\_pt\] the results for several more choices of spins and parities in $^{181,183,185,187}$Hg and $^{175,177,179,181}$Pt, respectively. These are nuclei that are currently being considered as candidates to measure their GT strength distributions at ISOLDE/CERN, using the TAS technique [@algora], and will complement the measurements already taken in Pb and Hg isotopes [@algora; @submitted; @thesis]. Obviously, the decay patterns should be affected by the spin and parity of the odd nucleon because they determine to a large extent the allowed spin and parities that can be reached in the daughter nucleus. This is especially true in the case of 1qp transitions where the odd nucleon involved determines the low-lying spectrum. Thus, one expects the low-lying GT strength to be especially sensitive to the spin-parity of the odd nucleon. This sensitivity translates immediately to the half-lives that depend on the strength contained below $Q_{EC}$. In the case $^{181}$Hg it is found experimentally that the ground state corresponds to $J^\pi=1/2^-$ with band heads at $J^\pi=7/2^-$ and $J^\pi=13/2^+$. The ground state in $^{183}$Hg is $J^\pi=1/2^-$ with another $J^\pi=7/2^-$ band head and a $J^\pi=13/2^+$ isomer at 266 keV. $^{185}$Hg has again a $J^\pi=1/2^-$ ground state with a $J^\pi=7/2^-$ band head at 34 keV, a $J^\pi=9/2^+$ at 213 keV, and a $J^\pi=13/2^+$ state at 99 keV. $^{187}$Hg is a $J^\pi=3/2^-$ nucleus with a $J^\pi=13/2^+$ band head and a $J^\pi=9/2^+$ state at 162 keV. In our calculations the experimental ground states $J^\pi=1/2^-$ correspond to prolate states with asymptotic quantum Nilsson numbers $[N n_z \Lambda ]K$ given by $[521]1/2$. We also consider prolate $7/2^-$ ($[514]7/2$) states, very close in energy an observed experimentally, as well as prolate $9/2^+$ ($[624]9/2$) states. Finally, we show the results for oblate shapes corresponding to $13/2^+$ ($[606]13/2$) states that are also seen experimentally. In $^{187}$Hg the experimental ground state $J^\pi=3/2^-$ corresponds to the oblate state $[521]3/2$. Besides the prolate $7/2^-$ ($[514]7/2$) with origin in the $f_{7/2}$ spherical orbital, we also consider a second prolate $7/2^-$ ($[503]7/2$) state in $^{183,185,187}$Hg with origin in the $h_{9/2}$ spherical orbital and labeled with an asterisk in Figs. \[fig\_odd\]-\[fig\_odd\_pt\]. These two states lead to quite different profiles of the GT strength distributions. Similarly, the ground state of $^{175}$Pt is experimentally found to be $J^\pi=7/2^-$ with a band head $J^\pi=13/2^+$ at an undetermined energy. The ground state of $^{177}$Pt is $J^\pi=5/2^-$ with a $J^\pi=7/2^+$ at 95 keV. $^{179}$Pt ($^{181}$Pt) has a $J^\pi=1/2^-$ ground state with a $J^\pi=9/2^+$ excited state at 299 keV (276 keV) and a $J^\pi=7/2^-$ excited state at 145 keV (117 keV). In addition to the states considered for Hg isotopes, calculations for Pt isotopes are also shown for prolate $5/2^-$ ($[512]5/2$) states and oblate $7/2^+$ ($[604]7/2$) and $9/2^+$ ($[604]9/2$) states.
As we can see in the figures, the sensitivity of the distributions to the spin-parity is large because of the selection rules of allowed transitions. In these examples it is comparable to the effect from deformation and therefore, one can conclude that odd-$A$ isotopes may not be the best candidates to look for deformation signatures on the $\beta$-decay patterns. On the other hand, this sensitivity could be helpful to get information on the nuclear shape based on the spin and parity of the decaying nucleus, which are characteristic and very different for oblate or prolate shapes. As a matter of fact, the possibility of measuring the GT strength distributions in odd-$A$ nuclei corresponding to the ground and isomeric states separately [@algora], would represent a breakthrough in the sense that the decay patterns of prolate and oblate configurations could be disentangled by selecting properly the spin-parity of the decaying isotope. This information could be used thereafter to infer information on the shape of the ground state of the even-even isotopes.
[cccccllr]{}isotope & && & $J{^\pi}$ & $Q_{EC}$ & $T_{1/2}^{\beta^+/EC}$ && $[N n_z \Lambda ]K^{\pi}$ && $T_{1/2}^{\beta^+/EC}$ $^{181}$Hg & $1/2^-$ & 7.210 & 4.9 && $[521]1/2^-$ & pro & 7.53 &&&&& $[514]7/2^-$ & pro & 3.39 &&&&& $[606]13/2^+$ & obl & 8.13 $^{183}$Hg & $1/2^-$ & 6.385 & 10.7 && $[521]1/2^-$ & pro & 21.55 &&&&& $[514]7/2^-$ & pro & 5.71 &&&&& $[503]7/2^-$ & pro & 45.21 &&&&& $[624]9/2^+$ & pro & 86.73 &&&&& $[606]13/2^+$ & obl & 36.18 $^{185}$Hg & $1/2^-$ & 5.690 & 52.2 && $[521]1/2^-$ & pro & 62.54 &&&&& $[514]7/2^-$ & pro & 9.95 &&&&& $[503]7/2^-$ & pro & 75.38 &&&&& $[624]9/2^+$ & pro & 71.74 &&&&& $[606]13/2^+$ & obl & 84.30 $^{187}$Hg & $3/2^-$ & 4.910 & 114 && $[521]3/2^-$ & obl & 83.19 &&&&& $[514]7/2^-$ & pro & 19.44 &&&&& $[503]7/2^-$ & pro & 194.4 &&&&& $[624]9/2^+$ & pro & 379.8 &&&&& $[606]13/2^+$ & obl & 464.4
[cccccllr]{}isotope & && & $J{^\pi}$ & $Q_{EC}$ & $T_{1/2}^{\beta^+/EC}$ && $[N n_z \Lambda ]K^{\pi}$ && $T_{1/2}^{\beta^+/EC}$ $^{175}$Pt & $7/2^-$ & 7.694 & 7.20 && $[514]7/2^-$ & pro & 2.33 &&&&& $[503]7/2^-$ & pro & 6.96 &&&&& $[512]5/2^-$ & pro & 10.06 &&&&& $[606]13/2^+$ & obl & 3.63 $^{177}$Pt & $5/2^-$ & 6.677 & 11.24 && $[521]1/2^-$ & pro & 15.98 &&&&& $[514]7/2^-$ & pro & 5.53 &&&&& $[503]7/2^-$ & pro & 23.61 &&&&& $[512]5/2^-$ & pro & 19.14 &&&&& $[604]7/2^+$ & obl & 30.91 $^{179}$Pt & $1/2^-$ & 5.811 & 21.25 && $[521]1/2^-$ & pro & 45.02 &&&&& $[514]7/2^-$ & pro & 10.38 &&&&& $[503]7/2^-$ & pro & 86.30 &&&&& $[512]5/2^-$ & pro & 53.49 &&&&& $[604]9/2^+$ & obl & 63.11 $^{181}$Pt & $1/2^-$ & 5.097 & 52.0 && $[521]1/2^-$ & pro & 64.23 &&&&& $[514]7/2^-$ & pro & 17.66 &&&&& $[503]7/2^-$ & pro & 143.6 &&&&& $[512]5/2^-$ & pro & 39.77 &&&&& $[604]9/2^+$ & obl & 51.73
Half-lives
----------
As we have seen above, the sensitivity of the GT strength distributions to the nuclear deformation could be used to get information about the nuclear shape in the neutron-deficient Hg and Pt isotopes. Unfortunately, these measurements are not yet available. However, we have experimental information on the $\beta^+/$EC-decay half-lives that summarize in a single quantity the detailed structure of the GT strength distribution profiles. As we can see from Eq. (\[t12\]), half-lives are no more that integral quantities obtained as sums of the GT strengths weighted with the energy-dependent phase-space factors given by Eq. (\[phase\]). Therefore, it is natural to contrast our calculations with this information.
The experimental half-lives of the neutron-deficient Hg and Pt isotopes can be seen in Figs. \[fig\_t12\_hg\] and \[fig\_t12\_pt\], respectively. The total half-lives taken from [@audi12] contain also contributions from the competing $\alpha$ decay. Using the experimental percentage of the $\beta^+/$EC involved in the total decay, we have extracted the $\beta^+/$EC half-lives, which are displayed in the figures. These half-lives are compared to our calculations using the two shapes (oblate and prolate) that minimize the energy in each isotope. We have joined with dashed lines the results corresponding to the absolute energy minimum in our calculations. The spins and parities of the odd-$A$ isotopes are those considered in Figs. \[fig\_bgt\_hg\_odd\] and \[fig\_bgt\_pt\_odd\]. In both cases, Hg and Pt isotopes, we obtain fair agreement with the trend observed experimentally.
In the heavier Hg isotopes oblate shapes reproduce better the experimental trend, whereas in the lighter Hg isotopes the results are more spread around the data and there is no clear advantage of one shape over the other. No firm conclusions can be extracted on preferences about the shape, except for the higher masses above $A=186$. In the case of Pt isotopes, the prolate shape looks more consistent with the data for $172<A<180$. The spread of results is somehow expected taking into account the uncertainties in the calculations coming from Skyrme forces, pairing gap parameters, residual interactions, $Q_{EC}$ values, and quenching factors included in the calculations. They were discussed in Refs. [@sarri05prc; @moreno06]. In the case of the heavier isotopes the agreement with experiment is somewhat worse, but one has to take into account that in these cases we are dealing with very large half-lives that are the natural consequence of very small $Q_{EC}$ energies as we approach the stable isotopes. Therefore, the half-lives are only sensitive to the very low-energy tail of the GT strength distribution and little changes in this tail can change the half-lives dramatically. In any case, the half-lives of almost stable nuclei can only constrain a tiny portion of the whole GT strength distribution and therefore their significance is minor.
Table \[table.1\] (\[table.2\]) shows the half-lives in the odd-$A$ Hg (Pt) isotopes considered in Fig. \[fig\_odd\] (\[fig\_odd\_pt\]). We show the experimental $J^\pi$, $Q_{EC}$, and $T_{1/2}^{\beta^+/EC}$ values [@audi12] together with the calculated QRPA(SLy4) results obtained for various states and deformations. The dispersion of the results due to the spin and parity of the odd nucleus is apparent.
CONCLUSIONS
===========
In this work we have studied bulk and decay properties of even and odd neutron-deficient Hg and Pt isotopes using a deformed pnQRPA formalism with spin-isospin ph and pp separable residual interactions. The quasiparticle mean field is generated from a deformed HF approach with two-body Skyrme effective interactions, taking SLy4 as a reference and comparing with results from Sk3 and SGII. The formalism includes pairing correlations in the BCS approximation, using fixed gap parameters extracted from the experimental masses. The equilibrium deformations in each isotope are derived self-consistently within the HF procedure obtaining oblate and prolate coexisting shapes in most isotopes. These results are very robust and different schemes including non-relativistic self-consistent treatments with either Skyrme or Gogny interactions, as well as relativistic mean field approaches produce similar results. The isotopic evolution in Hg and Pt chains show a shape transition in agreement with experimental findings. In addition, we have calculated mean square charge radii differences and have compared them to data from laser spectroscopy with reasonable agreement.
Then, we have focused on the main objective in this work, studying the decay properties of these isotopes. We have payed special attention to the deformation dependence of these properties in a search for additional fingerprints of nuclear shapes that would complement the information extracted by other means, such as rotational bands built on low-lying states and quadrupole transition rates among them. We have evaluated the energy distributions of the GT strength for the possible equilibrium shapes and have shown their energy profiles that will be compared with experiments already carried out on Hg isotopes that are being currently analyzed [@algora]. It is also highly encouraged to investigate experimentally the decay of odd-$A$ isotopes from both ground and shape-isomeric states. Measuring separately the decay patterns of states characterized by rather different spins and parities corresponding to different nuclear shapes would be a significant piece of information regarding deformation effects that can be later exploited to learn about the deformation in even systems.
The $\beta^+/$EC half-lives have been calculated and compared to the available experimental information. The reasonable agreement achieved validates the quality of our results. These calculations contribute to extend our knowledge of this interesting mass region characterized by shape coexistence by describing their decay properties in terms of the deformation.
We are grateful to E. Moya de Guerra, A. Algora, E. Nácher, and L. M. Fraile for useful discussions. This work was supported by Ministerio de Economía y Competitividad (Spain) under Contract No. FIS2011–23565 and the Consolider-Ingenio 2010 Programs CPAN CSD2007-00042.
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| ArXiv |
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abstract: 'We study the fine scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position, in $2$ and $3$ dimensions. In $2$d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In $3$d the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.'
address:
- |
IW: Department of Mathematics\
King’s College London\
Strand\
London WC2R 2LS\
England, UK
- |
NY: Department of Mathematics\
King’s College London\
Strand\
London WC2R 2LS\
England, UK
author:
- Igor Wigman
- Nadav Yesha
title: CLT for Planck scale mass distribution of toral Laplace eigenfunctions
---
Introduction
============
Given a smooth compact $d$-manifold ${\mathcal{M}}$ we are interested in the spectral properties of the Laplace-Beltrami operator $\Delta$ on ${\mathcal{M}}$. It is well-known that the eigenvalue spectrum of $\Delta$ is purely discrete, i.e., the set of numbers $E$ admitting a solution to the Helmholtz equation $$\Delta \phi + E\phi = 0$$ is a sequence $\{E_{j}\}_{j\ge 1}$ of numbers ordered with multiplicity in a non-decreasing order such that $ E_j \to \infty $. We denote the corresponding sequence $\{\phi_{j} \}_{j\ge 1}$ of (real-valued) eigenfunctions constituting an orthonormal basis of the square-integrable functions $L^{2}({\mathcal{M}})$ on ${\mathcal{M}}$; the sequence $\{\phi_{j} \}_{j\ge 1}$ is uniquely determined up to the spectral degeneracies (i.e., up to orthogonal transformations in each eigenspace of dimension $\ge 2$).
Shnirelman’s Theorem and Small-Scale Equidistribution
-----------------------------------------------------
Assuming w.l.o.g. that ${\mathcal{M}}$ is unit volume ${\operatorname{Vol}}({\mathcal{M}})=1$, the celebrated Shnirelman’s Theorem [@Sn; @Ze; @CdV] asserts that if ${\mathcal{M}}$ is chaotic (i.e., the geodesic flow on $ \mathcal{M} $ is ergodic), then “most" of the $\{\phi_{j}\}$ are $L^{2}$-equidistributed. In particular, they are equidistributed in position space, i.e., there exists a density $1$ sequence $j_{k}$ such that for all “nice" domains ${\mathcal{A}}\subseteq{\mathcal{M}}$ we have $$\label{eq:L2 mass phi->A/M}
\lim\limits_{k\rightarrow\infty}\int\limits_{{\mathcal{A}}}\phi_{j_{k}}(x)^{2}dx= {\operatorname{Vol}}({\mathcal{A}}).$$ Beyond Shnirelman’s Theorem, Berry’s universality conjecture [@Berry1; @Berry2] implies that for a [*generic*]{} chaotic manifold holds for ${\mathcal{A}}$ shrinking with $k$, slower than the Planck’s scale $E_{j_{k}}^{-1/2}$. More precisely, it states that there exists a density $1$ sequence $\{j_{k}\}_{k}$ so that if $r_{0}(E):{\mathbb{R}}_{> 0}\rightarrow{\mathbb{R}}_{> 0}$ satisfies $r_{0}(E)\cdot E^{1/2}\rightarrow\infty$ diverging arbitrarily slowly, then, for $B_{x}(r)$ the radius $r$ geodesic ball in ${\mathcal{M}}$ centred at $x$, we have $$\label{eq:|mass-exp|=o(r^d)}
\left|\int\limits_{B_{x}(r)}\phi_{j_{k}}(y)^{2}dy - {\operatorname{Vol}}(B_{x}(r)) \right|
= o_{k\rightarrow\infty} (r^{d})$$ uniformly for all $x\in{\mathcal{M}}$ and $r>r_{0}(E_{j_{k}})$, i.e., $$\label{eq:L2mass shrinking uniform}
\sup\limits_{\substack{x\in{\mathcal{M}}\\ r>r_{0}(E_{j_{k}})}}
\left|\frac{\int\limits_{B_{x}(r)}\phi_{j_{k}}(y)^{2}dy}{{\operatorname{Vol}}(B_{x}(r))} - 1\right| \rightarrow 0.$$
The following recent results are rigorous manifestations of the small-scale (“shrinking balls") statement . Luo and Sarnak [@Luo-Sarnak Theorem 1.2] established the small-scale equidistribution for Laplace eigenfunctions on the modular surface (assuming in addition that they are Hecke eigenfunctions) where $r>E^{-\alpha}$ with a small $ \alpha>0 $, and Young [@Young], conditionally on GRH, refined this estimate for $r>E^{-1/6+o(1)}$ holding for *all* such eigenfunctions. Hezari and Rivière [@Hezari-Riviere1], and independently Han [@Han1] established the equidistribution for Laplace eigenfunctions on manifolds of negative curvature on logarithmic scale (i.e., $r>(\log{E})^{-\alpha}$, for some $\alpha>0$), and Han [@Han2] considered random Laplace eigenfunctions on “symmetric" manifolds, of high spectral degeneracy; here the higher the spectral degeneracy is the smaller the allowed scale is. More recently, Han and Tacy [@HanTacy] proved small-scale equidistribution for random Gaussian combinations of eigenfunctions on compact manifolds for $ r>E^{-1/2+o(1)} $, and de Courcy-Ireland [@DeCourcyIreland] showed that, with high probability, the $L^{2}$-mass of random Gaussian spherical harmonics is, up to a small error, equidistributed, slightly above Planck scale.
Toral Laplace eigenfunctions
----------------------------
For the $d$-dimensional torus ${\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$, $d\ge 2$, there are high spectral degeneracies; in this case Lester and Rudnick [@LeRu Theorem 1.1] proved that the small-scale equidistribution is satisfied by a generic Laplace eigenfunction (also considered by Hezari and Rivière [@Hezari-Riviere2]). More precisely, they showed that every o.n.b. $\{\phi_{j}\}$ admits a density one subsequence $\{\phi_{j_{k}}\}$ of Laplace eigenfunctions obeying , with $r_{0}(E)=E^{-\alpha(d)}$, where $\alpha(d)$ is any number smaller than $$\label{eq:alpha(d) LeRu}
\alpha(d)<\frac{1}{2(d-1)},$$ an (almost) optimal Planck-scale result for $d=2$, yet somewhat weaker than Berry’s conjecture for $d> 2$.
One can express the real toral Laplace eigenfunctions explicitly as a sum of exponentials $$\label{eq:fn sum exp}
f_{n}\left(x\right)=\sum_{\lambda\in\mathcal{E}_{n}}c_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right), \hspace{10pt} (c_{-\lambda}=\overline{c_\lambda})$$ for $$\label{eq:S_d}
n\in S_{d}:=\{n=a_{1}^{2}+\ldots +a_{d}^{2}:\: a_{1},\ldots,a_{d}\in{\mathbb{Z}}\}$$ expressible as a sum of $d$ integer squares, and the corresponding frequencies $\lambda$ are the standard lattice points $$\label{eq:E_n}
{\mathcal{E}}_{n} = {\mathcal{E}}_{d;n}=\{\lambda\in{\mathbb{Z}}^{d}:\: \|\lambda\|^{2}=n\}$$ lying on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$; in this case the energy is $E=E_{n}=4\pi^{2}n$. We will assume w.l.o.g. that $f_{n}$ is $L^{2}$-normalised, equivalent to $$\label{eq:BasicNormalization}
\|f_{n}\|_{L^{2}({\mathbb{T}}^{d})}^{2} = \sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=1.$$
For every $ n\in S_d $, denote $$\label{eq:N}
N=N_{d;n}=\#\mathcal{E}_n.$$
When $d=2$, by Landau’s theorem, $
\left\{ n\le x:\,n\in S_{2}\right\} \sim K\frac{x}{\sqrt{\log x}}
$ where $ K>0 $ is the “Landau-Ramanujan constant". On average $N=N_{2;n}$ is of order of magnitude $\sqrt{\log n}$; however, for a density one sequence in $S_{2}$ we have $
N=\left(\log n\right)^{\log2/2+o\left(1\right)}.
$ In general, for $n\in S_{2}$ we have $$N=n^{o\left(1\right)}.$$
For $d=3$, Siegel’s theorem asserts that for $n\not\equiv0,4,7\,\left(8\right)$, $$N=N_{3;n}=n^{1/2+o\left(1\right)};$$ since $x\mapsto2^{a}x$ is a bijection between the solutions to $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=n$ and $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=4^{a}n$, we can always assume that $n\not\equiv0,4,7\,\left(8\right)$ with no loss of generality.
Granville and Wigman [@GranvilleWigman Theorem 1.2] refined the aforementioned estimate by Lester-Rudnick for $d=2$. They proved that in this case, is valid slightly above Planck-scale $r_{0}(E)=E^{-1/2+o(1)}$, for [*all*]{} eigenfunctions $f_{n}$ as in , corresponding to numbers $n$ so that the lattice points ${\mathcal{E}}_{n}$ are well-separated (“Bourgain-Rudnick sequences"), a condition satisfied [@Bourgain-Rudnick Lemma 5] by “generic" integers $n\in S_{2}$ in a strong quantitative sense, subsequently refined in [@GranvilleWigman Theorem 1.4], see section \[sec:quasi corr\].
Averaging mass w.r.t. ball centre
---------------------------------
For both the $2$-dimensional and the higher-dimensional tori it is possible to construct exceptional examples of sequences of toral eigenfunctions where the equidistribution condition is not satisfied: for $d\ge 2$ thin sequences [@LeRu Theorem $3.1$] $\{\phi_{j_{k}}\}$ of eigenfunctions violating condition at Planck-scale $r \cdot E_{j_{k}}^{1/2} \rightarrow \infty$, around the origin $x=0$, and even stronger, for $d\ge 3$ [@LeRu Theorem $4.1$ (construction by J. Bourgain)] eigenfunctions violating with $r \gg E^{-\alpha(d)}$ where $\alpha(d)> \frac{1}{2(d-1)}$, again around the origin $x=0$. In these cases, rather than keeping the ball centre $x=0$ at the origin, one may vary $x$, and study whether the “typical" discrepancy on the l.h.s. of is [*small*]{}, even if the existence of $x$ so that the l.h.s. of is [*not small*]{} is known, so that, in particular, is not satisfied.
A natural way to vary $x$ is to think of $x$ as [*random*]{}, drawn uniformly in ${\mathbb{T}}^{d}$. We define the random variable $$\label{eq:X_RV}
X_{f_{n},r}=X_{f_{n},r;x}:= \int\limits_{B_{x}(r)}f_{n}(y)^{2}dy,$$ and are interested in the distribution of $X_{f_{n},r}$ where $x$ is drawn randomly uniformly in ${\mathbb{T}}^{d}$. The relevant moments are: expectation $$\label{eq:Expectation}
{\mathbb{E}}[X_{f_{n},r}] = \int\limits_{{\mathbb{T}}^{d}}X_{f_{n},r;x}dx,$$ higher centred moments $$\label{eq:centred moments}
{\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{k}] = \int\limits_{{\mathbb{T}}^{d}}\left(X_{f_{n},r;x}-{\mathbb{E}}[X_{f_{n},r}]\right)^{k}dx, \hspace{1em}k\ge2,$$ and in particular the variance $$\label{eq:Variance}
{\mathcal{V}}(X_{f_{n},r}) = {\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{2}].$$
This approach of averaging the $L^{2}$-mass with respect to the ball centre (and keeping $f_{n}$ fixed) was pursued by Granville-Wigman [@GranvilleWigman] in the $2$-dimensional case, again slightly above the Planck scale $r>E^{-1/2+o(1)}$. In this regime, by proving an upper bound for ${\mathcal{V}}(X_{f_{n},r})$ beyond $({\mathbb{E}}[X_{f_{n},r}])^{2} = O(r^4)$, valid for [*all*]{} $n\in S_{2}$, under some flatness assumption on $f_{n}$ (cf. Definition \[def:ultraflat\] below), they established for [*“typical"*]{}, if [*not all*]{} $x\in{\mathbb{T}}^{2}$. It would be desirable to find a regime where it is possible to analyse the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{f_{n},r})$ of $X_{f_{n},r}$, and, if possible, determine the limit distribution law for $X_{f_{n},r}$; our principal results below achieve both of these in the $2$-dimensional case, and the former in the $3$-dimensional one (see theorems \[thm:VarMain\] and \[thm:Var3D\]). Such an approach of bounding the discrepancy variance while averaging over ball centres was recently used by Sarnak [@Sa] for mass distribution of forms on symmetric spaces, and P. Humphries [@Humphries] for mass distribution of automorphic forms.
Statement of the main results for $d=2,3$: asymptotics for the variance, CLT
----------------------------------------------------------------------------
Our principal results below are applicable to “flat" functions for $d=2,3$, understood in suitable, more and less restrictive, senses. For example, “Bourgain’s eigenfunction" [@Bourgain] $$\label{eq:BourgainEF}
f_{n}\left(x\right)=\frac{1}{\sqrt{N}}\sum_{\lambda\in\mathcal{E}_{n}}\varepsilon_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$$ with $\varepsilon_{\lambda}=\pm1$ for every $\lambda\in\mathcal{E}_{n}$, i.e. corresponding to $\left|c_{\lambda}\right|=N^{-1/2},$ satisfies any of the flatness conditions in the most restrictive sense. Denote ${\mathcal{B}}_{n}$ to be the class of Bourgain’s eigenfunctions.
Our first principal result determines the precise asymptotic behaviour of the variance ${\mathcal{V}}(X_{f_n,r})$ for the $2$-dimensional case, and moreover asserts that the moments of the standardized random $L^{2}$-mass of $f_{n}$ are asymptotically Gaussian; we subsequently deduce a Central Limit Theorem (see Corollary \[cor:CLT\_result\]). For the sake of elegance of presentation, it is formulated for Bourgain’s eigenfunctions ; below we formulate a more general result which holds for a larger class of flat eigenfunctions (see Theorem \[thm:VarMainGeneralized\] in section \[sec:statement results strong\]), and later a result where the averaging over the ball centre $x$ is itself restricted to shrinking balls (Theorem \[thm:VarMainExplRestricted\] in section \[sec:RestrictedAverages\]).
\[thm:VarMain\] There exists a density one sequence $S_{2}'\subseteq S_{2}$ so that the following holds. Let $r_{0}=r_{0}\left(n\right)=n^{-1/2}T_{0}\left(n\right)$ with $T_{0}\left(n\right)\to\infty $.
1. Fix a number $\epsilon>0$, and suppose that $ T_0(n) < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} $. Then as $n\to\infty$ along $S_{2}'$ we have $$\label{eq:var asympt d=2 Bourgain}
{\mathcal{V}}\left(X_{f_{n},r}\right)\sim\frac{16}{3 \pi}r^{4}T^{-1}$$ uniformly for all $$\label{eq:r0<r<n^-1/2*discr}
r_{0} < r <n^{-1/2}\left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon}$$ and $ f_n \in {\mathcal{B}}_{n} $, where $T:=n^{1/2}r.$
2. Under the above notation let $$\label{eq:standardizedX}
\hat{X}_{f_{n},r}:=\frac{X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}]}{\sqrt{{\mathcal{V}}\left(X_{f_{n},r}\right)}}$$ be the standardized random $L^{2}$-mass of $f_{n}$, $r_{1}=r_{1}(n)=n^{-1/2}T_{1}\left(n\right)$, and suppose further that the sequence of numbers $T_{1}(n)>T_{0}(n)$ satisfies $T_{1}(n)=O\left(N^{\xi}\right)$ for every $\xi>0$. Then for all $k\ge 3$ the $k$-th the moment of $\hat{X}_{f_{n},r}$ converges, for $n\rightarrow\infty$ along $S_{2}'$, to the standard Gaussian moment $$\label{eq:moments Gaussian lim}
{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] \rightarrow {\mathbb{E}}[Z^{k}],$$ uniformly for $r_{0}<r<r_{1}$ and $ f_n \in {\mathcal{B}}_{n} $, where $Z\sim N(0,1)$ is the standard Gaussian variable.
The claimed uniform asymptotics of the variance means explicitly that, as $n\rightarrow\infty$ along $S_{2}'$, one has $$\label{eq:var unif asymp d=2}
\sup\limits_{\substack{r_0 < r < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} \\ f_n \in {\mathcal{B}}_{n}}} \left|\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{\frac{16}{3 \pi}r^{4}T^{-1}} - 1\right| \rightarrow 0$$ and the uniform convergence of the moments means that for every $k\ge 3$, $$\sup\limits_{\substack{r_0 < r < r_1 \\ f_n \in {\mathcal{B}}_{n}}} \left|{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] - {\mathbb{E}}[Z^{k}]\right| \rightarrow 0.$$ Concerning the restricted range in Theorem \[thm:VarMain\] (and ) for the possible radii, it is directly related to a well-known result on the angular distribution of lattice points in ${\mathcal{E}}_{n}$, for generic $n\in S_{2}$. Namely, it was shown [@ErdosHall] that ${\mathcal{E}}_{n}$, projected by homothety to the unit circle, is equidistributed, and moreover, a quantitative measure for the discrepancy is asserted (see section \[sec:ang distr\] below, and, in particular, ), satisfied by [*generic*]{} $n\in S_{2}$. Bourgain [@Bourgain] observed that $f_{n}\in {\mathcal{B}}_{n}$, when averaged over $x\in {\mathbb{T}}^{d}$, exhibits Gaussianity in the following sense. Let $T>0$ be a fixed number, and define the scaled function $\varphi_{x}:[-1,1]^{2}\rightarrow{\mathbb{R}}$ around $x$ as $$\label{eq:varphi rand x def}
\varphi_{x}(y):= f_{n}\left( x+ \frac{T}{\sqrt{n}}\cdot y\right),$$ i.e. the trace of $f_{n}$ on the side-$2\frac{T}{\sqrt{n}}$ square centred at $x$. It was found [@Bourgain], that, upon thinking of $x\in{\mathbb{T}}^{2}$ as [*random*]{}, and $\varphi_{x}(\cdot)$ as a [*random field*]{} indexed by $[-1,1]^{2}$, it converges, in a suitable sense, to a particular [*Gaussian*]{} field (“monochromatic isotropic waves") on ${\mathbb{R}}^{2}$, restricted to $[-1,1]^{2}$. This allows one to infer some results on the (deterministic) functions $f_{n}\in {\mathcal{B}}_{n}$ from the analogous results on the limit Gaussian random field. We may then reinterpret the quantitative version of the angular equidistribution of lattice points as allowing the parameter $T$ in to grow as a (positive) logarithmic power of $n$, while still retaining the said asymptotic Gaussianity, also allowing for the comparison between the mass distribution of $f_{n}$ w.r.t. the position and mass distribution of monochromatic isotropic waves. Our intuition regarding the possibility of carrying on the explained “de-randomisation" argument for establishing results of similar nature to the presented results was recently validated by Sartori [@Sartori].
An application of the standard theory [@Feller §XVI.3 Lemma 2] allows us to infer a uniform Central Limit Theorem for the random variables $\hat{X}_{f_{n},r}$ from the convergence of their respective moments to the Gaussian ones.
\[cor:CLT\_result\] In the setting of Theorem \[thm:VarMain\] part (2), the distribution of the random variables $\{\hat{X}_{f_{n},r}\}$ converges uniformly to the standard Gaussian distribution: as $n\rightarrow\infty$ along $S_{2}'$ $${\operatorname{meas}}\{ x\in \mathbb{T}^2:\: \hat{X}_{f_{n},r;x} \le t\} \rightarrow
\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{t}e^{-z^{2}/2}dz,$$ uniformly for $t\in{\mathbb{R}}$, $r_{0}<r<r_{1}$ and $ f_n \in {\mathcal{B}}_{n} $.
For the $3$-dimensional case, for Bourgain’s eigenfunctions, we only claim precise asymptotic result on $\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)$, the good news being that the claimed results are valid for [*all*]{} energies satisfying the natural congruence assumptions.
\[thm:Var3D\] There exists a number $\eta>0$ such that if $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$, then for all $n\not\equiv0,4,7\,\left(8\right)$ we have $$\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)\sim r^{6}T^{-2},$$ uniformly for $r_{0} < r < n^{-1/2+\eta}$ and $ f_n \in {\mathcal{B}}_{n} $.
The meaning of the uniform statement in Theorem \[thm:Var3D\] is that $$\sup_{\begin{subarray}{c}
r_{0} < r < n^{-1/2+\eta} \\ f_n \in {\mathcal{B}}_{n} \end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{6}T^{-2}}-1\right|\to0\label{eq:AympVar3D}$$ as $n\to\infty$ along $n\not\equiv0,4,7\,\left(8\right)$, cf. in the $2$-dimensional case.
Statement of the main results for $d=2,3$: more general upper and lower bounds {#sec:statement results weak}
------------------------------------------------------------------------------
Let $f_{n}$ be as in , and consider the vector $$\label{eq:v_def}
\underline{v}:=(|c_{\lambda}|^{2})_{\lambda \in{\mathcal{E}}_{n}} \in {\mathbb{R}}^{\mathcal{E}_n} $$ of the squared absolute values of its coefficients; we denote its normalised $\ell_{\infty}$-norm $$\label{eq:vnorm inf}
[\underline{v}]_{\infty} := N \cdot \max\limits_{\lambda\in{\mathcal{E}}_{n}}|c_{\lambda}|^{2}.$$
\[def:ultraflat\]
We say that an eigenfunction $f_{n}$ in is $\epsilon$-ultraflat if its coefficients satisfy $$\label{eq:ultra_flat_cond}
[\underline{v}]_{\infty} \le N^{\epsilon}.$$ Denote ${\mathcal{U}}_{n;\epsilon}$ to be the class of $\epsilon$-ultraflat functions.
The following couple of theorems establish more general upper and lower bounds on ${\mathcal{V}}(X_{f_n,r})$ in the $2$ and $3$-dimensional cases respectively.
\[thm:UpperBound2d\] There exists a density $1$ sequence $S_{2}'\subseteq S_{2}$ and an absolute constant $C>0$ such that for every $ \epsilon>0 $, $ \eta >0 $, $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$ arbitrarily slowly, and $r=n^{-1/2}T>r_{0}$, as $n\to\infty$ along $S_{2}'$ we have $$\label{eq:bounds var ultraflat d=2}
T^{-1}N^{-2\epsilon}\ll \frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \ll N^{\epsilon}\cdot \left(T^{-1}+(\log{n})^{-\frac{1}{2}\log{\frac{\pi}{2}}+\eta} \right)$$ uniformly for $r_{0} < r< Cn^{-1/2}N^{1-\epsilon}$ and $f_{n}\in {\mathcal{U}}_{n;\epsilon}$, with the constant involved in the “$\ll$"-notation in is absolute for the lower bound, and depends only on $\eta$ for the upper bound. Moreover, the upper bound is valid for the extended range $r > r_{0}$ (with no upper bound on $r$ imposed), and the lower bound is valid for every $ n\in S_2 $.
\[thm:UpperBound3d\] There exists a number $\eta>0$ and a constant $C>0$ such that for every $\epsilon>0$, $r_{0}=r_{0}(n)=n^{-1/2}T_{0}(n)$ with $T_{0}(n)\rightarrow\infty$ arbitrarily slowly, $r=n^{-1/2}T>r_{0}$, and $n\not\equiv 0,4,7\,\left(8\right)$ we have $$\label{eq:bounds var ultraflat d=3}
T^{-2}N^{-2\epsilon} \ll \frac{{\mathcal{V}}(X_{f_{n},r})}{r^{6}} \ll N^{\epsilon} \left( T^{-2}+n^{-\eta}\right),$$ uniformly for $r_{0} < r < Cn^{-1/2}N^{1-\epsilon}$ and $f_{n}\in {\mathcal{U}}_{n;\epsilon}$, where the constants involved in the “$\ll$"-notation are absolute. Moreover, the upper bound in is valid for the extended range $r > r_{0}$.
For Bourgain’s eigenfunctions, the proofs of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\] yield slightly stronger bounds compared to and , namely $$T^{-1}\ll \frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \ll T^{-1}+(\log{n})^{-\frac{1}{2}\log{\frac{\pi}{2}}+\epsilon}$$ for $ d=2 $, and $$T^{-2} \ll \frac{{\mathcal{V}}(X_{f_{n},r})}{r^{6}} \ll T^{-2}+n^{-\eta}$$ for $ d=3 $.
Outline of the paper
--------------------
The rest of the paper is organised as follows. In section \[sec:statement results strong\] we formulate Theorem \[thm:VarMainGeneralized\], which, on one hand generalizes Theorem \[thm:VarMain\] for a larger class of flat eigenfunctions, and on the other hand, explicates a sufficient condition on $ n\in S_2 $ for its statements to hold; a few examples of application of Theorem \[thm:VarMainGeneralized\], corresponding to different asymptotic behaviour of the variance , are also discussed. Section \[sec:Proof\_Main\_thm\_part1\] is dedicated to giving a proof of the first part of Theorem \[thm:VarMain\] (resp. $1$st part of Theorem \[thm:VarMainGeneralized\]), whereas the second part of Theorem \[thm:VarMain\] (resp. $2$nd part of Theorem \[thm:VarMainGeneralized\]) is proved in section \[sec:Proof\_Main\_thm\_part2\]. Theorem \[thm:Var3D\], claiming the precise asymptotics for the $L^{2}$-mass variance for Bourgain’s eigenfunctions in $3$d, is proved in section \[sec:Proof\_3d\_theorem\].
In section \[sec:ProofOfBoundsThm\] we prove the various upper and lower bounds asserted by theorems \[thm:UpperBound2d\] and \[thm:UpperBound3d\]. A refinement of Theorem \[thm:VarMainGeneralized\], where rather than draw $x$ w.r.t. the uniform measure on the full torus, $x$ is drawn on balls slightly above Planck scale, is presented in section \[sec:RestrictedAverages\], and the additional subtleties of its proof as compared to the proof of Theorem \[thm:VarMainGeneralized\] are highlighted. Finally, section \[sec:AuxLemmasProof\] contains the proofs of all auxiliary lemmas, postponed in course of the proofs of the various results.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors of this manuscript wish to express their gratitude to J. Benatar, A. Granville, P. Kurlberg, Z. Rudnick, P. Sarnak and M. Sodin for numerous stimulating and fruitful discussions concerning various aspects of our work, and their interest in our research. It is a pleasure to thank the anonymous referee for his comments on an earlier version of this manuscript. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement n$^{\text{o}}$ 335141.
On Theorem \[thm:VarMain\]: CLT for mass distribution, $d=2$ {#sec:statement results strong}
============================================================
In this section we focus on Theorem \[thm:VarMain\]. Our first goal is to formulate a result, that on one hand generalises the statement of Theorem \[thm:VarMain\] to a larger class of eigenfunctions, and, on the other hand, provides a more explicit control over the generic numbers $n\in S_{2}$. To this end we discuss the angular distribution of $\lambda\in{\mathcal{E}}_{n}$ (section \[sec:ang distr\]), and the spectral correlations (section \[sec:quasi corr\]), also used in the course of the proof of the $3$-dimensional Theorem \[thm:Var3D\]; we will be able to formulate Theorem \[thm:VarMainGeneralized\], as prescribed above, by appealing to these. In section \[eq:examples varying theta\] we consider a few scenarios when Theorem \[thm:VarMainGeneralized\] is applicable, prescribing different asymptotic behaviour for the variance .
Angular equidistribution of lattice points {#sec:ang distr}
------------------------------------------
For every $\lambda=\left(\lambda_{1},\lambda_{2}\right)\in\mathcal{E}_{n}$, write $\lambda_{1}+i\lambda_{2}=\sqrt{n}e^{i\phi}$, and denote the various angles by $$0\le\phi_{1}<\phi_{2}<\dots<\phi_{N}<2\pi.$$ Recall that the discrepancy of the sequence $\phi_{j}$ is defined by $$\label{eq:Discrepancy_2d}
\Delta\left(n\right)=\sup_{0\le a\le b\le2\pi}\left|\frac{1}{N}\cdot \#\left\{ 1\le j\le N:\,\phi_{j}\in\left[a,b\right]\,\text{mod\,}2\pi\right\} -\frac{\left(b-a\right)}{2\pi}\right|.$$
For every $ \epsilon>0 $, we say that $ n\in S_2 $ satisfies the hypothesis $ \mathcal{D}(n,\epsilon) $ if $$\label{eq:D_n_epsilon}
\Delta\left(n\right)\le \left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}.$$ By Erdős-Hall [@ErdosHall Theorem 1], there exists a density one sequence $S_2'(\epsilon)\subseteq S_2 $ such that $ \mathcal{D}(n,\epsilon) $ is satisfied for every $n\in S_2'(\epsilon) $. By a standard diagonalization argument, there exists a density one sequence $ S_2'\subseteq S_2 $ such that $\mathcal{D}(n,\epsilon) $ is satisfied for *every* $ \epsilon>0 $ and $n\in S_2' $ sufficiently large. In particular, the angles $\left\{ \phi_{j}\right\} $ are equidistributed mod $2\pi$ along this sequence, i.e., the lattice points are equidistributed on the corresponding circles.
Spectral correlations in $2d$ (and $3d$) {#sec:quasi corr}
----------------------------------------
For $d=2$, while computing the moments of $X_{f_{n},r}$ (e.g. for Bourgain’s eigenfunction ), with $x$ drawn uniformly on the whole of ${\mathbb{T}}^{2}$, one exploits the orthogonality relations $$\int\limits_{{\mathbb{T}}^{2}}e(\langle \lambda , x \rangle)dx = \begin{cases}
0 &\lambda\ne 0 \\ 1 &\lambda=0
\end{cases}$$ for $\lambda\in{\mathbb{Z}}^{2}$ to naturally encounter the length-$l$ spectral correlation problem. That is, for $l\ge 2$ and $n\in S_{2} $ one is interested in the size of the length-$ l $ spectral correlation set $$\label{eq:Sc correlations def}
{\mathcal{S}}_{n}(l) = \left\{ (\lambda^{1},\ldots,\lambda^{l})\in({\mathcal{E}}_{n})^{l}:\: \sum\limits_{i=1}^{l}\lambda^{i}=0 \right\},$$ which, by an elementary congruence obstruction argument modulo $ 2 $, is only non-empty for $l=2k$ even.
In this case $l=2k$ we further define the [*diagonal*]{} correlations set to be all the permutations of tuples of the form $(\lambda^{1},-\lambda^{1},\ldots, \lambda^{k},-\lambda^{k})$: $$\label{eq:Dc diag def}
{\mathcal{D}}_{n}(l) = \left\{ \pi(\lambda^{1},-\lambda^{1},\ldots,\lambda^{k},-\lambda^{k}): \lambda^{1},\ldots,\lambda^{k}\in ({\mathcal{E}}_{n})^{k},\,\pi\in S_{l}\right\}.$$ The set ${\mathcal{D}}_{n}$ is dominated by non-degenerate tuples (i.e. $\lambda^{i}\ne \pm\lambda^{j}$ for $ i\ne j $), hence its size is asymptotic to $$|{\mathcal{D}}_{n}(l)|= \frac{(2k)!}{2^{k}\cdot k!}N^{k}\cdot \left(1 + O_{N\rightarrow\infty}\left( \frac{1}{N} \right) \right).$$
Clearly, ${\mathcal{D}}_{n}(l)\subseteq {\mathcal{S}}_{n}(l)$ so that, in particular ${\mathcal{S}}_{n}(l) \gg N^{l/2}$. To the other end, we have ${\mathcal{S}}_{n}(2) = {\mathcal{D}}_{n}(2)$ by the definition, and both the precise statement $$\label{eq:Zygmund 4-corr}
{\mathcal{S}}_{n}(4) = {\mathcal{D}}_{n}(4)$$ (used for the variance computation below) and the bound $$|{\mathcal{S}}_{n}(l)| = O_{N\rightarrow\infty}(N^{l-2})$$ follow from Zygmund’s elementary observation [@Zygmund]. For $ l=6 $, Bourgain (published in [@K-K-W]) improved Zygmund’s bound to $$|{\mathcal{S}}_{n}(6)| = o_{N\rightarrow\infty}(N^{4});$$ this was improved [@BombieriBourgain] to $$|{\mathcal{S}}_{n}(6)| = O_{N\rightarrow\infty}(N^{7/2}),$$ valid for [*all*]{} $n\in S_{2}$.
If one is willing to excise a thin sequence in $S_{2}$, then the more striking estimate [@BombieriBourgain] $$|{\mathcal{S}}_{n}(6)| = |{\mathcal{D}}_{n}(6)| + O(N^{3-\gamma}),$$ with some $\gamma >0$, is valid for a density $1$ sequence $S_{2}'\subseteq S_{2}$. More generally [@Bourgain], for every $l\ge 6$ even, there exists a density $1$ sequence $S_{2}'(l)\subseteq S_{2}$ and a number $\gamma_{l}>0$ such that $$\label{eq:corr diag dom l}
|{\mathcal{S}}_{n}(l)| = |{\mathcal{D}}_{n}(l)| + O(N^{l/2-\gamma_l})$$ along $n\in S_{2}'(l)$. A standard diagonal argument then yields the existence of a density $1$ sequence $S_2'\subseteq S_{2}$ so that is valid for [*all*]{} even $l\ge 4$.
Given an even number $l=2k\ge 2$ we say that a sequence $S_{2}'\subseteq S_{2}$ satisfies the length-$l$ [**diagonal domination**]{} assumption if there exists a number $\gamma=\gamma_l >0$ so that holds.
For the $3$-dimensional case under the consideration of Theorem \[thm:Var3D\] the analogous estimates to are required to evaluate the relevant moments of $X_{f_{n},r}$. We define ${\mathcal{S}}_{3;n}$ and ${\mathcal{D}}_{3;n}$ analogously to and respectively, this time the $\lambda^{i}$ are lying on the $2$-sphere of radius $\sqrt{n}$. Unlike the lattice points lying on circles, Zygmund’s argument is not applicable for the $2$-sphere, so that an analogue of is not valid; luckily the asymptotic statement $$\label{eq:3d 4-corr}
|{\mathcal{S}}_{3;n}(4)| = |{\mathcal{D}}_{3;n}(4)| + O\left( N^{7/4 +\epsilon} \right),$$ a key input to the variance computation in Theorem \[thm:Var3D\], was recently established [@BenatarMaffucci]. It was also shown in [@BenatarMaffucci] that the asymptotic diagonal domination for the higher length correlations sets does not hold in the $ 3 $-dimensional case.
A more general version of Theorem \[thm:VarMain\], with explicit control over $S_{2}'$ {#sec:VarMainExpl}
--------------------------------------------------------------------------------------
We are interested in extending Theorem \[thm:VarMain\] to a larger class of eigenfunctions. To this end, we introduce the following notation:
Let $f_{n}$ be an eigenfunction on the $2$-torus corresponding to coefficients $(c_{\lambda})_{\lambda\in{\mathcal{E}}_{n}}$ via , and $\underline{v}\in{\mathbb{R}}^{\mathcal{E}_n}\simeq {\mathbb{R}}^{N}$ as above.
1. Denote $$\label{eq:A4Def}
A_{4} = A_{4}(\underline{v}) = N\sum\limits_{\lambda\in{\mathcal{E}}_{n}} |c_{\lambda}|^{4} = N\cdot \|\underline{v}\|^{2}.$$
2. Given $\lambda\in{\mathcal{E}}_{n}$ let $\lambda_{+}$ be the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$, and $$\label{eq:BasicVariablesV}
V\left(\underline{v} \right):=
N\sum_{\lambda\in\mathcal{E}_{n}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|.$$
3. Let $$\label{eq:alphaDef}
\widetilde{V}(\underline{v}) = \frac{[\underline{v}]_{\infty} \cdot V(\underline{v})}{A_{4}(\underline{v})}.$$
The following Lemma, proved in section \[sec:AuxLemmasProof\], summarizes some basic properties of the quantities in (\[eq:vnorm inf\]), (\[eq:A4Def\]), (\[eq:BasicVariablesV\]) and (\[eq:alphaDef\]):
\[lem:BasicVarProp\] We have
1. $1\le A_{4} \le[\underline{v}]_{\infty}.$
2. $[\underline{v}]_{\infty} \le1+V\left(\underline{v} \right)$.
3. $V\left(\underline{v} \right) \le\widetilde{V}(\underline{v})\le V\left(\underline{v} \right)\left(1+V\left(\underline{v} \right)\right)$.
By we have that $$\label{eq:A4<->theta}
A_{4} = \cos(\theta)^{-2},$$ where $\theta = \theta_{f_n} = \theta(\underline{v},\underline{v_{0}})$ is the angle between $\underline{v}$ and the vector $\underline{v_{0}}= (\frac{1}{N})_{\lambda\in{\mathcal{E}}_{n}}$ corresponding to Bourgain’s eigenfunctions, hence $ \theta $ reflects the proximity of $f_{n}$ to Bourgain’s eigenfunction; by the first part of Lemma \[lem:BasicVarProp\], the angle $\theta$ is restricted to the interval $ \left[0,\arccos \left(1/\sqrt{N}\right)\right] \subseteq [0,\pi/2) $.
Given a sequence $T(n)\rightarrow\infty$ and a sequence $\eta(n)>0$ we define:
1. A sequence $\{{\mathcal{F}}_{1}(n;T(n),\eta(n))\}_{n}$ of families of functions consisting for $n\in S_2$ of all functions $f_{n}$ as in satisfying $$\label{eq:F_1_def}{\mathcal{F}}_{1}(n;T(n),\eta(n)) = \left\{f_{n}:\: \widetilde{V}(\underline{v})< \eta(n) \cdot \frac{T(n)}{\log T(n)} \right\}.$$
2. A sequence $\{{\mathcal{F}}_{2}(n;T(n),\eta(n))\}_{n}$ of families of functions consisting for $n\in S_2$ of all functions $f_{n}$ as in satisfying $$\label{eq:F_2_def}{\mathcal{F}}_{2}(n;T(n),\eta(n)) = \left\{f_{n}:\: [\underline{v}]_{\infty} < T(n)^{\eta(n)} \right\},$$ where we recall the notation for $[\underline{v}]_{\infty}$.
We are now in a position to state the generalized version of Theorem \[thm:VarMain\]:
\[thm:VarMainGeneralized\] Let $r_{0}=r_{0}\left(n\right)=n^{-1/2}T_{0}\left(n\right)$ with $T_{0}\left(n\right)\to\infty$, and $\eta(n)>0$ any vanishing sequence $\eta(n)\rightarrow 0$.
1. Fix a number $\epsilon>0$, and suppose that $ T_0(n) < \left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon} $. Then, if $S_{2}'\subseteq S_{2}$ is a sequence satisfying $ \mathcal{D}(n,\epsilon/2)$ for all $n\in S_{2}'$, as $n\rightarrow\infty$ along $S_{2}'$, we have $$\label{eq:var asympt d=2 precise}
{\mathcal{V}}\left(X_{f_{n},r}\right)\sim\frac{16}{3 \pi \cos^{2}\theta_{f_{n}}}r^{4}T^{-1}$$ with $\theta_{f_{n}}$ as in , uniformly for all $r_{0} < r <n^{-1/2}\left(\log n\right)^{\frac{1}{2}\log\frac{\pi}{2}-\epsilon}$ and $f_{n}\in{\mathcal{F}}_{1}(n;T(n),\eta(n))$, where $T:=T(n)=n^{1/2}r.$
2. Let $k\ge 3$ be an integer, $r_{1}=r_{1}(n)=n^{-1/2}T_{1}\left(n\right)$, and suppose further that the sequence of numbers $T_{1}(n)>T_{0}(n)$ satisfies $T_{1}(n)=O\left(N^{\xi}\right)$ for every $\xi>0$. Suppose that $S_{2}'\subseteq S_{2}$ is a sequence satisfying the length-$2k$ diagonal domination assumption and the hypothesis $\mathcal{D}(n,\epsilon)$ for all $n\in S_{2}'$. Then the $k$-th the moment of $\hat{X}_{f_{n},r}$ converges, as $n\rightarrow\infty$ along $S_{2}'$, to the standard Gaussian moment $${\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] \rightarrow {\mathbb{E}}[Z^{k}],$$ uniformly for $r_{0}<r<r_{1}$ and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$ where $Z\sim N(0,1)$ is the standard Gaussian variable.
Section \[eq:examples varying theta\] exhibits a few scenarios when Theorem \[thm:VarMainGeneralized\] is applicable; as in these the true asymptotic behaviour of the variance genuinely varies together with $\theta_{f_{n}}$, this demonstrates that $\theta_{f_{n}}$ (and hence $A_{4}$) is the proper flatness measure of $f_{n}$, see also examples \[ex:Bourgain\] and \[ex:flat vs nonflat\].
In the setting of Theorem \[thm:VarMainGeneralized\] part (2), the distribution of the random variables $\{\hat{X}_{f_{n},r}\}$ converges uniformly to the standard Gaussian distribution: as $n\rightarrow\infty$ along $S_{2}'$ $${\operatorname{meas}}\{ x\in\mathbb{T}^2 :\: \hat{X}_{f_{n},r;x} \le t\} \rightarrow
\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{t}e^{-z^{2}/2}dz,$$ uniformly for $t\in{\mathbb{R}}$, $r_{0}<r<r_{1}$, and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$.
Some examples of application of Theorem \[thm:VarMainGeneralized\] {#eq:examples varying theta}
------------------------------------------------------------------
\[ex:Bourgain\] Let $ f_n $ be Bourgain’s eigenfunction, so that $ [\underline{v}]_{\infty} =A_4 =1 $ and $ V\left(\underline{v} \right) = \widetilde{V}(\underline{v}) =0 $. For every $ \eta(n)>0,T(n)>1 $ we have $${\mathcal{B}}_{n} \subseteq {\mathcal{F}}_{1}(n;T(n),\eta(n)) \cap {\mathcal{F}}_{2}(n;T(n),\eta(n)).$$ Hence Theorem \[thm:VarMainGeneralized\] implies Theorem \[thm:VarMain\].
The following example exhibits a scenario when an application of Theorem \[thm:VarMainGeneralized\] yields a Central Limit Theorem for $X_{f_{n},r}$, corresponding to asymptotic behaviour of the respective variance ${\mathcal{V}}(X_{f_{n},r})$ which is very different from the behaviour in Theorem \[thm:VarMain\].
\[ex:flat vs nonflat\]
Let $\epsilon>0$, $r_{0}$, and $ T_0(n) $ as in Theorem \[thm:VarMainGeneralized\], and $r_{1}= r_1(n)=n^{-1/2} T_{1}(n) > r_{0}$ with $T_{1}(n) \le (\log{n})^{\frac{1}{2}\log{\frac{\pi}{2}}-\epsilon}$. There exists a density $1$ sequence $S_{2}'\subseteq S_{2}$ so that the following holds. Let $t=t(n)\in (0,1)$ be a number satisfying $t(n) \gg \frac{1}{T_{0}(n)^{\xi}}$ for every $\xi>0$, such that $N\cdot t$ is an integer. We choose an ordering $\lambda^{1},\lambda^{2},\ldots \lambda^{N}\in{\mathcal{E}}_{n}$ such that for every $1\le i \le N-1$ we have that $\lambda^{i+1}$ is the (clockwise) nearest neighbour $\lambda^{i+1}=\lambda^{i}_{+}$, and set $$\left(\left|c_{\lambda^{1}}\right|^{2},,\dots,\left|c_{\lambda^{N}}\right|^{2}\right)=(\underset{\begin{subarray}{c}
Nt\end{subarray}\text{\,times}}{\underbrace{\left(Nt\right)^{-1},\dots\dots,\left(Nt\right)^{-1}}},0\dots,0).$$ Then $$\label{eq:var asympt nonflat}
{\mathcal{V}}(X_{f_{n},r}) \sim \frac{16}{3 \pi}r^{4}t^{-1}T^{-1},$$ uniformly for $r_{0}<r=n^{-1/2}T<r_{1}$, and $f_{n}$ with coefficients $c_{\lambda}$ as above. If, in addition, we have $T_{1}(n)=O(N^{\xi})$ for every $\xi>0$, then the distribution of the standardised random variable $\hat{X}_{f_{n},r}$ converges to standard Gaussian uniformly.
Comparing to we observe that the asymptotic behaviour of the variance for the flat and the non-flat functions respectively is genuinely different, provided that we choose $t(n)\rightarrow 0$; we infer that the proposed flatness measure is the natural choice for this problem. One can also generalise Theorem \[thm:VarMain\] as follows:
\[cor:VarAsympGen\]
Let $\epsilon$, $r_{0}$, $T_{0}(n)$, $r_{1}$ and $T_{1}(n)$ be as in Theorem \[thm:VarMainGeneralized\], and $g:{\mathcal{S}}^{1}\rightarrow{\mathbb{R}}$ a non-negative function of bounded variation such that $ \|g\|_{L^{1}({\mathcal{S}}^{1})}=1$. For $n\in S_{2}$ and $\lambda\in {\mathcal{E}}_{n}$ we set $|\widetilde{c_{\lambda}}|^{2} := g(\lambda/\sqrt{n})$, and normalise the vector $\widetilde{\underline{v}}:=(|\widetilde{c_{\lambda}}|^{2})_{\lambda\in{\mathcal{E}}_{n}}$ by setting $\underline{v} := \frac{\widetilde{\underline{v}}}{\|\widetilde{\underline{v}}\|_{1}}$, i.e. $$\label{eq:v BV norm}
v:=(|{c_{\lambda}}|^{2})_{\lambda\in{\mathcal{E}}_{n}} =
\left(\frac{|\widetilde{c_{\lambda}}|^{2}}{\sum\limits_{\mu\in{\mathcal{E}}_{n}}|\widetilde{c_{\mu}}|^{2}}\right)_{\lambda\in{\mathcal{E}}_{n}}.$$
Then along a generic sequence $S_{2}'\subseteq S_{2}$ we have $${\mathcal{V}}(X_{f_{n},r}) \sim \frac{16}{3 \pi}\|g\|_{L^{2}({\mathcal{S}}^{1})}^{2}r^{4}T^{-1},$$ uniformly for $r_{0}<r=n^{-1/2}T<r_{1}$, and $f_{n}$ with coefficients $c_{\lambda}$ as in . If, in addition, we have $T_{1}(n)=O(N^{\xi})$ for every $\xi>0$, then the distribution of the standardised random variable $\hat{X}_{f_{n},r}$ converges to standard Gaussian.
By Koksma’s inequality (see e.g. [@KuipersNiederreiter]), $A_{4}\left(\underline{v} \right)\sim\left\Vert g\right\Vert _{2}^{2}$ along a density one sequence in $S_{2}$. Also note that $$V(\underline{v}) \ll V\left(g\right),$$ with the l.h.s. as in , and r.h.s. the variation of $g$ on ${\mathcal{S}}^{1}$. In light of Lemma \[lem:BasicVarProp\], both parts of Corollary \[cor:VarAsympGen\] follow from Theorem \[thm:VarMainGeneralized\].
Notation
========
For the convenience of the reader, we summarize here the notation used in our paper.
$ S_d=\{n=a_{1}^{2}+\ldots +a_{d}^{2}:\: a_{1},\ldots,a_{d}\in{\mathbb{Z}}\} $: the set of integers expressible as a sum of $ d $ squares, see .\
${\mathcal{E}}_{n} = {\mathcal{E}}_{d;n}=\{\lambda\in{\mathbb{Z}}^{d}:\: \|\lambda\|^{2}=n\}$: the standard lattice points lying on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$, see .\
$f_{n}\left(x\right)=\sum\limits_{\lambda\in\mathcal{E}_{n}}c_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$: the toral Laplace eigenfunctions, see .\
$ N=N_{d;n}=\#\mathcal{E}_n$: the number of lattice points on the $(d-1)$-dimensional sphere (a circle for $ d=2 $) of radius-$\sqrt{n}$, see .\
$B_{x}(r)$: the radius $r$ geodesic ball in $\mathbb{T}^d$ centred at $x$.\
$X_{f_{n},r}=X_{f_{n},r;x}= \int\limits_{B_{x}(r)}f_{n}(y)^{2}dy$: the $L^{2}$-mass of $ f_n $ restricted to $ B_{x}(r) $, where $ x $ is drawn randomly uniformly in $ \mathbb{T}^d $, see .\
${\mathbb{E}}[X_{f_{n},r}] = \int\limits_{{\mathbb{T}}^{d}}X_{f_{n},r;x}dx$: the expected value of $X_{f_{n},r} $, see .\
${\mathcal{V}}(X_{f_{n},r}) = {\mathbb{E}}[(X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}])^{2}]$: the variance of $X_{f_{n},r} $, see .\
$\hat{X}_{f_{n},r}:=\frac{X_{f_{n},r}-{\mathbb{E}}[X_{f_{n},r}]}{\sqrt{{\mathcal{V}}\left(X_{f_{n},r}\right)}}$: the standardized random $L^{2}$-mass of $ f_n $, see .\
$ T = n^{1/2}r $.\
$\underline{v}=(|c_{\lambda}|^{2})_{\lambda \in{\mathcal{E}}_{n}} \in {\mathbb{R}}^{\mathcal{E}_n} $: the vector of the squared absolute values of the coefficients of $f_n $, see .\
$[\underline{v}]_{\infty} = N \cdot \max\limits_{\lambda\in{\mathcal{E}}_{n}}|c_{\lambda}|^{2}$: the normalised $\ell_{\infty}$-norm of $ \underline{v} $, see .\
$\mathcal{B}_n$: the class of Bourgain’s eigenfunctions $f_{n}\left(x\right)=\frac{1}{\sqrt{N}}\sum\limits_{\lambda\in\mathcal{E}_{n}}\varepsilon_{\lambda}e\left(\left\langle x,\lambda\right\rangle \right)$, where $\varepsilon_{\lambda}=\pm1$ for every $\lambda\in\mathcal{E}_{n}$, see .\
${\mathcal{U}}_{n;\epsilon}$: the class of $ \epsilon $-ultraflat functions, where $ [\underline{v}]_{\infty} \le N^{\epsilon} $, see .\
$A_{4} = A_{4}(\underline{v}) = N\sum\limits_{\lambda\in{\mathcal{E}}_{n}} |c_{\lambda}|^{4} = N\cdot \|\underline{v}\|^{2}$, see .\
$\theta = \theta_{f_n} = \theta(\underline{v},\underline{v_{0}})$: the angle between $\underline{v}$ and the vector $\underline{v_{0}}= (\frac{1}{N})_{\lambda\in{\mathcal{E}}_{n}}$ corresponding to Bourgain’s eigenfunctions, see .\
$V\left(\underline{v} \right)=
N\sum\limits_{\lambda\in\mathcal{E}_{n}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|$, where $\lambda_{+}$ is the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$, see .\
$\widetilde{V}(\underline{v}) = \frac{[\underline{v}]_{\infty} \cdot V(\underline{v})}{A_{4}(\underline{v})}$, see .\
${\mathcal{F}}_{1}(n;T(n),\eta(n)) = \left\{f_{n}:\: \widetilde{V}(\underline{v})< \eta(n) \cdot \frac{T(n)}{\log T(n)} \right\}$, see .\
${\mathcal{F}}_{2}(n;T(n),\eta(n)) = \left\{f_{n}:\: [\underline{v}]_{\infty} < T(n)^{\eta(n)} \right\}$, see .\
$\widehat{\lambda}=\lambda/\sqrt{n}$: the projection of $\lambda \in \mathcal{E}_n$ onto $\mathcal{S}^{d-1}.$\
$\Delta\left(n\right)=\sup\limits_{0\le a\le b\le2\pi}\left|\frac{1}{N}\cdot \#\left\{ 1\le j\le N:\,\phi_{j}\in\left[a,b\right]\,\text{mod\,}2\pi\right\} -\frac{\left(b-a\right)}{2\pi}\right|$: the discrepancy of the angles $ \phi_j $ corresponding to the lattice points $ {\mathcal{E}}_{2;n} $, see .\
Hypothesis $ \mathcal{D}(n,\epsilon) $ holds if $\Delta\left(n\right)\le \left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}$, see .\
$\Delta_{3}\left(n\right)=\sup\limits_{\begin{subarray}{c}
x\in\mathcal{S}^{2}\\
0<r\le2
\end{subarray}}\left|\frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{3;n}:\,\left|\widehat{\lambda}-x\right|\le r\right\} -\frac{r^{2}}{4}\right|$: the spherical cap discrepancy of the points $ \mathcal{E}_{3;n} $, see .\
${\mathcal{S}}_{n}(l) = \left\{ (\lambda^{1},\ldots,\lambda^{l})\in({\mathcal{E}}_{n})^{l}:\: \sum\limits_{i=1}^{l}\lambda^{i}=0 \right\}$: the length-$ l $ spectral correlation set, see .\
${\mathcal{D}}_{n}(l) = \left\{ \pi(\lambda^{1},-\lambda^{1},\ldots,\lambda^{k},-\lambda^{k}): \lambda^{1},\ldots,\lambda^{k}\in ({\mathcal{E}}_{n})^{k},\,\pi\in S_{l}\right\}$: the diagonal correlations set, see .\
$\mathcal{A}_n (2k) = \left\{\left(\lambda_{1},\dots,\lambda_{2k}\right)\in{\mathcal{D}}_{n}(2k): \; \forall 1\le i\le k \; \lambda_{2i-1}\ne-\lambda_{2i} \right\}$: the set of “admissible” $ 2k $-tuples of lattice points, see .\
$S\left(\lambda_{1},\dots,\lambda_{2k}\right)$: the structure set of an admissible $ 2k $-tuple $\left(\lambda_{1},\dots,\lambda_{2k}\right)$, see .\
$J_{\alpha}\left(x\right)$: the Bessel function of the first kind of order $\alpha$.\
$g_{d}\left(x\right)=\frac{J_{d/2}\left(2 \pi x\right)}{(2 \pi x)^{d/2}}$: the Fourier transform of the characteristic function of the unit ball in $\mathbb{R}^{d}$, see .\
$ h_{2}\left(x\right)=\frac{J_{1}\left(2 \pi x\right)^{2}}{(2\pi x)^{2}}$, see .\
$h_{3}\left(x\right)=2\pi^{-1}(2 \pi x)^{-4}\left(\frac{\sin 2 \pi x}{2\pi x}-\cos 2 \pi x\right)^{2}$, see .\
$F_{\lambda_{0}}\left(s\right)=\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{2;n}:\,\left\Vert \widehat{\lambda}-\widehat{\lambda_{0}}\right\Vert \le s\right\}$, see .\
$F\left(s\right)=F_{f_{n}}\left(s\right)=\sum\limits_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{2;n}\\
0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}$, see .\
$F_{3}\left(s\right)=\frac{1}{N^{2}} \cdot \#\left\{ \lambda\ne\lambda'\in\mathcal{E}_{3;n}:\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}$, see .\
${\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}X_{f_{n},r;x}dx$: the “restricted” expected value of $X_{f_{n},r} $, see .\
${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = {\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{2}]$: the restricted variance of $X_{f_{n},r} $, see .\
$\mathcal{C}_{n}(l;K) =
\left\{(\lambda^{1},\ldots,\lambda^{l})\in\mathcal{E}_{n}^{l}:\: 0 < \left\| \sum\limits_{j=1}^{l}\lambda^{j} \right\| \le K \right\}$: the set of length-$l$ spectral quasi-correlations, see .\
Hypothesis $\mathcal{A}(n;l,\delta)$ holds if $\mathcal{C}_{n}(l;n^{1/2-\delta}) = \varnothing$, see .
Proof of Theorem \[thm:VarMainGeneralized\], part 1: asymptotics for the variance, $d=2$. {#sec:Proof_Main_thm_part1}
=========================================================================================
Expressing the variance
-----------------------
We begin with some preliminary expressions for the variance. Note that if $x$ is drawn randomly, uniformly on $\mathbb{T}^{d}$, then $$\mathbb{E}\left[X_{f_{n},r}\right]=\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d},\label{eq:ExpectationEquality}$$ and therefore in this case, we have $$\label{eq:variance_integral_form}
{\mathcal{V}}(X_{f_{n},r}) = \int\limits_{{\mathbb{T}}^{d}}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- \frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d} \right)^{2}dx.$$ Let $J_{\alpha}\left(x\right)$ be the Bessel function of the first kind of order $\alpha$. The following lemma, proved in section \[sec:AuxLemmasProof\], explicates the inner integral in :
\[lem:InnerIntegral\]We have $$\begin{aligned}
& \int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d} =\left(2\pi\right)^{d/2}r^{d}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}c_{\lambda}\overline{c_{\lambda'}}e\left(\left\langle x,\lambda-\lambda'\right\rangle \right)g_{d}\left(r\left\Vert \lambda-\lambda'\right\Vert \right),\label{eq:IntegrandVar}
\end{aligned}$$ where $$\label{eq:g_d}
g_{d}\left(x\right):=\frac{J_{d/2}\left(2 \pi x\right)}{(2 \pi x)^{d/2}}$$ is the Fourier transform of the characteristic function of the unit ball in $\mathbb{R}^{d}$.
The following formula for the variance follows from Lemma \[lem:InnerIntegral\], and :
\[lem:VarExpd2\]
1. (Granville-Wigman [@GranvilleWigman Lemma 2.1]) For $d=2$ we have $$\label{eq:VarFormula2d}
\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=8\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)$$ where $$\label{eq:h_2}
h_{2}\left(x\right):=\frac{J_{1}\left(2 \pi x\right)^{2}}{(2\pi x)^{2}}.$$
2. For $d=3$ and for every $\epsilon>0$, we have $$\begin{aligned}
\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right) & =16\pi^{3}r^{6}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{3}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)\label{eq:VarFormula3d}\\
& +O\left([\underline{v}]_{\infty}^2 r^{6}N^{-1/4+\epsilon}\right),\nonumber
\end{aligned}$$ where $$\label{eq:h_3}
h_{3}\left(x\right):=2\pi^{-1}(2 \pi x)^{-4}\left(\frac{\sin 2 \pi x}{2\pi x}-\cos 2 \pi x\right)^{2}.$$
Note that functions $g_{2}$ and $h_{2}$ satisfy the following properties:
\[lem:H2Formulas\]We have
1. $\int_{0}^{\infty}h_{2}\left(s\right)\,\text{d}s=\frac{2}{3\pi^2}$.
2. $g_{2}\left(s\right)\sim\frac{1}{2}\hspace{1em}\left(s\to0\right)$.
3. $g_{2}\left(s\right)\ll s^{-3/2}\hspace{1em}\left(s\to\infty\right)$.
4. $g_{2}'\left(s\right)=-\frac{J_{2}\left(2\pi s\right)}{s}\ll\left(1+s\right)^{-3/2}.$
Proof of Theorem \[thm:VarMainGeneralized\], part 1:
----------------------------------------------------
For $\lambda\in\mathcal{E}_{n}$ let $\widehat{\lambda}=\lambda/\sqrt{n}$ be the projection of $\lambda$ onto the unit circle $\mathcal{S}^{1}.$
1. For $ \lambda_0 \in \mathcal{E}_n $ and $0\le s\le2$, denote $$\label{eq:F_Lambda_Def}
F_{\lambda_{0}}\left(s\right)=\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\left\Vert \widehat{\lambda}-\widehat{\lambda_{0}}\right\Vert \le s\right\} .$$
2. For $0\le s\le2$ denote $$F\left(s\right)=F_{f_{n}}\left(s\right)=\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}.\label{eq:F_Function}$$
Recall that $\widetilde{V}(\underline{v})=\cos^{2}\theta \cdot [\underline{v}]_{\infty} V(\underline{v})$ by and .
\[prop:DiscreteProp\]We have $$\begin{aligned}
F\left(s\right) & =\frac{s}{\pi\cos^{2}\theta}\left(1+O\left(s^{2}+s^{-1}\Delta\left(n\right)+\widetilde{V}(\underline{v}) s+\widetilde{V}(\underline{v}) s^{-1}\Delta\left(n\right)^{2}\right)\right).
\end{aligned}$$
We postpone the proof of Proposition \[prop:DiscreteProp\] until section \[sec:ProofOfLemmaDiscrete\] to present the proof of the first part of Theorem \[thm:VarMainGeneralized\] (that yields the first part of Theorem \[thm:VarMain\]):
Assume that $ n\in S_2 $ satisfies the hypothesis $ \mathcal{D}(n,\epsilon/2) $. We may rewrite (\[eq:VarFormula2d\]) as $$\mathcal{V}\left(X_{f_{n},r}\right)=8\pi^{2}r^{4}\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F\left(s\right).\label{eq:VarianceintegralForm}$$ We apply integration by parts to (\[eq:VarianceintegralForm\]) twice, in opposite directions: first, by integration by parts and Proposition \[prop:DiscreteProp\], we get $$\begin{aligned}
8\pi^{2}r^{4}\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F\left(s\right) & =8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi^{2}r^{4}\int_{0}^{2}F\left(s\right)\,\text{d}h_{2}\left(Ts\right)\label{eq:VarEqAfterIntByParts}\\
& =8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right)\nonumber \\
& +Err\left(X_{f_n,r}\right)\nonumber
\end{aligned}$$ where $$\begin{aligned}
& Err\left(X_{f_n,r}\right)\ll r^{4}\cos^{-2}\theta\int_{0}^{2}\left(s^{3}+\Delta\left(n\right)+\widetilde{V}(\underline{v}) s^{2}+\widetilde{V}(\underline{v})\Delta\left(n\right)^{2}\right)T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s.
\end{aligned}$$ Integrating by parts again, the first two terms on the r.h.s of (\[eq:VarEqAfterIntByParts\]) satisfy $$\begin{aligned}
& 8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right)=8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)\label{eq:MainTermsVar}\\
& -16\pi r^{4}h_{2}\left(2T\right)\cos^{-2}\theta+8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}s.\nonumber
\end{aligned}$$ By the first and the third parts of Lemma \[lem:H2Formulas\], $$\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}s=\frac{1}{T}\int_{0}^{2T}h_{2}\left(s\right)\,\text{d}s=\frac{2}{3\pi^2}T^{-1}+O\left(T^{-3}\right),\label{eq:h2Integral}$$ and therefore, substituting (\[eq:h2Integral\]) into (\[eq:MainTermsVar\]), we obtain $$\begin{aligned}
\label{eq:VarianceMainTerms}
8\pi^{2}r^{4}h_{2}\left(2T\right)F\left(2\right)-8\pi r^{4}\cos^{-2}\theta\int_{0}^{2}s\,\text{d}h_{2}\left(Ts\right) & =\frac{16}{3\pi}\cos^{-2}\theta r^{4}T^{-1} +O\left(\cos^{-2}\theta r^{4}T^{-3}\right).
\end{aligned}$$ By the fourth part of Lemma \[lem:H2Formulas\], $$\begin{aligned}
\int_{0}^{2}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s & =\int_{0}^{2T}\left|h_{2}'\left(s\right)\right|\,\text{d}s\le\int_{0}^{\infty}\left|h_{2}'\left(s\right)\right|\,\text{d}s<\infty,
\end{aligned}$$ $$\int_{0}^{2}s^{2}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s=T^{-2}\int_{0}^{2T}s^{2}\left|h_{2}'\left(s\right)\right|\,\text{d}s\ll T^{-2}\log T$$ and $$\int_{0}^{2}s^{3}T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s=T^{-3}\int_{0}^{2T}s^{3}\left|h_{2}'\left(s\right)\right|\,\text{d}s\ll T^{-2},$$ and therefore for $ n $ satisfying $ \mathcal{D}(n,\epsilon/2), $ $$\begin{aligned}
Err\left(X_{f_n,r}\right) & \ll\cos^{-2}\theta r^{4}\left(T^{-2}+\Delta\left(n\right)+\widetilde{V}(\underline{v}) T^{-2}\log T+\widetilde{V}(\underline{v})\Delta\left(n\right)^{2}\right)\nonumber \\
& \ll\cos^{-2}\theta r^{4}\left(T^{-2}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\frac{\epsilon}{2}}+\widetilde{V}(\underline{v}) T^{-2}\log T+\widetilde{V}(\underline{v})\left(\log n\right)^{-\log\frac{\pi}{2}+\epsilon}\right),\label{eq:ErrorTerm}
\end{aligned}$$ and follows from , and .
Note that by (\[eq:ErrorTerm\]), for Bourgain’s eigenfunctions, for almost all $n\in S_{2}$ we have $$\sup_{\begin{subarray}{c}
r > r_{0} \\ f_n\in {\mathcal{B}}{n}
\end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{4}}-\frac{16}{3\pi}T^{-1}\right|=O\left(T_{0}^{-2}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}\right)$$ for every $\epsilon>0$, and in particular $$\label{eq:o_r4_2d}
\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=o\left(r^{4}\right)$$ uniformly for $r > r_{0}$ for a density one sequence in $S_{2}$. Therefore, serves as a refinement of [@GranvilleWigman Corollary 1.10] for this specific case (for a density one sequence in $S_{2}$), since [@GranvilleWigman Corollary 1.10] yields $\mathcal{V}\left(X_{f_n,r}\right)=o\left(r^{4}\right)$ under the additional assumption $T_{0}\gg n^{4\epsilon}$.
Proof of Proposition \[prop:DiscreteProp\] {#sec:ProofOfLemmaDiscrete}
------------------------------------------
In this section we prove Proposition \[prop:DiscreteProp\]. First, we define a binary relation on $\mathcal{E}_n$:
for $\lambda\ne-\lambda'\in\mathcal{E}_{n}$, we say that $\lambda\prec\lambda'$ if the arc on the circle $\sqrt{n}\mathcal{S}^{1}$ that connects $\lambda$ to $\lambda'$ counter-clockwise to $\lambda'$ is shorter than the arc that connects them clockwise to $\lambda'$. Recall that $\lambda_{+}$ is the clockwise nearest neighbour of $\lambda$ on $\sqrt{n}\mathcal{S}^{1}$. The proof of Proposition \[prop:DiscreteProp\] employs the following auxiliary lemma to be proved at section \[sec:AuxLemmasProof\], establishing (\[eq:F\_Function\]) in the particular case $\left|c_{\lambda}\right|^{2}=1$ for every $\lambda\in\mathcal{E}_{n}$:
\[lem:CosToDist\]
Fix $\lambda'\in\mathcal{E}_{n}.$ For $0\le s<2$, we have $$\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\succeq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} =\frac{s}{2\pi}+O\left(s^{3}+\Delta\left(n\right)\right)\label{eq:CosToDistEq}$$ where the constant involved in the ’O’-notation in (\[eq:CosToDistEq\]) is absolute.
The estimate (\[eq:CosToDistEq\]) is also valid with either ‘$\succ$’, ‘$\preceq$’ or ‘$\prec$’ in place of ‘$\succeq$’.
We are now in a position to prove Proposition \[prop:DiscreteProp\]:
First, we write $$\begin{aligned}
F\left(s\right) & =\sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\preceq\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}+\sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\succeq\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\label{eq:PartialSummation} +O\left(\frac{A_{4}}{N}\right).
\end{aligned}$$ Using summation by parts, we get that for every $\lambda'\in\mathcal{E}_{n}$ $$\begin{aligned}
\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\preceq\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2} & =\left|c_{\lambda'}\right|^{2} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\preceq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} \label{eq:PartialSumEq}\\
& -\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\prec\lambda'
\end{subarray}}\left(\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right) \cdot \#\left\{ \mu\in\mathcal{E}_{n}:\,\mu\preceq\lambda,\,\left\Vert \widehat{\mu}-\widehat{\lambda'}\right\Vert \le s\right\} .\nonumber
\end{aligned}$$ By Lemma \[lem:CosToDist\], the contribution of the first term on the r.h.s of (\[eq:PartialSumEq\]) to $F\left(s\right)$ is $$\begin{aligned}
\label{eq:first_term_discrete}
\sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{4} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\preceq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}
=A_{4} \cdot
\left(s/2\pi+O\left(s^{3}+\Delta\left(n\right)\right)\right).
\end{aligned}$$ The contribution of the sum on the r.h.s of (\[eq:PartialSumEq\]) to $F\left(s\right)$ is $$\begin{aligned}
\label{eq:second_term_discrete}
& \sum_{\lambda'\in\mathcal{E}_{n}}\left|c_{\lambda'}\right|^{2}\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\prec\lambda'
\end{subarray}}\left(\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right) \cdot \#\left\{ \mu\in\mathcal{E}_{n}:\,\mu\preceq\lambda,\,\left\Vert \widehat{\mu}-\widehat{\lambda'}\right\Vert \le s\right\} \\
& \ll N\left(s+\Delta\left(n\right)\right)\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\end{subarray}}\left|\left|c_{\lambda_{+}}\right|^{2}-\left|c_{\lambda}\right|^{2}\right|\sum_{\begin{subarray}{c}
\lambda'\in\mathcal{E}_{n} \nonumber \\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\prec\lambda'
\end{subarray}}\left|c_{\lambda'}\right|^{2}\ll\left(s+\Delta\left(n\right)\right)^{2}[\underline{v}]_{\infty} V(\underline{v}) .
\end{aligned}$$ By and , we have $$\label{eq:first_summation_discrete}
\sum_{\begin{subarray}{c}
\lambda\in\mathcal{E}_{n}\\
\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\\
\lambda\preceq\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2} = A_{4} \cdot
\left(s/2\pi+O\left(s^{3}+\Delta\left(n\right)\right)\right) + O\left(\left(s+\Delta\left(n\right)\right)^{2}[\underline{v}]_{\infty}V(\underline{v}) \right).$$ By symmetry, the second summation in (\[eq:PartialSummation\]) obeys with ‘$ \succ $’, ‘$ \succeq $’ and $ |c_{\lambda_-}|^2 $ in place of ‘$ \prec $’, ‘$ \preceq $’ and $ |c_{\lambda_+}|^2 $, where $ \lambda_- $ is the counter-clockwise nearest neighbour to $ \lambda $. The statement of Proposition \[prop:DiscreteProp\] follows from the analogues of the estimates , and .
Proof of Theorem \[thm:VarMainGeneralized\], part 2: Gaussian moments, $d=2$. {#sec:Proof_Main_thm_part2}
=============================================================================
In this section we study the higher moments of $\hat{X}_{f_{n},r}$ defined in , and prove the second part of Theorem \[thm:VarMainGeneralized\], also implying the second part of Theorem \[thm:VarMain\].
The proof of the following lower bound for $ {\mathcal{V}}\left(X_{f_{n},r}\right) $ with $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$ goes along the same lines as the proof of the lower bound in Theorem \[thm:UpperBound2d\] below:
\[lem:Lower\_Bound\_F2\] In the setting of Theorem \[thm:VarMainGeneralized\] part (2), we have $$\frac{{\mathcal{V}}(X_{f_n,r})}{r^{4}} \gg T(n)^{-1-2\eta(n)}$$ uniformly for $r_{0} < r < r_1$ and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$.
Before proceeding to the proof of Theorem \[thm:VarMainGeneralized\], we introduce some notation:
1. Define the set of “admissible” $2k$-tuples of lattice points by $$\label{eq:admissible_tuples}
\mathcal{A}_n (2k) = \left\{\left(\lambda_{1},\dots,\lambda_{2k}\right)\in{\mathcal{D}}_{n}(2k): \; \forall 1\le i\le k \; \lambda_{2i-1}\ne-\lambda_{2i} \right\}.$$
2. Given an admissible $2k$-tuple of lattice points $\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)$, let $\sim$ be the equivalence relation on the set $\left\{ 1,\dots,2k\right\} $, generated by:
1. $2i-1\sim2i$ for every $1\le i\le k$.
2. $j\sim j'$ if $\lambda_{j}+\lambda_{j}'=0.$
Let $\left\{ \Lambda_{1},\dots,\Lambda_{m}\right\} $ be the partition of $\left\{ 1,\dots,2k\right\} $ into equivalence classes of $\sim$, and denote $l_{j}=\#\Lambda_{m}/2$ for $1\le j\le m$, so that $\sum_{j=1}^{m}l_{j}=k$; clearly, $2\le l_{j}\in\mathbb{Z}$ for every $1\le j\le m$. We call the multiset $$\label{eq:structure_set}
S\left(\lambda_{1},\dots,\lambda_{2k}\right):=\left\{ l_{1},\dots,l_{m}\right\}$$ the structure set of the $2k$-tuple $\left(\lambda_{1},\dots,\lambda_{2k}\right).$
Recall that the moments of a standard Gaussian random variable $Z\sim N(0,1)$ are $${\mathbb{E}}[Z^{k}]=\begin{cases}
\left(k-1\right)!! & k\,\text{even}\\
0 & k\,\text{odd}.
\end{cases}$$ We are now in a position to prove the second part of Theorem \[thm:VarMainGeneralized\].
By the length-$2k$ diagonal domination assumption, we have $$\begin{aligned}
{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] & =(2\pi)^k r^{2k}{\mathcal{V}}\left(X_{f_{n},r}\right)^{-k/2}\sum_{\begin{subarray}{c}
\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\end{subarray}}\label{eq:kthMoment}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right) \\ &+O\left({\mathcal{V}}\left(X_{f_{n},r}\right)^{-k/2} [\underline{v}]_{\infty}^k r^{2k}N^{-\gamma}\right)\nonumber
\end{aligned}$$ for some $ \gamma>0. $ We can rearrange the summation in (\[eq:kthMoment\]), first summing over all possible structure sets $\mathcal{L}=\left\{ l_{1},\dots,l_{m}\right\} $ and then summing over the admissible $2k$-tuples $\left(\lambda_{1},\dots,\lambda_{2k}\right)\in\mathcal{E}_{n}^{2k}$ with the given structure set $S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L} $: let $$\begin{aligned}
S_{\mathcal{L}} & :=\sum_{\begin{subarray}{c}
\begin{subarray}{c}
\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\\
S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L}
\end{subarray}\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right),
\end{aligned}$$ so that we may rewrite the summation on the r.h.s. of as $$\begin{aligned}
\label{eq:Inner_Sum_with_SL}
& \sum_{\begin{subarray}{c}
\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{2j-1}}c_{\lambda_{2j}}g_{2}\left(r\left\Vert \lambda_{2j-1}+\lambda_{2j}\right\Vert \right)=\sum_{\begin{subarray}{c}
l_{1}+\dots+l_{m}=k\\
l_{1},\dots,l_{m}\ge2
\end{subarray}}\sum_{\begin{subarray}{c}
\begin{subarray}{c}
\left(\lambda_{1},\dots,\lambda_{2k}\right)\in \mathcal{A}_n (2k)\\
S\left(\lambda_{1},\dots,\lambda_{2k}\right)=\mathcal{L}
\end{subarray}\end{subarray}}S_{\mathcal{L}}.
\end{aligned}$$ For a fixed structure set $\mathcal{L}=\left\{ l_{1},\dots,l_{m}\right\} $, we have $$\begin{aligned}
S_{\mathcal{L}} =a\left(\mathcal{L}\right)\prod_{j=1}^{m}\sum_{\lambda_{1},\dots,\lambda_{l_{j}}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}g_{2}\left(r\left\Vert \lambda_{l_{j}}-\lambda_{1}\right\Vert \right)\label{eq:RearrangeInPartition}\prod_{i=1}^{l_{j}-1}\left|c_{\lambda_{i+1}}\right|^{2}g_{2}\left(r\left\Vert \lambda_{i}-\lambda_{i+1}\right\Vert \right)+O\left([\underline{v}]_{\infty}^k N^{-1}\right)
\end{aligned}$$ where $a\left(\mathcal{L}\right)$ is a constant depending on $\mathcal{L}$; omitting the condition that the lattice points are distinct on the r.h.s of (\[eq:RearrangeInPartition\]) is absorbed within the error term in (\[eq:RearrangeInPartition\]). Thus, $$\begin{aligned}
\label{eq:SL_upper_bound}
S_{\mathcal{L}} & \ll [\underline{v}]_{\infty}^k N^{-k}\prod_{j=1}^{m}\sum_{\lambda_{1}\in\mathcal{E}_{n}}\sum_{\lambda_{2}\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda_{2}-\lambda_{1}\right\Vert \right)\right|\cdots\sum_{\lambda_{l_{j}}\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda_{l_{j}-1}-\lambda_{l_{j}}\right\Vert \right)\right|+[\underline{v}]_{\infty}^k N^{-1}.
\end{aligned}$$ Recall the definition of $ F_{\lambda_0} $ in . By Lemma \[lem:CosToDist\], we have $$\label{eq:f_lambda_zero}
F_{\lambda_{0}}\left(s\right)=\frac{s}{\pi}+O\left(s^{3}+\Delta\left(n\right)\right)=O\left(s+\Delta\left(n\right)\right).$$ Thus, by Lemma \[lem:H2Formulas\] and , we have that $$\begin{aligned}
\label{eq:abs_g2_bound}
\frac{1}{N}\sum_{\lambda\in\mathcal{E}_{n}}\left|g_{2}\left(r\left\Vert \lambda-\lambda_{0}\right\Vert \right)\right| & =\int_{0}^{2}\left|g_{2}\left(Ts\right)\right|\,\text{d}F_{\lambda_{0}}\left(s\right)\\
& =\left|g_{2}\left(2T\right)\right|-\frac{1}{2N}+O\left(\int_{0}^{2}\left(s+\Delta\left(n\right)\right)T\left|g_{2}'\left(Ts\right)\right|\,\text{d}s\right) \nonumber \\
& =O\left(T^{-3/2}+\left(\Delta\left(n\right)+T^{-1}\right)\int_{0}^{2T}\left|g_{2}'\left(s\right)\right|\,\text{d}s\right) \nonumber \\
& =O\left(T^{-1}\right) \nonumber
\end{aligned}$$ for $ n $ satisfying the hypothesis $ \mathcal{D}(n,\epsilon) $. Applying to each of the $l_{j}-1$ inner summations in , we obtain
$$\begin{aligned}
S_{\mathcal{L}} & \ll [\underline{v}]_{\infty}^k N^{-k+m}\prod_{j=1}^{m}\left(NT^{-1}\right)^{l_{j}-1} + [\underline{v}]_{\infty}^k N^{-1} \ll [\underline{v}]_{\infty}^k T^{-k+m}.
\end{aligned}$$
Let $ \mathcal{L}_0 = \left\{ 2,2,\dots2,\right\} $. Note that if $\mathcal{L}\ne \mathcal{L}_0 $ then $m\le\frac{k-1}{2}$ and therefore $$\label{eq:SL_Estimate}
S_{\mathcal{L}}=O\left([\underline{v}]_{\infty}^k T^{-\frac{k+1}{2}}\right).$$ If $\mathcal{\mathcal{L}}=\mathcal{L}_0 $ (this is a viable option for $k$ even), then $$\label{eq:SL_formula}
S_{\mathcal{L}_0}=2^{k/2}\left(k-1\right)!!\left[\sum_{\lambda_{1}\ne\lambda_{2}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}\left|c_{\lambda_{2}}\right|^{2}h_{2}\left(r\left\Vert \lambda_{1}-\lambda_{2}\right\Vert \right)\right]^{k/2}+O\left([\underline{v}]_{\infty}^k N^{-1}\right).$$ By (\[eq:VarFormula2d\]), $$\label{eq:SL_Variance_asympt}
\sum_{\lambda_{1}\ne\lambda_{2}\in\mathcal{E}_{n}}\left|c_{\lambda_{1}}\right|^{2}\left|c_{\lambda_{2}}\right|^{2}h_{2}\left(r\left\Vert \lambda_{1}-\lambda_{2}\right\Vert \right)=\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{8\pi^2 r^4}.$$ Hence, and yield $$\label{eq:SL_final_form}
S_{\mathcal{L}_0}=\left(k-1\right)!!\left(\frac{{\mathcal{V}}\left(X_{f_{n},r}\right)}{4\pi^2r^4}\right)^{k/2}+O\left([\underline{v}]_{\infty}^k N^{-1}\right).$$ Substituting and into and applying Lemma \[lem:Lower\_Bound\_F2\], we finally obtain that for $k$ even $$\begin{aligned}
\left|{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}]-\left(k-1\right)!!\right| & \ll T^{k\eta(n)} [\underline{v}]_{\infty}^k\left(T^{-1/2}+T^{k/2}N^{-\min\{1,\gamma \}}\right) \ll T^{-1/2+2k\eta(n)}
\end{aligned}$$ and since for $k$ odd $ \mathcal{L}=\mathcal{L}_0 $ is not a viable option, we obtain $$\begin{aligned}
{\mathbb{E}}[\hat{X}_{f_{n},r}^{k}] & \ll T^{k\eta(n)} [\underline{v}]_{\infty}^k\left(T^{-1/2}+T^{k/2}N^{-\min\{1,\gamma \}}\right) \ll T^{-1/2+2k\eta(n) },
\end{aligned}$$ and the second part of Theorem \[thm:VarMainGeneralized\] follows.
Proof of Theorem \[thm:Var3D\]: asymptotics for the variance, $d=3$ {#sec:Proof_3d_theorem}
===================================================================
Proof of Theorem \[thm:Var3D\]
------------------------------
Denote $$\label{eq:F_3}
F_{3}\left(s\right)=\frac{1}{N^{2}} \cdot \#\left\{ \lambda\ne\lambda'\in\mathcal{E}_{n}:\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}$$(cf. ), and recall that the spherical cap discrepancy for the points in $\mathcal{E}_{n}$ is defined by $$\label{eq:Discrepancy_3d}
\Delta_{3}\left(n\right)=\sup_{\begin{subarray}{c}
x\in\mathcal{S}^{2}\\
0<r\le2
\end{subarray}}\left|\frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\left|\widehat{\lambda}-x\right|\le r\right\} -\frac{r^{2}}{4}\right|.$$
\[cor:DistMeasure3d\]We have $$\label{eq:3d_Discrepancy}
F_{3}\left(s\right)=\frac{s^{2}}{4}+O\left(\Delta_{3}\left(n\right)\right).$$
The estimate follows immediately from the definition of spherical cap discrepancy, since $$\begin{aligned}
F_{3}\left(s\right) & =\frac{1}{N}\sum_{\lambda'\in\mathcal{E}_{n}}\#\left\{ \lambda\in\mathcal{E}_{n}:\,0<\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\} =\frac{s^{2}}{4}+O\left(\Delta_{3}\left(n\right)\right).
\end{aligned}$$
The discrepancy $\Delta_{3}\left(n\right)$ satisfies $ \Delta_3(n) \le n^{-\eta} $ for some small $\eta>0$, see [@BourgainRudnickSarnak]. We are now in a position to prove Theorem \[thm:Var3D\]:
By we have $$\begin{aligned}
\label{eq:var_3d_asymp_formula}
\text{\ensuremath{\mathcal{V}}}\left(X_{f_n,r}\right) & =16\pi^{3}r^{6}\frac{1}{N^{2}}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)+O\left(r^{6}N^{-1/4+\epsilon}\right).
\end{aligned}$$ For the summation in we have, $$\frac{1}{N^{2}}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)=\int_{0}^{2}h_{3}\left(Ts\right)\,\text{d}F_{3}\left(s\right).$$ Thus, integrating by parts and using Lemma \[cor:DistMeasure3d\], $$\begin{aligned}
\frac{1}{N^{2}}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}h_{3}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right) & =h_{3}\left(2T\right)F_{3}\left(2\right)-\int_{0}^{2}F_{3}\left(s\right)\,\text{d}h_{3}\left(Ts\right)\label{eq:VarAftIntParts3D}\\
& =h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)+Err\left(X_{f_n,r}\right)\nonumber
\end{aligned}$$ where $$\label{eq:error_term_3d}
Err\left(X_{f_n,r}\right)\ll\Delta_{3}\left(n\right)\int_{0}^{2}T\left|h_{3}'\left(Ts\right)\right|\,\text{d}s.$$ Note that $h_{3}\left(s\right)\ll s^{-4}$ as $s\to\infty.$ Thus, integrating by parts, the main term on the r.h.s of (\[eq:VarAftIntParts3D\]) satisfies $$\label{eq:main_terms_after_intbyparts_3d}
h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)=\frac{1}{2}\int_{0}^{2}s \cdot h_{3}\left(Ts\right)\,\text{d}s+O\left(T^{-4}\right),$$ so that $$\label{eq:main_int_after_intbyparts}
\int_{0}^{2}s \cdot h_{3}\left(Ts\right)\,\text{d}s=\frac{1}{T^{2}}\int_{0}^{2T}s \cdot h_{3}\left(s\right)\,\text{d}s=\frac{1}{T^{2}}\int_{0}^{\infty}s \cdot h_{3}\left(s\right)\,\text{d}s+O\left(T^{-4}\right).$$ A direct computation shows that $$\label{eq:final_calc_main_term_3d}
\int_{0}^{\infty}s \cdot h_{3}\left(s\right)\,\text{d}s=\frac{1}{2\pi^3}\int_{0}^{\infty}\frac{1}{s^{3}}\left(\frac{\sin s}{s}-\cos s\right)^{2}\,\text{d}s=\left(2\pi\right)^{-3},$$ and therefore, substituting into and then into we get $$\label{eq:main_term_final_form_3d}
h_{3}\left(2T\right)F_{3}\left(2\right)-\frac{1}{4}\int_{0}^{2}s^{2}\,\text{d}h_{3}\left(Ts\right)=\frac{1}{16\pi^3}T^{-2}+O\left(T^{-4}\right).$$ Note that $h_{3}'\left(s\right)\ll\left(1+s^{4}\right)^{-1}$. Thus, $$\label{eq:err_upper_bound_3d}
\int_{0}^{2}T\left|h_{3}'\left(Ts\right)\right|\,\text{d}s=\int_{0}^{2T}\left|h_{3}'\left(s\right)\right|\,\text{d}s\le\int_{0}^{\infty}\left|h_{3}'\left(s\right)\right|\,\text{d}s<\infty$$ and therefore, substituting into we obtain $$Err\left(X_{f_n,r}\right)=O\left(\Delta_{3}\left(n\right)\right).\label{eq:3DErrorVar}$$ Substituting into and finally into we obtain (\[eq:AympVar3D\]).
Note that by (\[eq:3DErrorVar\]), $$\sup_{\begin{subarray}{c}
r > r_{0} \\ f_n\in {\mathcal{B}}{n}
\end{subarray}}\left|\frac{\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)}{r^{6}}-T^{-2}\right|=O\left(T_{0}^{-4}+n^{-\eta}\right)$$ for every $n\not\equiv0,4,7\,\left(8\right)$, and in particular $$\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right)=o\left(r^{6}\right)$$ uniformly for $r > r_{0}$ for every $n\not\equiv0,4,7\,\left(8\right)$.
Proofs of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\] {#sec:ProofOfBoundsThm}
=======================================================================
By substituting the bound $\left|c_{\lambda}\right|^{2}\le N^{-1+\epsilon}$ into (\[eq:VarFormula2d\]), we have
$$\begin{aligned}
\label{eq:var_ultraflat_bound}
\text{\ensuremath{\mathcal{V}}}\left(X_{f_{n},r}\right) & \ll r^{4} N^{-1+\epsilon}\sum_{\lambda_0 \in \mathcal{E}_n}\left|c_{\lambda_0}\right|^{2}\sum_{\lambda \in \mathcal{E}_n} h_{2}\left(r\left\Vert \lambda-\lambda_0\right\Vert \right).\end{aligned}$$
By Lemma \[lem:H2Formulas\] and by , we have $$\begin{aligned}
\label{eq:H2_sum}
\frac{1}{N}\sum_{\lambda\in\mathcal{E}_{n}}h_{2}\left(r\left\Vert \lambda-\lambda_{0}\right\Vert \right) & =\int_{0}^{2}h_{2}\left(Ts\right)\,\text{d}F_{\lambda_{0}}\left(s\right)\\
& =h_{2}\left(2T\right)-\frac{1}{4N}+O\left(\int_{0}^{2}\left(s+\Delta\left(n\right)\right)T\left|h_{2}'\left(Ts\right)\right|\,\text{d}s\right) \nonumber \\
& =O\left(T^{-1}+\left(\log n\right)^{-\frac{1}{2}\log\frac{\pi}{2}+\epsilon}\right) \nonumber\end{aligned}$$ for $ n $ satisfying the hypothesis $ \mathcal{D}(n,\epsilon) $. Substituting in , we get the upper bound in Theorem \[thm:UpperBound2d\]. The upper bound in Theorem \[thm:UpperBound3d\] follows along similar lines.
We now turn to proving the claimed lower bounds for the variance of $X_{f_n,r}$. First, we need the following lemma, proved at the end of section \[sec:ProofOfBoundsThm\]:
\[lem:LowerBoundPairs\]
1. Let $\left\{ x_{m}\right\} _{m=1}^{M}$ be $M$ points on the unit circle $\mathcal{S}^{1}.$ For every $1<T<M/2$ we have $$\#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} \gg M^{2}/T.$$
2. Let $\left\{ x_{m}\right\} _{m=1}^{M}$ be $M$ points on the unit sphere $S^{2}.$ For every $1<T<\sqrt{M}/2$ we have $$\#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} \gg M^{2}/T^{2}.$$
We are now in a position to prove the lower bounds , of Theorem \[thm:UpperBound2d\] and Theorem \[thm:UpperBound3d\]:
For $d=2$, we let $$R=\#\left\{ \lambda\in\mathcal{E}_{n}:\,\left|c_{\lambda}\right|^{2}\ge \frac{1}{2N}\right\},$$ so that $$1=\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=\sum_{\lambda\in R}\left|c_{\lambda}\right|^{2}+\sum_{\lambda\notin R}\left|c_{\lambda}\right|^{2}\le N^{-1+\epsilon} \cdot \#R+1/2,$$ and hence $\#R\ge 2N^{1-\epsilon}.$ By the second part of Lemma \[lem:H2Formulas\], for $c>0$ sufficiently small we have $$\begin{aligned}
{\mathcal{V}}(X_{f_n,r}) & =8\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right) \gg r^{4}N^{-2}\sum_{\lambda\ne\lambda'\in R}h_{2}\left(T\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \right)\\
& \gg r^{4}N^{-2}\cdot \#\left\{ \lambda\ne\lambda'\in R:\,\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le c/T\right\} .
\end{aligned}$$ By the first part of Lemma \[lem:LowerBoundPairs\], $${\mathcal{V}}(X_{f_n,r})\gg r^{4}N^{-2}\left(\#R\right)^{2}T^{-1}\gg r^{4}N^{-2\epsilon}T^{-1}.$$
The lower bound of Theorem \[thm:UpperBound3d\] follows along the same lines as the above, this time using the second part of Lemma \[lem:LowerBoundPairs\] in place of the first one.
Note that in the proof of of the lower bound in Theorem \[thm:UpperBound2d\] we have used the abundance of close-by pairs of lattice points with **$\left|c_{\lambda}\right|^{2}\ge \frac{1}{2N}$**; in the absence of such close-by lattice points, the bound does not hold. For example, for $d=2$, fix $\lambda_{0}\in\mathcal{E}_{n}$ and let $\left|c_{\pm\lambda_{0}}\right|^{2}=1/2$ and $c_{\lambda}=0$ for every $\lambda\ne\pm\lambda_{0}.$ Then $${\mathcal{V}}(X_{f_{n},r})=4\pi^{2}r^{4}h_{2}\left(2T\right)\ll r^{4}T^{-3}.$$
For the first part of Lemma \[lem:LowerBoundPairs\], divide $S^{1}$ into $k=O\left(T\right)$ arcs $I_{1},I_{2},\dots,I_{k}$ of length $<1/T$. For every $1\le j\le k,$ let $n_{j}=\#\left\{ m:\,x_{m}\in I_{j}\right\} ,$ so $\sum_{j=1}^{k}n_{j}=M$. By the Cauchy-Schwarz inequality, $$M^{2}=\left(\sum_{j=1}^{k}n_{j}\right)^{2}\le k\sum_{j=1}^{k}n_{j}^{2}\ll T\sum_{j=1}^{k}n_{j}^{2}.$$ Thus, $$\begin{aligned}
\#\left\{ x_{i}\ne x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} & =\#\left\{ x_{i},x_{j}:\,\left|x_{i}-x_{j}\right|\le1/T\right\} -M\\
& \gg\sum_{j=1}^{k}n_{j}^{2}-M\gg M^{2}/T-M\gg M^{2}/T.
\end{aligned}$$ The second part of Lemma \[lem:LowerBoundPairs\] is proved similarly.
Restricted averages {#sec:RestrictedAverages}
===================
Restricted moments
------------------
For $ d=2 $, most of our principal results above are also valid in the more difficult scenario where $x$ is drawn in $B_{x_{0}}(\rho)$ for some $x_{0}\in{\mathbb{T}}^{2}$ and $\rho\gg n^{-1/2+o(1)}$. In this case, the restricted moments are: expectation $$\label{eq:restricted_expectation}
{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}X_{f_{n},r;x}dx,$$ higher centred moments $$\label{eq:centred moments rest}
{\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{k}] = \frac{1}{{\operatorname{Vol}}(B_{x_0}(\rho))}\int\limits_{B_{x_{0}}(\rho)}\left(X_{f_{n},r;x}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}]\right)^{k}dx, \hspace{1em}k\ge2,$$ and in particular the variance $$\label{eq:restricted_variance}
{\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = {\mathbb{E}}_{x_{0},\rho}[(X_{f_{n},r}-{\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}])^{2}].$$
We reinterpret the statement of Granville-Wigman’s [@GranvilleWigman Theorem 1.2] as evaluating the expected mass $${\mathbb{E}}_{x_{0},\rho}[X_{f_{n},r}] \sim \pi r^{2},$$ valid for almost all $n\in S_{2}$, uniformly for $\rho\gg n^{-1/2+o(1)}$, $x_{0}\in{\mathbb{T}}^{2}$, and $ r>0 $ (see the first part of Lemma \[lem:ExpVarShrinking\]).
Quasi-correlations
------------------
For the restricted moments of $X_{f_{n},r}$ one also needs to cope with [*quasi-correlations*]{}, i.e. tuples $(\lambda^{1},\ldots,\lambda^{l}) \in {\mathcal{E}}_{n}^{l}$ with the sum $\sum\limits_{i=1}^{l}\lambda^{i}$ unexpectedly small, e.g. given a (small) fixed number $\delta >0$, $$\label{eq:sum tuple small l,delta}
\left\|\sum\limits_{i=1}^{l}\lambda^{i}\right\| < n^{1/2-\delta};$$ unlike the correlations , here there are no congruence obstructions, so that makes sense with $l$ odd or even.
\[def:separatedness Ac\]
1. For $n\in S_2$, $l\in\mathbb{Z}_{\ge 2}$, and $0 < K=K(n) < l\cdot n^{1/2}$ define the set of length-$l$ spectral quasi-correlations $$\label{eq:quasi_correlations}
\mathcal{C}_{n}(l;K) =
\left\{(\lambda^{1},\ldots,\lambda^{l})\in\mathcal{E}_{n}^{l}:\: 0 < \left\| \sum\limits_{j=1}^{l}\lambda^{j} \right\| \le K \right\}.$$
2. Given $\delta>0$ we say that $n \in S_{2}$ satisfies the $(l,\delta)$-separateness hypothesis $\mathcal{A}(n;l,\delta)$ if $$\label{eq:sep_hypothesis}
\mathcal{C}_{n}(l;n^{1/2-\delta}) = \varnothing.$$
For example, ${\mathcal{A}}(n;2,\delta)$ is equivalent to the aforementioned Bourgain-Rudnick separateness, satisfied [@Bourgain-Rudnick Lemma 5] by a density $1$ sequence $S_{2}'\subseteq S_{2}$. More generally, it was shown in the forthcoming paper [@BenatarBuckleyWigman], that for every $\delta>0$ and $l\ge 2$, the assumption ${\mathcal{A}}(n;l,\delta)$ is satisfied by generic $n\in S_{2}'(l,\delta)$, and hence a standard diagonal argument yields a density $1$ sequence $S_{2}'\subseteq S_{2}$ so that ${\mathcal{A}}(n;l,\delta)$ is satisfied for [*all*]{} $l\ge 2$ and $ \delta>0 $ for $ n\in S_2' $ sufficiently large.
\[thm:quasi-corr small\] For every fixed $l\ge 2$ and $\delta>0$ there exist a set $S_2'(l,\delta)\subseteq S_2$ such that:
1. The set $S_2'(l,\delta)$ has density $1$ in $S_2$.
2. For every $n\in S_2'(l,\delta)$ the length-$l$ spectral quasi-correlation set $$\mathcal{C}_{n}(l;n^{1/2-\delta})=\varnothing$$ is empty, i.e., the $(l,\delta)$-separateness hypothesis $\mathcal{A}(n;l,\delta)$ is satisfied.
A version of Theorem \[thm:VarMainGeneralized\] with restricted averages
------------------------------------------------------------------------
We have the following analogue of Theorem \[thm:VarMainGeneralized\]:
\[thm:VarMainExplRestricted\]
1. If $S_{2}'\subseteq S_{2}$ is a sequence satisfying the hypotheses $ \mathcal{D}(n,\epsilon/2),$ $ {\mathcal{A}}(n;2,\epsilon)$, and ${\mathcal{A}}(n;4,\epsilon)$ for all $n\in S_{2}'$, then in the setting of Theorem \[thm:VarMainGeneralized\] part (1), $${\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)\sim\frac{16}{3\pi \cos^{2}\theta_{f_{n}}}r^{4}T^{-1}$$ uniformly for all $x_{0}\in\mathbb{T}^{2}$, $n^{-1/2+\delta}\le\rho\le 1$ and $r_{0}< r< r_{1}$, and $f_{n}\in{\mathcal{F}}_{1}(n;T(n),\eta(n))$.
2. Let $k\ge 3$ be an integer. If $S_{2}'\subseteq S_{2}$ is a sequence satisfying the length-$2k$ diagonal domination assumption and the hypotheses $ \mathcal{D}(n,\epsilon),$ $ {\mathcal{A}}(n;2,\epsilon)$, $ {\mathcal{A}}(n;4,\epsilon)$, and ${\mathcal{A}}(n;2k,\epsilon)$ for all $n\in S_{2}'$, then in the setting of Theorem \[thm:VarMainGeneralized\] part (2), $$\mathbb{E}_{x_{0},\rho}\left[\hat{X}_{f_{n},r}^{k}\right] \to {\mathbb{E}}[Z^{k}]$$ uniformly for $x_{0}\in{\mathbb{T}}^{2}$, $r_{0} < r <r_{1}$, $n^{-1/2+\delta} \le \rho \le 1$, and $f_{n}\in {\mathcal{F}}_{2}(n;T(n),\eta(n))$, where $Z\sim N(0,1)$ is the standard Gaussian variable.
Theorem \[thm:VarMainExplRestricted\] follows along similar lines as the proof of Theorem \[thm:VarMainGeneralized\], where we use the expressions for the restricted moments below (cf. equation , Lemma \[lem:VarExpd2\] and equation ). We remark that Theorem \[thm:UpperBound2d\] can also be extended to $ {\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) $, however the lower bound will only hold for a generic $ n\in S_2 $.
\[lem:ExpVarShrinking\]
For $d=2$ let $0<\delta<1/2$, $0<\epsilon<\delta/5$, and $S_{2}'\subseteq S_{2}$.
1. If $n\in S_{2}'$ satisfy the hypothesis ${\mathcal{A}}(n;2,\epsilon) $, then $$\mathbb{E}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\pi r^{2}+O\left(r^{2}n^{-\frac{3}{5}\delta+3\epsilon}\right)$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$.
2. \[lem:VarFormulaShrinking\]
If $n\in S_{2}'$ satisfy the hypotheses $ {\mathcal{A}}(n;2,\epsilon) $ and ${\mathcal{A}}(n;4,\epsilon)$, then $$\begin{aligned}
\text{\ensuremath{\mathcal{V}}}_{x_{0},\rho}\left(X_{f_{n},r}\right) & =8\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}h_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right) +O\left(r^{4} n^{-\frac{3}{5}\delta+4\epsilon}\right)
\end{aligned}$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$.
\[lem:KthMoment\]For $d=2$ let $ k\ge3 $, $0<\delta<1/2$, $0<\epsilon< \delta/5$, and $ S_2' \subseteq S_2 $ satisfying $ {\mathcal{A}}(2;n,\epsilon), $ $ {\mathcal{A}}(4;n,\epsilon), $ and ${\mathcal{A}}(n;2k,\epsilon) $ for every $ n\in S_2' $ . We have $$\begin{aligned}
{\mathbb{E}}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] & =(2\pi)^k r^{2k}{\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2}\sum_{\begin{subarray}{c}
\forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\
\sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)=0
\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\
& +O\left({\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} r^{2k} n^{-\frac{3}{5}\delta+4\epsilon}\right)\nonumber
\end{aligned}$$ uniformly for $x_{0}\in\mathbb{T}^{2},$ $n^{-1/2+\delta}\le\rho\le1$ and $r>0$.
Proofs of Lemma \[lem:ExpVarShrinking\] and Lemma \[lem:KthMoment\]
-------------------------------------------------------------------
We have $$\label{eq:exp_basic_formula}
{\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y\,\text{d}x.$$
Granville-Wigman’s [@GranvilleWigman Theorem 1.2] asserts that for $ \epsilon_1 > \epsilon_2 > 0$, $ 0 < \epsilon_3 < \epsilon_1 - \epsilon_2 $ and $ n\in S_2 $ satisfying ${\mathcal{A}}(n;2,\epsilon_2) $, we have $$\label{eq:Granville_Wigman_theorem}
\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y = \pi r^2 \left(1+ O\left(n^{-3\epsilon_3 /2}\right)\right)$$ uniformly in $ x\in \mathbb{T}^2 $ and $ r>n^{-1/2 + \epsilon_1} $. If $r>n^{-1/2+\frac{2}{5}\delta}$, then by substituting with $ \epsilon_1 = \frac{2}{5}\delta $, $ \epsilon_2 = \epsilon $ and $ \epsilon_3 = \frac{2}{5}\delta - 2\epsilon $ into , we have $${\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\pi r^{2}\left(1+O\left(n^{-\frac{3}{2}\left( \frac{2}{5}\delta - 2\epsilon\right)}\right)\right)$$ for every $\rho$.
Otherwise, note that $${\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]=\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho+r\right)}f_{n}\left(y\right)^{2}\int_{B_{x_{0}}\left(\rho\right)\cap B_{y}\left(r\right)}\,\text{d}x\,\text{d}y,$$ so $$\label{eq:Exp_upper_lower_bnds}
\frac{r^{2}}{\rho^{2}}\int_{B_{x_{0}}\left(\rho-r\right)}f_{n}\left(y\right)^{2}\,\text{d}y\le{\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\le\frac{r^{2}}{\rho^{2}}\int_{B_{x_{0}}\left(\rho+r\right)}f_{n}\left(y\right)^{2}\,\text{d}y.$$ Since $r/\rho \le n^{-\frac{3}{5}\delta}$, we can use with $ \epsilon_1 = \delta $, $ \epsilon_2 = \epsilon $ and $ \epsilon_3 = \delta - 2\epsilon $ to deduce that $$\label{eq:Exp_Inner_integral}
\int_{B_{x_{0}}\left(\rho\pm r\right)}f_{n}\left(y\right)^{2}\,\text{d}y=\pi\rho^{2}\left(1+O\left(n^{-\frac{3}{5}\delta}\right)\right),$$ and the statement of the first part of Lemma \[lem:ExpVarShrinking\] follows upon substituting into .
We have $${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- {\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\right)^{2}\,\text{d}x.$$
By (\[eq:IntegrandVar\]), $$\begin{aligned}
& \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{2}\,\text{d}x\\
& =4\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\
\lambda\ne\lambda'\\
\lambda''\ne\lambda'''
\end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\
& \times\frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}e\left(\left\langle x,\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)\,\text{d}x\\
& =8\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}\left|c_{\lambda}\right|^{2}\left|c_{\lambda'}\right|^{2}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)^{2}\\
& +8\pi^{2}r^{4}\sum_{\begin{subarray}{c}
\lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\
\lambda\ne\lambda'\\
\lambda''\ne\lambda'''\\
\lambda-\lambda'+\lambda''-\lambda'''\ne0
\end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\
& \times e\left(\left\langle x_{0},\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)g_{2}\left(\rho\left\Vert \lambda-\lambda'+\lambda''-\lambda'''\right\Vert \right).
\end{aligned}$$ By the hypothesis ${\mathcal{A}}(n;4,\epsilon)$ and Lemma \[lem:H2Formulas\], we have $$\begin{aligned}
& \sum_{\begin{subarray}{c}
\lambda,\lambda',\lambda'',\lambda'''\in\mathcal{E}_{n}\\
\lambda\ne\lambda'\\
\lambda''\ne\lambda'''\\
\lambda-\lambda'+\lambda''-\lambda'''\ne0
\end{subarray}}c_{\lambda}\overline{c_{\lambda'}}c_{\lambda''}\overline{c_{\lambda'''}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right)g_{2}\left(r\left\Vert \lambda''-\lambda'''\right\Vert \right)\\
& \times e\left(\left\langle x_{0},\lambda-\lambda'+\lambda''-\lambda'''\right\rangle \right)g_{2}\left(\rho\left\Vert \lambda-\lambda'+\lambda''-\lambda'''\right\Vert \right)\\
& \ll\left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{4}\frac{1}{\left(n^{\delta-\epsilon}\right)^{3/2}}\ll N^{2}n^{-\frac{3}{2}(\delta-\epsilon)} \ll n^{-\frac{3}{2}\delta + 2\epsilon}.
\end{aligned}$$
Next, note that $$\begin{aligned}
\label{eq:inner_int_estimate}
\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2} &= 2\pi r^{2}\sum_{\lambda\ne\lambda'\in\mathcal{E}_{n}}
c_{\lambda}\overline{c_{\lambda'}}g_{2}\left(r\left\Vert \lambda-\lambda'\right\Vert \right) \ll r^2 \left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{2} \ll N r^2.
\end{aligned}$$ By and the first part of Lemma \[lem:ExpVarShrinking\],
$${\mathcal{V}}_{x_{0},\rho}(X_{f_{n},r}) = \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{2}\,\text{d}x + O\left(r^4 n^{-\frac{3}{5}\delta+4\epsilon}\right)$$
and the statement of Lemma \[lem:VarExpd2\] follows.
We have $${\mathbb{E}}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] = {\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} \cdot \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y- {\mathbb{E}}_{x_{0},\rho}\left[X_{f_{n},r}\right]\right)^{k}\,\text{d}x.$$ By (\[eq:IntegrandVar\]), we have $$\begin{aligned}
\frac{1}{\pi\rho^{2}}&\int_{B_{x_{0}}\left(\rho\right)} \left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^{2}\right)^{k}\,\text{d}x = \left(2\pi\right)^{k}r^{2k} \sum_{\begin{subarray}{c}
\forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\
\sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)=0
\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\
& +\left(2\pi\right)^{k}r^{2k}\sum_{\begin{subarray}{c}
\forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\
\sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)\ne0
\end{subarray}}\prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right)\\
& \times2e\left(\left\langle x_{0},\sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\rangle \right)g_{2}\left(\rho\left\Vert \sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\Vert \right).
\end{aligned}$$ By the hypothesis ${\mathcal{A}}(n;2k,\epsilon)$, $$\begin{aligned}
\sum_{\begin{subarray}{c}
\forall1\le i\le k,\,\lambda_{i}\ne\lambda_{i}'\in\mathcal{E}_{n}\\
\sum_{i=1}^{k}\left(\lambda_{i}-\lambda_{i}'\right)\neq0
\end{subarray}}& \prod_{j=1}^{k}c_{\lambda_{j}}\overline{c_{\lambda_{j}'}}g_{2}\left(r\left\Vert \lambda_{j}-\lambda_{j}'\right\Vert \right) e\left(\left\langle x_{0},\sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\rangle \right)g_{2}\left(\rho\left\Vert \sum_{j=1}^{k}\left(\lambda_{j}-\lambda_{j}'\right)\right\Vert \right)\\
& \ll\left(\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|\right)^{2k}\frac{1}{\left(n^{\delta - \epsilon}\right)^{3/2}}\ll N^k n^{-\frac{3}{2}(\delta-\epsilon)} \ll n^{-\frac{3}{2}+2\epsilon}.
\end{aligned}$$ By and the first part of Lemma \[lem:ExpVarShrinking\], $$\begin{aligned}
\mathbb{E}_{x_{0},\rho}[\hat{X}_{f_{n},r}^{k}] & = {\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} \cdot \frac{1}{\pi\rho^{2}}\int_{B_{x_{0}}\left(\rho\right)}\left(\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y-\pi r^2 \right)^{k}\,\text{d}x \\ &+ O\left({\mathcal{V}}_{x_{0},\rho}\left(X_{f_{n},r}\right)^{-k/2} r^{2k} n^{-\frac{3}{5}\delta+4\epsilon}\right),
\end{aligned}$$ and the statement of Lemma \[lem:KthMoment\] follows.
\[sec:AuxLemmasProof\]Proofs of auxiliary lemmas
================================================
In this section we provide the proofs for lemmas \[lem:BasicVarProp\], \[lem:InnerIntegral\], and \[lem:CosToDist\]:
1. The upper bound is straightforward, and the lower bound follows from (\[eq:BasicNormalization\]) by invoking the Cauchy-Schwarz inequality on .
2. By partial summation, for every $\lambda_{0}\in\mathcal{E}_{n}$ we have $$1=\sum_{\lambda\in\mathcal{E}_{n}}\left|c_{\lambda}\right|^{2}=N\left|c_{\lambda_{0}}\right|^{2}+E,$$ where $\left|E\right|\le V\left(\underline{v} \right)$. Since $\lambda_{0}$ is arbitrary, we deduce that $$[\underline{v}]_{\infty} \le1+V\left(\underline{v} \right).$$
3. Follows directly from parts $1$ and $2$ of this lemma.
We have $$\begin{aligned}
\label{eq:inner_int_expansion}
\int_{B_{x}\left(r\right)}f_{n}\left(y\right)^{2}\,\text{d}y&=\int_{B_{x}\left(r\right)}\sum_{\lambda,\lambda'\in\mathcal{E}_{n}}c_{\lambda}\overline{c_{\lambda'}}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y\\
& =\frac{\pi^{d/2}}{\Gamma\left(d/2+1\right)}r^{d}+\sum_{\begin{subarray}{c}
\lambda,\lambda'\in\mathcal{E}_{n}\\
\lambda\ne\lambda'
\end{subarray}}c_{\lambda}\overline{c_{\lambda'}}\int_{B_{x}\left(r\right)}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y.\nonumber
\end{aligned}$$ Transforming the variables $y=rz+x$, we obtain $$\label{eq:var_transformation}
\int_{B_{x}\left(r\right)}e\left(\left\langle y,\lambda-\lambda'\right\rangle \right)\,\text{d}y=r^{d}e\left(\left\langle x,\lambda-\lambda'\right\rangle \right)\int_{B_{0}\left(1\right)}e\left(\left\langle z,r\left(\lambda-\lambda'\right)\right\rangle \right)\,\text{d}z.$$ Note that $$\begin{aligned}
\label{eq:Fourier_ball}
\int_{B_{0}\left(1\right)}e\left(\left\langle z,r\left(\lambda-\lambda'\right)\right\rangle \right)\,\text{d}z & =\frac{\left(2\pi\right)^{d/2}J_{d/2}\left(2 \pi r\left\Vert \lambda-\lambda'\right\Vert \right)}{\left(2 \pi r\left\Vert \lambda-\lambda'\right\Vert \right)^{d/2}},
\end{aligned}$$ and (\[eq:IntegrandVar\]) follows upon substituting into and finally into .
Let $\theta_{\lambda}$ be the angle between $\lambda$ and $\lambda'$. Then $$\begin{aligned}
\frac{1}{N} \cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\lambda\succeq\lambda',\left\Vert \widehat{\lambda}-\widehat{\lambda'}\right\Vert \le s\right\}
& =\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\theta_{\lambda}\ge0,\,\sqrt{2\left(1-\cos\theta_{\lambda}\right)}\le s\right\} \\
& =\frac{1}{N}\cdot \#\left\{ \lambda\in\mathcal{E}_{n}:\,\theta_{\lambda}\in\left[0,\arccos\left(1-s^{2}/2\right)\right]\right\} \\
& =\frac{1}{2\pi}\arccos\left(1-s^{2}/2\right)+O\left(\Delta\left(n\right)\right)\\
& =\frac{s}{2\pi}+O\left(s^{3}+\Delta\left(n\right)\right)
\end{aligned}$$ which is the statement (\[eq:CosToDistEq\]) of Lemma \[lem:CosToDist\].
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| ArXiv |
---
abstract: 'Cerium-doped manganite thin films were grown epitaxially by pulsed laser deposition at $720\,^\circ$C and oxygen pressure $p_{O_2}=1-25\,$Pa and were subjected to different annealing steps. According to x-ray diffraction (XRD) data, the formation of CeO$_2$ as a secondary phase could be avoided for $p_{O_2}\ge 8\,$Pa. However, transmission electron microscopy shows the presence of CeO$_2$ nanoclusters, even in those films which appear to be single phase in XRD. With O$_2$ annealing, the metal-to-insulator transition temperature increases, while the saturation magnetization decreases and stays well below the theoretical value for electron-doped La$_{0.7}$Ce$_{0.3}$MnO$_3$ with mixed Mn$^{3+}$/Mn$^{2+}$ valences. The same trend is observed with decreasing film thickness from 100 to 20nm, indicating a higher oxygen content for thinner films. Hall measurements on a film which shows a metal-to-insulator transition clearly reveal holes as dominating charge carriers. Combining data from x-ray photoemission spectroscopy, for determination of the oxygen content, and x-ray absorption spectroscopy (XAS), for determination of the hole concentration and cation valences, we find that with increasing oxygen content the hole concentration increases and Mn valences are shifted from 2+ to 4+. The dominating Mn valences in the films are Mn$^{3+}$ and Mn$^{4+}$, and only a small amount of Mn$^{2+}$ ions can be observed by XAS. Mn$^{2+}$ and Ce$^{4+}$ XAS signals obtained in surface-sensitive total electron yield mode are strongly reduced in the bulk-sensitive fluorescence mode, which indicates hole-doping in the bulk for those films which do show a metal-to-insulator transition.'
author:
- 'R. Werner'
- 'C. Raisch'
- 'V. Leca'
- 'V. Ion'
- 'S. Bals'
- 'G. Van Tendeloo'
- 'T. Chassé'
- 'R. Kleiner'
- 'D. Koelle'
bibliography:
- 'References.bib'
title: 'Transport, magnetic, and structural properties of La$_{0.7}$Ce$_{0.3}$MnO$_3$ thin films. Evidence for hole-doping'
---
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–> 61.05.cj X-ray absorption spectroscopy: EXAFS, NEXAFS, XANES, etc.
61.05.cp X-ray diffraction
–> 68.37.Lp Transmission electron microscopy (TEM)
68.55.J- Morphology of films
–> 71.30.+h Metal–insulator transitions and other electronic transitions
–> 72.60.+g Mixed conductivity and conductivity transitions
72.80.Ga Transition-metal compounds
–> 75.47.Lx Manganites
75.60.Ej Magnetization curves, hysteresis, Barkhausen and related effects
75.70.Ak Magnetic properties of monolayers and thin films
–> 81.15.Fg Laser deposition
Introduction {#Sec:Introduction}
============
Hole-doped manganese perovskite oxides La$_{1-x}A_x$MnO$_3$, where $A$ is a divalent alkaline earth metal, have been intensively studied over the last years due to the interesting interplay between charge, spin, orbital and structural degrees of freedom.[@Imada98; @Coey99; @Salamon01] Without doping, LaMnO$_3$ is an antiferromagnetic insulator due to the super-exchange between the Mn$^{3+}$ ions.[@Millis98] In the hole-doped manganites, the divalent ion introduces holes by changing some Mn valences from Mn$^{3+}$ to Mn$^{4+}$. The properties of the hole-doped manganites are determined by the interplay of Hund´s rule coupling and the Jahn-Teller distortion of the Mn$^{3+}$ ions.[@Millis95] Their behavior can be qualitatively described by the double-exchange model,[@Zener51; @Anderson55] describing the interaction between manganese ions with mixed valences (Mn$^{3+}$ and Mn$^{4+}$). The strong spin-charge coupling via the double-exchange interaction explains the correlation between the metal-to-insulator (MI) and ferromagnet-to-paramagnet (FP) transition. Close to the MI transition temperature $T_{MI}$ an external magnetic field can reduce the spin disorder and therefore enhance the electron hopping between the manganese ions with mixed valences. This results in a large resistivity drop, called colossal magnetoresistance.[@Jonker50]
By substitution of La with a tetravalent ion, like Ce,[@Mandal97; @Gebhardt99; @Ganguly00] Sn,[@Li99a] or Te,[@Tan03a] instead of a divalent one, some of the Mn$^{3+}$ ions become Mn$^{2+}$ with electronic structure t$^3_{2g}$e$^2_g$ (compared to the t$^3_{2g}$e$^1_g$ electronic structure for Mn$^{3+}$). Hence, an extra electron may be induced in the e$_g$-band. Since Mn$^{2+}$ is a non-Jahn-Teller ion, like Mn$^{4+}$, one might expect a similar magnetic interaction between the Mn$^{3+}$ and Mn$^{2+}$ ions as for the well known hole-doped case.[@Mitra03a]
The first attempts to achieve electron-doping by substituting La with Ce were reported by Mandal and Das.[@Mandal97] However, they found hole-doping in their bulk samples. Later on, it was revealed that the bulk samples are a multiphase mixture which leads to the hole-doped behavior.[@Ganguly00; @Philip99] Single phase [La$_{0.7}$Ce$_{0.3}$MnO$_3$]{} (LCeMO) thin films have been prepared without any CeO$_2$ impurities [@Mitra01a; @Raychaudhuri99] regarding x-ray diffraction (XRD) data. The films showed FP and MI transitions similar to the hole-doped manganites. Surface-sensitive X-ray photoemission spectroscopy revealed the existence of Mn$^{2+}$ and Mn$^{3+}$ valences,[@Mitra03a; @Han04] which was interpreted as evidence of electron-doping. However, Hall measurements and thermopower measurements on comparable samples showed a hole-type character. [@Wang06; @Zhao00; @Yanagida04; @Yanagida05] By Ganguly [*et al.*]{} [@Ganguly00] it was further questioned whether LaMnO$_3$ accepts Ce-doping at all. Those authors questioned the reports on single phase LCeMO-films and claimed the presence of multi-phase mixtures, consisting of hole doped La-deficient phases with cerium oxide inclusions. Certainly, the existence of electron-doped manganites could enable new types of spintronic devices, such as $p-n$ junctions based on doped manganites.[@Mitra01] This motivates further research in order to improve understanding of the basic properties of those materials.
In this paper we present the results of studies on transport, magnetic and structural properties of LCeMO thin films grown by pulsed laser deposition (PLD) and their dependence on deposition parameters, annealing procedures and film thickness. We combine a variety of different characterization techniques in order to clarify the nature of the FP and MI transition in our LCeMO thin films.
Experimental Details {#Sec:Experiment}
====================
A commercially available stoichiometric polycrystalline La$_{0.7}$Ce$_{0.3}$MnO$_3$ target was used for thin film growth by PLD on (001) SrTiO$_3$ (STO) substrates (unless stated otherwise). The target was ablated by using a KrF ($\lambda$ = 248 nm) excimer laser at a repetition rate of $2-5\,$Hz. The energy density on the target was $E_d=2\,{\rm J/cm}^2$, while the substrate temperature during deposition was kept at $T_s=720\,^{\circ}$C for all films for which data are presented below, except for sample K with slightly lower $T_s$ and $E_d$ (cf. Tab. \[tab:overview\]). The oxygen pressure $p_{O_2}$ during film growth was varied in the 1–25Pa range with the aim of yielding single phase films with optimum morphology. We used a relatively low deposition pressure as compared to some literature data [@Mitra03a; @Wang06; @Mitra01; @Chang04] in order to avoid over-oxygenation of the films. This is important, as it is known that perovskite rare-earth manganites can accept a large excess of oxygen via the formation of cation vacancies, inducing hole-doping in the parent compound LaMnO$_3$.[@Toepfer97] In-situ high-pressure reflection high energy electron diffraction (RHEED) was used to monitor the growth mode and film thickness. After deposition, most of the films were in-situ annealed for 1h at $T=700\,^{\circ}$C and $p_{O_2}=1\,$bar and then cooled down with 10$^{\circ}$C per minute. In the following, those samples will be called ”in-situ annealed“ films, in contrast to the ”as-deposited“ films which were just cooled down to room temperature under deposition pressure. Some of the samples have been additionally annealed ex-situ at $p_{O_2}=1\,$bar in one or two steps ($1^{\rm st}$ step at $700\,^{\circ}$C; $2^{\rm nd}$ step at $750\,^{\circ}$C; each step for one hour). Table \[tab:overview\] summarizes the fabrication conditions and some characteristics of the LCeMO films described below.
[ccccccc]{}
------------------------------------------------------------------------
\# & $p_{O_2}$ (Pa) & & $d$ (nm) & $c$-axis ($\AA$) & $T_{MI}$ (K)\
& & in-situ & ex-situ & & &\
A & 1 & no & no & 100 & 3.921 & –\
B1 & & & no & & 3.905 & 175\
B2 & \[-1.5ex\]3 & \[-1.5ex\][no]{} & $1\times$ & \[-1.5ex\][90]{} & – & 250\
C & 8 & yes & no & 100 & 3.897 & 190\
D & 25 & no & no & 100 & 3.880 & 180\
E1 & & & no & & 3.894 & 210\
E2 & 8 & yes & $1\times$ & 65 & 3.887 & 216\
E3 & & & $2\times$ & & 3.872 & 230\
F & 8 & yes & no & 40 & 3.879 & 223\
G & 8 & yes & no & 20 & 3.870 & 232\
H & 3 & $(^*)$ & no & 100 & 3.876 & 260\
K & 3$(^{**})$ & no & no & 50 & 3.894 & 180\
$(^*)$ Cooled in 1bar O$_2$ without 1hour in-situ annealing $(^{**})$ deposited at $T_s=700\,^{\circ}$C with $E_d=1.75\,{\rm J/cm}^2$
The surface morphology was checked by atomic force microscopy (AFM) in contact mode. The crystal structure of the films was characterized by XRD and by high-resolution (HR) transmission electron microscopy (TEM). Transport properties were measured with a four probe technique, and a superconducting quantum interference device (SQUID) magnetometer was used to determine the magnetic properties of the samples. Hall measurements were performed in order to obtain information on the dominating type of charge carriers, and x-ray photoemission spectroscopy (XPS) was performed in order to obtain information on the oxygen content of different samples. The valences of the manganese and cerium ions were evaluated by x-ray absorption spectroscopy (XAS). XAS measurements in surface-sensitive total electron yield (TEY) mode and bulk sensitive fluorescence yield (FY) mode were carried out at the WERA dipole beamline (ANKA, Karlsruhe, Germany) with typical energy resolutions set between 100 and 400meV.
Structural Analysis
===================
Figure \[XRDCeO2\] shows the XRD $\Theta-2\Theta$ scans of four LCeMO thin films A, B, C, D (with similar thickness $d=$ 90–100nm) grown under different oxygen pressure $p_{O_2}=$ 1, 3, 8 and 25Pa, respectively. Sample C was in-situ annealed while the other samples were ”as-deposited“ films. According to the XRD data shown in Fig. \[XRDCeO2\], single phase LCeMO films were obtained for $p_{O_2}\ge 8\,$ Pa (samples C and D). For a lower deposition pressure, impurity peaks of CeO$_2$ appear (sample A and B). The substrate temperature $T_s$ also played a crucial role for the phase stability of the LCeMO films. By increasing $T_s$ up to $800\,^{\circ}$C, CeO$_2$ also appears for deposition pressures $p_{O_2}\ge 8\,$Pa. Such a behavior was also observed by Chang [*et al.*]{}.[@Chang04] As shown in the inset of Fig. \[XRDCeO2\], the $c$-axis decreases with increasing deposition pressure $p_{O_2}$. This can be explained by a decreasing concentration of oxygen vacancies with increasing $p_{O_2}$, as it is well known that oxygen vacancies tend to expand the lattice constants.[@Murugavel03]
![(Color online) XRD patterns of samples grown under different deposition pressures: $p_{O_2}=1,\,3,\,8$ and $25\,$Pa for sample A, B, C and D, respectively. CeO$_2$ can be identified in samples A and B. XRD scans are offset for clarity. The inset shows a detailed view around the (001) substrate peak including the (001) film peaks.[]{data-label="XRDCeO2"}](XRDCeO2){width="45.00000%"}
The surface roughness of the films depends strongly on deposition pressure, as shown by AFM and RHEED images on 100nm thick films in Fig. \[AFM-RHEED\] for (a) sample C ($p_{O_2}=8\,$Pa) with an rms roughness of 0.35nm and (b) sample D ($p_{O_2}=25\,$Pa;), with a much larger rms value of 2.15nm. The RHEED images show strong streaky patterns for the film deposited at $p_{O_2}=8\,$Pa \[Fig. \[AFM-RHEED\](a) right\], an indication of an atomically flat surface, while for higher deposition pressure \[here $p_{O_2}=25\,$Pa; Fig. \[AFM-RHEED\](b) right\] an increased surface roughness results in a combination of weaker streaks, together with the formation of a 3D RHEED pattern as a result of island growth. We note that sample C has an extremely smooth surface, showing unit-cell high terrace steps in the AFM image \[c.f. Fig. \[AFM-RHEED\](a) left\], which is quite unusual for such a thick LCeMO film. A similar morphology as for sample C was observed for all films deposited at an oxygen pressure in the range of 1-8Pa. For those conditions the films followed a 2D growth mode, as suggested by the RHEED and AFM data. Increasing the deposition pressure resulted in an increased step density during growth due to lower surface mobility, with the formation of 3D islands. Altogether, we found that $p_{O_2}=8\,$Pa was the optimum pressure for growing films without measurable CeO$_2$ concentration, as detected by XRD, and good surface morphology (rms roughness below 0.4nm).
![(Color online) AFM images (left; frame size $5\times 5\,\mu{\rm m}^2$) and RHEED images (right) of 100nm thick films: (a) sample C, grown at $p_{O_2}=8\,$Pa, and (b) sample D, grown at $p_{O_2}=25\,$Pa.[]{data-label="AFM-RHEED"}](AFM-RHEED){width="43.00000%"}
In order to evaluate the relation between CeO$_2$ formation and the substrate induced strain, 50nm thick LCeMO films were deposited on (001) STO, (110) NdGaO$_3$ and (001) NdGaO$_3$ substrates in the same deposition run.[^1] Here, we used a deposition pressure $p_{O_2}=3\,$Pa, in order to obtain a measurable amount of CeO$_2$. The XRD data showed no discernible difference in the amount of CeO$_2$ for the different substrates.
The growth and phase stability of some complex oxide materials may depend on the type of termination layer of the substrate.[@Huijbregtse01] Therefore, we have grown several LCeMO films on (001) STO substrates with different termination (either SrO or TiO$_2$) in order to determine whether the substrate termination influences the microstructure of the films. The SrO terminated substrates were obtained by annealing at $950\,^{\circ}$C, for 1h in an oxygen flow, while the TiO$_2$-terminated STO substrates were obtained by chemical etching in a BHF solution, following the procedure described in Ref. \[\]. The results showed no correlation between the substrate termination and the CeO$_2$ impurity phase formation. These results suggest that, for the conditions used in this study, the level of strain and the type of substrate termination do not have an important effect on the phase stability in the LCeMO system and that, most probably, the deposition conditions (in particular $T_s$ and $p_{O_2}$) are the determining factors.
Figure \[annealing\] shows the evolution of the $c$-axis with additional ex-situ annealing steps as obtained from XRD data for the (001) peak on sample E. As a result, the $c$-axis decreased from $c=3.894\,$[Å]{} to $c=3.872\,$[Å]{}. As the $a$- and $b$-axis bulk values for LCeMO are smaller than the ones of the STO substrate, the observed shrinking of the $c$-axis cannot be related to strain relaxation effects (which would increase $c$), but most probably to the incorporation of extra oxygen in the film. As another result of the annealing experiments, we did not find a correlation between ex-situ annealing and the CeO$_2$ concentration in our films. This is in contrast to the observations presented by Yanagida [*et al.*]{}[@Yanagida04] and Chang [*et al.*]{};[@Chang04] however, in their work, much longer annealing times (up to 10 hours) have been used. In our case, samples without secondary phase stayed single phase regarding the XRD data. However, while XRD data indicate that films deposited at 8-25Pa O$_2$ are single phase, HRTEM analysis showed evidence for phase separation even in these samples. The results of the microstructural TEM analysis are discussed in the following.
![(Color online) XRD pattern at the (001) peak for sample E, showing the evolution of the $c$-axis with ex-situ annealing steps: after in-situ annealing (1), first (2) and second (3) ex-situ annealing.[]{data-label="annealing"}](XRDrr33annealing){width="35.00000%"}
TEM
===
To obtain a better understanding on the relation between the microstructure and the physical properties of our LCeMO thin films, a few samples grown at different oxygen pressure were selected for TEM analysis. Here, we show results obtained from two films: sample E prepared at $p_{O_2}=8\,$Pa, which appears single phase at XRD, and sample K prepared at $p_{O_2}=3\,$Pa, containing CeO$_2$ as secondary phase. TEM studies were carried out using a JEOL 4000EX microscope operated at 400kV. The instrument has a point-to-point resolution of 0.17nm. Planview TEM specimens were prepared by mechanical polishing of the samples down to a thickness of $30\,\mu$m, followed by Ar ion-milling at grazing incidence to reach electron transparency.
Figure \[TEM\](a) shows a HRTEM plan view image of the LCeMO thin film grown at 8Pa O$_2$ (sample E). Several CeO$_2$ nanoclusters are indicated by arrows. A more detailed HRTEM image of one of the clusters is shown in Fig. \[TEM\](b). Figure \[TEM\](c) shows a TEM plan view image of the LCeMO thin film grown at 3Pa O$_2$ (sample K). In this sample, a higher density of CeO$_2$ nanoclusters in comparison to sample E is observed. Furthermore, the size of the clusters is also larger (although still within the nanometer region). The interface between the CeO$_2$ nanoclusters and the matrix is better defined in comparison to sample E.
![(a) Planview HRTEM images of (a) sample E grown at 8Pa O$_2$; arrows indicate the CeO$_2$ inclusions. An example of an inclusion \[cf. left arrow in (a)\] is shown in more detail in (b). (c) sample K grown at 8Pa O$_2$.[]{data-label="TEM"}](TEM){width="45.00000%"}
HRTEM data for the analyzed samples prove the presence of CeO$_2$ nanoclusters in the perovskite matrix (LCeMO) and show that CeO$_2$ segregation in the 3Pa sample is larger than in the 8Pa sample. In case of the 8Pa sample (and for another 25Pa film not shown) the small total volume of CeO$_2$ clusters made them untraceable by XRD. As an important consequence, our TEM data show that even LCeMO films which appear to be single phase from XRD data contain CeO$_2$ nanoclusters. This observation is important, as it has been shown [@Yanagida04] that the valence state of Mn in LCeMO is sensitive to the degree of Ce segregation, which drives the valences from Mn$^{3+}$ to Mn$^{4+}$, even in the presence of Ce$^{4+}$.
Transport and magnetic properties
=================================
![(Color online) Resistivity vs. temperature for samples A, B and E (with deposition pressure $p_{O_2}$ in parenthesis). The behavior after ex-situ annealing is shown for sample B and E (B1, E1: without ex-situ annealing; B2, E2: after $1^{\rm st}$ ex-situ annealing step; E3: after $2^{\rm nd}$ ex-situ annealing step).[]{data-label="RT"}](RT){width="45.00000%"}
Figure \[RT\] shows resistivity $\rho$ versus temperature $T$ for samples A, B and E. Sample A was ”as-deposited“ at $p_{O_2}=1\,$Pa and shows no metal-to-insulator transition at all. Due to its high resistivity we could not trace out $\rho(T)$ below $T\approx 150\,$K. Sample B, grown at 3Pa (also ”as-deposited“) shows a slight indication of a metal-to-insulator transition, i. e. a maximum in $\rho(T)$ at $T_{MI}= 175\,$K, with a strong increase in resistivity at $T{{\scriptscriptstyle\stackrel{<}{\sim}}}130\,$K, which can be explained by charge localization. Sample E, grown at 8Pa (annealed in-situ) shows a transition at $T_{MI}=210\,$K.
For sample E, the evolution of the $\rho(T)$ curves after two annealing steps (c. f. Sec.\[Sec:Experiment\]) is additionally shown. The $T_{MI}$ transition temperature increases to 230K, which is accompanied by a decreasing resistivity, presumably due to an increasing charge carrier density. This observation is consistent with results obtained by Yanagida [*et al.*]{}[@Yanagida04] and contradicts the picture of an electron-doped manganite: Oxygen annealing should decrease the concentration of Mn$^{2+}$ ions, hence, reduce the density of electrons as charge carriers and therefore lower $T_{MI}$ and increase resistivity.[@Wang06] The annealing steps seem to create more Mn$^{4+}$ in the samples, and the double-exchange between Mn$^{3+}$ and Mn$^{4+}$ gets stronger, which leads to an increase of $T_{MI}$. This interpretation is also supported by the results from measurements of the saturation magnetization ($M_s$) and the spectroscopic analysis, which will be discussed further below.
Figure \[RT\] also shows that $T_{MI}$ of sample B increases more drastically than sample E, even after only a single ex-situ annealing step. This might be due to the higher concentration of a secondary phase (CeO$_2$) in sample B (c. f. Fig. \[XRDCeO2\]), which may favor oxygen diffusion into the film due to crystal defects.
![(Color online) Magnetization vs. applied magnetic field at $T=20\,$K for the as-grown ($p_{O_2}=3\,$Pa) sample B (B1) and after ex-situ annealing (B2).[]{data-label="MvsH1"}](MvsH1){width="35.00000%"}
In Fig. \[MvsH1\] the magnetization $M$ (in units of $\mu_B/$Mn site) vs. applied field $\mu_0 H$ at $T=20\,$K is shown for sample B, measured ”as grown“ (B1) and after ex-situ annealing (B2). The ex-situ annealing step caused a decrease in the saturation magnetization $M_s$, from 2.93 to $2.40\,\mu_B$/Mn-site, while $T_{MI}$ increased from 175 to 250K. With the magnetic moments $m=$5, 4 and $3\,\mu_B$ for Mn$^{2+}$, Mn$^{3+}$ and Mn$^{4+}$, respectively, the theoretical value for the saturation magnetization of electron-doped LCeMO is $M_s=4.3\,\mu_B$/Mn-site.[@Zhang03] Until now, this value has never been achieved. However, for the hole-doped manganites, it is known that excess oxygen increases the valences from Mn$^{3+}$ to Mn$^{4+}$, and therefore decreases the magnetization. Hence, the observed decrease in $M_s$ with oxygen annealing can be explained by the decrease in Mn$^{2+}$ and concomitant increase in Mn$^{4+}$ concentration.
![(Color online) Comparison of samples with different thickness $d$, grown under the same deposition conditions ($p_{O_2}=8\,$Pa; in-situ annealed). (a) XRD $\Theta - 2\Theta$ scans; the inset shows that the $c$-axis value increases with increasing $d$. (b) Resistivity vs. temperature; the inset shows that the transition temperature $T_{MI}$ decreases and the saturation magnetization $M_s$ (from $M(H)$ data; not shown) increases with increasing $d$.[]{data-label="thickness"}](thickness){width="45.00000%"}
In order to study the dependence of structural, transport and magnetic properties on film thickness $d$, four samples (C, E, F, G with $d$=100, 65, 40 and 20nm, respectively) were grown under the same conditions, i.e., at $p_{O_2}=8\,$Pa with in-situ annealing. The $\Theta-2\Theta$ XRD scans of the (001) peak in Fig. \[thickness\](a) show that with decreasing film thickness the $c$-axis shrinks \[see inset\]. Assuming a fixed unit cell volume, this observation might be explained by increasing tensile strain with decreasing $d$, as the bulk in-plane lattice parameters of LCeMO are smaller than those for the STO substrate. However, as oxygen vacancies tend to expand the lattice parameters, an increasing lack of oxygen with increasing $d$ has the same effect. The transport properties shown in Fig. \[thickness\](b) indicate exactly this lack of oxygen with increasing film thickness. Sample C, with largest $d$, shows again charge localization at low $T$, while the thinnest film has the highest $T_{MI}$ \[c. f. inset\] and lowest $\rho$. >From magnetization measurements on samples C, E, F and G we also find that $M_s$ increases with $d$ \[c. f. inset in Fig. \[thickness\](b)\]. The lowest saturation magnetization for the thinnest sample G is another indication for the higher oxygen concentration compared to the others.
Hall measurements
=================
In order to determine the type of majority charge carriers via the Hall effect, we chose one of our films (sample H, $d=100\,$nm) which was deposited at relatively low oxygen pressure ($p_{O_2}=3\,$Pa) and cooled in 1bar, without an annealing step. >From measurements of the longitudinal resistivity $\rho(T)$ of the patterned film we find a clear MI transition with rather high $T_{MI}=260\,$K. The Hall resistivity $\rho_H$ was measured at $T=10$, 50 and 100K in magnetic fields up to 14T. The sign of the Hall voltage was carefully checked by using an $n$-doped silicon reference sample. Figure \[Hall\] shows $\rho_H$ vs. applied magnetic field $H$. The drop of $\rho_H$ in the low-field range reflects the so-called anomalous Hall Effect, $\rho_{aH}=R_{aH}\mu_0M$, which is due to spin orbit interaction.[@Karplus54] Here, $R_{aH}$ is the Hall coefficient for the anomalous Hall effect. With further increasing field, the data show the expected linear behavior of the normal Hall effect $\rho_{nH}=R_{nH}\mu_0H$ with Hall coefficient $R_{nH}=1/ne$ and charge carrier density $n$. The main feature in Fig. \[Hall\] is the positive slope $\partial\rho_H/\partial H$ at high fields, which reveals the majority of the carriers to be holes with $n=1.57$, 1.60 and $1.78\times 10^{22}\,{\rm cm}^{-3}$, for $T=10$, 50 and 100K, respectively. This corresponds to 0.94–1.07 holes/Mn-site. The observation of hole-doping is consistent with the results from transport and magnetization measurements discussed above and also with the spectroscopic analysis, which will be presented in the following section.
![(Color online) Field dependence of the Hall resistivity of sample H. The positive slope at high magnetic field identifies the majority of the carriers to be holes. The solid lines are linear fits to the high-field data.[]{data-label="Hall"}](Hall){width="35.00000%"}
Spectroscopic Analysis
=======================
X-ray Absorption Spectroscopy (XAS) was performed on LCeMO thin films prepared under different conditions, in order to investigate the relation between the manganese valences, the oxygen content and transport and magnetic properties. In total electron yield (TEY) detection mode only the uppermost 5 - 10nm are probed, depending on the electron escape depth, while in fluorescence yield (FY) mode x-ray photons are detected. They have typical attenuation lengths from 100nm (Ce M edge) to 200nm (O K and Mn L edge), thus giving insight into the bulk structure of the samples.
Here we compare two films, D (as-deposited) and G (in-situ annealed), which were deposited at different oxygen pressure $p_{O_2}=25\,$Pa and 8Pa, respectively. >From XPS measurements we find that the oxygen content of G is higher than the one of D. This shows that the higher deposition pressure (for sample D) is not the key to higher oxygen concentration, but that annealing is most relevant. Sample G shows a MI transition at 232K \[c. f. Fig. \[thickness\](b)\], while sample D shows a weak transition at 180K and charge localization at lower temperatures.
A typical spectrum of the O K edge of LCeMO, measured in bulk sensitive fluorescence yield (FY) mode, is seen in Fig. \[XAS-DvsG\] (left). The first structure at about 530eV arises from transitions from the O1s level to states, which are commonly understood to be of mixed Mn3d-O2p character and as being a measure of the hole concentration.[@Abbate92; @Manella05; @Chang05] In fact we found that this prepeak is stronger in sample G, i.e. the more oxidized sample. The second, rather broad and asymmetric feature at 532 to 537eV is attributed to La5d (Ce), La4f (Ce) states hybridized with O2p states. A third set of states (not shown here) is found at about 543eV and is widely believed to derive from hybridization of O2p with higher energy metal-states like Mn 4sp and La 6sp.[@Abbate92]
![(Color online) XA spectra of samples D and G. On the left side the oxygen K edge (FY mode) is shown with the prepeak increasing with higher oxygen content. The right side shows the manganese L$_3$ edge (TEY mode) with different amounts of Mn$^{2+}$ for differently oxidized samples.[]{data-label="XAS-DvsG"}](XAS-DvsG){width="42.00000%"}
The corresponding spectrum at the Mn L edge taken in TEY detection mode is shown in Fig. \[XAS-DvsG\] (right). Both, the L$_3$ edge at 642eV and the L$_2$ edge at 653eV (not shown here) are strongly broadened, indicating the presence of a variety of valence states. The most important feature is the shoulder at 640eV, which is a clear indication of divalent Mn, as can be shown by a comparison with XAS data from MnO.[@Nagel07] In Fig. \[XAS-DvsG\] (right) this shoulder is more pronounced in sample D, i.e., the less oxidized sample. The relative spectral weight of this feature in combination with the relative intensity of Mn3d-O2p states taken from the O K edge is essential to explain the properties of the different samples. A higher degree of oxidation leads to a higher relative spectral weight of the O K prepeak and a lower amount of Mn$^{2+}$. By introducing more oxygen, more holes are created and the manganese valence is increased. This finding is further supported by measurements on three additional samples (not shown here), also showing the effect of film thickness, oxygen pressure during growth and duration of post-growth annealing in oxygen.
The remaining issue is the oxidation state of the Ce ions, which is important for the type of doping. Looking at the Ce absorption M$_5$ edge both in surface-sensitive TEY detection mode and bulk-sensitive FY mode, we found striking differences in the spectral shapes of the measured spectra, as shown in Fig. \[XAS-Ce-Mn\](a). Cerium reference data for CeO$_2$ and CeF$_3$ were taken from Ref. \[\]. In total electron yield detection mode the edge is identical to a pure CeO$_2$ edge, i.e. cerium in a Ce$^{4+}$ state. However, when increasing the information depth by switching to bulk sensitive FY detection,the edge changes drastically. The FY signal contains contributions from Ce$^{4+}$ and Ce$^{3+}$. Note that thermodynamically the reducing power of cerium is not sufficient for the Mn$^{3+}$ - Mn$^{2+}$ transition. The same trends are seen in the FY spectra of the Mn and O edges. Manganese reference data were taken from Ref. \[\]. In case of the Mn edge \[Fig. \[XAS-Ce-Mn\](b)\] a decrease of the Mn$^{2+}$ related feature at 640eV is visible in the FY data, and the edges are broadened towards higher energies than in TEY mode. This indicates an increased amount of Mn$^{3+}$ (642eV) and Mn$^{4+}$ (644eV) species within the film as compared to the near surface region. Finally, at the O K edge (not shown here) the relative prepeak intensity at 530eV increases with growing information depth from TEY to FY mode. As this feature is proportional to the hole concentration, this finding further emphasizes the point that the bulk is more oxidized than the surface and that the majority charge carriers are indeed holes.
![(Color online) XA spectra of sample D in TEY and FY mode (scaled to the TEY intensity) at different absorption edges: (a) Cerium M$_5$ edge; reference spectra of CeO$_2$ and CeF$_3$ were added for comparison. Please note the mixture of Ce$^{3+}$ and Ce$^{4+}$ in FY mode. (b) Manganese L$_3$ edge. The FY data are self-absorption corrected following a procedure by Ref. \[\]. Reference spectra of MnO (blue), Mn$_2$O$_3$ and MnO$_2$ were added for comparison. Please note the missing Mn$^{2+}$ shoulder in FY mode.[]{data-label="XAS-Ce-Mn"}](XAS-Ce-Mn){width="42.00000%"}
Conclusions {#Sec:Conclusions}
===========
We investigated La$_{0.7}$Ce$_{0.3}$MnO$_3$ thin films of variable thickness, grown epitaxially at different oxygen pressure $p_{O_2}$ and subjected to different oxygen annealing procedures. We find that thin film growth at low deposition pressure favors phase separation via the formation of CeO$_2$ inclusions. For higher deposition pressure, still CeO$_2$ nanoclusters are found, as shown by transmission electron microscopy, even for those films which appear to be single phase in x-ray diffraction analysis. Combining electric transport, magnetization and Hall measurements with x-ray photoemission and absorption spectroscopy we obtain a consistent picture in the sense that the appearance of a metal-to-insulator transition in electric transport measurements is always associated with hole doping and the presence of a mixed system of Mn$^{2+}$, Mn$^{3+}$ and Mn$^{4+}$, despite finding Ce$^{4+}$ as a sign of electron doping. The hole-doped behavior of our films may be explained by the presence of cation vacancies (due to CeO$_2$ clustering), which can be occupied by excess oxygen that shifts the valences from Mn$^{2+}$ to Mn$^{3+}$ or Mn$^{4+}$. In particular, oxidation states are well reproduced in the x-ray absorption spectra and fit to the transport properties. Upon oxidizing the samples, the system goes towards Mn$^{3+}$ / Mn$^{4+}$ as expected, while reducing the films forms more Mn$^{2+}$ species. In particular for less oxidized films, we find a reduced layer at the surface with a more oxidized bulk underneath. This explains some of the peculiarities of this system, namely the discrepancy between finding Mn$^{2+}$ and Ce$^{4+}$ and still having holes as majority carriers. Furthermore, this demonstrates that one has to be very careful in relating surface sensitive spectroscopy data to bulk sensitive transport and magnetization data.
We gratefully acknowledge Kathrin Dörr for helpful discussions and Matthias Althammer and Sebastian Gönnenwein for their support with the Hall measurements. Furthermore, we acknowledge the ANKA Angstroemquelle Karlsruhe for the provision of beamtime and we would like to thank P. Nagel, M. Merz and S. Schuppler for the skillful technical assistance using beamline WERA and for valuable discussions. This work was funded by the Deutsche Forschungsgemeinschaft (project no. KO 1303/8-1) and by the European Union under the Framework 6 program for an Integrated Infrastructure Initiative, ref. 026019 ESTEEM. S. B. thanks the Fund for Scientific Research – Flanders.
[^1]: From the bulk values for the LCeMO lattice constants $a=3.821\,\AA$ and $b=3.902\,\AA$ [@Chang04] one obtains an in-plane lattice mismatch ranging from -2% (tensile strain) to +1% (compressive strain) for the different substrates used here.
| ArXiv |
---
abstract: 'We investigate the notion of Connes-amenability, introduced by Runde in [@Runde1], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [@Runde2], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group. This latter point was first shown by Gr[ö]{}nbæk in [@Gron1], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C$^*$-algebras.'
author:
- Matthew Daws
title: '<span style="font-variant:small-caps;">Connes-amenability of bidual and weighted semigroup algebras</span>'
---
*2000 Mathematics Subject Classification:* 22D15, 43A20, 46H25, 46H99 (primary), 46E15, 46M20, 47B47.
Introduction
============
We first fix some notation, following [@Dales]. For a Banach space $E$, we let $E'$ be its dual space, and for $\mu\in E'$ and $x\in E$, we write ${{\langle {\mu} , {x} \rangle}} = \mu(x)$ for notational convenience. We then have the canonical map $\kappa_E:E\rightarrow E''$ defined by ${{\langle {\kappa_E(x)} , {\mu} \rangle}} = {{\langle {\mu} , {x} \rangle}}$ for $\mu\in E',x\in E$. For Banach spaces $E$ and $F$, we write ${\mathcal{B}}(E,F)$ for the Banach space of bounded linear maps between $E$ and $F$. We write ${\mathcal{B}}(E,E) = {\mathcal{B}}(E)$. For $T\in{\mathcal{B}}(E,F)$, the *adjoint* of $T$ is $T'\in{\mathcal{B}}(F',E')$, defined by ${{\langle {T'(\mu)} , {x} \rangle}} =
{{\langle {\mu} , {T(x)} \rangle}}$, for $\mu\in F'$ and $x\in E$.
Let ${\mathcal{A}}$ be a Banach algebra. A *Banach left ${\mathcal{A}}$-module* is a Banach space $E$ together with a bilinear map ${\mathcal{A}}\times E \rightarrow E; (a,x) \mapsto a\cdot x$, such that $\|a\cdot x\|\leq \|a\|\|x\|$ and $a\cdot(b\cdot x)=
ab\cdot x$ for $a,b\in{\mathcal{A}}$ and $x\in E$. Similarly, we have the notion of a *Banach right ${\mathcal{A}}$-module* and a *Banach ${\mathcal{A}}$-bimodule*. If $E$ is a Banach ${\mathcal{A}}$-bimodule (resp. left or right module) then ${\mathcal{A}}'$ is a Banach ${\mathcal{A}}$-bimodule (resp. right or left module) with module action given by $${{\langle {a\cdot\mu} , {x} \rangle}} = {{\langle {\mu} , {x\cdot a} \rangle}}
\qquad {{\langle {\mu\cdot a} , {x} \rangle}} = {{\langle {\mu} , {a\cdot x} \rangle}}
\qquad (a\in{\mathcal{A}}, x\in E).$$ Notice that as ${\mathcal{A}}$ is certainly a bimodule over itself (with module action induced by the algebra product) we also have that ${\mathcal{A}}'$, ${\mathcal{A}}''$ etc. are Banach ${\mathcal{A}}$-bimodules. Given a Banach ${\mathcal{A}}$-bimodule $E$, a subspace $F$ of $E$ is a *submodule* if $a\cdot x, x\cdot a\in F$ for each $a\in{\mathcal{A}}$ and $x\in F$. For Banach ${\mathcal{A}}$-bimodules $E$ and $F$, $T\in{\mathcal{B}}(E,F)$ is an *${\mathcal{A}}$-bimodule homomorphism* when $$a \cdot T(x) = T(a\cdot x) \qquad
T(x) \cdot a = T(x\cdot a) \qquad (a\in{\mathcal{A}}, x\in E).$$
A linear map $d:{\mathcal{A}} \rightarrow E$ between a Banach algebra ${\mathcal{A}}$ and a Banach ${\mathcal{A}}$-bimodule $E$ is a *derivation* if $d(ab) = a \cdot d(b) + d(a)\cdot b$ for $a,b\in{\mathcal{A}}$. For $x\in E$, we define $\delta_x : {\mathcal{A}}\rightarrow E$ by $\delta_x(a) =
a\cdot x-x\cdot a$. Then $\delta_x$ is a derivation, called an *inner derivation*.
A Banach algebra ${\mathcal{A}}$ is said to be *super-amenable* or *contractable* if every bounded derivation $d:{\mathcal{A}}\rightarrow E$, for every Banach ${\mathcal{A}}$-bimodule $E$, is inner. For example, a C$^*$-algebra ${\mathcal{A}}$ is super-amenable if and only if ${\mathcal{A}}$ is finite-dimensional. It is conjectured that there are no infinite-dimensional, super-amenable Banach algebras.
If we restrict to derivations to $E'$ for Banach ${\mathcal{A}}$-bimodules $E$ then we arrive at the notion of *amenability*. For example, a C$^*$-algebra ${\mathcal{A}}$ is amenable if and only if ${\mathcal{A}}$ is nuclear; a group algebra $L^1(G)$ is amenable if and only if the locally compact group $G$ is amenable (which is the motivating example). See [@RundeBook] for further discussions of amenability and related notions.
Let $E$ be a Banach space and $F$ a closed subspace of $E$. Then we naturally, isometrically, identify $F'$ with $E' / F^\circ$, where $$F^\circ = \{ \mu\in E' : {{\langle {\mu} , {x} \rangle}}=0 \ (x\in F) \}.$$
\[dual\_means\] Let $E$ be a Banach space and $E_*$ be a closed subspace of $E'$. Let $\pi_{E_*}: E'' \rightarrow E''/E_*^\circ$ be the quotient map, and suppose that $\pi_{E_*} \circ \kappa_E$ is an isomorphism from $E$ to $E_*'$. Then we say that $E$ is a *dual Banach space* with *predual $E_*$*.
When ${\mathcal{A}}$ is a dual Banach space with predual ${\mathcal{A}}_*$ which is also a submodule of ${\mathcal{A}}'$ we say that ${\mathcal{A}}$ is a *dual Banach algebra*. [$\square$]{}
For a dual Banach algebra ${\mathcal{A}}$ with predual ${\mathcal{A}}_*$, we henceforth identify ${\mathcal{A}}$ with ${\mathcal{A}}_*'$. Thus we get a weak$^*$-topology on ${\mathcal{A}}$, which we denote by $\sigma({\mathcal{A}},{\mathcal{A}}_*)$. It is a simple exercise to show that ${\mathcal{A}}$ is a dual Banach algebra if and only if ${\mathcal{A}}$ is a dual Banach space such that the algebra product is separately $\sigma({\mathcal{A}},{\mathcal{A}}_*)$-continuous (see [@Runde1]). The following lemma is standard.
\[weak\_star\_cts\] Let $E$ and $F$ be dual Banach spaces with preduals $E_*$ and $F_*$ respectively, and let $T\in{\mathcal{B}}(E,F)$. Then the following are equivalent:
1. $T$ is $\sigma(E,E_*) - \sigma(F,F_*)$ continuous;
2. $T'(\kappa_{F_*}(F_*)) \subseteq \kappa_{E_*}(E_*)$;
3. there exists $S\in{\mathcal{B}}(F_*,E_*)$ such that $S'=T$.
[$\square$]{}
As noticed by Runde (see [@Runde1]), there are very few Banach algebras which are both dual and amenable. For von Neumann algebras, which are the motivating example of dual Banach algebras, there is a weaker notion of amenablity, called Connes-amenability, which has a natural generalisation to the case of dual Banach algebras.
Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Let $E$ be a Banach ${\mathcal{A}}$-bimodule. Then $E'$ is a *w$^*$-Banach ${\mathcal{A}}$-bimodule* if, for each $\mu\in E'$, the maps $${\mathcal{A}}\rightarrow E',\quad a\mapsto
\begin{cases} a\cdot \mu, \\ \mu\cdot a \end{cases}$$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*) - \sigma(E',E)$ continuous.
Then $({\mathcal{A}},{\mathcal{A}}_*)$ is Connes-amenable if, for each w$^*$-Banach ${\mathcal{A}}$-bimodule $E'$, each derivation $d:{\mathcal{A}}\rightarrow E'$, which is $\sigma({\mathcal{A}},{\mathcal{A}}_*) -
\sigma(E',E)$ continuous, is inner. [$\square$]{}
Given a Banach algebra ${\mathcal{A}}$, we define bilinear maps ${\mathcal{A}}''\times{\mathcal{A}}'\rightarrow{\mathcal{A}}'$ and ${\mathcal{A}}'\times{\mathcal{A}}''\rightarrow{\mathcal{A}}'$ by $${{\langle {\Phi\cdot\mu} , {a} \rangle}} = {{\langle {\Phi} , {\mu\cdot a} \rangle}}
\quad {{\langle {\mu\cdot\Phi} , {a} \rangle}} = {{\langle {\Phi} , {a\cdot\mu} \rangle}}
\qquad (\Phi\in{\mathcal{A}}'', \mu\in{\mathcal{A}}', a\in{\mathcal{A}}).$$ We then define two bilinear maps ${\Box},{\Diamond}:{\mathcal{A}}''\times{\mathcal{A}}''
\rightarrow{\mathcal{A}}''$ by $${{\langle {\Phi{\Box}\Psi} , {\mu} \rangle}} = {{\langle {\Phi} , {\Psi\cdot\mu} \rangle}}
\quad {{\langle {\Phi{\Diamond}\Psi} , {\mu} \rangle}} = {{\langle {\Psi} , {\mu\cdot\Phi} \rangle}}
\qquad (\Phi,\Psi\in{\mathcal{A}}'', \mu\in{\mathcal{A}}').$$ We can check that ${\Box}$ and ${\Diamond}$ are actually algebra products, called the *first* and *second Arens products* respectively. Then $\kappa_A:{\mathcal{A}}\rightarrow{\mathcal{A}}''$ is a homomorphism with respect to either Arens product. When ${\Box}= {\Diamond}$, we say that ${\mathcal{A}}$ is *Arens regular*. In particular, when ${\mathcal{A}}$ is Arens regular, we may check that ${\mathcal{A}}''$ is a dual Banach algebra with predual ${\mathcal{A}}'$.
\[ca\_facts\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra. When ${\mathcal{A}}$ is amenable, ${\mathcal{A}}''$ is Connes-amenable. If ${\kappa_{{\mathcal{A}}}({\mathcal{A}})}$ is an ideal in ${\mathcal{A}}''$ and ${\mathcal{A}}''$ is Connes-amenable, then ${\mathcal{A}}$ is amenable.
Let ${\mathcal{A}}$ be a C$^*$-algebra. Then ${\mathcal{A}}$ is Arens regular, and ${\mathcal{A}}''$ is Connes-amenable if and only if ${\mathcal{A}}$ is amenable.
The first statements are [@Runde1 Corollary 4.3] and [@Runde1 Theorem 4.4]. The statement about C$^*$-algebras is detailed in [@RundeBook Chapter 6].
Another class of Connes-amenable dual Banach algebras is given by Runde in [@Runde4], where it is shown that $M(G)$, the measure algebra of a locally compact group $G$, is amenable if and only if $G$ is amenable.
The organisation of this paper is as follows. Firstly, we study intrinsic characterisations of amenability, recalling a result of Runde from [@Runde2]. We then simplify these conditions in the case of Arens regular Banach algebras. We recall the notion of an *injective* module, and quickly note how Connes-amenability can be phrased in this language. The final section of the paper then applies these ideas to weighted semigroup algebras. We finish with some open questions.
Characterisations of amenability
================================
Let $E$ and $F$ be Banach spaces, and form the algebraic tensor product $E\otimes F$. We can norm $E\otimes F$ with the *projective tensor norm*, defined as $$\|u\|_\pi = \inf\Big\{ \sum_{k=1}^n \|x_k\|\|y_k\| :
u = \sum_{k=1}^n x_k \otimes y_k \Big\}
\qquad (u\in E\otimes F).$$ Then the completion of $(E\otimes F, \|\cdot\|_\pi)$ is $E {{\widehat{\otimes}}}F$, the *projective tensor product* of $E$ and $F$.
Let ${\mathcal{A}}$ be a Banach algebra. Then ${\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}$ is a Banach ${\mathcal{A}}$-bimodule for the module actions given by $$a\cdot (b\otimes c) = ab \otimes c,
\quad (b\otimes c) \cdot a = b\otimes ca
\qquad (a\in{\mathcal{A}}, b\otimes c\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ Define $\Delta_{{\mathcal{A}}} : {\mathcal{A}} {{\widehat{\otimes}}}{\mathcal{A}} \rightarrow {\mathcal{A}}$ by $\Delta_{{\mathcal{A}}}(a\otimes b)=ab$. Then $\Delta_{{\mathcal{A}}}$ is an ${\mathcal{A}}$-bimodule homomorphism.
\[when\_amen\] Let ${\mathcal{A}}$ be a Banach algebra. Then the following are equivalent:
1. ${\mathcal{A}}$ is amenable;
2. ${\mathcal{A}}$ has a *virtual diagonal*, which is a functional $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ such that $a\cdot M =M\cdot a$ and $\Delta_{{\mathcal{A}}}''(M) \cdot a = \kappa_{{\mathcal{A}}}(a)$ for each $a\in{\mathcal{A}}$.
[$\square$]{}
Runde introduced, in [@Runde2], the following notion in order to prove a version of the above theorem for Connes-amenability.
Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$, and let $E$ be a Banach ${\mathcal{A}}$-bimodule. Then $x\in \sigma WC(E)$ if and only if the maps ${\mathcal{A}}\rightarrow E$, $$a \mapsto \begin{cases} a\cdot x, \\ x\cdot a \end{cases}$$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*) - \sigma(E,E')$ continuous. [$\square$]{}
It is clear that $\sigma WC(E)$ is a closed submodule of $E$. The ${\mathcal{A}}$-bimodule homomorphism $\Delta_{{\mathcal{A}}}$ has adjoint $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}' \rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. In [@Runde2 Corollary 4.6] it is shown that $\Delta'_{{\mathcal{A}}}({\mathcal{A}}_*) \subseteq \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )$. Consequently, we can view $\Delta_{{\mathcal{A}}}'$ as a map ${\mathcal{A}}_* \rightarrow \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )$, and hence view $\Delta''_{{\mathcal{A}}}$ as a map $\sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )' \rightarrow {\mathcal{A}}_*'
= {\mathcal{A}}$, denoted by $\tilde \Delta_{{\mathcal{A}}}$.
\[Runde\_Thm\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Then the following are equivalent:
1. ${\mathcal{A}}$ is Connes-amenable;
2. ${\mathcal{A}}$ has a *$\sigma WC$-virtual diagonal*, which is $M\in \sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )'$ such that $a\cdot M = M\cdot a$ and $a \tilde \Delta_{{\mathcal{A}}}(M) = a$ for each $a\in{\mathcal{A}}$.
This is [@Runde2 Theorem 4.8].
In particular, we see that a Connes-amenable Banach algebra is unital (which can of course be shown in an elementary fashion, as in [@Runde1 Proposition 4.1]).
Connes-amenability for biduals of algebras {#con_amen_bidual}
==========================================
Recall Gantmacher’s theorem, which states that a bounded linear map $T:E\rightarrow F$ between Banach spaces $E$ and $F$ is *weakly-compact* if and only if $T''(E'') \subseteq \kappa_F(F)$. We write ${\mathcal{W}}(E,F)$ for the collection of weakly-compact operators in ${\mathcal{B}}(E,F)$.
\[wsw\_cty\] Let $E$ be a dual Banach space with predual $E_*$, let $F$ be a Banach space, and let $T\in{\mathcal{B}}(E,F')$. Then the following are equivalent, and in particular each imply that $T$ is weakly-compact:
1. $T$ is $\sigma(E,E_*) - \sigma(F',F'')$ continuous;
2. $T'(F'') \subseteq \kappa_{E_*}(E_*)$;
3. there exists $S\in{\mathcal{W}}(F,E_*)$ such that $S' = T$.
That (1) and (2) are equivalent is standard (compare with Lemma \[weak\_star\_cts\]).
Suppose that (2) holds, so that we may define $S\in{\mathcal{B}}(F,E_*)$ by $\kappa_{E_*}\circ S = T'\circ\kappa_F$. Then, for $x\in E$ and $y\in F$, we have $${{\langle {x} , {S(y)} \rangle}} = {{\langle {T'(\kappa_F(y))} , {x} \rangle}}
= {{\langle {T(x)} , {y} \rangle}},$$ so that $S'=T$. Then $S''(F'') = T'(F'') \subseteq
\kappa_{E_*}(E_*)$, so that $S$ is weakly-compact, by Gantmacher’s Theorem, so that (3) holds.
Conversely, if (3) holds, as $S$ is weakly-compact, we have $\kappa_{E*}(E_*) \supseteq S''(F'') = T'(F'')$, so that (2) holds.
It is standard that for Banach spaces $E$ and $F$, we have $(E{{\widehat{\otimes}}}F)' = {\mathcal{B}}(F,E')$ with duality defined by $${{\langle {T} , {x\otimes y} \rangle}} = {{\langle {T(y)} , {x} \rangle}}
\qquad (T\in{\mathcal{B}}(F,E'), x\otimes y\in E{{\widehat{\otimes}}}F).$$ Then we see, for $a,b,c\in{\mathcal{A}}$ and $T\in({\mathcal{A}}
{{\widehat{\otimes}}}{\mathcal{A}})' = {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, that ${{\langle {a\cdot T} , {b\otimes c} \rangle}} = {{\langle {T(ca)} , {b} \rangle}}$ and that ${{\langle {T\cdot a} , {b\otimes c} \rangle}} = {{\langle {T(c)} , {ab} \rangle}}
= {{\langle {T(c)\cdot a} , {b} \rangle}}$ so that $$(a\cdot T)(c) = T(ca), \quad (T\cdot a)(c) = T(c)\cdot a
\qquad (a,c\in{\mathcal{A}}, T:{\mathcal{A}}\rightarrow{\mathcal{A}}').
\label{eq:one}$$
Notice that we could also have defined $(E{{\widehat{\otimes}}}F)'$ to be ${\mathcal{B}}(E,F')$. This would induce a different bimodule structure on ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and we shall see in Section \[Inj\_predual\] that our chosen convention seems more natural for the task at hand.
\[first\_wap\_prop\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$, define maps $\phi_r, \phi_l : {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}\rightarrow{\mathcal{A}}'$ by $$\phi_r(a\otimes b) = T'\kappa_{{\mathcal{A}}}(a) \cdot b,
\quad \phi_l(a\otimes b) = a \cdot T(b)
\qquad (a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ Then $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact and have ranges contained in ${\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)}$.
For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$, define $R_T,L_T : {\mathcal{A}}\rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$ by $R_T(a) = a\cdot T$ and $L_T = T\cdot a$, for $a\in{\mathcal{A}}$. By definition, $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $R_T$ and $L_T$ are $\sigma({\mathcal{A}},{\mathcal{A}}_*)-\sigma({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'),({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'')$ continuous. By Lemma \[wsw\_cty\], this is if and only if there exist $\varphi_r, \varphi_l \in {\mathcal{W}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},{\mathcal{A}}_*)$ such that $\varphi_r' = R_T$ and $\varphi_l' = L_T$.
For $a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}$ and $c\in{\mathcal{A}}$, we see that $$\begin{aligned}
{{\langle {c} , {\varphi_r(a\otimes b)} \rangle}} &= {{\langle {R_T(c)} , {a\otimes b} \rangle}}
= {{\langle {c\cdot T} , {a\otimes b} \rangle}} = {{\langle {T(bc)} , {a} \rangle}} \\
&= {{\langle {T'\kappa_{{\mathcal{A}}}(a)} , {bc} \rangle}}
= {{\langle {T'\kappa_{{\mathcal{A}}}(a)\cdot b} , {c} \rangle}}
= {{\langle {\phi_r(a\otimes b)} , {c} \rangle}}, \\
{{\langle {c} , {\varphi_l(a\otimes b)} \rangle}} &= {{\langle {L_T(c)} , {a\otimes b} \rangle}}
= {{\langle {T\cdot c} , {a\otimes b} \rangle}} = {{\langle {T(b)} , {ca} \rangle}} \\
&= {{\langle {a\cdot T(b)} , {c} \rangle}} = {{\langle {\phi_l(a\otimes b)} , {c} \rangle}}.\end{aligned}$$ Thus $\kappa_{{\mathcal{A}}_*}\circ\varphi_r = \phi_r$ and $\kappa_{{\mathcal{A}}_*}\circ\varphi_l = \phi_l$. Consequently, we see that $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact and take values in ${\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)}$.
The following definition is [@Runde2 Definition 4.1].
Let ${\mathcal{A}}$ be a Banach algebra and let $E$ be a Banach ${\mathcal{A}}$-bimodule. An element $x\in E$ is *weakly almost periodic* if the maps $${\mathcal{A}}\rightarrow E,\quad
a \mapsto \begin{cases} a\cdot x, \\ x\cdot a \end{cases}$$ are weakly-compact. The collection of weakly almost periodic elements in $E$ is denoted by ${\operatorname{WAP}}(E)$. [$\square$]{}
\[wap\_to\_maps\] Let ${\mathcal{A}}$ be a Banach algebra, and let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')
= ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. Let $\phi_r,\phi_l : {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}
\rightarrow{\mathcal{A}}'$ be as above. Then $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$ if and only if $\phi_r$ and $\phi_l$ are weakly-compact.
Let $R_T,L_T:{\mathcal{A}}\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ be as in the above proof. By definition, $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$ if and only if $L_T$ and $R_T$ are weakly-compact. We can verify that $$\phi_r'\circ\kappa_{{\mathcal{A}}} = R_T,\
\phi_l'\circ\kappa_{{\mathcal{A}}} = L_T,\
R_T'\circ\kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}} = \phi_r,\
L_T'\circ\kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}} = \phi_l,$$ which completes the proof.
\[unital\_dual\_sigma\] Let ${\mathcal{A}}$ be a unital, dual Banach algebra with predual ${\mathcal{A}}_*$, and let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') = ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$. The following are equivalent, and, in particular, each imply that $T$ is weakly-compact:
1. $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$;
2. $T({\mathcal{A}}) \subseteq
\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}}))
\subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, and $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$;
3. $T({\mathcal{A}}) \subseteq
\kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}}))
\subseteq \kappa_{{\mathcal{A}}_*}({\mathcal{A}}_*)$, and $T \in {\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') )$.
Let $e_{{\mathcal{A}}}$ be the unit of ${\mathcal{A}}$, so that for $a\in{\mathcal{A}}$, we have $T(a) = \phi_l(e_{{\mathcal{A}}} \otimes a)$ and $T'\kappa_{{\mathcal{A}}}(a) = \phi_r(a \otimes e_{{\mathcal{A}}})$, which shows that (1) implies (2); clearly (2) implies (1).
As ${\mathcal{A}}_*$ is an ${\mathcal{A}}$-bimodule, (2) and (3) are equivalent by an application of Lemma \[wap\_to\_maps\] and Proposition \[first\_wap\_prop\].
\[When\_Con\_Amen\] Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. Then ${\mathcal{A}}$ is Connes-amenable if and only if ${\mathcal{A}}$ is unital and there exists $M\in({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ such that:
1. ${{\langle {M} , {a\cdot T-T\cdot a} \rangle}}=0$ for $a\in{\mathcal{A}}$ and $T\in\sigma WC( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$;
2. $\kappa_{{\mathcal{A}}_*}' \Delta''_{{\mathcal{A}}}(M) = e_{{\mathcal{A}}}$, where $e_{{\mathcal{A}}}$ is the unit of ${\mathcal{A}}$.
As $\sigma WC( ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})' )'$ is a quotient of $({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$, this is just a re-statement of Theorem \[Runde\_Thm\].
When ${\mathcal{A}}$ is an Arens regular Banach algebra, ${\mathcal{A}}''$ is a dual Banach algebra with canonical predual ${\mathcal{A}}'$. In this case, we can make some significant simplifications in the characterisation of when ${\mathcal{A}}''$ is Connes-amenable.
For a Banach algebra ${\mathcal{A}}$, we define the map $\kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}}: {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}
\rightarrow {\mathcal{A}}'' {{\widehat{\otimes}}}{\mathcal{A}}''$ by $$( \kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}} ) (a\otimes b)
= \kappa_{{\mathcal{A}}}(a) \otimes \kappa_{{\mathcal{A}}}(b)
\qquad (a\otimes b\in {\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}).$$ We turn ${\mathcal{A}}'' {{\widehat{\otimes}}}{\mathcal{A}}''$ into a Banach ${\mathcal{A}}$-bimodule in the canonical way. Then $\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}}$ is an ${\mathcal{A}}$-bimodule homomorphism. The following is a simple verification.
\[can\_proj\] Let ${\mathcal{A}}$ be a Banach algebra. The map $$\iota_{{\mathcal{A}}}:{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}') \rightarrow
{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}'''); \ T\mapsto T'',$$ is an ${\mathcal{A}}$-bimodule homomorphism which is an isometry onto its range. Furthermore, we have that $(\kappa_{{\mathcal{A}}} \otimes \kappa_{{\mathcal{A}}})' \circ \iota_{{\mathcal{A}}} =
I_{{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')}$. Define $\rho_{{\mathcal{A}}}:
{\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}''\rightarrow ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ by $${{\langle {\rho_{{\mathcal{A}}}(\tau)} , {T} \rangle}} = {{\langle {T''} , {\tau} \rangle}}
\qquad (\tau\in {\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'',
T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')=({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})').$$ Then $\rho_{{\mathcal{A}}}$ is a norm-decreasing ${\mathcal{A}}$-bimodule homomorphism which satisfies $\rho_{{\mathcal{A}}} \circ
(\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}}) = \kappa_{{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}}$. [$\square$]{}
For a Banach algebra ${\mathcal{A}}$, it is clear that ${\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ is a sub-${\mathcal{A}}$-bimodule of ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')=({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'$.
\[arens\_wap\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is unital, and let $T\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') =
({\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'')'$. Then the following are equivalent:
1. $T \in \sigma WC({\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}'''))$, where we treat ${\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$ as an ${\mathcal{A}}''$-bimodule;
2. $T = S''$ for some $S\in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$, where now we treat ${\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ as an ${\mathcal{A}}$-bimodule.
We apply Corollary \[unital\_dual\_sigma\] to ${\mathcal{A}}''$, so that (1) is equivalent to $T$ being weakly-compact, $T({\mathcal{A}}'')\subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, $T'(\kappa_{{\mathcal{A}}''}({\mathcal{A}}'')) \subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, and $T\in{\operatorname{WAP}}( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. Thus, if (1) holds, then there exists $T_0 \in {\mathcal{W}}({\mathcal{A}}'',{\mathcal{A}}')$ such that $T = \kappa_{{\mathcal{A}}'} \circ T_0$, and there exists $T_1 \in {\mathcal{W}}({\mathcal{A}}'',{\mathcal{A}}')$ such that $T'\circ\kappa_{{\mathcal{A}}''} = \kappa_{{\mathcal{A}}'} \circ T_1$. Let $S = T_0\circ\kappa_{{\mathcal{A}}} \in {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. Then, for $a\in{\mathcal{A}}$ and $\Psi\in{\mathcal{A}}''$, we have $$\begin{aligned}
{{\langle {S'(\Psi)} , {a} \rangle}} &= {{\langle {\Psi} , {T_0(\kappa_{{\mathcal{A}}}(a))} \rangle}}
= {{\langle {T(\kappa_{{\mathcal{A}}}(a))} , {\Psi} \rangle}}
= {{\langle {T'(\kappa_{{\mathcal{A}}''}(\Psi))} , {\kappa_{{\mathcal{A}}}(a)} \rangle}} \\
&= {{\langle {\kappa_{{\mathcal{A}}}(a)} , {T_1(\Psi)} \rangle}}
= {{\langle {T_1(\Psi)} , {a} \rangle}},\end{aligned}$$ so that $S' = T_1$. Thus, for $\Phi,\Psi\in{\mathcal{A}}''$, we have $$\begin{aligned}
{{\langle {S''(\Phi)} , {\Psi} \rangle}} &= {{\langle {\Phi} , {T_1(\Psi)} \rangle}}
= {{\langle {T'(\kappa_{{\mathcal{A}}''}(\Psi))} , {\Phi} \rangle}}
= {{\langle {T(\Phi)} , {\Psi} \rangle}},\end{aligned}$$ so that $S''=T$. We know that the maps $L_T,R_T:
{\mathcal{A}}''\rightarrow{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$, defined by $L_T(\Phi) = T\cdot\Phi$ and $R_T(\Phi) = \Phi\cdot T$ for $\Phi\in{\mathcal{A}}''$, are weakly-compact. Define $L_S,R_S:
{\mathcal{A}}\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ is an analogous manner, using $S\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. For $a\in{\mathcal{A}}$, $S\cdot a
\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$, so for $\Psi\in{\mathcal{A}}''$ and $b\in{\mathcal{A}}$, $${{\langle {(S\cdot a)'(\Psi)} , {b} \rangle}} = {{\langle {\Psi} , {(S\cdot a)(b)} \rangle}}
= {{\langle {\Psi} , {S(b)\cdot a} \rangle}} = {{\langle {a\cdot\Psi} , {S(b)} \rangle}}
= {{\langle {S'(a\cdot\Psi)} , {b} \rangle}}.$$ Thus, for $a\in{\mathcal{A}}$ and $\Phi,\Psi\in{\mathcal{A}}''$, we have that $$\begin{aligned}
{{\langle { \iota_{{\mathcal{A}}}(L_S(a))(\Phi) } , {\Psi} \rangle}} &=
{{\langle { (S\cdot a)''(\Phi) } , {\Psi} \rangle}} = {{\langle { \Phi } , { S'(a\cdot\Psi) } \rangle}}
= {{\langle { S''(\Phi) \cdot a } , {\Psi} \rangle}},\end{aligned}$$ so that $\iota_{{\mathcal{A}}}(L_S(a))(\Phi) = S''(\Phi)\cdot a$, and hence that $\iota_{{\mathcal{A}}}(L_S(a)) = S''\cdot a
= T\cdot a = T\cdot\kappa_{{\mathcal{A}}}(a) = L_T(\kappa_{{\mathcal{A}}}(a))$. Thus we have that $L_S = (\kappa_{{\mathcal{A}}}\otimes\kappa_{{\mathcal{A}}})'
\circ R_T \circ \kappa_{{\mathcal{A}}}$, so that $L_S$ is weakly-compact. A similar calculation shows that $R_S$ is also weakly-compact, so that $S \in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$. This shows that (1) implies (2).
Conversely, if (2) holds, then $L_S$ and $R_S$ are weakly-compact. As $S$ is weakly-compact, $T({\mathcal{A}}'') = S''({\mathcal{A}}'') \subseteq
\kappa_{{\mathcal{A}}'}({\mathcal{A}}')$ and $T'(\kappa_{{\mathcal{A}}''}({\mathcal{A}}''))
= S'''(\kappa_{{\mathcal{A}}''}({\mathcal{A}}'')) = \kappa_{{\mathcal{A}}'}(S'({\mathcal{A}}''))
\subseteq \kappa_{{\mathcal{A}}'}({\mathcal{A}}')$, and $T$ is weakly-compact. Thus, to show (1), we are required to show that $L_T$ and $R_T$ are weakly-compact.
For $a,b\in{\mathcal{A}}$ and $\Phi\in{\mathcal{A}}'$, we have $${{\langle {(a\cdot S)'(\Phi)} , {b} \rangle}} = {{\langle {\Phi} , {S(ba)} \rangle}}
= {{\langle {a \cdot S'(\Phi)} , {b} \rangle}}.$$ Then, for $\Phi,\Psi\in{\mathcal{A}}''$ and $a\in{\mathcal{A}}$, we thus have $$\begin{aligned}
{{\langle {R_S'(\rho_{{\mathcal{A}}}(\Phi\otimes\Psi))} , {a} \rangle}} &=
{{\langle {(a\cdot S)''} , {\Phi\otimes\Psi} \rangle}} =
{{\langle {(a\cdot S)''(\Psi)} , {\Phi} \rangle}} = {{\langle {\Psi} , {a\cdot S'(\Phi)} \rangle}} \\
&= {{\langle {\Psi\cdot a} , {S'(\Phi)} \rangle}} =
{{\langle {\Psi{\Box}\kappa_{{\mathcal{A}}}(a)} , {S'(\Phi)} \rangle}}
= {{\langle {S'(\Phi)\cdot\Psi} , {a} \rangle}}.\end{aligned}$$ Hence we see that $R_S'(\rho_{{\mathcal{A}}}(\Phi\otimes\Psi))
= S'(\Phi)\cdot\Psi$. Let $U=R_S'\circ\rho_{{\mathcal{A}}}:
{\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'' \rightarrow {\mathcal{A}}'$, so that as $R_S$ is weakly-compact, so is $U$. Then, for $\Phi,\Psi,
\Gamma\in{\mathcal{A}}''$, we have that $$\begin{aligned}
{{\langle {U'(\Gamma)} , {\Phi\otimes\Psi} \rangle}} &=
{{\langle {\Gamma} , {S'(\Phi)\cdot\Psi} \rangle}} = {{\langle {\Psi{\Diamond}\Gamma} , {S'(\Phi)} \rangle}}
= {{\langle {S''(\Psi{\Box}\Gamma)} , {\Phi} \rangle}}
= {{\langle {(\Gamma\cdot S'')(\Psi)} , {\Phi} \rangle}},\end{aligned}$$ so that $U'(\Gamma) = \Gamma\cdot T$, that is, $U' = R_T$, so that $R_T$ is weakly-compact. Similarly, we can show that $L_T$ is weakly-compact, completing the proof.
\[Connes\_Amen\] Let ${\mathcal{A}}$ be an Arens regular Banach algebra. Then ${\mathcal{A}}''$ is Connes-amenable if and only if ${\mathcal{A}}''$ is unital and there exists $M \in ({\mathcal{A}} {{\widehat{\otimes}}}{\mathcal{A}})''$ such that:
1. $\Delta_{{\mathcal{A}}}''(M) = e_{{\mathcal{A}}''}$, the unit of ${\mathcal{A}}''$;
2. ${{\langle {M} , {a\cdot T-T\cdot a} \rangle}}=0$ for each $a\in{\mathcal{A}}$ and each $T\in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$.
By Theorem \[When\_Con\_Amen\], we wish to show that the existence of such an $M$ is equivalent to the existence of $N\in({\mathcal{A}}''{{\widehat{\otimes}}}{\mathcal{A}}'')''$ such that:
(N1)
: $\kappa_{{\mathcal{A}}'}'\Delta_{{\mathcal{A}}''}''(N) = e_{{\mathcal{A}}''}$;
(N2)
: ${{\langle {N} , {\Phi\cdot S - S\cdot\Phi} \rangle}}=0$ for each $\Phi\in{\mathcal{A}}''$ and each $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$.
We can verify that $\iota_{{\mathcal{A}}} \circ \Delta_{{\mathcal{A}}}'
= \Delta'_{{\mathcal{A}}''} \circ \kappa_{{\mathcal{A}}'}$, so that (N1) is equivalent to $\Delta_{{\mathcal{A}}}'' \iota'_{{\mathcal{A}}}(N) =
e_{{\mathcal{A}}''}$. For $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$, we know that $S=T''$ for some $T \in {\operatorname{WAP}}( {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}') )$, by Theorem \[arens\_wap\]. That is, the maps $\phi_r$ and $\phi_l$, formed using $T$ as in Proposition \[first\_wap\_prop\], are weakly-compact. Then, for $\Phi\in{\mathcal{A}}''$, $\phi_r'(\Phi), \phi_l'(\Phi) \in
{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and we can check that $$\phi_r'(\Phi)(a) = \kappa_{{\mathcal{A}}}'T''(a\cdot\Phi),
\quad \phi_l'(\Phi)(a) = T(a)\cdot\Phi
\qquad (a\in{\mathcal{A}}).$$ Then $\phi_r'(\Phi)', \phi_l'(\Phi)'\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}')$ are the maps $$\phi_r'(\Phi)'(\Psi) = \Phi\cdot T'(\Psi),
\quad \phi_l'(\Phi)'(\Psi) = T'(\Phi{\Box}\Psi)
\qquad (\Psi\in{\mathcal{A}}''),$$ where we remember that $T''({\mathcal{A}}'') \subseteq
\kappa_{{\mathcal{A}}'}({\mathcal{A}}')$. Consequently $\phi_r'(\Phi)'', \phi_l'(\Phi)''\in{\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''')$ are given by $$\phi_r'(\Phi)''(\Psi) = T''(\Psi{\Box}\Phi),
\quad\phi_l'(\Phi)''(\Psi) = T''(\Psi) \cdot \Phi
\qquad (\Psi\in{\mathcal{A}}''),$$ where ${\mathcal{A}}'''$ is an ${\mathcal{A}}''$-bimodule, as ${\mathcal{A}}''$ is Arens regular. That is, $\phi_r'(\Phi)'' = \Phi\cdot S$ and $\phi_l'(\Phi)'' = S\cdot\Phi$. Hence (N2) is equivalent to $$0 = {{\langle {N} , {\phi_r'(\Phi)'' - \phi_l'(\Phi)''} \rangle}}
= {{\langle {N} , {\iota_{{\mathcal{A}}}( \phi_r'(\Phi) - \phi_l'(\Phi) )} \rangle}}
= {{\langle {\iota_{{\mathcal{A}}}'(N)} , {\phi_r'(\Phi) - \phi_l'(\Phi)} \rangle}},$$ for each $\Phi\in{\mathcal{A}}''$ and $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. That is, (N2) is equivalent to $$\phi_r''\iota_{{\mathcal{A}}}'(N) - \phi_l''\iota_{{\mathcal{A}}}'(N) = 0
\qquad ( S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') ) ).$$
As $\phi_r$ and $\phi_l$ are weakly-compact, $\phi_r''$ and $\phi_l''$ take values in ${\kappa_{{\mathcal{A}}'}({\mathcal{A}}')}$, and so (N2) is equivalent to $$0 = {{\langle {\phi_r''\iota_{{\mathcal{A}}}'(N) - \phi_l''\iota_{{\mathcal{A}}}'(N)} , {\kappa_{{\mathcal{A}}}(a)} \rangle}} =
{{\langle {\iota_{{\mathcal{A}}}'(N)} , { \phi_r'(\kappa_{{\mathcal{A}}}(a)) -
\phi_l'(\kappa_{{\mathcal{A}}}(a)) } \rangle}},$$ for each $a\in{\mathcal{A}}$ and each $S\in\sigma WC( {\mathcal{B}}({\mathcal{A}}'',{\mathcal{A}}''') )$. However, $\phi_r'(\kappa_{{\mathcal{A}}}(a)) -
\phi_l'(\kappa_{{\mathcal{A}}}(a)) = a\cdot T - T\cdot a$, so that (N2) is equivalent to $$0 = {{\langle {\iota_{{\mathcal{A}}}'(N)} , {a\cdot T - T \cdot a} \rangle}}
\qquad (a\in{\mathcal{A}}),$$ for each $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$ such that $\phi_r$ and $\phi_l$ are weakly-compact.
Thus we have established that (N1) holds for $N$ if and only if (1) holds for $M=\iota_{{\mathcal{A}}}'(N)$, and that (N2) holds for $N$ if and only if (2) holds for $M=\iota_{{\mathcal{A}}}'(N)$, completing the proof.
We immediately see that ${\mathcal{A}}$ amenable implies that ${\mathcal{A}}''$ is Connes-amenable. Furthermore, if ${\mathcal{A}}$ is itself a dual Banach algebra, then Corollary \[unital\_dual\_sigma\] shows that if ${\mathcal{A}}''$ is Connes-amenable, then ${\mathcal{A}}$ is Connes-amenable: notice that if $e_{{\mathcal{A}}''}$ is the unit of ${\mathcal{A}}''$, then $${{\langle {\kappa_{{\mathcal{A}}_*}'(e_{{\mathcal{A}}''})a} , {\mu} \rangle}}
= {{\langle {e_{{\mathcal{A}}''} \cdot a} , {\kappa_{{\mathcal{A}}_*}(\mu)} \rangle}}
= {{\langle {\kappa_{{\mathcal{A}}}(a)} , {\kappa_{{\mathcal{A}}_*}(\mu)} \rangle}}
= {{\langle {a} , {\mu} \rangle}}
\qquad (a\in{\mathcal{A}}, \mu\in{\mathcal{A}}_*),$$ so that $\kappa_{{\mathcal{A}}_*}'(e_{{\mathcal{A}}''})$ is the unit of ${\mathcal{A}}$.
Injectivity of the predual module {#Inj_predual}
=================================
Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ and $F$ be Banach left ${\mathcal{A}}$-modules. We write ${_{{\mathcal{A}}}}{\mathcal{B}}(E,F)$ for the closed subspace of ${\mathcal{B}}(E,F)$ consisting of left ${\mathcal{A}}$-module homomorphisms, and similarly write ${\mathcal{B}}_{{\mathcal{A}}}(E,F)$ and ${_{{\mathcal{A}}}}{\mathcal{B}}_{{\mathcal{A}}}(E,F)$ for right ${\mathcal{A}}$-module and ${\mathcal{A}}$-bimodule homomorphisms, respectively. We say that $T\in{_{{\mathcal{A}}}}{\mathcal{B}}(E,F)$ is *admissible* if both the kernel and image of $T$ are closed, complemented subspaces of, respectively, $E$ and $F$. If $T$ is injective, this is equivalent to the existence of $S\in{\mathcal{B}}(F,E)$ such that $ST=I_E$.
Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ be a Banach left ${\mathcal{A}}$-module. Then $E$ is *injective* if, whenever $F$ and $G$ are Banach left ${\mathcal{A}}$-modules, $\theta\in{_{{\mathcal{A}}}}{\mathcal{B}}(F,G)$ is injective and admissible, and $\sigma\in{_{{\mathcal{A}}}}{\mathcal{B}}(F,E)$, there exists $\rho\in{_{{\mathcal{A}}}}{\mathcal{B}}(G,E)$ with $\rho\circ\theta = \sigma$.
We say that $E$ is *left-injective* when we wish to stress that we are treating $E$ as a left module. Similar definitions hold for right modules and bimodules (written *right-injective* and *bi-injective* where necessary).
Let ${\mathcal{A}}$ be a Banach algebra, let $E$ be a Banach left ${\mathcal{A}}$-module, and turn ${\mathcal{B}}({\mathcal{A}},E)$ into a left ${\mathcal{A}}$-module by setting $$(a\cdot T)(b) = T(ba) \qquad (a,b\in{\mathcal{A}}, T\in{\mathcal{B}}({\mathcal{A}},E)).$$ Then there is a canonical left ${\mathcal{A}}$-module homomorphism $\iota:E\rightarrow {\mathcal{B}}({\mathcal{A}},E)$ given by $$\iota(x)(a) = a\cdot x\qquad (a\in{\mathcal{A}},x\in E).$$ Notice that if $E$ is a closed submodule of ${\mathcal{A}}'$, then ${\mathcal{B}}({\mathcal{A}},E)$ is a closed submodule of $(A{{\widehat{\otimes}}}A)' =
{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, and $\iota$ is the restriction of $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}'\rightarrow {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ to $E$.
Similarly, we turn ${\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E)$ into a Banach ${\mathcal{A}}$-bimodule by $$(a\cdot T)(b\otimes c) = T(ba\otimes c), \
(T\cdot a)(b\otimes c) = T(b\otimes ac) \quad
( a,b,c\in{\mathcal{A}}, T\in{\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E) ).$$ We then define (with an abuse of notation) $\iota:E\rightarrow{\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E)$ by $$\iota(x)(a\otimes b) = a\cdot x \cdot b\qquad (x\in E,
a\otimes b\in{\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}}),$$ so that $\iota$ is an ${\mathcal{A}}$-bimodule homomorphism.
We can also turn ${\mathcal{B}}({\mathcal{A}},E)$ into a right ${\mathcal{A}}$-module by reversing the above (in particular, we need to take the other possible choice in Section \[con\_amen\_bidual\] leading to different module actions as compared to those in (\[eq:one\]).)
Let ${\mathcal{A}}$ be a Banach algebra, and let $E$ be a *faithful* Banach left ${\mathcal{A}}$-module (that is, for each non-zero $x\in E$ there exists $a\in{\mathcal{A}}$ with $a\cdot x\not=0$). Then $E$ is injective if and only if there exists $\phi\in{_{{\mathcal{A}}}}{\mathcal{B}}( {\mathcal{B}}({\mathcal{A}},E), E )$ such that $\phi \circ \iota = I_E$.
Similarly, if $E$ is a left and right faithful Banach ${\mathcal{A}}$-bimodule (that is, for each non-zero $x\in E$ there exists $a,b\in{\mathcal{A}}$ with $a\cdot x\not=0$ and $x\cdot b\not=0$). Then $E$ is injective if and only if there exists $\phi\in
{_{{\mathcal{A}}}}{\mathcal{B}}_{{\mathcal{A}}}( {\mathcal{B}}({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}},E), E)$ such that $\phi\circ\iota = I_E$.
The first claim is [@DP Proposition 1.7], and the second claim is an obvious generalisation.
Again, there exists a similar characterisation for right modules.
Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$. It is simple to show (see [@Runde2]) that if ${\mathcal{A}}_*$ is bi-injective, then ${\mathcal{A}}$ is Connes-amenable. Helemskii showed in [@Hel2] that for a von Neumann algebra ${\mathcal{A}}$, the converse is true. However, Runde (see [@Runde2]) and Tabaldyev (see [@Tab]) have shown that $M(G)$, the measure algebra of a locally compact group $G$, while being a dual Banach algebra with predual $C_0(G)$, has that $C_0(G)$ is a left-injective $M(G)$-module only when $G$ is finite. Of course, Runde (see [@Runde4]) has shown that $M(G)$ is Connes-amenable if and only if $G$ is amenable.
Similarly, it is simple to show (using a virtual diagonal) that if ${\mathcal{A}}$ is a Banach algebra with a bounded approximate identity, then ${\mathcal{A}}$ is amenable if and only if ${\mathcal{A}}'$ is bi-injective.
Let $E$ and $F$ be Banach left ${\mathcal{A}}$-modules, and let $\phi:E\rightarrow F$ be a left ${\mathcal{A}}$-module homomorphism which is bounded below. Then $\phi(E)$ is a closed submodule of $F$, so that $F / \phi(E)$ is a Banach left ${\mathcal{A}}$-module. Hence we have a *short exact sequence*: $$\spreaddiagramrows{4ex} \spreaddiagramcolumns{6ex}
\xymatrix{ 0 \ar[r] &
E \ar@<1ex>[r]^{\phi} &
F \ar@{->>}[r] \ar@<1ex>@{-->}[l]^{P} &
F/\phi(E) \ar[r] & 0 }.$$ If there exists a bounded linear map $P:F\rightarrow E$ such that $P\circ\phi = I_E$, then we say that the short exact sequence is *admissible*. If, further, we may choose $P$ to be a left ${\mathcal{A}}$-module homomorphism, then the short exact sequence is said to *split*. Similar definitions hold for right modules and bimodules.
Let ${\mathcal{A}}$ be a Banach algebra, let $E$ be a Banach left ${\mathcal{A}}$-module, and consider the following admissible short exact sequence: $$\spreaddiagramrows{4ex} \spreaddiagramcolumns{6ex}
\xymatrix{ 0 \ar[r] &
E \ar@<1ex>[r]^{\iota} &
{\mathcal{B}}({\mathcal{A}},E) \ar@{->>}[r] \ar@<1ex>@{-->}[l]^{P} &
{\mathcal{B}}({\mathcal{A}},E) / \iota(E) \ar[r] & 0 }.$$ Then $E$ is injective if and only if this short exact sequence splits.
See, for example, [@RundeBook Section 5.3].
Let ${\mathcal{A}}$ be a unital dual Banach algebra with predual ${\mathcal{A}}_*$, and consider the following admissible short exact sequence of ${\mathcal{A}}$-bimodules: $$\spreaddiagramcolumns{3ex}
\xymatrix{ 0 \ar[r] &
{\mathcal{A}}_* \ar@<1ex>[r]^(.35){\Delta_{{\mathcal{A}}}'} &
\sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})') \ar@{->>}[r] \ar@<1ex>@{-->}[l]^(0.65){P} &
\sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})')/\Delta_{{\mathcal{A}}}'({\mathcal{A}}_*) \ar[r] & 0 }.
\label{con_amen_ses}$$ Then ${\mathcal{A}}$ is Connes-amenable if and only if this short exact sequence splits.
Notice that $\Delta'_{{\mathcal{A}}}$ certainly maps ${\mathcal{A}}_*$ into $\sigma WC(({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})') = \sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, and that Corollary \[unital\_dual\_sigma\] shows that we can define $P:\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))\rightarrow{\mathcal{A}}_*$ by $P(T) =
T(e_{{\mathcal{A}}})$ for $T\in \sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$.
Suppose that we can choose $P$ to be an ${\mathcal{A}}$-bimodule homomorphism. Then let $M = P'(e_{{\mathcal{A}}})$, so that for $a\in{\mathcal{A}}$ and $T\in\sigma
WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, $${{\langle {a\cdot M-M\cdot a} , {T} \rangle}} = {{\langle {e_{{\mathcal{A}}}} , {P(T\cdot a-a\cdot T)} \rangle}}
= {{\langle {a-a} , {P(T)} \rangle}} = 0,$$ so that $a\cdot M - M\cdot a$. Also $\Delta''_{{\mathcal{A}}}(M) =
(P\circ\Delta'_{{\mathcal{A}}})'(e_{{\mathcal{A}}}) = e_{{\mathcal{A}}}$, so that $M$ is a $\sigma WC$-virtual diagonal, and hence ${\mathcal{A}}$ is Connes-amenable by Runde’s theorem.
Conversely, let $M$ be a $\sigma WC$-virtual diagonal and define $P:\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))\rightarrow{\mathcal{A}}'$ by $${{\langle {P(T)} , {a} \rangle}} = {{\langle {M} , {a\cdot T} \rangle}} \qquad (a\in{\mathcal{A}},
T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')).$$ Let $(a_\alpha)$ be a bounded net in ${\mathcal{A}}$ which tends to $a\in{\mathcal{A}}$ in the $\sigma({\mathcal{A}},{\mathcal{A}}_*)$-topology. By definition, $a_\alpha\cdot T\rightarrow a\cdot T$ weakly, for each $T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))$, so that ${{\langle {P(T)} , {a_\alpha} \rangle}} \rightarrow {{\langle {P(T)} , {a} \rangle}}$. This implies that $P$ maps into ${\mathcal{A}}_*$, as required. Then, for $\mu\in{\mathcal{A}}_*$, $${{\langle {a} , {P\Delta_{{\mathcal{A}}}'(\mu)} \rangle}} = {{\langle {M} , {a\cdot\Delta_{{\mathcal{A}}}'(\mu)} \rangle}}
= {{\langle {M} , {\Delta_{{\mathcal{A}}}'(a\cdot\mu)} \rangle}} = {{\langle {e_{{\mathcal{A}}}} , {a\cdot \mu} \rangle}}
= {{\langle {a} , {\mu} \rangle}} \qquad (a\in{\mathcal{A}}),$$ so that $P\Delta_{{\mathcal{A}}}'=I_{{\mathcal{A}}_*}$. Finally, we note that $$\begin{aligned}
{{\langle {P(a\cdot T\cdot b)} , {c} \rangle}} &= {{\langle {M} , {ca\cdot T\cdot b} \rangle}}
= {{\langle {b\cdot M} , {ca\cdot T} \rangle}}= {{\langle {M\cdot b} , {ca\cdot T} \rangle}} \\
&= {{\langle {P(T)} , {bca} \rangle}} = {{\langle {a\cdot P(T) \cdot b} , {c} \rangle}}
\qquad (a,b,c\in{\mathcal{A}}, T\in\sigma WC({\mathcal{B}}({\mathcal{A}},{\mathcal{A}}'))),\end{aligned}$$ so that $P$ is an ${\mathcal{A}}$-bimodule homomorphism, as required.
Let ${\mathcal{A}}$ be an Arens regular Banach algebra. By reversing the argument Theorem \[arens\_wap\], we can show that $\Delta'_{{\mathcal{A}}}:{\mathcal{A}}'\rightarrow{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ actually maps into ${\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$. Furthermore, if ${\mathcal{A}}''$ is unital, then we may define $P:{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))
\rightarrow {\mathcal{A}}'$ by $${{\langle {P(T)} , {a} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {P(a)} \rangle}}
\qquad (a\in{\mathcal{A}}, T\in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))).$$ Then we have that $${{\langle {P\Delta'_{{\mathcal{A}}}(\mu)} , {a} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {a\cdot\mu} \rangle}}
= {{\langle {\mu} , {a} \rangle}} \qquad (a\in{\mathcal{A}}, \mu\in{\mathcal{A}}').$$
Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is unital, and consider the following admissible short exact sequence of ${\mathcal{A}}$-bimodules: $$\spreaddiagramcolumns{3ex}
\xymatrix{ 0 \ar[r] &
{\mathcal{A}}' \ar@<1ex>[r]^(0.3){\Delta_{{\mathcal{A}}}'} &
{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) \ar@{->>}[r] \ar@<1ex>[l]^(0.7){P} &
{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) / \Delta_{{\mathcal{A}}}'({\mathcal{A}}') \ar[r] & 0 }.
\label{eq:two}$$ Then ${\mathcal{A}}''$ is Connes-amenable if and only if this short exact sequence splits.
This follows in the same manner as the above proof, using Theorem \[Connes\_Amen\].
Beurling algebras
=================
Let $S$ be a discrete semigroup (we can extend the following definitions to locally compact semigroups, but for the questions we are interested in, the results for non-discrete groups are trivial). A *weight* on $S$ is a function $\omega: S
\rightarrow \mathbb R^{>0}$ such that $$\omega(st)\leq \omega(s)\omega(t) \qquad (s,t\in S).$$ Furthermore, if $S$ is unital with unit $u_S$, then we also insist that $\omega(u_S)=1$. This last condition is simply a normalisation condition, as we can always set $\hat\omega(s)
= \sup\{ \omega(st) \omega(t)^{-1} : t\in S \}$ for each $s\in S$. For $s,t\in S$, we have that $\omega(st) \leq \hat\omega(s)\omega(t)$, so that $$\hat\omega(st) = \sup\{ \omega(str)\omega(r)^{-1} : r\in S \}
\leq \sup \{ \hat\omega(s)\omega(tr)\omega(r)^{-1} : r\in S \}
= \hat\omega(s) \hat\omega(t).$$ Clearly $\hat\omega(u_S)=1$ and $\hat\omega(s)\leq\omega(s)$ for each $s\in S$, while $\hat\omega(s) \geq \omega(s)\omega(u_S)^{-1}$, so that $\hat\omega$ is equivalent to $\omega$.
We form the Banach space $$l^1(S,\omega) = \Big\{ (a_g)_{g\in S} \subseteq\mathbb C :
\|(a_g)\| := \sum_{g\in S} |a_g| \omega(g) < \infty \Big\}.$$ Then $l^1(S,\omega)$, with the convolution product, is a Banach algebra, called a *Beurling algebra*. See [@CY] and [@DL] for further information on Beurling algebras and, in particular, their second duals.
It will be more convenient for us to think of $l^1(S,\omega)$ as the Banach space $l^1(S)$ together with a weighted algebra product. Indeed, for $g\in S$, let $\delta_g\in l^1(S)$ be the standard unit vector basis element which is thought of as a point-mass at $g$. Then each $x\in l^1(S)$ can be written uniquely as $x = \sum_{g\in S} x_g \delta_g$ for some family $(x_g) \subseteq \mathbb C$ such that $\|x\| = \sum_{g\in S}
|x_g| <\infty$. We then define $$\delta_g \star_\omega \delta_h = \delta_g \star \delta_h
= \delta_{gh} \Omega(g,h) \qquad (g,h\in S),$$ where $\Omega(g,h) = \omega(gh)\omega(g)^{-1}\omega(h)^{-1}$, and extend $\star$ to $l^1(S)$ by linearity and continuity.
For example, if $\omega$ and $\hat\omega$ are equivalent weights on $S$, the define $\psi: l^1(S,\omega) \rightarrow l^1(S,\hat\omega)$ by $\psi(\delta_s) = \hat\omega(s) \omega(s)^{-1} \delta_s$. As $\omega$ and $\hat\omega$ are equivalent, $\psi$ is an isomorphism of Banach spaces. Then $\psi(\delta_s \star \delta_t) = \omega(st) \omega(s)^{-1} \omega(t)^{-1}
\hat\omega(st) \omega(st)^{-1} \delta_{st} =
\psi(\delta_s) \star \psi(\delta_t)$, so that $\psi$ is a homomorphism.
For a set $I$, we define the space $c_0(I)$ as $$c_0(I) = \Big\{ (a_i)_{i\in I} : \forall\, \epsilon>0,
|\{ i\in I : |a_i|\geq\epsilon \}|<\infty \Big\},$$ where $| \cdot |$ is the cardinality of a set. We equip $c_0(I)$ with the supremum norm; then $c_0(I)' = l^1(I)$. For $i\in I$, we let $e_i\in c_0(I)$ be the point mass at $i$, that is, ${{\langle {\delta_j} , {e_i} \rangle}} = \delta_{i,j}$, the Kronecker delta, for $\delta_j\in l^1(I)$. Then $c_0(I)$ is the closed linear span of $\{ e_i : i\in I \}$. We let $l^\infty(I)$ be the Banach space of all bounded families $(a_i)_{i\in I}$, with the supremum norm. Then $l^1(I)'=l^\infty(I)$, we can treat $c_0(I)$ as a subspace of $l^\infty(I)$, and the map $\kappa_{c_0(I)}: c_0(I)\rightarrow
l^\infty(I)$ is just the inclusion map.
For a semigroup $S$ and $s\in S$, we define maps $L_s,R_s:S
\rightarrow S$ by $$L_s(t) = st, \quad R_s(t) = ts \qquad (t\in S).$$ If, for each $s\in S$, $L_s$ and $R_s$ are finite-to-one maps, then we say that $S$ is *weakly cancellative*. When $L_s$ and $R_s$ are injective for each $s\in S$, we say that $S$ is *cancellative*. When $S$ is abelian and cancellative, a construction going back to Grothendieck shows that $S$ is a sub-semigroup of some abelian group. However, this can fail to hold for non-abelian semigroups.
Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$. Then $c_0(S) \subseteq l^\infty(S) = {\mathcal{A}}'$ is a sub-${\mathcal{A}}$-module of ${\mathcal{A}}'$, so that $l^1(S,\omega)$ is a dual Banach algebra with predual $c_0(S)$.
For $g,h\in S$ and $a=(a_s)_{s\in S} \in l^1(S,\omega)$, we have $${{\langle {e_g \cdot \delta_h} , {a} \rangle}} = {{\langle {e_g} , {\delta_h \star a} \rangle}}
= {{\langle {e_g} , {\sum_{s\in S} a_s \delta_{hs} \Omega(h,s)} \rangle}}
= \sum_{\{ s\in S : hs = g\}} a_s \Omega(h,s).$$ As $S$ is weakly cancellative, there exists at most finitely many $s\in S$ such that $hs=g$, so that $e_g \cdot \delta_h$ is a member of $c_0(S)$. Thus we see that $c_0(S)$ is a right sub-${\mathcal{A}}$-module of ${\mathcal{A}}'$. The argument on the left follows in an analogous manner.
Notice that the above result will hold for some semigroups $S$ which are not weakly cancellative, provided that the weight behaves in a certain way. However, it would appear that the later results do not easily generalise to the non-weakly cancellative case.
Following [@DL Definition 2.2], we have the following definition.
Let $I$ and $J$ be non-empty infinite sets, and let $f:I\times J\rightarrow \mathbb C$ be a function. Then *$f$ clusters on $I\times J$* if $$\lim_{n\rightarrow\infty} \lim_{m\rightarrow\infty} f(x_m,y_n)
= \lim_{m\rightarrow\infty} \lim_{n\rightarrow\infty} f(x_m,y_n),$$ whenever $(x_m)\subseteq I$ and $(y_n)\subseteq J$ are sequences of distinct elements, and both iterated limits exist.
Furthermore, *$f$ $0$-clusters on $I\times J$* if $f$ clusters on $I\times J$, and the iterated limits are always $0$, when they exist. [$\square$]{}
From now on we shall exclude the trivial case when our (semi-)group is finite.
\[B\_AR\] Let $S$ be a discrete, weakly cancellative semigroup, and let $\omega$ be a weight on $S$. Then the following are equivalent:
1. $l^1(S,\omega)$ is Arens regular;
2. for sequences of distinct elements $(g_j)$ and $(h_k)$ in $S$, we have $$\lim_{j\rightarrow\infty} \lim_{k\rightarrow\infty}
\Omega(g_j,h_k) = 0,$$ whenever the iterated limit exists;
3. $\Omega$ $0$-clusters on $S\times S$.
That (1) and (2) are equivalent for cancellative semigroups is [@CY Theorem 1]. Close examination of the proof shows that this holds for weakly cancellative semigroups as well. That (1) and (3) are equivalent follows by generalising the proof of [@DL Theorem 7.11], which is essentially an application of Grothendieck’s criterion for an operator to be weakly-compact. Alternatively, it follows easily that (2) and (3) are equivalent by considering the *opposite semigroup* to $S$ where we reverse the product.
In [@CY] it is also shown that if $G$ is a discrete, uncountable group, then $l^1(G,\omega)$ is not Arens regular for any weight $\omega$. Furthermore, by [@CY Theorem 2], if $G$ is a non-discrete locally compact group, then $L^1(G,\omega)$ is never Arens regular.
We shall consider both the Connes-amenability of $l^1(S,\omega)''$ and $l^1(S,\omega)$ (with respect to the canonical predual $c_0(S)$) as, with reference to Corollary \[unital\_dual\_sigma\] and Theorem \[arens\_wap\], the calculations should be similar.
\[weak\_comp\_infty\] Let $I$ be a non-empty set, and let $X\subseteq l^\infty(I)$ be a subset. Then the following are equivalent:
1. $X$ is relatively weakly-compact;
2. $X$ is relatively sequentially weakly-compact;
3. the absolutely convex hull of $X$ is relatively weakly-compact;
4. if we define $f:I \times X \rightarrow \mathbb C$ by $f(i,x) = {{\langle {x} , {\delta_i} \rangle}}$ for $i\in I$ and $x\in X$, then $f$ clusters on $I\times X$;
That (1) and (2) are equivalent is the Eberlien-Smulian theorem; that (1) and (3) are equivalent is the Krein-Smulian theorem. That (1) and (4) are equivalent is a result of Grothendieck, detailed in, for example, [@DL Theorem 2.3].
It is standard that for non-empty sets $I$ and $J$, we have that $l^1(I) {{\widehat{\otimes}}}l^1(J) = l^1(I \times J)$, where, for $i\in I$ and $j\in J$, $\delta_i \otimes \delta_j\in l^1(I){{\widehat{\otimes}}}l^1(J)$ is identified with $\delta_{(i,j)}\in l^1(I\times J)$. Thus we have $( l^1(I) {{\widehat{\otimes}}}l^1(J) )' = {\mathcal{B}}(l^1(I), l^\infty(J))
= l^1(I\times J)' = l^\infty(I\times J)$, where $T\in
{\mathcal{B}}(l^1(I), l^\infty(J))$ is identified with $(T_{(i,j)})
\in l^\infty(I\times J)$, where $T_{(i,j)} =
{{\langle {T(\delta_i)} , {\delta_j} \rangle}}$.
**Is this paragraph used?** Let $S$ be a countable, discrete, unital semigroup, and let $\omega$ be a weight on $S$. Then $l^1(S\times S)$ is a Banach $l^1(S,\omega)$-bimodule, with module actions $$\delta_k \cdot \delta_{(g,h)} = \delta_{(kg,h)} \Omega(k,g)
\quad , \quad
\delta_{(g,h)} \cdot \delta_k = \delta_{(g,hk)} \Omega(h,k)
\qquad (g,h,k\in S).$$
For a non-empty set $I$, the unit ball of $l^1(I)$ is the closure of the absolutely-convex hull of the set $\{ \delta_i : i\in I\}$, so that for a Banach space $E$, by the Krein-Smulian theorem, a map $T:l^1(I)\rightarrow E$ is weakly-compact if and only if the set $\{ T(\delta_i) : i\in I \}$ is relatively weakly-compact in $E$.
\[wap\_c\_zero\] Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$. Let $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ be such that $T({\mathcal{A}}) \subseteq \kappa_{c_0(S)}(c_0(S))$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq \kappa_{c_0(S)}(c_0(S))$. Then $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$, and $T\in{\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}'))$ if and only if, for each sequence $(k_n)$ of distinct elements of $S$, and each sequence $(g_m,h_m)$ of distinct elements of $S\times S$ such that the repeated limits $$\begin{gathered}
\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}}, \
\lim_n \lim_m \Omega(k_n,g_m) \label{cond_one} \\
\lim_n \lim_m {{\langle {T(\delta_{h_mk_n})} , {\delta_{g_m}} \rangle}}, \
\lim_n \lim_m \Omega(h_m,k_n) \label{cond_two}\end{gathered}$$ all exist, we have that at least one repeated limit in each row is zero.
That $T$ is weakly-compact follows from Gantmacher’s Theorem (compare with Corollary \[unital\_dual\_sigma\]). To show that $T\in{\operatorname{WAP}}$, by Lemma \[wap\_to\_maps\], we are required to show that the maps $\phi_r$ and $\phi_l$ are weakly-compact. We shall show that $\phi_l$ is weakly-compact if and only if one of the repeated limits in the first line (\[cond\_one\]) is zero; the proof that $\phi_r$ is related to (\[cond\_two\]) follows in a similar way. We have that $$\phi_l(\delta_{(g,h)}) = \phi_l(\delta_g \otimes \delta_h) =
\delta_g \cdot T(\delta_h) \qquad (g,h\in S).$$ By Proposition \[weak\_comp\_infty\], $\phi_l$ is weakly-compact if and only if the function $$S \times (S \times S) \rightarrow \mathbb C ; \
(k,(g,h)) \mapsto {{\langle {\delta_g\cdot T(\delta_h)} , {\delta_k} \rangle}}
= {{\langle {T(\delta_h)} , {\delta_{kg}} \rangle}} \Omega(k,g) \qquad (g,h,k\in S)$$ clusters on $S \times (S \times S)$. As $T$ is weakly-compact, the function $$S \times S\rightarrow\mathbb C; \quad
(g,h) \mapsto {{\langle {T(\delta_g)} , {\delta_h} \rangle}} \qquad (g,h\in S)$$ does cluster on $S\times S$.
Let $(k_n)$ be a sequence of distinct elements of $S$, and let $(g_m,h_m)$ be a sequence of distinct elements of $S\times S$ such that the iterated limits $$\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m)
, \quad
\lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m)
\label{eq:five}$$ exist. We now investigate when these iterated limits are equal.
Suppose firstly that, by moving to a subsequence if necessary, we have that $g_m = g$ for all $m$. Further, by moving to a subsequence if necessary, we may suppose that $\lim_n \Omega(k_n,g) = \alpha$, say, and that $(k_ng)$ is a sequence of distinct elements (as $S$ is weakly cancellative). Then $$\begin{aligned}
\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} & \Omega(k_n,g_m)
= \lim_n \Omega(k_n,g) \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}} \\
&= \alpha \lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}}
= \alpha \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng}} \rangle}} \\
&= \lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} \Omega(k_n,g_m),\end{aligned}$$ where we can swap the order of taking limits, as $T$ is weakly-compact.
Alternatively, if we cannot move to a subsequence such that $(g_m)$ is constant, then we may move to subsequence such that $(g_m)$ is a sequence of distinct elements, and such that the iterated limits $$\begin{gathered}
\lim_m \lim_n \Omega(k_n,g_m),\quad
\lim_n \lim_m \Omega(k_n,g_m), \\
\lim_m \lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}},\quad
\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}}\end{gathered}$$ all exists. As $T({\mathcal{A}})\subseteq\kappa_{c_0(S)}(c_0(S))$, we have that $$\{ g\in S : |{{\langle {T(\delta_h)} , {\delta_g} \rangle}}|\geq\epsilon \}
\text{ is finite } \qquad (\epsilon>0, h\in S).$$ Consequently, and using the fact that $S$ is weakly cancellative, we see that $$\lim_n {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}} = 0$$ for each $m$. Hence the iterated limits in (\[eq:five\]) are equal if and only if we have that at least one repeated limit in (\[cond\_one\]) is zero.
\[auto\_semi\_wap\] Let $S$ be a discrete, unital, weakly cancellative semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular. Then ${\operatorname{WAP}}({\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')) = {\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$.
Let $T\in{\mathcal{W}}({\mathcal{A}},{\mathcal{A}}')$. We can follow the above proof through until the point at which we use the fact that $T({\mathcal{A}})\subseteq \kappa_{c_0(S)}(c_0(S))$. However, as $l^1(S,\omega)$ is Arens regular, by Theorem \[B\_AR\], we have that $$\lim_m \lim_n \Omega(k_n,g_m) =
\lim_n \lim_m \Omega(k_n,g_m) = 0,$$ so that the iterated limits in (\[eq:five\]) must be $0$, implying that $\phi_l$ is weakly-compact. In a similar manner, $\phi_r$ is weakly-compact.
\[when\_loneg\_cam\] Let $S$ be a discrete weakly cancellative semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular and ${\mathcal{A}}''$ is unital with unit $e_{{\mathcal{A}}''}$. Then ${\mathcal{A}}''$ is Connes-amenable if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that:
1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = {{\langle {e_{{\mathcal{A}}''}} , {f} \rangle}}$ for each bounded family $(f_g)_{g\in S}$;
2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_
{(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$ which clusters on $S\times S$.
We use Theorem \[Connes\_Amen\] and Proposition \[auto\_semi\_wap\]. For $f=(f_g)_{g\in S} \in l^\infty(S)$, we have $${{\langle { \Delta'_{{\mathcal{A}}}(f) } , {\delta_g\otimes \delta_h} \rangle}}
= {{\langle {f} , {\delta_{gh}} \rangle}}\Omega(g,h) \qquad (g,h\in S),$$ so that $\Delta'_{{\mathcal{A}}}(f) = ( {{\langle {f} , {\delta_{gh}} \rangle}}\Omega(g,h) )_{(g,h)\in S\times S} \in l^\infty(S\times S)$. As $f\in l^\infty(S)$ was arbitrary, we have condition (1).
For $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$, we treat $T$ as being a member of $l^\infty(S\times S)$. Then $T$ is weakly-compact if and only if the family $({{\langle {T(\delta_g)} , {\delta_h} \rangle}})_{(g,h)\in S\times S}$ clusters on $S\times S$. For $k\in S$, we have $${{\langle { \delta_k\cdot T - T\cdot \delta_k } , {\delta_g\otimes \delta_h} \rangle}}
= {{\langle {T(\delta_{hk})} , {\delta_g} \rangle}} \Omega(h,k)
- {{\langle {T(\delta_h)} , {\delta_{kg}} \rangle}} \Omega(k,g).$$ Thus we have condition (2).
Notice that if $S$ is unital with unit $u_S$, then the unit of ${\mathcal{A}}$ (and hence ${\mathcal{A}}''$) is $\delta_{u_S}$. In this case, condition (1) reduces to ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}}
= f_{u_S}$.
\[when\_loneg\_am\] Let $S$ be a discrete unital semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. Then ${\mathcal{A}}$ is amenable if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that:
1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} = f_{u_S}$, where $u_S\in S$ is the unit of $S$, for each bounded family $(f_g)_{g\in S}$;
2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_
{(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$.
This follows from Theorem \[when\_amen\] in the same way that the above follows from Theorem \[Connes\_Amen\].
Notice that condition (2) of Theorem \[when\_loneg\_am\] is strictly stronger than condition (2) of Theorem \[when\_loneg\_cam\].
\[C\_amen\_predual\] Let $S$ be a discrete, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}} = l^1(S,\omega)$ be unital with unit $e_{{\mathcal{A}}}$. Then ${\mathcal{A}}$ is Connes-amenable, with respect to the predual $c_0(S)$, if and only if there exists $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})'' = l^\infty(S\times S)'$ such that:
1. ${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in S\times S}} \rangle}} =
{{\langle {e_{{\mathcal{A}}}} , {f} \rangle}}$ for each family $(f_g)_{g\in S} \in c_0(S)$;
2. ${{\langle {M} , {( f(hk,g)\Omega(h,k) - f(h,kg)\Omega(k,g) )_
{(g,h)\in S\times S}} \rangle}} = 0$ for each $k\in S$, and each bounded function $f:S\times S\rightarrow\mathbb C$ which satisfies the conclusions of Proposition \[wap\_c\_zero\].
We now use Theorem \[When\_Con\_Amen\]. By $f$ satisfying the conclusions of Proposition \[wap\_c\_zero\], we identify $f:S\times S\rightarrow\mathbb C$ with $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}')$ by ${{\langle {T(\delta_g)} , {\delta_h} \rangle}} = f(g,h)$, for $g,h\in S$.
We shall now establish when $l^1(S,\omega)$ and $l^1(S,\omega)''$ are Connes-amenable. For a discrete group $G$, a weight $\omega$ on $G$ and $h\in G$, define $J_h\in{\mathcal{B}}(l^\infty(G))$ by $$J_h(f) = \big( f_{hg} \Omega(h,g)\omega(h) \Omega(g^{-1},h^{-1})
\omega(h^{-1}) \big)_{g\in G} \qquad (f=(f_g)_{g\in G}\in l^\infty(G)).$$ Notice then that, for $f\in l^\infty(G)$, we have $$\| J_h(f) \| = \sup_g |f_{hg}| \omega(hg)\omega(g)^{-1}
\omega(g^{-1}h^{-1})\omega(g^{-1})^{-1}
\leq \|f\| \omega(h) \omega(h^{-1}),$$ so that $J_h$ is bounded.
\[group\_omega\_amen\] Let $G$ be a discrete group, and let $\omega$ be a weight on $G$. We say that $G$ is *$\omega$-amenable* if there exists $N\in l^\infty(G)'$ such that:
1. ${{\langle {N} , {(\Omega(g,g^{-1}))_{g\in G}} \rangle}}=1$, where $\Omega$ is defined by $\omega$, and hence $(\Omega(g,g^{-1}))_{g\in G}$ is a bounded family forming an element of $l^\infty(G)$;
2. $J_h'(N)=N$ for each $h\in G$.
[$\square$]{}
Notice that if $\omega$ is identically $1$, then this condition reduces to the usual notion of a group being amenable (we usually require that $N$ is a *mean*, in that $N$ is a positive functional on $l^\infty(G)$, but by forming real and imaginary parts, and then positive and negative parts, we can easily generate a non-zero scalar multiple of a mean from a functional $N$ satisfying the definition above).
Let $G$ be a discrete group, let $\omega$ be a weight on $G$, and let ${\mathcal{A}}=l^1(G,\omega)$. Then the following are equivalent:
1. ${\mathcal{A}}$ is Connes-amenable, with respect to the predual $c_0(G)$;
2. ${\mathcal{A}}$ is amenable;
3. $G$ is $\omega$-amenable.
Furthermore, if ${\mathcal{A}}$ is Arens regular, then these conditions are equivalent to ${\mathcal{A}}''$ being Connes-amenable.
It is clear that (2) implies (1). When ${\mathcal{A}}$ is Arens regular, (2) implies that ${\mathcal{A}}''$ is Connes-amenable, and ${\mathcal{A}}''$ Connes-amenable implies (1). We shall thus show that (1) implies (3), and that (3) implies (2).
If (1) holds, then let $M\in l^\infty(G\times G)'$ be given as in Theorem \[C\_amen\_predual\]. Define $\phi:l^\infty(G)
\rightarrow l^\infty(G\times G)$ by $${{\langle {\phi(f)} , {\delta_{(g,h)}} \rangle}} = \begin{cases}
f_g & : g = h^{-1}, \\ 0 &: g\not=h^{-1}, \end{cases}
\quad\qquad ( f=(f_g)_{g\in G} \in l^\infty(G) ).$$ Let $N = \phi'(M) \in l^\infty(G)'$. Then we have $$\phi( (\Omega(g,g^{-1}))_{g\in G} ) =
( \delta_{h,g^{-1}} \Omega(g,h) )_{(g,h)\in G\times G}
= ( \delta_{gh,e_G} \Omega(g,h) )_{(g,h)\in G\times G},$$ where $\delta$ is the Kronecker delta, so that $${{\langle {N} , {(\Omega(g,g^{-1}))_{g\in G}} \rangle}} = \delta_{e_G,e_G} = 1,$$ by condition (1) on $M$ from Theorem \[C\_amen\_predual\]; clearly $(\delta_{e_G,g})_{g\in G} \in c_0(G)$.
Fix $k\in G$ and $f\in l^\infty(G)$. Define $F:G\times G\rightarrow
\mathbb C$ by $$F(h,g) = \delta_{gh,k} f_g \omega(k) \omega(hk^{-1})
\omega(h)^{-1}. \qquad (g,h\in G).$$ Then we have $|F(h,g)| \leq |f_g| |\omega(k)| |\omega(hk^{-1})|
|\omega(h)|^{-1} \leq \|f\|_\infty |\omega(k)| |\omega(k^{-1})|$, so that $F$ is bounded. Let $T:{\mathcal{A}}\rightarrow{\mathcal{A}}'$ be the operator associated with $F$. For $g,h\in G$, we have that $F(h,g)\not=0$ only when $gh=k$, so that $T({\mathcal{A}})\subseteq c_0(S)$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}})) \subseteq c_0(S)$. Furthermore, if $(k_n)$ is a sequence of distinct elements in $G$, and $(g_m,h_m)$ is a sequence of distinct elements in $G\times G$, then $\lim_n \lim_m F(h_m,k_ng_m) = 0$. This follows, as for $n_0$ fixed, $k_{n_0}g_mh_m = k$ only if $g_mh_m = k_{n_0}^{-1} k$, so if this holds for all sufficiently large $m$, we have that $k_ng_mh_m\not=k$ for sufficiently large $m$ and $n\not=n_0$. Similarly, $\lim_n \lim_m F(h_mk_n,g_m) = 0$, so that $F$ satisfies the conditions of Proposition \[wap\_c\_zero\].
Notice that $${{\langle { \phi(J_k(f)) } , { \delta_{(g,h)} } \rangle}} =
\delta_{gh,e_G} {{\langle { J_k(f) } , { \delta_g } \rangle}}
= \delta_{gh,e_G} f_{kg} \omega(kg)\omega(g)^{-1}
\omega(g^{-1}k^{-1})\omega(g^{-1})^{-1}.$$ Thus we have $$\begin{aligned}
F(hk,g)&\Omega(h,k) - F(h,kg)\Omega(k,g) \\
&= \delta_{ghk,k} f_g \omega(k) \omega(hkk^{-1})
\omega(hk)^{-1} \Omega(h,k) -
\delta_{kgh,k} f_{kg} \omega(k) \omega(hk^{-1})
\omega(h)^{-1} \Omega(k,g) \\
&= \delta_{gh,e_G} f_g - \delta_{gh,e_G} f_{kg}
\omega(hk^{-1}) \omega(h)^{-1} \omega(kg) \omega(g)^{-1} \\
&= {{\langle {\phi(f) - \phi(J_k(f)) } , { \delta_{(g,h)} } \rangle}}.\end{aligned}$$ So, by condition (2) from Theorem \[C\_amen\_predual\], we have that $${{\langle {N} , { f - J_k(f) } \rangle}} = 0,$$ which, as $f$ was arbitrary, shows that $N = J_k'(N)$, as required.
Now suppose that $G$ is $\omega$-amenable. We shall show that ${\mathcal{A}}$ is amenable, which completes the proof. Define $\psi : l^\infty(G\times G)\rightarrow l^\infty(G)$ by $${{\langle { \psi(F) } , { \delta_g } \rangle}} = F(g,g^{-1})
\qquad ( F\in l^\infty(G\times G), g\in G).$$ Let $N\in l^\infty(G)'$ be as in Definition \[group\_omega\_amen\], and let $M =\psi'(N)$. Then let $(f_g)_{g\in G}$ be a bounded family in $\mathbb C$, so that $${{\langle {M} , {(f_{gh}\Omega(g,h))_{(g,h)\in G\times G}} \rangle}} =
{{\langle {N} , {(f_{e_G}\Omega(g,g^{-1}))_{g\in G}} \rangle}} = f_{e_G},$$ verifying condition (1) of Theorem \[when\_loneg\_am\] for $M$.
Let $f:G\times G\rightarrow\mathbb C$ be a bounded function, and let $k\in G$. Then $$\begin{aligned}
\psi\big( (f(hk,g) & \Omega(h,k) - f(h,kg)\Omega(k,g))_{(g,h)\in G\times G} \big) \\
&= \big( f(g^{-1}k,g)\Omega(g^{-1},k) - f(g^{-1},kg)\Omega(k,g) \big)_{g\in G}.\end{aligned}$$ Define $F:G\times G\rightarrow\mathbb C$ by $$F(g,h) = f(hk,g) \Omega(h,k) \qquad (g,h\in G),$$ so that $F$ is bounded. For $g\in G$, we have that $$\begin{aligned}
&{{\langle { \psi(F)-J_k(\psi(F)) } , { \delta_g } \rangle}} \\ &=
f(g^{-1}k,g) \Omega(g^{-1},k)
- f((kg)^{-1}k,kg) \Omega((kg)^{-1},k) \omega(kg) \omega(g)^{-1}
\omega(g^{-1}k^{-1}) \omega(g^{-1})^{-1} \\
&= f(g^{-1}k,g) \Omega(g^{-1},k)
- f(g^{-1},kg) \omega(k)^{-1} \omega(kg) \omega(g)^{-1} \\
&= f(g^{-1}k,g) \Omega(g^{-1},k)
- f(g^{-1},kg) \Omega(k,g).\end{aligned}$$ Consequently, using condition (2) of Definition \[group\_omega\_amen\], we have established condition (2) of Theorem \[when\_loneg\_am\] for $M$. This shows that $l^1(G,\omega)$ is amenable.
If $S$ is a semigroup which is not cancellative, then it is possible for $l^1(S)$ to be unital while $S$ is not. For example, let $S$ be $(\mathbb N,\max)$ (where $\mathbb N=\{1,2,3,\ldots\}$ say) with adjoined idempotents $u$ and $v$ such that $uv=vu=1$ and $un = nu = vn = nv = n$ for $n\in\mathbb N$. Then $S$ is a weakly cancellative, commutative semigroup without a unit, but $e = \delta_u+\delta_v-\delta_1$ is easily seen to be a unit for $l^1(S)$. Indeed, $S$ is seen to be a finite semilattice of groups, so by the result of [@Gron2], $l^1(S)$ is amenable.
In [@Gron1 Theorem 2.3] it is shown that if $l^1(S,\omega)$ is amenable for a cancellative, unital semigroup $S$ and some weight $\omega$, then $S$ is actually a group. We shall now show that this holds for Connes-amenability as well.
For a cancellative, unital semigroup $S$, with unit $u_S$, if $g\in S$ is invertible, then $g$ has a unique inverse, denoted by $g^{-1}$. Furthermore, if $g$ has a left inverse, say $hg=u_S$, then $ghg =
g = u_Sg$ so that $gh=u_S$; similarly, if $gh=u_S$ then $hg=u_S$.
Let $S$ be a weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. Suppose that ${\mathcal{A}}$ is Connes-amenable with respect to the predual $c_0(S)$. If $S$ is cancellative or unital, then $S$ is a group.
As ${\mathcal{A}}$ is Connes-amenable, let $M\in ({\mathcal{A}}{{\widehat{\otimes}}}{\mathcal{A}})''$ be as in Theorem \[C\_amen\_predual\]. Then ${\mathcal{A}}$ is unital, with unit $e_{{\mathcal{A}}} = (a_s)_{s\in S} \in l^1(S,\omega)$ say. For now, we shall not assume that $e_{{\mathcal{A}}}$ has norm one, as the standard renorming to ensure this will not (a priori) necessarily yield an $l^1(S,\hat\omega)$ algebra for some weight $\hat\omega$. Suppose that $S$ is cancellative. Fix $h\in S$, so that $$\sum_{s\in S} a_s \delta_{sh} \Omega(s,h)
= e_{{\mathcal{A}}}\star \delta_h =
\delta_h = \delta_h \star e_{{\mathcal{A}}} =
\sum_{s\in S} a_s \delta_{hs} \Omega(h,s).$$ In particular, for each $h\in S$ there is a unique $u_h\in S$ such that $h u_h = h$ (so that $h u_h h = h^2$ implying that $u_h h=h$), and we have that $a_{u_h} \omega(u_h)^{-1} =1$. We also see that $a_s=0$ for each $s\in S$ such that $sh\not=h$, that is, $s\not=u_h$. However, $h$ was arbitrary, so that $S$ is unital with unit $u_S$, and $e_{{\mathcal{A}}} =
\omega(u_S) \delta_{u_S}$, where we can now assume that $\omega(u_S)=1$ by a renorming.
Now suppose that $S$ is a unital, weakly cancellative semigroup, so that the unit of ${\mathcal{A}}$ is $\delta_{u_S}$. Suppose that $s\in S$ has no right inverse. Define $F:S\times S\rightarrow\mathbb C$ by $$F(h,sg)=0, \quad F(hs,g) = \begin{cases}
\Omega(g,hs) &: gh=u_S, \\ 0 &: \text{otherwise.} \end{cases}
\qquad (g,h\in S).$$ To show that this is well-defined, suppose that for $g,h,j,k\in S$, we have that $h=js$, $sg=k$ and $kj=u_S$. Then $s(gj) = kj = u_S$, so that $s$ has a right inverse, a contradiction. Then $F$ is bounded, so let $T:{\mathcal{A}}\rightarrow
{\mathcal{A}}'$ be the operator associated with $F$. Then $F(a,b)\not=0$ only when $ba=s$, so as $S$ is weakly cancellative, we see that $T({\mathcal{A}})\subseteq c_0(S)$ and $T'(\kappa_{{\mathcal{A}}}({\mathcal{A}}))
\subseteq c_0(S)$.
Suppose that for sequences of distinct elements $(k_n)\subseteq S$ and $(g_m,h_m)\subseteq S\times S$, we have that $$\lim_n \lim_m {{\langle {T(\delta_{h_m})} , {\delta_{k_ng_m}} \rangle}}
= \lim_n \lim_m F(h_m,k_ng_m) \not=0.$$ Then, for some $N>0$ and $\epsilon>0$, for each $n\geq N$, $\lim_m F(h_m,k_ng_m)\geq \epsilon$. Hence, for $n\geq N$, there exists $M_n>0$ such that if $m\geq M_n$, then $k_ng_mh_m = s$ (as otherwise $F(h_m,k_ng_m)=0$). This, however, contradicts $S$ being weakly cancellative. Similarly, if $\lim_n \lim_m {{\langle {T(\delta_{h_mk_n})} , {\delta_{g_m}} \rangle}}\not=0$, then we need $g_mh_mk_n=s$ for all $n,m$ sufficiently large, which is a contradiction. Thus $T$ satisfies all the conditions of Proposition \[wap\_c\_zero\].
Then, for $g,h\in S$, if $gh=u_S$, we have that $\Omega(h,s)\Omega(g,hs)
= \omega(h)^{-1} \omega(g)^{-1} = \Omega(g,h)$, so that $$F(hs,g)\Omega(h,s) - F(h,sg)\Omega(s,g)
= \begin{cases} \Omega(g,h) &: gh=u_S, \\ 0 &: \text{otherwise}.
\end{cases}$$ Hence condition (2) of Theorem \[C\_amen\_predual\] implies that ${{\langle {M} , {(\delta_{gh,u_S} \Omega(g,h))_{(g,h)\in S\times S}} \rangle}}=0$, which contradicts condition (1) of this theorem. Hence every element of $S$ has a right inverse.
By symmetry (or by repeating the argument on the left) we see that every element of $S$ has a left inverse, and that hence $S$ must be a group.
We hence have the following theorem, which shows that weighted semigroup algebras behave like C$^*$-algebras with regards to Connes-amenability.
\[main\_thm\] Let $S$ be a discrete cancellative semigroup, and let $\omega$ be a weight on $S$. The following are equivalent:
1. $l^1(S,\omega)$ is amenable;
2. $l^1(S,\omega)$ is Connes-amenable, with respect to the predual $c_0(S)$;
If $l^1(S,\omega)$ is Arens regular, then these conditions are equivalent to $l^1(S,\omega)''$ being Connes-amenable. These equivalent conditions imply that $S$ is a group. [$\square$]{}
This result extends the result of [@Runde3], where it is shown that $M(G)$, the *measure algebra* of a locally compact group $G$, is Connes-amenable if and only if $G$ is amenable. This follows as, for discrete groups $G$, $M(G) = l^1(G)$.
Let $\omega$ be the weight on $\mathbb Z$ defined by $\omega(n) = 1+|n|$ for $n\in\mathbb Z$. By Theorem \[B\_AR\], ${\mathcal{A}} = l^1(\mathbb Z,\omega)$ is Arens regular. For $m,n\in\mathbb Z$ and $f = (a_k)_{k\in\mathbb Z}\in
l^\infty(\mathbb Z)$, we have that $${{\langle { \delta_m \cdot f } , { \delta_n } \rangle}} =
{{\langle {f} , {\delta_{n+m}\Omega(n,m)} \rangle}}
= f_{n+m} \frac{1+|n+m|}{(1+|n|)(1+|m|)}.$$ Suppose that $M {\Box}\kappa_{{\mathcal{A}}}(\delta_m) = \kappa_{{\mathcal{A}}}(a)$ for some $m\in\mathbb Z$, $M \in l^\infty(\mathbb Z)'$ and $a\in{\mathcal{A}}$. Then ${{\langle {M} , {\delta_m\cdot f} \rangle}} = {{\langle {f} , {a} \rangle}}$ for each $f\in l^\infty(\mathbb Z)$, so by letting $f = \kappa_{c_0(\mathbb Z)}(e_k)
\in c_0(\mathbb Z)$, we see that $a = \sum_{k\in\mathbb Z} a_k \delta_k$, where $a_k = {{\langle {M} , {\delta_m\cdot\kappa_{c_0(\mathbb Z)}(e_k)} \rangle}}$. However, $\delta_m\cdot\kappa_{c_0(\mathbb Z)}(e_k) \in
\kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$ for each $k\in\mathbb Z$, so if $M \in c_0(\mathbb Z)^\circ$, then $a=0$.
Consequently, if $M {\Box}\kappa_{{\mathcal{A}}}(\delta_m) \in \kappa_{{\mathcal{A}}}({\mathcal{A}})$ for each $m\in\mathbb Z$ and $M \in l^\infty(\mathbb Z)'$, then $\delta_m \cdot f \in \kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$ for each $m\in\mathbb Z$ and $f \in l^\infty(\mathbb Z)$. However, if $\mathbf{1} \in l^\infty(\mathbb Z)$ is the constant $1$ sequence, then $$\lim_n {{\langle {\delta_m\cdot\mathbf{1}} , {\delta_n} \rangle}}
= \lim_n \frac{1+|n+m|}{(1+|n|)(1+|m|)}
= \frac{1}{1+|m|},$$ so that $\delta_m\cdot\mathbf{1} \not\in
\kappa_{c_0(\mathbb Z)}(c_0(\mathbb Z))$.
We hence conclude that ${\mathcal{A}}$ is not an ideal in ${\mathcal{A}}''$, and so we cannot apply Theorem \[ca\_facts\] in this case. [$\square$]{}
Unfortunately, it is not possible for $l^1(S,\omega)$ to be both amenable and Arens regular.
\[Gron\_Thm\] Let $G$ be discrete group, and let $\omega$ be a weight on $G$. Then $l^1(G,\omega)$ is amenable if and only if $G$ is an amenable group, and $\sup\{ \omega(g) \omega(g^{-1})
: g\in G \} < \infty$.
This is [@Gron1 Theorem 3.2].
Let $S$ be a discrete, unital semigroup, and let $\omega$ be a weight on $S$ such that ${\mathcal{A}} = l^1(S,\omega)$ is Arens regular. Let $K>0$ and $B\subseteq S$ be such that for each $g\in B$, $g$ has a right inverse $g^{-1}$ (which need not be unique), and $\omega(g)\omega(g^{-1})\leq K$. Then $B$ is finite.
For $g\in B$ and $h\in S$, we have $$\omega(g)\omega(h) = \omega(g) \omega(hgg^{-1}) \leq
\omega(g) \omega(hg) \omega(g^{-1}) \leq K \omega(hg),$$ so that $\Omega(h,g) \geq K^{-1}$. Suppose now that $B$ is infinite. Then we can easily construct sequences which violate condition (2) of Theorem \[B\_AR\], showing that ${\mathcal{A}}$ is not Arens regular. This contradiction shows that $B$ must be finite.
Injectivity of the predual module {#injectivity-of-the-predual-module}
---------------------------------
Let $S$ be a unital, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$, ${\mathcal{A}}_* = c_0(S)$. Then ${\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*) = {\mathcal{B}}(l^1,c_0) = l^\infty(c_0) \subseteq
l^\infty(S\times S)$, where we identify $T:{\mathcal{A}}\rightarrow{\mathcal{A}}_*$ with the bounded family $({{\langle {\delta_s} , {T(\delta_t)} \rangle}})_{(s,t)\in S\times S}$. Let $\phi : {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*) \rightarrow {\mathcal{A}}_*$, so that $\phi$ is represented by a bounded family $(M_s)_{s\in S} \subseteq
{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$ using the relation $${{\langle {\delta_s} , {\phi(T)} \rangle}} = {{\langle {M_s} , {T} \rangle}}
\qquad (s\in S, T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)).$$ Suppose further that $\phi$ is a left ${\mathcal{A}}$-module homomorphism. Then $${{\langle {\delta_s} , {\phi(T)} \rangle}} = {{\langle {\delta_{u_s}} , {\phi(\delta_s\cdot T)} \rangle}}
= {{\langle {M_{u_S}} , {\delta_s\cdot T} \rangle}} = {{\langle {M_s} , {T} \rangle}}
\qquad (s\in S, T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)), \label{eq:three}$$ so that $M_s = M_{u_S} \cdot \delta_s$ for each $s\in S$. We see also that $\phi$ maps into $c_0(S)$ (and not just $l^\infty(S)$) if and only if $$\lim_{s\rightarrow\infty} {{\langle {M_{u_S}} , {\delta_s\cdot T} \rangle}}
= 0 \qquad (T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)).$$
Conversely, if condition (\[eq:three\]) holds, then for $s,t\in S$ and $T\in{\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)$, we have that $$\begin{aligned}
{{\langle {\delta_s} , {\phi(\delta_t\cdot T)} \rangle}} &= {{\langle {M_s} , {\delta_t\cdot T} \rangle}}
= {{\langle {M_{u_S}} , {\delta_s\cdot\delta_t\cdot T} \rangle}}
= \Omega(s,t) {{\langle {M_{st}} , {T} \rangle}} \\
&= \Omega(s,t){{\langle {\delta_{st}} , {\phi(T)} \rangle}}
= {{\langle {\delta_s} , {\delta_t\cdot \phi(T)} \rangle}}.\end{aligned}$$ Hence $\phi$ is a left ${\mathcal{A}}$-module homomorphism.
Notice that $c_0(S\times S) \subseteq {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)$, so that $c_0(S\times S)^\circ \subseteq {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$.
Let $G$ be a group and $\omega$ be a weight on $G$ such that for each $\epsilon>0$, the set $\{ g\in G : \omega(g) \omega(g^{-1})
< \epsilon^{-1} \}$ is finite. Then we say that the weight $\omega$ is *strongly non-amenable*.
Let $G$ be a group, and let $\omega$ be a weight on $G$ such that $\omega$ is not strongly non-amenable, and let $\phi:{\mathcal{B}}({\mathcal{A}},c_0(G))\rightarrow c_0(G)$ be a left ${\mathcal{A}}$-module homomorphism. If $\phi$ is represented by $(M_g)_{g\in G}$ as above, then $M_{u_G} \in c_0(S\times S)^\circ$.
We adapt the methods of [@DP] to the weighted, discrete case. As $\omega$ is not strongly non-amenable, there exists some $K>0$ such that the set $X_K = \{ g\in G : \omega(g)\omega(g^{-1})\leq K \}$ is infinite. Let $M=M_{u_G}$, and suppose that $M\not\in c_0(G\times G)^\circ$, so that for some $g,h\in G$, we have that $\delta:={{\langle {M} , {e_{(g,h)}} \rangle}}\not=0$. We shall henceforth treat $e_{(g,h)}$ as a member of ${\mathcal{B}}({\mathcal{A}},c_0(G))$, noting that for $k\in G$, $${{\langle {\delta_s} , {(\delta_k\cdot e_{(g,h)})(\delta_t)} \rangle}}
= \begin{cases} \Omega(t,k) &: s=g, t=hk^{-1}, \\
0 &: \text{otherwise.} \end{cases}$$ We claim that we can find a sequence $(g_n)_{n\in\mathbb N}$ of distinct elements in $G$ such that $$\begin{gathered}
|{{\langle {M\cdot\delta_{g_m^{-1}g_n}} , {e_{(g,h)}} \rangle}}| \leq
K^{-1} 2^{-2-|m-n|} \qquad (n\not=m), \\
\omega(g_n) \omega(g_n^{-1}) \leq K \qquad (n\in\mathbb N).\end{gathered}$$ We can do this as $\phi$ must map into $c_0(G)$, so that for any $T:{\mathcal{A}}\rightarrow c_0(G)$, we have $\lim_{g\rightarrow\infty}
{{\langle {M\cdot\delta_g} , {T} \rangle}}=0$. Explicitly, let $g_1\in X_K$ be arbitrary, and suppose that we have found $g_1,\ldots,g_k$. Then notice that the sets $$\begin{gathered}
\{ s\in G : |{{\langle {M\cdot\delta_{s^{-1}g_n}} , {e_{(g,h)}} \rangle}}| > K^{-1}
2^{-2-|k+1-n|} : 1\leq n\leq k \}, \\
\{ s\in G : |{{\langle {M\cdot\delta_{g_m^{-1}s}} , {e_{(g,h)}} \rangle}}| > K^{-1}
2^{-2-|k+1-m|} : 1\leq m\leq k \}\end{gathered}$$ are finite, so as $X_K$ is infinite, we can certainly find some $x_{k+1}$.
Then, for $x=(x_n)\in l^\infty(\mathbb N)$, define $T_x:
{\mathcal{A}}\rightarrow c_0(G)$ by setting ${{\langle {\delta_g} , {T_x(\delta_{hg_n^{-1}})} \rangle}} = x_n \Omega(hg_n^{-1},g_n)$ for $n\geq 1$, and ${{\langle {\delta_s} , {T_x(\delta_t)} \rangle}}=0$ otherwise. Then clearly $T_x$ does map into $c_0(G)$, and $\|T_x\| \leq
\|x\|$. Notice that for $s,t\in G$, we have $$\begin{aligned}
{{\langle {\delta_s} , {T_x(\delta_t)} \rangle}}
&= \begin{cases} x_n \Omega(t,g_n) &:
s=g, t=hg_n^{-1}, \\ 0 &: \text{otherwise,} \end{cases} \\
&= \sum_n x_n {{\langle {\delta_s} , {(\delta_{g_n}\cdot e_{(g,h)})(\delta_t)} \rangle}}.\end{aligned}$$ Define $Q:l^\infty(\mathbb N)\rightarrow c_0(\mathbb N)$ by $${{\langle {\delta_n} , {Q(x)} \rangle}} = {{\langle {M} , {\delta_{g_n^{-1}}\cdot T_x} \rangle}}
\qquad (n\in\mathbb N),$$ so that $Q$ is bounded and linear.
Let $n_0\geq 1$ and let $x = e_{n_0} \in c_0(\mathbb N)
\subseteq l^\infty(\mathbb N)$. Then, $T_x = \delta_{g_{n_0}}
\cdot e_{(g,h)}$, so that $$\begin{aligned}
{{\langle {\delta_n} , {Q(x)} \rangle}} &= {{\langle {M} , {\delta_{g_n^{-1}}\cdot T_x} \rangle}}
= {{\langle {M} , {\delta_{g_n^{-1}}\cdot(\delta_{g_{n_0}}\cdot e_{(g,h)})} \rangle}} \\
&= \begin{cases} \delta\, \Omega(g_{n_0}^{-1},g_{n_0}) &: n=n_0, \\
\Omega(g_n^{-1},g_{n_0})
{{\langle {M\cdot e_{g_n^{-1}g_{n_0}}} , {e_{(g,h)}} \rangle}} &: n\not=n_0. \end{cases}\end{aligned}$$ Define $Q_1 \in{\mathcal{B}}(c_0(\mathbb N))$ by $$Q_1(x) = \big( \Omega(g_n^{-1},g_n) x_n \big)_{n\in\mathbb N}
\qquad ( x=(x_n) \in c_0(\mathbb N) ).$$ Then, as each $g_n\in X_K$, $Q_1$ is an invertible operator. Let $Q_2$ be the restriction of $Q$ to $c_0(\mathbb N)$, so that $Q_2\in{\mathcal{B}}(c_0(\mathbb N))$ and $Q_2 = \delta Q_1 +
\delta Q_3Q_1$ for some $Q_3\in{\mathcal{B}}(c_0(\mathbb N))$. Thus $Q_3 = \delta^{-1}Q_2Q_1^{-1} - I_{c_0(\mathbb N)}$, so that for $x\in c_0(\mathbb N)$, we have that $$\begin{aligned}
\|Q_3(x)\| &= \sup_n |{{\langle {\delta_n} , {\delta^{-1}Q_2Q_1^{-1}(x) - x} \rangle}}|
= \sup_n \Big| \sum_m x_m {{\langle {\delta_n} , {\delta^{-1}Q_2Q_1^{-1}(e_m) - e_m} \rangle}} \Big| \\
&= \sup_n \Big| \sum_{m\not=n} x_m \Omega(g_m^{-1},g_m)^{-1}
\Omega(g_n^{-1},g_m) {{\langle {M\cdot\delta_{g_n^{-1}g_m}} , {e_{(g,h)}} \rangle}} \Big| \\
&\leq K^{-1} \sup_n \sum_{m\not=n} |x_m| 2^{-2-|m-n|} \omega(g_m)\omega(g_m^{-1})
\leq \|x\| / 2.\end{aligned}$$ Consequently $Q_3-I_{c_0(\mathbb N)}$ is invertible, so that $Q_2Q_1^{-1}$ is invertible, showing that $Q_2$ is invertible. However, this implies that $Q_2^{-1}Q:l^\infty(\mathbb N)\rightarrow
c_0(\mathbb N)$ is a projection, which is a well-known contradiction, completing the proof.
\[amen\_not\_inj\] Let $G$ be a countable group, let $\omega$ be a weight which is not strongly non-amenable, and let ${\mathcal{A}}=l^1(G,\omega)$. Then $c_0(G)$ is not left-injective.
Suppose, towards a contradiction, that $c_0(G)$ is left-injective, so that there exists $M=M_{u_G}
\in {\mathcal{B}}({\mathcal{A}},{\mathcal{A}}_*)'$ as above, with the additional condition that $$\begin{aligned}
\delta_{g,h} &= {{\langle {\delta_g} , {\phi\Delta'_{{\mathcal{A}}}(e_h)} \rangle}}
= {{\langle {M} , {\delta_g \cdot \Delta'_{{\mathcal{A}}}(e_h)} \rangle}}
= \Omega(hg^{-1},g) {{\langle {M} , {\Delta'_{{\mathcal{A}}}(e_{hg^{-1}})} \rangle}} \\
&= \Omega(hg^{-1},g) {{\langle {M} , {\big( \delta_{st,hg^{-1}}
\Omega(s,t) \big)_{(s,t)\in G\times G}} \rangle}}
\qquad (g,h\in G). \end{aligned}$$ This clearly reduces to $$\delta_{g,u_G} = {{\langle {M} , {\big( \delta_{st,g}
\Omega(s,t) \big)_{(s,t)\in G\times G}} \rangle}} \qquad (g\in G).$$
As $G$ is countable, we can enumerate $G$ as $G = \{ g_n : n\in\mathbb N\}$. Then, for $g_n\in G$, let $X_{g_n} = \{ g_1, \ldots, g_n \}
\subseteq G$. Define $Q:l^\infty(G)\rightarrow{\mathcal{B}}({\mathcal{A}},c_0(G))$ by $${{\langle {\delta_s} , {Q(x)(\delta_t)} \rangle}} =
\Omega(s,t) \sum_{g\in X_t} x_g \delta_{st,g}
\qquad (s,t\in G, x\in l^\infty(G)).$$ Then, for each $t\in G$, as $X_t$ is finite, we see that $Q(x)(\delta_t)\in c_0(G)$, so $Q$ is well-defined. Clearly $Q$ is linear, and we see that for $x\in l^\infty(G)$, $$\|Q(x)\| = \sup_{s,t\in G} \Omega(s,t) \Big|\sum_{g\in X_t} x_g \delta_{st,g}\Big|
\leq \sup_{s,t\in G} \sum_{\{g\in X_t : g=st\}} |x_g|
= \|x\|,$$ so that $Q$ is norm-decreasing. Then, for $h\in G$, we have that $${{\langle {\delta_s} , {Q(e_h)(\delta_t)} \rangle}}
= \Omega(s,t) \sum_{g\in X_t} \delta_{g,h} \delta_{st,g}
= \begin{cases} {{\langle {\delta_s} , {\Delta_{{\mathcal{A}}}'(e_h)(\delta_t)} \rangle}}
&: h\in X_t, \\ 0 &: h\not\in X_t. \end{cases}$$ Let $h=g_{n_0}$, so that $\{ t\in G : h\not\in X_t \} =
\{ g_n\in G : h\not\in X_{g_n} \} = \{ g_1, g_2, \ldots, g_{n_0-1}\}$. We hence see that $Q(e_{g_0}) - \Delta'_{{\mathcal{A}}}(e_{g_0}) \in
c_0(G\times G)$. By the preceding proposition, we hence have that $I_{c_0(G)} = \phi\circ\Delta_{{\mathcal{A}}}' = \phi\circ (Q|_{c_0(G)})$. However, this implies that $\phi\circ Q:l^\infty(G)\rightarrow
c_0(G)$ is a projection onto $c_0(G)$, giving us the required contradiction.
\[sg\_not\_inj\] Let $S$ be a discrete, weakly cancellative semigroup, let $\omega$ be a weight on $S$, and let ${\mathcal{A}}=l^1(S,\omega)$. When $S$ is unital, or $S$ is cancellative, $c_0(G)$ is not a bi-injective ${\mathcal{A}}$-bimodule.
Suppose, towards a contradiction, that $c_0(G)$ is bi-injective. Then ${\mathcal{A}}$ is Connes-amenable, so that Theorem \[main\_thm\] implies that ${\mathcal{A}}$ is amenable, and that $S=G$ is a group. By Theorem \[Gron\_Thm\], we know that $\omega$ is not strongly non-amenable. Suppose that $G$ is countable, so that the above theorem shows that $c_0(G)$ is not left-injective, and that hence $c_0(G)$ is certainly not bi-injective, a contradiction.
Suppose that $G$ is not countable. Then let $H$ be some countably infinite subgroup of $G$. Let $K = \sup\{ \omega(g)\omega(g^{-1})
: g\in G \}<\infty$, and let $g,h\in G$. Then $$\Omega(g,h) = \frac{\omega(gh)}{\omega(g)\omega(h)}
= \frac{\omega(gh)}{\omega(g)\omega(g^{-1}gh)}
\geq \frac{\omega(gh)}{\omega(g)\omega(g^{-1})\omega(gh)}
= \frac{1}{\omega(g)\omega(g^{-1})} \geq K^{-1},$$ so that $\Omega$ is bounded below on $G\times G$, and hence on $H\times H$.
Then we can find $X\subseteq G$ such that $G = \bigcup_{x\in X} Hx$ and $Hx\cap Hy = \emptyset$ for distinct $x,y\in X$. Notice that if $g\in Hx$ then $g^{-1}\in x^{-1}H$, so that $G = \bigcup_{x\in X}
x^{-1}H$ as well.
By the proof of Theorem \[amen\_not\_inj\], we see that $c_0(H)$ is not a left-injective $l^1(H,\omega)$-module. Suppose, towards a contradiction, that we do have some left ${\mathcal{A}}$-module homomorphism $\phi:{\mathcal{B}}(l^1(G,\omega),c_0(G)) \rightarrow c_0(G)$ with $\phi
\Delta_{{\mathcal{A}}}' = I_{{\mathcal{A}}'}$. Notice that certainly ${\mathcal{B}}(l^1(G,\omega),c_0(G))$ and $c_0(G)$ are Banach left $l^1(H,\omega)$-modules, by restricting the action from $l^1(G,\omega)$.
Define a map $\psi : {\mathcal{B}}(l^1(H,\omega),c_0(H)) \rightarrow
{\mathcal{B}}(l^1(G,\omega),c_0(G))$ by, for $g,k\in G$, $${{\langle {\delta_g} , {\psi(T)(\delta_k)} \rangle}} =
\begin{cases} \frac{\omega(s) \omega(t)}{\omega(tx) \omega(k)} {{\langle {\delta_t} , {T(\delta_s)} \rangle}}
&: g=tx, k=x^{-1}s \text{ for some } x\in X, s,t\in H, \\
0 &: \text{otherwise.} \end{cases}$$ Certainly $\psi$ is linear, while $$\|\psi(T)\|
\leq \|T\| \sup_{s,t\in H, x\in X}
\frac{\omega(s)\omega(t)}{\omega(tx)\omega(x^{-1}s)}
\leq \|T\| \sup_{s,t\in H, x\in X}
\frac{\omega(s)\omega(t)}{\omega(txx^{-1}s)}
= \|T\| \sup_{s,t\in H} \Omega(t,s)^{-1},$$ so that $\psi$ is bounded. For $h,s,t\in H$, and $x\in X$, we have $$\begin{aligned}
&{{\langle {\delta_{tx}} , {(\delta_h\cdot\psi(T))(\delta_{x^{-1}s})} \rangle}}
= \Omega(x^{-1}s,h) {{\langle {\delta_{tx}} , {\psi(T)(\delta_{x^{-1}sh})} \rangle}} \\
&= \frac{\Omega(x^{-1}s,h) \omega(sh) \omega(t)}{\omega(tx)
\omega(x^{-1}sh)} {{\langle {\delta_t} , {T(\delta_s)} \rangle}}
= \frac{\omega(sh) \omega(t)}{\omega(x^{-1}s) \omega(h) \omega(tx)}
{{\langle {\delta_t} , {T(\delta_s)} \rangle}} \\
&= \omega(s) \omega(x^{-1}s)^{-1} \omega(t) \omega(tx)^{-1}
{{\langle {\delta_t} , {(\delta_h\cdot T)(\delta_s)} \rangle}}
= {{\langle {\delta_{tx}} , {\psi(\delta_h\cdot T)(\delta_{x^{-1}s})} \rangle}}.\end{aligned}$$ Thus $\psi$ is a left $l^1(H,\omega)$-module homomorphism. For $h,s,t\in H$ and $x\in X$, we then have that $$\begin{aligned}
{{\langle {\delta_{tx}} , {\psi(\Delta'_{l^1(H,\omega)}(e_h))(\delta_{x^{-1}s})} \rangle}}
&= \frac{\omega(t) \omega(s)}{\omega(x^{-1}s) \omega(tx)}
{{\langle {\delta_t} , {\delta_s\cdot e_h} \rangle}}
= \Omega(tx,x^{-1}s) \delta_{ts,h} \\
&= {{\langle {\delta_{tx}} , {\delta_{x^{-1}s} \cdot e_h} \rangle}}
= {{\langle {\delta_{tx}} , {\Delta'_{{\mathcal{A}}}(e_h)(\delta_{x^{-1}s})} \rangle}}.\end{aligned}$$ If $g,k\in G$ are such that $gk\not\in H$ then $g=tx$ and $k=y^{-1}s$ for some $s,t\in H$ and distinct $x,y\in X$. Then, for $h\in H$, we have that $gk\not=h$, so that $${{\langle {\delta_g} , {\Delta'_{{\mathcal{A}}}(e_h)(\delta_k)} \rangle}}
= \Omega(g,k) \delta_{gk,h}
= 0 = {{\langle {\delta_g} , {\psi(\Delta'_{l^1(H,\omega)}(e_h))(\delta_k)} \rangle}}.$$ Hence $\psi\circ \Delta'_{l^1(H,\omega)}$ is equal to $\Delta'_{{\mathcal{A}}}$ restricted to $l^1(H,\omega)$.
Let $P : c_0(G) \rightarrow c_0(H)$ be the natural projection, which is obviously an $l^1(H,\omega)$-module homomorphism. Then $Q = P \circ \phi \circ \psi : {\mathcal{B}}( l^1(H,\omega), c_0(H) )
\rightarrow c_0(H)$ is a bounded left $l^1(H,\omega)$-module homomorphism, and $Q \circ \Delta'_{l^1(H,\omega)} = I_{c_o(H)}$. This contradiction completes the proof.
We note that just because $\Omega$ is bounded below does not imply that $\omega$ is bounded, so that $l^1(G,\omega)$ is not necessarily isomorphic to $l^1(G)$, and hence we cannot simply apply the results of [@DP].
We have not been able to establish if $c_0(S)$ can every be a left-injective $l^1(S,\omega)$-module for some semigroup $S$ and weight $\omega$.
Open questions
==============
We state a few open questions of interest:
1. Let ${\mathcal{A}}$ be an Arens regular Banach algebra such that ${\mathcal{A}}''$ is Connes-amenable. Need ${\mathcal{A}}$ be amenable?
2. This is true for C$^*$-algebras. Can we find a “simple” proof?
3. Let ${\mathcal{A}}$ be a dual Banach algebra with predual ${\mathcal{A}}_*$, and suppose that ${\mathcal{A}}_*$ is bi-injective. If ${\mathcal{A}}$ necessarily a von Neumann algebra or the bidual of an Arens regular Banach algebra ${\mathcal{B}}$ such that ${\mathcal{B}}$ is an ideal in ${\mathcal{A}}$?
4. Let $S$ be a (weakly cancellative) semigroup, and let $\omega$ be a weight on $S$. Classify (up to isomorphism) the preduals of $l^1(S,\omega)$, and calculate which preduals yield a Connes-amenable Banach algebra.
5. This question was asked by Niels Gr[ø]{}nbæk. In most of our examples, it is obvious that when ${\mathcal{A}}$ is a Connes-amenable dual Banach algebra, there is ${\mathcal{B}}\subseteq{\mathcal{A}}$ which is weak$^*$-dense and amenable. Is this always true?
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St. John’s College,\
Oxford\
OX1 3JP\
United Kingdom
`[email protected]`
| ArXiv |
---
abstract: 'In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the *pseudo-backstepping* method, which uses a *pseudo-kernel* that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear term(s). At the end of the paper, we give numerical simulations to illustrate the main result.'
address: 'Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, TURKEY'
author:
- 'Türker Özsarı^\*^ & Ahmet Batal'
bibliography:
- 'myreferences.bib'
title: 'Pseudo-backstepping and its application to the control of Korteweg-de Vries equation from the right endpoint on a finite domain'
---
\#1
Introduction
============
This article is devoted to the study of the boundary feedback stabilization of the Korteweg-de Vries (KdV) equation on a bounded domain $\Omega=(0,L)\subset \mathbb{R}$. The linear version of the model under consideration is given by $$\label{KdVBurgers}
\begin{cases}
\displaystyle u_{t} + u_{x} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = U(t), u_{x}(L,t)=V(t) & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega,
\end{cases}$$ whereas the nonlinear version of this model is written with the main equation in replaced with $$\label{nonlinearKdV}u_t+u_x+u_{xxx}+uu_x=0.$$
In , $u=u(x,t)$ is a real valued function that can model the evolution of the amplitude of a weakly nonlinear shallow dispersive wave in space and time [@KdVPaper]. The inputs $U(t)=U(u(t,\cdot))$ and $V(t)=V(u(t,\cdot))$ at the right endpoint of the boundary are feedbacks. The goal is to choose these boundary feedbacks so that the solutions of and decay to zero as $t\rightarrow \infty$, at an exponential rate in the mean-square sense.
Controlling the behavior of solutions of evolution equations is an important topic, and many approaches have been proposed. One method is to use local or global interior controllers. Another method is to use external (boundary) controllers, especially in those models where it is difficult to access the domain. Using feedback type controls is a common tactic to stabilize the solutions. However, non-feedback type controls (open loop control systems) are also used for steering solutions to or near a desired state. Exact, null, or approximate controllability models have been developed for almost all well-known PDEs.
Exact boundary controllability of linear and nonlinear KdV equations with the same type of boundary conditions as in was studied by [@Rosier1997], [@Cor2004], [@Zhang99], [@Glass08], [@Cerpa07], [@Cerpa09], [@RosZha09], and [@Glass10]. In these papers, the boundary inputs are chosen in advance to steer solutions to a desired final state at a given time. This results in an open loop model. In contrast, the boundary inputs in our model depend on the solution itself, and is therefore closed loop.
Stabilization of solutions of the KdV equation with a localised interior damping was achieved by [@Perla2002], [@Pazo05], [@Mass07], and [@Balogh2000]. There are also some results achieving stabilization of the KdV equation by using predetermined local boundary feedbacks; see for instance [@Liu2002], and [@Jia2016].
Motivation
----------
and with homogeneous boundary conditions ($U=V\equiv 0$) are both dissipative since $\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2\le 0.$ However, this does not always guarantee exponential decay. It is well-known that if $\displaystyle L\in \mathcal{N}\equiv \left\{2\pi\sqrt{\frac{k^2+kl+l^2}{3}}, k,l\in \mathbb{N}\right\}$ (so called *critical lengths* for KdV), then the solution does not need to decay to zero at all. For example, if $L=2\pi$, $u=1-\cos(x)$ is a (time independent) solution of on $\Omega=(0,2\pi)$, but its $L^2-$norm is constant in $t$. On the other hand, if $L$ is not critical, one can show the exponential stabilization of solutions for under homogeneous boundary conditions; see for example [@Perla2002 Theorem 2.1].
Recently, [@Cerpa2013] studied the boundary feedback stabilization of the KdV equation with the boundary conditions $$\label{BCforKdVLeft}
u(0,t) = U(t), u_{x}(L,t) = 0, u(L,t) = 0$$ by using the back-stepping technique (see for example [@KrsBook]). [@Cerpa2013] proved that given any $r>0$, there corresponds a smooth kernel $k=k(x,y)$ such that the boundary feedback controller $U(t)=U(u(t,\cdot))=\int_0^Lk(0,y)u(y,t)dy$ steers the solution of the linear KdV equation to zero with the decay rate estimate $\|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-r t}.$ Moreover, the same result also holds true for the nonlinear KdV equation provided that $u_0$ is sufficiently small in the $L^2-$sense. Here, $k=k(x,y)$ is an appropriately chosen kernel function satisfying a third order PDE model on a triangular domain that involves three boundary conditions. In [@Cerpa2013], the control acts on the Dirichlet boundary condition at the *left* endpoint of the domain. However, the situation is very different if the control acts at the *right* endpoint of the domain, because then the kernel of the backstepping controller has to satisfy an overdetermined PDE model whose solution may or may not exist. Therefore, the problem of finding backstepping controllers acting at the *right* endpoint of the domain is interesting.
Coron & Lü [@Cor14] studied this problem with a single controller acting from the Neumann boundary condition on domains of *uncritical* lengths. They prove the rapid exponential stabilization of solutions for the KdV equation under a smallness assumption on the initial datum. The method of [@Cor14] is based on using a rough kernel function in the backstepping integral transformation. The construction of the rough kernel relies on the exact controllability of the linear KdV equation by the Neumann boundary control acting at the right endpoint of the domain. However, the exact controllability was proved only for the domains of uncritical lengths. On the other hand, the exponential decay of solutions for the linearized KdV equation holds even without adding any control to the system when the length of the domain does not belong to the set of critical lengths [@Perla2002]. Therefore, the following remains as an important problem:
\[mainprob\] Let $L>0$ (not necessarily uncritical). Can you find a kernel $k=k(x,y)$ such that the solution of and satisfies $$\label{decayest}\|u(\cdot,t)\|_{L^2(\Omega)}=\mathcal{O}(e^{-rt})$$ for some $r>0$ with boundary feedback controllers given by $$\label{controller}
U(t)=\int_0^Lk(L,y)u(y,t)dy \text{ and } V(t)=\int_0^Lk_x(L,y)u(y,t)dy\,?$$
A stronger version of the above problem is the following:
\[mainprob1\] Given $r>0$, can you find a kernel $k=k(x,y)$ such that the solution of and satisfies the $L^2-$decay estimate with the boundary feedback controllers given in ?
This paper and the method proposed address only Problem \[mainprob\], and the latter problem still remains open for domains of critical length.
In order to understand the nature of the problem and the difficulty here, let us consider the linearised KdV equation in . A backstepping controller for this linear model is generally constructed by using a transformation given by $$\label{transform}w(x,t)\equiv u(x,t)-\int_0^xk(x,y)u(y,t)dy,$$ where the unknown kernel function $k(x,y)$ is chosen in such a way that if $u$ is a solution of with the boundary feedback controllers given in , then $w$ is a solution of the damped homogeneous initial-boundary value problem (so called “*target system*”) $$\label{HomKdVBurgers}
\begin{cases}
\displaystyle w_{t} + w_{x} + w_{xxx} + \lambda w = 0 & \text { in } \Omega\times \mathbb{R_+},\\
w(0,t) = w(L,t) = w_{x}(L,t) = 0 & \text { in } \mathbb{R_+},\\
w(x,0)=w_0(x)\equiv u_0-\int_0^xk(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ The reason is that the solution of satisfies $\|w(t)\|_{L^2(\Omega)}=O(e^{-\lambda t})$, and if the given transformation is invertible, one can hope to get a similar decay property for $u$.
The essence of the back-stepping algorithm is to find an appropriate kernel function $k$ which serves the purpose. In order to do this, one simply assumes that $u$ solves and plugs in $u(x,t)-\int_0^xk(x,y)u(y,t)dy$ into the main equation in wherever one sees $w$. This gives a set of sufficient conditions that the kernel has to satisfy. Note that $w$ satisfies the given homogeneous boundary conditions $w(0,t)=w(L,t)=w_x(L,t)=0$ by the transformation in and the choice of the feedback controllers in . In order for the main equation in to be satisfied, one can impose a few conditions on $k$. Indeed, computing the derivative of $w$ with respect to the temporal and spatial derivatives and putting these together, we obtain the following: $$\begin{gathered}
\nonumber w_{t}(x,t) + w_{x}(x,t) + w_{xxx}(x,t) + \lambda w(x,t)= k_y(x,0)u_x(0,t)\\
-\int_{0}^{x}u(y,t)\left[k_{xxx}(x,y) + k_{x}(x,y) + k_{yyy}(x,y) + k_{y}(x,y) + \lambda k(x,y)\right]dy \label{23} \\
\nonumber -k(x,0)u_{xx}(0,t) - u_{x}(x,t)\left[k_{y}(x,x) + k_{x}(x,x) + 2\frac{d}{dx}k(x,x)\right] \\
\nonumber + u(x,t)\left[\lambda - k_{xx}(x,x) + k_{yy}(x,x) - \frac{d}{dx}k_{x}(x,x) - \frac{d^{2}}{dx^{2}}k(x,x)\right].\end{gathered}$$ The above equation is the same as that of the target system if $k$ solves the third order partial differential equation together with the set of boundary conditions given by $$\begin{aligned}
\label{kEq}
% \nonumber to remove numbering (before each equation)
\nonumber k_{xxx} + k_{yyy} + k_{y} + k_{x} &=& -\lambda k, \\
k(x,x) = k(x,0)=k_y(x,0) &=& 0, \\
\nonumber k_{x}(x,x) &=& \frac{\lambda}{3}x,\end{aligned}$$ where the PDE model is considered on the triangular spatial domain $\mathcal{T}\equiv \{(x,y)\in \mathbb{R}^2\,|\,x\in [0,L], y\in [0,x]\}\,\,\text{(see Figure \ref{regT} below)}.$
![Triangular region $T$ for $L=2\pi$[]{data-label="regT"}](RecT)
In order to solve the problem , one generally first applies a change of variables. Here, an appropriate choice would be to define $t\equiv y$, $s\equiv x-y$, and $G(s,t)\equiv k(x,y)$. Then, $G$ satisfies the boundary value problem given by $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{Geq}G_{ttt} - 3G_{stt}+ 3G_{sst} + G_{t} &=& -\lambda G, \\
\label{Geqb}G(s,0) = G_t(s,0) = G(0,t) &=& 0, \\
\label{Geqc}G_s(0,t) &=& \frac{\lambda}{3}t\end{aligned}$$ on the triangular domain $\mathcal{T}_{0}\equiv \left\{ (s,t) \,|\, t \in [0,L], s \in [0,L-t]\right\}\,\,\text{(see Figure \ref{regT0} below)}.$
Unfortunately, it is not easy to decide whether - has a solution. Note that there is also a mismatch between the boundary conditions $ G_t(s,0)=0$ and $G_s(0,t) = \displaystyle \frac{\lambda}{3}t$ in the sense that $G_{ts}(0,0)=0\neq G_{st}(0,0)=\displaystyle \frac{\lambda}{3}.$ Hence, the standard back-stepping algorithm fails because it enforces us to solve an overdetermined singular PDE model. This issue does not arise if one tries to control the system from the left endpoint of the domain as in [@Cerpa2013].
![Triangular region $T_0$ for $L=2\pi$[]{data-label="regT0"}](RecT0)
The adverse effect of the nonhomogeneous boundary condition in the kernel PDE model was eliminated by expanding the domain from a triangle into a rectangle in [@Cor14]. However, this approach brings a dirac delta term to the right hand side of the main equation; see the kernel model in [@Cor14 Section 1]. The cost of this is that the constructed kernel cannot be expected to be very smooth. However, the higher regularity is crucial to rigorously justify the calculations in that show the equivalence of the original plant and the exponentially stable target system. Therefore, we rely on a different idea based on constructing an imperfect but smooth kernel. The details of this construction are given below.
Pseudo-backstepping
-------------------
We introduce a new backstepping technique which eliminates the difficulties explained in the previous section. In the standard backstepping method, the plant model is transformed into the most desirable (e.g., exponentially stable) target system with a transformation as in . This is called forward transformation. The target system is then transformed back into the plant model via an inverse transformation, generally in the form $$\label{backwardt}u(x,t)=w(x,t)+\int_0^xp(x,y)w(y,t)dy.$$ This is called backward transformation. A combination of these two steps allows one to conclude that the plant is stable if and only if the target system is stable in the same sense (see Figure \[Backstepping\]).
![Standard back-stepping[]{data-label="Backstepping"}](Backstepping)
Unfortunately, applying this algorithm to our problem forces kernels $p$ and $k$ to be solutions of overdetermined boundary value problems, and thus the method fails.
Our strategy uses a pseudo-kernel which is chosen as a solution of a corrected version of the gain control PDE given by: $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{Geqeps1}\tilde{G}_{ttt} - 3\tilde{G}_{stt}+ 3\tilde{G}_{sst} + \tilde{G}_{t} &=& -\lambda \tilde{G}, \\
\label{Geqeps2}\tilde{G}(s,0) = \tilde{G}(0,t) &=& 0, \\
\label{Geqeps3}\tilde{G}_s(0,t) &=& \frac{{\lambda}}{3}t\end{aligned}$$ on the triangular domain $\mathcal{T}_{0}$. Unlike in the previous model -, here the boundary condition $\tilde{G}_t(s,0)=0$ is completely disregarded. One advantage of using this modified model is that we can solve it. Another is that, even though the boundary condition $\tilde{G}_t(s,0)=0$ is disregarded, we can control the size of this boundary condition by choosing ${\lambda}$ sufficiently small. The cost of using a pseudo-kernel is that the target system changes, (see the modified target system in ), which causes a slower rate of decay. Nevertheless, this new method (henceforth referred to as *pseudo-backstepping*) allows us to obtain physically reasonable exponential decay rates for some choice of $\lambda$ (see Table \[table1\] for sample decay rates for some values of $\lambda$ on a domain of length $L=2\pi$).
Another aspect of our method is that instead of using a concrete backward transformation as in , we rely on the existence of an abstract inverse transformation that maps the solution of the modified target system back into the original plant. The existence of such a transformation is proved via succession (see Lemma \[inverselem\] below). This type of backward transformation was previously used in the stabilization of the heat equation with a localized source of instability [@Liu03]. We do not search for an inverse of type to avoid a highly overdetermined system that would result from computing the temporal and spatial derivatives of the given transformation and finding the conditions that $p$ has to satisfy.
![Pseudo-backstepping[]{data-label="pseudo-bs"}](pseudo-bs)
Main results
------------
Applying the pseudo-backstepping method explained above to the linearized and nonlinear KdV models given in and , we are able to prove the following wellposedness and stabilization theorems:
\[Linthm0\] Let $T>0$, $u_0\in L^2(\Omega)$ and $$\label{controllers}U(t) = \int_0^L\tilde{k}(L,y)u(y,t)dy,\,V(t)= \int_0^L\tilde{k}_x(L,y)u(y,t)dy,$$ where $\tilde{k}$ is a smooth kernel given by . Then, has a unique solution $u\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $u_x\in C([0,L];L^2(0,T)).$ Moreover, the same result also holds true for the nonlinear KdV equation if $\|u_0\|_{L^2(\Omega)}$ is sufficiently small.
Indeed, our analysis in this paper also shows that if $u_0\in H^3(\Omega)$ and satisfies the compatibility conditions $$\label{compcond}u_0(0)=0, u_0(L)=\int_0^L\tilde{k}(L,y)u_0(y)dy, u_0'(L)=\int_0^L\tilde{k}_x(L,y)u_0(y)dy,$$ then the solution of or the local solution of satisfies $u\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$ One can also interpolate to get regularity in the fractional spaces. For example, let $u_0\in H^s(\Omega)$ ($s\in [0,3]$) so that it satisfies the compatibility conditions $u_0(0)=0, u_0(L)=\int_0^L\tilde{k}(L,y)u_0(y)dy$ if $s\in [0,3/2]$ and the compatibility conditions if $s\in (3/2,3].$ Then, the solution of or the local solution of satisfies $u\in C([0,T];H^s(\Omega))\cap L^2(0,T;H^{s+1}(\Omega)).$
\[Linthm\] Let $u_0\in L^2(\Omega)$. Then, for sufficiently small $\lambda>0$, one has $\alpha=\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2>0$, where $\tilde{k}$ is given by , and the corresponding solution of with the boundary feedback controllers satisfies $ \|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-\alpha t}.$ Moreover, the same decay property is also true for the nonlinear KdV equation if $\|u_0\|_{L^2(\Omega)}$ is sufficiently small.
The proof of Theorem \[Linthm\] is given in the next section. Table \[table1\] gives some examples where exponential stabilization can be achieved. For example, when $\lambda=0.03$, the decay rate is approximately of order $\mathcal{O}(e^{-0.18 t})$ on a domain of length $L=2\pi$. The exponential decay rate is substantially small (see Table \[table1\]) relative to the decay rates one can get by controlling the equation from the left end-point with the same type of boundary conditions. Indeed, what matters is is not where the controller is located,but rather the number of boundary conditions specified on the opposite side of the boundary. For example, if one specified two boundary conditions at the left and only one boundary condition at the right, then it would be easier to control from right and more difficult to control from left, in contrast to the problem studied in this paper.
$\lambda$ $\alpha=\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2$
----------- ----------------------------------------------------------------------
0.01
0.02
0.03
0.04
0.05
0.10
1
: \[table1\]Numerical experiments on a domain of critical length $L=2\pi$
Stabilization
=============
In this section, we prove Theorem \[Linthm\]. First, we prove the existence of the pseudo-kernel and the abstract inverse transformation. Secondly, by using the multiplier method, we obtain the stabilization for suitable $\lambda.$ The multiplier method is applied only formally, but the calculations can be justified by a standard density argument and the regularity results proved in the next section.
Linearised model {#ProofofThm}
----------------
The sought-after solution of - can be constructed by applying the successive approximations technique to the integral equation $$\label{GepsInt}
\tilde{G}(s,t) = \frac{{\lambda}}{3}st+\frac{1}{3}\int_0^t\int_0^s\int_0^\omega (-\tilde{G}_{ttt} + 3\tilde{G}_{stt} - \tilde{G}_{t} -\lambda \tilde{G})(\xi,\eta)d\xi d\omega d\eta.$$ Indeed, we have the following lemma.
\[lemback\] There exists a $C^\infty$-function $\tilde{G}$ such that $\tilde{G}$ solves the integral equation as well as the boundary value problem given in -.
Let $P$ be defined by $$\label{aP}
(P f)(s,t) = \frac{1}{3}\int_0^t\int_0^s\int_0^\omega (-f_{ttt} + 3f_{stt} - f_{t} -\lambda f)(\xi,\eta)d\xi d\omega d\eta.$$ By , we need to solve the equation $\tilde{G}(s,t)=\frac{{\lambda}}{3}st+P\tilde{G}(s,t).$ Define $\tilde{G}^0\equiv 0,$ $\displaystyle\tilde{G}^1(s,t)=\frac{{\lambda}}{3}st,$ and $\tilde{G}^{n+1}=\tilde{G}^1+P \tilde{G}^n.$ Then for $n\geq 1$, $\tilde{G}^{n+1}-\tilde{G}^{n} = P(\tilde{G}^{n}-\tilde{G}^{n-1}).$ So if we define $H^0(s,t)=st$ and $H^{n+1}=PH^n$, we get $H^n=\frac{3}{\lambda}(\tilde{G}^{n+1}-\tilde{G}^{n}).$ Moreover, for $j>i,$ $$\label{aCauchy}
\tilde{G}^j-\tilde{G}^i= \sum_{n=i}^{n=j-1}\tilde{G}^{n+1}-\tilde{G}^{n}=\frac{\lambda}{3}\sum_{n=i}^{n=j-1}H^{n}.$$ Let $\| \cdot \|_{\infty}$ denote the supremum norm of a function on the triangle $T_0$. It follows from that to prove $\tilde{G}_n$ (and its partial derivatives) is Cauchy with respect to the norm $\| \cdot \|_{\infty}$ it is enough to show $H^n$ (and its partial derivatives) is an absolutely summable sequence with respect to the same norm.
To show $H^n$’s are absolutely summable, let us first write $P$ as the sum of four operators $P= P_{-2}+P_{-1}+P_0+P_1,$ where $$P_{-2}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -f_{ttt}(\xi,\eta) d\xi d\omega'd\eta,\,P_{-1}f= \int_0^t\int_0^s\int_0^\omega f_{stt}(\xi,\eta) d\xi d\omega'd\eta,$$ $$P_{0}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -f_{t}(\xi,\eta) d\xi d\omega'd\eta,\,P_{1}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -\lambda f(\xi,\eta) d\xi d\omega'd\eta.$$ Then $$\label{aproduct}
H^n=P^nH^0=(P_{-2}+P_{-1}+ P_0+P_1)^nst=\sum_{r=1}^{4^n}R_{r,n}st$$ where $R_{r,n}:=P_{j_{r,n}}P_{j_{r,n-1}}\cdot\cdot\cdot P_{j_{r,1}}$, $j_{r,i} \in \{-2,-1,0,1\}$. Observe that for positive integers $m$ and nonnegative integers $k$ $$\label{aPi}
P_{-1}s^m t^k= c_{-1}s^{m+1}t^{k-1} \; \text{and} \; P_{i}s^m t^k= c_{i}s^{m+2} t^{k+i} \; \text{for} \; i=-2,0,1,$$ where $$\label{acm2}
c_{-2}=
\begin{cases}
0 & \text { if } k\leq 2,\\
-\frac{k(k-1)}{3(m+1)(m+2)} & \text { if } k > 2,\\
\end{cases}$$ $$\label{acm1}
c_{-1}=
\begin{cases}
0 & \text { if } k\leq 1,\\
\frac{k}{(m+1)} & \text { if } k > 1,\\
\end{cases}$$ $$\label{ac0}
c_{0}=-\frac{1}{3(m+1)(m+2)},$$ $$\label{ac1}
c_{1}=-\frac{\lambda}{3(m+1)(m+2)(k+1)}.$$ Let $\sigma=\sigma(n,r)=\sum_{i=1}^n j_{r,i}$. From - one can easily see that for each $n$ and $r$ $$\label{amonomials}
R_{r,n}st=
\begin{cases}
0 & \text { if } \sigma <-1,\\
C_{r,n}s^\beta t^{\sigma+1} & \text { if } \sigma \geq -1\\
\end{cases}$$ where $n+1\leq \beta\leq 2n+1$ and $C_{r,n}$ is a constant which only depends on $n$ and $r$.
Let $\tilde{\lambda}=\max\{1,\lambda\}$. We claim that for each $n$ and $r$, $$\label{aclaim}
|C_{r,n}|\leq \frac{\tilde{\lambda}^n}{(n+1)!(\sigma+1)!}.$$ Taking $m=1$, $k=1$ in -, one can check that the claim holds for $n=1$. Suppose it holds for $n=\ell-1$ and for all $r \in \{1,2,.. ,4^{\ell -1}\}$. Then for $n=\ell$ and $r^* \in \{1,2,.. ,4^{\ell}\}$, using and , we obtain $R_{r^*,\ell}st=P_i R_{r,\ell-1}st= C_{r,\ell-1}P_i s^\beta t^{\sigma+1}=C_{r,\ell-1}c_i s^{\beta^*} t^{\sigma^*+1}$ for some $i\in\{-2,-1,0,1\}$ and $r \in \{1,2,.. ,4^{\ell -1}\}$, where $\beta^*$ is either $\beta+1$ or $\beta+2$, $\sigma^*=\sigma +i$. By the induction assumption $C_{r,\ell-1}\leq \frac{\tilde{\lambda}^{\ell-1}}{\ell!(\sigma+1)!}.$ Moreover - and the fact that $\beta\geq \ell$ imply $|c_i|\leq \frac{\sigma+1}{\ell+1}$ for $i=-1,-2$, $|c_0| <\frac{1}{\ell+1}$, and $|c_1|< \frac{\lambda}{(\sigma+2)(\ell+1)}$. Hence for each $i\in \{-2,-1,0,1\}$ we get $|C_{r^*,\ell}|= |C_{r,(\ell-1)}c_i| \leq \frac{\tilde{\lambda}^{\ell}}{(\ell+1)!(\sigma+i+1)!}=\frac{\tilde{\lambda}^{\ell}}{(\ell+1)!(\sigma^*+1)!}$, which proves that the claim holds for $n=\ell$ as well.
By , , and the fact that $0\leq s, t \leq L$ in the triangle $T_0$, we obtain $$\label{Hnest0}\|H^n\|_{\infty}\leq \frac{4^n\tilde{\lambda}^nL^{3n+2}}{(n+1)!}$$ which is summable. Moreover, since $H^n$ is a linear combination of $4^n$ monomials of the form $s^\beta t^{\sigma+1}$ with $\beta\leq 2n+1$ and $\sigma\leq n$, any partial derivative $\partial^a_s \partial^b_t H^n$ of $H^n$ will be absolutely less than $$\label{Hnest}\displaystyle\frac{(2n+1)^a (n+1)^b 4^n\tilde{\lambda}^nL^{3n+2-a-b}}{(n+1)!}$$ which is also summable.
Now, we define the pseudo-kernel by $$\label{ktilde}\tilde{k}(x,y):=\tilde{G}(x-y,y)$$ and consider the transformation given by $$\label{mod-transform}\tilde{w}(x,t)\equiv u(x,t)-\int_0^x\tilde{k}(x,y)u(y,t)dy.$$
![Pseudo-kernel $\tilde{k}$ when $\lambda=0.01$ ($L=2\pi$)[]{data-label="Impk"}](Impk)
![Control effort at the Dirichlet b.c. for different $\lambda$ ($L=2\pi$)[]{data-label="effortk1y"}](effortk1y)
![Control effort at the Neumann b.c. for different $\lambda$ ($L=2\pi$)[]{data-label="effortky1y"}](effortky1y)
Note that we have $\tilde{u}_x(0,t)=\tilde{w}_x(0,t)$ by the boundary conditions of $\tilde{k}$. Using this fact, we can rewrite the modified target system as $$\label{HomKdVBurgers-1}
\begin{cases}
\displaystyle \tilde{w}_{t} +\tilde{ w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w} = \tilde{k}_y(x,0)\tilde{w}_x(0,t) & \text { in } \Omega\times \mathbb{R_+},\\
\tilde{w}(0,t) = \tilde{w}(L,t) = \tilde{w}_{x}(L,t) = 0 & \text { in } \mathbb{R_+},\\
\tilde{w}(x,0)=\tilde{w}_0(x):= u_0-\int_0^x\tilde{k}(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ Multiplying the above model by $\tilde{w}$ and integrating over $(0,1)$, using the Cauchy-Schwarz inequality, we obtain $$\begin{gathered}
\label{HomKdVBurgers-2}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+\lambda\|\tilde{w}(t)\|_{L^2(\Omega)}^2
\le -\frac{1}{2}|\tilde{w}_x(0,t)|^2+\int_0^L\tilde{k}_y(x,0)\tilde{w}_x(0,t)\tilde{w}(x,t)dx\\
\le \cancel{-\frac{1}{2}|\tilde{w}_x(0,t)|^2}+\cancel{\frac{1}{2}|\tilde{w}_x(0,t)|^2}+\frac{1}{2}\left(\int_0^L|\tilde{k}_y(x,0)||\tilde{w}(x,t)|dx\right)^2.\end{gathered}$$ Since $\tilde{k}$ is smooth on the compact set $\mathcal{T}$, we have $$\label{HomKdVBurgers-3}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+ \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le 0.$$ It follows that $$\label{HomKdVBurgers-4}
\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le \|\tilde{w}_0\|_{L^2(\Omega)}^2e^{-2{\alpha}t} ,$$ where ${\alpha}\equiv \lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2.$ The graph of the function $\tilde{k}_y(\cdot,0)$ is depicted in Figure \[ky0tilde\] on a domain of length $L=2\pi$.
![Pseudo-kernel $\tilde{k}$ when $\lambda=0.03$ ($L=2\pi$)[]{data-label="ky0tilde"}](ky0tilde)
By taking $L^2(\Omega)$ norms of both sides of (with $t=0$) and using the Cauchy-Schwarz inequality, we get $$\label{w0u0} \|\tilde{w}_0\|_{L^2(\Omega)}\le \left(1+\|\tilde{k}\|_{L^2(T)}\right) \|u_0\|_{L^2(\Omega)}.$$
Let $K:H^l(\Omega)\rightarrow H^l(\Omega)$ ($l\ge 0$) be the integral operator defined by $(K\varphi)(x):=\int_0^x\tilde{k}(x,y)\varphi(y)dy.$ It is not difficult to prove that the operator $I-K$ is invertible from $H^l(\Omega)\rightarrow H^l(\Omega)$ (for $l\ge 0$) with a bounded inverse. This is proved in a general setting in the lemma below:
\[inverselem\] $I-K$ is invertible with a bounded inverse from $H^l(\Omega)\rightarrow H^l(\Omega)$ ($l\ge 0$).
The above lemma can be expressed in a sharper form. Indeed, the proof below shows that $\Phi$ is a bounded operator from $L^2(\Omega)\rightarrow H^l(\Omega)$ ($l=0,1,2$) and it is a bounded operator from $H^{l-2}(\Omega)\rightarrow H^{l}(\Omega)$ ($l> 2$).
The above lemma can be proven by slightly modifying the proof of [@Liu03 Lemma 2.4]. However, we will still give a brief proof here since we will need to refer to some crucial details of the proof of this lemma later in the proofs of the stabilization and well-posedness results.
To this end, let us first consider the case $l=0$ and let $\psi=(I-K)\varphi$ for some $\varphi\in L^2(\Omega)$. The idea is to first write $\psi=\varphi-v$ where $v=K\varphi.$ Note that then, $$\psi(x)=\varphi(x)-[K\varphi](x)=(\psi(x)+v(x))-\int_0^x\tilde{k}(x,y)(\psi(y)+v(y))dy.$$ This gives $$v(x)=\int_0^x\tilde{k}(x,y)\psi(y)dy+\int_0^x\tilde{k}(x,y)v(y)dy.$$ Given a fixed $\psi$, one can solve this equation via succession (see [@Liu03 Lemma 2.4] for the details of the succession argument). This implicitly defines a linear operator $\Phi:\psi\mapsto v$ on $L^2(\Omega)$ with the property that $\Phi$ is bounded, i.e., there exists $C_0>0$ such that $$\label{l0}\|v\|_{L^2(\Omega)}\le C_0\|\psi\|_{L^2(\Omega)},$$ where $C_0$ depends only on $\|\tilde{k}\|_{L^\infty(\mathcal{T})}$. But then, $\varphi$ is simply equal to $(I+\Phi)\psi$, and therefore $(I-K)^{-1}$ exists, equals $I+\Phi$, and is bounded. By differentiating and using the smoothness of $\tilde{k}$, $(I-K)^{-1}$ extends to a linear bounded operator also on Sobolev spaces $H^l(\Omega)$ ($l\ge 1$). Indeed, since $\tilde{k}(x,x)=0$, we have $$\label{vxx}v_x(x)=\int_0^x\tilde{k}_x(x,y)(\psi(y)+v(y))dy,$$ which implies $\|v_x\|_{L^2(\Omega)}\le \|\tilde{k}_x\|_{L^2(\mathcal{T})}\left(\|\psi\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\right).$ Hence, using , we have $$\label{l1}\|v\|_{H^1(\Omega)}\le C_1\|\psi\|_{L^2(\Omega)},$$ where $C_1$ depends on $\|\tilde{k}_x\|_{L^2(\mathcal{T})}$ and $C_0$. This shows that $\Phi$ is bounded from $L^2(\Omega)$ into $H^1(\Omega)$, a fortiori bounded from $H^1(\Omega)$ into $H^1(\Omega)$. Now for $l=2$, using $k_x(x,x)=\frac{\lambda}{3}x$, $(\partial_x^2v)(x)=\frac{\lambda}{3}x(\psi(x)+v(x))+\int_0^x(\partial_x^2\tilde{k})(x,y)(\psi(y)+v(y))dy.$ Taking $L^2(\Omega)$ norms of both sides and using the previous inequalities, we get $\|v\|_{H^2(\Omega)}\le C_2\|\psi\|_{L^2(\Omega)},$ where $C_2$ depends on $\|\partial_x^2\tilde{k}\|_{L^2(\mathcal{T})}$, $C_1$, and $\lambda$. This shows that $\Phi$ is bounded from $L^2(\Omega)$ into $H^2(\Omega)$, a fortiori bounded from $H^1(\Omega)$ or $H^2(\Omega)$ into $H^2\Omega)$. Proceeding in the same fashion, one can show that $\|v\|_{H^3(\Omega)}\le C_3\|\psi\|_{H^1(\Omega)},$ where $C_3$ is a fixed constant depending on various norms of $\tilde{k}$. More generally, $\|v\|_{H^l(\Omega)}\le C_l\|\psi\|_{H^{l-2}(\Omega)},$ where $l> 2$ and $C_l$ depends on various norms of $\tilde{k}$. Hence, for $l> 2$, $\Phi$ is a bounded operator from $H^{l-2}(\Omega)$ into $H^l(\Omega)$, and a fortiori bounded from $H^{l}(\Omega)$ into $H^l(\Omega)$.
Another important estimate that follows from via is that $$\label{linftyrem}\|v_x\|_{L^\infty(\Omega)}\le C\|\psi\|_{L^2(\Omega)}$$ for some $C>0$ that depends on $\tilde{k}$.
From the above lemma, it follows in particular that $u(x,t)=[(I-K)^{-1}\tilde{w}](x,t)$, and moreover $$\label{ulessw}\|u(t)\|_{L^2(\Omega)}\le \|(I-K)^{-1}\|_{B[L^2(\Omega)]}\cdot \|\tilde{w}(t)\|_{L^2(\Omega)},$$ where $\|\cdot\|_{B[L^2(\Omega)}$ is the operator norm of $(I-K)^{-1}$ from $L^2(\Omega)$ into $L^2(\Omega)$.
Combining with and , we conclude that
$$\label{linshot}\|u(t)\|_{L^2(\Omega)}\le \left(1+\|\tilde{k}\|_{L^2(T)}\right)\|(I-K)^{-1}\|_{B[L^2(\Omega)]}\,\|u_0\|_{L^2(\Omega)}e^{-\alpha t}.$$
We can prove that the parameter $\alpha$ in the above estimate is positive if $\lambda$ is sufficiently small. Indeed, we have the following lemma.
\[alemlambda\]For a given $L$, there exists sufficiently small $\lambda$ such that $\alpha=\lambda-\frac{1}{2}\|k_y(\cdot, 0)\|^2_{L^2}>0.$
Taking the partial derivative of both sides of with respect to $t$ and taking $i=0$ we see that $\tilde{G}^j_t(s,t)=\frac{\lambda}{3}\sum_{n=0}^{j-1}H^n_t(s,t).$ Passing to the limit we obtain $\tilde{G}_t(s,t)=\frac{\lambda}{3}\sum_{n=0}^{\infty}H^n_t(s,t).$ Note that for $\lambda<1$, $\tilde{\lambda}=1$. Therefore by the summation term is absolutely less than some constant $M$ that only depends on $L$. Hence we get $\|\tilde{G}_t\|_{\infty}\leq\frac{\lambda M}{3}.$ Since $k_y(x,0)=\tilde{G}_t(s,0)$, in particular we have $\|k_y(\cdot, 0)\|^2_{L^2}\leq L \|k_y(\cdot, 0)\|^2_{\infty}\leq L \|\tilde{G}_t\|^2_{\infty}\leq \frac{\lambda^2 M^2 L}{9}.$ As a result, $\alpha=\lambda-\frac{1}{2}\|k_y(\cdot, 0)\|^2_{L^2}\geq \lambda-\frac{\lambda^2 M^2 L}{18}=\lambda^2(\frac{1}{\lambda}-\frac{ M^2 L}{18})$ which is positive for sufficiently small $\lambda$.
The inequality together with Lemma \[alemlambda\] proves the linear part of Theorem \[Linthm\].
Nonlinear model {#ProofofThm2}
---------------
In this section, we consider the nonlinear KdV model with the feedback controllers given in . By using the transformation given in , we obtain the following PDE from , noting that $\tilde{k}(x,x)=0$: $$\label{ch414}
\tilde{w}_{t} + \tilde{w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w}
= \tilde{k}_y(\cdot,0)\tilde{w}_x(0,\cdot)-(I-K)[\left(\tilde{w}+ v\right)\left(\tilde{w}_{x} + v_x\right)]$$ with homogeneous boundary conditions $$\tilde{w}(0,t) = 0 \; , \; \tilde{w}(L,t) = 0, \quad \textrm{and} \quad \tilde{w}_{x}(L,t) = 0,$$ where $v(x,t)=[\Phi\tilde{w}](x,t)$, with $\Phi$ being the linear operator defined in Section \[ProofofThm\] in the proof of Lemma \[inverselem\]. Multiplying by $\tilde{w}(x,t)$ and integrating over $\Omega=(0,L)$, we obtain $$\begin{gathered}
\label{ch416}
\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{t}(x,t)dx = \int_0^L\tilde{k}_y(x,0)\tilde{w}_x(0,t)\tilde{w}(x,t)dx-\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{x}(x,t)dx \\
- \int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{xxx}(x,t)dx
-\lambda\int_{0}^{L}\tilde{w}^{2}(x,t)dx - \int_{0}^{L}\tilde{w}^2(x,t)\tilde{w}_x(x,t)dx - \int_{0}^{L}\tilde{w}^2(x,t)v_x(x,t)dx \\
-\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_x(x,t)v(x,t)dx-\int_{0}^{L}\tilde{w}(x,t)v(x,t)v_x(x,t)dx\\
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{v}_y(y,t)dy\right)\tilde{w}(x,t)dx\\
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{v}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{v}(y,t)\tilde{v}_y(y,t)dy\right)\tilde{w}(x,t)dx.\end{gathered}$$
We estimate the last four terms at the right hand side of as follows: $$\begin{gathered}
\label{four-1}
\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx = \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}(x,y)\frac{\partial}{\partial y}\tilde{w}^2(y,t)dy\right)\tilde{w}(x,t)dx\\
=\frac{1}{2}\int_0^L\left.\tilde{k}(x,y)\tilde{w}^2(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}_y(x,y)\tilde{w}^2(y,t)dy\right)\tilde{w}(x,t)dx\\
\le \frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|\tilde{w}(t)\|_{L^2(\Omega)}^3,\end{gathered}$$
$$\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t){v}_y(y,t)dy\right)\tilde{w}(x,t)dx \le \|\tilde{k}\|_{L^2(T_0)}\|v_x(t)\|_{L^\infty(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2,$$
$$\begin{gathered}
\int_0^L\left(\int_0^x\tilde{k}(x,y){v}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx \\
=\int_0^L\left.\tilde{k}(x,y){v}(y,t)\tilde{w}(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \int_0^L\left(\int_0^x\tilde{k}_y(x,y){v}(y,t)\tilde{w}(y,t)dy\right)\tilde{w}(x,t)dx\\
- \int_0^L\left(\int_0^x\tilde{k}(x,y){v}_y(y,t)\tilde{w}(y,t)dy\right)\tilde{w}(x,t)dx\le \sqrt{L}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2\\
+\|\tilde{k}\|_{L^2(T_0)}\|v_x(t)\|_{L^\infty(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2,\end{gathered}$$
$$\begin{gathered}
\label{four-4}
\int_0^L\left(\int_0^x\tilde{k}(x,y)v(y,t)v_y(y,t)dy\right)\tilde{w}(x,t)dx = \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}(x,y)\frac{\partial}{\partial y}v^2(y,t)dy\right)\tilde{w}(x,t)dx\\
=\frac{1}{2}\int_0^L\left.\tilde{k}(x,y)v^2(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}_y(x,y)v^2(y,t)dy\right)\tilde{w}(x,t)dx\\
\le \frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}^2\|\tilde{w}(t)\|_{L^2(\Omega)}.\end{gathered}$$
Now, estimating the other terms using integration by parts and the Cauchy-Schwarz inequality, and combining these with -, it follows that $$\begin{gathered}
\label{ch417}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2 + \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2 \\
\le \left(\frac{3}{2}+2\|\tilde{k}\|_{L^2(T_0)}\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\|v_x(t)\|_{L^\infty(\Omega)}+\|\tilde{w}(t)\|_{L^2(\Omega)}\|v(t)\|_{L^2(\Omega)}\|v_x(t)\|_{L^\infty(\Omega)}\\
+\frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|\tilde{w}(t)\|_{L^2(\Omega)}^3
+\sqrt{L}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2\\
+\frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}^2\|\tilde{w}(t)\|_{L^2(\Omega)}.\end{gathered}$$
Using and , we deduce the following inequality: $$\label{bernoulli}
y'+2\alpha y-cy^\frac{3}{2}\le 0,$$ where $y(t)\equiv \|\tilde{w}(t)\|_{L^2(\Omega)}^2$, and $c$ is a constant that depends on $L$ and various norms of $\tilde{k}$. Solving the inequality and assuming $\displaystyle \|\tilde{w}_0\|_{L^2(\Omega)}<\frac{\alpha}{c}$, we get $$\label{Nonlinw}
\|\tilde{w}(t)\|_{L^2(\Omega)}^2=y(t)\le \frac{1}{\left[\left(\frac{1}{\|\tilde{w}_0\|_{L^2(\Omega)}}-\frac{c}{2\alpha}\right)e^{\alpha t}+\frac{c}{2\alpha}\right]^2}<\frac{1}{\left[\frac{e^{\alpha t}}{2\|\tilde{w}_0\|_{L^2(\Omega)}}\right]^2}.$$ Recall that $\|\tilde{w}_0\|_{L^2(\Omega)}\lesssim\|u_0\|_{L^2(\Omega)}$. Combining this with and , we deduce $$\|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-\alpha t}, \text{ for } t\ge 0.$$ Hence, the proof of Theorem \[Linthm\] for the nonlinear KdV equation is also complete. Note that the smallness assumption on the initial datum $\tilde{w}_0$ implies a smallness assumption on $u_0$ due to the fact that we also have $\|u_0\|_{L^2(\Omega)}\lesssim\|\tilde{w}_0\|_{L^2(\Omega)}$ thanks to Lemma \[inverselem\].
Well-posedness
==============
In this section, we prove the well-posedness of the PDE models studied in the previous sections. For simplicity, we assume $L=1$ throughout this section. This assumption has no consequence as far as wellposedness is concerned, and all results proved here are also true for any $L>0$. Thanks to Lemma \[lemback\], it is enough to prove the well-posedness of the respective modified target systems in order to obtain well-posedness of and .
Linearised model {#linearised-model}
----------------
Consider the following linear KdV equation with homogeneous boundary conditions. $$\label{KdV-wp}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) & \text { in } \Omega\times \mathbb{R_+},\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } \mathbb{R_+},\\
{y}(x,0)=y_0\in L^2(\Omega) & \text { in } \Omega.
\end{cases}$$ We have the following result.
\[wellposednessprop1\]
1. Let $T'>0$ be arbitrary and $y_0,a\in L^2(\Omega)$. Then, there exists $T\in (0,T')$ independent of the size of $y_0$ such that has a unique local solution $y\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $y_x\in C([0,1];L^2(0,T)).$ Moreover, if $a\in L^\infty(\Omega)$, then $y$ extends as a global solution. In other words, $T$ can be taken as $T'$.
2. Let $a\in H^1(\Omega)$ and let $y_0\in H^3(\Omega)$ satisfy the compatibility conditions $y_0(0)=y_0(1)=y_0'(1)=0.$ Then, the (local/global) solution in part (i) enjoys the extra regularity $y\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$
**Step 1 - Local wellposedness:** Let us define the linear operator $A:D(A)\subset L^2(\Omega)\rightarrow L^2(\Omega)$ by $A\varphi := -\varphi'-\varphi''',$ where $D(A):=\{\varphi\in H^3(\Omega):\varphi(0)=\varphi(1)=\varphi'(1)=0\}.$ Then, the initial boundary value problem can be rewritten in the abstract operator theoretic form $$\label{KdV-wp-abs}
\begin{cases}
\displaystyle \dot{y} = Ay+Fy,\\
{y}(0)=y_0,
\end{cases}$$ where $F\varphi:=-\lambda \varphi + a(\cdot)\gamma_1^0\varphi$. Here, $\gamma_1^0$ is the first order trace operator at the left endpoint, i.e., $\gamma_1^0\varphi:=\varphi'(0)$. This operator is well-defined for $\varphi\in H^{\frac{3}{2}+\epsilon}(\Omega)\supset D(A)$.
It is not difficult to see that the adjoint of $A$ is defined by $A^*\varphi:= \varphi'+\varphi'''$ with $D(A^*):=\{\varphi\in H^3(\Omega):\varphi(0)=\varphi(1)=\varphi'(0)=0\}.$
$A$ is a densely defined closed operator, and moreover, $A$ and $A^*$ are dissipative [@Rosier1997 Proposition 3.1]. Therefore, $A$ generates a strongly continuous semigroup of contractions $\displaystyle\{S(t)\}_{t\ge 0}$ on $L^2(\Omega)$ [@Pazy Corollary I.4.4]. Now we construct the operator $$\label{soloperator}y=[\Psi z](t):= S(t)y_0+\int_0^tS(t-s)Fz(s)ds.$$
Let us define the space (see e.g., [@BSZ03]) $$\label{ourspace}Y_T:=\{z\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))\,|\,z_x\in C([0,1];L^2(0,T))\}$$ equipped with the norm $\|z\|_{Y_T}:=\left(\|z\|_{C([0,T];L^2(\Omega))}^2+\|z\|_{L^2(0,T;H^1(\Omega))}^2+\|z_x\|_{C([0,1];L^2(0,T))}^2\right)^{\frac{1}{2}}.$ Then, for $z\in Y_T$, by using the semigroup estimates [@BSZ03 Prop 2.1, Prop 2.4, Prop 2.16, Prop 2.17], we have $$\begin{gathered}
\label{estimate01}\|y\|_{Y_T}=\|\Psi z\|_{Y_T}\le \|S(t)y_0\|_{Y_T}+\left\|\int_0^tS(t-s)Fz(s)ds\right\|_{Y_T}\\
\le c_0(1+T)^\frac{1}{2}\|y_0\|_{L^2(\Omega)}+c_1(1+T)^\frac{1}{2}\left\|-\lambda z+az_x(0,\cdot)\right\|_{L^1(0,T;L^2(\Omega))}\\
\le c_0(1+T)^\frac{1}{2}\|y_0\|_{L^2(\Omega)}+c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z\|_{Y_T}\,,\end{gathered}$$ where $c_0$ and $c_1$ are positive constants which do not depend on the varying parameters. It follows that $\Psi$ maps $Y_T$ into itself. Now, let $z_1,z_2\in Y_T$ and $y_1=\Psi z_1$, $y_2=\Psi z_2$. By using similar arguments, we have $$\|y_1-y_2\|_{Y_T}=\|\Psi z_1-\Psi z_2\|_{Y_T}\le c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z_1-z_2\|_{Y_T}.$$ Let $T\in (0,T')$ be such that $0< (1+T)^\frac{1}{2}\sqrt{T}< \left(\frac{1}{c_1\left(1+\|a\|_{L^2(\Omega)}\right)}\right).$ Then, $\Psi$ is a contraction on $Y_T$, and this gives us a unique local solution $y\in Y_T$. Here, the size of $T$ is independent of the size of the initial datum. This contrasts with the corresponding nonlinear model in which the size of $T$ is related to the size of the initial datum.
**Step 2 - Global wellposedness:** Let $T_{\max}\le T'$ be the maximal time of existence for the local solution found in Step 1 in the sense that $y\in Y_T$ for all $T<T_{\max}$. In order to prove that $y$ is global, and deduce that $T$ can be taken as $T'$, it is enough to show that $\displaystyle \lim_{T\rightarrow T_{\max}^-}\|y\|_{Y_T}<\infty.$ This will be proved via multipliers, which will be done only formally, but the calculations can always be justified by a density argument which relies on the regularity result in part (ii) of this proposition. To this end, we multiply by $y$ and integrate over $\Omega$ to obtain $$\label{Iden01}
\frac{1}{2}\frac{d}{dt}\|y(t)\|_{L^2(\Omega)}^2+\frac{1}{2}|y_x(0,t)|^2 + \lambda \|y(t)\|_{L^2(\Omega)}^2 = \int_0^1 a(x)y_x(0,t)y(x,t)dx.$$ Using $\epsilon$-Young’s inequality with $\displaystyle\epsilon=\frac{1}{4}$, we have $$\label{Iden02}
\frac{1}{2}\frac{d}{dt}\|y(t)\|_{L^2(\Omega)}^2+\frac{1}{4}|y_x(0,t)|^2 + \lambda \|y(t)\|_{L^2(\Omega)}^2 \le \|a(x)\|_{L^\infty(\Omega)}^2\|y(t)\|_{L^2(\Omega)}^2.$$ Integrating the above inequality over $(0,t)$, we get $$\label{Iden03}
\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt + \le 2\|y_0\|_{L^2(\Omega)}^2+4(\|a(x)\|_{L^\infty(\Omega)}^2-\lambda)\int_0^t\|y(s)\|_{L^2(\Omega)}^2ds.$$ Let $E_0(t):=\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt.$ Then, from , we get $$E_0(t)\le 2\|y_0\|_{L^2(\Omega)}^2+4\left|\|a(x)\|_{L^\infty}^2-\lambda\right|\int_0^tE_0(s)ds.$$ Now, thanks to the Gronwall’s lemma, we have $$\label{Iden04}
E_0(t)=\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt \le 2\|y_0\|_{L^2(\Omega)}^2e^{4\left|\|a(x)\|_{L^\infty}^2-\lambda\right|t},\, t\in [0,T_{\max}).$$ We in particular deduce that $$\label{Firstimpest}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{C([0,T];L^2(\Omega))}\le\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}<\infty.$$ By using , we also deduce that $$\label{Iden05}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{L^2(0,T;L^2(\Omega))}\le \sqrt{2T_{max}}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}.$$ Secondly, we multiply by $xy$ and integrate over $\Omega\times (0,t)$ and get $$\begin{gathered}
\label{Iden06}
\int_{0}^1xy^2(x,s)dx+3\int_0^t\int_0^1y_x^2(x,s)dxds+\lambda\int_0^t\int_0^1xy^2(x,s)dxds\\
= \int_0^1xy_0^2(x)dx+\int_0^t\int_0^1y^2(x,s)dxds+ \int_0^t\int_0^1xay_x(0,s)y(x,s)dxds.\end{gathered}$$ From the above identity, it follows that
$$\label{Iden07}
\|y_x\|_{L^2(0,t;L^2(\Omega))}^2\le \frac{1}{3}\|y_0\|_{L^2(\Omega)}^2+\left(\frac{1}{2}+\frac{\|a\|_{L^\infty(\Omega)}^2}{18}\right)\int_0^tE_0(s)ds.$$
Combining the above inequality with , we deduce that
$$\begin{gathered}
\label{Iden08}
\lim_{T\rightarrow T_{\max}^-}\|y_x\|_{L^2(0,T;L^2(\Omega))}\\
\le \frac{1}{\sqrt{3}}\|y_0\|_{L^2(\Omega)}+\left(\frac{1}{\sqrt{2}}+\frac{\|a\|_{L^\infty(\Omega)}}{3\sqrt{2}}\right)\sqrt{2T_{\max}}\left(\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}\right).\end{gathered}$$
Using and , we deduce that $$\begin{gathered}
\label{Iden09}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{L^2(0,T;H^1(\Omega))}\\
\le \frac{1}{\sqrt{3}}\|y_0\|_{L^2(\Omega)}+\left(1+\frac{1}{\sqrt{2}}+\frac{\|a\|_{L^\infty(\Omega)}}{3\sqrt{2}}\right)\sqrt{2T_{\max}}\left(\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}\right)<\infty.\end{gathered}$$
Since $y$ is the fixed point in , we have $$y=S(t)y_0+\int_0^tS(t-s)Fy(s)ds.$$ Using [@BSZ03 Prop 2.16 and Prop 2.17], we know that the semigroup enjoys the properties $$\label{semi-prop1}
\sup_{x\in \Omega}\left\|\partial_x [S(t)y_0](x)\right\|_{L^2(0,T)}\le c_2\|y_0\|_{L^2(\Omega)}$$ and $$\label{semi-prop2}
\sup_{x\in \Omega}\left\|\partial_x \left[\int_0^tS(t-s)Fy(s)ds\right](x)\right\|_{L^2(0,T)}\le c_3\int_0^T\left\|[Fy](\cdot,t)\right\|_{L^2(\Omega)}dt$$ for some $c_2,c_3>0.$ From the definition of $Fy$ we have $$\|[Fy](\cdot,t)\|_{L^2(\Omega)}\le \lambda\|y(\cdot,t)\|_{L^2(\Omega)}+\|a\|_{L^2(\Omega)}|y_x(0,t)|.$$ Therefore, by and the Cauchy-Schwarz inequality, we have the estimate $$\begin{gathered}
\label{semi-prop3}\int_0^T\left\|[Fy](\cdot,t)\right\|_{L^2(\Omega)}dt\le \lambda\int_0^T\sqrt{E_0(t)}dt+\|a\|_{L^2(\Omega)}\sqrt{T}\sqrt{E_0(T)}\\
\le \left(\lambda T_{\max}+\|a\|_{L^2(\Omega)}\sqrt{T_{\max}}\right)\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}.\end{gathered}$$ Now, it follows from - that $$\begin{gathered}
\lim_{T\rightarrow T_{\max}^-}\|y_x\|_{C([0,1];L^2(0,T))}\\
\le c_2\|y_0\|_{L^2(\Omega)}+c_3\left(\lambda T_{\max}+\|a\|_{L^2(\Omega)}\sqrt{T_{\max}}\right)\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}<\infty.\end{gathered}$$
**Step 3 - Regularity:** Regarding the regular solutions, assume that $y_0\in D(A)$ and consider the following problem: $$\label{KdV-wp-regular}
\begin{cases}
\displaystyle q_{t} +q_{x} + q_{xxx} + {\lambda} q = a(\cdot)q_x(0,\cdot) & \text { in } \Omega\times (0,T),\\
{q}(0,t) = {q}(1,t) = {q}_{x}(1,t) = 0 & \text { in } (0,T),\\
{q}(x,0)=q_0\equiv -y_0'(x)-y_0'''(x)-\lambda y_0(x)+y_0'(0)a(x) & \text { in } \Omega.
\end{cases}$$ Note that $q_0\in L^2(\Omega)$, and we can solve in $Y_T$ as before. Now, we set $y(x,t):=y_0(x)+\int_0^tq(x,s)ds.$ Then, $$\begin{gathered}
\label{regular01}
y_t(x,t)+y_x(x,t)+y_{xxx}(x,t)+\lambda y(x,t)-a(x)y_x(0,t)
=q(x,t)+y_0'(x)+y_0'''(x)+\lambda y_0-y_0'(0)a(x)\\
+\int_0^t\left(q_x(x,s)+q_{xxx}(x,s)+\lambda q(x,t)-a(x)q_x(0,s)\right)ds=0,\end{gathered}$$ and moreover $y(x,0)=y_0$ and $y(0,t)=y(1,t)=y_x(1,t)=0.$ Therefore, $y$ solves . Writing $$y_{xxx}(x,t)=-q(x,t)-y_x(x,t)-\lambda y(x,t)+a(x)y_x(0,t)$$ and taking $L^2(\Omega)$ norms of both sides we get $$\|\partial_x^3 y(t)\|_{L^2(\Omega)}\le \|q(t)\|_{L^2(\Omega)}+\|\partial_x y(t)\|_{L^2(\Omega)}+\lambda\|y(t)\|_{L^2(\Omega)}+|y_x(0,t)|\|a\|_{L^2(\Omega)}.$$ Recall that we have the Gargliardo-Nirenberg inequalities $$\|\partial_x y(t)\|_{L^2(\Omega)}\lesssim \|y\|_{L^2(\Omega)}^\frac{2}{3}\|\partial_x^3y\|_{L^2(\Omega)}^\frac{1}{3}\text{ and }\|\partial_x^2 y(t)\|_{L^2(\Omega)}\lesssim \|y\|_{L^2(\Omega)}^\frac{1}{3}\|\partial_x^3y\|_{L^2(\Omega)}^\frac{2}{3},$$ and the trace inequality (remember that $y_x(1,t)=0$): $|y_x(0,t)|\le \|\partial_x^2 y\|_{L^2(\Omega)}.$
Using these estimates, we get $\|\partial_x^3 y(t)\|_{L^2(\Omega)}\lesssim \|q(t)\|_{L^2(\Omega)}+\|y(t)\|_{L^2(\Omega)}.$ By taking the sup norm with respect to the temporal variable, we deduce that $y\in C([0,T];H^3(\Omega)).$
Similarly, writing out $\partial_x^4y(x,t)=-q_x(x,t)-y_{xx}(x,t)-\lambda y_x(x,t)+a'(x)y_x(0,t),$ using the Gagliardo-Nirenberg and trace inequality, we get $\|\partial_x^4 y(t)\|_{L^2(\Omega)}\lesssim \|q_x(t)\|_{L^2(\Omega)}+\|y_x(t)\|_{L^2(\Omega)}.$ Taking $L^2(0,T)$ norms of both sides we deduce that $y\in L^2(0,T;H^4(\Omega)).$
Global well-posedness of the linearized model now follows from the Proposition \[wellposednessprop1\] that we have just proved.
One can interpolate between part (i) and part (ii) of the above proposition with respect to the smoothness of initial data and get the corresponding well-posedness and regularity result in fractional spaces. For example, let $y_0\in H^s(\Omega)$ ($s\in [0,3]$) so that it satisfies the compatibility conditions $y_0(0)=y_0(1)=0$ if $s\in [0,3/2]$ and the compatibility conditions $y_0(0)=y_0(1)=y_0'(1)=0$ if $s\in (3/2,3].$ Then, with $a=a(x)$ sufficiently smooth, one has $$y\in Y_T^s:= \{\psi\in C([0,T];H^s(\Omega))\cap L^2(0,T;H^{s+1}(\Omega))\,|\,\psi_x\in C([0,1];L^2(0,T))\}.$$ The arguments in Step 3 of the proof of the above proposition can be easily extended to the nonhomogeneous equation $\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot)+f.$ One can first study this equation with $s=0$, $f\in L^1(0,T;L^2(\Omega)),$ and secondly with $s=3$, $f\in W^{1,1}(0,T;L^2(\Omega)).$ Then, by interpolation, for $s\in (0,3)$, one can get $y\in Y_T^s$ if $f\in W^{s/3,1}(0,T;L^2(\Omega))$. Moreover, the following estimates are true: $$\label{YsT}
\|y\|_{Y_{T}^s}\lesssim \|y_0\|_{H^s(\Omega)}+\|f\|_{W^{s/3,1}(0,T;L^2(\Omega))},$$ and for $s=3$, $$\label{YsT3}
\|y_t\|_{Y_{T}}\lesssim \|y_0\|_{H^3(\Omega)}+\|f\|_{W^{1,1}(0,T;L^2(\Omega))}.$$
Nonlinear model {#nonlinear-model}
---------------
Consider the following nonlinear KdV equation with homogeneous boundary conditions. $$\label{nonlinKdV-wp}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) -(I-K)[\left(y+v\right)\left(y_{x} + v_x\right)] & \text { in } \Omega\times \mathbb{R_+},\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } \mathbb{R_+},\\
{y}(x,0)=y_0\in L^2(\Omega) & \text { in } \Omega,
\end{cases}$$ where $v=\Phi(y)$, $\Phi$ being the linear operator defined in Section \[ProofofThm\] in the proof of Lemma \[inverselem\].
\[wellposednessprop2\]
1. Let $T'>0$ be arbitrary and $y_0,a\in L^2(\Omega)$. Then, there exists $T\in (0,T')$ depending on the size of $y_0$ such that has a unique local solution $y\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $y_x\in C([0,1];L^2(0,T)).$ Moreover, if $a\in L^\infty(\Omega)$ and $\|y_0\|_{L^2(\Omega)}$ is sufficiently small, then $y$ extends as a global solution. In other words, $T$ can be taken as $T'$.
2. Let $a\in H^1(\Omega)$ and let $y_0\in H^3(\Omega)$ satisfy the computability conditions $y_0(0)=y_0(1)=y_0'(1)=0.$ Then, the local solution in part (i) enjoys the extra regularity $y\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$
**Step 1 - Local wellposedness:** At first, we set a nonlinear operator $\Upsilon$ as follows: $$\label{soloperator2}y=[\Upsilon z](t):= S(t)y_0+\int_0^tS(t-s)Fz(s)ds,$$ where $Fz:= -\lambda z+ a(\cdot)z_x(0,\cdot)-(I-K)[\left(z+v\right)\left(z_{x} + v_x\right)]$ with $v=\Phi(z).$ Here, we consider $\Upsilon$ on a set given by $S_{T,r}:=\{z\in Y_T,\,\|z\|_{Y_T}\le r\},$ where $Y_T$ is as in . The parameters $T,r>0$ will be determined later. $S_{T,r}$ is a complete metric subspace of $Y_T$ with respect to the metric induced by the norm of $Y_T$. Since $v=\Phi z$, due to we have $$\label{vzrel2}
\|v\|_{C([0,T];L^2(\Omega))}\le C_0\|z(t)\|_{C([0,T];L^2(\Omega))}.$$ Similarly, using we deduce $$\label{vzrel4}
\|v\|_{L^2(0,T);H^1(\Omega))}\le C_1\|z(t)\|_{L^2(0,T);L^2(\Omega))}.$$ Finally,
$$\begin{gathered}
\label{vzrel5}
\|v_x(x)\|_{L^2(0,T)}^2=\int_0^T\left|\int_0^x\tilde{k}_x(x,y)z(y,t)dy\right|^2dt
\le \left(\int_0^1|\tilde{k}_x(x,y)|^2dy\right)\|z\|_{L^2(0,T);L^2(\Omega))}^2\,,
\end{gathered}$$
from which it follows that
$$\label{vzrel6}
\sup_{x\in (0,1)}\|v_x(x)\|_{L^2(0,T)}\\
\le \|z\|_{L^2(0,T);L^2(\Omega))}\sup_{x\in (0,1)}\left(\int_0^1|\tilde{k}_x(x,y)|^2dy\right)^\frac{1}{2}.$$
Combining , , and , we have $$\label{vzrel7}
\|v\|_{Y_T}\le c_{\tilde{k}}\|z\|_{Y_T}\,,$$ where $c_{\tilde{k}}>0$ is a constant which only depends on various finite norms of $\tilde{k}$. Taking the $Y_{T}$ norm of both sides of , using the same semigroup estimates on $Y_T$ and the boundedness of $I-K$, we obtain $$\begin{gathered}
\label{vzrel8}
\|\Upsilon z\|_{Y_T}\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left\|[Fz](\cdot,s)\right\|_{L^2(\Omega)}ds\\
\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left\|a(\cdot)z_x(0,s)-\lambda z-(I-K)[(z+v)(z_x+v_x)]\,\right \|_{L^2(\Omega)}ds\\
\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left[\left\|a(\cdot)z_x(0,s)-\lambda z\right\|_{L^2(\Omega)}+\left\|zz_x\right \|_{L^2(\Omega)}+\left\|zv_x\right \|_{L^2(\Omega)}+\left\|vz_x\right \|_{L^2(\Omega)}+\left\|vv_x\right \|_{L^2(\Omega)}\right]ds\\
\le c_0\|y_0\|_{Y_T}
+c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z\|_{Y_T}+(T^\frac{1}{2}+T^\frac{1}{3})
\left(\|z\|_{Y_T}^2+2\|z\|_{Y_T}\|v\|_{Y_T}+\|v\|_{_{Y_T}}^2\right)\right]\\
\le c_0\|y_0\|_{Y_T}+c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})+(1+3c_{\tilde{k}})(T^\frac{1}{2}+T^\frac{1}{3})
\|z\|_{Y_T}\right]\|z\|_{Y_T},
\end{gathered}$$ where the fourth inequality follows from [@BSZ03 Lemma 3.1]. Let us set $r=2c_0\|y_0\|_{Y_T}$, and choose $T>0$ to be small enough that $$c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})+(1+3c_{\tilde{k}})(T^\frac{1}{2}+T^\frac{1}{3})
r\right]\le\frac{1}{2}.$$ With such choice of $(r,T)$, we get $\|\Upsilon z\|_{Y_T}\le r$ for all $z\in S_{T,r}$. Therefore, $\Upsilon$ is a map from $S_{T,r}$ into $S_{T,r}$.
Now, we claim that $\Upsilon$ is indeed a contraction on $S_{T,r}$ if $T$ is sufficiently small. In order to see this, let $z,z'\in S_{T,r}.$ Then, similar to , we have $$\begin{gathered}
\label{vzrel9}
\|\Upsilon z-\Upsilon z'\|_{Y_T}\le c_1\int_0^{T}\left\|[Fz-Fz'](\cdot,s)\right\|_{L^2(\Omega)}ds\\
\le c_1\int_0^{T}\left\|a(\cdot)(z_x(0,s)-z'_x(0,s))-\lambda (z-z')\right\|_{L^2(\Omega)}ds\\
+c_1\int_0^T\left[\left\|zz_x-z'z'_x\right \|_{L^2(\Omega)}+\left\|zv_x-z'v'_x\right \|_{L^2(\Omega)}+\left\|vz_x-v'z'_x\right \|_{L^2(\Omega)}+\left\|vv_x-v'v'_x\right \|_{L^2(\Omega)}\right]ds\\
\le c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z-z'\|_{Y_T}+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z\|_{Y_T}+\|z'\|_T)\|z-z'\|_{Y_T}\\
+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z'\|_{Y_T}\|v-v'\|_{Y_T}+\|v\|_T\|z-z'\|_{Y_T})+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z\|_{Y_T}\|v-v'\|_{Y_T}+\|v\|_{Y_T}\|z-z'\|_{Y_T})\\
+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|v\|_{Y_T}+\|v'\|_{Y_T})\|v-v'\|_{Y_T}.
\end{gathered}$$
Now, using , for the same $r$ as before, but choosing $T$ smaller if necessary, we obtain $$\|\Upsilon z-\Upsilon z'\|_{Y_T}\le\rho \|z-z'\|_{Y_T}$$ for some $\rho\in (0,1)$. Then, by the Banach contraction theorem, we get the existence and uniqueness of a local solution in $S_{T,r}$.
**Step 2 - Regularity:** Let $y_0\in D(A)$. We define the closed space $$B_{T,r}:=\{(\psi,\varphi)\in Y_T^3\times Y_T\,|\,\psi\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega)), \varphi=\psi_t, \|\psi\|_{Y_T^3}+\|\varphi\|_{Y_T}\le r\}.$$ Now, given $(z,\tilde{z})\in B_{T,r}$ let $q$ be a solution of
$$\label{nonlinKdV-wp-q-2}
\begin{cases}
\displaystyle q_{t} +q_{x} + q_{xxx} + {\lambda} q \\
= a(x)q_x(0,\cdot) - (I-K)[(\tilde{z}+\tilde{v})(z_x+v_{x})-(z+v)(\tilde{z}_{x}+\tilde{v}_{x})] & \text { in } \Omega\times (0,T),\\
{q}(0,t) = {q}(1,t) = {q}_{x}(1,t) = 0 & \text { in } (0,T),\\
{q}(x,0)= q_0:=-y_0'-y_0'''-\lambda y_0+a(x)y_0'(0)-(y_0+v_0)(y_0'+v_0') & \text { in } \Omega,
\end{cases}$$
where $v=\Phi(z)$, $v_0=\Phi(y_0)$, $\tilde{v}=\Phi(\tilde{z})$. Set $y=y_0+\int_0^tqds$. Then, $y_t=q$ and $y$ solves
$$\label{nonlinKdV-wp-q-1}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) - (I-K)[(z+v)(z_x+v_x)] & \text { in } \Omega\times (0,T),\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } (0,T),\\
{y}(x,0)=y_0 & \text { in } \Omega.
\end{cases}$$
We set an operator $\Theta: (z,\tilde{z})\mapsto (y,q)$ associated with the system of equations given by -. One can show that for suitable $r$ and small $T$, the operator $\Theta$ maps $B_{T,r}$ onto itself in a contractive manner. This can be done by obtaining the same type of estimates given in Step 1 for both the solution of and . Therefore, it has a unique fixed point whose first component is the regular local solution we are looking for.
**Step 3 - Global solutions:** Global wellposedness in $Y_T$ with small initial datum follows directly from the stabilization estimate proved in Section \[ProofofThm2\].
Global well-posedness of the nonlinear modified target system now follows from the Proposition \[wellposednessprop2\] that we just proved.
Using a single controller
=========================
Smaller decay rate
------------------
Although we studied the model with two controls at the right hand side, it is also possible to use only one control. For example [@Cor14] proves exponential stability with the control acting only from the Neumann boundary condition when $L$ is not of critical length. When $L$ is not restricted to uncritical lengths, we can still obtain exponential stability with a single Dirichlet control rather than a Neumann control by using the pseudo-backstepping method above. However, this causes a smaller rate of decay. Consider for instance the plant $$\label{singlecontrol}
\begin{cases}
\displaystyle u_{t} + u_{x} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = U(t), u_{x}(L,t)=0 & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega.
\end{cases}$$ Then the backstepping transformation gives the following target system $$\label{HomKdVBurgers-single-1}
\begin{cases}
\displaystyle \tilde{w}_{t} +\tilde{ w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w} = \tilde{k}_y(x,0)\tilde{w}_x(0,t) & \text { in } \Omega\times \mathbb{R_+},\\
\tilde{w}(0,t) = \tilde{w}(L,t) = 0, \tilde{w}_{x}(L,t) = -\int_0^L\tilde{k}_x(L,y)u(y,t)dy & \text { in } \mathbb{R_+},\\
\tilde{w}(x,0)=\tilde{w}_0(x):= u_0-\int_0^x\tilde{k}(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ If we multiply the above system by $\tilde{w}$, integrate over $(0,L)$, and use integration by parts, the Cauchy-Schwarz inequality, and boundary conditions we obtain $$\label{HomKdVBurgers-single-2}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+ \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2-\frac{1}{2}\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)}^2\|(I-K)^{-1}\|_{B[L^2(\Omega)]}^2 \right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le 0.$$ Comparing and we see that we still achieve where $\alpha= \lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2$ is replaced by $\beta=\left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2-\frac{1}{2}\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)}^2\|(I-K)^{-1}\|_{B[L^2(\Omega)]}^2 \right)$. Recall that in Lemma \[alemlambda\] we showed $\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)} \sim \lambda $. One can also get $\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)} \sim \lambda $ by using similar arguments. Moreover, using the calculations in [@Liu03] we deduce that $\|(I-K)^{-1}\|_{B[L^2(\Omega)]}\sim 1+\lambda e^{C\lambda}$ where $C>0$ depends only on $L$. Hence positivity of $\beta$ is guaranteed for a sufficiently small choice of $\lambda$. As a result, the decay rate decreases while the exponential stability still holds.
Glass and Guerrero [@Glass10] proved that is exactly controllable if and only if $L$ does not belong to a set of critical lengths $\mathcal{O}$, defined by them, which is different than $\mathcal{N}$. They showed that if $L\in \mathcal{O}$, then the following problem has a nontrivial solution: $$\label{phieq1}
\begin{cases}
\varphi'''+\varphi'=\lambda \varphi & \text { in } (0,L),\\
\varphi(0) = \varphi(L) = \varphi'(0)=\varphi''(L)=0.
\end{cases}$$ Moreover, it was found that for any $u_0\in L^2(\Omega)$ and control input $U\in L^2(0,T)$, the function $$\label{ftcons}
e^{-\lambda t}\int_0^Lu(x,t;U)\varphi(x)dx$$ remains constant in time, where $u=u(x,t;U)$ is the corresponding trajectory for with control input $U$. Regarding the same KdV system, we construct a boundary feedback which yields stabilizability with a certain decay rate for all $L>0$ including $L\in \mathcal{O}$. This shows that for a certain control input $U$, the integral $\int_0^Lu(x,t;U)\varphi(x)dx$ decays in time because $\left|\int_0^Lu(x,t;U)\varphi(x)dx\right|\le \|u(t;U)\|_{L^2(\Omega)}\|\varphi\|_{L^2(\Omega)}$, where $\|u(t;U)\|_{L^2(\Omega)}$ decays exponentially. Therefore, since is invariant in time, it follows that $|e^{-\lambda t}|$ must increase, which is only possible if $Re(\lambda)<0$. However, if $Re(\lambda)<0$, then $|e^{-\lambda t}|\rightarrow \infty$. Since is valid for all control inputs, we conclude in particular that $\int_0^Lu(x,t;0)\varphi(x)dx\rightarrow 0$ $(U(t)\equiv 0$). This shows that any uncontrollable trajectory corresponding to some initial state $u_0$ with no feedback is attracted to the orthogonal complement of the span of the set consisting of the real and imaginary parts of the nontrivial solutions of .
Regarding the nonlinear KdV equation with same boundary conditions, Glass and Guerrero [@Glass10] obtained the exact controllability given that the domain is of uncritical length and the initial and final states are small. This was not an if and only if statement unlike the linear problem. Therefore, we do not know whether the exact controllability on domains of critical lengths is true with the control acting at the right Dirichlet b.c. On the other hand, our feedback control design can be easily extended to the nonlinear system for small data as in Section 2.2. The nonlinear case is quite interesting because maybe the nonlinear term $uu_x$ is creating a further stability effect on the solutions, which might drive them to zero by themselves. As far as we know, the exact controllability for the (nonlinear) KdV equation as well as the decay of solutions to zero by themselves in the presence of the nonlinear term $uu_x$ remain as open problems on critical length domains, see for instance Cerpa [@cer14].
A second order feedback law
---------------------------
In this section, we check whether it is possible to stabilize the solutions of the KdV equation by using the feedback law $u(L,t)=U(t)$ with the input $U(t)=u_{xx}(L,t)$. In order to gain some intuition regarding this problem, let us consider the linearized model with the input $U(t)=u_{xx}(L,t)$. Multiplying the main equation in by $u$ and integrating over $\Omega$ by using the given boundary conditions, one obtains $$\label{newL2iden}
\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2 = -\frac{3}{2}|u(L,t)|^2-\frac{1}{2}|u_x(0,t)|^2\le 0.$$ The inequality shows that $\|u(t)\|_{L^2(\Omega)}$ is non-increasing, but it is not clear whether it decays to zero. In order to better understand the behavior of the solution, one generally must study the spectral properties of the corresponding evolution. Regarding , one can study the operator $$Au=-u'''-u',\,\,\,D(A)=\{u\in H^3(\Omega)\,|\,u(0)=u'(L)=u(L)-u''(L)=0\}.$$ However, it is quite difficult to analyze the eigenvalues of this operator. This is because the characteristic equation corresponding to the eigenvalue problem $Au=\lambda u$ takes the form $r^3+r+\lambda =0$, which is not easy to study. Therefore, we will consider this problem on the rather simplified model given below, neglecting the first order term $u_x$: $$\label{simplemodel}
\begin{cases}
\displaystyle u_{t} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = u_{xx}(L,t), u_{x}(L,t)=0 & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega,
\end{cases}$$ where the inequality takes the form $$\label{newL2iden2}
\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2 = -|u(L,t)|^2-\frac{1}{2}|u_x(0,t)|^2\le 0.$$
Spectral properties {#spectral-properties .unnumbered}
-------------------
The operator which generates the evolution corresponding to is a third order dissipative differential operator given by $$Au=-u''',\,\,\,D(A)=\{u\in H^3(\Omega)\,|\,u(0)=u'(L)=u(L)-u''(L)=0\},$$ which has compact resolvent and spectrum involving countably many eigenvalues $\{\lambda_k\}_{k\in \mathbb{Z}}$ satisfying $\text{Re}\lambda_k\le 0$. Moreover, these eigenvalues satisfy the properties given in the lemma below, whose proof uses the approach presented in [@Zhang001 Prop 2.2] and [@Zhang002 Prop 3.1].
$\lambda_k = -\frac{8\pi^3}{3\sqrt{3}L^3}|k|^3 $ as $|k|\rightarrow \infty$. Moreover, $\exists \eta<0$ s.t. $\text{Re}\lambda_k<\eta$ $\forall k\in \mathbb{Z}$.
Let $\lambda$ be an eigenvalue of $A$. Then, $\text{Re}\lambda \le 0$, and we can assume wlog that $\text{Im}\lambda\le 0$ since $\bar{\lambda}$ is also an eigenvalue of $A$. Let us first see that $\text{Re}\lambda$ cannot be equal to zero. To this end, let $i\xi$ be an eigenvalue with $\xi\in \mathbb{R}$ and $u$ be the corresponding eigenvector. Then, we note that $$\text{Re}(Au,u)_{L^2(\Omega)}=-|u|^2(L)-\frac{1}{2}|u'(0)|^2=\text{Re}(i\xi\|u\|_{L^2(\Omega)}^2)=0.$$ We get $u(L)=u'(0)=0$, which implies together with other boundary conditions $u\equiv 0$. This contradicts the fact that $u$ was an eigenvector. Hence, $\text{Re}\lambda<0$.
Let $r_i$ $(i=0,1,2)$ be the three roots of the characteristic equation $r^3+\lambda=0$ corresponding to the ode $$\label{Aulambdau}
Au=\lambda u,$$with $r_1$ being the root in the first quadrant. Note that we have $r_i=\alpha^i r_1$, $\alpha=e^{\frac{2\pi i}{3}}$. The solution of is $u(x)=\sum_{i=0}^2c_ie^{r_i x}$ where $\{c_i, i=0,1,2\}$, due to given boundary conditions, satisfy the system of equations $$\begin{cases} \sum_{i=0}^2c_i =0 \\ \sum_{i=0}^2c_ir_ie^{r_i L} =0 \\ \sum_{i=0}^2 c_i(1-r_i^2e^{r_i L})=0, \end{cases}$$ which has a nontrivial solution if $$\sum_{i=0}^2a_{ij}r_i(1-r_j^2)e^{(r_i+r_j)L}=0,$$ where $a_{12}=a_{20}=a_{01}=1$ and $a_{21}=a_{02}=a_{10}=-1$. Multiplying the above equation by $e^{-r_0L}$ and neglecting the relatively small terms which involve $e^{(r_1+r_2-r_0)L}$, we get $$(1+\alpha)(1+\alpha^2r_0^2)e^{r_2L}+(1+\alpha r_0^2)e^{r_1L}=0.$$ Now, we see that we can further neglect the terms $(1+\alpha)e^{r_2L}$ and $e^{r_1L}$ since they are much smaller than $(1+\alpha)\alpha^2r_0^2e^{r_2L}$ and $\alpha r_0^2e^{r_1L}$, respectively. Therefore, asymptotically, we get $$(1+\alpha)\alpha e^{r_2L}+e^{r_1L}=0.$$ Observe that $\alpha(1+\alpha)=-1$. Therefore, the asymptotic relation is reduced to $$e^{(r_2-r_1)L}=1,$$ from which, together with the definition of $\alpha,r_1$, and $r_2$, it follows that $\lambda_k=-r_{0,k}^3=-\frac{8\pi^3}{3\sqrt{3}L^3}|k|^3$ asymptotically. This property combined with the fact that $\text{Re}\lambda_k<0$ proves the second part of the lemma.
It is easy to see that $A^*$ is defined by $$A^*w=w''',\,\,\,D(A^*)=\{w\in H^3(\Omega)\,|\,w(0)=w'(0)=w(L)+w''(L)=0\}.$$ Now, the following result follows classically from the spectral properties of $A$ above.
([@Zhang001 Prop 2.1, 2.2], [@Zhang002 Prop 3.2]) $A$ is a discrete spectral operator, and all but a finite number of eigenvalues of $A$ correspond to one dimensional projections $E(\lambda;T)$. Both $A$ and $A^*$ have complete sets of eigenvectors, $\{\phi_k\}_{k\in \mathbb{Z}}$ and $\{\psi_j\}_{j\in \mathbb{Z}}$, respectively, satisfying $(\phi_k,\psi_j)_{L^2(\Omega)}=\delta_{kj}$ and forming dual Riesz bases for $L^2(\Omega)$.
A special multiplier and stabilization {#a-special-multiplier-and-stabilization .unnumbered}
--------------------------------------
We set $Y:=\sum_{k}Y_k$ where $Y_k$ is defined by $$Y_k(u)=(u,\psi_k)_{L^2(\Omega)}\psi_k.$$ Then $Y$ is bounded and positive definite by the uniform $\ell^2$ convergence property of $\{\phi_k\}_{k\in \mathbb{Z}}$ and $\{\psi_j\}_{j\in \mathbb{Z}}$. Second, we define the symmetric, bounded, nonnegative operator $X=\sum_{k}\xi_kY_k$, where $\xi_k=-\frac{1}{2\text{Re}\lambda_k}$, which satisfies the additional property $$A^*X+XA+Y=0.$$ Using $Xu$ as a multiplier, we compute $$\label{XYcomp}
\frac{d}{dt}(Xu,u)_{L^2(\Omega)} = (\frac{d}{dt}Xu,u)_{L^2(\Omega)}+(Xu,u_t)_{L^2(\Omega)}
=(XAu,u)_{L^2(\Omega)}+(Xu,Au)_{L^2(\Omega)}
=-(Yu,u)_{L^2(\Omega)}.$$
together with imply that $$\label{gronprep1}
\frac{d}{dt}((I+X)u,u)_{L^2(\Omega)}\le -(Yu,u)_{L^2(\Omega)}.$$ Now, integrating in time and using the positive definiteness of $(I+X)$ and $Y$, applying Gronwall’s inequality, we obtain the exponential decay of solutions in $L^2(\Omega)$, and the following theorem follows.
\[specthm\] Let $u$ be a solution of the linearized KdV equation in . Then, there exists some $\gamma>0$ independent of $u_0$ such that $$\|u(t)\|_{L^2(\Omega)} \lesssim \|u_0\|_{L^2(\Omega)}e^{-\gamma t}$$ for $t\ge 0$.
Note that Theorem \[specthm\] was proved for the simplified linearized model . The situation is more challenging for more general models involving other terms such as $u_x$ and/or the nonlinear term $uu_x$. Moreover, the approaches of [@Zhang001] and [@Zhang002] do not seem to directly apply to these more general problems under the boundary conditions $u(0)=u'(L)=u(L)-u''(L)=0$. This is due to the fact that the eigenvalue analysis gets much more challenging with a more complicated third order characteristic equation. In order to simplify the eigenvalue analysis, one can still use the simpler operator $Au=-u'''$ treating $-u_x$ and/or $uu_x$ as source term(s). But then, one needs the adjoint operator $A^*$ to satisfy very desirable boundary conditions so that the trace terms are cancelled out when one applies the special multiplier. However, this does not become the case with the given boundary conditions in the model. This issue is not present with the boundary conditions used in [@Zhang001] and [@Zhang002]. Therefore, the case of more general equations with the first order term $u_x$ and/or the nonlinear term $uu_x$ remain interesting open problems.
Numerical simulations
=====================
We modify the finite difference scheme given in [@Pazato] to fit it into the present situation, where we have first order trace terms in the main equations of the target systems and inhomogeneous boundary inputs of feedback type in the original plant. We numerically solve the KdV equation both in the controlled and uncontrolled cases. We are also able to verify our main result numerically. First, we simulate an uncontrolled solution of the KdV equation and then we simulate the controlled solution. From our simulations, one can see that the boundary controllers constructed using a pseudo-kernel effectively stabilize the solutions with a suitable choice of $\lambda$. The calculations are performed in Wolfram Mathematica^^11.
For simplicity, we consider only the linearised problem. The nonlinear problem can be treated in a similar way by including an additional fixed point argument to the algorithm we describe here. We use the notation given in [@Pazato]. To this end, we set the discrete space $$X_J:=\{\tilde{w}=(\tilde{w}_0,\tilde{w}_1,...,\tilde{w}_J)\in \mathbb{R}^{J+1}\,|\,\tilde{w}_0=\tilde{w}_{J-1}=\tilde{w}_J=0\},$$ and the difference operators $\displaystyle (D^+\tilde{w})_j:=\frac{\tilde{w}_{j+1}-\tilde{w}_j}{\delta x}$, $\displaystyle (D^-\tilde{w})_j:=\frac{\tilde{w}_{j}-\tilde{w}_{j-1}}{\delta x}$ for $j=1,...,J-1$, and $\displaystyle D=\frac{1}{2}(D^++D^-)$. We will call the space and time steps $\delta x$ and $\delta t$ for $j=0,...,J,$ and $n=0,1,...,N$, respectively. Using this notation, the numerical approximation of the linearised target system takes the form $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{wjn1}\frac{\tilde{w}_{j}^{n+1}-\tilde{w}_j^n}{\delta t}+(\mathcal{A}\tilde{w}^{n+1})_j+\lambda \tilde{w}_j^{n+1}&=& \tilde{k}_y(x_j,0)\frac{\tilde{w}_{1}^{n}}{\delta x},\hspace{.1in} j=1,...,J-1\\
\label{wjn2}\tilde{w}_0=\tilde{w}_{J-1}=\tilde{w}_J &=& 0, \\
\label{wjn3} \tilde{w}_0 &=&\int_{x_{j-\frac{1}{2}}}^{x_{j^+\frac{1}{2}}}\tilde{w}_0(x)dx,\hspace{.1in} j=1,...,J-1,\end{aligned}$$ where $x_{j\mp\frac{1}{2}}=(j\mp\frac{1}{2})\delta x$, $x_j=j\delta x$. The $(J-1)\times (J-1)$ matrix $\mathcal{A}$ approximates $\tilde{w}_x+\tilde{w}_{xxx}$ and is defined by $\mathcal{A}:=D^+D^+D^-+D$. Let us set $\mathcal{C}:=(1+\delta t\lambda)I+\delta t A$. Then, from the main equation, we obtain $\tilde{w}_{j}^{n+1}=\mathcal{C}^{-1}\left(\tilde{w}_j^n+\frac{\delta t}{\delta x}\tilde{k}_y(x_j,0)\tilde{w}_{1}^{n}\right)$ for $j=1,...,J-1$.
In order to approximate the solution of the original plant with feedback controllers, we use the succession idea in the proof of Lemma \[inverselem\]. Note that given $\tilde{w}$, $v$ is the fixed point of the equation $v=K(\tilde{w}+v)$. For numerical purposes, let $m$ denote the number of iterations in the succession and set $v^0=\mathcal{K}\tilde{w}$, $v^{k}:=\mathcal{K}(\tilde{w}+v^{k-1})$ for $1\le k\le m$, where $\mathcal{K}$ is the numerical approximation of the integral in the definition of $K$. Then, $v^m$ is an approximation of $v=\Phi(\tilde{w})$, and one gets an approximation of the original plant by setting $u(x_j,t_n):=\tilde{w}(x_j,t_n)+v^m(x_j,t_n)$.
On a domain of critical length, one can find time-independent solutions, as we mentioned in the introduction. Figure \[uncont-sol\] below shows such a solution on a domain of length $L=2\pi$ whose $L^2$-norm is preserved in time.\
![Uncontrolled solution with initial datum $u_0=1-\cos(x)$ on a domain of length $2\pi$.[]{data-label="uncont-sol"}](uncont-sol)
If one applies the boundary controllers constructed with the same initial profile that the uncontrolled solution has in Figure \[uncont-sol\], then the new solution will decay to zero as we illustrate in Figure \[controlled\].
![Controlled solution with initial datum $u_0=1-\cos(x)$, $\lambda=0.03$, with a controller using the pseudo-kernel $\tilde{k}$ on a domain of length $2\pi$.[]{data-label="controlled"}](controlled)
Figure \[u1t\] shows the controller behavior on the Dirichlet boundary condition at the right endpoint. As one can see, less control is needed as the wave gets supressed.\
![Dirichlet controller at the right endpoint ($\lambda=0.03$)[]{data-label="u1t"}](u1t "fig:")\
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to express our gratitude to the anonymous referee whose valuable insights significantly improved the quality of this article. We would also like to thank Katherine H. Willcox from Izmir Institute of Technology for her English editing of this paper.
| ArXiv |
---
abstract: 'Scintillating CaWO$_4$ single crystals are a promising multi-element target for rare-event searches and are currently used in the direct Dark Matter experiment CRESST (Cryogenic Rare Event Search with Superconducting Thermometers). The relative light output of different particle interactions in CaWO$_4$ is quantified by Quenching Factors (QFs). These are essential for an active background discrimination and the identification of a possible signal induced by weakly interacting massive particles (WIMPs). We present the first precise measurements of the QFs of O, Ca and W at mK temperatures by irradiating a cryogenic detector with a fast neutron beam. A clear energy dependence of the QFs and a variation between different CaWO$_4$ single crystals were observed for the first time. For typical CRESST detectors the QFs in the region-of-interest (10-40keV) are $QF_O^{ROI}=(11.2{\pm}0.5)$%, $QF_{Ca}^{ROI}=(5.94{\pm}0.49)$% and $QF_W^{ROI}=(1.72{\pm}0.21)$%. The latest CRESST data (run32) is reanalyzed using these fundamentally new results on light quenching in CaWO$_4$ having moderate influence on the WIMP analysis. Their relevance for future CRESST runs and for the clarification of previously published results of direct Dark Matter experiments is emphasized.'
author:
- 'R. Strauss'
- 'G. Angloher'
- 'A. Bento'
- 'C. Bucci'
- 'L. Canonica'
- 'A. Erb'
- 'F.v.Feilitzsch'
- 'P.Gorla'
- 'A. Gütlein'
- 'D. Hauff'
- 'J. Jochum'
- 'H. Kraus'
- 'J.-C. Lanfranchi'
- 'J. Loebell'
- 'A. Münster'
- 'F.Petricca'
- 'W. Potzel'
- 'F. Pröbst'
- 'F. Reindl'
- 'S. Roth'
- 'K. Rottler'
- 'C. Sailer'
- 'K. Schäffner'
- 'J.Schieck'
- 'S. Scholl'
- 'S. Schönert'
- 'W. Seidel'
- 'M.v.Sivers'
- 'L. Stodolsky'
- 'C. Strandhagen'
- 'A.Tanzke'
- 'M. Uffinger'
- 'A. Ulrich'
- 'I. Usherov'
- 'S. Wawoczny'
- 'M. Willers'
- 'M. Wüstrich'
- 'A. Zöller'
- 'W. Carli'
- 'C. Ciemniak'
- 'H. Hagn'
- 'D. Hellgartner'
bibliography:
- 'quenching\_final.bib'
title: 'Precision Measurements of Light Quenching in CaWO$_4$ Crystals at mK Temperatures'
---
Rare-event searches for Dark Matter (DM) in the form of weakly interacting massive particles (WIMPs) [@Bertone:2004pz; @Jungman:1995df] have reached impressive sensitivities during the last decade [@Cushman:2013zza]. Well motivated WIMP candidates with masses $m_\chi$ between a few GeV/$c^2$ and a few TeV/$c^2$ might be detectable via nuclear recoils of few keV in terrestrial experiments [@Lewin:1995rx]. While the DAMA/LIBRA [@Bernabei:2010mq], and recently the CoGeNT [@Aalseth:2010vx], CRESST [@Angloher:2012vn], and the CDMS(Si) [@PhysRevLett.111.251301] experiments observed excess signals that might be interpreted as induced by DM particles with $m_\chi{\sim10}$GeV/$c^2$ at WIMP-nucleon cross-sections of ${\sim}10^{-4}$pb, this scenario is ruled out by the LUX [@Akerib:2013tjd] and XENON100 [@Aprile:2012nq] experiments, and almost excluded by the CDMS(Ge) [@Ahmed:2009zw; @Ahmed:2010wy] and EDELWEISS [@Armengaud:2011cy; @Armengaud:2012pfa] experiments. It is strongly disfavoured by accelerator constraints [@ATLAS:2012ky; @Chatrchyan:2012me] and in mild tension with an extended analysis [@PhysRevD.85.021301] of published CRESST data [@Angloher2009270].\
The CRESST experiment [@Angloher:2012vn] employs scintillating CaWO$_4$ crystals [@edison; @PhysRevB.75.184308] as a multi-element target material. The key feature of a CRESST detector module is the simultaneous measurement of the recoil energy $E_r$ by a particle interaction in the crystal (operated as cryogenic calorimeter at mK temperatures [@Probst:1995fk]) and the corresponding scintillation-light energy $E_l$ by a separate cryogenic light absorber. Since the relative light yield $LY{=}E_l/E_r$ is reduced for highly ionizing particles compared to electron recoils (commonly referred to as quenching) nuclear-recoil events can be discriminated from e$^-$/$\gamma$ and $\alpha$ backgrounds. The phenomenological Birks model [@birks1964theory] predicts this quenching effect to be stronger the higher the mass number $A$ of the recoiling ion, which allows to distinguish, in general, between O ($A{\approx}16$), Ca ($A{\approx}40$) and W ($A{\approx}184$) recoils. The expected WIMP-recoil spectrum - assuming coherent scattering - is completely dominated by W-scatters for $m_\chi\,\gtrsim 20$GeV/c$^2$. However, the light targets O and Ca make CRESST detectors particularly sensitive to low-mass WIMPs of $1$GeV$\,\lesssim\, m_\chi\,\lesssim$20GeV. Furthermore, the knowledge of the recoil composition of O, Ca and W allows a test of the assumed $A^2$-dependence of the spin-independent WIMP-nucleon cross-section [@Jungman:1995df]. In addition, background neutrons, which are mainly visible as O-scatters (from kinematics [@scholl_paper]), can be discriminated statistically.\
The mean LY of e$^-$/$\gamma$ events ($LY_{\gamma}$) is energy dependent and phenomenologically parametrized as ${LY_{\gamma}(E_r){=}(p_0+p_1E_r)(1-p_2\exp(-E_r/p_3)}$ [@strauss_PhD]. By convention, $LY_{\gamma}$(122keV) is normalized to unity. The parameters $p_{0}$, $p_1$, $p_2$ and $p_3$ are derived from a maximum-likelihood (ML) fit for every detector module individually. For the module used in this work the fit yields: $p_0\,{=}\,1.07$, $p_1\,{=}\,{-}1.40\cdot 10^{-5}$keV$^{-1}$, $p_2\,{=}\,6.94\cdot 10^{-2}$ and $p_3\,{=}\,147$keV (errors are negligible for the following analysis). The exponential decrease towards lower recoil energies (quantified by $p_2$ and $p_3$) accounts for the scintillator non-proportionality [@Lang:2009uh]. The Quenching Factor (QF) of a nucleus $x$ - in general energy dependent - is defined as $QF_x(E_r)=LY_x(E_r)/LY_{\gamma,corr}(E_r)$ where $LY_x$ is the mean LY of a nuclear recoil x. For normalization, the LY of e$^{-}$/$\gamma$ events corrected for the scintillator non-proportionality (which is not observed for nuclear recoils) is used by convention: $LY_{\gamma,corr}=p_0+p_1E_r$. For typical CRESST detector modules, the uncertainties in energy and LY are well described by gaussians [@Angloher:2012vn] consistent with photon-counting statistics in the energy range considered in this work.\
Since the resolution of light-detectors operated in the CRESST setup at present is not sufficient to disentangle O, Ca and W recoils unambiguously, dedicated experiments to measure the QFs of CaWO$_4$ are necessary. Earlier attempts yield inconclusive results, in particular for the value of $QF_W$ [@Jagemann:2006sx; @Ninkovic:2006xy; @Bavykina:2007ze].\
At the accelerator of the Maier-Leibnitz-Laboratorium (MLL) in Garching a dedicated neutron-scattering facility for precision measurements of QFs at mK temperatures was set up (see FIG.\[fig:setupMLL\]). A pulsed $^{11}$B beam of ${\sim}65$MeV in bunches of 2-3ns (FWHM) produces monoenergetic neutrons of ${\sim}11$MeV via the nuclear reaction p($^{11}$B,n)$^{11}$C in a pressurized H$_2$ target [@Jagemann2005245]. These neutrons are irradiated onto a CRESST-like detector module consisting of a ${\sim}$10g cylindrical CaWO$_4$ single crystal (20mm in diameter, 5mm in height) and a separated Si light absorber (20mm in diameter, 500$\mu$m thick) [@Strauss:2012fk]. Both are operated as cryogenic detectors in a dilution refrigerator at ${\sim}20\,$mK [@Lanfranchi20091405]. Undergoing elastic (single) nuclear scattering in CaWO$_4$ the neutrons are tagged at a fixed scattering angle $\Theta$ in an array of 40 liquid-scintillator (EJ301) detectors which allow fast timing (${\sim}2$ns) and n/$\gamma$ discrimination. Depending upon which of the three nuclei is hit a distinct amount of energy is deposited by the neutron in the crystal. Triple-coincidences between (1) a $^{11}$B pulse on the H$_2$ target, (2) a neutron pulse in a liquid-scintillator detector and (3) a nuclear-recoil event in the CaWO$_4$ crystal can be extracted from the data set. A neutron time-of-flight (TOF) measurement between neutron production and detection combined with a precise phononic measurement of the energy deposition in the crystal (resolution ${\sim}1$keV (FWHM)) allows an identification of the recoiling nucleus. To derive the individual QF the corresponding scintillation-light output is measured simultaneously by the light detector. Since the onset uncertainty of cryodetector pulses is large (${\sim}5\,\mu$s) compared to typical neutron TOFs (${\sim}$50ns) an offline coincidence analysis has to be performed [@strauss_PhD].\
![Schematic experimental setup of the neutron-scattering facility. Neutrons produced by the accelerator are scattered off a CRESST-like detector module (operated at 20mK) and tagged in liquid-scintillator neutron detectors at a fixed scattering angle $\Theta$.[]{data-label="fig:setupMLL"}](facility.pdf){width="38.00000%"}
The experiment was optimized for the measurement of $QF_W$ [@strauss_PhD; @strauss_ltd15]. To enhance the number of W-scatters a scattering angle of ${\Theta\,{=}\,80^\circ}$ was chosen due to scattering kinematics [@Jagemann:2006sx]. For this specific angle, the expected recoil energy of triple-coincident events is ${\sim}\,100$keV for W, ${\sim}\,450$keV for Ca, and ${\sim}\,1.1$MeV for O . In ${\sim}\,3$ weeks of beam time a total of ${\sim}\,10^8$ cryodetector pulses were recorded. FIG.\[fig:timing\] shows the time difference $\Delta t$ between neutron events with the correct TOF identified in one of the liquid-scintillator detectors and the closest W-recoil (in time) in the CaWO$_4$ crystal ($E_r=100{\pm}20$keV). A gaussian peak of triple-coincidences on W (dashed red line) at $\Delta t\,{\approx}\,0.016$ms and a width of $\sigma_t\,{\approx}\,4.8\,\mu$s (onset resolution of the cryodetector) is observed above a background due to accidental coincidences uniformly distributed in time (shaded area). Within the $2\sigma$-bounds of the peak 158 W-scatters are identified with a signal-to-background (S/B) ratio of ${\sim}\,7:1$.\
![Histogram of the time difference $\Delta t$ between neutron events with the correct TOF and the closest W-recoil in the CaWO$_4$ crystal ($E_r\,{=}\,100\,{\pm}\,20$keV). A fit to the distribution (solid black line) including a constant for the accidental background (shaded area) and a gaussian for the triple-coincidences on W (dashed red line) is shown. 158 W-scatters are identified with a signal-to-background ratio of ${\sim}\,7:1$. []{data-label="fig:timing"}](time_final.eps){width="48.00000%"}
The mean LY of the extracted W-scatters is found at a lower value compared to the mean LY of all nuclear recoils, i.e., the (overlapping) contributions of O, Ca and W if no coincidence measurement is involved. The accidental coincidences have a LY-distribution equal to that which is modelled by a probability-density function (background-pdf) [@strauss_PhD]. A simultaneous maximum-likelihood (ML) fit is performed including (1) the timing distribution which fixes the S/B ratio and the number of identified W-events, and (2) the LY distribution described by a gaussian (W-events) and the background-pdf. The final results are $LY_W\,{=}\,0.0208\,{\pm}\,0.0024$ and $QF_W\,{=}\,(1.96\,{\pm}\,0.22)$%, correspondingly (errors are dominated by statistics). FIG.\[fig:LY\] shows the LY histogram of the identified events and the fit by the gaussian (dashed red line) and the background-pdf (shaded area).\
![LY histogram of the 158 events identified as triple-coincidences on W. A fit to the distribution (solid black line) is shown which includes a gaussian (dashed red line) accounting for W-scatters and the background-pdf (shaded area) describing accidental coincidences. The simultaneous ML fit including the timing distribution yields $QF_W\,{=}\,(1.96\,{\pm}\,0.22)$%. []{data-label="fig:LY"}](LY_final.eps){width="48.00000%"}
For the measurement of $QF_{Ca}$ and $QF_O$ no coincidence signals are necessary, instead, an analysis of the nuclear-recoil data alone is sufficient. Commonly CRESST data is displayed in the energy-LY plane [@Angloher:2012vn] giving rise to nearly horizontal bands which correspond to different types of particle interactions ($LY\,{\approx}\,1$ for electron and $LY\,{\lesssim}\,0.2$ for nuclear recoils). The nuclear-recoil bands of the data recorded during ${\sim}1$ week of beam time (${\sim}\,5\cdot 10^5$ pulses) are shown in FIG.\[fig:energyDependence\] bottom (grey dots). From kinematics using ${\sim}11$MeV neutrons as probes the O-recoil band extends up to ${\sim}2.4$MeV while the Ca- and W-bands extend up to ${\sim}1.05$MeV and ${\sim}240$keV, respectively [@Jagemann2005245]. Despite the strong overlap of the 3 nuclear-recoil bands the contributions of O and Ca fitted by two gaussians can be disentangled at $E_r\,{\gtrsim}\,350$keV (see FIG.\[fig:energyDependence\] top) due to high statistics and a good light-detector resolution. In FIG.\[fig:qf\_results\] the results for $QF_O$ and $QF_{Ca}$ (red error bars) derived by these independent one-dimensional (1-dim) fits are shown for selected recoil-energy slices of 20keV in width. All parameters in the fit are left free except for the LY-resolutions which are fixed by a ML fit of the electron-recoil band [@strauss_PhD]. While $QF_O$ clearly rises towards lower recoil energies, this effect is less pronounced for $QF_{Ca}$.\
![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](canLy300.eps "fig:"){width="48.00000%"} ![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](canLy40.eps "fig:"){width="48.00000%"} ![ Top/Middle: LY histograms of two energy slices at 350 and 40keV (of 20keV in width) fitted by gaussians. Bottom: Neutron-induced nuclear-recoil events of O, Ca, and W plotted in the LY-energy plane (grey dots). The corresponding 1$\sigma$ acceptance bounds as derived from the correlated ML fit are indicated (see text). []{data-label="fig:energyDependence"}](LY_allNR_v2.eps "fig:"){width="48.60000%"}
Below ${\sim}350$keV, due to the strong overlap of the nuclear-recoil bands, this simple approach fails. Instead, a correlated ML fit was performed based on the following assumptions: (1) for the mean LY of O- and Ca-scatters the phenomenological parametrization $LY_x(E_r)\,{=}\,LY_x^\infty\left(1+f_x\cdot \exp{(-E_r/\lambda_x)}\right)$ is proposed with the free parameters $LY_x^\infty$ (LY at $E_r\,{=}\,\infty$), $f_x$ (fraction of energy-dependent component) and $\lambda_x$ (exponential decay with energy), and (2) the mean LY of W-scatters is approximated to be constant in the relevant energy range (up to $\sim$240keV) at the value precisely measured with the triple-coincidence technique ($LY_W\,{=}\,0.0208\,{\pm}\,0.0024$). These assumptions are supported by the result of the 1-dim fits (see FIG.\[fig:qf\_results\]), by Birks’ model [@birks1964theory], and by a recent work [@sabine_phD] which predict the strength of the energy-dependence to decrease with $A$. The nuclear-recoil bands are cut into energy intervals of 10keV (20keV to 1MeV), of 20keV (1MeV to 1.4MeV) and 50keV (above 1.4MeV) and fitted with up to 3 gaussians depending on the recoil energy (e.g., shown in FIG.\[fig:energyDependence\] middle for $E_r{=}40$keV). Except for the assumptions mentioned above and the LY-resolution all parameters are left free in the fit. The fit converges over the entire energy range (20-1800keV). In TABLE\[tab:energydependence\] the results for $LY_x^\infty$, $f_x$ and $\lambda_x$ are presented which correspond, e.g. at 40keV, to $QF_O{=}(12.6{\pm}0.5)$%, $QF_{Ca}{=}(6.73{\pm}0.43)$% (here, the errors are dominated by systematics) and $QF_W{=}(1.96{\pm}0.22)$% at $1\sigma$ C.L. The 1-$\sigma$ acceptance bounds of O, Ca and W recoils as obtained in the correlated ML fit are shown in FIG.\[fig:energyDependence\] bottom. The final results for $QF_O$, $QF_{Ca}$ and $QF_W$ are presented in and are found to be in perfect agreement with the outcome of the 1-dim fits (red error bars). These are the first experimental results which clearly show a rise of the QFs of O (Ca) of ${\sim}28$% (${\sim}6$%) towards the ROI (10-40keV) compared to that at a recoil energy of 500keV.\
--------- --------------------- ------------------- ----------------
nucleus $LY_x^\infty$ $f_x$ $\lambda_x$
O $0.07908\pm0.00002$ $0.7088\pm0.0008$ $567.1\pm0.9$
Ca $0.05949\pm0.00078$ $0.1887\pm0.0022$ $801.3\pm18.8$
--------- --------------------- ------------------- ----------------
: \[tab:energydependence\]Results for the free parameters $LY_x^\infty$, $f_x$ and $\lambda_x$ of the ML analysis. The statistical errors are given at $1\sigma$ C.L.
![Results of the correlated ML analysis for $QF_O$, $QF_{Ca}$ and $QF_W$ (solid lines). The shaded areas indicate the 1$\sigma$ and 2$\sigma$ bounds. For the first time a clear energy dependence of $QF_O$ and $QF_{Ca}$ is observed. These results are in agreement with that of the 1-dim fits of discrete energy intervals (see text) shown as red error bars. $QF_W$ is fixed (in the correlated fit) at the value measured by the triple-coincidence technique. []{data-label="fig:qf_results"}](bands_v7.eps){width="48.00000%"}
In previous works, the QFs of CaWO$_4$ were assumed to be constant over the entire energy range [@Angloher:2012vn]. A statistical analysis shows that this simple model is clearly disfavoured. Employing a likelihood-ratio test in combination with Monte-Carlo simulations gives a p-value of $p<10^{-5}$ for the data presented here to be consistent with constant QFs. Furthermore, the derived energy spectra of the individual recoiling nuclei agree with the expectation from incident 11MeV neutrons while the constant QF approach provides non-physical results.\
In the present paper, using the 8 detector modules operated in the last CRESST measurement campaign (run32) an additional aspect was investigated: the variation of the quenching behaviour among *different* CaWO$_4$ crystals [@strauss_PhD]. Nuclear recoils acquired during neutron-calibration campaigns of CRESST run32 are completely dominated by O-scatters at $E_r\gtrsim 150$keV (from kinematics) [@Angloher:2012vn]. Despite low statistics (a factor of ${\sim}100$ less compared to the measurement presented here) in the available data, the mean LY of O-events can be determined by a gaussian fit with a precision of $\mathcal{O}$(1%) for every module. In this way, the mean QF of O between 150 and 200keV was determined individually for the 8 detector modules operated in run32 ($\overline{QF_O^\ast}$) and for the reference detector operated at the neutron-scattering facility ($\overline{QF_O}$). Different values of $\overline{QF_O^\ast}$ are observed for the CRESST detector crystals (variation by ${\sim}11$%) and for the reference crystal (${\sim}12$% higher than the mean of $\overline{QF_O^\ast}$). This variation appears to be correlated with the crystal’s optical quality. The QF - which is a relative quantity - is found to be lower if a crystal has a smaller defect density and thus a higher absolute light output, i.e., the LY of nuclear recoils is less affected by an increased defect density. This is in agreement with the prediction described in a recent work [@sabine_phD]. In the present paper, a simple model to account for this variation is proposed: For every detector module which is to be calibrated a scaling factor $\epsilon$ is introduced, $\epsilon\,{=}\,\overline{QF_O^\ast}/\overline{QF_O}$. Then, within this model the QFs of the nucleus $x$ can be calculated for every module by $QF_{x}^\ast(E_r)\,{=}\,\epsilon\cdot QF_{x}(E_r)$ where $QF_{x}$ is the value precisely measured within this work. The nuclear-recoil behaviour of CRESST modules is well described by energy-dependent QFs. In the QFs, averaged over the ROI (10-40keV), and the scaling factor $\epsilon$ are listed for two selected detector modules (Rita and Daisy, with the lowest and highest absolute light output, respectively) and the mean of all 8 detector modules of run32 (Ø), $QF_O^{ROI}\,{=}\,(11.2{\pm}0.5)$%, $QF_{Ca}^{ROI}\,{=}\,(5.94{\pm}0.49)$% and $QF_W^{ROI}\,{=}\,(1.72{\pm}0.21)$%.\
------- ------------------- -------------------------- ----------------------------- --------------------------
$\epsilon$ $QF_O^\mathrm{ROI}$\[%\] $QF_{Ca}^\mathrm{ROI}$\[%\] $QF_W^\mathrm{ROI}$\[%\]
Rita $0.844{\pm}0.006$ $10.8{\pm}0.5$ $5.70{\pm}0.44$ $1.65{\pm}0.19$
Daisy $0.939{\pm}0.021$ $12.0{\pm}0.7$ $6.33{\pm}0.58$ $1.84{\pm}0.24$
Ø $0.880{\pm}0.011$ $11.2{\pm}0.5$ $5.94{\pm}0.49$ $1.72{\pm}0.21$
------- ------------------- -------------------------- ----------------------------- --------------------------
: \[tab:CRESST\] QF results averaged over the ROI (10-40keV) and adjusted by the scaling factor $\epsilon$ for the modules Rita and Daisy, and the mean (Ø) of all run32 detectors ($1\sigma$ errors).
We now turn to the effect of energy-dependent quenching since constant QFs as assumed in earlier CRESST publications do not sufficiently describe the behaviour of the nuclear-recoil bands. The value of $QF_O$ in the ROI was underestimated by ${\sim}8$% while the room-temperature measurements overestimated the values of $QF_{Ca}$ and $QF_W$ by ${\sim}7$% and ${\sim}130$%, respectively [@Angloher:2012vn]. Therefore, the parameter space of accepted nuclear recoils is larger than assumed in earlier publications (by ${\sim}46$%) requiring a re-analysis of the published CRESST data.\
During the latest measuring campaign (run32) a statistically significant signal ($4.2\sigma$) above known backgrounds was observed. If interpreted as induced by DM particles two WIMP solutions were found [@Angloher:2012vn], e.g. at a mass of $m_\chi=11.6\,$GeV/c$^2$ with a WIMP-nucleon cross section of $\sigma_\chi=3.7\cdot10^{-5}$pb. The dedicated ML analysis was repeated using the new QF values ($\O$ in TABLE\[tab:CRESST\]) yielding $m_\chi=12.0$GeV/c$^2$ and $\sigma_\chi=3.2\cdot10^{-5}$pb at 3.9$\sigma$. Beside this moderate change of the WIMP parameters also the background composition ($e^-$, $\gamma$, neutrons, $\alpha$’s and $^{206}$Pb) is influenced. This is mainly due to the significantly lower value of $QF_W$ which increases the leakage of $^{206}$Pb recoils into the ROI (by ${\sim}$18%). The other WIMP solution is influenced similarly: $m_\chi$ changes from 25.3 to 25.5GeV/c$^2$, $\sigma_\chi$ from $1.6\cdot10^{-6}$ to $1.5\cdot10^{-6}$pb and the significance drops slightly from 4.7 to 4.3$\sigma$.\
In conclusion, the first precise measurement of $QF_W$ at mK temperatures and under conditions comparable to that of the CRESST experiment was obtained at the neutron-scattering facility in Garching by an extensive triple-coincidence technique. Furthermore, the QFs of all three nuclei in CaWO$_4$ were precisely determined by a dedicated maximum-likelihood analysis over the entire energy range (${\sim}20{-}1800$keV). The observed energy dependence of the QFs, which is more pronounced for lighter nuclei, has significant influence on the determination of the ROI for DM search with CRESST. The observed variation of the QFs between different CaWO$_4$ crystals is related to the optical quality and can be adapted to every individual crystal by the simple model proposed above. The updated values of the QFs are highly relevant to disentangle the recoil composition (O, Ca and W) of a possible DM signal and, therefore, to determine the WIMP parameters. Since the separation between the O and W recoil bands is higher by ${\sim}46$% compared to earlier assumptions, background neutrons which are mainly visible as O-scatters [@scholl_paper] can be discriminated more efficiently from possible WIMP-induced events. A reanalysis of the run32 data shows a moderate influence of the new QF values on the WIMP parameters.\
The results obtained here are of importance for the current CRESST run (run33) and upcoming measuring campaigns. Providing a highly improved background level run33 has the potential to clarify the origin of the observed excess signal and to set competitive limits for the spin-independent WIMP-nucleon cross section in the near future. For the planned multi-material DM experiment EURECA (European Underground Rare Event Calorimeter Array) [@Kraus:2011zz] the neutron-scattering facility will be an important tool to investigate the light quenching of alternative target materials in the future.\
This research was supported by the DFG cluster of excellence: “Origin and Structure of the Universe”, the DFG “Transregio 27: Neutrinos and Beyond”, the “Helmholtz Alliance for Astroparticle Phyiscs”, the “Maier-Leibnitz-Laboratorium” (Garching) and by the BMBF: Project 05A11WOC EURECA-XENON.
| ArXiv |
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abstract: 'We investigated the electrostatic interaction between two identical dust grains of an infinite mass immersed in homogeneous plasma by employing first-principles N-body simulations combined with the Ewald method. We specifically tested the possibility of an attractive force due to overlapping Debye spheres (ODSs), as was suggested by Resendes et al. (1998). Our simulation results demonstrate that the electrostatic interaction is repulsive and even stronger than the standard Yukawa potential. We showed that the measured electric field acting on the grain is highly consistent with a model electrostatic potential around a single isolated grain that takes into account a correction due to the orbital motion limited theory. Our result is qualitatively consistent with the counterargument suggested by Markes and Williams (2000), indicating the absence of the ODS attractive force.'
author:
- 'H. Itou'
- 'T. Amano'
- 'M. Hoshino'
title: 'First-principles simulations of electrostatic interactions between dust grains'
---
Introduction
============
Dust grains are quite common in astrophysical environments. They are thought to exist in, for example, interstellar molecular clouds, protoplanetary disks, planetary rings, the Earth’s magnetosphere, and tails of comets. In addition, in laboratories, the lattice formation of dust grains, known as Coulomb crystallization, is a well-known phenomenon that has fascinated many researchers. Dust grains immersed in plasmas usually acquire a large amount of charge through several charging processes, such as collisions with plasma particles and photoemission. Such charged grains and the ambient plasma are electromagnetically coupled with each other, forming so-called dusty plasmas or complex plasmas. Dusty plasma has been studied for both industrial and astrophysical applications, largely motivated by the in-situ detection of dust grains in the Solar System and Ikezi’s prediction, and subsequent experimental verification of Coulomb crystallization.[@Goertz89; @Angelis92; @Shukla09; @Shukla01; @text; @Ikezi86]
When collisions between dust grains and plasma particles are dominant among the charging processes, dust grains become negatively charged because the thermal velocity of electrons is generally higher than that of ions, resulting in a larger electron current. Therefore, one would expect a repulsive shielded electrostatic Coulomb potential (or Yukawa potential) to exist. In reality, however, forces acting on dust grains may be much more complex because the interaction forces between charged dust grains are mediated by the ambient plasma in a complicated manner. There has been much discussion on forces acting between dust grains, including attractive forces for which the ambient plasma response plays the essential role.[@Shukla09; @Lampe00] It is necessary to understand the nature of such attractive interactions among dust grains because they may play a role in the aggregation or crystallization of dust grains observed in laboratories, as well as the formation of stars and planets in the dense cores of interstellar molecular clouds.
One such attractive force acting between two grains, and on which we focus in the present study, is the force due to overlapping Debye spheres (ODSs).[@Resendes98] According to Resendes et al. (1998), when two charged dust grains (each having charge $q$) exist in a plasma, their interaction potential, including the electrostatic energy of ambient plasma particles, may be modified from the simple Yukawa potential. The potential in this case may be written as $$\label{l-j}
q{\phi_{{\rm ODS}}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{1}{2} \right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right)+{\rm constant},$$ where $\lambda_{{\rm D}}$ is the Debye length and $d$ is the intergrain distance. This is similar to the Lennard-Jones potential, which is repulsive at short distances and weakly attractive at longer distances. It is clear that a Lennard-Jones-like potential can assist the processes of aggregation and crystallization, and in fact, it has been shown that the attractive force due to ODSs has a drastic effect on aggregation and crystallization in dusty plasmas if indeed effective.[@Hou09] It has also been suggested that the ODS attractive force may enhance gravitational instability and assist the formation of stars and planets in astrophysical environments.[@Shukla06] On the other hand, the derivation of this attractive potential is based on several nontrivial assumptions that need to be verified. For instance, one must assume that the electrostatic potential around a dust grain is given by the Yukawa potential: $$\label{yukawa}
q{\phi}\left(r\right)=\frac{q^{2}}{r}\exp\left(-\frac{r}{\lambda_{{\rm D}}}\right).$$ In addition, linear superposition of the potential around two dust grains (with ODSs) should be valid in order for such an attractive force to exist. Since the concept of Debye shielding is the key to understanding the attractive force, one must be careful in adopting these assumptions. Furthermore, the derivation of the ODS attractive ${\it force}$ from Eq. (\[l-j\]) assumes that the force operating between the grains is given by the derivative of Eq. (\[l-j\]) with respect to the intergrain distance $d$. We note that Markes and Williams (2000) pointed out that this assumption is incorrect in that it does not take into account energy exchange with the ambient plasma.[@Markes00] Lampe et al. (2000) also suggested that, on the basis of orbital motion limited (OML) theory, such an attractive force would not exist.[@Lampe00] Nevertheless, those counterarguments are also based on some non-trivial assumptions. Consequently, the existence or nonexistence of the ODS attractive force has yet remained a controversial issue.
The purpose of our study is thus to investigate the validity of the theory of the ODS attractive force from first principles. We employ the direct N-body simulation method in which all particle-particle interactions acting through the electrostatic Coulomb force are calculated. This first-principles approach allows us to investigate the electrostatic potential structure of sub-Debye scales without making any assumptions, and thus provides a direct answer to the problem.
It is demonstrated herein that the electric field acting on a charged grain actually deviates from the standard Yukawa-type field in general. We find that the electrostatic force acting between two dust grains is repulsive rather than attractive, which may be well explained by OML theory for an isolated test charge. There is no noticeable signature of the net attractive force due to the effect of ODSs around dust grains. Our result is qualitatively consistent with the analysis given by Markes and Williams (2000). Although the simulations were performed within a limited range of plasma parameters, this strongly indicates the ODS attractive force is absent in reality.
Simulation method
=================
Our N-body simulations are performed in a periodic system (surrounded by a virtual perfectly conducting medium at the infinite distance). The system consists of the simulation box and its replicas, and the box contains many plasma particles (ions and electrons) and two charged dust grains. For the time integration, the Coulomb force acting on each particle must be evaluated by taking the summation over all particles. Since the Coulomb interaction is a long-range interaction, convergence of the summation is very slow and the calculation of contributions from many particles at long distances significantly increases the number of operations required.
We thus adopt the Ewald method, which allows us to accelerate the summation by dividing it into two parts: one in real space and the other in wavenumber space. For instance, the electrostatic potential may be calculated as follows: $$\label{ewald}
U=U_{{\rm real}}+U_{{\rm wave}}-U_{{\rm self}},$$ $$\label{real}
U_{{\rm real}}=\frac{1}{2}\sum_{i,j}\sum_{n}{\frac{q_{i}q_{j}}{r_{ijn}} {\rm erfc}\left(\frac{r_{ijn}}{\sigma} \right)},$$ $$\label{wave}
U_{{\rm wave}}=\frac{1}{2}\sum_{i,j}\sum_{{\bm k}\neq 0}{q_{i}q_{j}\frac{\exp\left[{-\pi^{2}\sigma^{2}k^{2}+2\pi i{\bm k}\cdot\left({\bm r}_{i}-{\bm r}_{j}\right)}\right]}{\pi V k^{2}}},$$ $$\label{self}
U_{{\rm self}}=\frac{1}{\sqrt{\pi}\sigma}\sum_{i}{q_{i}^{2}}.$$ Here, $n$ represents the labels of boxes, $r_{ijn}$ is the distance between particles $i$ and $j$ in box $n$, $q_{i}$ is the charge of particle $i$, ${\bm k}$ is the wavenumber vector, and $V$ is the volume of the box. The parameter $\sigma$ gives a cut-off radius beyond which the direct summation in real space, Eq. (\[real\]), is replaced by that in wavenumber space, Eq. (\[wave\]). Note that in Eq. (\[real\]), the term $n=0$ has to be excluded for $i=j$. This method approximates long-wavelength modes associated with the long-range nature of the Coulomb interaction in wavenumber space with the aid of the Fourier transform, whereas short-wavelength components arising from close encounters between particles are accurately calculated. The electric field is given by the spatial derivatives of Eqs. (\[real\]) and (\[wave\]) and is calculated in the same way.[@Deserno98-1; @Pollock96]
In calculating Eq. (\[real\]), we introduce a small softening parameter $\epsilon$ and rewrite Eq. (\[real\]) as $$\label{ereal}
U_{{\rm real}}=\frac{1}{2}\sum_{i,j}\sum_{n}{\frac{q_{i}q_{j}}{\sqrt{r_{ijn}^{2}+\epsilon^{2}}} {\rm erfc}\left(\frac{\sqrt{r_{ijn}^{2}+\epsilon^{2}}}{\sigma} \right)}.$$ With the softening technique, we ignore large-angle scatterings between particles at distances ${\protect\raisebox{-0.5ex}{$\:\stackrel{\textstyle <}{\sim}\:$}}\epsilon$ because resolving such scatterings would require very small time steps. Since we are interested in weakly coupled space and astrophysical plasmas that are defined by a large plasma parameter $\Lambda$ (where small-angle scatterings play the dominant role), we think this technique is reasonable for our purpose.
Having calculated the electric fields acting on particles, we can solve the equations of motion for each particle: $$\label{newton1}
m_{i}\frac{d}{dt}{\bm v}_{i}=q_{i}{\bm E},$$ $$\label{newton2}
\frac{d}{dt}{\bm r}_{i}={\bm v}_{i},$$ where $m_{i}$, ${\bm v}_{i}$, and ${\bm r}_{i}$ are the mass, velocity, and position of particle $i$, respectively, and ${\bm E}$ is the electric field at each particle position ${\bm r}_{i}$. In Eq. (\[newton1\]), assuming nonrelativistic plasma temperatures, $v_{i}/c\ll1$, we ignore the effect of magnetic fields.
Throughout the present paper, the masses of ions and electrons are assumed to be equal to allow the system to relax quickly to an equilibrium state. This assumption may be justified because the mass ratio affects only the time scale, and structures of the equilibrium state can be assumed to be independent of the mass ratio. Therefore, we only discuss the properties of equilibrium states. Note that because of the symmetry of ion and electron masses, the sign of the grain charge is irrelevant. Simulations are performed with two identical dust grains of infinite mass in the box. That is to say, the grain mass is so large that the change in positions can be ignored on the simulation time scale, which is typically limited to a few plasma oscillation periods. The effect of finite grain size is also ignored. These assumptions are made to simplify the problem as much as possible for our purpose of investigating the electrostatic interactions between plasma particles and dust grains.
simulation result
=================
Simulations were initialized with plasma particles distributed randomly in space, and two dust grains placed at fixed distances in the box. The velocity distribution was initialized to a Maxwellian distribution for a given temperature. Time integration was carried out until the system reached an equilibrium state, at which point we measured the properties of the system. The simulation box was a cuboid whose dimensions were $2L$ in the $x$ direction and $L$ in the $y$ and $z$ directions. Throughout this paper, we use a softening parameter of $\epsilon=0.03L$ in simulations. Each grain was located at $\left(y,z\right)=\left(L/2,L/2\right)$, and the intergrain distance along the $x$ axis was varied in each simulation run. By comparing the equilibrium states of different runs, we measured the dependence on the intergrain distance.
![Temporal evolution of total electrostatic potential energy for a run with $d=0.1L$, $q=1000e$, $2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}0}\simeq0.11L$, $\lambda_{{\rm D}}\simeq0.12L$, and $\Lambda\simeq16$. The dotted line indicates the equilibrium value.[]{data-label="ene"}](enenene.eps){width="90mm"}
The system is characterized by the dust charge $q$ and the intergrain distance $d$. The number densities of electrons and ions are denoted $n_{{\rm e}}$ and $n_{{\rm p}}$, which are chosen so that charge neutrality (including dust charges) is satisfied. In the following, time and space are respectively normalized by the inverse plasma frequency $1/\omega_{p}$, where $\omega_{p}=\left(4\pi n_{{\rm e}} e^{2}/m_{{\rm e}}+4\pi n_{{\rm p}} e^{2}/m_{{\rm p}}\right)^{1/2}$, and the Debye length $\lambda_{{\rm D}}$. Note that the Debye length is defined as $\lambda_{{\rm D}}=\left(4\pi n_{{\rm e}} e^{2}/kT_{{\rm e}}+4\pi n_{{\rm p}} e^{2}/kT_{{\rm p}}\right)^{-1/2}$, including both ion and electron contributions, and the temperatures of the resultant equilibrium states are used. Here, $e$ is the elementary charge, and $m_{{\rm e}}$, $m_{{\rm p}}$, $T_{{\rm e}}$, and $T_{{\rm p}}$ are the electron mass, proton mass, electron temperature, and proton temperature, respectively. Note that we always assumed that the initial electron and proton temperatures were the same for simplicity.
In Fig.\[ene\], the time variation of the potential energy integrated over the simulation box is shown for the example of a run with an intergrain distance of $d=0.1L$. The energy is normalized by $q^2/\lambda_{{\rm D}0}$, where $\lambda_{{\rm D}0}$ is the Debye length defined by the initial temperature. In this run, $q=1000e$, $2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}0}\simeq0.11L$, $\lambda_{{\rm D}}\simeq0.12L$, and $\Lambda\equiv\left(n_{{\rm e}}+n_{{\rm p}}\right)\lambda_{{\rm D}}^{3}\simeq16$. Generally speaking, the Debye length in the final equilibrium state, denoted $\lambda_{{\rm D}}$, actually differs from $\lambda_{{\rm D}0}$, as explained below. We see from Fig.\[ene\] that the potential energy decreases during the first $\sim1/\omega_{p}$, and then fluctuates around the equilibrium value. This initial decrease in the potential may be explained by the redistribution of plasma particles due to Debye shielding. This decrease in the potential energy is compensated by an increase in the plasma temperature, changing the Debye length from the initial value accordingly. All runs discussed in this paper showed essentially the same trend. We thus assume that the equilibrium was achieved by the time $\omega_{{\rm p}}t\sim8$, and physical quantities averaged after this time were regarded as equilibrium values.
![Summary of simulation results. The normalized electric field acting on the grain multiplied by $d^{2}$ is shown as a function of the intergrain distance $d$. Note that the distance is normalized by the Debye length defined with the kinetic energy measured at the equilibrium states rather than the initial temperature. The red and green lines are the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential, respectively. Blue triangles and magenta diamonds show the results of our simulations with $\Lambda\simeq13$ and $\Lambda\simeq16$, respectively.[]{data-label="res_ele"}](reseses.eps){width="90mm"}
Figure \[res\_ele\] summarizes the results of our simulations. Blue triangles show the results for $2L^{3}n_{{\rm e}}=5000$, $2L^{3}n_{{\rm p}}=3000$, $\lambda_{{\rm D}}/L\simeq0.15L$, and $\Lambda\simeq13$. Individual triangles represent intergrain distances of $d=0.2L, 0.4L, 0.5L, 0.6L, 0.8L$. Simulations with a different set of parameters ($2L^{3}n_{{\rm e}}=10000$, $2L^{3}n_{{\rm p}}=8000$, $\lambda_{{\rm D}}/L\simeq0.12L$, and $\Lambda\simeq16$) were also run, and the results are shown by magenta diamonds; in this case, the intergrain distances were $d=0.1L, 0.25L, 0.4L, 0.5L, 0.6L, 0.75L, 0.9L$. In all runs, $q=1000e$ and $\lambda_{{\rm D}0}\simeq0.11L$. Note that $\lambda_{{\rm D}}$, which normalizes the intergrain distances in Fig.\[res\_ele\], was defined at the equilibrium states, and thus not necessarily the same in each run because the self-consistent increase in temperature depends on plasma densities, plasma parameters and intergrain distances $d$. The red and green lines in Fig.\[res\_ele\] show the theoretical curves expected from the standard Yukawa potential and the ODS attractive potential of Resendes et al. (1998), respectively, which are written as $$\label{yukawa-e}
qE_{{\rm Yukawa}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac{\lambda_{{\rm D}}}{d}\right)^{2}+\frac{\lambda_{{\rm D}}}{d}\right]\exp\left(-\frac{d}{\lambda_{{\rm D}}}\right)$$ and $$\label{l-j-e}
qE_{{\rm ODS}}\left(d\right)=\frac{q^{2}}{\lambda_{{\rm D}}^{2}}\left[\left(\frac{\lambda_{{\rm D}}}{d}\right)^{2}+\frac{\lambda_{{\rm D}}}{d}-\frac{1}{2}\right]\exp\left(-\frac{d}{\lambda_{{\rm D}}}\right).$$ Eq. (\[l-j-e\]) assumes that the force on the grain is given by the derivative of Eq. (\[l-j\]) with respect to $d$. The error bars represent the standard deviation ($1\sigma$) of temporal fluctuations after the system has reached an equilibrium state. Note that when calculating the electric field acting on the grain, we used a softening parameter of $\epsilon=d/12$, which is different from that used in the simulation to reduce the variance of the measured electric fields. That is to say, the softening parameter $\epsilon$ is chosen to be proportional to the intergrain distance, whereas it is constant in all simulations. This choice is mainly motivated by the conjecture that the equilibrium electrostatic structure will not strongly depend on the softening parameter. However, some remarks must be made before discussing the results.
First, the effect of softening is not seen even at $d \lesssim 0.03 L\left(\simeq0.2-0.25\lambda_{{\rm D}}\right)$ because the softening parameters used in the calculations are smaller than the simulation value at $d < 0.36L\simeq2.5-3\lambda_{{\rm D}}$. In the region where the softening effect is significant, it is obvious that the potential approaches the Coulomb potential because the softening parameter in the simulations is chosen to be smaller than the mean particle distance. Therefore, this will not change our conclusions.
Second, the error bars may be underestimated at $d > 0.36L$ because the softening parameter used in the calculation becomes larger than that in the simulations. (Note that large error bars are caused by close encounters with plasma particles.) In any case, the error bars are so large that it is difficult to extract a physically meaningful argument in this regime.
Third, we have confirmed that calculation with a constant softening parameter of $\epsilon=0.03L$ (i.e., consistent with the simulations) does not change the result substantially. Although the error bars in the far regions, i.e., $d > 0.36L$, tend to increase, the average electric fields stay within the error bars shown in Fig.\[res\_ele\].
Based on these discussions, we believe that the simulation results are reliable at least in the intermediate regime, i.e., $1\lambda_{{\rm D}}\lesssim d\lesssim2.5\lambda_{{\rm D}}$. In this region, it is evident from Fig.\[res\_ele\] that the simulation results deviate from the theoretical prediction of the ODS attractive potential beyond $2\sigma$. The result also suggests that the electric fields acting on the grain are even larger than the standard Yukawa potential prediction. Although the large error bars make it difficult to draw conclusions from this result alone, the systematic deviation from the theoretical predictions suggests that the underlying assumptions made in the derivation of (\[yukawa-e\]) and (\[l-j-e\]) may be violated. In the next section, we discuss possible reasons for this discrepancy between the theory and simulations.
Discussion
==========
Our simulation results show that the force between two dust grains is repulsive and stronger than that predicted by the standard Yukawa potential Eq. (\[yukawa\]). At first, we discuss the validity of Eq. (\[yukawa\]). When the grain radius is negligible, the functional form of the Yukawa potential itself must be correct at large distances, where the shielding is nearly complete and the first-order expansion of the Boltzmann-type density distribution is appropriate. In fact, Poisson’s equation and the linearized Boltzmann distributions give $$\label{long}
q\phi\left(r\right)=\alpha\frac{q^{2}}{r}\exp{\left(-\frac{r}{\lambda_{{\rm D}}}\right)}.$$ However, the coefficient $\alpha$ (integration constant) in Eq. (\[long\]) is unknown and must be determined by the inner boundary condition. In standard textbooks, it is determined by assuming that the outer solution smoothly connects to the bare Coulomb potential at $r \rightarrow 0$, which gives $\alpha=1$.
On the other hand, according to OML theory, $\alpha\neq1$ in general. In OML theory, when particle absorption by dust grains is ignored, the density distribution of ions around a negatively charged dust grain may be written as [@Lampe00] $$\label{oml}
n_{{\rm p}}=n_{0}\left[\exp\left(-\frac{e\phi}{kT_{{\rm p}}}\right){\rm erfc}\left(\sqrt{-\frac{e\phi}{kT_{{\rm p}}}} \right)+\frac{2}{\sqrt{\pi}}\sqrt{-\frac{e\phi}{kT_{{\rm p}}}} \right]$$ instead of the Boltzmann distribution $$\label{boltz-p}
n_{{\rm p}}=n_{0}\exp\left(-\frac{e\phi}{kT_{{\rm p}}}\right),$$ whereas the electron density distribution is written as $$\label{boltz-e}
n_{{\rm e}}=n_{0}\exp\left(\frac{e\phi}{kT_{{\rm e}}}\right)$$ in both cases. It is easy to show that Eqs. (\[oml\]) and (\[boltz-p\]) give the same dependence on $e\phi/kT$ when expanded to first order in $e\phi/kT\ll1$, meaning that the functional form is the same far from the grain.
Since the OML correction given by Eq. (\[oml\]) gives an ion density much lower than that suggested by the Boltzmann distribution given by Eq. (\[boltz-p\]) close to the grain, the shielding of the potential becomes weaker. We may thus expect $\alpha\geq1$ in general if the OML correction is taken into account.[@Lampe00] The parameter $\alpha$ may be determined by the solution in the inner region, where the OML correction may become important. On the other hand, the OML solution must also be connected to the bare Coulomb potential $$\label{short}
q\phi\sim \frac{q^{2}}{r},$$ at distances on the order of the mean interparticle distance $a$, which is defined as $$\label{wig2}
\frac{a}{\lambda_{{\rm D}}}\equiv \sqrt[3]{\frac{3}{4\pi \Lambda}},$$ where $\Lambda$ is the plasma parameter. While it is difficult to analyze the potential structure analytically in the inner region with the OML correction, we expect $\alpha\left(\Lambda\right)$ to be a decreasing function of $\Lambda$ because a larger $\Lambda$ narrows the region in which the OML correction should be taken into account and strengthens the shielding effect.
![Comparison between simulation results and theoretical models including the OML correction for the electric field acting on the grain. Only results with $\Lambda\simeq16$ are shown.[]{data-label="com"}](comcom.eps){width="90mm"}
To determine the value of $\alpha$, Figure \[com\] compares the simulation results and a theoretical electric field around ${\it a}$ ${\it single}$ ${\it isolated}$ ${\it grain}$ including the OML correction. That is to say, the potential $\phi$ was determined by solving Poisson’s equation, $$\label{poi}
\nabla^{2}\phi=-4\pi e\left(n_{{\rm p}}-n_{{\rm e}}\right),$$ with the ion and electron densities given by Eqs. (\[oml\]) and (\[boltz-e\]), respectively. $n_{0}$ in Eqs. (\[oml\]) and (\[boltz-e\]) was approximated as $n_{0}=(n_{{\rm e}0}+n_{{\rm p}0})/2$ for simplicity. The plasma parameter was $\Lambda\simeq16$, which is almost the same as that in our simulations. The electric field $E$ was calculated by taking spatial derivatives of $\phi$. As we have already mentioned, the functional form of Eq. (\[long\]) should be valid far from the grain even if the OML correction is included. Therefore, Poisson’s equation was integrated from a large radial distance toward the inner region by taking $\alpha$ as a free parameter. We then tried to find the values of $\alpha$ for which this theoretical solution reasonably matched the simulation results. It is readily seen from Fig.\[com\] that the simulation results are well explained by this model with $\alpha\simeq 1.8-2.0$. Note again that the theoretical curve is for an isolated grain, whereas the simulation results are obtained with two dust grains. This means that the effect of ODSs is not observed, at least to a detectable level beyond the error bars of our simulations. This result is qualitatively consistent with the suggestion by Markes and Williams (2000). They have shown explicitly that the electrostatic force acting between two grains surrounded by a plasma is repulsive by solving Poisson’s equation. The critical assumption in their model is that the ion and electron densities can be written as a function of the local electrostatic potential alone. Although this assumption sounds reasonable for instance in the collisionless limit where OML theory should apply, its validity must be tested carefully. On the other hand, our first principles approach free from such an assumption also demonstrates a repulsive nature for the electrostatic interaction. Furthermore, the fact that the electric field around the grain is consistent with the OML theory indicates the assumption made by Markes and Williams (2000) is indeed reasonable.
One might argue that the fact that $\alpha\neq1$ explains the discrepancy between the simulation results and ODS theory, but this is not the case. Assuming that linear superposition of the potential is also possible for $\alpha\neq1$, we can easily calculate the ODS attractive force for this case as well. The resulting attractive potential force may be written as $$\label{ml-j}
q{\phi_{{\rm ODS}}}\left(d\right)=\alpha\frac{q^{2}}{\lambda_{{\rm D}}}\left(\frac{\lambda_{{\rm D}}}{d}-\frac{\alpha}{2}\right)\exp\left(-\frac{d}{\lambda_{{\rm D}}} \right),$$ which is shown in Fig.\[mod\] for $\alpha=1, 1.2, 1.4$. It can be easily understood that the potential minimum moves inward and the depth increases as $\alpha$ increases. In fact, an easy analytical calculation confirms this tendency. Clearly, $\alpha\neq1$ does not help to explain the discrepancy.
![Modified ODS attractive potential given by Eq. (\[ml-j\]). Red, green, and blue lines represent $\alpha = 1, 1.2, 1.4$, respectively.[]{data-label="mod"}](modod.eps){width="90mm"}
Although it is not easy to analytically determine the value of $\alpha$ in general, we can estimate the upper and lower bounds as follows. We define $r_{{\rm c}}$ as a solution to the equation $$\label{exp-equ}
\frac{\alpha}{4\pi\Lambda}\frac{q}{e}\frac{\lambda_{{\rm D}}}{r_{{\rm c}}}\exp{\left(-\frac{r_{{\rm c}}}{\lambda_{{\rm D}}}\right)}=1,$$ where the left-hand side is the normalized outer potential. An analytic solution to this equation is given by $$\label{sol}
r_{{\rm c}}=\lambda_{{\rm D}} W\left(\frac{\alpha}{4\pi \Lambda}\frac{q}{e}\right),$$ where $W\left(x\right)$ is the inverse function of $x=W\exp\left(W\right)$, which is also known as the Lambert W-function. The potential at $r=r_{{\rm c}}$ may be approximated by $\phi\left(r_{{\rm c}}\right)=\alpha q \exp{\left(-r_{{\rm c}}/\lambda\right)}/r_{{\rm c}}$ and should be bounded by $q \exp{\left(-a/\lambda\right)}/a$ and the bare Coulomb potential $q/a$, leading to the inequality $1\leq\alpha\leq \exp{\left(r_{{\rm c}}/\lambda\right)}$. Using $\Lambda$ and $q/e$, we can rewrite this inequality as $$\label{ine1}
1\leq\alpha\leq \exp{\left(\frac{1}{4\pi \Lambda} \frac{q}{e}\right)}.$$ This estimate must be modified when $\Lambda$ is much larger than the critical value $\Lambda_{{\rm c}}$ for which the condition $a=r_{{\rm c}}$ is satisfied. When $a\gg r_{{\rm c}}$, $\phi\left(a\right)$ rather than $\phi\left(r_{{\rm c}}\right)$ must be used for a similar comparison, yielding $$\label{ine2}
1\leq\alpha\leq \exp{\left(\sqrt[3]{\frac{3}{4\pi\Lambda}}\right)}.$$ The condition $r_{{\rm c}}=a$ leads to $\Lambda_{{\rm c}}\sim\left(q/e\right)^{3/2}$, which can also be expressed as $kT_{{\rm c}}\sim eq/a$ with a critical temperature $T_{{\rm c}}$. From this, it is clear that when the temperature is above the critical value, the plasma is weakly coupled even with dust grains having relatively large charge. This indicates that the OML correction in this regime is only a minor modification, and essentially, the Yukawa-type potential in the far zone directly connects to the bare Coulomb potential.
In our simulations, since we used large dust charges with relatively small numbers of particles, the plasma parameter is smaller than the critical value. Note that the plasma parameter of dusty plasmas in space is usually huge, and so is almost always above the critical value. Our choice of dust charge was motivated by the fact that the theoretical ODS attractive force is proportional to $q$, and the effect is expected to be more pronounced for larger dust charges. As a drawback, we were forced to use sub-critical plasma parameters owing to limited computational resources. Because of this, it was not possible to draw a final conclusion. Nevertheless, the qualitative consistency between our results and the counterargument against the ODS attractive force strongly indicates that the ODS attractive force may not operate in reality. In particular, we believe the assumption that the derivative of the potential energy of the whole system with respect to the intergrain distance provides a net force acting on the grain is incorrect as was pointed out by Markes and Williams (2000). Equation (\[ine2\]) shows that, when the plasma parameter is sufficiently large, $\alpha$ becomes almost unity and the potential structure around the grain approaches Eq. (\[yukawa\]), on which the derivation of the ODS attractive potential is based. Even in this parameter regime, our results suggest that the electric field acting on the grain is given by the spatial derivative of the potential at the grain’s position rather than that of the potential of the whole system with respect to the intergrain distance. In this case, the electrostatic force acting between two dust grains is always repulsive.
Of course, our results should apply only to the simplest situation where two infinitely small dust grains remain at rest with respect to an ambient fully ionized collisionless plasma. There has been a lot of discussion on the force acting on dust grains that may be affected by, e.g., finite grain size, relative streaming between the plasma and grains. Comprehensive understanding of the net force due the combined effect of those contributions is needed for, e.g., star and planet formation in astrophysical environments.
Conclusion
==========
We investigated the electrostatic interaction between dust grains surrounded by a plasma by employing first-principles N-body simulations combined with the Ewald method. It was shown that the interaction between two charged dust grains is repulsive and its magnitude is somewhat larger than that derived from the Yukawa potential. The force acting on the dust grains was explained by OML theory for a single isolated grain quite well. The result is consistent with the analysis given by Markes and Williams (2000). Consequently, we think that the electrostatic force acting between dust grains are always repulsive. Nevertheless, since our simulations have been performed only in a limited parameter range, a final conclusion awaits simulations with much higher plasma parameters, which will be made possible by adopting modern numerical schemes such as particle-particle particle-mesh and special-purpose GRAPE (GRAvity-piPE) computers for N-body simulations. [@text2; @Yamamoto06]
Acknowledgement
===============
We are grateful to the anonymous referee for his/her critical and constructive comments on the manuscript.
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| ArXiv |
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abstract: 'The magnetic and transport properties have been investigated for the composite polycrystalline manganites, (1-x)$La_{2/3}Ca_{1/3}MnO_3$/(x)yttria-stabilized zirconia ( (1-x)LCMO/(x)YSZ ), at various YSZ fractions, x, ranging from 0 % to 15 %. The ac magnetic susceptibility, $\chi(T)$, DC magnetization, $M(T)$, temperature dependent resistivity, $\rho(T)$ and thermoelectric power (TEP), $S(T)$, have been measured. It was found, surprisingly, that a TEP peak showed up in the magnetic transition region for the sample with the x even as little as 0.75 %. The magnetic transition temperature reaches the minimum value as x increases from 0 % to 4.5 % and goes up as x increases further. Several possible factors such as the effect of strain, the finite size effect, and the effect of magnetic tunnelling coupling, [*etc.*]{}, in affecting the above physical properties of the composite manganites have been studied carefully. The strain induced by the YSZ/LCMO boundary layer (BL) was identified as the leading factor to account for the x dependence of these properties. It demonstrates that the effect of strain could be important in the bulk manganites as in the film sample.'
address: |
(1) Department of Physics and State Key Lab for Mesoscopic Physics, Peking University, Beijing 100871, P.R.China\
(2) Electron microscopy laboratory, Peking University, Beijing 100871, P.R.China
author:
- 'Wei Liu$^{1}$, Chinping Chen$^{1}$[@chen], Xinfeng Wang$^{1}$, Jun Zhao$^{1}$, Xiangyu Xu$^{1}$, Jun Xu$^{1,2}$, and Shousheng Yan$^{1}$'
title: ' Effect of strain on the magnetic and transport properties of the composite manganites, La$_{2/3}$Ca$_{1/3}$MnO$_3$/yttria-stabilized zirconia'
---
Introduction
============
The perovskite manganites, $A_{1-y}B_yMnO_3$, in which A is for the rare earth trivalent cation and B for the alkaline divalent one, exhibit complicated phases at various temperatures and hole doping concentrations, y. Due to the important magnetic application potentials and fundamental research interests, tremendous activities in the physics community have focused in this area during the past decade. With an appropriate doping concentration y, an FM transition takes place at the Curie point, $T_C$, with the decreasing temperature. It is accompanied with the metal-insulator (MI) transition at $T_P$. A colossal magneto-resistance (CMR) slao occurs around this temperature. These properties can be explained with the double exchange (DE) mechanism, [@Zener51; @Anderson55; @Goodenough55; @deGennes60].
Recently, growing attentions have been focused on the effect of strain arising from the interface or surface states of the CMR thin film [@Dulli00; @Bibes01; @Bibes03]. The variation of the magnetic transition temperature, $T_C$, was interpreted as attributed to the strain. However, it is usually difficult to separate the strain-induced effect from the finite size confinement one with the thin films [@Zhang01; @Ziese02; @Andres03]. On the other hand, the effect of strain is usually overlooked in the bulk samples. We would like to demonstrate that the effect of the BL strain with the polycrystalline composite manganites, (1-x)LCMO/(x)YSZ, is a very important factor for the variation of the physical properties such as $T_C$, the temperature dependent resistivity, $\rho(T)$, and the $S(T)$ behavior, [*etc.*]{}.
Sample preparation
==================
A double-staged process was applied in preparing the LCMO/YSZ samples [@Yuan02]. In the first stage, the LCMO nano-sized powder was produced by the sol-gel method and then sintered at 1300$\ ^{o}C$ for 10 hours to form crystals of about 20 $\mu$m. In the second stage, the thus-obtained LCMO powders were mixed with the YSZ powders of about 2 $\mu$m for the heat treatment at 1350$\
^{o}C$ for another 10 hours. The X-ray diffraction (XRD) was performed by a Philip x’ pert diffractometer using the Cu K$_{\alpha}$ line (1.54056 $\AA$). The XRD spectra are presented in Fig. \[XRD\]. The YSZ phase was identified for the samples with x exceeding 4.5 %. The lattice constants of the LCMO phase remain unchanged within 0.001 $\AA$ for all of the YSZ mixing concentration, x. This indicates that, within the detection sensitivity of XRD, the bulk LCMO composition was not modified during the heat treatment due to any possible diffusion of ions from the YSZ composition. We have prepared the samples of LCMO annealed at 1400$\ ^{o}C$ without the mixing of YSZ as well. This would demonstrate the widely investigated disorder effect resulting from different annealing conditions.
The morphology of the sample was investigated by a scanning electron microscope (SEM) performed on the system of FEI STRATA DB235 focus ion beam (FIB) electron microscope. It revealed that at low mixing concentration, x $<$ 4.5 %, the LCMO phase formed large cluster surrounded by a thin layer of YSZ component at a thickness of the order of 10 nm. The BL area increases while the cluster size of the LCMO phase decreases with the increasing YSZ fraction. The interconnecting paths between the adjacent LCMO clusters separated by the YSZ layers would reduce accordingly. On the other hand, at the mixing fraction, x, exceeding about 4.5 %, the YSZ phase forms cluster-like structure by itself. Thus, the BL area decreases correspondingly. This observation is consistent with the previous report[@Yuan02]. Within the detection sensitivities of the XRD and SEM, no indication of the inter diffusion between the LCMO and the YSZ phases exists. The two phases, hence, form solid mixture with the existence of the BL strain in between. Since the YSZ phase is insulating and non-magnetic. the LCMO/YSZ composites are, therefore, appropriate for the investigation of the magnetic and transport properties under the BL strain.
Experiment
==========
The dc magnetization, $M(T)$, and ac magnetic susceptibility, $\chi(T) = \chi'+ i\chi''$, were performed using Quantum Design PPMS and MPMS, respectively. The ac susceptibility measurement was carried out with the excitation field of 10 Oe at 113 Hz under a few Oe of dc background field. The applied field for the $M(T)$ measurement, including the field-cooled (FC) and zero-field-cooled (ZFC), is 50 Oe, while the field applied during the cooling stage before the FC measurement is 3000 Oe. The magnetic transition temperatures, $T_C(dc)$ and $T_C(ac)$, are obtained using the definition, $dM/dT$ and $d\chi'/dT$. These two transition temperatures agree with each other well within a few Kelvin, as plotted in Fig. \[mag1\]. Also plotted in the same figure are the metal-insulator (MI) transition temperature, $T_P$, determined by the $\rho(T)$ measurement, and the magnetic transition temperature width, $\Delta T$, defined by the difference of temperatures at which $d\chi'/dT$ = 0. The correlation of the $T_C$ with the BL effect is apparent in the figure. The lowest $T_C$ occurs at x = 4.5 %, corresponding to the minimum BL inferred from the SEM observation. In order to investigate the effect of thermal treatment on $T_C$, the sample of x = 0 annealed at 1400$\ ^{o}C$ was measured as well for the magnetic transition temperature, $T_C$. It is 260 K using the same maximum slope criteria described above. The depression of $T_C$ down to about 245 K with the sample annealed at 1300$\ ^{o}C$ accounts for about 6 % effect. This is a well studied effect and is attributed to the disorder associated with the polycrystalline grain size distribution, [@Sanchez96; @Gupta96].
The out-of-phase component, $\chi''(T)$, is shown in Fig. \[chi2\]. Two characteristic dissipation features, the peak at high temperature, $T_{DH}$, and the bump at low temperature, $T_{DL}$, appear for each sample and are depicted in the inset of Fig. \[chi2\] as a function of the YSZ fraction, x. $T_{DH}$ is lower than $T_C$ by a few Kelvins. It is resulting from the energy dissipation process associated with the critical spin fluctuation near the FM phase transition. The x-dependence of $T_{DH}$ is therefore similar to that of $T_C$. On the other hand, $T_{DL}$ appears around 70 K for all of the samples, including the one with x = 0. This indicates that the LT bump is independent of the BL. In fact, similar bumps in $\chi''(T)$ at $T < T_C$ have been observed in many previous experiments, attributed to the magnetic inhomogeneity [@Moreira98]. Hence, the x independence of $T_{DL}$ is an indication that the characteristic crystalline grain size associated with the LCMO phase stays unaltered with the mixing of the YSZ component. The field-cooled (FC) and zero-field-cooled (ZFC) dc magnetization measurements on the chosen samples, x = 0 %, 1.25 %, 4.5 % and 15 %, were performed to investigate the magnetic inhomogeneity. Fig. \[FC-ZFC\] shows the normalized magnetization, M(T)/M$_m$, versus the reduced temperature, T/$T_C$, where M$_m$ is the maximum magnetization occurring near the freezing temperature at which the FC and ZFC branches of the $M(T)$ curves diverge. There is no appreciable difference for the x = 0 % sample from the other ones, indicating that the magnetic disorder revealed by this measurement is ascribed to the polycrystalline grains, independent of the BL. This is of the same origin causing the LT bump in the $\chi''(T)$ measurements. Note that, there is no structure in the $M(T)$ curve or the in-phase component, $\chi'(T)$ (not shown here), at the temperature near the LT bump. It indicates that it is not resulting from a magnetic phase transition.
The $\rho(T)$ measurement was carried out from 80 K to 300 K in a home-made insert-probe by a standard 4-probe dc techniques using cold-pressed indium as the electrical contacts. The typical contact resistances is on the order of a few $\Omega$ with the applied current on the order of a few mA. The $\rho(T)$ curves are published in Fig. 1 by Liu [*et al*]{} [@Liu03]. In the region below $T_C$, the $\rho(T)$ behaviors are analyzed with the various scattering processes of the electrons by the function $\rho(T) =
\rho_0 + AT^a + BT^b$, where a = 2 or 3/2, and b = 5 or 9/2. The maximum fitting range in temperature for a stable result is from the lowest temperature of the measurement to $T \sim$ 0.8 $T_C$. The $AT^a$ term with a = 2 or 3/2 gives equally good fitting of the experimental data. The coefficient of the fitting, A, corresponding to a = 3/2 or 2 also exhibits identical x dependence. It is difficult to distinguish which of the following two processes is the more important one, the $T^2$ term for the electron-electron scattering or the $T^{3/2}$ term for the scattering of electrons by the disordered spin glass component [@SpinGlass93]. The x dependence of this term is represented by the coefficient $A$, calculated using a = 2, versus x and plotted in Fig. \[RT\]. The variation with different samples is within a factor of 4. Similarly, an equally good fitting is obtained with the $BT^b$ term using b = 5 for the electron-phonon scattering or b = 9/2 for the electron-magnon scattering within the framework of DE mechanism [@Kubo72]. Since the $B$-variation versus x is the same using b = 9/2 or 5, the ratio of $B/A$ is plotted in the inset of Fig. \[RT\] using b = 5. Also plotted in the same inset is the $\rho_0/A$ ratio, by the open squares. The x dependence of $\rho_0$ associated with the residual or disorder scattering process is only slightly higher than that of the $AT^a$ term. From the above analysis for the various scattering processes, the $BT^b$ term is affected most profoundly by the presence of the LCMO/YSZ BL layer, indicating that the BL-induced strain has a strong effect on the electron-phonon coupling strength.
The temperature dependent TEP, $S(T)$, was measured with the series of LCMO/YSZ samples by the home-made insert-probe using the dc differential technique. The electrical contacts were made by the cold-pressed indium. The sample was installed across two temperature platforms. One was regulated at temperature $T$, while the other controlled to vary within $T$ + $\Delta T$. A continuous voltage output, $\Delta V$, taken by Keithley 2010 multimeter was recorded with the corresponding $\Delta T$, typically a few tenths of a kelvin, changing slowly. The slope of the linear relation between $\Delta V$ and $\Delta T$, with the correction of the contribution from the Cu leads, would give the measured TEP of the sample. Abrupt TEP jumps occur during the magnetic phase transition for the x $>$ 0 % samples, but not for the sample with x = 0 %, see Fig. \[TEP\]. This demonstrates clearly a strong correlation of the jumps with the existence of the BL, and was interpreted as due to the magnetic inhomogeneity induced by the BL strain [@Liu03]. Similar TEP jump with the magneto-TEP effect has been observed in the thin film CMR manganites also [@Jaime96; @Jaime99]. The substrate strain unavoidably caused the magnetic inhomogeneity in the sample. Under the applied magnetic field, the inhomogeneous magnetic component has been reduced. The TEP jump was therefore suppressed to show the magneto-TEP effect.
Discussion
==========
The non-magnetic, insulating YSZ component intermixing in the LCMO causes variations in the magnetic and transport properties of the manganites. Most of the interesting features in the physical properties under current investigations are strongly correlated with the LCMO/YSZ BL. Several effects would be introduced on the samples by the existence of the BL. However, only a leading one is accounted for the observed x dependence. The strain induced by the BL would be the major factor identified in the present work.
Usually, the effect of strain on the physical properties of the manganites, especially on the $T_C$ variations, is studied with the films. However, for thickness under a few hundred nanometers, the finite size confinement effect would become important to superimpose on the effect of strain. With the substrate strain, the interface magnetic inhomogeneity has been directly observed by the techniques of NMR [@Bibes01] or X-ray photoemission spectroscopy [@Dulli00]. Nonetheless, for these films, the finite size effect seems to be the leading factors in the depression of $T_C$, dominating the effect of strain under discussion. Fig. \[FiniteSize\] displays the shift of $T_C$ versus film thickness, $d$, summarized from many of the previous experiments for various thin films grown on different substrates[@Bibes01; @Andres03; @Gupta96; @Bibes02; @Campillo01; @Snyder96; @Huhtinen02; @Xiong96; @Rao98]. The results follows very well the law of finite size confinement[@Fisher72; @FiniteSize00], $|T_C(\infty)-T_C(d)|/T_C(\infty) = (\xi_0/d)^\lambda$ with $\lambda$ = 1 in consistent with the result from the mean field theory[@Zhang01], where $\xi_0 =$ 6 nm is the correlation length at T = 0 K, $T_C(d)$ is the transition temperature for a film of thickness, $d$, and $T_C(\infty)$ is that for the corresponding bulk samples. Note that the dispersion of the data points in Fig. \[FiniteSize\] indicates that the effect of the substrate strain and the crystallinity of the films superimposed on top of the confinement effect are non-negligible. This is reasonable since these points are summarized from various experiments performed by different laboratories. The above result indicates that the finite size effect is the leading factor for the suppression of $T_C$ with the thin film samples at a thickness less than a few hundred nanometers, even with the obvious coexistence of the substrate strain often suggested as the solely factor.
For an LCMO cluster enclosed by the YSZ component at the small YSZ fraction, x $\leq$ 4.5 %, $T_C$ drops dramatically with the increasing x. In this region, the cluster size is on the order of several tens of micrometers. This is simply too large for the finite size confinement effect to occur according to the analysis for the thin films. For the YSZ serving as a non-magnetic separation between the LCMO phase, the reduction in the effective magnetic coupling is unlikely the major factor for the x dependence of the $T_C$ depression either. This effect would more or less level off at a layer thickness of a few nm according to the previous experiment[@Sirena03]. In the present work, the non-magnetic YSZ layer is at least 10 nm in thickness, reaching the region of saturation for such an effect. The effect of intergranular magnetic tunnelling coupling, which is beyond the DE mechanism, is unlikely the major factor either responsible for the observed properties. In this case, the depression of $T_C$ is less than 5 % with $T_P$ lower than $T_C$ by a temperature depending on the extent of the intergranular coupling strength. [@Sanchez96; @Gupta96; @Mahesh96; @Mahendrian96; @Hwang96; @Yuan03]. The main features of the present results, see Fig. \[mag1\], do not fit the description above since $T_C$ is depressed by more than 20 % with the varying x and $T_P$ follows it closely, see Fig. \[mag1\]. Furthermore, the insulating YSZ layer with a thickness of more than 10 nm is too thick for the electrical current to tunnel through at LT to show metallic property.
In the polycrystalline LCMO/YSZ composite system, the LCMO cluster is larger than 10 $\mu$m. The ratio of the boundary strained layer over the volume depends on the thickness of the strained layer. It is possible for the spatial relaxation of an interfacial strain to extend over a distance of 1 $\mu$m [@Soh02], and results in a non-negligible volume fraction of the boundary strained layer in the bulk LCMO component. According to the previous reports, $T_C$ would be seriously depressed by the biaxial strain resulting from the substrate lattice mismatch. Merely 1 % of the biaxial strain would cause an order of 10 % variation in $T_C$[@Millis98b], as demonstrated by the experiment of ultrasound spectroscopy[@Darling98]. Yet, such a low level of strain would go undetected by the usual experimental techniques such as the XRD analysis. The magnetic anisotropy or inhomogeneity caused by the strain would explain the depression of $T_C$, and the corresponding broadening of the magnetic transition, $\Delta T$.
In the analysis of $\rho(T)$ at $T < T_C$, the residual term, $\rho_0$, and the $AT^a$ term exhibit much less structure dependence than the $BT^b$ term. This is a strong evidence supporting that the electron-phonon coupling strength is modified by the existence of the BL. Since the Jahn-Teller (J-T) phonon mode depends strongly on the biaxial strain of the lattice [@Millis98b], it is reasonable to infer that the x dependence of the $BT^b$ term is caused by the biaxial strain, which affects the magnetic transition, $T_C$ as well.
The strong correlation of the TEP jump during the magnetic transition with the presence of the BL is interpreted as of magnetic origin[@Liu03], and can be reasonably explained by the magnetic inhomogeneity induced by the strain. At LT, $S(T)$ shows a typical metallic behavior with a small absolute value. As the temperature increases toward the HT region, the fraction of the PM component having the semiconducting property increases. For the x = 0 sample, the change in the PM fraction relative to the FM one is smooth, showing a smooth transition in $S(T)$. On the other hand, the introduction of the BL with the x $>$ 0 samples would cause an extra contribution from the inhomogeneous magnetism, resulting in an abrupt deviation of $S(T)$ from the metallic region. Interestingly, in the previou work on the series of samples with constant valence, $Pr_{0.7}Ca_{0.3-x}Sr_xMnO_3$ [@Hejtmanek96], the temperature-dependent TEP behavior has been demonstrated to correlate strongly with the magnetic transition. Especially, a TEP jump begins at the temperature near the ferromagnetic-antiferromagnetic (AF) transition as shown in Fig.4 by Hejtmanek [*et al*]{} [@Hejtmanek96]. According to the present picture in explaining the TEP behavior, the occurrence of the AF component within the FM matrix is responsible for the abrupt jump of the TEP. A noteworthy point, however, is that the cause of the inhomogeneous distribution of the magnetism for the $Pr_{0.7}Ca_{0.3-x}Sr_xMnO_3$ samples is attributed to the FM-AF transition, quite different from the existence of the BL-induced strain in the presence work.
Conclusion
==========
In conclusion, the YSZ component in the LCMO/YSZ composite materials induces a large effect on the various physical properties such as the variations of $T_C$ and $T_P$, the broadening of magnetic transition, the pronounced TEP jump during the magnetic transition, and the resistivity variation in the LT FM phase, [*etc.*]{}. The BL-induced strain plays a crucial role in the explanation of the observed properties. In this respect, the effect of strain is not only important in the manganite film already reported by some of the recent works, but also has a profound effect in the bulk sample, as demonstrated by the present work.
Acknowledgement
===============
We are grateful to Prof. Songliu Yuan of the Department of Physics, Huazhong University of Science and Technology, Wuhan, China, for providing us with the samples and for the helpful discussions. One of the authors, C.P. Chen, would also like to appreciate Prof. Tong-han Lin of the Department of Physics, Peking University, Beijing, China, for some of the points raised and the fruitful discussions.
e-mail address : [email protected], TEL : +86-10-62751751
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| ArXiv |
---
author:
- 'C.R. Cowley[^1]'
- 'S. Hubrig'
title: 'The absorption and emission spectrum of the magnetic Herbig Ae star HD 190073[^2]'
---
Introduction {#sec:intro}
============
The intriguing spectrum of the magnetic Herbig Ae star HD190073 (V1295Aql) has attracted the attention of classical as well as modern spectroscopists (Merrill 1933; Swings & Struve 1940; Catala et al. 2007, henceforth CAT). Pogodin, Franco & Lopes (2005, henceforth P05) give a detailed description of the spectrum along with a historical resume of investigations from the 1930’s. We note that the nature of HD190073 as a young, Herbig Ae star became widely recognized some three decades after Herbig’s (1960) seminal paper.
HD190073 was included among the 24 young stars studied for abundances by Acke & Waelkens (2004, henceforth, AW). In this important paper, the authors made the bold assumption that abundances could be determined for stars of this nature using standard techniques-plane parallel, one dimensional models, in hydrostatic equilibrium. The models were used to obtain abundances from absorption lines with equivalent widths less than 150 mÅ. These assumptions might very well be questioned. Material is being accreted by these young stars, and the infall velocities are thought to be near free-fall, several hundred kms$^{-1}$. Does this infall produce shocks and heating of the atmospheres that could invalidate models that neglect such complications?
AW nevertheless proceeded. Although they did not state this explicitly, the justification for their assumptions is empirical, and may be found in their results. Basically, these are the fact that their approach yields entirely reasonable stellar parameters and abundances, including agreement from lines of neutral and ionized elements. Stated simply, their assumptions led them to self-consistent results. We make these same assumptions in the present work, taking some comfort in the fact that self consistency is all one ever has in science.
While AW’s studies were both competent and thorough, better observational material is currently available, making it possible to use systematically weaker lines, and to study more elements. We have also made use of the wings of the Balmer lines, not used by AW.
The lower Balmer lines have central emissions. In the case of H$\alpha$, the emission dominates the feature. The Balmer lines and especially H$\alpha$, have been extensively studied (e.g. P05; Cuttela & Ringuelet 1990).
In the present paper we also study the weaker metallic emission lines, to provide information on the physical conditions where this emission occurs. This was discussed by CAT who were primarily concerned with the magnetic field of HD 190073. They also give a detailed qualitative description of the metallic emission lines (primarily Ti , Fe / (cf. Sect. \[sec:emiss\] and following).
Hubrig et al. (2006, 2009) reported a longitudinal magnetic field of 84$\pm$30 Gauss, up to 104$\pm$19 Gauss, while CAT found a longitudinal field of 74$\pm$10 Gauss.
Observations {#sec:obs}
============
We downloaded 8 HARPS spectra from the ESO archive, all obtained on 11 November 2008 within 74 minutes of one another. These were averaged, binned to 0.02Å and mildly Fourier filtered. The resulting spectrum has a signal-to-noise ratio of 350 or more. The resolution of HARPS spectra usually cited as over 100000, is not significantly modified for our purposes on the averaged spectrum, as Fig. \[fig:line4481\] illustrates.
![ The HARPS spectrum (ADP.HARPS.2008-11-10T23:43:14.386\_2\_SID\_A) and averaged spectra in the region of the Mg doublet $\lambda$4481Å. The HARPS (gray and red in online version) spectrum has been displaced slightly downward for display purposes.[]{data-label="fig:line4481"}](4481.ps){width="55mm" height="83mm"}
UVES spectra, obtained on 18 September 2005 cover the region from 3044 to 10257 Å. They were used for special features (e.g. the \[Ca \] lines), but not for abundances.
Reduction {#sec:reduction}
=========
![The region from $\lambda\lambda$4000 to 4050 Å shows numerous measurable absorption features. Note the broad absorption at $\lambda$4026 Å, which is He . Fe $\lambda$4045 Å shows strong emission as well as absorption.[]{data-label="fig:line4050"}](4050.ps){width="54mm" height="83mm"}
The averaged HARPS spectrum was measured for 1796 lines. The UVES spectrum was also measured for line identifications in the region $\lambda\lambda$3054–3867 Å. We measured 760 absorption lines, which were often severely affected by emission.
Many absorption lines, especially weak ones, were not significantly affected by emission, and suitable for abundance determinations. Figure \[fig:line4050\] shows a typical region with many relatively unperturbed absorptions. Preliminary, automated identifications were made, and wavelength coincidence statistics (WCS, Cowley & Hensberge 1981) were performed. A few spectra not investigated by AW were analyzed: He , Na , Al , Si , S , S , Co , Mn , Mn , Ni , Zn , and Sr . We found no exotic elements, such as lanthanides, or unusual 4d or 5d elements.
![ A comparison of equivalent width measurements by AW and the present study (UMICH).[]{data-label="fig:pltdif"}](pltdif.ps){width="54mm" height="83mm"}
Lines were chosen for equivalent width measurement with the help of the automated identification list, which lists plausible identifications within 0.03 Å of any measured feature. Blends were rejected. Usually, we avoided lines with equivalent widths greater than 20 mÅ but in order to compare our measurements with those of AW, we included a few stronger lines.
A comparison of measurements is given in Fig. \[fig:pltdif\]. Generally, the measurements agree well with one another, and differences can usually be explained by judgments of where to draw the continuum when a line is partially in emission, or there is emission close by. Differences in the case of one of the solid circles is surely due to emission, as Ti $\lambda$4398 Å falls between two strong emission lines. The other solid point is for O $\lambda$3947 Å. This is apparently a misidentification. Note that Fig. \[fig:pltdif\] is not logarithmic.
The model atmosphere and abundance methods {#sec:model}
==========================================
The methods used to obtain abundances from the equivalent widths, including model atmosphere construction are explained in some detail in two previous papers (Cowley et al. 2010a,b). Briefly, the $T(\tau_{5000})$ from Atlas 9 (Kurucz 1993) as implemented by Sbordone et al. (2004) was used with Michigan software to product depth-dependent models. The effective temperature and gravity were selected from ionization and excitation equilibrium as well as fits to the wings of H$\beta$–H$\delta$.
We have adopted a somewhat lower temperature than used by AW, 8750 K, and $\log g = 3.0$. The former used 9250 K, and $\log g=3.5$. We also used a lower microturbulence, 2 kms$^{-1}$, compared to AW’s 3 kms$^{-1}$, but this is not important for most of our weaker lines. Oscillator strengths were taken from the modern literature when possible, or from compilations by NIST (Ralchenko 2010, preferred) or VALD (Kupka et al. 1999). Default damping constants were used as in the studies cited, but they are unimportant for weak lines.
Abundances
==========
--------------- ------------------------- ------ ----- ------- -------
Ion $\log({\rm El}/{\Sum})$ sd $N$ Sun AW
\[1.5pt\] He –1.15 0.38 2 –1.11
C –3.40 0.23 36 –3.61 –3.55
[**N** ]{}\* –3.50 0.38 9 –4.21 –3.40
O –3.29 0.10 12 –3.35 –3.38
[**Na** ]{} –5.25 0.24 5 –5.80
Mg –4.29 0.23 3 –4.44 –4.52
Mg –4.54 0.16 8 –4.44
[**Al** ]{}\* –6.07 1 –5.59 –6.01
Si –4.43 0.36 7 –4.53 –4.41
Si –4.61 0.13 10 –4.53
S –4.62 0.06 3 –4.92
S –4.40 0.45 6
Ca –5.78 0.11 2 –5.70 –5.41
Ca –5.63 0.19 6 –5.70
[**Sc** ]{}\* –9.16 0.13 10 –8.89 –9.00
Ti –7.18 0.19 32 –7.09 –7.07
V –8.07 0.11 14 –8.11 –7.93
Cr –6.54 0.12 6 –6.40 –6.35
Cr –6.37 0.17 22 –6.40
Mn –6.60 0.28 3 –6.61
Mn –6.53 0.24 13 –6.61
Fe –4.54 0.15 182 –4.54 –4.53
Fe –4.54 0.21 145 –4.54
Co –7.13 1 –7.05
Ni –5.86 0.15 18 –5.82 –5.73
Ni –5.68 0.23 5 –5.82
Zn –7.48 0.08 2 –7.48
Sr –8.53 0.54 2 –9.17
Y –10.05 0.19 8 –9.83 –9.79
Zr –9.43 0.15 13 –9.46 –9.12
Ba –9.88 0.12 3 –9.86 –9.72
\[1.5pt\]
--------------- ------------------------- ------ ----- ------- -------
: Abundances in HD190073 from the current study and AW.[]{data-label="tab:abund"}
The AW abundances have been converted from differential values, using Anders & Grevesse (1989) abundances, which AG adopted. Our abundances (see Table \[tab:abund\]) refer to the Asplund et al. (2009) scale. The case is not strong that any of these abundances differ significantly from the solar abundance. Nevertheless, we have highlighted in bold face some elements that deserve additional attention. Nitrogen, in particular, deserves attention, as it has been found in excess in the Herbig Ae star HD 101412 (Cowley et al. 2010a). Asterisks mark cases where AW and the present work agree on possibly significant departures from solar abundances. NLTE effects could also be responsible for some non-solar abundances (Kamp et al. 2001).
Neutral helium {#sec:he1}
--------------
The helium abundance is from $\lambda\lambda$4026 (see Fig. \[fig:line4050\]) and 4713 Å. Both lines are weak, but in excellent agreement with one another. However $\lambda$4471 Å was also found in absorption, and analyzed. The value of $\log({\rm He}/\Sum)$ from this line was found to be $-$1.53, some 0.4 dex below the mean of the other two lines. We have chosen to disregard this value, as possibly weakened by partial emission. Should it be included in the average, we find $\log({\rm He}/\Sum) = -1.29\pm 0.21$(sd), still solar, within the errors. The D$_3$ line ($\lambda$5876 Å) of He is in emission, and included in Table \[tab:lindat\]. It was observed at numerous phases by P05, whose observations (see their Fig. 3) do not show the central reversal clearly seen in our Fig. \[fig:d3\]. This feature was measured at $\lambda^*$5875.60, virtually unshifted from the expected photospheric position $\lambda$5875.64. Moreover, an LTE synthesis using Voigt profiles and an assumed solar abundance matches the observed absorption in shape and strength. Given the likelihood of NLTE, it is unclear how seriously to take this result. Nevertheless, the D$_3$ absorption is consistent with photospheric absorption, and a solar abundance.
![ He D$_3$ line in the spectrum of HD190073. The central absorption is arguably photospheric, and agrees in shape and strength with a calculated absorption profile.[]{data-label="fig:d3"}](d3.ps){width="54mm" height="83mm"}
The D$_3$ emission is remarkably similar in morphology to the metallic emissions (with unshifted absorptions), though it is much broader. It resembles the P05 illustrations at phases ‘a’ or ‘e’, with a maximum shifted somewhat to the red, and a longer violet tail.
The emission spectrum {#sec:emiss}
=====================
A second focus of the current paper is the emission spectrum, in particular, permitted and forbidden metallic lines.
![The emission/absorption, metallic-line spectrum of two Herbig Ae stars, contrasted. Note the proclivity of the intrinsically stronger lines to be in emission in HD 190073.[]{data-label="fig:p2dat"}](p2dat.ps){width="54mm" height="83mm"}
Previous studies (P05, CAT) provide detailed descriptions of numerous metallic emission lines, primarily of Fe and , Ti , and Cr , as well as the Na D-lines, and have illustrations similar to the upper spectrum of Fig. \[fig:p2dat\]. As the P05 work has a temporal dimension lacking in the present study, we briefly summarize their findings. The emissions show mild temporal variations both in strengths and widths. The profiles are somewhat asymmetric, with their peak intensities generally very slightly red shifted with respect to the photospheric absorption spectrum. The widths of the features are significantly larger than these shifts, and a considerable fraction of the emission is shifted to the blue.
The P05 observations are all in good agreement with the current findings, which we take to be a representative sample. Table \[tab:lindat\] gives measurements of the peak intensities, equivalent widths, and FWHM for the strong relatively unblended emission lines as they appear on our averaged HARPS spectra. Multiplet numbers follow the spectrum designation. The intensities are in units of the continuum, and the equivalent widths are the areas of the emissions above the continuum, which is assumed to have unit intensity. The measurements are from segments fitted by eye to the emission lines, as illustrated in Fig. \[fig:5183\].
![ Segment fits (black) to the emission from Mg-2 $\lambda$5183.60 Å, the strongest of the Mgb lines. Observations: gray (red online) with dots. The portion of the fitted curve near the central absorption is a by-eye estimate of the missing part of the profile. The maximum designated $I^0$, and other properties of similar interpolations are given in Table \[tab:lindat\].[]{data-label="fig:5183"}](5183.ps){width="54mm" height="83mm"}
All lines in the table had central absorptions. In measuring the $I^0$-values, an attempt was made to interpolate over this absorption, so the $I^0$-value is a few per cent higher than the maximum of the emission. The accuracy of the measurements vary. Repeated measurements show consistency for FWHM are generally within 10%. Underlying emission from other lines is the prime cause of the uncertainty.
------------------- ------------- ------- ---------- ------- --------------
$\lambda$ Ion/Mult. $I^0$ $W$\[Å\] FWHM FWHM
\[Å\] \[kms$^{-1}$
\[1.5pt\] 4045.81 Fe I-43 1.11 0.130 1.02 75.6
4063.59 Fe I-43 1.08 0.065 0.83 61.2
4071.74 Fe I-43 1.11 0.113 0.82 60.4
4077.71 Sr II-1 1.12 0.109 0.87 64.0
4143.87 Fe I-43 1.05 0.032 0.61 44.1
4163.65 Ti II-105 1.05 0.056 0.93 67.0
4173.46 Fe I-27 1.12 0.095 0.80 57.5
4178.86 Fe II-28 1.12 0.180 1.26 90.4
4215.52 Sr II-1 1.06 0.472 0.81 57.6
4233.17 Fe II-27 1.27 0.402 1.29 91.4
4246.82 Sc II-7 1.07 0.062 0.82 57.9
4271.76 Fe I-42 1.06 0.620 0.82 57.5
4290.22 Ti II-42 1.12 0.112 0.87 60.8
4294.10 Ti II-20 1.13 0.119 0.82 57.2
4300.05 Ti II-41 1.23 0.308 1.06 73.9
4307.86 Ti II-41 1.15 0.179 1.04 72.4
4351.77 Fe I-27 1.28 0.420 1.31 90.2
4383.55 Fe I-41 1.14 0.119 0.83 56.8
4404.75 Fe I-41 1.08 0.077 0.86 58.5
4443.79 Ti II-19 1.24 0.311 1.06 71.5
4450.48 Ti II-19 1.06 0.075 1.04 70.1
4468.51 Ti II-31 1.25 0.281 0.89 59.7
4491.41 Fe II-37 1.12 0.155 1.15 76.8
4501.27 Ti II-31 1.24 0.319 1.03 68.6
4508.29 Fe II-38 1.17 0.198 1.04 69.2
4515.34 Fe II-37 1.12 0.176 1.25 83.0
4533.97 Ti II-50 1.19 0.261 1.18 78.0
4541.52 Fe II-38 1.06 0.048 0.80 52.8
4558.65 Cr II-44 1.14 0.186 1.17 76.9
4563.76 Ti II-50 1.16 0.183 0.97 63.8
4571.97 Ti II-82 1.26 0.332 1.07 70.2
4576.34 Fe II-38 1.07 0.081 0.97 63.5
4588.20 Cr II-44 1.09 0.105 0.96 62.7
4618.80 Cr II-44 1.11 1.350 1.07 69.4
4629.34 Fe II-37 1.17 0.211 1.00 64.8
4634.07 Cr II-44 1.04 0.043 0.87 56.3
4731.45 Fe II-43 1.05 0.058 0.93 58.9
4805.09 Ti II92 1.04 0.332 0.86 53.7
4824.13 Cr II-30 1.08 0.953 1.06 65.9
4923.92 Fe II-42 1.65 1.340 1.70 103.5
4957.60 Fe I-318 1.74 1.590 1.82 110.1
5169.03 Fe II-42 1.77 1.890 2.14 124.1
5183.60 Mg I-2 1.23 0.303 1.10 63.6
5197.58 Fe II-49 1.23 0.231 1.23 70.9
5234.63 Fe II49 1.22 0.437 1.22 69.9
5264.81 Fe II-48 1.04 0.030 0.66 37.6
5284.11 Fe II-41 1.08 0.117 1.20 68.1
5362.87 Fe II-48 1.15 0.216 1.22 68.2
5534.85 Fe II-55 1.10 0.140 1.10 59.6
5875.64 He I-4 1.09 0.390 4.17 212.7
5889.95 Na I-D$_2$ 1.76 1.380 1.38 70.2
5895.92 Na I-D$_1$ 1.70 1.130 1.36 69.2
5991.38 Fe II-55p 1.03 0.043 1.38 69.1
6238.39 Fe II-74 1.07 0.078 1.14 54.8
6247.56 Fe II-74 1.13 0.217 1.46 70.1
6347.11 Si II-2 1.11 0.325 2.50 118.1
6371.37 Si II-2 1.08 0.228 2.86 134.6
6416.92 Fe II-74 1.06 0.098 1.42 65.7
6432.68 Fe II-40 1.05 0.073 1.16 54.1
6562.82 H I 6.82 32.20 5.34 243.9
5158.78 Fe II-19F\] 1.04 0.012 0.282 16.4
5577.35 O I-3F\] 1.01 0.004 0.44 23.4
6300.30 O I-1F\] 1.11 0.056 0.41 19.3
6363.78 O I-1F\] 1.04 0.019 0.42 19.7
7291.47 Ca II-1F\] 1.11 0.046 0.43 17.8
7323.89 Ca II-1F\] 1.10 0.033 0.28 11.3
\[1.5pt\]
------------------- ------------- ------- ---------- ------- --------------
: Maximum intensity measurements, equivalent widths, and full widths at half maximum for selected emission lines.[]{data-label="tab:lindat"}
Resumé: the permitted emissions
-------------------------------
We summarize salient properties of the permitted emission lines:
- The centers of gravity are shifted by ca. 5 to the red.
- The profiles are somewhat asymmetric, with a longer violet than red tail.
- The central absorption wavelengths are photospheric within the errors of measurement. That is, the radial velocities of the weaker photospheric absorptions agree with those of the central absorption components of the emission lines.
- The emission lines are the intrinsically strongest lines. Weaker lines show weaker emission, until, for the weakest lines, the observed features are all in absorption.
CAT suggested the emissions arose in conditions similar to those of a photosphere. They suggested a heated region with densities and temperatures in the range of $10^{13}$–$10^{14}$ cm$^{-3}$ and $15\,000$–$20\,000$K.
Their value of 65 kms$^{-1}$ as a typical FWHM for the emissions agrees well with our measurements (Table \[tab:lindat\]). The origin of this velocity, however, is not readily apparent. They speculate that these velocities are due to a supersonic turbulence. Such “turbulence” might arise from the roil of accreting material settling onto the photosphere.
Forbidden lines\[sec:forbidln\]
-------------------------------
### The \[O \] lines\[sec:ForO1\]
![The nebular ($\rm ^3P_2$–$^1\rm D_2$) \[O \] transition $\lambda$6300 Å. The long arrow points to the laboratory position at 6300.30 Å. The stellar feature seems slightly red shifted. The short arrow points to a narrow, blue-shifted satellite feature that is also seen at the same displacement in \[O \] $\lambda\lambda$6363 and 5577 Å. The strong absorption lines are atmospheric.[]{data-label="fig:6300"}](6300.ps){width="55mm" height="83mm"}
Both $\lambda\lambda$6300 and 6363 \[O \] are present as well as the auroral transition $\lambda$5577 Å. In addition to what we shall call the main features, all three lines show faint, sharp, “satellite” components shifted to the violet by ca. 25. This structure is illustrated in Figs. \[fig:6300\] and \[fig:5577d\]. The main \[O \] features are roughly one third the width of the typical permitted metallic-line features, but their peak intensities have comparably small red shifts of ca. 5 . It is plausible to assume the \[O \] arises in a region further from the star, and therefore of lower density than the gas giving rise to the permitted metallic lines.
The satellite emissions may arise in a polar stream, if we assume the system is viewed pole on. The velocity, however, is not high. The satellite features of all three \[O \] lines is ca. $-25$ .
### The \[Ca -F1\] doublet {#sec:forbidca}
![The \[Ca \] lines $\lambda\lambda$7291 (black) and 7324 Å (gray, red in online version) from multiplet 1-F. Wavelengths for $\lambda$7324 Å are at the top abscissa. The arrow indicates a possible narrow component of $\lambda$7291 Å corresponding to those seen in \[O \].[]{data-label="fig:forca2"}](forca2.ps){width="55mm" height="83mm"}
Hamann (1994) has noted the presence of \[Ca \] in a number of young stars, including the Herbig Ae V380Ori. We are not aware that the lines have been previously noted in HD190073. They are also seen in supernovae (Kirshner & Kwan 1975) and extragalactic spectra. (Donahue & Voit 1993). Merrill (1943) reported \[Ca II\] lines in emission in the peculiar hydrogen-poor binary $\upsilon$ß,Sgr (see also Greenstein & Merrill 1946).
We find definite emissions at the positions of the forbidden $\rm ^2S_{1/2}$–$ ^2\rm D_{3/2,5/2}$ transitions. The air wavelengths, determined from the energy levels, are 7291.47 and 7323.89 Å. These features were identified on a the UVES spectrogram. The spectrum (Fig. \[fig:forca2\]) is too noisy or blended to show the presence of satellite features of $\lambda$7324 Å, but it might be present for $\lambda$7291 Å. Measurements of the main features are included in Table \[tab:lindat\]. The maxima are shifted to the red by 2 to 4 , in general agreement with the \[O\] and metallic lines. The FWHM agree with those of other forbidden lines.
### \[Fe \]\[sec:fe2forbid\]
![ Forbidden Fe lines from multiplet 19-F. The wavelength scale for the weaker of the two lines, $\lambda$5376 Å (gray, red in online version) is at the top of the figure. Vertical lines mark the positions of wavelengths derived from the atomic energy levels.[]{data-label="fig:5158"}](5158.ps){width="55mm" height="83mm"}
Several workers have discussed \[Fe II\] emission lines in Herbig Ae/Be stars (Finkenzeller 1985; Donati et al. 1997). We find a definite, sharp emission feature with a maximum measured at $\lambda^*$5158.84 Å. This wavelength is close to that of \[Fe -19F\], $\lambda$5158.78 Å. (Laboratory positions for \[Fe \] are from Fuhr & Wiese (2006) rather than the RMT). Another line from this multiplet is weakly present (Fig. \[fig:5158\]). Both features are seen on the unaveraged HARPS spectra as well as UVES spectra taken some 3 years previously (see Sect. \[sec:obs\]). Several other lines in Multiplet 19-F are arguably present ($\lambda\lambda$5261, 5296, 5072 Å), other lines are masked by blends or in a HARPS order gap. The line $\lambda$5158 Åis entered in Table \[tab:lindat\]. It has a FWHM comparable to that of the other forbidden lines.
We found no other \[Fe \] features that could be said to be unambiguously present. The well-known \[Fe \] line $\lambda$4244 Å is at best, marginally present.
### Physical conditions from the forbidden lines
“Critical electron densities” are obtained from observed forbidden transitions by equating the Einstein spontaneous decay rate to the collisional deexcitation rate. Typical values are given by Draine (2011) for $T_{\rm e} = 10\,000$K. For the \[O\] $\rm ^1D_2$-level (the upper level of $\lambda\lambda$6300 and 6363 Å), the critical $N_{\rm e}$ is 1.6$\times 10^6$ cm$^{-3}$. We calculate a similar critical density for the \[Ca \] lines with the help of rates calculated by Burgess et al. (1995), and a lifetime of the $\rm ^2D$ term given by NIST (Ralchenko et al. 2010).
We see no evidence of the \[S -2F\] pair at $\lambda\lambda$6717 and 6731 Å, though they have been observed in Herbig Ae/Be stars (Corcoran & Ray 1997). For this pair, Draine gives critical densities of $10^3$–$10^4$ cm$^{-3}$. We conclude the forbidden lines we do see arise in a region where the electron density is between $10^4$ and $10^{6-7}$ cm$^{-3}$.
![ The auroral ($\rm ^1D_2$–$ ^1\rm S_0$) \[O \] transition $\lambda$5577 (gray, red in online version). The long arrow points to 5577.34 Å, the NIST wavelength. The maximum, and center of gravity of the line shifted slightly to the red, as are the nebular lines (Fig. \[fig:6300\]). The shorter arrow points to the sharp, blue-shifted feature with the same displacement as the sharp component of \[O \] $\lambda\lambda$6300 and 6363 Å.[]{data-label="fig:5577d"}](5577d.ps){width="55mm" height="83mm"}
When the nebular as well as auroral transitions of \[O \] are available, the ratio may allow one to determine values of $T_{\rm e}$ and $N_{\rm e}$ compatible with the observation. The average of three measurements on $\lambda$5577 gives $W = 0.0042$Å, which with $W = 0.056\,$Å for $\lambda$6300 yields a ratio of 13.3. If we assume the excited levels of O arise from electron excitation, we may interpolate in the plot of Gorti et al. (2011) to find acceptable the values given in Table \[tab:Gorti\]. With the electron density constraint given above, we find temperatures in the range 7500 to 10000 K for the volume where the forbidden lines are formed.
An alternate interpretation of \[O \] emission in Herbig Ae/Be systems is discussed by Acke, van den Ancker & Dullemond (2005). In their model, the excited O levels arise primarily from the photodissociation of the OH molecule.
---------------- ------------------
$T_{\rm e}$ $\log N_{\rm e}$
\[1.5pt\] 5000 8
7500 7
10000 6.5
12000 6.2
\[1.5pt\]
---------------- ------------------
: Values of $T_{\rm e}$ (in K) and $N_{\rm e}$ (in cm$^{-3}$) compatible with the observed $\lambda$6300/5577-ratio = 13.3.[]{data-label="tab:Gorti"}
We are grateful for the availability of the ESO archive. This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. Our calculations have made extensive use of the VALD atomic data base (Kupka et al. 1999), as well as the facilities provided by NIST (Ralchenko et al. 2010). CRC thanks colleagues at Michigan for many helpful suggestions. Jesús Hernández suggested that we examine the forbidden lines.
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[^1]: Corresponding author: [[email protected]]{}
[^2]: Based on ESO Archival data, from ESO programme076.B-0055(A) and programme 082.D-0833(A)
| ArXiv |
---
abstract: '[The formation of singularities on a free surface of a conducting ideal fluid in a strong electric field is considered. It is found that the nonlinear equations of two-dimensional fluid motion can be solved in the small-angle approximation. This enables us to show that for almost arbitrary initial conditions the surface curvature becomes infinite in a finite time. ]{}'
---
[**Formation of Root Singularities on the Free Surface\
of a Conducting Fluid in an Electric Field**]{}
[**N. M. Zubarev**]{}
Electrohydrodynamic instability of a free surface of a conducting fluid in an external electric field \[1,2\] plays an essential role in a general problem of the electric strength. The interaction of strong electric field with induced charges at the surface of the fluid (liquid metal for applications) leads to the avalanche-like growth of surface perturbations and, as a consequence, to the formation of regions with high energy concentration which destruction can be accompanied by intensive emissive processes.
In this Letter we will show that the nonlinear equations of motion of a conducting fluid can be effectively solved in the approximation of small perturbations of the boundary. This allows us to study the nonlinear dynamics of the electrohydrodynamic instability and, in particular, the most physically meaningful singular solutions.
Let us consider an irrotational motion of a conducting ideal fluid with a free surface, $z=\eta(x,y,t)$, that occupies the region $-\infty<z\leq\eta(x,y,t)$, in an external uniform electric field $E$. We will assume the influence of gravitational and capillary forces to be negligibly small, which corresponds to the condition $$E^2\gg 8\pi\sqrt{g\alpha\rho},$$ where $g$ is the acceleration of gravity, $\alpha$ is the surface tension coefficient, and $\rho$ is the mass density.
The potential of the electric field $\varphi$ satisfies the Laplace equation, $$\Delta\varphi=0,$$ with the following boundary conditions, $$\varphi\to -Ez, \qquad z\to\infty,$$ $$\varphi=0, \qquad z=\eta.$$ The velocity potential $\Phi$ satisfies the incompressibility equation $$\Delta\Phi=0,$$ which one should solve together with the dynamic and kinematic relations on the free surface, $$\frac{\partial\Phi}{\partial t}+\frac{(\nabla\Phi)^2}{2}=
\frac{(\nabla\varphi)^2}{8\pi\rho}+F(t), \qquad z=\eta,$$ $$\frac{\partial\eta}{\partial t}=\frac{\partial\Phi}{\partial z}
-\nabla\eta\cdot\nabla\Phi,
\qquad z=\eta,$$ where $F$ is some function of variable $t$, and the boundary condition $$\Phi\to 0, \qquad z\to-\infty.$$ The quantities $\eta(x,y,t)$ É $\psi(x,y,t)=\Phi|_{z=\eta}$ are canonically conjugated, so that the equations of motion take the Hamiltonian form \[3\], $$\frac{\partial\psi}{\partial t}=-\frac{\delta H}{\delta\eta},
\qquad
\frac{\partial\eta}{\partial t}=\frac{\delta H}{\delta\psi},$$ where the Hamiltonian $$H=\int\limits_{z\leq\eta}\frac{(\nabla\Phi)^2}{2} d^3 r
-\int\limits_{z\geq\eta}\frac{(\nabla\varphi)^2}{8\pi\rho} d^3 r$$ coincides with the total energy of a system. With the help of the Green formula it can be rewritten as the surface integral, $$H=\int\limits_{s}\left[\frac{\psi}{2}\,\frac{\partial\Phi}{\partial n}+
\frac{E\eta}{8\pi\rho}\,\frac{\partial\tilde\varphi}{\partial n}\right]ds,$$ where $\tilde\varphi=\varphi+Ez$ is the perturbation of the electric field potential; $ds$ is the surface differential.
Let us assume $|\nabla\eta|\ll 1$, which corresponds to the approximation of small surface angles. In such a case we can expand the integrand in a power series of canonical variables $\eta$ and $\psi$. Restricting ourselves to quadratic and cubic terms we find after scale transformations $$t\to t E^{-1}(4\pi\rho)^{1/2},
\quad
\psi\to\psi E/(4\pi\rho)^{1/2},
\quad
H\to HE^2/(4\pi\rho)$$ the following expression for the Hamiltonian, $$H=\frac{1}{2}\int\left[\psi\hat k\psi+
\eta\left((\nabla\psi)^2-(\hat k\psi)^2\right)\right] d^2 r$$ $$-\frac{1}{2}\int\left[\eta\hat k\eta-\eta\left((\nabla\eta)^2-
(\hat k\eta)^2\right)\right] d^2 r.$$ Here $\hat k$ is the integral operator with the difference kernel, whose Fourier transform is the modulus of the wave vector, $$\hat{k}f=-\frac{1}{2\pi}\!\int\limits_{-\infty}^{+\infty}
\int\limits_{-\infty}^{+\infty}
\frac{f(x',y')}{\left[(x'-x)^2+(y'-y)^2\right]^{3/2}}\,dx'dy'.$$ The equations of motion, corresponding to this Hamiltonian, take the following form, $$\psi_t-\hat k\eta=\frac{1}{2}\left[(\hat k\psi)^2-(\nabla\psi)^2+
(\hat k\eta)^2-(\nabla\eta)^2\right]+
\hat k(\eta\hat k\eta)+\nabla(\eta\nabla\eta),$$ $$\eta_t-\hat k\psi=-\hat k(\eta\hat k\psi)-\nabla(\eta\nabla\psi).$$ Subtraction of Eqs. (2) and (1) gives in the linear approximation the relaxation equation $$(\psi-\eta)_t=-\hat k(\psi-\eta),$$ whence it follows that we can set $\psi=\eta$ in the nonlinear terms of Eqs. (1) and (2), which allows us to simplify the equations of motion. Actually, adding Eqs. (1) and (2) we obtain an equation for a new function $f=(\psi+\eta)/2$, $$f_t-\hat k f=\frac{1}{2}\,(\hat k f)^2-\frac{1}{2}\,(\nabla f)^2,$$ which corresponds to the consideration of the growing branch of the solutions. As $f=\eta$ in the linear approximation, Eq. (3) governs the behavior of the elevation $\eta$.
First we consider the one-dimensional case when function $f$ depends only on $x$ (and $t$) and the integral operator $\hat k$ can be expressed in terms of the Hilbert transform $\hat H$, $$\hat k=-\frac{\partial}{\partial x}\,\hat H,
\qquad
\hat{H}f=\frac{1}{\pi}\,\mbox{P}\!\!\int\limits_{-\infty}^{+\infty}
\frac{f(x')}{x'-x}\,dx',$$ where P denotes the principal value of the integral. As a result, Eq. (3) can be rewritten as $$f_t+\hat H f_x=\frac{1}{2}\,(\hat H f_x)^2-\frac{1}{2}\,(f_x)^2.$$ It should be noted that if one introduces a new function $\tilde f=\hat H f$, then Eq. (4) transforms into the equation proposed in Ref. \[4\] for the description of the nonlinear stages of the Kelvin-Helmholtz instability.
For further consideration it is convenient to introduce a function, analytically extendable into the upper half-plane of the complex variable $x$, $$v=\frac{1}{2}\,(1-i\hat H)f_x.$$ Then Eq. (4) takes the form $$\mbox{Re}\left(v_t+iv_x+2vv_x\right)=0,$$ that is, the investigation of integro-differential equation (4) amounts to the analysis of the partial differential equation $$v_t+iv_x+2vv_x=0,$$ which describes the wave breaking in the complex plane. Let us study this process in analogy with \[5,6\], where a similar problem was considered. Eq. (5) can be solved by the standard method of characteristics, $$v=Q(x'),$$ $$x=x'+it+2Q(x')t.$$ where the function $Q$ is defined from initial conditions. It is clear that in order to obtain an explicit form of the solution we must resolve Eq. (7) with respect to $x'$. A mapping $x\to x'$, defined by Eq. (7), will be ambiguous if $\partial x/\partial x'=0$ in some point, i.e. $$1+2Q_{x'}t=0.$$ Solution of (8) gives a trajectory $x'=x'(t)$ on the complex plane $x'$. Then the motion of the branch points of the function $v$ is defined by an expression $$x(t)=x'(t)+it+2Q(x'(t))t.$$ At some moment $t_0$ when the branch point touches the real axis, the analiticity of $v(x,t)$ at the upper half-plane of variable $x$ breaks, and a singularity appears in the solution of Eq. (4).
Let us consider the solution behavior close to the singularity. Expansion of (6) and (7) at a small vicinity of $x=x(t_0)$ up to the leading orders gives $$v=Q_0-\delta x'/(2t_0),$$ $$\delta x=i\delta t+2Q_0\delta t+Q''t_0(\delta x')^2,$$ where $Q_0=Q(x'(t_0))$, $Q''=Q_{x'x'}(x'(t_0))$, $\delta x\!=\!x\!-\!x(t_0)$, $\delta x'\!=\!x'\!-\!x'(t_0)$, and $\delta t\!=\!t\!-\!t_0$. Eliminating $\delta x'$ from these equations, we find that close to singularity $v_x$ can be represented in the self-similar form ($\delta x\sim\delta t$), $$v_x=-\left[16Q''t_0^3
(\delta x-i\delta t-2Q_0\delta t)\right]^{-1/2}.$$ As $\mbox{Re}(v)=\eta/2$ in the linear approximation, we have at $t=t_0$ $$\eta_{xx}\sim|\delta x|^{-1/2},$$ that is the surface curvature becomes infinite in a finite time. It should be mentioned that such a behavior of the charged surface is similar to the behavior of a free surface of an ideal fluid in the absence of external forces \[5,6\], though the singularities are of a different nature (in the latter case the singularity formation is connected with inertial forces).
Let us show that the solutions corresponding to the root singularity regime are consistent with the applicability condition of the truncated equation (3). Let $Q(x')$ be a rational function with one pole in the lower half-plane, $$Q(x')=-\frac{is}{2(x'+iA)^2},$$ which corresponds to the spatially localized one-dimensional perturbation of the surface ($s>0$ and $A>0$). The characteristic surface angles are thought to be small, $\gamma\approx s/A^2\ll 1$.
It is clear from the symmetries of (9) that the most rapid branch point touches the real axis at $x=0$. Then the critical moment $t_0$ can be found directly from Eqs. (7) and (8). Expansion of $t_0$ with respect to the small parameter $\gamma$ gives $$t_0\approx A\left[1-3(\gamma/4)^{1/3}\right].$$ Taking into account that the evolution of the surface perturbation can be described by an approximate formula $$\eta(x,t)=\frac{s(A-t)}{(A-t)^2+x^2},$$ we have for the dynamics of the characteristic angles $$\gamma(t)\approx\frac{s}{(A-t)^2}.$$ Then, substituting the expression for $t_0$ (10) into this formula, we find that at the moment of the singularity formation with the required accuracy $$\gamma(t_0)\sim\gamma^{1/3},$$ that is, the angles remain small and the root singularities are consistent with our assumption about small surface angles.
In conclusion, we would like to consider the more general case where the weak dependence of all quantities from the spatial variable $y$ is taken into account. One can find that if the condition $|k_x|\ll|k_y|$ holds for the characteristic wave numbers, then the evolution of the fluid surface is described by an equation $$\left[v_t+iv_x+2vv_x\right]_x=-iv_{yy}/2,$$ which extends Eq. (5) to the two-dimensional case.
An interesting group of particular solutions of this equation can be found with the help of substitution $v(x,y,t)=w(z,t)$, where $$z=x-\frac{i}{2}\,\frac{(y-y_0)^2}{t}.$$ The equation for $w$ looks like $$w_t+iw_z+2ww_z=-w/(2t).$$ It is integrable by the method of characteristics, so that we can study the analyticity violation similarly to the one-dimensional case. Considering a motion of branch points in the complex plane of the variable $z$ we find that a singularity arises at some moment $t_0<0$ at the point $y_0$ along the $y$-axis. Close to the singular point at the critical moment $t=t_0$ we get $$\left.\eta_{xx}\right|_{\delta y=0}\sim|\delta x|^{-1/2},
\qquad
\left.\eta_{xx}\right|_{\delta x=0}\sim|\delta y|^{-1}.$$ This means that in the examined quasi-two-dimensional case the second derivative of the surface profile becomes infinite at a single isolated point.
Thus, the consideration of the behavior of a conducting fluid surface in a strong electric field shows that the nonlinearity determines the tendency for the formation of singularities of the root character, corresponding to the surface points with infinite curvature. We can assume that such weak singularities serve as the origin of the more powerful singularities observed in the experiments \[7,8\].
I would like to thank A.M. Iskoldsky and N.B. Volkov for helpful discussions, and E.A. Kuznetsov for attracting my attention to Refs. \[5,6\]. This work was supported by Russian Foundation for Basic Research, Grant No. 97–02–16177.
[**References**]{}
1. L. Tonks, Phys. Rev. 48 (1935) 562.
2. Ya.I. Frenkel, Zh. Teh. Fiz. 6 (1936) 347.
3. V.E. Zakharov, J. Appl. Mech. Tech. Phys. 2 (1968) 190.
4. S.K. Zhdanov and B.A. Trubnikov, Sov. Phys. JETP 67 (1988) 1575.
5. E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, Phys. Lett. A 182 (1993) 387.
6. E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, Phys. Rev. E 49 (1994) 1283.
7. M.D. Gabovich and V.Ya. Poritsky, JETP Lett. 33, (1981) 304.
8. A.V. Batrakov, S.A. Popov, and D.I. Proskurovsky, Tech. Phys. Lett. 19 (1993) 627.
| ArXiv |
---
abstract: 'In the framework of the search of dark matter in galactic halos in form of massive compact halo object (MACHOs), we discuss the status of microlensing observations towards the Magellanic Clouds and the Andromeda galaxy, M31. The detection of a few microlensing events has been reported, but an unambiguous conclusion on the halo content in form on MACHOs has not been reached yet. A more detailed modelling of the expected signal and a larger statistics of observed events are mandatory in order to shed light on this important astrophysical issue.'
author:
- 'S. Calchi Novati'
title: Microlensing in Galactic Halos
---
Introduction
============
Gravitational microlensing, as first noted in [@ref:pacz86], is a very efficient tool for the detection and the characterisation of massive astrophysical halo compact objects (MACHOs), a possible component of dark matter halos. Following the first exciting detection of microlensing events [@ref:macho93; @ref:eros93; @ref:ogle93], by now the detection of $\sim 30$ events have been reported towards the Magellanic Clouds and our nearby galaxy, M31, and first interesting conclusions on this issue have been reported (Section \[sec:LMC\] and Section \[sec:M31\]). Soon enough, however, the Galactic bulge probed to be an almost an interesting target [@ref:pacz91], and indeed by now the number of observed microlensing events along this line of sight exceeds by two order of magnitudes that observed towards the Magellanic Clouds and M31. In that case the contribution from the dark matter halo is expected to be extremely small compared to that of either bulge or disc (faint) stars [@ref:griest91]. Microlensing searches towards the Galactic bulge are therefore important as they allow to constrain the inner Galactic structure [@ref:pacz94]. Recently, the MACHO [@ref:popowski05], OGLE [@ref:sumi06] and EROS [@ref:hamadache06] collaborations presented the results of their observational campaigns towards this target. A remarkable conclusion is the agreement among these different searches as for the observed value of the optical depth and the agreement with the theoretical expectations [@ref:evans02; @ref:hangould03]. For a more recent discussion see also [@ref:novati07], where the issue of the bulge mass spectrum is treated.
The microlensing quantities {#sec:ml}
===========================
Microlensing events are due to a lensing object passing near the line of sight towards a background star. Because of the event configuration, the observable effect during a microlensing event is an apparent transient amplification of the star’s flux (for a review see e.g. [@ref:roulet97]).
The *optical depth* is the instantaneous probability that at a given time a given star is amplified so to give rise to an observable event. This quantity is the probability to find a lens within the “microlensing tube”, a tube around the line of sight of (variable) radius equal to the *Einstein radius*, $R_\mathrm{E}=\sqrt{4G\mu_l/c^2\, D_l D_{ls}/D_s}$, where $\mu_l$ is the lens mass, $D_l,\,D_s$ are the distance to the lens and to the source, respectively, and $D_{ls}=D_s-D_l$. The optical depth reads $$\label{eq:tau}
\tau = \frac{4\pi G D_s^2}{c^2}\int_{0}^{D_s} \mathrm{d}x \rho(x) x(1-x)\,,$$ where $\rho$ is the *mass* density distribution of lenses and $x\equiv D_l/D_s$. The optical depth provides valuable informations on the overall density distribution of the lensing objects, but it can not be used to further characterise the events, in particular, it does not depend on the lens mass. This is because lighter (heavier) objects are, for a given total mass of the lens population, more (less) numerous but their lensing cross section is smaller (larger), and the two effects cancel out. The optical depth turns out to be an extremely small quantity, of order of magnitude $\sim 10^{-6}$. This implies that one has to monitor extremely large sets of stars to achieve a reasonable statistics.
The experiments measure the number of the events and their characteristics, in particular their durations. To evaluate these quantities one makes use of the microlensing *rate* that expresses the number of lenses that pass through the volume element of the microlensing tube $\mathrm{d}^3x$ in the time interval $\mathrm{d}t$ for a given lens number density distribution $n(\vec{x})$ and velocity distribution $f(\vec{v})$ $$\label{eq:rate}
\mathrm{d} \Gamma = \frac{n_l\,\mathrm{d}^3 x}{\mathrm{d}t}
\times f(\vec{v}_l) \mathrm{d}^3 v_l\,.$$ The volume element of the microlensing tube is $\mathrm{d}^3 x=(\vec{v}_{r\bot} \cdot \hat{\vec{n}}) \mathrm{d}t \mathrm{d}S$. $\mathrm{d}S=\mathrm{d}l\mathrm{d}D_l$ is the portion of the tube external surface, and $\mathrm{d}l=u_t R_\mathrm{E} \mathrm{d}\alpha$, where $u_t$ is the maximum impact parameter, $\vec{v}_{r}$ is the lens relative velocity with respect to the microlensing tube and $\vec{v}_{r\bot}$ its component in the plane orthogonal to the line of sight, and $\hat{\vec{n}}$ is the unit vector normal to the tube inner surface at the point where the microlensing tube is crossed by the lens. The velocity of the lenses entering the tube is $\vec{v}_l=\vec{v}_r+\vec{v}_t$. $\vec{v}_t$ is the tube velocity.
The differential rate is directly related to the number of expected microlensing events as $\mathrm{d}N=N_\mathrm{obs} T_\mathrm{obs} \mathrm{d}\Gamma$, where $N_\mathrm{obs}, T_\mathrm{obs}$ are the number of monitored sources and the whole observation time, respectively. Furthermore, the distribution for the duration of the microlensing events, the *Einstein time*, $t_\mathrm{E}=R_\mathrm{E}/v_{r\bot}$, can also be deduced from the differential microlensing rate, as $\mathrm{d}\Gamma/\mathrm{d}t_\mathrm{E}$. Besides on the lens mass, the key quantity one is usually interested into, $t_\mathrm{E}$ depends also on other usually unobservable quantities. It is therefore suitable to observe a large enough number of events so to be able to deal statistically with the degeneracies intrinsic to the parameter space of microlensing events.
Eventually note that, in calculating the microlensing quantities, the optical depth and the rate, one can also take into account the source spatial and velocity distributions.
Microlensing towards the Magellanic Clouds {#sec:LMC}
==========================================
![Top left: projection on the sky plane of the column density of the LMC disc and bar. The numerical values on the contours are in $10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$ units. The three innermost contours correspond to 10, 20 and $30\times 10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$. The location of the MACHO (black stars and empty diamonds) and EROS (triangles) microlensing candidates are shown. The $x-y$ axes are directed towards West and North respectively. From top right to bottom left: contours maps of the optical depth for lenses in the Galactic halo, LMC halo and self lensing, respectively. The numerical values are in $10^{-8}$ units. Also shown, the location of the fields observed by the MACHO collaboration. (Figures adapted from [@ref:mancini04].)[]{data-label="fig:lmc-tau"}](lmc1 "fig:"){width="7cm"} ![Top left: projection on the sky plane of the column density of the LMC disc and bar. The numerical values on the contours are in $10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$ units. The three innermost contours correspond to 10, 20 and $30\times 10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$. The location of the MACHO (black stars and empty diamonds) and EROS (triangles) microlensing candidates are shown. The $x-y$ axes are directed towards West and North respectively. From top right to bottom left: contours maps of the optical depth for lenses in the Galactic halo, LMC halo and self lensing, respectively. The numerical values are in $10^{-8}$ units. Also shown, the location of the fields observed by the MACHO collaboration. (Figures adapted from [@ref:mancini04].)[]{data-label="fig:lmc-tau"}](lmc2 "fig:"){width="7cm"} ![Top left: projection on the sky plane of the column density of the LMC disc and bar. The numerical values on the contours are in $10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$ units. The three innermost contours correspond to 10, 20 and $30\times 10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$. The location of the MACHO (black stars and empty diamonds) and EROS (triangles) microlensing candidates are shown. The $x-y$ axes are directed towards West and North respectively. From top right to bottom left: contours maps of the optical depth for lenses in the Galactic halo, LMC halo and self lensing, respectively. The numerical values are in $10^{-8}$ units. Also shown, the location of the fields observed by the MACHO collaboration. (Figures adapted from [@ref:mancini04].)[]{data-label="fig:lmc-tau"}](lmc3 "fig:"){width="7cm"} ![Top left: projection on the sky plane of the column density of the LMC disc and bar. The numerical values on the contours are in $10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$ units. The three innermost contours correspond to 10, 20 and $30\times 10^7~\mathrm{M}_\odot~\mathrm{kpc}^{-2}$. The location of the MACHO (black stars and empty diamonds) and EROS (triangles) microlensing candidates are shown. The $x-y$ axes are directed towards West and North respectively. From top right to bottom left: contours maps of the optical depth for lenses in the Galactic halo, LMC halo and self lensing, respectively. The numerical values are in $10^{-8}$ units. Also shown, the location of the fields observed by the MACHO collaboration. (Figures adapted from [@ref:mancini04].)[]{data-label="fig:lmc-tau"}](lmc4 "fig:"){width="7cm"}
The first survey aimed at the detection of microlensing events have been carried out towards the Large and Small Magellanic Clouds (LMC and SMC, respectively), so to probe the MACHO content within the Galactic halo. The main results have been obtained by the MACHO [@ref:macho00] and the EROS [@ref:eros07] collaborations.
MACHO reported the detection of 13-17 microlensing events towards the LMC, and concluded that a rather significant (mass) fraction of the Galactic halo, $f\sim~20\%$, is made up of dark mass objects of $\sim~0.4~\textrm{M}_\odot$. On the other hand, EROS reported the detection of 1 event towards the SMC and no events towards the LMC, whereas they evaluated, for a full halo of $0.4~\textrm{M}_\odot$ MACHO, an expected number of microlensing events $\sim 39$. Correspondingly, EROS put a rather severe *upper* limit on the halo fraction in form of MACHOs, $f<0.08$ for $0.4~\textrm{M}_\odot$ MACHO objects.
The disagreement between the results obtained by the MACHO and the EROS collaboration leaves the issue of the halo content in form of MACHOs open. A possible issue is the nature of the flux variations reported by the MACHO collaboration. Indeed, microlensing searches are plagued by variable stars (that represent the overwhelmingly majority of the flux variations detected) masquerading as microlensing events. However, Bennet [@ref:bennet05] performed a new analysis on the MACHO data set concluding that “\[…\] the main conclusions of the MACHO LMC analysis are unchanged by the variable star contamination\[…\]”.
Looking for MACHO events, the second background is constituted by “self-lensing” events, where, besides the source, also the lens belongs to some luminous star population (either in the LMC itself or possibly, along the line of sight, in the Galactic disc). This possibility was first addressed in [@ref:sahu94; @ref:wu94] and has been further discussed by several authors (e.g. [@ref:gould95; @ref:gyuk00]).
Besides these possible background contaminations, a few aspects of the EROS analysis are worth being mentioned. First, while the fields observed by MACHO towards the LMC are all concentrated around the central region, EROS monitored an extremely larger region. This alleviates the issue of self lensing but also that of a possible clumpiness of the Galactic halo right along the line of sight towards the LMC (this argument is balanced, however, by the much smaller expected rate in the outer with respect to the inner LMC regions). Second, EROS restricted his analysis to a subsample of bright sources, this choice being motivated by the superior photometric precision of the corresponding light curves (so to reduce possible contamination from variable stars), and by the possibility of a better understanding of the so-called “blending” effect. The latter issue is of particular relevance in microlensing analyses and it concerns the ability to correctly evaluate the source flux, in absence of amplification, in crowded fields where the observed objects can be, to some extent, the blend of several stars. It is worth stressing that a similar approach was crucial to get to the agreement between the theoretical expectations and the measured values of the optical depth in the case of observations towards the Galactic bulge.
![Scatter plot of the observed values (empty boxes) of the Einstein time and of the expected values of the median duration (filled stars) with respect to the self-lensing optical depth evaluated along the direction of the events. The dashed line for $\tau_\mathrm{SL}=2$ approximately delimits the inner LMC region, where a good agreement is found between the two values for most of the observed events, and the outer region, where the rise in the expected duration is clearly not observed. (Figure adapted from [@ref:mancini04].)[]{data-label="fig:lmc-rate"}](lmc-te){width="11cm"}
![Galactic (top) and LMC dark matter halo fraction, median value with 68% CL error, as a function of the MACHO mass. For values in the mass range $(0.1-0.3)~\mathrm{M}_\odot$, preferred for LMC lenses on the basis of an analysis of the event duration and spatial distributions, the LMC halo dark matter fraction turns out to be significantly larger than the Galactic one. (Figure adapted from [@ref:novati06].) []{data-label="fig:lmc-halo"}](lmc-2f){width="9cm"}
More recently, new analyses of the MACHO results have been undertaken. In [@ref:jetzer02] it is shown that the observed events are probably distributed among different components (disc, Galactic halo, the LMC halo and self lensing). Taking advantage of a new modelling of both the luminous and the dark components of the LMC [@ref:vdm02], in [@ref:mancini04] the self-lensing issue has been once more addressed considering the set of microlensing events reported by the MACHO collaboration. In Figure \[fig:lmc-tau\] the density profile of the luminous components of the LMC, disc and bar, is shown together with the optical depth profiles for the Galactic halo, the LMC halo and self lensing. Furthermore, through an analysis of the differential microlensing rate it has been shown that self-lensing events cannot contribute to all of the $\sim~10$ observed events. First, the expected number for self lensing turns out to be significantly smaller, about 1-2 events at most. Second (Figure \[fig:lmc-rate\]), for self-lensing events one expects a peculiar signature in the relationship between the event duration and their spatial distribution, with longer durations expected in the outer LMC region (where, correspondingly, the self-lensing optical depth turns out to be smaller). Such a relationship, however, is not observed. The same set of MACHO events has also been analysed in [@ref:novati06], where in particular the question of a possible significant contribution to the observed events of lenses belonging to the dark matter halo of the LMC, as opposed to those of the Galactic halo population, has been addressed (this possibility had previously been discussed in [@ref:gould93; @ref:kerins99]). In particular, studying both the spatial and the duration distributions, it is shown that only a fraction of the events have characteristics that match those expected for the latter population, hinting that a population of somewhat lighter, $\sim~0.2~\mathrm{M}_\odot$, LMC halo MACHO may indeed contribute to the observed events. Challenging the usual assumption of equal halo fractions in form of MACHO for the Galactic and the LMC halo it was then shown, Figure \[fig:lmc-halo\], that indeed for MACHO masses in the range $(0.1-0.3)~\mathrm{M}_\odot$ the LMC halo mass fraction can be significantly larger than the Milky Way’s so that up to about half of the observed events could indeed be attributed to the LMC MACHO dark matter halo.
Microlensing towards M31 {#sec:M31}
========================
The contradictory results of the microlensing campaigns towards the Magellanic Clouds challenge to probe the MACHO distribution along different line of sights. The Andromeda galaxy, M31, nearby and similar to the Milky Way, is a suitable target for this search [@ref:crotts92; @ref:baillon93; @ref:jetzer94]. First, it allows us to explore the Galactic halo along a different line of sight. Second, it has its own dark matter halo that, as we look at it from outside, can be studied globally. We stress that this is a fundamental advantage with respect to the Magellanic Clouds searches. The analysis shows that, as an order of magnitude, for a given MACHO mass and halo fraction, one expects about 3 microlensing events by MACHOs in the M31 halo for each event due to a MACHO in the Galactic halo (in fact, in the latter case the number of available lenses is enormously smaller, by about 4 orders of magnitude, but this fact is, almost, balanced by the much larger value of the Einstein radius). Eventually, the high inclination of the M31 disc is expected to provide a strong gradient in the spatial distribution of microlensing events, that can in principle give an unmistakable signature for M31 microlensing halo events.
![Projected on M31, the boundaries of the observed INT fields are shown together with the position of the microlensing candidates (circles) reported by the POINT-AGAPE collaboration. Note in particular the position of the microlensing event N2, located rather far away from the M31 centre, where the expected self-lensing signal is very low. Note also that S4, the M31-M32 microlensing event, and S5, a possible binary event not included in the selection, are not included in the analysis for the determination of the halo fraction in form of MACHOs. (Figure adapted from [@ref:novati05].)[]{data-label="fig:int-fields"}](INTevts4){width="8cm"}
As compared to that to the LMC and the SMC, $\sim 50~\mathrm{kpc}$, the distance to M31 is, by more than an order of magnitude, larger, $\sim 770~\mathrm{kpc}$. As a consequence, the potential sources of microlensing events are not going to be, as for the Magellanic Clouds, resolved objects. This calls for a peculiar technique, usually referred to as “pixel-lensing”, whose key feature is the fact that one monitors flux variations of unresolved objects in each pixel element of the image [@ref:gould96]. Although, in principle, *all* stars in the pixel field are possible sources one can only detect lensing events due either to bright enough stars or to extremely high amplification events (in any case all stars in the pixel field contribute to the overall flux background). As it has been first shown by the AGAPE group [@ref:agape97], the former case is by far the most likely, with, in any case, a number of potential sources per arsec-square that can easily exceed, in the more crowded region, a few hundreds. With respect to the analysis towards the Magellanic Clouds this means that a huge number of sources is potentially available. However, one must handle with the difficulty that the source flux is not a directly observable quantity. This adds a further degeneracy in the microlensing parameter space, in particular it does not allow one to unambiguously determine, out of the observed event duration, the Einstein time. Instead, what is directly observable is the so-called $t_{1/2}$, the full-width-at-half-maximum of the flux variation visible above the background. It turns out that, for self-lensing events as well for MACHO events in the mass range preferred by the Magellanic Clouds searches, $t_{1/2}$ is of order of only a few days. This is shorter than the typical durations observed towards the Magellanic Clouds, and this is relevant in the analysis as it makes easier the distinction between the contaminating background of variable stars and the truly microlensing flux variations.
The first convincing detection of a microlensing event towards M31 has been reported by the AGAPE group [@ref:agape99]. At the same time, other collaborations have undertaken searches for microlensing towards M31 and the detection of a few more candidate events has been discussed: Columbia-VATT [@ref:crotts96], POINT-AGAPE [@ref:auriere01; @ref:paulin03], SLOTT-AGAPE [@ref:novati02; @ref:novati03], WeCAPP [@ref:riffeser03], MEGA [@ref:mega04], NainiTal [@ref:joshi05].
Beyond the detection of viable microlensing candidates, in order to draw conclusions on the physical issue of the nature of the lenses, and therefore on the halo content in form of MACHOs, one must develops models able to predict the expected signal. With respect to the Magellanic Clouds searches a few important differences arise. First, as a direct consequence of the unresolved-source issue, for M31 experiments one has to model the luminosity functions of the sources. A related issue is that the intrinsic M31 surface brightness shows a strong gradient moving towards the galaxy centre, and this introduces a spatial dependent noise level one has to take into account in order to correctly predict the level of amplification needed, for a given source magnitude, to give rise to a detectable microlensing event [@ref:kerins01].
The second important difference is the ratio of expected self lensing versus MACHO lensing, that in the case of observations towards M31 is much larger than for observations towards the LMC. This is a consequence of the fact that the M31 luminous components are much more massive than the LMC one’s. The exact figure depends on the observed field of view, the MACHO mass and halo fraction (and also on the not-so-well-known self-lensing contribution). However, comparing for instance the MACHO and the POINT-AGAPE analyses (to be discussed below) for full halos of $\sim~0.5~\mathrm{M}_\odot$ this ratio turns out to be *larger*, in the M31 case, by about one order of magnitude. As for the search of the MACHO lensing signal, this expected self-lensing signal constitutes therefore an unwanted background one must be able to deal with and eventually to get rid of. On the other hand, a relatively large self-lensing signal is important as it allows one to study the characteristics of the M31 stellar populations.
A complete analysis of the microlensing, observed and expected, signal has been performed by the POINT-AGAPE [@ref:novati05] and the MEGA [@ref:mega06] collaborations. The two groups shared the same set of data, taken at the 2.5m INT telescope over a period of 4 years (1999-2002), but carried out completely independent analyses. As for the data analysis, in particular, POINT-AGAPE used the so-called “pixel-photometry” [@ref:agape97], while MEGA used the DIA (difference image analysis) photometry [@ref:tomaney96]. Furthermore, the two groups followed different strategies as for the determination of the efficiency of their analysis pipeline, this step being the fundamental link between the theoretical predictions and the results of the data analysis.
![The POINT-AGAPE results: Most probable values, upper and lower 95% CL limit for the halo fraction as a function of the MACHO mass. (Figure adapted from [@ref:novati05].)[]{data-label="fig:pa-res"}](pa){width="9cm"}
The conclusions out of these two experiments turned out to be in disagreement. POINT-AGAPE claims for an evidence of a MACHO contribution to Galactic halos, whereas MEGA finds his detected signal to be compatible with self lensing.
POINT-AGAPE [@ref:novati05] restricted his search to bright microlensing events, and reported the detection of 6 microlensing candidates, for which the possible variable stars contamination was throughly discussed and eventually discarded. Among these 6, one was found to be located right along the line of sight to M32, a M31 satellite galaxy, and attributed to an intergalactic M31-M32 microlensing event [@ref:paulin02] and therefore excluded from the analysis of the halo content in form of MACHOs. Through a Monte Carlo analysis of the expected signal it was then shown that the expected self-lensing signal, for viable M31 luminous models, was at most $\sim~1$ event, to be compared with $\sim~7$ events expected for full halos (both the Galactic and the M31) of $0.5~\mathrm{M}_\odot$ MACHOs. Taking into account the spatial distribution of the observed events, in that respect it turned out to be particularly relevant the position of 1 event, located rather far away from the M31 centre (Figure \[fig:int-fields\]), POINT-AGAPE concluded claiming that “\[…\] the observed signal is much larger than expected from self lensing alone and we conclude, at the 95% confidence level, that at least 20% of the halo mass in the direction of M31 must be in the form of MACHOs if their average mass lies in the range $0.5-1~\mathrm{M}_\odot$ \[…\]” (Figure \[fig:pa-res\]).
MEGA [@ref:mega06] identified 14 microlensing candidates reaching, however, an altogether different conclusion : “\[…\] the observed event rate is consistent with the rate predicted for self-lensing - a MACHO halo fraction of 30% or higher can be ruled out at the 95% confidence level \[…\]”.
These contradictory results give rises to a through debate. One of the main point of disagreement is, in fact, the prediction of the expected self-lensing signal and its characterisation with respect to MACHO lensing (e.g. [@ref:kerins01; @ref:baltz05; @ref:riffeser06]). The results of the MEGA experiment have been further analysed in [@ref:ingrosso06; @ref:ingrosso07], where in particular the spatial and the duration distributions of the observed events has been considered, with the conclusion that self lensing can not explain all the reported microlensing candidates.
Conclusions
===========
Microlensing searches towards the Magellanic Clouds and the Andromeda galaxy, M31, have given, in recent years, first exciting though somewhat contradictory conclusions. The detection of microlensing events has been reported, however their interpretation with respect to the halo dark matter issue is still open to debate. Along both line of sights “evidence” for a MACHO signal as well as null results have been reported. (We may stress that, in the framework of galaxy formation theory, even a relatively “small” halo fraction contribution in form of MACHOs, say at the 10%-20% level, may turn out to be relevant). New observational campaigns are currently under way to further address this interesting issue. The SuperMACHO collaboration [@ref:rest05] is observing the LMC with a much larger field of view than previous campaigns. Towards M31, the Angrstom [@ref:kerins06] and the PLAN [@ref:loiano07] campaigns are currently underway. Both, in different ways, aims rather to the central M31 region, so to properly characterise the self-lensing signal. Furthermore, as opposed to previous analyses, there is an effort to get to a more suitable sampling of the observational data so to allow a better reconstruction of the microlensing event parameters, in particular of the Einstein time.
It is a pleasure to thank the organisers of this I Italian-Pakistan Workshop of Relativistic Astrophysics for the warm ambience we enjoyed through this interesting meeting. Thanks to Jean Kaplan for carefully reading the manuscript.
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| ArXiv |
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abstract: |
Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language $\mathcal{L}_{QD}^{P}$ of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically *justified* or *unjustified* in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained.
**Key words:** pragmatics, quantum logic, quantum mechanics, justifiability, decidability, global pluralism.
author:
- |
Claudio Garola\
Dipartimento di Fisica, Università di Lecce e Sezione INFN\
73100 Lecce, Italy\
E-mail: [email protected]
title: A Pragmatic Interpretation of Quantum Logic
---
Introduction
============
The formal structure called *quantum logic* (QL) springs out in a natural way from the formalism of quantum mechanics (QM). Scholars have debated for a long time on it, wondering whether it subtends a concept of quantum truth which is typical of QM, and a huge literature exists on this topic. We limit ourselves here to quote the classical book by Jammer,$^{(1)}$ which provides a general review of QL from its birth to the early seventies, and the recent books by Rèdei$^{(2)}$ and Dalla Chiara *et al.*,$^{(3)}$ which contain updated bibliographies.
Whenever the existence of a quantum concept of truth is accepted, one sees at once that it has to be radically different from the classical (Tarskian) concept, since the set of propositions of QL has an algebraic structure which is different from the structure of classical propositional logic. Thus, a new problem arises, *i.e*. the problem of the “correct” logic to be adopted when reasoning in QM.
We want to show in the present paper that the above problem can be avoided by adopting an *integrated perspective*, which preserves both the globality of logic (in the sense of *global pluralism*, which admits the existence of a plurality of mutually compatible logical systems, but not of systems which are mutually incompatible$^{(4)}$) and the classical notion of *truth as correspondence*, which we consider as explicated rigorously by Tarski’s semantic theory.$^{(5,6)}$ This perspective reconciliates non-Tarskian theories of truth with Tarski’s theory by reinterpreting them as theories of metalinguistic concepts that are different from truth, and can be fruitfully applied to QL. Indeed, we prove in this paper that QL can be interpreted as a theory of the concept of *empirical justification* within QM.
In order to grasp intuitively our results, let us anticipate briefly some remarks that will be discussed more extensively in Sec. 2.
First of all, it must be noted that QM usually avoids making statements about properties of individual samples of a physical system (*physical objects*). Rather, it is concerned with probabilities of results of measurements on physical objects (*standard interpretation*, as espounded in any manual of QM; see, *e.g*., Refs. 7, 8 and 9), or with statistical predictions about ensembles of identically prepared physical objects (*statistical interpretation*; see, *e.g*., Refs. 1, 10 and 11). Yet, QM also distinguishes between properties that are real (or *actual*) and properties that are not real (or *potential*) in a given state $S$ of the physical system that is considered (briefly, the property $E$ is actual in $S$ whenever a test of $E$ on any physical object $x$ in $S$ would show that $E$ is possessed by $x$ without changing $S^{(12)}$). This amounts to introduce implicitly a concept of truth that also applies to statements about individuals. Indeed, asserting that a property $E$ is actual in the state $S$ is equivalent to asserting that the statement $E(x)$ that attributes $E$ to a physical object $x$ is *true* for every $x$ in the state $S$. Moreover, according to QM, $E(x)$ is true, for a given $x$ in the state $S$, if and only if (briefly, *iff*) $E$ is actual in the state $S$.$^{(12)}$ Falsity is then defined by considering a complementary property $E^{\bot }$ of $E$, so that $E(x)$ is false for a given $x$ in the state $S$ iff $E^{\bot }$ is actual in $S$. It follows in particular that $E(x)$ is true (false) for a given $x$* *in the state* *$S$ iff it is true (false) for every $x$ in $S$, or, equivalently, iff it is *certainly true* (*certainly false*) in $S$. This result explains the notion of true as *certain* introduced in some well known approaches to QM$^{(13,14)}$. More important, it shows that the notion of truth has very peculiar features in QM. Indeed, the truth and falsity of a statement $E(x)$ about an individual are equivalent to the truth of two universally quantified statements. Both these statements may be false. In this case $E(x)$ has no truth value, hence it is meaningless. The existence of meaningless statements implies, in particular, that no Tarskian set-theoretical semantics can be introduced in QM.
The quantum notion of truth and meaning pointed out above is typical of the standard interpretation of QM, and it is inspired by a verificationist position which identifies truth and verifiability, meaning and verifiability conditions. These identifications are rather doubtful from an epistemological viewpoint, yet it is commonly maintained in the literature that the standard quantum conception of truth has no alternatives, since it seems firmly rooted in the formalism of QM itself. The mathematical apparatus of QM would imply indeed the impossibility of defining an assignment function associating a truth value with every individual statement of the form $E(x)$ by referring only to the property $E$ and the state $S$ of $x$. The outcomes obtained in a concrete experiment whenever $E$ or $E^{\bot }$ are not actual in $S$ would depend on the set of observations that are carried out simultaneously, not only on $S$ (*contextuality*).$^{(15-18)}$
Notwithstanding the arguments supporting it, the standard viewpoint can be criticized, and an alternative *SR interpretation* of QM can be constructed based on an epistemological position (*semantic realism*, or, briefly *SR*) which allows one to define a truth value for every statement of the form $E(x)$ according to a Tarskian set-theoretical model.$^{(19-26)}$ Of course, all statements that are certainly true (equivalently, true) or certainly false (equivalently, false) according to the standard interpretation with its quantum concept of truth, are also certainly true or certainly false, respectively, according to the SR interpretation with its Tarskian concept of truth. The remaining statements are meaningless according to the former interpretation, while they have truth values according to the latter: these values, however, may change when different objects in the same state are considered, and cannot be predicted in QM (which is, in this sense, an incomplete theory).
Because of its intuitive, philosophical and technical advantages, we adopt the SR interpretation in the present paper. It is then important to observe that our definitions and reasonings take into account only statements that are certainly true (certainly false) in the sense explained above, hence they actually do not depend on the choice of the interpretation of QM (standard or SR). Thus, our reinterpretation of QL should be acceptable also for logicians and physicists who do not agree with our epistemological position. Of course, if the SR interpretation is not accepted one loses all philosophical advantages of the integrated perspective mentioned at the beginning of this section.
Let us come now to empirical justification. Whenever a statement $E(x)$ is certainly true (certainly false), its truth (falsity) can be predicted within QM if the property $E$ and the state $S$ of $x$ are known, and can be checked (by means of nontrivial physical procedures, see Sec. 2.6). Hence, we can say that the assertion of $E(x)$ ($E^{\bot }(x)$) is empirically justified, since we can both deduce the truth of $E(x)$ ($E^{\bot }(x)$) inside QM and provide an empirical proof of it. More formally, one can introduce an assertion sign $\vdash $ and say that $E(x)$ is certainly true (certainly false) iff $\vdash E(x)$ ($\vdash E^{\bot }(x)$) is empirically justified. In this way a semantic notion (certainty of truth) is translated into a pragmatic notion (empirical justification). Now, we remind that a pragmatic extension of a classical language and some general properties of the concept of justification have been studied by Dalla Pozza and by the author$^{(27)}$ and note that all results obtained in this research apply to the notion of empirical justification introduced above. Moreover, further results can be obtained which are typical of the case under consideration, since the notion of justification is now specified (empirical justification in QM). Thus, a pragmatic language $\mathcal{L}_{Q}^{P}$ can be constructed (Sec. 3) in which assertions of the form $\vdash E(x)$ are taken as elementary *assertive formulas* (*afs*) and pragmatic connectives are introduced, for which a *set-theoretical pragmatics* is defined basing on the concept of empirical justification in QM. This pragmatics defines a justification value for every elementary or complex af of $\mathcal{L}_{Q}^{P}$, yet not all complex afs of $\mathcal{L}_{Q}^{P}$ are *pragmatically decidable*, that is, such that an empirical procedure of justification exists (it obviously exists for all elementary afs of $\mathcal{L}_{Q}^{P}$ because of our arguments above). However, one can single out a subset of pragmatically decidable afs of $\mathcal{L}_{Q}^{P}$ and consider a sublanguage $\mathcal{L}_{QD}^{P}$ of $\mathcal{L}_{Q}^{P}$ which contains only afs in this subset. It is then easy to see that our set-theoretical pragmatics, when restricted to $\mathcal{L}_{QD}^{P} $, endows it with the structure of QL.
The above result is highly interesting in our opinion. Indeed, it provides the desired reinterpretation of QL as a theory of the metalinguistic concept of empirical justification in QM, allowing us to place it within an integrationist perspective that avoids any conflict with classical logic (we stress again that this conclusion can be accepted also by scholars who want to maintain the standard interpretation of QM).
We conclude this Introduction by observing that our results suggest that the standard partition of properties in two subsets (actual properties and potential properties) should be substituted by a partition in three subsets, as follows.
*Actual properties*. A property $E$ is actual in the state $S$ iff the assertion $\vdash E(x)$, with $x$ in $S$, is justified.
*Nonactual properties*. A property $E$ is nonactual in the state $S$ iff the assertion $\vdash E^{\bot }(x)$, with $x$ in $S$, is justified.
*Potential properties*. A property $E$ is potential in the state $S$ iff both assertions $\vdash E(x)$ and $\vdash E^{\bot }(x)$, with $x$ in $S$, are unjustified.
Physical preliminaries
======================
We introduce in this section a number of symbols, definitions and physical concepts that will be extensively used in Sec. 3 in order to supply an intuitive support and an intended interpretation for the pragmatic language that will be introduced there.
Basic notions and mathematical representations
----------------------------------------------
The following notions will be taken as primitive.
*Physical system* $\Omega $*.*
*Pure state* $S$* of* $\Omega $, and *set $\mathcal{S}$ of all pure states of* $\Omega $ (the word *pure* will be usually implied in the following).
*Testable property* $E$* of* $\Omega $, and *set $\mathcal{E}$ of all testable properties of* $\Omega $ (the word *testable* will be usually implied in the following).[^1]
States and properties will be interpreted operationally as follows.
A state $S\in \mathcal{S}$ is a class of physically equivalent[^2] preparing devices (briefly, *preparations*) which may prepare individual samples of $\Omega $ (*physical objects*). A physical object $x$ *is in the state* $S$ iff it is prepared by a preparation $\pi \in S$.
A property $E\in \mathcal{E}$ is a class of physically equivalent ideal dichotomic (outcomes 1, 0) registering devices (briefly, *registrations*) which may test physical objects.[^3]
The above notions do not distinguish between classical and quantum mechanics. The mathematical representation of physical systems, states and properties are different, however, in the two theories. Let us resume these representations in the case of QM.
Every physical system $\Omega $ is associated with a Hilbert space $\mathcal{H}$ over the field of complex numbers (we use the Dirac notation $\mid \cdot
\rangle $ in order to denote vectors of $\mathcal{H}$ in the following).
Let us denote by $(\mathcal{L(H)},\subset )$ the partially ordered set (briefly, *poset*) of all closed subspaces of $\mathcal{H}$ (here $\subset $ denotes set-theoretical inclusion), and let $\mathcal{A}\subset
\mathcal{L(H)}$ be the set of all one-dimensional subspaces of $\mathcal{H}$. Then (in absence of superselection rules) a mapping
$\varphi :S\in \mathcal{S\longrightarrow \varphi }(S)\in \mathcal{A}$
exists which maps bijectively the set $\mathcal{S}$ of all states of $\Omega
$ onto $\mathcal{A}$,[^4] and a mapping
$\chi :E\in \mathcal{E\longrightarrow \chi }(E)\in \mathcal{L(H)}$
exists which maps bijectively the set $\mathcal{E}$ of all properties of $\Omega $ onto $\mathcal{L(H)}$.[^5]
Physical Quantum Logic
----------------------
The poset $(\mathcal{L(H)},\subset )$ is characterized by a set of mathematical properties. In particular, it is a complete, orthocomplemented, weakly modular, atomic lattice which satisfies the covering law$^{(13,27-30)} $. We denote by $^{\bot }$, $\Cap $ and $\Cup $ orthocomplementation, meet and join, respectively, in $(\mathcal{L(H)},\subset )$, and remind that $\Cap $ coincides with the set-theoretical intersection $\cap $ of subspaces of $\mathcal{H}$, while $^{\bot }$ does not generally coincide with the set-theoretical complementation $^{\prime }$, nor $\Cup $ coincides with the set-theoretical union $\cup $. Furthermore, we denote the minimal element $\{\mid 0\rangle \}$ and the maximal element $\mathcal{H}$ of $(\mathcal{L(H)},\subset )$ by $O$ and $I$, respectively. Finally, we note that $\mathcal{A}$ obviously coincides with the set of all atoms of $(\mathcal{L(H)},\subset )$.
Let us denote by $\prec $ the order induced on $\mathcal{E}$, via the bijective representation $\chi $, by the order $\subset $ defined on $\mathcal{L(H)}$. Then, the poset $(\mathcal{E},\prec )$ is order-isomorphic to $(\mathcal{L(H)},\subset )$, hence it is characterized by the same mathematical properties characterizing $(\mathcal{L(H)},\subset )$. In particular, the unary operation induced on it, via $\chi $, by the orthocomplementation defined on $(\mathcal{L(H)},\subset )$, is an orthocomplementation, and $(\mathcal{E},\prec )$ is an orthomodular (i.e., orthocomplemented and weakly modular) lattice, usually called *the lattice of properties* of $\Omega $. By abuse of language, we denote the lattice operations on $(\mathcal{E},\prec )$ by the same symbols used above in order to denote the corresponding lattice operations on $(\mathcal{L(H)},\subset )$.
Orthomodular lattices are said to characterize semantically *orthomodular QLs* in the literature.$^{(3)}$ The lattice of properties has a less general structure in QM, since it inherits a number of further properties from $(\mathcal{L(H)},\subset )$. Therefore, $(\mathcal{E},\prec
) $ will be called *physical QL* in this paper.
A further lattice, isomorphic to $(\mathcal{E},\prec )$, will be used in the following. In order to introduce it, let us consider the mapping
$\rho :E\in \mathcal{E}\longrightarrow \mathcal{S}_{E}=\{S\in \mathcal{S}\mid \varphi (S)\subset \chi (E)\}\in \mathcal{L(S)}$,
where $\mathcal{L(S)}=\{\mathcal{S}_{E}\mid E\in \mathcal{E}\}$ is the range of $\rho $, which generally is a proper subset of the power set $\mathcal{P(S)}$ of $\mathcal{S}$. The poset $(\mathcal{L(S)},\subset )$ is order-isomorphic to $(\mathcal{L(H)},\subset )$, hence to $(\mathcal{E},\prec )$, since $\varphi $ and $\chi $ are bijective, so that $\rho $ is bijective and order-preserving. Therefore $(\mathcal{L(S)},\subset )$ is characterized by the same mathematical properties characterizing $(\mathcal{E},\prec )$. In particular, the unary operation induced on it, via $\rho $, by the orthocomplementation defined on $(\mathcal{E},\prec )$, is an orthocomplementation, and $(\mathcal{L(S)},\subset )$ is an orthomodular lattice. We denote orthocomplementation, meet and join on $(\mathcal{L(S)},\subset )$ by the same symbols $^{\bot }$, $\Cap $, and $\Cup $, respectively, that we have used in order to denote the corresponding operations on $(\mathcal{L(H)},\subset )$ and $(\mathcal{E},\prec )$, and call $(\mathcal{L(S)},\subset )$ *the lattice of closed subsets of* $\mathcal{S}$ (the word *closed* refers here to the fact that, for every $\mathcal{S}_{E}\in $ $\mathcal{L(S)}$, $(\mathcal{S}_{E}^{\bot
})^{\bot }=\mathcal{S}_{E}$). We also note that the operation $\Cap $ coincides with the set-theoretical intersection $\cap $ on $\mathcal{L(S)}$ because of the analogous result holding in $(\mathcal{L(H)},\subset )$.[^6]
To close up, let us observe that the unary operation $^{\bot }$ defined on $\mathcal{L(S)}$ can be extended to $\mathcal{P(S)}$ by setting, for every $\mathcal{T\in P(S)}$,
$\mathcal{T}^{\bot }=($*min*$\{\mathcal{S}_{E}\in \mathcal{L(S)\mid
T\subset S}_{E}\})^{\bot }$
(the symbol *min* obviously refers to the order $\subset $ defined on $\mathcal{L(S)}$). This extension will be needed indeed in Sec. 3.2.
Actual and potential properties
-------------------------------
We say that a property $E$ is* actual* (*nonactual*) in the state $S$ iff one can perform a test of $E$ on any physical object $x$ in the state $S$ by means of a registration $r\in E$, obtaining outcome 1 (0) without modifying $S$.[^7]
Basing on the above definition, for every state $S\in \mathcal{S}$ three subsets of $\mathcal{E}$ can be introduced.
$\mathcal{E}_{S}$ : the set of all properties that are actual in $S$.
$\mathcal{E}_{S}^{\bot }$ : the set of all properties that are nonactual in $S$.
$\mathcal{E}_{S}^{I}$ : the set $\mathcal{E}\setminus \mathcal{E}_{S}\cup
\mathcal{E}_{S}^{\bot }$ (called the set of all properties that are *indeterminate*, or* potential*, in $S$).
By using the mathematical apparatus of QM, the sets $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ can be characterized as follows.
$\mathcal{E}_{S}=\{E\in \mathcal{E}\mid \varphi (S)\subset \chi (E)\}=\{E\in
\mathcal{E}\mid S\in \mathcal{S}_{E}\}$.
$\mathcal{E}_{S}^{\bot }=\{E\in \mathcal{E}\mid \varphi (S)\subset \chi
(E)^{\bot }\}=\{E\in \mathcal{E}\mid S\in \mathcal{S}_{E}^{\bot }\}$.
It can also be proved that $\mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$) coincides with the set of all properties that have probability 1 (0), according to QM, for every $x$ in the state $S$, and that the set $\mathcal{E}_{S}^{I}$ (which is non-void in QM, while it would be void in classical physics) coincides with the set of all properties that have probability different from 0 and 1 for every $x$ in the state $S$.
Further characterizations of the above sets can be obtained as follows.$^{(12)}$
Since the mapping $\rho $ is bijective, while every singleton $\{S\}$, with $S\in \mathcal{S}$, obviously is an atom of $\mathcal{L(S)}$, one can associate a property $E_{S}=\rho ^{-1}(\{S\})$ (equivalently, $E_{S}=\chi
^{-1}(\varphi (S))$) with every $S\in \mathcal{S}$. This property is an atom of $(\mathcal{E},\prec )$, and is usually called the *support* of $S$. The mapping $\rho ^{-1}$ thus induces a one-to-one correspondence between (pure) states and atoms of $(\mathcal{E},\prec )$. Then, one can prove the following equalities.
$\mathcal{E}_{S}=\{E\in \mathcal{E}\mid E_{S}\prec E\}$.
$\mathcal{E}_{S}^{\bot }=\{E\in \mathcal{E}\mid E\prec E_{S}^{\bot }\}$.
$\mathcal{E}_{S}^{I}=\{E\in \mathcal{E}\mid E_{S}\nprec E$ and $E\nprec
E_{S}^{\bot }\}$.
Finally, the following equality also follows from the above definitions.
$\mathcal{S}_{E}=\{S\in \mathcal{S}\mid E_{S}\prec E\}.$
Truth in standard QM
--------------------
No mention has been done of truth values (*true/false*) in the foregoing sections. However, we will be concerned with logical structures in Sec. 3, hence it is natural to wonder what QM says about the truth of a sentence as “the physical object $x$ has the property $E$” (briefly, $E(x)$ in the following).
We have already noted in the Introduction that QM usually avoids making explicit statements regarding individual samples of physical systems. Yet, a sentence as “the property $E$ is actual in the state $S$” (Sec. 2.3) intuitively means that all physical objects in the state $S$ have the property $E$. Hence, it can be translated, in terms of truth, into the sentence “for every physical object $x$ in the state $S$, $E(x)$ is true”. This translation shows that QM is concerned also with truth values of individual statements. Moreover, by considering the literature on the subject, one can argue that QM more or less implicitly adopts the following verificationist criterion of truth.$^{(12)}$
EV (empirical verificationism). *The sentence* $E(x)$* has truth value* true* (*false*) for a physical object* $x$* in the state* $S$* iff* $E$* is actual (nonactual) in* $S$*, while it is meaningless otherwise.*
Criterion EV is obviously at odds with standard definitions in classical logic (CL), and is suggested by the fact that $E$ can be attributed (not attributed) to a physical object $x$ in the state S on the basis of an experimental procedure only when it is actual (nonactual) for $x$ (see Sec. 2.6). Hence, we say that $E(x)$ is *Q-true* (*Q-false*) whenever its truth value is true (false) according to criterion EV, in order to stress the difference between the truth values introduced in QM and those introduced in CL.
Because of the foregoing translation, criterion EV implies the following proposition.
TF. *The sentence* $E(x)$* is* Q-true* (*Q-false*) for a physical object x in the state* $S$* iff it is* Q-true* (*Q-false*) for every physical object x in the state* $S$*.*
Loosely speaking, proposition TF can be rephrased by saying that $E(x)$ is true (false) in the sense established by criterion EV iff it is *certainly* true (*certainly* false) in the same sense, which explains the intuitive terminology that we have adopted in the Introduction.
Furthermore, criterion EV implies that $E(x)$ has a truth value in standard QM iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (of course, $E(x)$ is Q-true iff $E\in \mathcal{E}_{S}$, Q-false iff $E\in \mathcal{E}_{S}^{\bot }$). It is then important to observe that the characterizations of $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ provided in Sec. 2.3 show that, for every $S\in \mathcal{S}$, one can deduce from theoretical laws of QM whether a property $E$ belongs to $\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$. In particular, $E$ belongs to $\mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$) iff it has probability 1 (0) for every $x$ in the state $S$. Hence, one can predict, for every $E\in \mathcal{E}$ and $x$ in the state $S$, whether $E(x)$ is Q-true, Q-false or meaningless. This result shows that standard QM is a semantically complete theory$^{(12)}$ and, together with proposition TF, explains the definition of *true* as *certain*, or *predictable*, which occurs in some approaches to QM.$^{(13,14)}$
Nonobjectivity versus objectivity in QM
---------------------------------------
The position expounded in Sec. 2.4 about the truth value of sentences of the form $E(x)$, with $E\in \mathcal{E}$, is sometimes summarized by saying, briefly, that physical properties are *nonobjective* in standard QM (to be precise, only the properties in $\mathcal{E}_{S}^{I}$ should be classified as nonobjective for a given $S\in \mathcal{S}$).
Nonobjectivity of properties is supported by a number of arguments. Some of them are based on empirical results (e.g., the two-slits experiment), some follow from seemingly reasonable epistemological choices (e.g., the adoption of a verificationist position, together with the indeterminacy principle) and some take the form of theorems deduced from the mathematical apparatus of QM. These last arguments are usually considered conclusive in the literature. We remind here the Bell-Kochen-Specker and Bell’s theorems$^{(15-18)}$ which seem to prove that it is impossible to assign classical truth values to all sentences of the form $E(x)$, with $E\in \mathcal{E}$, without contradicting the predictions of QM.
However, all arguments which show that nonobjectivity of properties is an unavoidable feature of QM can be criticized (this of course does not make the claim of nonobjectivity wrong, but only proves that there are alternatives to it). In particular, one can observe that a *no-go theorem* as Bell-Kochen-Specker’s is certainly correct from a mathematical viewpoint, but rests on implicit assumptions that are problematic from a physical and epistemological viewpoint.$^{(22-25)}$ Basing on this criticism, an alternative interpretation (*semantic realism*, or *SR*, interpretation) has been propounded by the author, together with other authors.$^{(19-23,25,26)}$ As we have already observed in the Introduction, the SR interpretation adopts a Tarskian theory of truth as correspondence, and all properties are objective according to it (equivalently, the sentence $E(x)$ has a truth value defined in a classical way for every physical object $x$ and property $E$). According to this interpretation $E(x)$ is *certainly true* (*certainly false*) in the state $S$, that is, it is true (false) in a classical sense for every $x$ is in the state $S$, iff $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$), hence iff it is Q-true (Q-false) according to the standard interpretation.
The SR interpretation of QM has some definite advantages. Firstly, it makes QM compatible with a realistic perspective without requiring any change of its mathematical apparatus and preserving all statistical predictions following from the standard interpretation, hence it provides a solution of the quantum measurement problem.$^{(26)}$ Secondly, it rests on a classical conception of truth and meaning. Thirdly, it leads one to consider QM as an incomplete theory,$^{(12)}$ and provides some suggestions about the way in which a more general theory embodying QM could be constructed.
Also within the SR interpretation one can deduce from theoretical laws of QM whether $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$), for a given $S\in \mathcal{S}$. Moreover, for every $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, the sentence $E(x)$ obviously is certainly true, hence true (certainly false, hence false) iff $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$). On the contrary, no prediction of the truth value of $E(x)$ can be done if $E\notin \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$. Thus, the difference between the standard and the SR interpretation reduces to the fact that, whenever $E\in \mathcal{E}_{S}^{I}$, $E(x)$ is meaningless within the former, while it has a truth value that cannot be predicted by QM within the latter.
Empirical proof in QM
---------------------
The results at the end of Secs. 2.4 and 2.5 show that, whenever $x$ is in the state $S$, the truth value of the sentence $E(x)$ can be predicted (or *theoretically proved*) iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, both in the standard and in the SR interpretation. One is thus led to wonder whether and when the truth value of $E(x)$ can be determined empirically. At first glance, it seems sufficient to test $x$ by means of a registering device belonging to $E$ (Sec. 2.1). This is untrue according to the standard as well as the SR interpretation. Indeed, both interpretations maintain that a single test modifies, whenever $E\notin
\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, the state $S$ of the physical object $x$, so that its result refers to the final state after the test, which is different from $S$ (moreover, within the standard interpretation, $E(x)$ has no truth value whenever $E\notin \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$). Thus, a test of $E(x)$ is physically meaningful iff $E\in
\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, since only in this case it does not modify the state $S$. It follows that an *empirical proof* of the truth value of $E(x)$ can be given iff a theoretical proof of this value exists, and it consists in checking whether $E\in \mathcal{E}_{S}$ or $E\in \mathcal{E}_{S}^{\bot }$. Then, the characterizations of $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ in Sec. 2.3 suggest the empirical procedures to be adopted. Indeed, they show that $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$) iff $E_{S}\prec E$ ($E\prec E_{S}^{\bot }$), or, equivalently, iff $S\in \mathcal{S}_{E}$ ($S\in \mathcal{S}_{E}^{\bot }$). Hence, one can get an empirical proof that $E(x)$ is Q-true (Q-false) within the standard interpretation, or equivalently, that $E(x)$ is certainly true, hence true (certainly false, hence false) within the SR interpretation, by checking whether the state $S$ of $x$ belongs to the set $\mathcal{S}_{E}$ ($\mathcal{S}_{E}^{\bot }$). The empirical procedure required by this check is rather complex, since it does not reduce to a test of $E$ on the physical object $x$, but consists in testing a huge number of physical objects in the state $S$ by means of registrations belonging to $E$, in order to show that all of them yield outcome 1 (0) (it has been proven elsewhere$^{(26)}$ that this procedure actually tests a quantified statement, or a second order physical property).
We conclude by noticing that truth and empirical provability of truth coincide within the standard interpretation of QM, which expresses the verificationist position that characterizes this interpretation. On the contrary, within the SR interpretation of QM the concepts of truth and empirical provability of truth are different, in accordance with the well known distinction between truth and epistemic accessibility of truth in classical logic.
QL as a pragmatic language
==========================
We aim to show in this section that physical QL can be recovered as a pragmatic language in the sense established in Ref. 27. It is noteworthy that, by weakening slightly the assumptions introduced in Ref. 27, one could perform this task without choosing between the standard and the SR interpretation of QM (see footnotes 8 and 9). We adopt however the SR interpretation in this section, since we maintain that the verificationist attitude of the standard interpretation is epistemologically and philosophically doubtful, but we point out by means of footnotes the simple changes to be introduced in order to attain the same results within the standard interpretation.
The general pragmatic language $\mathcal{L}^{P}$
------------------------------------------------
Let us summarize briefly the construction of the general pragmatic language $\mathcal{L}^{P}$ introduced in Ref. 27.
The alphabet $\mathcal{A}^{P}$ of $\mathcal{L}^{P}$ contains as *descriptive signs* the propositional letters $p$, $q$, $r$,...; as*logical-semantic signs* the connectives $\urcorner $, $\wedge $, $\vee $, $\rightarrow $ and $\leftrightarrow $; as * logical-pragmatic signs* the assertion sign $\vdash $ and the connectives $N$, $K$, $A$, $C$ and* *$E$; as* auxiliary signs* the round brackets $(.)$.* *The set $\psi _{R}$ of all *radical formulas* (*rfs*) of $\mathcal{L}^{P}$ is made up by all formulas constructed by means of descriptive and logical-semantic signs, following the standard recursive rules of classical propositional logic (a rf consisting of a propositional letter only is then called *atomic*). The set $\psi _{A}$ of all *assertive formulas* (*afs*) of $\mathcal{L}^{P}$ is made up by all rfs preceded by the assertive sign $\vdash $ (*elementary* afs), plus all formulas constructed by using elementary afs and following standard recursive rules in which $N$, $K$, $A$, $C$ and $E$ take the place of $\urcorner $, $\wedge $, $\vee $, $\rightarrow $ and $\leftrightarrow $, respectively.
A *semantic interpretation* of $\mathcal{L}^{P}$ is then defined as a pair $(\{1,0\},\sigma )$, where $\sigma $ is an* assignment function* which maps $\psi _{R}$ onto the set $\{1,0\}$ of* truth values* (1 standing for *true* and 0 for *false*), following the standard truth rules of classical propositional calculus.
Whenever a semantic interpretation $\sigma $ is given, a *pragmatic interpretation* of $\mathcal{L}^{P}$ is defined as a pair $(\{J,U\},\pi
_{\sigma })$, where $\pi _{\sigma }$ is a *pragmatic evaluation function* which maps $\psi _{A}$ onto the set $\{J,U\}$ of *justification values* following * justification rules* which refer to $\sigma $ and are based on the informal properties of the metalinguistic concept of proof in natural languages. In particular, the following justification rules hold.
JR$_{1}$. *Let* $\alpha \in \psi _{R}$*; then,* $\pi _{\sigma
}(\vdash \alpha )=J$* iff a proof exists that* $\alpha $* is true, i.e., that* $\sigma (\alpha )=1$* (hence,* $\pi _{\sigma
}(\vdash \alpha )=U$* iff no proof exists that* $\alpha $*is true).*
JR$_{2}$.* Let* $\delta \in \psi _{A}$*; then,* $\pi _{\sigma
}(N\delta )=J$* iff a proof exists that* $\delta $* is unjustified, i.e., that* $\pi _{\sigma }(\delta )=U$*.*
JR$_{3}$.* Let* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}$*; then,*
*(i)* $\pi _{\sigma }(\delta _{1}K\delta _{2})=J$* iff* $\pi
_{\sigma }(\delta _{1})=J$* and* $\pi _{\sigma }(\delta _{2})=J$*,*
*(ii)* $\pi _{\sigma }(\delta _{1}A\delta _{2})=J$* iff* $\pi
_{\sigma }(\delta _{1})=J$* or* $\pi _{\sigma }(\delta _{2})=J$*,*
*(iii)* $\pi _{\sigma }(\delta _{1}C\delta _{2})=J$* iff a proof exists that* $\pi _{\sigma }(\delta _{2})=J$* whenever* $\pi
_{\sigma }(\delta _{1})=J$*,*
*(iv)* $\pi _{\sigma }(\delta _{1}E\delta _{2})=J$* iff* $\pi
_{\sigma }(\delta _{1}C\delta _{2})=J$* and* $\pi _{\sigma }(\delta
_{2}C\delta _{1})=J$*.*
Furthermore, the following *correctness criterion* holds in $\mathcal{L}^{P}$.
CC\. *Let* $\alpha \in \psi _{R}$*; then,* $\pi _{\sigma
}(\vdash \alpha )=J$* implies* $\sigma (\alpha )=1.$
Finally, the set of all pragmatic evaluation functions that can be associated with a given semantic interpretation $\sigma $ is denoted by $\Pi
_{\sigma }$.
The quantum pragmatic language $\mathcal{L}_{Q}^{P}$
----------------------------------------------------
The quantum pragmatic language $\mathcal{L}_{Q}^{P}$ that we want to introduce here is obtained by specializing syntax, semantics and pragmatics of $\mathcal{L}^{P}$. Let us begin with the syntax. We introduce the following assumptions on $\mathcal{L}_{Q}^{P}$.
A$_{1}$. *The propositional letters* $p$*,* $q$*, ... are substituted by the symbols* $E(x)$*,* $F(x)$*, ..., with* $E$*,* $F$*, ...* $\in \mathcal{E}$*.*
A$_{2}$. *The set* $\psi _{R}^{Q}$* of all rfs of* $\mathcal{L}_{Q}^{P}$* reduces to the set of all atomic rfs of* $\mathcal{L}_{Q}^{P}$* (in different words, if* $\alpha $* is a rf of* $\mathcal{L}_{Q}^{P}$*, then* $\alpha =E(x)$*, with* $E\in
\mathcal{E}$*).*
A$_{3}$. *Only the logical-pragmatic signs* $\vdash $*,* $N$*,* $K$* and* $A$* appear in the afs of* $\mathcal{L}_{Q}^{P}$*.*
The substitution in A$_{1}$ aims to suggest the *intended interpretation* that we adopt in the following. To be precise, the rfs $E(x)$, $F(x)$, ... are interpreted as sentences stating that the physical object $x$ has the properties $E$, $F$, ..., respectively (Sec. 2.4).
The restriction in A$_{2}$ aims to select rfs that are interpreted as *testable* sentences, i.e., sentences stating testable physical properties (Sec. 2.1), so that physical procedures exist for testing their truth values (which may not occur in the case of a rf of the form, say, $E(x)\vee F(x)$; note that a similar restriction has been introduced in Ref. 27 when recovering intuitionistic propositional logic within $\mathcal{L}^{P} $).
The restriction in A$_{3}$ is introduced for the sake of simplicity, since only the pragmatic connectives $N$, $K$ and $A$ are relevant for our goals in this paper.
Because of A$_{1}$, A$_{2}$ and A$_{3}$, the set $\psi _{A}^{Q}$ of afs of $\mathcal{L}_{Q}^{P}$ is made up by all formulas constructed by means of the following recursive rules.
\(i) *Let* $E(x)$* be a rf. Then* $\vdash E(x)$* is an af.*
\(ii) *Let* $\delta $* be an af. Then,* $N\delta $* is an af.*
\(iii) *Let* $\delta _{1}$* and* $\delta _{2}$* be afs. Then,* $\delta _{1}K\delta _{2}$* and* $\delta _{1}A\delta _{2}$* are afs.*
Let us come now to the semantics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$.
A$_{4}$. *Every assignment function* $\sigma $* defined on* $\psi _{R}^{Q}$* is induced by an interpretation* $\xi $* of the variable x that appears in the rfs into a universe* $\mathcal{U}$* of physical objects, hence* $\sigma =\sigma (\xi )$* and the values of* $\sigma $ *on* $\psi _{R}^{Q}$ *are consistent with (not necessarily determined by) the laws of QM within the intended interpretation established above.*
Let us comment briefly on assumption A$_{4}$.
Firstly, note that the interpretation $\xi $ was understood in Sec. 2.1, when we introduced the informal expression “the physical object $x$ is in the state $S$”.
Secondly, observe that the requirement that $\sigma =\sigma (\xi )$ be consistent with the laws of QM (briefly, *QM-consistent*) obviously follows from the fact that these laws, via intended interpretation, establish relations among the truth values of elementary rfs of $\mathcal{L}_{Q}^{P}$ whenever a specific physical object is considered. We denote by $\Sigma $ in the following the set of all QM-consistent assigment functions.
Thirdly, note that, since $\sigma =$ $\sigma (\xi )$, there may be many interpretations of the variable x that lead to the same assigment function.
Finally, observe that the universe $\mathcal{U}$ can be partitioned into (disjoint) subsets of physical objects, each of which consists of physical objects in the same state (different subsets corresponding to different states). Thus, specifying the state $S$ of $x$ means requiring that the interpretation $\xi $ of $x$ that is considered maps $x$ on a physical object in the subset corresponding to the state $S$, hence it singles out a subclass $\Sigma _{S}\subset \Sigma $ of assigment functions. All functions in $\Sigma _{S}$ assign truth value 1 (0) to a sentence $E(x)\in \psi
_{R}^{Q}$ whenever $E\in \mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$), while the truth values assigned by different functions in $\Sigma _{S}$ to $E(x)$ may differ if $E$ $\notin $ $\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$.[^8]
Let us come now to the pragmatics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$.
A$_{5}$. *Let a mapping* $\xi $* be given which interpretes the variable* $x$* in the rfs of* $\mathcal{L}_{Q}^{P}$* on a physical object in the state* $S$*. A proof that the rf* $E(x)$* is true (false) consists in performing one of the empirical procedures mentioned in Sec. 2.6 and showing that* $E\in \mathcal{E}_{S}$* (*$E\in \mathcal{E}_{S}^{\bot }$*).*
Assumption A$_{5}$ is obviously suggested by the intended interpretation discussed above. Taking into account A$_{1}$ and JR$_{1}$ in Sec. 3.1, it implies the following statement.
P.* Let* $E(x)$* be a rf of* $\mathcal{L}_{Q}^{P}$*, let* $\xi $* be an interpretation of the variable* $x$* on a physical object in the state* $S$*, and let* $S_{E}$* be defined as in Sec. 2.2. Then,*
$\pi _{\sigma (\xi )}(\vdash E(x))=J$* iff* $S\in $* *$S_{E}$*,*
$\pi _{\sigma (\xi )}(\vdash E(x))=U$* iff* $S\notin $* *$S_{E}$*.*
The above result specifies $\pi _{\sigma (\xi )}$ on the set of all elementary afs of $\mathcal{L}_{Q}^{P}$ and shows that it depends only on the state $S$. Hence, we write $\pi _{S}$ in place of $\pi _{\sigma (\xi )}$ in the following (for the sake of brevity, we also agree to use the intuitive statement “the physical object $x$ is in the state $S$” introduced in Sec. 2.1 in place of the more rigorous statement “the variable $x$ is interpreted on a physical object in the state $S$”).
Statement P provides the starting point for introducing a *set-theoretical pragmatics* for $\mathcal{L}_{Q}^{P}$, as follows.
Firstly, we introduce a mapping
$f:\delta \in \psi _{A}^{Q}\longrightarrow \mathcal{S}_{\delta }\in \mathcal{P(S)}$
which associates a *pragmatic extension* $\mathcal{S}_{\delta }$ with every assertive formula $\delta \in \psi _{A}^{Q}$, defined by the following recursive rules.
\(i) *For every* $E(x)\in \psi _{R}^{Q}$*,* $f(\vdash
E(x))=S_{\vdash E(x)}=S_{E}$*.*
\(ii) *For every* $\delta $* *$\in \psi _{A}^{Q}$*,* $f(N\delta )=S_{N\delta }=S_{\delta }^{\bot }$*.*
(iii)* For every* $\delta _{1}$*,* $\delta _{2}\in \psi
_{A}^{Q}$*,* $f(\delta _{1}K$ $\delta _{2})=\mathcal{S}_{\delta
_{1}K\delta _{2}}=\mathcal{S}_{\delta _{1}}\cap \mathcal{S}_{\delta _{2}}$.
\(iv) *For every* $\delta _{1}$*,* $\delta _{2}\in \psi
_{A}^{Q} $*,* $f(\delta _{1}A$* *$\delta _{2})=S_{\delta
_{1}A\delta _{2}}=S_{\delta _{1}}\cup S_{\delta _{2}}$*.*
Secondly, we rewrite statement P above substituting $\mathcal{S}_{\vdash
E(x)}$ to $\mathcal{S}_{E}$ in it.
P$^{\prime }$.* Let* $\vdash E(x)$* be an elementary af of* $\mathcal{L}_{Q}^{P}$* and let* $x$ *be in the state* $S$*. Then,*
$\pi _{S}(\vdash E(x))=J$* iff* $S\in $* *$S_{\vdash E(x)}$*,*
$\pi _{S}(\vdash E(x))=U$* iff* $S\notin $* *$S_{\vdash
E(x)} $*.*
Thirdly, we note that statement P$^{\prime }$ defines the pragmatic evaluation function $\pi _{S}$ on all elementary afs of $\mathcal{L}_{Q}^{P}$.
Finally, for every $S\in \mathcal{S}$, we extend $\pi _{S}$ from the set of all elementary afs of $\mathcal{L}_{Q}^{P}$ to the set $\psi _{A}^{Q}$ of all afs of $\mathcal{L}_{Q}^{P}$ bearing in mind JR$_{2}$ and JR$_{3}$ in Sec. 3.1, hence introducing the following recursive rules.
\(i) *For every* $\delta $* *$\in \psi _{A}^{Q}$*,* $\pi _{S}(N\delta )=J$* iff* $S\in S_{N\delta }=S_{\delta }^{\bot }$*.*
(ii)* For every* $\delta _{1}$*,* $\delta _{2}\in \psi
_{A}^{Q}$*,* $\pi _{S}(\delta _{1}K$* *$\delta _{2})=J$* iff* $S\in S_{\delta _{1}K\delta _{2}}=S_{\delta _{1}}\cap
S_{\delta _{2}}$*.*
\(iii) *For every* $\delta _{1}$*,* $\delta _{2}\in \psi
_{A}^{Q}$*,* $\pi _{S}(\delta _{1}A$* *$\delta _{2})=J$* iff* $S\in S_{\delta _{1}A\delta _{2}}=S_{\delta _{1}}\cup
S_{\delta _{2}}$*.*
The above procedure defines, for every $S\in \mathcal{S}$, a pragmatic evaluation function
$\pi _{S}:\delta \in \psi _{A}^{Q}\longrightarrow \pi _{S}(\delta )\in
\{J,U\}$
which provides a set-theoretical pragmatics for $\mathcal{L}_{Q}^{P}$, as stated.
On the notion of justification in $\mathcal{L}_{Q}^{P}$
-------------------------------------------------------
The notion of justification introduced in Sec. 3.2 is basic in our approach and must be clearly understood. So we devote this section to comments on it.
Whenever an elementary af $\vdash E(x)$ of $\mathcal{L}_{Q}^{P}$ is considered, the notion of justification obviously coincides with the notion of existence of an empirical proof of the truth of $E(x)$ because of assumption A$_{5}$ and proposition P in Sec. 3.2, which fits in with JR$_{1}$ in Sec. 3.1.
Whenever molecular afs of $\mathcal{L}^{P}$ are considered, one can grasp intuitively the meaning of the notion of justification for them by considering simple instances. Indeed, let $E(x)$ be a rf and let $x$ be in the state $S$. We get
$\pi _{S}(N\vdash E(x))=J$ iff $S\in \mathcal{S}_{E}^{\bot }$,
which means, shortly, that it is justified to assert that $E(x)$ cannot be asserted iff MQ entails that the truth value of $E(x)$ is *false* for every $x$ in the state $S$. This result, of course, fits in with JR$_{2}$ in Sec. 3.1.
Furthermore, let $E(x)$ and $F(x)$ be rfs, and let $x$ be in the state $S$. We get
$\pi _{S}(\vdash E(x)K\vdash F(x))=J$ iff $S\in \mathcal{S}_{E}\cap \mathcal{S}_{F}$,
$\pi _{S}(\vdash E(x)A\vdash F(x))=J$ iff $S\in \mathcal{S}_{E}\cup \mathcal{S}_{F}$.
The first equality shows that asserting $E(x)$ and $F(x)$ conjointly is justified iff both assertions are justified. The second equality shows that asserting $E(x)$ or asserting $F(x)$ is justified iff one of these assertions is justified. Both these results, of course, fit in with JR$_{3}$ in Sec. 3.1.
We add that
$\pi _{S}(\vdash E(x))=J$ implies $\pi _{S}(N\vdash E(x))=U$
and
$\pi _{S}(N\vdash E(x))=J$ implies $\pi _{S}(\vdash E(x))=U$
since $\mathcal{S}_{E}\cap \mathcal{S}_{E}^{\bot }=\emptyset $. Nevertheless,
$\pi _{S}(\vdash E(x))=U$ and $\pi _{S}(N\vdash E(x))=U$ iff $S\notin
\mathcal{S}_{E}\cup \mathcal{S}_{E}^{\bot }$,
which shows that a *tertium non datur* principle does not hold for the pragmatic connective $N$ in $\mathcal{L}_{Q}^{P}$ (it has already been proved in Ref. 27 that this principle does not hold in the general language $\mathcal{L}^{P}$).
It is also interesting to note that the justification values of different elementary afs, say $\vdash E(x)$ and $\vdash F(x)$, must be different for some state $S$, since $\mathcal{S}_{E}\neq \mathcal{S}_{F}$ if $E\neq F$ (Sec. 2.2), hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash F(x)}$.
Finally, we remind that the general theory of $\mathcal{L}^{P}$ associates an assignment function $\sigma $ with a set $\Pi _{\sigma }$ of pragmatic evaluation functions (Sec. 3.1), hence this also occurs within $\mathcal{L}_{Q}^{P}$. One may then wonder whether $\Pi _{\sigma }$ is necessarily nonvoid and, if this is the case, whether it may contain more than one pragmatic evaluation function. In order to answer these questions, let us consider an interpretation $\xi $ of the variable $x$ that maps $x$ on a physical object in the state $S$. Then, $\xi $ determines a unique assignment function $\sigma (\xi )$ and a unique pragmatic evaluation function associated with it, that we have denoted by $\pi _{S}$, for it depends only on the state $S$. Since every assigment function in $\Sigma $ is induced by an interpretation $\xi $ because of A$_{4}$ in Sec. 3.2, this proves that $\Pi _{\sigma }$ is necessarily nonvoid for every $\sigma \in
\Sigma $. Moreover, note that an interpretation $\xi ^{\prime }$ of $x$ may exist within the SR interpretation of QM that maps $x$ on a physical object in the state $S^{\prime }$, with $S^{\prime }\neq S$, yet such that $\sigma
(\xi ^{\prime })=\sigma (\xi )$. The pragmatic evaluation functions $\pi
_{S} $ and $\pi _{S^{\prime }}$ are then different, but they are both associated with the assignment function $\sigma =\sigma (\xi )=\sigma (\xi
^{\prime })$, so that they both belong to $\Pi _{\sigma }$. Hence, $\Pi
_{\sigma }$ may contain many pragmatic evaluation functions.[^9]
Pragmatic validity and order in $\mathcal{L}_{Q}^{P}$
-----------------------------------------------------
Coming back to the general language $\mathcal{L}^{P}$, we remind that a notion of pragmatic validity (invalidity) is introduced in it by means of the following definition.
*Let* $\delta \in \psi _{A}$*. Then,* $\delta $* is* pragmatically valid*, or* p-valid* (*pragmatically invalid*, or* p-invalid*) iff for every* $\sigma \in \Sigma $* and* $\pi _{\sigma }\in \Pi _{\sigma }$*,* $\pi _{\sigma }(\delta
)=J$* (*$\pi _{\sigma }(\delta )=U$*).*
By using the notions of justification in $\mathcal{L}_{Q}^{P}$, one can translate the notion of p-validity (p-invalidity) within $\mathcal{L}_{Q}^{P} $ as follows.
*Let* $\delta \in \psi _{A}^{Q}$*. Then,* $\delta $*is p-valid (p-invalid) iff, for every* $S\in S$*,* $\pi _{S}(\delta
)=J$* (*$\pi _{S}(\delta )=U$*).*
The notion of p-validity (p-invalidity) can then be characterized as follows.
*Let* $\delta \in \psi _{A}^{Q}$*. Then,* $\delta $*is p-valid (p-invalid) iff* $S_{\delta }=S$* (*$S_{\delta
}=\emptyset $*).*
The set of all p-valid afs plays in $\mathcal{L}_{Q}^{P}$ a role similar to the role of tautologies in classical logic, and some afs in it can be selected as axioms if one tries to construct a p-correct and p-complete calculus for $\mathcal{L}_{Q}^{P}$. We will not deal, however, with this topic in the present paper.
Furthermore, let us observe that a binary relation can be introduced in the general language $\mathcal{L}^{P}$ by means of the following definition.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff a proof exists that* $\delta _{2}$* is justified whenever* $\delta _{1}$*is justified (equivalently,* $\delta _{1}\prec \delta _{2}$* iff* $\delta _{1}C\delta _{2}$* is justified*).
The set-theoretical pragmatics introduced in Sec. 3.2 allows one to translate the above definition in $\mathcal{L}_{Q}^{P}$ as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff for every* $S\in S$*,* $\pi _{S}(\delta _{1})=J$* implies* $\pi
_{S}(\delta _{2})=J$*.*
The binary relation $\prec $ can then be characterized as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.*
The relation $\prec $ is obviously a pre-order relation on $\psi _{A}^{Q}$, hence it induces canonically an equivalence relation $\approx $ on $\psi
_{A}^{Q}$, defined as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $\delta _{1}\prec $* *$\delta _{2}$* and* $\delta _{2}\prec $* *$\delta _{1}$*.*
The equivalence relation $\approx $ can then be characterized as follows.
*For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $S_{\delta _{1}}=S_{\delta _{2}}$*.*
Decidability versus justifiability in $\mathcal{L}_{Q}^{P}$
-----------------------------------------------------------
We have commented rather extensively in Sec. 3.3 on the notion of justification formalized in $\mathcal{L}_{Q}^{P}$, for every $S\in \mathcal{S}$, by the pragmatic evaluation function $\pi _{S}$. It must still be noted, however, that the definition of $\pi _{S}$ on all afs in $\psi _{A}^{Q}$ does not grant that an empirical procedure of proof exists which allows one to establish, for every $S\in \mathcal{S}$, the justification value of every af of $\mathcal{L}_{Q}^{P}$. In order to understand how this may occur, note that the notion of empirical proof is defined by A$_{5}$ for atomic rfs of $\mathcal{L}_{Q}^{P}$ and makes explicit reference, for every $E(x)\in \psi
_{R}^{Q}$, to the closed subset $\mathcal{S}_{E}\in \mathcal{L(S)}$ associated with $E$ by the function $\rho $ introduced in Sec. 2.2. Basing on this notion, the justification value $\pi _{S}(\vdash E(x))$ of an elementary af $\vdash E(x)\in \psi _{A}^{Q}$ can be determined by means of the same empirical procedure, making reference to the closed subset $\mathcal{S}_{\vdash E(x)}=\mathcal{S}_{E}$ associated to $\vdash E(x)$ by the function $f$ (Sec. 3.2). Yet, whenever $\pi _{S}$ is recursively defined on the whole $\psi _{A}^{Q}$, new subsets of states are introduced (as $\mathcal{S}_{\delta _{1}}\cup \mathcal{S}_{\delta _{2}}$) which do not necessarily belong to $\mathcal{L(S)}$. If an af $\delta $ is associated by $f$ with a subset that does not belong to $\mathcal{L(S)}$, no empirical procedure exists in QM which allows one to determine the justification value $\pi _{S}(\delta )$.
We are thus led to introduce the subset $\psi _{AD}^{Q}$ $\subset \psi
_{A}^{Q}$ of all *pragmatically decidable*, or *p-decidable*, afs of $\mathcal{L}_{Q}^{P}$. An af $\delta $ of $\mathcal{L}_{Q}^{P}$ is p-decidable iff an empirical procedure of proof exists which allows one to establish whether $\delta $ is justified or unjustified, whatever the state $S$ of $x$ may be.
Because of the remark above, the subset of all p-decidable afs of $\mathcal{L}_{Q}^{P}$ can be characterized as follows.
$\psi _{AD}^{Q}=\{\delta \in \psi _{A}^{Q}\mid $ $\mathcal{S}_{\delta }\in
\mathcal{L(S)}\}$.
Let us discuss some criteria for establishing whether a given af $\delta \in
$ $\psi _{A}^{Q}$ belongs to $\psi _{AD}^{Q}$.
C$_{1}$. *All elementary afs of* $\psi _{A}^{Q}$* belong to* $\psi _{AD}^{Q}$*.*
C$_{2}$. *If* $\delta \in $* *$\psi _{AD}^{Q}$*, then* $N\delta \in $* *$\psi _{AD}^{Q}$* *
Indeed, $S_{\delta }\in \mathcal{L(S)}$ implies $S_{\delta }^{\bot }\in
\mathcal{L(S)}$.
C$_{3}$. *If* $\delta _{1}$*,* $\delta _{2}\in $* *$\psi _{AD}^{Q}$*, then* $\delta _{1}K$* *$\delta _{2}\in $* *$\psi _{AD}^{Q}$
Indeed, $S_{\delta _{1}}\in \mathcal{L(S)}$ and $S_{\delta _{2}}\in
\mathcal{L(S)}$ imply $S_{\delta _{1}}\cap S_{\delta _{2}}\in \mathcal{L(S)}
$, since $S_{\delta _{1}}\cap S_{\delta _{2}}=S_{\delta _{1}}\Cap S_{\delta
_{2}}$ because of known properties of the lattice $(\mathcal{L(S)},\subset
) $ (Sec. 2.2).
C$_{4}$. *If* $\delta _{1}$*,* $\delta _{2}\in $* *$\psi _{AD}^{Q}$*, then* $\delta _{1}A$* *$\delta _{2}$* may belong or not to* $\psi _{AD}^{Q}$*. To be precise, it belongs to* $\psi _{AD}^{Q}$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$* or* $S_{\delta _{2}}\subset S_{\delta _{1}}$
Indeed, $S_{\delta _{1}}\cup S_{\delta _{2}}\in \mathcal{L(S)}$ or, equivalently, $S_{\delta _{1}}\cup S_{\delta _{2}}=S_{\delta _{1}}\Cup
S_{\delta _{2}}$, iff one of the conditions in C$_{4}$ is satisfied.
It is apparent from criteria C$_{2}$ and C$_{3}$ that $\psi _{AD}^{Q}$ is closed with respect to the pragmatic connectives $N$ and $K$, in the sense that $\delta \in \psi _{AD}^{Q}$ implies $N\delta \in \psi _{AD}^{Q}$, and $\delta _{1}$, $\delta _{2}\in \psi _{AD}^{Q}$ implies $\delta _{1}K\delta
_{2}\in \psi _{AD}^{Q}$. On the contrary, $\psi _{AD}^{Q}$ is not closed with respect to $A$, since it may occur that $\delta _{1}A$ $\delta
_{2}\notin \psi _{AD}^{Q}$ even if $\delta _{1}$, $\delta _{2}\in \psi
_{AD}^{Q}$. In order to obtain a closed subset of afs of $\mathcal{L}_{Q}^{P} $, one can consider the set
$\phi _{AD}^{Q}=\{\delta \in \psi _{A}^{Q}\mid $ the pragmatic connective $A$ does not occur in $\delta \}$.
The set $\phi _{AD}^{Q}$ obviously contains all elementary afs of $\mathcal{L}_{Q}^{P}$, plus all afs of $\psi _{A}^{Q}$ in which only the pragmatic connectives $N$ and $K$ occur. We can thus consider a sublanguage of $\mathcal{L}_{Q}^{P}$ whose set of afs reduces to $\phi _{AD}^{Q}$. This new language is relevant since all its afs are p-decidable, hence we call it* the* *p-decidable sublanguage* of $\mathcal{L}_{Q}^{P}$ and denote it by $\mathcal{L}_{QD}^{P}$.
The p-decidable sublanguage $\mathcal{L}_{QD}^{P}$
--------------------------------------------------
As we have anticipated in the Introduction, we aim to show in this paper that the sublanguage $\mathcal{L}_{QD}^{P}$ has the structure of a physical QL, hence it provides a new pragmatic interpretation of this relevant physical structure. However, this interpretation will be more satisfactory from an intuitive viewpoint if we endow $\mathcal{L}_{QD}^{P}$ with some further derived pragmatic connectives which can be made to correspond with connectives of physical QL. To this end, we introduce the following definitions.
D$_{1}$. *We call* quantum pragmatic disjunction *the connective* $A_{Q}$* defined as follows.*
*For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*,* $\delta _{1}A_{Q}\delta _{2}=N((N\delta _{1})K(N\delta
_{2}))$*.*
D$_{2}$. *We call* quantum pragmatic implication* the connective* $I_{Q}$* defined as follows.*
*For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*,* $\delta _{1}I_{Q}\delta _{2}=(N\delta _{1})A_{Q}(\delta
_{1}K\delta _{2})$*.*
Let us discuss the justification rules which hold for afs in which the new connectives $A_{Q}$ and $I_{Q}$ occur.
By using the function $f$ introduced in Sec. 3.2 we get (since the set-theoretical operation $\cap $ coincides with the lattice operation $\Cap
$ in $(\mathcal{L(S)},\subset )$, see Sec. 2.2),
$\mathcal{S}_{\delta _{1}A_{Q}\delta _{2}}=\mathcal{S}_{(N\delta
_{1})K(N\delta _{2})}^{\bot }=(\mathcal{S}_{N\delta _{1}}\cap \mathcal{S}_{N\delta _{2}})^{\bot }=(\mathcal{S}_{\delta _{1}}^{\bot }\Cap \mathcal{S}_{\delta _{2}}^{\bot })^{\bot }=(\mathcal{S}_{\delta _{1}}\Cup \mathcal{S}_{\delta _{2}})$.
Hence, for every $S\in \mathcal{S}$,
$\pi _{S}(\delta _{1}A_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{\delta
_{1}}\Cup \mathcal{S}_{\delta _{2}}$.
Let us come to the quantum pragmatic implication $I_{Q}$. By using the definition of $A_{Q}$, one gets
$\delta _{1}I_{Q}\delta _{2}=N((NN\delta _{1})K(N(\delta _{1}K\delta _{2}))$.
By using the function $f$ and the above result about $A_{Q}$, one then gets
$\mathcal{S}_{\delta _{1}I_{Q}\delta _{2}}=\mathcal{S}_{N\delta _{1}}\Cup
\mathcal{S}_{\delta _{1}K\delta _{2}}=\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$.
It follows that, for every $S\in \mathcal{S}$,
$\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{\delta
_{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$.
Let us observe now that $\mathcal{L}_{QD}^{P}$ obviously inherits the notions of p-validity and order defined in $\mathcal{L}_{Q}^{P}$ (Sec. 3.4). Hence, we can illustrate the role of the connective $I_{Q}$ within $\mathcal{L}_{QD}^{P}$ by means of the following *pragmatic deduction lemma*.
PDL.* Let* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi
_{AD}^{Q}$*. Then,* $\delta _{1}\prec $* *$\delta _{2}$* iff for every* $S\in S$*,* $\pi _{S}(\delta _{1}I_{Q}\delta
_{2})=J$* (equivalently, iff* $\delta _{1}I_{Q}\delta _{2}$*is p-valid).*
Proof. The following sequence of equivalences holds.
For every $S\in \mathcal{S}$, $\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff for every $S\in \mathcal{S}$, $S\in \mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$ iff $\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})=\mathcal{S}$ iff $\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}}=$ $\mathcal{S}_{\delta _{1}}$ iff $\mathcal{S}_{\delta
_{1}}\subset \mathcal{S}_{\delta _{2}}$ iff $\delta _{1}\prec $ $\delta _{2}$.$\blacksquare \smallskip $
PDL shows that the quantum pragmatic implication $I_{Q}$ plays within $\mathcal{L}_{QD}^{P}$ a role similar to the role of material implication in classical logic.
Interpreting QL onto $\mathcal{L}_{QD}^{P}$
-------------------------------------------
In order to show that the physical QL $(\mathcal{E},\prec )$ introduced in Sec. 2.2 can be interpreted into $\mathcal{L}_{QD}^{P}$, a further preliminary step is needed. To be precise, let us make reference to the preorder introduced on $\psi _{A}^{Q}$ in Sec. 3.4 and consider the pre-ordered set $(\phi _{AD}^{Q},\prec )$ of all afs of $\mathcal{L}_{QD}^{P} $. Furthermore, let us denote by $\approx $ (by abuse of language) the restriction of the equivalence relation introduced on $\psi _{A}^{Q}$ in Sec. 3.4 to $\phi _{AD}^{Q}$, and let us denote by $\prec $ (again by abuse of language) the partial order induced on $\phi _{AD}^{Q}/\approx $ by the preorder defined on $\phi _{AD}^{Q}$. Then, let us show that $(\phi
_{AD}^{Q}/\approx ,\prec )$ is order isomorphic to $(\mathcal{L(S)},\subset
) $.
Let us consider the mapping
$f_{\approx }:[\delta ]_{\approx }\in \psi _{AD}^{Q}/\approx
\;\longrightarrow $ $\mathcal{S}_{\delta }\in \mathcal{L(S)}$.
This mapping is obviously well defined because of the characterization of $\approx $ in Sec. 3.4. Furthermore, the following statements hold.
\(i) *For every* $\delta \in \phi _{AD}^{Q}$*, one and only one elementary af* $\vdash E(x)$* exists such that* $\vdash E(x)\in
\lbrack \delta ]_{\approx }$.
\(ii) *The mapping* $f_{\approx }$* is bijective.*
\(iii) *For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*,* $[\delta _{1}]_{\approx }\prec \lbrack \delta
_{2}]_{\approx }$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.*
Let us prove (i). Consider $[\delta ]_{\approx }$. Since $\mathcal{S}_{\delta }\in \mathcal{L(S)}$ and $\rho $ is bijective (Sec. 2.2), a property $E\in \mathcal{E}$ exists such that $E=\rho ^{-1}(\mathcal{S}_{\delta })$, hence $\mathcal{S}_{\delta }=\mathcal{S}_{E}$. It follows that $[\delta ]_{\approx }$ contains the af $\vdash E(x)$, for $\mathcal{S}_{\vdash E(x)}=\mathcal{S}_{E}$ (Sec. 3.2). Moreover, $[\delta ]_{\approx }$ does not contain any further elementary af. Indeed, let $\vdash F(x)$ be an elementary af of $\phi _{AD}^{Q}$ with $E\neq F$: then, $\mathcal{S}_{E}\neq
\mathcal{S}_{F}$, hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash
F(x)}$, which implies $\vdash F(x)\notin \lbrack \delta ]_{\approx }$. Thus, statement (i) is proved.
The proofs of statements (ii) and (iii) are then immediate. Indeed, statement (ii) follows from (i) and from the definition of $f_{\approx }$, while statement (iii) follows from (ii) and from the definition of $\prec $ on $\phi _{AD}^{Q}/\approx $.
Because of (ii) and (iii), the poset $(\phi _{AD}^{Q}/\approx ,\prec )$ is order-isomorphic to $(\mathcal{L(S)},\subset )$, as stated.
Let us come now to physical QL. We have seen in Sec. 2.2 that $(\mathcal{L(S)},\subset )$ is order-isomorphic to $(\mathcal{E},\prec )$. We can then conclude that $(\mathcal{E},\prec )$ is order-isomorphic to $(\phi
_{AD}^{Q}/\approx ,\prec )$, which provides the desired interpretation of a physical QL into $\mathcal{L}_{QD}^{P}$.
Let us comment briefly on the pragmatic interpretation of physical QL provided above.
Firstly, we note that our interpretation maps $\mathcal{E}$ on the quotient set $\phi _{AD}^{Q}/\approx $, not onto $\phi _{AD}^{Q}$. Yet, the set of the (well formed) formulas of the lattice $(\mathcal{E},^{\bot },\Cap ,\Cup
) $ can be mapped bijectively onto $\phi _{AD}^{Q}$ by means of the mapping induced by the following formal correspondence.
\(i) $E\in \mathcal{E}$ $\longleftrightarrow \vdash E(x)\in \phi _{AD}^{Q}$.
\(ii) $^{\bot }\longleftrightarrow N$
\(iii) $\Cap \longleftrightarrow K$
\(iv) $\Cup \longleftrightarrow A_{Q}$.
Thus, the formal language of QL, for which the lattice $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic semantics*,$^{(3)}$ can be substituted by the language $\mathcal{L}_{QD}^{P}$, for which $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic pragmatics* (by the way, we also note that the above correspondence makes $I_{Q}$ correspond to a *Sasaki hook*, the role of which is well known in QL). This reinterpretation is relevant from a philosophical viewpoint, since it avoids all problems following from the standard concept of quantum truth (Sec. 2.4) considering physical QL as formalizing properties of a quantum concept of justification rather than a quantum concept of truth. This makes physical QL consistent also with the classical concept of truth adopted with the SR interpretation of QM (Sec. 2.5). Furthermore, as we have already observed in the Introduction, it places physical QL within a general *integrated perspective*, according to which non-Tarskian theories of truth can be integrated with Tarski’s theory by reinterpreting them as theories of metalinguistic concepts that are different from truth (in the case of physical QL, the concept of *empirical justification* in QM).
Secondly, we observe that our interpretation has some consequences that are intuitively satisfactory. For instance, for every state $S\in \mathcal{S}$, it attributes a justification value to every af in $\phi _{AD}^{Q}$, while it is well known that there are formulas in physical QL which have no truth value according to the standard interpretation of QL (Sec. 2.4).
Some remarks on a possible calculus for $\mathcal{L}_{QD}^{P}$
--------------------------------------------------------------
One may obviously wonder whether a calculus can be given for the language $\mathcal{L}_{QD}^{P}$ which is *pragmatically correct* (*p-correct*) and *pragmatically complete* (*p-complete*). This is not a difficult task if we limit ourselves to the general lattice structure of $(\phi _{AD}^{Q}/\approx ,\prec )$. Indeed, a set of axioms and/or inference rules which endow $\phi _{AD}^{Q}/\approx $ of the structure of orthomodular lattice can be easily obtained by using the formal correspondence introduced in Sec. 3.7, since this correspondence allows one to translate the axioms and/or inference rules that are usually stated in order to provide a calculus for orthomodular QL into $\phi _{AD}^{Q}$ (of course, all the afs produced by this translation are p-valid afs of $\mathcal{L}_{QD}^{P}$). Here is a sample set of axioms of this kind (where, of course, $\delta $, $\delta _{1}$, $\delta _{2}$ and $\delta _{3}$ are afs of $\phi _{AD}^{Q}$) obtained by translating a set of rules provided by Dalla Chiara and Giuntini.$^{(32)}$
A$_{1}$. $\delta I_{Q}\delta $.
A$_{2}$. $($ $\delta _{1}K$ $\delta _{2})I_{Q}\delta _{1}$.
A$_{3}$. $($ $\delta _{1}K$ $\delta _{2})I_{Q}\delta _{2}$.
A$_{4}$. $\delta I_{Q}(NN\delta )$.
A$_{5}$. $(NN\delta )I_{Q}\delta $.
A$_{6}$. $((\delta _{1}I_{Q}\delta _{2})K(\delta _{1}I_{Q}\delta
_{3}))I_{Q}(\delta _{1}I_{Q}(\delta _{2}K\delta _{3}))$.
A$_{7}$. $((\delta _{1}I_{Q}\delta _{2})K(\delta _{2}I_{Q}\delta
_{3}))I_{Q}(\delta _{1}I_{Q}\delta _{3})$.
A$_{8}$. $(\delta _{1}I_{Q}\delta _{2})I_{Q}((N\delta _{2})I_{Q}(N\delta
_{1}))$.
A$_{9}$. $(\delta _{1}I_{Q}\delta _{2})I_{Q}(\delta _{2}I_{Q}(\delta
_{1}A_{Q}((N\delta _{1})K\delta _{2})))$.
However, in order to obtain physical QL one needs a number of further axioms, since the structure of $(\mathcal{L(H)},\subset )$ must be recovered (Sec. 2.2). Providing a complete calculus for such a structure is a much more complicate task, which must take into account a number of mathematical results in lattice theory (in particular, Soler’s theorem$^{(33)}$). Therefore we will not discuss this problem in the present paper.
**ACKNOWLEDGEMENT**
The author is greatly indebted to Carlo Dalla Pozza, Jaroslaw Pykacz and Sandro Sozzo for reading the manuscript and providing many useful suggestions.
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[^1]: It must be noted that the physical properties considered here are first order properties from a logical viewpoint.$^{(26)}$ Higher order properties obviously occur in physics and will be encountered later on (Sec. 2.6), but they need not be considered here.
[^2]: The notion of physical equivalence for preparing or registering devices is not trivial.$^{(11,21)}$ We do not discuss it here for the sake of brevity.
[^3]: Note that a registration may act as a new preparation of the physical object $x$, so that the state of $x$ may change after a test on it.
[^4]: It follows easily that every state S can also be represented by any vector $\mid \psi \rangle \in \varphi (S)\in \mathcal{A}$, which is the standard representation adopted in elementary QM. Moreover, a state S is usually represented by an (orthogonal) projection operator on $\varphi (S)$ in more advanced QM. However, the representation $\varphi $ introduced here is more suitable for our purposes in the present paper.
[^5]: Equivalently, a property is often represented in QM as a pair $(A,\Delta )$, where is $A$ a self-adjoint operator on $\mathcal{H}$ representing a physical observable, and $\Delta $ a Borel set on the real line.$^{(28)}$ We do not use this representation, however, in the present paper.
[^6]: Whenever the dimension of $\mathcal{H}$ is finite, the lattice $(\mathcal{L(H)},\subset )$ and/or the lattice $(\mathcal{L(S)},\subset )$ can be identified with Birkhoff and von Neumann’s lattice of *experimental propositions*, which was introduced in the 1936 paper that started the research on QL.$^{(31)}$ This identification is impossible, however, if $\mathcal{H}$ is not finite-dimensional, since Birkhoff and von Neumann’s lattice is modular, not simply weakly modular. The requirement of modularity has deep roots in the von Neumann concept of probability in QM according to some authors.$^{(2)}$
[^7]: One can provide an intuitive support to this definition by noticing that the result obtained in a test of $E$ on a physical object $x$ in the state $S$ can be attributed to $x$ only whenever $S$ is not modified by the test. Moreover, only in this case the test is *repeatable*, i.e., it can be performed again obtaining the same result.
It is well known that classical physics assumes that tests which do not modify the state $S$ are always possible, at least as ideal limits of concrete procedures, while this assumption does not hold in QM.
[^8]: Assumption A$_{4}$ can be stated unchanged whenever the standard interpretation of QM is adopted instead of the SR interpretation. In this case, however, for every $\xi $, $\sigma (\xi )$ is defined only on a subset of rfs, not on the whole $\psi _{R}^{Q}$ (which requires a weakening of the assumptions on $\sigma $ if one wants to recover this case within the general perspective in Sec. 3.1). Furthermore, $\Sigma _{S}$ reduces to a singleton. Indeed, for every interpretation $\xi $, a state $S=S(\xi )$ exists such that $\xi (x)\in S$. Then, $\sigma (\xi )$ is defined on a rf $E(x)$ iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (Sec. 2.4), and does not change if $\xi $ is substituted by an interpretation $\xi ^{\prime
} $ such that $\xi ^{\prime }(x)\in S$.
[^9]: Assumption A$_{5}$ in Sec. 3.2 can be stated unchanged if the standard interpretation of QM is adopted instead of the SR interpretation. In this case, however, it is impossible that a mapping $\xi ^{\prime }$ exists such that $\xi ^{\prime }(x)\in S^{\prime }$, with $S\neq S^{\prime }$ and $\sigma (\xi )=\sigma (\xi ^{\prime })$, since $\sigma (\xi )$ and $\sigma
(\xi ^{\prime })$ are defined on different domains ($\mathcal{E}_{S}\cup
\mathcal{E}_{S}^{\bot }$ and $\mathcal{E}_{S^{\prime }}\cup \mathcal{E}_{S^{\prime }}^{\bot }$, respectively). Hence, an assigment function $\sigma
$ is associated with a unique state $S$, and $\Pi _{\sigma }$ reduces to the singleton $\{\pi _{S}\}$.
| ArXiv |
---
author:
- 'R. K. Zamanov'
- 'K. A. Stoyanov'
- 'J. Martí'
- 'G. Y. Latev'
- 'Y. M. Nikolov'
- 'M. F. Bode'
- 'P. L. Luque-Escamilla'
date: 'Received April 20, 2016; accepted July 8, 2016'
title: 'Optical spectroscopy of Be/gamma-ray binaries'
---
Introduction
============
The rapid and sustained progress of high energy and very high energy astrophysics in recent years enabled the identification of a new group of binary stars emitting at TeV energies (e.g. Paredes et al. 2013). These objects, called $\gamma$-ray binaries, are high-mass X-ray binaries that consist of a compact object (neutron star or black hole) orbiting an optical companion that is an OB star. There are five confirmed $\gamma$-ray binaries so far: PSR B1259-63/LS 2883 (Aharonian et al. 2005), LS 5039/V479 Sct (Aharonian et al. 2006), (Albert et al., 2009), HESS J0632+057/MWC 148 (Aharonian et al. 2007), and 1FGL J1018.6-5856 (H.E.S.S. Collaboration et al. 2015). Their most distinctive fingerprint is a spectral energy distribution dominated by non-thermal photons with energies up to the TeV domain. Recently, Eger et al. (2016) proposed a binary nature for the $\gamma$-ray source HESS J1832-093/2MASS J18324516-092154 and this object probably belongs to the family of the $\gamma$-ray binaries as a sixth member.
The binary system, PSR B1259-63 is unique, since it is the only one where the compact object has been identified as a radio pulsar (Johnston et al. 1992, 1994). The nature of the compact object is known in PSR B1259-63 as a neutron star, and in AGL J2241+4454/MWC 656 as a black hole (Casares et al. 2014). Although not included in the confirmed list, MWC 656 was selected as a target here despite not having shown all the observational properties of a canonical $\gamma$-ray binary yet. It was only occasionally detected by the AGILE observatory at GeV energies and not yet detected in the TeV domain (see Aleksi[ć]{} et al. 2015). Nevertheless, the fact that the black hole nature of the compact companion is almost certain renders it very similar to the typical $\gamma$-ray binaries. In the other systems the nature of the compact object remains unclear (e.g. Dubus 2013). In addition to these objects, there are several other binary systems ($\eta$ Car, Cyg X-1, Cyg X-3, Cen X-3, and SS 433) that are detected as GeV sources, but not as TeV sources so far.
Here we report high-resolution spectral observations of , MWC 148, and MWC 656, and discuss circumstellar disc size, disc truncation, interstellar extinction, and rotation of their mass donors. The mass donors (primaries) of these three targets are emission-line Be stars. The Be stars are non-supergiant, fast-rotating B-type and luminosity class III-V stars which, at some point in their lives, have shown spectral lines in emission (Porter & Rivinius 2003). The material expelled from the equatorial belt of a rapidly rotating Be star forms an outwardly diffusing gaseous, dust-free Keplerian disc (Rivinius et al. 2013). In the optical/infrared band, the two most significant observational characteristics of Be stars and their excretion discs are the emission lines and the infrared excess. Moving along the orbit, the compact object passes close to this disc, and sometimes may even go through it causing significant perturbations in its structure. This circumstellar disc feeds the accretion disc around the compact object and/or interacts with its relativistic wind.
Observations
============
[c c c c l l l ]{} Date-obs & exp-time & S/N & Orb. phase &\
yyyymmdd...hhmm & & $H\alpha$ & &\
\
[** ** ]{} &\
20140217...1923 & 60 min & 20 & 0.455 &\
20140314...1746 & 60 min & 42 & 0.396 &\
20150805...0009 & 60 min & 45 & 0.579 &\
\
[**MWC 148** ]{} &\
20140113...1857 & 60 min & 56 & 0.758 &\
20140217...2031 & 60 min & 44 & 0.870 &\
20140218...1826 & 60 min & 62 & 0.872 &\
20140313...2002 & 60 min & 54 & 0.946 &\
20140314...1855 & 60 min & 81 & 0.949 &\
20140315...1833 & 60 min & 46 & 0.952 &\
\
[**MWC 656** ]{} &\
20150705...2259 & 30 min & 55 & 0.691 &\
20150804...0017 & 30 min & 45 & 0.173 &\
20150804...2229 & 30 min & 56 & 0.188 &\
\
High-resolution optical spectra of the three northern Be/$\gamma$-ray binaries were secured with the fibre-fed Echelle spectrograph [*ESpeRo*]{} attached to the 2.0 m telescope of the National Astronomical Observatory Rozhen, located in Rhodope mountains, Bulgaria. The spectrograph uses R2 grating with 37.5 grooves/mm, Andor CCD camera 2048 x 2048 px, 13.5x13.5 $\mu m$ px$^{-1}$ (Bonev et al. 2016). The spectrograph provides a dispersion of 0.06 Åpx$^{-1}$ at 6560 Å and 0.04 Åpx$^{-1}$ at 4800 Å.
The spectra were reduced in the standard way including bias removal, flat-field correction, and wavelength calibration. Pre-processing of data and parameter measurements are performed using various routines provided in IRAF. The journal of observations is presented in Table \[tab.J\], where the date, start of the exposure, exposure time, and signal-to-noise ratio at about $\lambda 6600$ Å are given. The orbital phases are calculated using $HJD_0= 2443366.775$, $HJD_0= 2454857.5,$ and $HJD_0= 2453243.7$ for , MWC 148, and MWC 656, respectively, and orbital periods given in Sect. \[sect.2\].
Emission line profiles of , MWC 148, and MWC 656 are plotted on Fig.\[f1.examp\]. Spectral line parameters equivalent width (W) and distance between the peaks ($\Delta V$) for the prominent lines ($H\alpha$, H$\beta$, H$\gamma$, $HeI \lambda 5876,$ and $FeII \lambda 5316$) are given in Table \[tab.2\]. The typical error on the equivalent width is below $\pm 10$ % for lines with $W > 1$ Å and up to $\pm 20$% for lines with $W \lesssim 1$ Å. The typical error on $\Delta V$ is $\pm 10$ . It is worth noting that [**(1)**]{} in FeII lines are not detectable; [**(2)**]{} In MWC 656 on spectrum 20150705 the HeI $\lambda5876$ line is not visible (probably emission fills up the absorption).
In addition to the Rozhen data we use 98 spectra of MWC 148 and 68 spectra of MWC 656 (analysed in Casares et al. 2012) from the archive of the 2.0 m Liverpool Telescope[^1] (Steele et al. 2004). These spectra were obtained using the Fibre-fed RObotic Dual-beam Optical Spectrograph (FRODOSpec; Morales-Rueda et al. 2004). The spectrograph is fed by a fibre bundle array consisting of $12\times12$ lenslets of 0.82 arcsec each, which is reformatted as a slit. The spectrograph was operated in a high-resolution mode, providing a dispersion of 0.8 Åpx$^{-1}$ at 6500 Å, 0.35 Åpx$^{-1}$ at 4800 Å, and typical $S/N \gtrsim 100$. FRODOSpec spectra were processed using the fully automated data reduction pipeline of Barnsley et al. (2012). The typical error on the equivalent width is $\pm 10$ % and on $\Delta V$ is $\pm 20$ .
[cccccccccccccccccccclll]{} & & & & & &\
date-obs & $W_\alpha$ & $\Delta V_\alpha$ & $W_\beta$ & $\Delta V_\beta$ & $W_\gamma$ & $\Delta V_\gamma$ & $W_{HeI5876}$ & $\Delta V_{HeI5876}$ & $W_{FeII5316}$ & $\Delta V_{FeII}$ &\
yyyymmdd.hhmm & Å & & Å & & Å & & Å & & Å & &\
\
[** ** ]{} &\
20140217.1923 & -8.6 & 316 & -0.71 & 416 & +0.5 & ... & +0.39 & 455 & ... & ... &\
20140314.1746 & -8.1 & 309 & -1.12 & 411 & +0.7 & ... & +0.17 & 456 & ... & ... &\
20150805.0009 & -8.2 & 337 & -1.16 & 421 & +0.8 & ... & +0.40 & 416 & ... & ... &\
\
[**MWC 148** ]{} &\
20140113.1857 & -29.5 & 105 & -4.19 & 182 & -1.38 & 188 & -0.49 & 231 & -0.49 & 210 &\
20140217.2031 & -30.9 & 92 & -4.27 & 162 & -1.24 & 178 & -0.37 & 243 & -0.37 & 177 &\
20140218.1826 & -29.3 & 87 & -4.12 & 161 & -1.12 & 171 & -0.39 & 244 & -0.39 & 182 &\
20140313.2002 & -29.0 & ... & -3.41 & 165 & -0.77 & 149 & -0.39 & 246 & -0.39 & 186 &\
20140314.1855 & -28.5 & ... & -3.79 & 170 & -1.04 & 159 & -0.37 & 251 & -0.37 & 185 &\
20140315.1833 & -26.8 & ... & -3.84 & 168 & -1.01 & 175 & -0.51 & 264 & -0.51 & 186 &\
\
[**MWC 656** ]{} &\
20150705.2259 & -23.3 & ... & -2.26 & 246 & -0.42 & 301 & +0.0 & ... & -0.40 & 275 &\
20150804.0017 & -21.9 & ... & -2.12 & 244 & -0.23 & 289 & +0.22 & ... & -0.48 & 240 &\
20150804.2229 & -21.2 & ... & -1.98 & 246 & -0.34 & 311 & +0.19 & ... & -0.42 & 227 &\
\
\[tab.2\]
Objects: System parameters {#sect.2}
==========================
(V615 Cas) was identified as a $\gamma$-ray source with the $COS B$ satellite 35 years ago (Swanenburg et al. 1981). For the orbital period of , we adopt $P_{orb}=26.4960 \pm 0.0028$ d, which was derived with Bayesian analysis of radio observations (Gregory 2002) and an orbital eccentricity $e=0.537$, which was obtained on the basis of the radial velocity of the primary (Casares et al. 2005; Aragona et al. 2009). For the primary, Grundstrom et al. (2007) suggested a B0V star with radius $R_1=6.7 \pm 0.9$ . A B0V star is expected to have on average $M_1 \approx 15$ (Hohle et al. 2010). We adopt $v \sin i = 349 \pm 6$ for the projected rotational velocity of the mass donor (Hutchings & Crampton 1981, Zamanov et al. 2013).
MWC 148 (HD 259440) was identified as the counterpart of the variable TeV source HESS J0632+057 (Aharonian et al. 2007). We adopt $P_{orb} = 315 ^{+6}_{-4}$ d derived from the X-ray data (Aliu et al. 2014), which is consistent with the previous result of $321 \pm 5$ days (Bongiorno et al. 2011). For this object Aragona et al. (2010) derived $T_{eff} = 27500 - 30000$ K, $\log g = 3.75 - 4.00$, $M_1 = 13.2 - 19.0$ , and $R_1 = 7.8 \pm 1.8 $ . For the calculations in Sect.\[Disc.size\], we adopt $e=0.83$, periastron at phase 0.967 (Casares et al. 2012), and $v \sin i = 230 - 240$ (Moritani et al. 2015).
MWC 656 (HD 215227) is the emission-line Be star that lies within the positional error circle of the $AGILE$ $\gamma$-ray source AGL J2241+4454 (Lucarelli et al. 2010). It is the first and until now the only detected binary composed of a Be star and a black hole (Casares et al. 2014). For the orbital period, we adopt $P_{orb}=60.37 \pm 0.04$ d obtained with optical photometry (Williams et al. 2010), $e=0.10 \pm 0.04$ estimated on the basis of the radial velocity measurements and $v \sin i = 330 \pm 30$ (Casares et al. 2014). For the primary, Williams et al. (2010) estimated $T_{eff} = 19000 \pm 3000$ K, $\log g = 3.7 \pm 0.2 $, $M_1 = 7.7 \pm 2.0$ , $R_1 = 6.6 \pm 1.9$ . Casares et al. (2014) considered that the mass donor is a giant (B1.5-2 III) and give a mass range $M_1 = 10 - 16$ . On average a B1.5-2 III star is expected to have about $R_1 \approx 8.3 - 8.8$ (Straizys & Kuriliene 1981). From newer values of the luminosity (Hohle et al. 2010), such a star is expected to have $M_1 \approx 8.0 - 10.0$ and radius $R_1 \approx 9.5 - 10$ . We adopt $R_1 \approx 10$ for the calculations in Sect.\[Disc.size\].
[cccc|cccccllllll]{} Date-obs & $R_{disc}(H\alpha)$ & $R_{disc}(H\alpha)$ & $R_{disc}(H\alpha)$ & $R_{disc}(H\beta)$ & $R_{disc}(H\gamma)$ & $R_{disc}(HeI5876)$ & $R_{disc}(FeII)$ &\
yyyymmdd.hhmm & & & & & & & &\
& () & () & () & & & & &\
\
[** ** ]{} &\
20140217.1923 & 33 & 32 & 36 & 19 & ... & 16 & ... &\
20140314.1746 & 34 & 33 & 33 & 19 & ... & 16 & ... &\
20150805.0009 & 29 & 31 & 34 & 18 & ... & 19 & ... &\
\
[**MWC 148** ]{} &\
20140113.1857 & 156 & 165 & 180 & 52 & 49 & 32 & 39 & &\
20140217.2031 & 205 & 208 & 190 & 66 & 54 & 29 & 55 & &\
20140218.1826 & 226 & 211 & 178 & 66 & 59 & 29 & 52 & &\
20140313.2002 & ... & 201 & 176 & 63 & 52 & 29 & 50 & &\
20140314.1855 & ... & 189 & 172 & 60 & 56 & 27 & 50 & &\
20140315.1833 & ... & 193 & 160 & 60 & 50 & 25 & 50 & &\
\
[**MWC 656** ]{} &\
20150705.2259 & ... & 213 & 174 & 63 & 42 & ... & 51 & &\
20150804.0017 & ... & 216 & 162 & 64 & 46 & ... & 66 & &\
20150804.2229 & ... & 213 & 156 & 63 & 40 & ... & 74 & &\
\
\[tab.D\]
Circumstellar disc
==================
Peak separation in different lines {#peak.sep}
----------------------------------
For the Be stars, the peak separations in different lines follow approximately the relations (Hanuschik et al. 1988) $$\begin{aligned}
\Delta V_\beta \approx 1.8 \Delta V_\alpha \label{H3.1} \\
\Delta V_\gamma \approx 1.2 \Delta V_\beta \approx 2.2 \Delta V_\alpha \label{H3.2} \\
\Delta V_{\rm FeII} \approx 2.0 \Delta V_\alpha \label{H3.3} \\
\Delta V_{\rm FeII} \approx 1.1 \Delta V_\beta \label{H3.4}
,\end{aligned}$$ where Eq. \[H3.4\] is derived from Eqs. \[H3.1\] and \[H3.3\].
For using the measurements in Table \[tab.2\], we obtain $\Delta V_\beta = 1.30 \pm 0.04 \, \Delta V_\alpha$ and $\Delta V_{HeI5876}= 1.38 \pm 0.13 \, \Delta V_\alpha $. The ratio $\Delta V_\beta / \Delta V_\alpha$ is considerably below the average value for the Be stars (see Eq.\[H3.1\]). We obtain $\Delta V_\beta = 1.78 \pm 0.06 \, \Delta V_\alpha$, $\Delta V_\gamma = 1.07 \pm 0.03 \, \Delta V_\beta$, and $\Delta V_{FeII5316} = 1.12 \pm 0.03 \, \Delta V_\beta$, $\Delta V_{HeI5876} = 1.47 \pm 0.10 \, \Delta V_\beta$ for MWC 148. We use only three spectra for $H\alpha$ (20140113, 20140217, and 20140218) when two peaks in are visible. The value of $\Delta V_\beta / \Delta V_\alpha \approx 1.78$ is very similar to 1.8 in Be stars, the ratio $\Delta V_{FeII5316} / \Delta V_\beta \approx 1.07 $ is similar to 1.1 in Be stars, and the value of $\Delta V_\gamma / \Delta V_\beta \approx 1.07$ is again similar to the value 1.2 for Be stars.
We estimate $\Delta V_\beta = 1.72 \pm 0.18 \, \Delta V_\alpha$ for MWC 656 (using six spectra from the Liverpool Telescope FRODOSpec, where two peaks are visible in both $H\alpha$ and $H\beta$), $\Delta V_\gamma = 1.22 \pm 0.04 \, \Delta V_\beta$, $\Delta V_{FeII5316} = 1.01 \pm 0.10 \, \Delta V_\beta$, where all three ratios are similar to the corresponding values (Eq. \[H3.1\], \[H3.2\], \[H3.4\]) in Be stars. We do not see two peaks on high-resolution Rozhen spectra of this object. However two peaks are clearly distinguishable on a few of the LT spectra. In every case of detection/non-detection of the double peak structure, $W\alpha$ is very similar $19 < W\alpha < 25$ Å.
The comparison of the peak separation of different emission lines shows that MWC 148 and MWC 656 have circumstellar disc that is similar to that of the normal Be stars. At this stage considerable deviation from the behaviour of the Be stars is only detected in . In this star the $H\alpha$-emitting disc is only 1.7 times larger than the H$\beta$-emitting disc, while in normal Be stars it is 3.3 times larger. This probably is one more indication that outer parts of the disc are truncated as a result of the relatively short orbital period.
Disc size {#Disc.size}
---------
For rotationally dominated profiles, the peak separation can be regarded as a measure of the outer radius ($R_{disc}$) of the emitting disc (Huang 1972) $$\left( \frac{\Delta V}
{2\,v\,\sin{i}} \;\right)
= \; \left( \frac {R_{disc}}{R_1}\;\right)^{-j} ,
\label{Huang}$$ where $j=0.5$ for Keplerian rotation, $j=1$ for angular momentum conservation, $R_1$ is the radius of the primary, and $v\,\sin{i}$ is its projected rotational velocity. Eq. \[Huang\] relies on the assumptions that (1) the Be star is rotating critically, and (2) that the line profile shape is dominated by kinematics and radiative transfer does not play a role.
When the two peaks are visible in the emission lines, we can estimate the disc radius using Eq. \[Huang\]. The calculated disc size for different emission lines are given in Table \[tab.D\].
In the $H\alpha$ emission line of two peaks are clearly visible on all of our spectra. However in MWC 656 the $H\alpha$ emission line seems to exhibit three peaks (see Fig. \[f1.examp\]). Two peaks in $H\alpha$ emission of MWC 148 are clearly detectable on January-February 2014 observations. Two peaks are not distinguishable on the spectra obtained in March 2014 (when the companion is at periastron), which probably indicates perturbations in the outer parts of the disc caused by the orbital motion of the compact object.
In the $H\beta$ line two peaks are visible on all the Rozhen spectra. We take this opportunity to obtain an estimation of the $R_{disc}(H\alpha)$ using $\Delta V_\beta$; the ratios $\Delta V_\beta / \Delta V_\alpha$ (as obtained in Sect. \[peak.sep\]), and Eq.\[Huang\]. The $R_{disc}(H\alpha)$ values calculated in this way are given in Table \[tab.D\] and indicated with $(^b)$.
Disc size and $W_\alpha$ {#RdWa}
------------------------
The disc size and $W_\alpha$ are connected (see also Hanuschik 1989; Grundstrom & Gies 2006). This simply expresses the fact that $R_{disc}$ grows as $W_\alpha$ becomes larger. In Fig. \[f3.EW.dV\], we plot $\log \Delta V_\alpha / 2 $ versus $\log W_\alpha$. In this figure 138 data points are plotted for Be stars taken from Andrillat (1983), Hanuschik (1986), Hanuschik et al. (1988), Dachs et al. (1992), Slettebak et al. (1992), and Catanzaro (2013). In this figure, the data for Be/$\gamma$-ray binaries are also plotted. The three Be/$\gamma$-ray binaries are inside the distribution of normal Be stars.
There is a moderate to strong correlation between the variables with Pearson correlation coefficient -0.63, Spearman’s (rho) rank correlation 0.64, and $p$-$value \sim 10^{-15}$. The dependence is of the form $$\; \; \log \; (\Delta V_{\alpha}/2 v \sin i ) = -a \; \log W_\alpha + b,
\label{Han2}$$ and the slope is shallower for stars with $ W_\alpha < 3$ Å as noted by Hanuschik et al. (1988). For 120 data points in the interval $3 \le W_\alpha \le 50$ Å, using a least-squares approximation we calculate $a=0.592 \pm 0.030$ and $b=0.165 \pm 0.036$. This fit as well as the correlation coefficients are calculated using only normal Be stars. Using Eq. \[Huang\] and Eq. \[Han2\] we then obtain $$\left( \frac {R_{disc}}{R_1}\;\right)^{-j} = 1.462 \; W_\alpha^{-0.592}.
\label{Rd.W1}$$ Having in mind that (1) the discs of the Be stars are near Keplerian (Porter & Rivinius 2003, Meilland et al. 2012); (2) the Be stars rotate at rates below the critical rate (e.g. Chauville et al. 2001), and (3) at higher optical depths the emission line peaks are shifted towards lower velocities (Hummel & Dachs 1992), we calculate the disc radius with the following formula: $$\frac {R_{disc}}{R_1} = \; \epsilon \; 0.467 \; \; W_\alpha^{1.184},
\label{Rd.W2}$$ where $\epsilon$ is a dimensionless parameter (see also Zamanov et al. 2013), for which we adopt $\epsilon = 0.9 \pm 0.1$. The disc sizes calculated with Eq. \[Rd.W2\] are given in Table \[tab.D\] and are denoted with $(^c)$. As can be seen, the values agree with those obtained with the conventional method. We estimate average values of the dimensionless quantity $R_{disc} / \epsilon R_1 = 8.7 \pm 1.9$ (for ), $R_{disc} / \epsilon R_1 = 43 \pm 5$ (for MWC 148), and $R_{disc} / \epsilon R_1 = 18.0 \pm 1.1$ (for MWC 656), respectively.
Disc truncation {#disc.trunc}
---------------
The orbit of the compact object, the average size of $H\alpha$ disc, the average size of $H\beta$ disc, and the Be star are plotted in Fig. 3. The coordinates X and Y are in solar radii. The histograms of $H\alpha$ disc size, calculated using Eq. \[Rd.W2\], are plotted in Fig. 4. For , we use our new data and published data from Paredes et al. (1994), Steele et al. (1996), Liu & Yan (2005), Grundstrom et al. (2007), McSwain et al. (2010), and Zamanov et al. (1999, 2013). We use Rozhen and Liverpool Telescope spectra for MWC 148 and MWC 656. In all three stars the distribution of $R_{disc}$ values has a very well pronounced peak. The tendency for the disc emission fluxes to cluster at specified levels is related to the truncation of the disc at specific disc radii by the orbiting compact object (e.g. Coe et al. 2006). Okazaki & Negueruela (2001) proposed that these limiting radii are defined by the closest approach of the companion in the high-eccentricity systems and by resonances between the orbital period and the disc gas rotational periods in the low-eccentricity systems. The resonance radii are given by $${R_{n:m}^{3/2}} = \frac{m \; (G \: M_1)^{1/2}}{2 \: \pi} \: \frac{P_{orb}}{n},
\label{eq.resona}$$ where $G$ is the gravitational constant, $n$ is the integer number of disc gas rotational periods, and $m$ is the integer number of orbital periods. The important resonances are not only those with $n:1$, but can also be $n:m$ in general. For (assuming $M_1 \approx 15$ , $M_2 \approx 1.4$ , $e \approx 0.537$), we estimate the distances between the components $a(1-e) \approx 44$ and $a(1+e)\approx 146$ for the periastron and apastron, respectively. As can be seen from Fig. \[f5.hist\], the disc size is $R_{disc} \sim a(1-e)$ and it never goes close to $a(1+e)$. The resonances that correspond to disc size are between 5:1 and 1:1, and the peak on the histogram corresponds to the 2:1 resonance.
For MWC 148, with the currently available data ($M_1 \approx 15$ , $M_2 \approx 4$ , $e \approx 0.83$), we estimate $a(1-e) \approx 88$ and $a(1+e)\approx 951$ for the periastron and apastron, respectively. In Fig. \[f5.hist\], it is apparent that $a(1-e) < R_{disc} < a(1+e)$. The 2:1 resonance is the closest to the peak of the distribution. We note in passing that the disc size in this star could have a bi-modal distribution (a second peak with lower intensity seems to emerge close to 4:1 resonance radius).
For MWC 656 (assuming $M_1 \approx 9$ , $M_2 \approx 4$ , $e \approx 0.1$), we estimate $a(1-e) \approx 137$ and $a(1+e)\approx 167$ for the periastron and apastron, respectively. In Fig. \[f5.hist\], it is seen that the 1:1 resonance is very close to the peak of the distribution and $a(1-e) \lesssim R_{disc} \lesssim a(1+e)$. The disc size rarely goes above $a(1+e)$.
Interstellar extinction: Estimates of $E(B-V)$ from interstellar lines
======================================================================
There is a strong correlation between equivalent width of the diffuse interstellar bands (DIBs) and reddening (Herbig 1975; Puspitarini et al. 2013). There is also a calibrated relation of reddening with the equivalent width of the interstellar line $KI \lambda7699$Å (Munari & Zwitter 1997). Aiming to estimate the interstellar extinction towards our objects, we measure equivalent widths of $KI \lambda7699$Å and DIBs $\lambda6613, \lambda5780, \lambda5797$.
[**LSI+61$^0$303:** ]{} For this object, Hunt et al. (1994) use $E(B-V)=0.93$ (Hutchings & Crampton 1981). Howarth (1983) using the 2200 Å extinction bump obtained $E(B-V)=0.75 \pm 0.1$. Steele et al. (1998) estimated $E(B-V)=0.70 \pm 0.40$ from Na I D$_2$ and $E(B-V)= 0.65 \pm 0.25$ from diffuse interstellar bands. For , we measure $0.17 \le W(KI \lambda7699) \le 0.19$ Å, $0.17 < W(DIB \lambda6613) < 0.19$ Å, $0.34 < W(DIB \lambda5780) < 0.41$ Å, $0.09 < W(DIB \lambda5797) < 0.16$ Å, which following Munari & Zwitter (1997) and Puspitarini et al. (2013) calibrations corresponds to $E(B-V)=0.84 \pm 0.08$.
[**MWC 148:** ]{} For this star, Friedemann (1992) estimated E(B-V)=0.85 from the 217 nm band. We measure $0.14 < W(KI \lambda7699) < 0.20$ Å, $0.13 < W(DIB \lambda6613) < 0.18$ Å, $0.28 < W(DIB \lambda5780) < 0.34$ Å, $0.13 < W(DIB \lambda5797) < 0.15$ Å, which corresponds to a slightly lower value $E(B-V)=0.77 \pm 0.06$.
[**MWC 656:** ]{} For this star, Williams et al. (2010) gave a low value E(B-V)=0.02. Casares et al. (2014) estimated E(B-V)=0.24. We measure $0.04 < W(DIB \lambda6613) < 0.06$ Å, $0.09 \le W(DIB \lambda5780) \le 0.10$ Å, $0.03 < W(DIB\lambda5797) < 0.05$ Å, and estimate $E(B-V)=0.25 \pm 0.02$.
Rotational period of the mass donors
====================================
In close binary systems the rotation of the companions of compact objects is accelerated by mass transfer and tidal forces (e.g. Ablimit & Lü 2012). Adopting the parameters given in Sect. \[sect.2\], we estimate the rotational periods of the mass donors $P_{rot} \approx 0.92$ d (for ), $P_{rot} \approx 0.91$ d (MWC 148), and $P_{rot} \approx 0.86$ d (MWC 656). For the $\gamma$-ray binary LS 5039, the mass donor is a O6.5V((f)) star with $R_{1} = 9.3 \pm 0.6 $ , inclination $i \approx 24.9^0$, $= 113 \pm 8$ (Casares et al. 2005). We calculate $P_{rot} = 1.764$ d. Because of the short orbital period $P_{orb} = 3.91$ d, the rotation of the mass donor could be pseudo-synchronized with the orbital motion (Casares et al. 2005).
Radio observations of the pulsar in PSR B1259-63/LS 2883 allow the orbital parameters to be precisely established: $P_{orb} = 1236.72$ d and eccentricity of e = 0.87 (Wang et al. 2004; Shannon et al. 2014). For the primary, Negueruela et al. (2011) estimated $R_1 = 9.0 \pm 1.5$ , $ = 260 \pm 15$ , and $i_{orb} \approx 23^0$, which give $P_{rot} = 0.689$ d.
In Fig. \[f2.Prot\] we plot $P_{rot}$ versus $P_{orb}$ for a number of high-mass X-ray binaries (see also Stoyanov & Zamanov 2009). High-mass X-ray binaries with a giant/supergiant component, Be/X-ray binaries, and Be/$\gamma$-ray binaries are plotted with different symbols in this figure. The solid line represents synchronization ($P_{rot} = P_{orb}$). Among the five $\gamma$-ray binaries with known rotational velocity of the mass donor, the rotation of the mass donor is close to synchronization with the orbital motion in only one (LS 5039). The four Be/$\gamma$-ray binaries are not close to the line of synchronization. They occupy the same region as the Be/X-ray binaries; in other words regarding the rotation of the mass donor the Be/$\gamma$-ray binaries are similar to the Be/X-ray binaries. In these, the tidal force of the compact object acts to decelerate the rotation of the mass donor. The spin-down of the Be stars due to angular momentum transport from star to disc (Porter 1998) is another source of deceleration.
In the well-known Be star $\gamma$ Cas, Robinson & Smith (2000) found that the X-ray flux varied with a period P = 1.1 d, which they interpreted as the rotational period of the mass donor. A similar period is detected in photometric observations (Harmanec et al. 2000; Henry & Smith 2012). This periodicity is probably due to the interaction between magnetic field of the Be star and its circumstellar disc or the presence of some physical feature, such as a spot or cloud, co-rotating with the star. The optical emission lines of MWC 148 are practically identical to those of $\gamma$ Cas. All detected lines in the observed spectral range (Balmer lines, HeI lines and FeII lines) have similar equivalent widths, intensities, profiles, and even a so-called wine-bottle structure noticeable in the $H\alpha$ line (see Fig. 1 in Zamanov et al. 2016). Bearing in mind the above estimations and the curious similarities between: (i) the mean 20-60 keV X-ray luminosity of $\gamma$ Cas and (Shrader et al. 2015) and (ii) the optical emission lines of $\gamma$ Cas and MWC 148, we consider that periodicity $\sim\!1$ day could be detectable in X-ray/optical bands in the Be/$\gamma$-ray binaries and could provide a direct measurement of the rotational period of the mass donor.
Discussion
==========
The three Be/$\gamma$-ray binaries discussed here have non-zero eccentricities and misalignment between the spin axis of the star and the spin axis of the binary orbit could be possible (Martin et al. 2009).
The inclination of the primary star in to the line of sight is probably $ i_{Be} \sim 70^0$ (Zamanov et al. 2013). Aragona et al. (2014) derived $a_1 \sin i_{orb} = 8.64 \pm 0.52$. Assuming $M_1 = 15$, $M_2 = 1.4$ , we estimate $ i_{orb} \sim 67^0 - 73^0$. It appears that there is no significant deviation of the orbital plane from the equatorial plane of the Be star.
The emission lines of MWC 148 are very similar to those of $\gamma$ Cas. The emission lines are most sensitive to the footpoint density and inclination angle (Hummel 1994). It means that in MWC 148 the Be star inclination should be similar to that of $\gamma$ Cas, for which the inclination is in the range $40^0 - 50^0$ (Clarke 1990, Quirrenbach et al. 1997). For MWC 148, Casares et al. (2012) estimated $a_1 \sin i_{orb} = 77.6 \pm 25.9$, which for $M_1 = 15$ and a 4 black hole gives $ 45^0 \lesssim i_{orb} \lesssim 65^0$.
For MWC 656, Casares et al. (2014) give $M_1 \sin ^3 i_{orb} = 5.83 \pm 0.70$. Bearing in mind the range of $8$ $\le M_1 \le 10$ , this gives $53^0 \lesssim i_{orb} \lesssim 59^0$. Inclination of the Be star can be evaluated from the full width at zero intensity of the FeII lines $FWZI/2 \sin i = (G M_1/ R_1)^{1/2}$. From FWZI of FeII lines (Casares et al. 2012) and using $R_1= 9.5 - 10.0$ we estimate $i_{Be} \approx 53 - 61^0$. It appears that both planes are almost complanar.
There are no signs of considerable deviation between the two planes. The opening half-angle of the Be stars’ circumstellar disc are $\sim \! 10^0$ (Tycner et al. 2006; Cyr et al. 2015) and it means that the compact object is practically orbiting in the plane of the circumstellar disc. The comparison between the orbit and circumstellar disc size (see Fig. \[f5.orbit\] and Sect. \[disc.trunc\]) shows that in these three objects we have three different situations:
- In the neutron star passes through the outer parts of the circumstellar disc at periastron, but it does not enter deeply in the disc;
- In MWC 148 the compact object goes into the innermost parts of the disc (passes through the innermost parts and penetrates deeply in the disc) during the periastron passage;
- In MWC 656 the black hole is constantly accreting from the outer edge of the circumstellar disc.
In MWC 656, this means that the compact object (black hole) is at the disc border at all times and as a consequence it will have a higher and stable mass accretion rate along the entire orbit. It seems to be a very clear case of disc truncation in which the circumstellar disc is cut almost exactly at the black hole orbit.
Because the $H\alpha$ peaks are connected with the outermost parts of the disc, the above three items probably explain the observational findings:
1\. In the $H\alpha$ emission line has a two-peak profile at all times (in all our spectra in the period 1987 - 2015),because the circumstellar disc size is relatively small and the neutron star passes only through the outermost parts of the circumstellar disc at periastron; 2. In MWC 148, the big jumps in the $H\alpha$ parameters, $W_\alpha$, full width at half maximum and radial velocity (see Fig. 4 of Casares 2012) occur because the compact object enters (reaches) the inner parts of the disc during the periastron passage.
3\. In MWC 656 the double-peak profile is not often visible because the black hole is at the outer edge at all times and makes perturbations exactly in the places where the $H\alpha$ peaks are formed.
It is worth noting that when the compact object causes large-scale perturbations, distorted profiles, such as those observed in 1A 535+262 (Moritani et al. 2011, 2013), will appear. If only a small portion of the outer disc is perturbed then it will appear in the central part of the emission line profile (e.g. in between the peaks or even filling the central dip) because the outer parts of the disc produce the central part of the $H\alpha$ emission line profile. Similar additional emission is already detected in the $H\alpha$ spectra of (Paredes et al. 1990; Liu et al. 2000; Zamanov & Marti 2000).
Gamma-ray emission has been repeatedly observed to be periodic in the system LSI+61303 (Albert et al. 2009; Saha et al. 2016) and also very likely in MWC148, where its periodic X-ray flares are highly correlated with TeV emission (Aliu et al. 2014). A similar situation occurs in the case of LS 5039 (Aharonian et al. 2006), 1FGL J1018.6-20135856 (H. E. S. S. Collaboration et al. 2015), and PSR B1259-63 (H.E.S.S. Collaboration et al. 2013), which has the longest orbital period observed in a $\gamma$-ray binary (about 3 yr). This is likely related to the very different physical conditions sampled by the compact companion as it revolves around the primary star in an eccentric orbit. Non-thermal emission (e.g. Dubus 2013) is produced from particles accelerated at the shock between the wind of the pulsar and matter flowing out from the primary star (the polar wind as well as the disc).
The X-ray and $\gamma$-ray light curve of MWC 148 shows two maxima at orbital phases 0.35 and 0.75 and minimum at apastron passage (Acciari et al. 2009; Aliu et al. 2014). X-ray and $\gamma$-ray fluxes are correlated as mentioned above, in agreement with leptonic emission models, where relativistic electrons lose energy by synchrotron emission and inverse Compton emission (Maier & for the VERITAS Collaboration 2015). The highly eccentric orbital geometry sketched in the central panel of Fig. \[f5.orbit\] is also in good agreement with a periodic flaring system.
In the case of MWC 656, $\gamma$-rays have been detected occasionally (Williams et al. 2010), but so far never reaching the TeV energy domain. The main differences from previous sources are the fact that the compact object has been shown to be a black hole and its orbit is only moderately eccentric (Casares et al. 2014). Based on our spectroscopic observations, the size of the excretion disc is such that the black hole is accreting matter only from its lower density outer edges. As mentioned before, this implies that the accretion rate is stable but at the same time low. Indeed, the quiescent X-ray emission level of the system is as weak as $\sim 10^{-8}$ Eddington luminosity (Munar-Adrover et al. 2014). Therefore, episodic $\gamma$-ray flares such as those detected by AGILE, likely require some enhancement of mass loss from the primary Be star or clumps in its circumstellar disc. We speculate here that the physical mechanism responsible for $\gamma$-ray emission in the MWC 656 black hole context could be related to the alternative microquasar-jet scenario also proposed for $\gamma$-ray binaries (see e.g. Romero et al. 2007). The fact that MWC 656 seems to adhere to the low-luminosity end of the X-ray/radio correlation for hard state compact jets also points in this direction (Dzib et al. 2015).
Conclusions
===========
From the spectroscopic observations of the three Be/$\gamma$-ray binaries we deduce that in the neutron star crosses the outer parts of the circumstellar disc at periastron, in MWC 148 the compact object passes deeply through the disc during the periastron passage, and in MWC 656 the black hole is accreting from the outer parts of the circumstellar disc during the entire orbital cycle. The histograms in all three stars show that the disc size clusters at specific levels, indicating the circumstellar disc is truncated by the orbiting compact object. We estimate the interstellar extinction towards , MWC 148, and MWC 656. The rotation of the mass donors is similar to that of the Be/X-ray binaries. We suggest that the three stars deserve to be searched for a periodicity of about $1.0$ day.
The authors are grateful to an anonymous referee for valuable comments and suggestions. This work was partially supported by grant AYA2013-47447-C3-3-P from the Spanish Ministerio de Economía y Competitividad (MINECO), and by the Consejería de Economía, Innovación, Ciencia y Empleo of Junta de Andalucía as research group FQM-322, as well as FEDER funds.
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[^1]: The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council.
| ArXiv |
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abstract: 'Let $k$ be a knot in $S^3$. In [@HS], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover of $(S^3, k)$ from a thin position of $k$. In this article, we will prove that if a thin position of $k$ induces a thin decomposition of double branched cover of $(S^3,k)$ by Howards and Schultens’ method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of $k$ on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width (i.e. for $k=k_1\#k_2$, $w(k)=w(k_1)\# w(k_2)-2$) in this case. Moreover, we will generalize the hypothesis to the property a thin position induces a manifold decomposition whose thick surfaces consists of strongly irreducible or critical surfaces (so topologically minimal.)'
author:
- Jungsoo Kim
date: '13, Jan, 2010'
title: A note on the nearly additivity of knot width
---
Introduction and result
=======================
Let $k$ be a knot in $S^3$. In [@HS], H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover (abbreviate it as *DBC*, and call the method *the H-S method*) of $(S^3, k)$ (see section \[section-DBC\]) and they proved that for $2$-bridge knots and $3$-bridge knots in thin position DBC inherits thin manifold decomposition (note that a knot in a thin position may not induce a thin manifold decomposition by the H-S method in general, see [@HS] and [@HRS].) Indeed, if $k$ is a non-prime $3$-bridge knot, then $k=k_1\#k_2$ for $2$-bridge knots $k_1$ and $k_2$, and thin position of $k$ is the sum of $k_1$ and $k_2$ by stacking a thin position of one of the knots on top of a thin position of the other (see Corollary 2.5 of [@HS].) So $w(k)=w(k_1)+w(k_2)-2$, i.e. $k$ holds the *nearly additivity* of knot width (for $k=k_1\#k_2$, $w(k)=w(k_1)+w(k_2)-2$, see [@ST2] for more details on “the nearly additivity of knot width”.)
So we get a question whether the property that a thin position of a knot induces a thin manifold decomposition of DBC by the H-N method implies the nearly additivity of knot width in general. If a thin position of $k$ is the sum of an ordered stack of prime summands of $k$ where each summand is in a thin position (“*a sum of an ordered stack of prime summands*” means like the left of Figure \[figure-thin\], where the bottom summand is a Montesinos knot $M(0; (2,1), (3,1), (3, 1), (5, 1))$ in a thin position (this figure is borrowed from Figure 5.2.(c) of [@HK]) and the top summands are trefoils in thin position,) then the sum of an ordered stack like that in a different order also determines a thin position of $k$. So $k$ must hold the nearly additivity of knot width by the uniqueness of prime factorization of $k$. In [@RS], Y. Rieck and E. Sedgwick proved that thin position of the sum of small knots is the sum of an ordered stack in that manner, i.e. it holds the nearly additivity of knot width. But M. Scharlemann and A. Thompson proposed a way to construct a example to contradict the nearly additivity of knot width (see [@ST2].) Although R. Blair and M. Tomova proved that most of Scharlemann and Thompson’s constructions do not produce counterexamples for the nearly additivity of knot width (see [@BT],) the question seems not obvious.
In this article, we will prove that the question is true.
\[theorem-1\] If a thin position of a knot $k$ induces a thin manifold decomposition of double branched cover of $(S^3,k)$ by the Howards and Schultens’ method, then the thin position is the sum of prime summands by stacking a thin position of one of prime summands of $k$ on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.
In section \[section-critical\], we will generalize Theorem 1.1 by using the concept *critical surface* originated from D. Bachman (see [@Bachman1] and [@Bachman3] for the original definition and the recently modified definition of “critical surface”.) So we will get the corollary.\
[**Corollary 5.3.**]{} *If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens’ method where each thick surface $H_+$ of the manifold decomposition of $M$ is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.*
Generalized Heegaard splittings
===============================
In this section, we will introduce some definitions about generalized Heegaard splittings. We use the notations and definitions by D. Bachman in [@Bachman3] through this section for convenience.
A *compression body* (a *punctured compression body* resp.) is a $3$-manifold which can be obtained by starting with some closed, orientable, connected surface, $H$, forming the product $H\times I$, attaching some number of $2$-handles to $H\times\{1\}$ and capping off all resulting $2$-sphere boundary components (some $2$-sphere boundaries resp.) that are not contained in $H\times\{0\}$ with $3$-balls. The boundary component $H\times\{0\}$ is referred to as $\partial_+$. The rest of the boundary is referred to as $\partial_-$.
A *Heegaard splitting* of a $3$-manifold $M$ is an expression of $M$ as a union $V\cup_H W$, where $V$ and $W$ are compression bodies that intersect in a transversally oriented surface $H=\partial_+V=\partial_+ W$. If $V\cup_H W$ is a Heegaard splitting of $M$ then we say $H$ is a *Heegaard surface*.
Let $V\cup_H W$ be a Heegaard splitting of a $3$-manifold $M$. Then we say the pair $(V,W)$ is a *weak reducing pair* for $H$ if $V$ and $W$ are disjoint compressing disks on opposite sides of $H$. A Heegaard surface is *strongly irreducible* if it is compressible to both sides but has no weak reducing pairs.
\[definition-GHS\] A *generalized Heegaard splitting* (GHS)[^1] H of a $3$-manifold $M$ is a pair of sets of pairwise disjoint, transversally oriented, connected surfaces, $\operatorname{Thick}(H)$ and $\operatorname{Thin}(H)$ (in this article, we will call the elements of each of both *thick surfaces* and *thin surfaces*, resp.), which satisfies the following conditions.
1. Each component $M'$ of $M-\operatorname{Thin}(H)$ meets a unique element $H_+$ of $\operatorname{Thick}(H)$ and $H_+$ is a Heegaard surface in $M'$. Henceforth we will denote the closure of the component of $M-\operatorname{Thin}(H)$ that contains an element $H_+\in\operatorname{Thick}(H)$ as $M(H_+)$.
2. As each Heegaard surface $H_+\subset M(H_+)$ is transversally oriented, we can consistently talk about the points of $M(H_+)$ that are “above” $H_+$ or “below” $H_+$. Suppose $H_-\in \operatorname{Thin}(H)$. Let $M(H_+)$ and $M(H'_+)$ be the submanifolds on each side of $H_-$. Then $H_-$ is below $H_+$ if and only if it is above $H'_+$.
3. There is a partial ordering on the elements of $\operatorname{Thin}(H)$ which satisfies the following: Suppose $H_+$ is an element of $\operatorname{Thick}(H)$, $H_-$ is a component of $\partial M(H_+)$ above $H_+$ and $H'_-$ is a component of $\partial M(H_+)$ below $H_+$. Then $H_->H'_-$.
Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is *strongly irreducible* if each element $H_+\in \operatorname{Thick}(H)$ is strongly irreducible in $M(H_+)$.
The manifold decomposition of double branched cover of $(S^3,k)$ from a thin position of $k$\[section-DBC\]
===========================================================================================================
In this section, we will describe the method to construct a manifold decomposition of DBC from a thin position of a knot originated from H.N. Howards and J. Schultens in [@HS] (see section 3 of [@HS] for more details.) We borrow the notions and definitions directly from [@HS].
Let $h:\{S^3 - (\text{two points})\} \to [0, 1]$ be a height function on $S^3$ that restricts to a Morse function on $k$. Choose a regular value $t_i$ between each pair of adjacent critical values of $h|k$. The *width* of $k$ with respect to $h$ is $\sum_i \#|k\cap h^{-1}(t_i)|$. Define the *width* of $k$ to be the minimum width of $k$ with respect to $h$ over all $h$. A *thin position* of $k$ is the presentation of $k$ with respect to a height function that realizes the width of $k$.
A *thin level* for $k$ is a $2$-sphere $S$ such that the following hold:
1. $S = h^{-1}(t_0)$ for some regular value $t_0$;
2. $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a minimum of $k$ lying above $t_0$ and $y$ is a maximum of $k$ lying below $t_0$.
A *thick level* is a $2$-sphere $S$ such that the following hold:
1. $S = h^{-1}(t_0)$ for some regular value $t_0$;
2. $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a maximum of $k$ lying above $t_0$ and $y$ is a minimum of $k$ lying below $t_0$.
A *manifold decomposition* is a generalized version of GHS in Definition \[definition-GHS\][^2], where we permit thin surfaces to be 2-spheres (this “manifold decomposition” is originated from [@ST] by M. Scharlemann and A. Thompson.) So adjacent thick and thin surfaces in a manifold decomposition cobound a (possibly, punctured-) compression body.
Let the *complexity* of a connected surface $S$ be $c(S)= 1-\chi(S) = 2 \operatorname{genus}(S)-1$ for $S$ of positive genus. Define $c(S^2) = 0$. For $S$ not necessarily connected define $c(S) = \sum\{c(S')| S'\text{ a connected component of }S\}$.
Let the *width* of the manifold decomposition $H$ of $M$ be the set of integers $\{c(S_i)| 1\leq i \leq k, \text{ each }S_i\text{ is the thick surface of }H \}$.
Order these integers in monotonically non-increasing order. Compare the ordered multi-sets lexicographically.
Define the *width* $w(M)$ of $M$ to be the minimal width over all manifold decompositions using the above ordering of the sets of integers.
A given manifold decomposition of $M$ is *thin* if the width of the manifold decomposition is the width of $M$.
Let $k$ be a knot and denote DBC of $(S^3,k)$ by $M$. If $k$ is in a thin position, then $M$ inherits a manifold decomposition as follows: Denote the thick levels of $k$ by $S_1,\cdots,S_n$ and the thin levels by $L_1,\cdots,L_{n-1}$. Each $S_i$ and each $L_i$ is a sphere that meets the knot some (even) number of times. More specifically, each $S_i$ meets $k$ at least $4$ times and each $L_i$ meets $k$ at least $2$ times. Denote the surface in $M$ corresponding to $S_i$ by $\tilde{S}_i$ and the surface in $M$ corresponding to $L_i$ by $\tilde{L}_i$ . Each $\tilde{S}_i$ is a closed orientable surface of genus at least $1$ and each $\tilde{L}_i$ is a closed orientable surface. More specifically, if $S_i$ meets $k$ exactly $2l$ times, then $\tilde{S}_i$ is a closed orientable surface of genus $l-1$. And if $L_i$ meets $k$ exactly $2l$ times, then $\tilde{L}_i$ is a closed orientable surface of genus $l-1$. See Figure \[figure-thin\].
![Thin and thick levels of $k$ and corresponding surfaces in $M$\[figure-thin\]](fig-thin.pdf){width="8.5cm"}
The $3$-ball bounded by $S_1$ ($S_n$ resp.) in $S^3$ corresponds to a handlebody $H_1$ ($H_n$ resp.) in $M$, where $H_1$ ($H_n$ resp.) is bounded by $\tilde{S}_1$ ($\tilde{S}_n$ resp.) in $M$. Moreover, the submanifold between $S_i$ and $L_{i-1}$ in $S^3$ (between $L_i$ and $S_{i}$ resp.) corresponds to a (possibly punctured) compression body $C_i^1$ ($C_i^2$ resp.) in $M$, where $\partial_+ C_i^1 = \tilde{S}_i$ ($\partial_+C_i^2=\tilde{S}_{i}$ resp.) and $\partial_- C_i^1 = \tilde{L}_{i-1}$ ($\partial_-C_i^2=\tilde{L}_{i}$ resp.) See Figure \[figure-md\]. Moreover, $C_i^1$ ($C_i^2$ resp.) is not a trivial compression body, i.e. homeomorphic to $F\times I$ for a closed surface $F$.
![The manifold decomposition of $M$\[figure-md\]](fig-md.pdf){width="4cm"}
Therefore, the manifold decomposition of $M$ by Howards and Schultens can be written as an ordered set of thin and thick surfaces, $$\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\},$$ where we denote the empty negative boundaries of $H_1$ and $H_n$ as $\tilde{L}_0$ and $\tilde{L}_n$ for convenience.
Proof of Theorem \[theorem-1\]\[section-main-theorem\]
======================================================
Let us consider the assumption that the given thin position of $k$ induces a thin manifold decomposition of DBC $M$ of $(S^3,k)$. In Rule 6 of [@ST], Scharlemann and Thompson proved that each thick surface $S$ in a thin manifold decomposition is *weakly incompressible*, i.e. any two compressing disks for $S$ on opposite sides of $S$ intersect along their boundary, so every thick surface in a thin manifold decomposition is strongly irreducible in $M(S)$ (we can also check it from Proposition 4.2.3 of [@SSS].)
Let $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method as in the end of section \[section-DBC\], where each $\tilde{S}_i$ for $i=1,\cdots,n$ ($\tilde{L}_i$’s for $i=1,\cdots, n-1$ resp.) comes from a thick level (thin level resp.) of the given thin position of $k$.
If there exist $m$-thin levels which intersect $k$ in two points, then we get $m$-thin spheres in $H$. Since the construction of the manifold decomposition allows only one surface for each thick or thin level, these $m$-thin spheres must be $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ for some $1\leq j_1< \cdots< j_m\leq {n-1}$. Let $\tilde{L}_{j_0}$ be $\tilde{L}_0$ and $\tilde{L}_{j_{m+1}}$ be $\tilde{L}_{n}$ for convenience (so they are empty sets.)
Now we cut DBC along the $m$-thin spheres, and cap off all $S^2$ boundaries with $3$-balls. Then we get $(m+1)$-manifolds, $M_0$, $M_1$, $\cdots$, $M_m$, where each $M_i$ comes from the submanifold of DBC bounded by $\tilde{L}_{j_{i}}$ and $\tilde{L}_{j_{i+1}}$ for $i=0,\cdots, m$. In addition, we can induce the canonical manifold decomposition $H_i$ of $M_i$ from $H$, where $H_i=\{\, \emptyset, \tilde{S}_{j_i+1}, \tilde{L}_{j_i+1},\cdots,\tilde{L}_{j_{i+1}-1},\tilde{S}_{j_{i+1}},\emptyset\}$. Now every thin surface of $H_i$ is not homeomorphic to $S^2$ for $i=0,\cdots,m$, so we can say that the manifold decomposition $H_i$ of $M_i$ is a GHS in Definition of \[definition-GHS\]. Moreover, it is obvious that the strongly irreducibility of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a strongly irreducible GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.
By Lemma 4.7 of [@Bachman3], $M_i$ is irreducible or homeomorphic to $S^3$ (obviously, it is also irreducible) for $i=0,\cdots,m$ , i.e. no one is $S^2\times S^1$. Since the thin spheres $\tilde{L}_{j_1},\cdots,\tilde{L}_{j_m}$ are separating in $M$, no one of them is an essential sphere in a $S^2\times S^1$ piece in the prime decomposition of $M$. So we can assume that there is no $S^2\times S^1$ piece in the prime decomposition of $M$. But it is not obvious whether the sum $M_0\#_{\tilde{L}_{j_1}}M_1\#_{\tilde{L}_{j_2}}\#\cdots\#_{\tilde{L}_{j_m}}M_m$ does not have any trivial summand. So we need the following claim.\
[**Claim.** *No $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.*\
]{}
Suppose that $M_l$ is homeomorphic to $S^3$ for some $l$.
Let the thin levels in the thin position of $k$ corresponding to $\tilde{L}_{j_l}$ and $\tilde{L}_{j_{l+1}}$ be $L'$ and $L''$ (one of both may be empty level if $l=0$ or $m$), and the thick level in the thin position of $k$ corresponding to $\tilde{S}_{j_l+1}$ be $S$. The thin levels $L'$ and $L''$ intersect $k$ in two points, i.e. each of both $L'$ and $L''$ realizes a connected sum of $k$, i.e. $k=k_1 \#_{L'} k' \#_{L''} k_2$. Moreover, the thick level $S$ must intersect $k$ in $4$ or more points. Since $M_l\cong S^3$ is DBC of $(k',S^3)$ and $S^3$ has the unique representation as DBC of $S^3$ branched along a knot or a link (see [@Wa2], and this is also true for the other lens spaces, see [@HR] or Problem 3.26 of [@K],) $k'$ is an unknot. Since $k$ is in a thin position, the unknot summand $k'$ in $k=k_1 \#_{L'} k' \#_{L''} k_2$ must intersect all levels between $L'$ and $L''$ in two points, this contradicts the existence of the thick level $S$. So we get a contradiction. In the cases of $l=0$ and $m$, we get a contradiction by similar arguments for each case. This completes the proof of Claim.
Now we can say that the thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ determine the prime decomposition of $M$. Let us consider the thin levels $L_{j_1}$, $\cdots$, $L_{j_m}$ corresponding to $\tilde{L}_{j_1}$, $\cdots$,$\tilde{L}_{j_m}$. Then they cut $k$ into $m+1$ summands, i.e $k=k_0\#\cdots\#k_m$. In particular, each $M_i$ is DBC of $(S^3,k_i)$ for $i=0,\cdots,m$ and it is also a prime manifold as already proved. Moreover, each $k_i$ must be prime by Corollary 4 of [@KT]. Since $k$ is in a thin position, each summand $k_i$ must be in a thin position. This complete the proof of Theorem \[theorem-1\].
A generalization of Theorem \[theorem-1\]\[section-critical\]
=============================================================
In this section, we will generalize Theorem \[theorem-1\] using the concept *critical surface*. D. Bachman introduced a concept “critical surface” in [@Bachman1] and prove several properties about critical surface and minimal common stabilization. Also he proved Gordon’s conjecture (see Problem 3.91 of [@K]) using critical surface theory in [@Bachman3]. In particular, he used the term *topological minimal surface* to denote the class of surfaces such that they are incompressible, strongly irreducible or critical in [@Bachman4].
\[definition-critical\] Let $H$ be a Heegaard surface in some $3$-manifold which is compressible to both sides. The surface $H$ is *critical* if the set of all compressing disks for $H$ can be partitioned into subsets $C_0$ and $C_1$ such that the following hold.
1. For each $i=0,1$ there is at least one weak reducing pair $(V_i,W_i)$, where $V_i$, $W_i\in C_i$.
2. If $V\in C_0$ and $W\in C_1$ then $(V,W)$ is not a weak reducing pair.
Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is *psudo-critical* if each thick surface $H_+\in \operatorname{Thick}(H)$ is strongly irreducible or critical in $M(H_+)$.[^3]
Now we introduce a generalization of Theorem \[theorem-1\].
\[corollary-1\] If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens’ method where each thick surface $H_+$ of the manifold decomposition is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.
Assume $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method, and $m$-thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ cut $M$ into $M_0,\cdots,M_m$ with the canonical GHS $H_0,\cdots,H_m$ as in section \[section-main-theorem\]. In particular, every thick surface $H_+\in \operatorname{Thick}{H}$ is strongly irreducible or critical in $M(H_+)$ by the hypothesis.
It is obvious that the strongly irreducibility or criticality of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a psudo-critical GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.
Since the proofs of Lemma 4.6 and Lemma 4.7 of [@Bachman3] do not depend on the number of critical thick levels, and only depend on the property that each thick surface of the GHS is strongly irreducible or critical, we can extend both lemmas to psudo-critical GHS. So we get each $M_i$ is irreducible or homeomorphic to $S^3$.
The remaining arguments of the proof of Corollary \[corollary-1\] are the same as those of Theorem \[theorem-1\]. This completes the proof of Corollary \[corollary-1\]
[10]{}
D. Bachman, *Critical heegaard surfaces*, Trans. Amer. Math. Soc. **354** (2002), no. 10, 4015–4042 (electronic).
D. Bachman, *Connected sums of unstabilized Heegaard splittings are unstabilized*, Geom. Topol. **12** (2008), no. 4, 2327–2378.
D. Bachman, *Barriers to topologically minimal surfaces*, arXiv:0903.1692v1
R. Blair and M. Tomova, *Companions of the unknot and width additivity*, arXiv:0908.4103v1
D.J. Heath and T. Kobayashi, *Essential tangle decomposition from thin position of a link*, Pacific J. Math. **179** (1997), no.1, 101–117.
H. Howards, Y. Rieck, and J. Schultens, *Thin position for knots and 3-manifolds: a unified approach; Workshop on Heegaard Splittings*, Geom. Topol. Monogr. **12** (2007), 89–120.
C. Hodgson and and J.H. Rubinstein, *Involutions and isotopies of lens spaces*, Knot theory and manifolds (Vancouver, B.C., 1983), 60–96, Lecture Notes in Math., 1144, Springer, Berlin, 1985.
H. Howards and J. Schultens, *Thin position for knots and $3$-manifolds*, Topology Appl., **155** (2008), no. 13, 1371–1381.
R. Kirby, *Problems in low-dimensional topology*, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. **2**, Amer. Math. Soc. (1997) 35–473.
P. Kim and J. Tollefson *Splitting the PL involutions of nonprime 3-manifolds*, Michigan Math. J. **27** (1980) no. 3, 259–274.
Y. Rieck and E. Sedgwick, *Thin position for a connected sum of small knots*, Algebr. Geom. Topol. **2** (2002), 297–309 (electronic).
T. Saito, M. Scharlemann, and J. Schultens, *Lecture notes on generalized Heegaard splittings*, arXiv:math/0504167.
M. Scharlemann and A. Thompson, *Thin position for $3$-manifolds*, A.M.S. Contemp. Math., 164:231-238, 1994.
M. Scharlemann and A. Thompson, *On the additivity of knot width*, Proceedings of the Casson Fest, 135–144 (electronic), Geom. Topol. Monogr., **7**, Geom. Topol. Publ., Coventry, 2004.
F. Waldhausen, *Über involutionen der 3-sphäre*, Topology **8** (1969) 81–91.
[^1]: Note that D. Bachman did not allow thin surfaces in a GHS to be $2$-spheres in [@Bachman3]. In particular, he introduced more generalized concept “pseudo-GHS” in [@Bachman3] to deal with thin spheres and trivial compression bodies.
[^2]: Many authors use the term “generalized Heegaard splitting” to denote “manifold decomposition” in [@ST] by M. Scharlemann and A. Thompson, but the author distinguished the terms “manifold decomposition” and “generalized Heegaard splitting” to use two different definitions at the same time.
[^3]: This definition is weaker than the definition “*critical GHS*” in [@Bachman3].
| ArXiv |
---
abstract: 'Caching at mobile devices, accompanied by device-to-device (D2D) communications, is one promising technique to accommodate the exponentially increasing mobile data traffic. While most previous works ignored user mobility, there are some recent works taking it into account. However, the duration of user contact times has been ignored, making it difficult to explicitly characterize the effect of mobility. In this paper, we adopt the alternating renewal process to model the duration of both the contact and inter-contact times, and investigate how the caching performance is affected by mobility. The *data offloading ratio*, i.e., the proportion of requested data that can be delivered via D2D links, is taken as the performance metric. We first approximate the distribution of the *communication time* for a given user by beta distribution through moment matching. With this approximation, an accurate expression of the data offloading ratio is derived. For the homogeneous case where the average contact and inter-contact times of different user pairs are identical, we prove that the data offloading ratio increases with the user moving speed, assuming that the transmission rate remains the same. Simulation results are provided to show the accuracy of the approximate result, and also validate the effect of user mobility.'
author:
- '[^1]'
bibliography:
- 'IEEEabrv.bib'
- 'report.bib'
title: Mobility Increases the Data Offloading Ratio in D2D Caching Networks
---
Introduction
============
The mobile data traffic is growing at an exponential rate, among which mobile video accounts for more than a half [@forecast2016cisco]. Caching popular contents at helper nodes or user devices is a promising approach to reduce the data traffic on the backhaul links, as well as improving the user experience of video streaming applications [@d2d-cache; @jcache]. In comparison with the commonly considered femto-caching system, caching at devices enjoys a unique advantage, i.e., the devices’ aggregate caching capacity grows with the number of devices [@d2d-cache]. Moreover, device caching can promote device-to-device (D2D) communications, where nearby mobile devices may communicate directly rather than being forced to communicate through the base station (BS)[@design].
Recently, caching in D2D networks has attracted lots of attentions. In [@scaling], the scaling behavior of the number of D2D collaborating links was identified. Three concentration regimes, classified by the concentration of the file popularity, were investigated. The outage-throughput tradeoff and optimal scaling laws of both the throughput and outage probability were studied in [@tradeoff]. Two coded caching schemes, i.e., centralized and decentralized, were proposed in [@fundamentallimits], where the contents are delivered via broadcasting.
So far, an important characteristic of mobile users, i.e., the user mobility, has been ignored in previous studies of D2D caching networks. There are some works starting to consider the effect of user mobility. Effective methodologies to utilize the user mobility information in caching design were discussed in [@magmobility]. In [@mobilitycodedcaching], the effect of mobility was evaluated in D2D networks with coded caching, with the conclusion that mobility can improve the scaling law of throughput. This result was based on the assumption that the user locations are random and independent in each time slot, which failed to take into account the temporal correlation.
The inter-contact model, which considers the temporal correlation of the user mobility, has been widely applied [@exintercontactmodel], where the timeline for an arbitrary pair of mobile users are divided into *contact times* and *inter-contact times*. Specifically, the *contact times* denote the time intervals when the mobile users are located within the transmission range. Correspondingly, the *inter-contact times* denote the time intervals between contact times [@pocket]. This model has been used to develop device caching schemes to exploit the user mobility pattern in [@mobilitycaching]. The throughput-delay scaling law was developed by characterizing the inter-contact pattern of the random walk model [@scalingmobility]. In these works, it was assumed that a fixed amount of data can be delivered within one contact time, while the duration of the contact times was not considered. However, as the user moving speed will affect the durations of both the contact and inter-contact times, it is critical to account for their effects when investigating the impact of user mobility on caching performance.
In this paper, we shall analytically evaluate the effect of mobility in D2D caching networks, by adopting an alternating renewal process to model the mobility pattern so that both the contact and inter-contact times are accounted for. The *data offloading ratio*, which is defined as the proportion of data that can be obtained via D2D links, is adopted as the performance metric. The main contribution is an approximate expression for the data offloading ratio, for which the main difficulty is to deal with multiple alternating renewal processes. We tackle it by first deriving the expectation and variance of the *communication time* of a given user, and then use a beta random variable to approximate it by moment matching. Furthermore, we investigate the effect of mobility in a homogeneous case, where the average contact and inter-contact times for all the user pairs are the same. In the low-to-medium mobility scenario, by assuming that the transmission rate is irrelevant to the user speed, it is proved that the data offloading ratio increases with the user speed for any caching strategy that does not cache the same contents at all devices. Simulation results validates the accuracy of the derived expression, as well as the effect of the user mobility.
System Model and Performance Metric
===================================
In this section, we will first introduce the alternating renewal process to model the user mobility pattern, and discuss the caching and file delivery models. Then, the performance metric, i.e., the data offloading ratio, will be defined.
User Mobility Model
-------------------
![The timeline for an arbitrary pair of mobile users.[]{data-label="intercontact"}](intercontact){width="3in"}
The inter-contact model, which captures the temporal correlation of the user mobility [@exintercontactmodel], is used to model the user mobility pattern. Specifically, the timeline of each pair of users is divided into *contact times*, i.e, the times when the users are in the transmission range, and *inter-contact times*, i.e., the times between consecutive contact times. Considering that contact times and inter-contact times appear alternatively in the timeline of a pair of users, similar to [@renewalmodel], an alternating renewal process is applied to model the pairwise contact pattern, as defined below [@renewalprocess].
Consider a stochastic process with state space $\{A,B\}$, and the successive durations for the system to be in states $A$ and $B$ are denoted as $\xi_k,k=1,2,\cdots$ and $\eta_k,k=1,2,\cdots$, respectively, which are i.i.d.. Specifically, the system starts at state $A$ and remains for $\xi_1$, then switches to state $B$ for $\eta_1$, then backs to state $A$ for $\xi_2$, and so forth. Let $\psi_k=\xi_k+\eta_k$. The counting process of $\psi_k$ is called as an *alternating renewal process*.
As shown in Fig. \[intercontact\], if the pair of users is in contact at $t=0$, $\xi_k$ and $\eta_k$ represent the contact times and inter-contact times, respectively; otherwise, $\xi_k$ and $\eta_k$ represent the inter-contact times and contact times, respectively. It was shown in [@exintercontact] that exponential curves well fit the distribution of inter-contact times, while in [@excontact], it was identified that exponential distribution is a good approximation for the distribution of the contact times. Thus, same as [@renewalmodel], we assume that the contact times and inter-contact times follows independent exponential distributions. For simplicity, the timelines of different user pairs are assumed to be independent. Specifically, we consider $N_u$ users in a network, and the index set of the users is denoted as $\mathcal{S}=\{1,2,\cdots,N_u\}$. The contact times and inter-contact times of users $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$ follow independent exponential distributions with parameters $\lambda^C_{i,j}$ and $\lambda^I_{i,j}$, respectively.
Caching and File Transmission Model
-----------------------------------
![A sample network with three mobile users.[]{data-label="model"}](model){width="2.6in"}
There is a library with $N_f$ files, whose index set is denoted as $\mathcal{F}=\{1,2,\cdots,N_f\}$, each with size $C$. Each user device has a limited storage capacity, and each file can be completely cached or not cached at all at each user device. Specifically, the caching placement is denoted as $$x_{j,f}=
\begin{cases}
1, \text{if user $j$ caches file $f$}, \\
0, \text{if user $j$ does not cache file $f$},
\end{cases}$$ where $j \in \mathcal{S}$ and $f \in \mathcal{F}$. User $i \in \mathcal{S}$, is assumed to request a file $f \in \mathcal{F}$ with probability $p^r_{i,f}$, where $\sum \limits_{f \in \mathcal{F} } p^r_{i,f} =1$. When a user requests a file $f$, it will first check its own cache, and then download the file from the users that are in contact and store file $f$, with a fixed transmission rate, denoted as $R$. If the user cannot get the whole file within a certain delay threshold, denoted as $T^d$, it will download the remaining part from the BS. We assume that the delay threshold is larger than the time required to download each content, i.e., $T^d>\frac{C}{R}$. Fig. \[model\] shows a sample network, where user $1$ gets part of the requested file during the contact time with user $2$, then gets the whole file after the contact time with user $3$.
Performance metric
------------------
![An illustration of the communication time.[]{data-label="transmissiontime"}](transmissiontime){width="3.5in"}
The *data offloading ratio*, which is defined as the percentage of requested content that can be obtained via D2D links rather than downloading from the BS, is used as the performance metric. Specifically, the data offloading ratio for user $i \in \mathcal{S}$ is defined as $$\mathcal{P}_i=\sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left\{ x_{i,f} + \frac{ (1-x_{i,f})\mathbb{E} _{D_{i,f}}\left[ \min \left( D_{i,f} ,C \right) \right]}{C} \right\},$$ where $D_{i,f}$ denotes the amount of requested data that can be delivered via D2D links when user $i$ requests file $f$. Since a fixed transmission rate is assumed, $D_{i,f}$ can be written as $D_{i,f}=R T^c_{i,f}$, where $T^c_{i,f}$ is the *communication time* for user $i$ to download file $f$ from other users caching file $f$ within time $T^d$. We assume that user $i$ can download file $f$ while at least one user caching file $f$ is in contact, where the handover time is ignored. Fig. \[transmissiontime\] shows the communication time of user $1$ in Fig. \[model\]. Then, the average data offloading ratio is $$\begin{aligned}
\label{define_ratio}
&\mathcal{P}= \notag \\
&\frac{1}{N_u}\sum \limits_{i \in \mathcal{S}} \sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left\{ x_{i,f} + \frac{ (1-x_{i,f}) \mathbb{E}_{T^c_{i,f} } \left[ \min \left( R T^c_{i,f} ,C \right) \right]}{C} \right\}.\end{aligned}$$
In the following, we will evaluate the data offloading ratio given in (\[define\_ratio\]) for any given caching strategy, and investigate the effect of user mobility on caching performance.
Data Offloading Ratio Analysis
==============================
The main difficulty of evaluating the data offloading ratio is to find the distribution of the communication time. As this distribution is highly complicated, instead of deriving it directly, we will develop an accurate approximation. In this section, we will first approximate the distribution of the communication time by a beta distribution, and then, an approximation of the data offloading ratio will be obtained.
Communication time analysis
---------------------------
To help analyze the communication time, we first define some stochastic processes.
Define $\mathbf{H}_{i,j}$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$, as the continuous-time random process, i.e., $\mathbf{H}_{i,j}=\{H_{i,j}(t), t \in (0,\infty)\}$ with state space $\{0,1\}$, where $H_{i,j}(t)=1$ means that users $i$ and $j$ are in contact at the time instant $t$; otherwise $H_{i,j}(t)=0$. The durations of staying in states $1$ and $0$ follow i.i.d. exponential distributions with parameter $\lambda^C_{i,j}$ and $\lambda^I_{i,j}$, respectively.
Define $\mathbf{H}_i^f$, where $i \in \mathcal{S}$ and $f \in \mathcal{F}$, as the continuous-time random process, i.e., $\mathbf{H}^f_{i}=\{H^f_{i}(t), t \in (0,\infty)\}$ with state space $\{0,1\}$, where $H^f_{i}(t)=1$ means that users $i$ can download file $f$ from any other user caching file $f$ at time instant $t$; otherwise $H^f_{i}(t)=0$.
At time $t$, since user $i$ can download file $f$ when at least one user caching file $f$ is in contact, we get $H^f_i(t)=1- \prod \limits_{j \in \mathcal{S} \backslash \{i\},x_{j,f}=1}\left[1-H_{i,j}(t) \right]$. Similar to [@renewalmodel], it is reasonable to assume that when a user requests a file, the alternating process between each pair of users has been running for a long time. Thus, denote $T^r_{i,f}$, $i \in \mathcal{S}$ and $f \in \mathcal{F}$, as the time of user $i$ requests file $f$, and the communication time $T^c_{i,f}$ can be derived as $T^c_{i,f}= \lim \limits_{T^r_{i,f} \to \infty} \int _{T^r_{i,f}}^{T^r_{i,f}+T^d} H^f_i(t) dt.$ In the following, we will derive the expectation and variance of the communication time.
\[ev\] When user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$, which is not stored at its own cache, the expectation and variance of its communication time is $$\label{expectation_t}
\mathbb{E}\left[T^c_{i,f} \right]=T^d\left( 1-\prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \right).$$ and $$\begin{aligned}
\label{var_t}
\mathrm{Var} \left[ T^c_{i,f} \right] = & 2 \int_0^{T^d} (T^d-u) \prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \frac{\lambda^C_{i,j}}{(\lambda^I_{i,j}+\lambda^C_{i,j})^2} \notag \\
&\times \left[ \lambda^C_{i,j} + \lambda^I_{i,j} e^{-u(\lambda_{i,j}^C+\lambda^I_{i,j})} \right] du \notag \\
&-(T^d)^2 \prod \limits_{j \in \mathcal{S}, x_{j,f}=1} \left(\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \right)^2.\end{aligned}$$
See Appendix A.
Since the communication time $T^c_{i,f}$ is a bounded random variable, we propose to approximate its distribution by a beta distribution, which is widely used to model the random variables limited to finite ranges. Specifically, we consider $T^c_{i,f} \approx T^d Y_{i,f}$, where $Y_{i,f} \sim B(\alpha_{i,f},\beta_{i,f})$, $i \in \mathcal{S}$ and $f \in \mathcal{F}$, if $\sum_{j \in \mathcal{S} \backslash \{i\}} x_{j,f}>0$, which means that user $i$ may download file $f$ from at least one user; otherwise, $T^c_{i,f}=0$. Let $\mathbb{E}[T^dY_{i,f}]=\mathbb{E}[T^c_{i,f}]$ and $\mathrm{Var}[T^dY_{i,f}]=\mathrm{Var}[T^c_{i,f}]$, and the parameters of the beta distribution to match the first two moments can be derived as[^2] $$\label{beta_p}
\begin{cases}
\alpha_{i,f}=\frac{\mathbb{E}[T^c_{i,f}]^2 (T^d-\mathbb{E}[T^c_{i,f}])}{ \mathrm{Var}[T^c_{i,f}]T^d}-\frac{\mathbb{E}[T^c_{i,f}]}{T^d} \\
\beta_{i,f}=\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f}
\end{cases}$$
Data offloading ratio approximation
-----------------------------------
Based on the above approximation, we get an approximate expression of the data offloading ratio in Proposition \[E\_od\]. Simulations will show that the approximation is quite accurate.
\[E\_od\] The data offloading ratio is approximated as $$\begin{aligned}
\label{ratio}
\mathcal{P} = &\frac{1}{N_u}\sum \limits_{i \in \mathcal{S}} \sum \limits_{f \in \mathcal{F}} p^r_{i,f} \left[ x_{i,f} + (1-x_{i,f}) \mathcal{P}_{i,f} \right],\end{aligned}$$ where $\mathcal{P}_{i,f}$ is the data offloading ratio when user $i$ requests file $f$, which is not in its own cache, approximated by $$\begin{aligned}
\label{ex_g}
&\mathcal{P}_{i,f} \approx 1-I_{\frac{C}{T^dR}} (\alpha_{i,f},\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f}) \notag \\
& +\frac{\mathbb{E}[T^c_{i,f}]R}{C} I_{\frac{C}{T^dR}}(\alpha_{i,f}+1,\frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \alpha_{i,f}) \Big] \Big\},\end{aligned}$$ if $\sum_{j \in \mathcal{S} \backslash \{i\}} x_{j,f}>0$ and $0$ elsewhere, where $I_r(\cdot,\cdot)$ is the incomplete beta function, and $\alpha_{i,f}$ is given in (\[beta\_p\]).
Following (\[define\_ratio\]), (\[expectation\_t\]), (\[var\_t\]), and (\[beta\_p\]), the expression in (\[ratio\]) can be obtained. Due to the space limitation, the detail is omitted.
Effect of Mobile User Speed
===========================
In this section, we will consider a homogeneous case, where the contact and inter-contact parameters among all pairs of users are the same, i.e., $\lambda^C=\lambda^C_{i,j}>0$ and $\lambda^I=\lambda^I_{i,j}>0$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$. We will investigate how the user moving speed affects the data offloading ratio for a given caching strategy. If all users cache the same contents, the D2D communications will not help the content delivery. Thus, in the following, we assume that the contents cached at different users are not all the same. This investigation will be based on the approximate expression in (\[ratio\]), and simulations will be provided later to verify the results.
Communication time analysis
---------------------------
Under the above assumptions, the expectation and variance of the communication time can be simplified, as in the following corollary.
\[sim\_ev\] When $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, where $i \in \mathcal{S}$ and $j \in \mathcal{S} \backslash \{i\}$, the expectation and variance of a user requests file $f$, which is not stored at its own cache, are given by $$\begin{aligned}
&\mathbb{E}[T^c_{i,f}]=T^d\left[ 1-\left( \frac{\lambda^C}{\lambda^C+\lambda^I} \right)^{n_f} \right], \label{expect} \\
&\mathrm{Var}[T^c_{i,f}]= \left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f}
\sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{l(\lambda^C+\lambda^I)} \notag \\
& \quad \times \left[ T^d-\frac{1}{l(\lambda^C+\lambda^I)}+\frac{e^{-l(\lambda^C+\lambda^I)T}}{l(\lambda^C+\lambda^I)} \right], \label{variance}\end{aligned}$$ where $i \in \mathcal{S}$ and $n_f=\sum \limits_{j \in \mathcal{S}} x_{j,f}$ denotes the number of users caching file $f$.
See Appendix A.
Mobile user speed
-----------------
We first characterize the relationship between the user speed and the parameters $\lambda^C$ and $\lambda^I$ in Lemma \[speed\].
\[speed\] When all the user speeds change by $s$ times, the contact and inter-contact parameters will also change by $s$ times, i.e., from $\lambda^C$ and $\lambda^I$ to $s\lambda^C$ and $s\lambda^I$, respectively.
The time for user $i$ to move along a certain path $L_i$ can be given as a curve integral $\int_{L_i} \frac{dz}{v_i(z)}$, where $v_i(z)$ is the speed of user $i$ when passing by a point $z$ on the path $L_i$. When the speed of user $i$ changes by $s$ times, the time for moving along the path $L_i$ changes to $\int_{L_i} \frac{dz}{s v_i(z)}=\frac{1}{s}\int_{L_i} \frac{dz}{ v_i(z)}$, which scales by $\frac{1}{s}$ times. During each contact or inter-contact time, users $i$ and $j$ move along certain paths. When all the user speeds change by $s$ times, each contact or inter-contact time changes by $\frac{1}{s}$ times, and thus, the average ones change by $\frac{1}{s}$ times. Since the contact and inter-contact times are assumed to be exponential distributed with mean $\frac{1}{\lambda^C}$ and $\frac{1}{\lambda^I}$, respectively, the parameters $\lambda^C$ and $\lambda^I$ scale by $s$ times.
Considering that a larger $s$ means that users are moving faster, in the following, we will investigate how changing $s$ will affect the data offloading ratio. For simplicity, we assume that the transmission rate is a constant, and will not change with the user speed. This is reasonable in the low-to-medium mobility regime. Firstly, the effect of user speed on the communication time is shown in Lemma \[s\_t\] .
\[s\_t\] When $s$ increases, which is equivalent to increasing the user speed, the expectation of the communication time when a user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$ that is not in its own cache, i.e., $\mathbb{E}[T^c_{i,f}]$, keeps the same, and the corresponding variance, i.e., $\mathrm{Var}[T^c_{i,f}]$, decreases, if the number of users caching file $f$ is larger than $0$, i.e., $n_f>0$. Accordingly, the parameter $\alpha_{i,f}$ of the beta distribution increases.
See Appendix B.
Then, we evaluate the relationship between $\alpha_{i,f}$ and the data offloading ratio when user $i$ requests file $f$ that is not in its own cache, i.e., $\mathcal{P}_{i,f}$ in (\[ex\_g\]), in Lemma \[s\_g\].
\[s\_g\] When user $i \in \mathcal{S}$ requests file $f \in \mathcal{F}$ and cannot find it in its own cache, the data offloading ratio, i.e., $\mathcal{P}_{i,f}$, increases with $\alpha_{i,f}$.
See Appendix C.
Base on Lemmas \[s\_t\] and \[s\_g\], we can specify the effect of user speed on the data offloading ratio in Proposition \[s\_d\].
\[s\_d\] If the transmission rate does not change with the user speed, and the average contact and inter-contact times among all the pairs are the same, the data offloading ratio increases with the user moving speed.
See Appendix D.
The result in Proposition \[s\_d\] is valid for any caching strategy, only excluding the case that all the users have the same cache contents.
Simulation results
==================
In the simulation, the content request probability follows a Zipf distribution with parameter $\gamma_r$, i.e., $p_f=\frac{f^{-\gamma_r}}{\sum \limits_{i \in \mathcal{F}} i^{-\gamma_r}}$, $f \in \mathcal{F}$ [@d2d-cache]. Meanwhile, each user caches 5 contents, and a random caching strategy is applied [@randomcache], where the probabilities of the contents cached at each user are proportional to the file request probabilities.
![Data offloading ratio with $N_f=100$, $T^d=300s$ and $\gamma_r=0.6$.[]{data-label="fig_scheme1"}](paper1){width="2.6in"}
Fig. \[fig\_scheme1\] validates the accuracy of the approximation in (\[ratio\]). The inter-contact parameters $\lambda^I_{i,j}, i \in \mathcal{S}, j \in \mathcal{S} \backslash \{i\}$ are generated according to a gamma distribution as $\Gamma(4.43,1/1088)$ [@aggregate_real]. Similar as [@renewalmodel], we assume the average of the contact parameters are $5$ times larger than the inter-contact parameters. Thus, the contact parameters are generated∂ according to $\Gamma(4.43 \times 25,1/1088/5)$. It is shown from Fig. \[fig\_scheme1\] that the theoretical results are very close to the simulation results, which means the approximate expression (\[ratio\]) is quit accurate. Furthermore, the data offloading ratio increases with the number of users, which is brought by the increasing aggregate caching capacity and the content sharing via D2D links.
![Data offloading ratio with $N_u=15$, $\lambda^C=0.001s$, $\lambda^I=0.0002s$, $N_f=100$, $T^d=300s$ and $\gamma_r=0.6$.[]{data-label="fig_scheme2"}](paper2){width="2.6in"}
In Fig. \[fig\_scheme2\], the effect of $s$ is demonstrated, where increasing $s$ is equivalent to increasing the user speed. Firstly, the small gap between the theoretical and simulation results again verifies the accuracy of the approximate expression in (\[ratio\]). It is also shown that the data offloading ratio increases with $s$, which confirms the conclusion in Proposition \[s\_d\]. Moreover, from Fig. \[fig\_scheme2\], the increasing rate of the data offloading ratio is decreases with the user moving speed.
conclusions
===========
In this paper, we investigated the effect of user mobility on the caching performance in a D2D caching network. The communication time of a given user was firstly approximated by a beta distribution, through matching the first two moments. Then, an approximate expression of the data offloading ratio was derived. For a homogeneous case, where the average contact and inter-contact times are the same for all the user pairs, we evaluated how the user moving speed affects the data offloading ratio. Specifically, it was proved that the data offloading ratio increases with the user speed, assuming that the transmission rate is irrelevant to the user speed. Simulation results validated the accuracy of the approximate expression of the data offloading ratio, and demonstrated that the data offloading ratio increases with the user speed, while the increasing rate decreases with the user speed.
Appendix {#appendix .unnumbered}
========
Proof of Lemma \[ev\] and Corollary \[sim\_ev\]
-----------------------------------------------
As the timeline of different user pairs are independent, the expectation of the communication time when user $i$ requests file $f$, which is not in its own cache, can be written as $$\mathbb{E} [ T^c_{i,f} ]=\lim \limits_{T^r_{i,f} \to \infty} \int _{T^r_{i,f}}^{T^r_{i,f}+T^d} \left[ 1- \prod \limits_{j \in \mathcal{S},x_{j,f}=1}\left(1- \mathbb{E} H_{i,j}(t) \right) \right] dt.$$ Since the timeline between each pair of users is modeled as an alternating renewal process, according to [@renewalprocess], we have $\lim \limits_{t \to \infty} \Pr[H_{i,j}(t)=1]=\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}$. Thus, $\lim \limits_{t \to \infty} \mathbb{E}[ H_{i,j}(t)]=\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}$, and then, the expectation in (\[expectation\_t\]) can be obtained. Let $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, and we can get the expression in (\[expect\]). The variance of the communication time is $$\begin{aligned}
\label{v}
&\mathrm{Var} [ T^c_{i,f} ]= \notag \\
&2 \lim \limits_{T^r_{i,f} \to \infty} \int_{T^r_{i,f}}^{T^r_{i,f}+T^d} \int_{T^r_{i,f}}^{\tau} \Pr[H^f_{i}(t)=1,H^f_{i}(\tau)=1] dtd\tau \notag \\
&-\left( \mathbb{E} [ T^c_{i,f} ] \right)^2\end{aligned}$$ According to [@renewalprocess], $\Pr[H_{i,j}(\tau)=0|H_{i,j}(t)=0]=\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}+\frac{\lambda^I_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}}e^{-({\lambda^C_{i,j}+\lambda^I_{i,j}})(\tau-t)}$. Then, when $T_{i,f}^r \to \infty$, we can get $$\begin{aligned}
\label{prob}
&\Pr[H^f_{i}(\tau)=1,H^f_{i}(t)=1]
=1-2 \prod \limits_{j \in \mathcal{S},x_{j,f}=1}\frac{\lambda^C_{i,j}}{\lambda^C_{i,j}+\lambda^I_{i,j}} \notag \\
& \quad + \prod \limits_{j \in \mathcal{S},x_{j,f}=1} \frac{\lambda^C_{i,j}}{(\lambda^I_{i,j}+\lambda^C_{i,j})^2}\left[ \lambda^C_{i,j} + \lambda^I_{i,j} e^{-(\lambda_{i,j}^C+\lambda^I_{i,j})(\tau-t)} \right]\end{aligned}$$ Let $u=\tau-t$ and substitute (\[prob\]) into (\[v\]), and we can get (\[var\_t\]). Let $\lambda^C=\lambda^C_{i,j}$ and $\lambda^I=\lambda^I_{i,j}$, and we can get (\[variance\]) with the binomial theorem.
Proof of Lemma \[s\_t\]
-----------------------
When the user speed changes by $s$ times, the expectation of the communication time in (\[expect\]) keeps the same, while the variance changes to $$\begin{aligned}
&\mathrm{Var}[T^c_{i,f}]= \left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f}
\sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{sl(\lambda^C+\lambda^I)} \notag \\
& \quad \times \left[ T^d-\frac{1}{sl(\lambda^C+\lambda^I)}+\frac{e^{-sl(\lambda^C+\lambda^I)T^d}}{sl(\lambda^C+\lambda^I)} \right],\end{aligned}$$ To prove that $\mathrm{Var}[T^c_{i,f}]$ decreases with $s$, we will prove that $\frac{
\partial \mathrm{Var}[T^c_{i,f}]}{\partial s}<0$. The partial derivation of $\mathrm{Var}[T^c_{i,f}]$ is $$\begin{aligned}
\label{dir}
&\frac{\partial \mathrm{Var}[T^c_{i,f}]}{\partial s}= \notag \\
&\left[ \frac{\lambda^C}{(\lambda^C+\lambda^I)^2} \right]^{n_f}
\sum \limits_{l=1}^{{n_f}} \binom{{n_f}}{l} \frac{(\lambda^C)^{n_f-l} (\lambda^I)^{l}}{s^3l^2(\lambda^C+\lambda^I)^2} \mathcal{A}_1(x),\end{aligned}$$ where $\mathcal{A}_1(x)=-x-x e^{-x}-2(e^{-x}-1)$ and $x=sl(\lambda^C+\lambda^I)T^d>0$. Since $\mathcal{A}'_1(x)=-1+(1+x) e^{-x} < -1+(1+x) \frac{1}{1+x}=0$, $\mathcal{A}_1(x)$ is a decreasing function of $x$. Thus, $\mathcal{A}_1(x)<\mathcal{A}_1(0)=0$. According to (\[dir\]), when $n_f>0$, we have $\frac{
\partial \mathrm{Var}[T^c_{i,f}]}{\partial s}<0$. The parameter $\alpha_{i,f}$ given in (\[beta\_p\]) is a decreasing function of $\mathrm{Var}[T^c_{i,f}]$, and thus increases with $s$.
Proof of Lemma \[s\_g\]
-----------------------
To simplify the expression in (\[ex\_g\]), denote $r \triangleq \frac{C}{T^d R} \in (0,1)$, $y \triangleq \frac{T^d-\mathbb{E}[T^c_{i,f}]}{\mathbb{E}[T^c_{i,f}]} \ge 0$, and $\alpha \triangleq \alpha_{i,f}$. The expression in (\[ex\_g\]) can be rewritten as a function of $\alpha$, given as $$\begin{aligned}
&\mathcal{P}_{i,f}=1- \frac{\int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1} du }{B(\alpha,y\alpha)}.\end{aligned}$$ Let $g(\alpha)=1-\mathcal{P}_{i,f}$, the derivation of $g(\alpha)$ is $$\begin{aligned}
&g'(\alpha)= \notag \\
&\frac{1}{B(\alpha,y\alpha)} \Bigg\{\int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\
&- \int_0^r (1-\frac{u}{r}) u^{\alpha-1} (1-u)^{y \alpha-1} du D(y,\alpha) \Bigg\},\end{aligned}$$ where $D(y,\alpha)=\psi(\alpha)+y \psi(y\alpha)-(1+y)\psi[(1+y)\alpha]$ and $\psi(\cdot)$ is the digamma function. If $r=1$, $g'(\alpha)=\frac{\partial [y/(1+y)]}{\partial \alpha}=0$. Denote $\mathcal{A}_2(r)=\frac{B(\alpha,y\alpha)}{r}g'(\alpha)$, $\mathcal{A}_2(1)=0$ and $$\begin{aligned}
&\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r)= \notag \\
&\lim \limits_{r \to 0^{+}} \int_0^r (r-u) u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\\end{aligned}$$ Since $ r \ge u \ge 0$ and $y \ge 0$, $(r-u) u^{\alpha-1} (1-u)^{y \alpha-1} \ge 0$ and $\ln u + y \ln (1-u) \le 0$, thus, $\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r) \le 0$. The derivation of $\mathcal{A}_2(r)$ is $$\begin{aligned}
\mathcal{A}'_2(r)= &\int_0^r u^{\alpha-1} (1-u)^{y \alpha-1}[\ln u + y \ln (1-u)] du \notag \\
&- \int_0^r u^{\alpha-1} (1-u)^{y \alpha-1} du D(y,\alpha).\end{aligned}$$ Thus, $\mathcal{A}'_2(1)=\frac{\partial B(\alpha,y \alpha)}{\partial \alpha}-\frac{\partial B(\alpha,y \alpha)}{\partial \alpha}=0$ and $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le 0$. Then, we can get $\mathcal{A}''_2(r)= r^{\alpha-1} (1-r)^{y \alpha-1}[\ln r + y \ln (1-r)-D(y,\alpha)]$. Let $\mathcal{A}_3(r)=r^{1-\alpha} (1-r)^{1-y \alpha} \mathcal{A}''_2(r)$, then, there is one zero point of $\mathcal{A}'_3(r)=\frac{1-(1+y)x}{x(1-x)}$ in $(0,1]$. Thus, there is one inflection point of $\mathcal{A}_3(r)$. Considering that $\lim \limits_{r \to 0^{+}}\mathcal{A}_3(r)=\lim \limits_{r \to 1^{-}}\mathcal{A}_3(r)=-\infty$, the sign of $\mathcal{A}_3(r)$ may be negative, or first negative, then positive, and then negative, while $r$ increases in $(0,1)$. If $\mathcal{A}_3(r)<0$, then $\mathcal{A}''_2(r)<0$ when $r \in (0,1)$. However, we have $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le \mathcal{A}'_2(1)$, which means that $\mathcal{A}'_2(r)$ can not be a decreasing function in $(0,1)$. Thus, the sign of $\mathcal{A}_3(r)$ is first negative, then positive, and then negative, while $r$ increases in $(0,1)$. Since $\mathcal{A}''_2(r)$ has the same sign with $\mathcal{A}_3(r)$ in $(0,1)$, $\mathcal{A}'_2(r)$ first decreases, then increases, and then decreases while $r$ increases in $(0,1)$. Considering that $\lim \limits_{r \to 0^{+}} \mathcal{A}'_2(r) \le 0$ and $\mathcal{A}'_2(1)=0$, the sign of $\mathcal{A}'_2(r)$ must be first negative, and then positive in $(0,1)$. Therefore, while $r$ increases in $(0,1)$, $\mathcal{A}_2(r)$ first decreases, and then increases. Considering that $\lim \limits_{r \to 0^{+}} \mathcal{A}_2(r) \le 0$ and $\mathcal{A}_2(1)=0$, we have $\mathcal{A}_2(r)<0$ in $(0,1)$ and $\mathcal{A}_2(r)=0$ when $r=1$. Since $g'(\alpha)=\frac{r}{B(\alpha,y\alpha)} \mathcal{A}_2(r)$, we get $g'(\alpha)<0$ in $(0,1)$. Thus, $g(\alpha)$ decreases with $\alpha$, and $\mathcal{P}_{i,f}=1-g(\alpha)$ increases with $\alpha$.
Proof of Proposition \[s\_d\]
-----------------------------
The data offloading ratio in (\[ratio\]) increases with the increasing of $\mathcal{P}_{i,f}$ if $x_{i,f}=0$, $i \in \mathcal{S}$, $f \in \mathcal{F}$. Then, based on Lemmas \[s\_t\] and \[s\_g\], we can get that the data offloading ratio when user $i$ requests file $f$ from other users, i.e., $\mathcal{P}_{i,f}$, decreases with the user speed when $n_f>0$, otherwise $\mathcal{P}_{i,f}=0$. Accordingly, the data offloading ratio when user $i$ requests file $f$, i.e., $x_{i,f}+ (1-x_{i,f}) \mathcal{P}_{i,f} $, increases with the user speed when $x_{i,f}=0$ and $n_f>0$; otherwise, it keeps the same, where $i \in \mathcal{S}$, $f \in \mathcal{F}$. Since we consider that not all the users cache the same contents, there must exists $i' \in \mathcal{S}$, $j' \in \mathcal{S}$ and $f' \in \mathcal{F}$, where $x_{i',f'}=0$ and $x_{j',f'}=1$, i.e, $n_{f'}>0$. Thus, the data offloading ratio increases with the user speed.
[^1]: This work was supported by the Hong Kong Research Grants Council under Grant No. 610113.
[^2]: The parameters of the beta distribution should be positive, and it can be proved that $\alpha_{i,f}>0$ and $\beta_{i,f}>0$, by $e^{-u(\lambda_{i,j}^I+\lambda_{i,j}^C)} \le 1$. The detail is omitted due to the space limitation.
| ArXiv |
---
abstract: 'We show that Sarnak’s conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.'
author:
- Jon Chaika
- Alex Eskin
title: Mobius disjointness for interval exchange transformations on three intervals
---
Introduction
============
Let $\mu: {{{\mathbb}N}}\to \{-1, 0, 1\}$ denote the Möbius function, namely, $\mu(n) = 0$ if $n$ is not square-free, $\mu(n) = 1$ if $n$ is square-free and has an even number of prime factors, and $\mu(n) = -1$ if $n$ is square-free and has an odd number of prime factors.
Let $X$ be a topological space, and let $T: X \to X$ be an invertible map. We think of the map $T$ as a dynamical system. Peter Sarnak made the following far-reaching conjecture:
\[conj:Sarnak\] Suppose the topological entropy of $T$ is $0$. Then, for any $x \in
X$, and any (continuous) function $f: X \to {{{\mathbb}R}}$, $$\label{eq:Mobius:disjointness}
\lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} f(T^n x) \mu(n) = 0.$$
\[def:IET\] An interval exchange transformation (IET) is given by a vector $\vec{\ell} = (\ell_1, \dots, \ell_d) \in {{{\mathbb}R}}^d_+$ and a permutation $\pi$ on $\{ 1, \dots, d \}$. From $\vec{\ell}$ we obtain $d$-subintervals of $[0, \sum_{i=1}^d \ell_i)$ as follows: $$I_1 = [0,\ell_1), \quad I_2 = [\ell_1, \ell_1 + \ell_2), \dots, I_d =
[\sum_{i=1}^{d-1} \ell_d, \sum_{i=1}^d \ell_d).$$ Now we obtain a $d$-Interval Exchange Transformation $T = T_{\pi,
\vec{\ell}} :[0, \sum_{i=1}^d \ell_i) \to [0, \sum_{i=1}^d \ell_i)$ which exchanges the intervals according to $\pi$. More precisely, if $x \in I_j$, then $$T(x) = x - \sum_{k < j} \ell_k + \sum_{\pi(k) < \pi(j)} \ell_k.$$
It is well known that the topological entropy of any interval exchange transformation is $0$. Thus, if Conjecture \[conj:Sarnak\] is true, then (\[eq:Mobius:disjointness\]) should hold for any interval exchange transformation.
In this paper, we consider only the case $d=3$. Extending our results e.g. to $d=4$ will require fundamental new ideas.
\[lemma:3iet:to:rotation\] If $T$ is a $3$-IET with permutation $\begin{pmatrix}1&2&3\\ 3 & 2 & 1\end{pmatrix}$, then $T$ is also the induced map of a rotation on an interval.
Let $\hat{R}: [0, \ell_1 + 2\ell_2 + \ell_3) \to [0,\ell_1 + 2 \ell_2 +
\ell_3)$ be given by $$\hat{R}(x) = \begin{cases}
x + \ell_2 + \ell_3 & \text{if $x \le \ell_1 + \ell_2$} \\
x + \ell_2 + \ell_3 - (\ell_1 + 2 \ell_2 + \ell_3) & \text{otherwise.}
\end{cases}$$ Then $\hat{R}$ is a $2$-IET (hence a rotation), and the induced map of $\hat{R}$ on the interval $[0,\ell_1+\ell_2+\ell_3)$ is $T$.
[[**The maps $T$ and $R$, the number $\alpha$ and the interval $J$.** ]{}]{} Let $R: [0,1) \to [0,1)$ denote rotation by $\alpha$ (i.e. $R(x) = x+\alpha$ mod $1$). Let $J =
[0,z)$ be a subset of $[0,1]$. In the rest of the paper, we assume that the $3$-IET $T$ is the induced map of $R$ to $J$ and that $x \notin J$ implies $Rx \in J$. [[**The numbers $a_k$, $p_k$ and $q_k$.** ]{}]{} Let $a_0, a_1, \dots, $ denote the continued fraction expansion of $\alpha$. Let $p_k/q_k$ denote the continued fraction convergents of $\alpha$. Then, $$q_{k+1} = a_{k+1} q_k + q_{k-1}.$$
[[**Connection to tori and tori with marked points:** ]{}]{} Let ${{\mathcal M}}_1$ denote the space of flat tori of area $1$. The space ${{\mathcal M}}_1$ admits a transitive action by the Lie group $SL(2,{{{\mathbb}R}})$. Let $\hat{Y} \in {{\mathcal M}}_1$ denote the square torus. Then, the stabilizer of $\hat{Y}$ is $SL(2,{{{{\mathbb}Z}}})$, and thus ${{\mathcal M}}_1$ can we identified with $SL(2,{{{\mathbb}R}})/SL(2,{{{{\mathbb}Z}}})$. Under this identification, a torus with a fundamental domain the parallelogram whose vertices are the points $0$, $v_1$, $v_2$ and $v_1 + v_2$ corresponds to the coset $M SL(2,{{{{\mathbb}Z}}})$ where $M \in
SL(2,{{{\mathbb}R}})$ is the matrix whose columns are $v_1$ and $v_2$. The $SL(2,{{{\mathbb}R}})$ action on ${{\mathcal M}}_1$ coincides with the left multiplication action on $SL(2,{{{\mathbb}R}})/SL(2,{{{{\mathbb}Z}}})$.
Let ${{\mathcal M}}_{1,2}$ denote the space of tori with two marked points. This space also admits an action by $SL(2,{{{\mathbb}R}})$. If $g \in SL(2,{{{\mathbb}R}})$ and $X \in {{\mathcal M}}_{1,2}$ is the torus with fundamental domain the parallelogram with vertices $0$, $v_1$, $v_2$ and $v_1 + v_2$ and with the marked points $p_1$, $p_2$, then $g X$ is the torus with fundamental domain the parallelogram with vertices $0$, $g v_1$, $g v_2$ and $g(v_1 + v_2)$, and with the marked points $g p_1$ and $g p_2$.
Recall that $R:[0,1] \to [0,1]$ denotes the rotation by $\alpha$. Let $\hat{X}=\begin{pmatrix} 1&-\alpha\\0&1 \end{pmatrix} \hat{Y} \in
{{\mathcal M}}_1$. Observe that the first return of the vertical flow on $\hat{X}$ to the horizontal side coincides with $R$. If $T$ is a 3-IET given by the induced map of $R$ to an interval $J=[0,z)$ then $T$ is also the first return of the vertical flow on $\hat{X}$ to a horizontal segment of length $|J|$. Let $X$ denote the torus $\hat{X}$ with two marked points, one at each endpoint of the horizontal segment of length $J$.
Let $g_{t} = \begin{pmatrix} e^{t} & 0 \\ 0 & e^{-t} \end{pmatrix} \in
SL(2,{{{\mathbb}R}})$. We refer to the action of the $1$-parameter subgroup $g_t$ as the geodesic flow on ${{\mathcal M}}_1$ (or ${{\mathcal M}}_{1,2}$). The action of $g_t$ on both ${{\mathcal M}}_1$ and ${{\mathcal M}}_{1,2}$ is ergodic.
[[**Renormalization.** ]{}]{} We will need to put a diophantine condition on the IET $T$. In terms of $X \in {{\mathcal M}}_{1,2}$, we want the geodesic ray $\{ g_t X {\;\: : \;\:}t > 0 \}$ to spend significant time in compact subsets of ${{\mathcal M}}_{1,2}$. Directly in terms of the IET data, our conditions are the following:
ASSUMPTIONS: There exist constants $C_1,C_2,C_3,C_4 >1$ such that the following holds: Suppose $\ell \in {{{\mathbb}N}}$ and $0<\eta$ are small enough. Then there exists $\hat{c}_\eta>0$ and $c_\ell$ (depending on $\eta$ and $\ell$ respectively) so that for every $0<c<c_\ell$ there exists a constant $k_c \in {{{\mathbb}N}}$ and infinite sequences $L_i,k_i$ so that:
1. $q_{k_i-1}<c{L_i}<q_{k_i}$.
2. $a_{k_i}<C_1$,
3. $a_{k_i+1}<C_2$ and
4. $a_{k_i+2}<C_3$.
5. The shortest vertical trajectory on the torus from one marked point to a ${\| q_k\alpha\|}$ neighborhood of the other has length at least $\frac{q_k}{C_4}$. \[Separated\]
6. There exists $u_i$ so that either $$\lambda(\psi_{L_i}^{-1}(u_i))>\hat{c}_{\eta} \text{ and } \lambda(\psi_{L_i}^{-1}((-\infty,u_i)))<\eta \hat{c}_\eta$$ or $$\lambda(\psi_{L_i}^{-1}(u_i))>\hat{c}_\eta \text{ and } \lambda(\psi_{L_i}^{-1}((u_i,\infty)))<\eta \hat{c}_\eta$$ where $\psi_r(x) = \sum_{\ell=0}^{r -1} \chi_J(R^\ell x)$ and our 3-IET is the first return map of $R$ to $J$.
7. $\underset{i \to \infty}{\lim}\, \int_0^1d(R^{L_i}x,x)d\lambda=0$.
8. We have $L_i>{q_{k_{i}+\ell}}$.
9. We have $\max \{j:\psi_{L_i}^{-1}(j)\neq \emptyset\}- \min \{j:\psi_{L_i}^{-1}(j)\neq \emptyset\}<k_c.$
10. There exists $v_i$ so that $q_{v_i}\leq L_i<q_{v_i+1}$ and either $L_i=q_{v_i}$ or $a_{v_i+1}>4$ and $L_i=pq_{v_i}$ where $p\leq \lfloor \frac{a_{v_i+1}}4\rfloor$.
Our main result is the following:
\[theorem:3iet:mobius:disjoint\] Suppose $T$ is a $3$-IET satisfying the assumptions (A0)-(A9). Then, Möbius disjointness, i.e. (\[eq:Mobius:disjointness\]) holds.
Assumptions (A0)-(A9) are reasonable in view of the following:
\[prop:good:assump\] Let $X \in \mathcal{M}_{1,2}$. Let $\nu_T$ be the measure on $\mathcal{M}_{1,2}$ given by $\int f d\nu_T=\frac 1 T\int_0^T f(g_tX)dt$ for all $f \in \mathcal{C}_c(\mathcal{M}_{1,2})$. If there exists a weak-\* limit of $\nu_T$ that is not the zero measure then the corresponding 3-IET satisfies assumptions (A0)-(A9).
Proposition \[prop:good:assump\] is proved in §\[sec:renorm\].
From the Birkhoff ergodic theorem and Proposition \[prop:good:assump\] it is clear that almost all $3$-IET’s satisfy the assumptions (A0)-(A9). Thus, an immediate corollary of Theorem \[theorem:3iet:mobius:disjoint\] is the following:
\[cor:almost:all:mobius\] For almost all 3-IET’s, Möbius disjointness (i.e. (\[eq:Mobius:disjointness\]) holds.
[[**Disjointness.** ]{}]{} As in e.g. [@Borgain:Sarnak:Ziegler] we derive the Möbius disjointness result from a result about joinings of powers of $T$. In fact, we prove the following:
\[theorem:disjoint:powers\] If $T$ is a 3-IET that satisfies assumptions (A0)-(A9) then there exists $\kappa>1$ so that for all $n>0$, $B_n=\{m<n:T^m \text{ is not disjoint from }T^n\}$ has the property that $m_1<m_2 \in B_n$ then $\frac{m_2}{m_1}>\kappa$.
See Appendix \[sec:appendix:A\] for a proof that Theorem \[theorem:disjoint:powers\] implies Theorem \[theorem:3iet:mobius:disjoint\]. This is a straightforward modification of a note of Harper [@Harper:note]. It is included for completeness.
In Appendix \[sec:appendix:B\], we prove that for almost every 3-IET, $T$, $T^n$ is disjoint from $T^m$ for all $0<n<m$. This gives an alternative (and much easier) proof of Corollary \[cor:almost:all:mobius\]. However, the proof in Appendix \[sec:appendix:B\] does not give a useful diophantine condition under which Möbius disjointness holds.
[[**Related work:** ]{}]{} Möbius disjointness has been shown for a variety of systems see for example [@ELR], [@GT] and [@W] among others. Most closely related to this work is [@D], where Vinogradov’s circle method is used to prove that every rotation (2-IET) is disjoint from Möbius; [@Bou] which shows a set of 3-IETs satisfying a certain measure 0 condition are disjoint from Möbius and [@Borgain:Sarnak:Ziegler] where a slightly stronger version of our criterion is introduced to show that the time 1 map of horocycle flows are disjoint from Möbius. This last paper motivated our approach.
[[**Further questions and conjectures.** ]{}]{}
What is the Hausdorff codimension of the set of $X\in \mathcal{M}_{1,2}$ so that any weak-\* limit point of $\nu_T$ is the zero measure?
For almost every IET that is not of rotation type and $n<m\in \mathbb{N}$ we have $T^n$ is disjoint from $T^m$. In fact if $U_T$ is the unitary operator associated to (composition with) $T$ on $L^2$ function of inttegral zero then there is a sequence $k_1,...$ so that $U_{T^{nk_i}}$ converges to the 0 operator in the weak operator topology and $U_{T^{mk_i}}$ converges to the identity operator in the strong operator topology.
[[**Outline:** ]{}]{} The Section 2 establishes an abstract disjointness criterion, Proposition \[prop:fact:qfact\]. Sections 4 uses this to prove Theorem \[theorem:disjoint:powers\]. Section 3 recalls standard facts about rotations used in Section 4. Section 5 proves Proposition \[prop:good:assump\]. Appendix A proves that Theorem \[theorem:disjoint:powers\] implies Theorem \[theorem:3iet:mobius:disjoint\]. Appendix B proves that almost every 3-IET has the property that all of its distinct positive powers are disjoint.
[[**Acknowledgments:** ]{}]{} J. C. was supported in part by NSF grants DMS-135500 and DMS-1452762 and the Sloan foundation. A. E. is supported in part by NSF grant DMS 1201422 and the Simons Foundation. The authors thank Adam Harper for graciously letting us modify his note and a helpful correspondence. A. E. thanks Princeton University and the Institute for Advanced Study for support during part of this work. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program Dynamics of Group Actions and Number Theory where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1.
Disjointness criterion {#sec:criterion}
======================
Let $(X,d)$ be a metric space. We set $X_1 = X_2 = X$, and write the product $X {\times}X$ as $X_1 {\times}X_2$. Let $\lambda$ be a measure on $X$, and let $T_1: X_1 {\times}X_1$ and $T_2: X_2 \to X_2$ be $\lambda$-preserving maps. Let $\sigma$ be a joining of $(X_1,T_1,
\lambda)$ and $(X_2, T_2,\lambda)$, i.e. $\sigma$ is an ergodic $T_1 {\times}T_2$-invariant measure on $X_1 {\times}X_2$ which projects to $\lambda$ in either factor.
Our basic strategy is due to Ratner [@Ratner:Joinings]. In fact, we use the following proposition:
\[prop:Ratner:disjointness\] Suppose $S: X \to X$ is a $\lambda$-preserving map which commutes with $T_1$ and $T_2$. Suppose $d_1 \ge 0$, $d_2 > 0$, and for every $\delta > 0$ and any compact set $K \subset X_1 {\times}X_2$ with $\sigma(K) > 1-\delta$ and for every $\delta > \epsilon > 0$ there exist points $(x,y) \in X_1 {\times}X_2$, $(x', y') \in X_1 {\times}X_2$ and $r \in {{{\mathbb}N}}$ so that the following conditions hold:
- $(T_1{\times}T_2)^r(x',y') \in K$.
- $(T_1{\times}T_2)^r(x,y) \in K$.
- $d(T_1^{r} x', S^{d_1} T_1^{r} x) +
d(T_2^{r} y', S^{-d_2} T_2^{r} y) < \epsilon$.
Then, $\sigma$ is $S^{d_1} {\times}S^{-d_2}$-invariant.
Suppose $\sigma$ is not $S^{d_1} {\times}S^{-d_2}$-invariant. Then $(S^{d_1} {\times}S^{-d_2})\sigma$ is an ergodic $T_1 {\times}T_2$-invariant measure which is distinct from $\sigma$. Thus, $(S^{d_1} {\times}S^{-d_2})\sigma$ and $\sigma$ are mutually singular. It follows that for any $\delta > 0$ there exists a compact set $K$ with $\sigma(K) > 1-\delta$ such that $(S^{d_1} {\times}S^{-d_2}) K
\cap K = \emptyset$. Then there exists $\epsilon$ such that $$d((S^{d_1} {\times}S^{-d_2}) K, K ) > \epsilon.$$ This is not consistent with conditions (a)-(c).
\[prop:fact:qfact\] Suppose $S$ is continuous except for finitely many points, and suppose $\lambda$ gives zero measure to the points of discontinuity of $S$. Assume
1. There exists a sequence of measurable partitions of $X_1$, $U^{(i)}_{-k},...,U^{(i)}_k$ and a sequence of numbers $r_i$ so that $$\underset{i \to \infty}{\lim}\,
\int_{U_j^{(i)}}d(T_1^{r_i}(x),S^jx)\, d\lambda(x)=0.$$
2. There exists $\ell \in \{1,2,3\}$, a sequence of measurable sets $A_i$, and functions $F_i$ preserving the measure $\lambda$, so that
1. $\underset{i \to \infty}{\lim}\int_{A_i}d(F_iy,y)\,d\lambda(y)=0$.
2. $\underset{i \to
\infty}{\lim}\int_{A_i}d(S^{-\ell}(T_2^{r_i}y),T_2^{r_i}F_iy)\, d\lambda(y)=0$.\
This implies
3. $\underset{i \to \infty}{\lim}\int_{F_i(A_i)}d(F_i^{-1}y,y)\,d\lambda(y)=0$.
4. $\underset{i \to
\infty}{\lim}\int_{F_i(A_i)}d(S^{\ell}(T_2^{r_i}y),T_2^{r_i}F_i^{-1}y)\, d\lambda(y)=0$.
3. There exists an absolute constant $\delta_0 > 0$ such that the following holds: for any $0 < \delta < \delta_0$, either there exists $a \in {{{{\mathbb}Z}}}$ so that for infinitely many $i$, $$\label{eq:a:is:good}
\sigma(\{(x,y):x\in U_{a}^{(i)} \text{ and } y \in A_i\})> 27 \delta +
14 \lambda( \bigcup_{\ell < a} U_\ell^{(i)})$$ or there exists $a' \in {{{{\mathbb}Z}}}$ so that for infinitely many $i$, $$\label{eq:a:prime:is:good}
\sigma(\{(x,y):x\in U_{a'}^{(i)} \text{ and } y \in F_i(A_i)\})> 27
\delta + 14 \lambda(\bigcup_{\ell>a'}U_\ell^{(i)}).$$
Under the assumptions (1)-(3) with (\[eq:a:is:good\]) there exists $d\geq 0$ so that $\sigma$ is $S^d\times S^{-\ell}$ invariant. Also, under the assumptions (1)-(3) with (\[eq:a:prime:is:good\]) there exists $d\leq 0$ so that $\sigma$ is $S^d\times S^{\ell}$ invariant.
\[lemma:find:friend\] Suppose $\epsilon' > 0$ and $\delta > 0$. Then, for any compact set $K \subset X_1 {\times}X_2$ with $\sigma(K) > 1-\delta$ and all $i \in {{{\mathbb}N}}$ sufficiently large, there exists a compact set $K_i'
\subset X_1 {\times}X_2$ with $\sigma(K_i') > 1- 7 \delta$ such that for all $(x,y) \in K_i'$ with $y \in A_i$, there exists $x' \in X_1$ with $(x', F_i y)
\in K$ and $d(x', x ) < \epsilon'$. Similarly, for all $(x,y) \in
K_i'$ with $y \in F(A_i)$, there exists $x' \in X_1$ with $(x',F_i^{-1} y) \in K$ and $d(x',x) < \epsilon'$.
Define a probability measure $\tilde{\sigma}$ on $X_1 {\times}X_2$ by $$\tilde{\sigma}(E) = \frac{\sigma(E \cap K)}{\sigma(K)}.$$ For $y \in X_2$, let $\tilde{\sigma}_y$ be the conditional measure of $\tilde{\sigma}$ along $X_1 {\times}\{ y\}$.
Let $B(x,\epsilon')$ denote the open ball of radius $\epsilon'$. For $\tilde{\sigma}$- almost all $(x,y) \in X_1 {\times}X_2$, $\tilde{\sigma}_y(B(x,\epsilon'/2)) > 0$. Therefore, there exists $\rho(\epsilon',\delta) > 0$ and a set $K_1 \subset K$ with $\tilde{\sigma}(K_1) > 1-\delta$ such that for all $(x,y) \in K_1$, $$\tilde{\sigma}_y(B(x,\epsilon'/2)) > \rho(\epsilon',\delta).$$ Let $\pi_2: X_1 {\times}X_2 \to X_2$ denote projection to the second factor. Since the function $y \to \tilde{\sigma}_y$ is measurable, by Lusin’s theorem there exists a compact set $K_2 \subset X_2$ with $\pi_2^*(\tilde{\sigma})(K_2) > 1 - \delta$ on which it is uniformly continuous relative to the Kantorovich-Rubinstein metric, $$d(\mu,\nu) = \sup_{f} \left|\int_{X_1} f \, d\nu -\int_{X_1} f \, d\nu
\right|,$$ where the sup is taken over all $1$-Lipshitz functions $f: X_1 \to
{{{\mathbb}R}}$ with $\sup |f(x)| \le 1$. Then, there exists $\delta' > 0$ such that for all $y,y' \in K_2$ with $d(y',y) < \delta'$ and for all $x \in X_1$ such that $(x,y) \in K_1$ $$\tilde{\sigma}_{y'}(B(x,\epsilon')) > \rho(\epsilon',\delta)/2.$$ Then, for all $y,y' \in K_2$, with $d(y,y') < \delta$ and all $x$ with $(x,y) \in K_1$, $$\label{eq:sigma:y:prime:B:x:epsilon}
\sigma_{y'}(B(x,\epsilon') \cap K) > 0.$$ We now estimate $\lambda(K_2)$. For $E \subset X_2$, $$\pi_2^*\tilde{\sigma}(E) = \tilde{\sigma}(\pi_2^{-1}(E)) =
\frac{\sigma(\pi_2^{-1}(E) \cap K)}{\sigma(K)} \le
\frac{\sigma(\pi_2^{-1}(E))}{1-\delta} = \frac{\lambda(E)}{1-\delta}.$$ Therefore, $$\lambda(K_2) \ge (1-\delta) \pi_2^*\tilde{\sigma}(K_2) \ge
(1-\delta)^2 \ge 1 - 2 \delta.$$
By condition (2a) in Proposition \[prop:fact:qfact\], for $i$ sufficiently large, there exists a compact set $K_{3,i} \subset X_1 {\times}X_2$ with $\sigma(K_{3,i}) > 1-\delta$ such that for $(x,y) \in K_{3,i}$, $y \in A_i$ and $i$ sufficiently large, $$\label{eq:T2Liy:close:to:y}
d(F_i y, y) < \delta'.$$ Now let $$K'_i = (id \times F_i)^{-1}\left(K_1 \cap (X_1 {\times}K_2) \right)
\cap K_{3,i} \cap (X_1 {\times}K_2).$$ Then, $$\sigma(K'_i)> 1 - 7 \delta.$$ Suppose $(x,y) \in K'_i$, with $y \in A_i$. For large enough $i$, (\[eq:T2Liy:close:to:y\]) holds, and also $(x,y) \in K_1 {\times}K_2$ and $F_i y \in K_2$. Thus (\[eq:sigma:y:prime:B:x:epsilon\]) holds (with $y' = F_i
y$). This implies the first statement of the lemma. The proof of the second statement is identical.
We establish the (\[eq:a:is:good\]) case. The (\[eq:a:prime:is:good\]) case is analogous. The basic strategy is to choose $(x,y) \in U_a^{(i)} {\times}A_i$, and apply Proposition \[prop:Ratner:disjointness\] with $r=r_i$ to the points $(x,y)$ and $(x',F y)$, where $x'$ is as in Lemma \[lemma:find:friend\]. We now give the details.
Suppose $\delta > 0$ and $0 < \epsilon < \delta$ are arbitrary. Let $\Delta$ denote the union of the points of discontinuity of $S^j$, $1 \le j \le k$. There exists $c_1(\epsilon) > 0$ such that if we let $$K_0 = \{ x \in X_1 {\;\: : \;\:}d(x,\Delta) > c_1(\epsilon) \}$$ then $\lambda(K_0) > 1-\epsilon > 1 - \delta$. Let $$K_{00} = \{ x \in X_1 {\;\: : \;\:}d(x,\Delta) > c_1(\epsilon)/2 \}.$$ Since $K_{00}$ is compact and $S$ is continuous on $K_{00}$, there exists $\epsilon' > 0$ such that if $x_1, x_2 \in K_{00}$, with $d(x_1,x_2) < \epsilon'$ then for all $1
\le j \le k$, $d(S^j x_1, S^j x_2) < \epsilon/6$. Without loss of generality, we may assume that $\epsilon' < c_1(\epsilon)/2$. Then, we have, for all $1 \le j \le k$, $$\label{eq:S:uniformly:cts}
d(S^{j} x, S^{j} x') < \epsilon/6 \qquad\text{if $x \in K_0$ and $d(x',x) <
\epsilon'$}.$$
Let $a$ be as in Proposition \[prop:fact:qfact\] (3). Write $$\gamma = \lambda(\bigcup_{\ell < a} U_\ell^{(i)})$$
We may assume that $i$ is large enough so that there exists a compact set $$K_{1b} \subset X_1 \setminus \bigcup_{\ell < a} U_\ell^{(i)}$$ with $\lambda(K_{1b}) > 1- 2\gamma$.
In view of assumption (1) of Proposition \[prop:fact:qfact\], there exists a compact set $K_{1a} \subset X_1$ with $\lambda(K_{1a}) >
1-\delta$ such that $$d(T_1^{r_i} x, S^j x) < \epsilon/6 \qquad\text{ for $x \in U^{(i)}_j$.}$$
In view of assumption (2b) of Proposition \[prop:fact:qfact\], there exists a compact set $K_{2b} \subset X_2$ with $\lambda(K_{2b}) > 1-\delta$ such that for $y \in K_{2b} \cap A_i$ and $i \in {{{\mathbb}N}}$ sufficiently large, $$|S(T_2^{r_i}y)-T_2^{r_i}F_iy| < \frac{\epsilon}{2}.$$
As in the proof of Proposition \[prop:Ratner:disjointness\], let $K$ be a compact set so that $\sigma(K)>1-\delta$ and $(T_1^{d}\times T_2^{-\ell})K$ are compact and disjoint for all $0\leq d\leq k$ and $0\leq \ell\leq 3$. Formally, $K$ may depend on $d$, but without loss of generality we may assume that the same $K$ works for all $0 \le d
\le k$.
Let $$K''_i = ((K_{1a} \cap
K_{1b}) {\times}X)\cap (T_1{\times}T_2)^{-r_i} K$$ Note that $\sigma(K''_i) > 1 - 3 \delta - 2\gamma$. Let $K'_i$ be as in Lemma \[lemma:find:friend\] for $K''_i$ instead of $K$. We have $$\sigma(K'_i) > 1 - 21 \delta - 14 \gamma.$$ Let $$\Omega= (T_1{\times}T_2)^{-r_i}(K \cap (K_0 {\times}K_0)) \cap K'_i \cap (X_1 {\times}K_{2b}) \cap (K_0 {\times}K_0).$$ Then, $$\sigma(\Omega) \ge 1 - 3\delta - (21 \delta + 14 \gamma) - \delta - 2\delta
= 1 - 27 \delta - 14 \gamma.$$ Let $$G_i = \{(x,y):x\in U_{a}^{(i)} \text{ and } y \in A_i\}.$$ By (\[eq:a:is:good\]), $\sigma(\Omega
\cap G_i) > 0$.
Now let $(x,y)$ be any point in $\Omega \cap G_i$. Then, by Lemma \[lemma:find:friend\], there exists $x' \in K_{1a} \cap K_{1b}$ with $(x',F_i y) \in (T_1{\times}T_2)^{-r_i} K$. Because $$(x',y') = (x', F y) \in ((K_{1a} \cap
K_{1b}) {\times}X)\cap (T_1{\times}T_2)^{-r_i} K$$ letting $r=r_i$ conditions (a) and (b) of Proposition \[prop:Ratner:disjointness\] hold. Also, since $x' \in
K_{1b}$, $x' \not\in \bigcup_{\ell < a} U_a^{(i)}$, and thus we may assume $x' \in U_b^{(i)}$ for some $b \ge a$. Then, since $x' \in
K_{1a}$, $$d(T_1^{r_i} x', S^{b} x') < \epsilon/6.$$ Also, in view of Lemma \[lemma:find:friend\], $$d(x', x) < \epsilon',$$ and since $x \in K_{1a}$, $$d(T_1^{r_i} x, S^a x) < \frac \epsilon 6.$$ We have $x \in K_0$ and $T_1^{r_i} x \in K_0$. Therefore, by (\[eq:S:uniformly:cts\]), $$\begin{gathered}
d(T_1^{r_i} x', S^{b-a} T_1^{r_i} x) \le d(T_1^{r_i} x', S^{b} x') + d(S^{b}
x', S^{b} x) + d(S^{b} x, S^{b-a} T_1^{r_i} x) \\
< \frac{\epsilon}{3} + \frac{\epsilon}{6} + \frac{\epsilon}{6} = \frac{\epsilon}{2}.\end{gathered}$$ Similarly, $$d(T_2^{r_i} y', S^{-r} T_2^{r_i} y) = d(T_2^{r_i} F_iy, S^{-r} T_2^{r_i} y) <
\frac{\epsilon}{2}.$$
Therefore, assumption (c) in Proposition \[prop:Ratner:disjointness\] also holds with $d_1=b-a$ and $d_2=\ell$, and Proposition \[prop:Ratner:disjointness\] can be applied.
Let $\lambda$ denote Lebesgue measure on $[0,1]$. Let $S: [0,1] \to [0,1]$ be a 3-IET, $T_1=S^n$ and $T_2=S^m$. Let $\sigma$ be an ergodic joining of $T_1$ and $T_2$.
\[cor:trivial:joining\] If $S$ is weakly mixing and the conditions of Proposition \[prop:fact:qfact\] are satisfied then $\sigma = \lambda {\times}\lambda$.
Note that by [@BN] all the 3-IETs we consider are weakly mixing.
This corollary uses the following standard result:
(See for example [@Rudolph:book Lemma 6.14].) \[lemma:rudolph\] If $(X,B,\mu,T)$ is ergodic and $\sigma$ is a joining of $(X,M_1,\mu,S_1)$ and $(Y,M_2,\nu_2,S_2)$ that is $T \times id$ invariant then $\sigma =\nu_1\times \nu_2$.
We include a proof because the statement in [@Rudolph:book] is slightly more specific.
Given $A \subset Y$ with positive $\nu_2$ measure let $\sigma_A(B)=\sigma (B \times A)$. This is a measure on $X$. Because $\sigma$ has marginals $\mu,\nu_2$ this measure is absolutely continuous with respect to $\mu$. So it has a Radon-Nikodym derivative $f_A$. By our assumption this is a $T$ invariant function and so it is constant. This implies any two rectangles with the same dimensions have the same measure and thus $\sigma$ is the product measure.
We show only the (\[eq:a:is:good\]) case, since the (\[eq:a:prime:is:good\]) case is similar. Because $\sigma$ is $S^{d_1} \times S^{-r}$ invariant it is $S^{md_1}\times S^{-m}$ invariant. Using the fact that $\sigma$ is a joining of $S^n$ and $S^m$, this implies that $\sigma$ is $S^{md_1+nr}\times id$ invariant. Since $S$ is weak mixing and thus totally ergodic, $S^{md_1+nr}$ is ergodic and so by Lemma \[lemma:rudolph\], $\sigma=\lambda {\times}\lambda$.
In §\[sec:facts:rotation\]-§\[sec:work\] we will show that Proposition \[prop:fact:qfact\] can be applied to prove Theorem \[theorem:disjoint:powers\].
Facts about rotations {#sec:facts:rotation}
=====================
Let $\|q_k\alpha\|=dist(q_k\alpha,\mathbb{Z})$. Let $\lambda$ be Lebesgue measure. $d(x,y)=\min\{|x-y|,1-|x-y|\}$
\[lemma:q:k:plus:1:ak:qk\] $q_{k+1}=a_{k+1}q_k+q_{k-1}$
\[lemma:good:bound\] $\frac 1 {q_{k+1}+q_k}<\|q_{k}\alpha\|<\frac 1 {a_{k+1}q_k}$
See [@Khinchin:book], 4 lines before equation 34.
\[lemma:orbit:dense\] $\{R^ix\}_{i=0}^{q_k-1}$ is $2\|q_{k-1}\alpha\|$ dense for all $x$.
Because $R$ is an isometry, it suffices to prove this for $x=0$. We approximate $\{R^n(0)\}_{n=0}^{q_k-1}$ by $\{n\frac{p_k}{q_k} \text{ mod 1}\}_{n=0}^{q_k-1}$, a set that is $\frac 1 {q_k}$ dense. Now $R^\ell(0)$ is within $\|q_k\alpha\|=q_k|\alpha-\frac{p_k}{q_k}|$ of $\ell \frac{p_k}{q_k}$ mod 1 for $0\leq \ell\leq q_k$. Since $\|q_k\alpha\|<\|q_{k-1}\alpha|$ and by Lemma \[lemma:good:bound\] we have $\frac{1}{q_k}<2\|q_{k-1}\alpha\|$. This establishes the lemma.
\[lemma:orbit:sep\] $\{R^ix\}_{i=0}^{q_k-1}$ is $\|q_{k-1}\alpha\|$ separated for all $x$.
\[lemma:ret:time\] If $x$ is in an interval $I$ of size $\|q_k\alpha\|$ then the return time of $x$ to $I$ is either $q_{k+1}$ or $q_{k+1}+q_k$. If $k$ is even and $I =
[-\|q_k \alpha \|,0)$ then the return time of $q_{k+1} + q_k$ takes place on $[-\|q_{k+1} \alpha\|,0)$. If $k$ is odd and $I=[0,\|q_k\alpha\|)$ then the return time of $q_{k+1}+q_k$ takes place on $[\|q_k\alpha\|-\|q_{k+1}\alpha\|,\|q_k\alpha\|)$.
First, we assume $k+1$ is odd. If $x\in I$ then $R^{q_{k+1}}x=x-\|q_{k+1}\alpha\|$. So if $x$ is not in the leftmost $\|q_{k+1}\|$ of $I$ then $R^{q_{k+1}}x\in I$. Otherwise, $R^{q_k}x=x+\|q_\alpha\|$ is on the right of $I$ and within $\|q_{k+1}\alpha|$ of $I$. So $R^{q_{k+1}+q_k}x\in I$. The case of $k+1$ even is similar.
\[lemma:DK\](Denjoy-Koksma) If $f$ is bounded variation then $|\sum_{i=0}^{q_k-1}f(R^ix)-q_k\int f \,
d\lambda|\leq var(f)$.
Note if $f$ is the characteristic function of an interval $var(f)=2$. This is the only case we use in the sequel and we present the proof of this case below. A similar argument will be used to prove the more general Lemma \[lemma:happy:times\].
Following the paragraph in the introduction ‘Connection to tori and tori with marked points’ we want to understand the intersections of a (half open) vertical line segment of length $q_k$ to a horizontal line segment of length $z$ on $\hat{X}$, see Figure \[fig:original\]. (Indeed, $R^{q_k}$ is given by a vertical trajectory of length $q_k$ and $\sum_{j=0}^{q_k-1}\chi_J(R^jx)$ is given by the intersection of the corresponding vertical trajectory of length $q_k$ with a horizontal trajectory of length $z$.) This is equivalent to understanding the intersections of a vertical segment of length $1$ to a horizontal line segment of length $q_kz$ on $g_{\log(q_k)}\hat{X}$. Call these segments $\gamma_1$ and $\gamma_2$ respectively, see Figure \[fig:renorm1\]. We close up these two curves as pictured in Figure \[fig:renorm2\] using the following observations:\
1. Any vertical trajectory of length $q_k$ on $\hat{X}$ has that its endpoints differ by a horizontal vector of length at most $\|q_k\alpha\|<\frac 1{a_{k+1}q_k}.$ This implies we can close $\gamma_1$ up by a horizontal segment, $\zeta_1$, of length less than $\frac 1 {a_{k+1}}\leq1$. Call the resulting closed curve $\hat{\gamma}_1$.
2. We may close up $\gamma_2$ by a vertical segment $\zeta_2'$ of length at most $1$, union a horizontal segment $\zeta_2$ of length at most $\frac 1 2 $ which is either contained in $\gamma_2$ or disjoint from it. Call the resulting closed curve $\hat{\gamma}_2$.
Any vertical segment of length 1 on $g_{\log(q_k)}\hat{X}$ is a translate of $\gamma_1$ and so we may close it up so that it is a translate of $\hat{\gamma}_1$. The intersection of any translate of $\hat{\gamma}_1$ with $\hat{\gamma}_2$ is constant (it is a topological invariant of these curves). So now we study the intersection of translates of $\hat{\gamma}_1$ and $\zeta_2\cup \zeta_2'$. $\gamma_1$ can intersect $\zeta_2$ either $0$ and $1$ times. $\gamma_1$ does not intersect $\zeta_2'$ and $\zeta_1$ does not intersect $\zeta_2$. Once again $\zeta_1$ intersects $\zeta_2'$ at most once. To summarize the intersections with $\gamma_2$ of any two translates of $\gamma_1$ differ by at most $2$.
![The torus $\hat{X}$. A vertical segment of length $q_k$ intersects a horizontal slit of length $z$.[]{data-label="fig:original"}](original){width="30.00000%"}
![The torus $g_{\log(q_k)} \hat{X}$: A vertical segment $\gamma_1$ of length $1$ (drawn in red) intersects a horizontal slit $\gamma_2$ of length $q^k z$ (drawn in blue).[]{data-label="fig:renorm1"}](renorm1){width="30.00000%"}
![Closing the curves. We complete the vertical segment $\gamma_1$ to a closed curve $\hat{\gamma_1}$ by adding a horizontal segment $\zeta_1$ (drawn in green). Simularly, we close up the horizontal slit $\gamma_2$ to obtain a closed curve $\hat{\gamma}_2$ by adding in a horizontal segment $\zeta_2$ and a vertical segment $\zeta_2'$ (drawn in purple).[]{data-label="fig:renorm2"}](renorm2){width="30.00000%"}
\[lemma:happy:times\] For all $k\in \mathbb{N}$ with $a_{k+1}>4$, and $i \in \mathbb{N}$ with $i \leq \lfloor \frac {a_{k+1}}{4}\rfloor$ we have that there exists $j$ with $\lambda(\psi_{iq_k}^{-1}(j))>\frac 1 {12}$ and either $j-\min\{\ell: \psi_{jq_k}^{-1}(\ell) \neq \emptyset\}\leq 1$ or $\max\{\ell:\psi_{jq_k}^{-1}(\ell)\neq \emptyset\}-j\leq 1$. Moreover $\psi_{iq_k}$ is at most $i+2$ valued.
This is similar to the proof of Lemma \[lemma:DK\], but the vertical segment on $\hat{X}$ has length $iq_k$. We once again work on $g_{\log(q_k)}\hat{X}$, where the vertical segment $\gamma_1$ has length $i$, and the slit $\gamma_2$ has length $q_k z$ (See Figure \[fig:renorm2\]). Thus, we need to estimate the number of intersections between $\gamma_1$ and $\gamma_2$. As in the proof of Lemma \[lemma:DK\], we make the following observations (see Figure \[fig:renorm2\]):
1. Any vertical trajectory of length $iq_k$ on $\hat{X}$ has that its endpoints differ by a horizontal vector of length at most $i\|q_k\alpha\|<\frac i{a_{k+1}q_k}.$ This implies we can close $\gamma_1$ up by a horizontal segment, $\zeta_1$, of length less than $\frac i {a_{k+1}}<\frac 1 4 $. Call the resulting closed curve $\hat{\gamma}_1$.
2. We may close up $\gamma_2$ by a vertical segment of length at most $1$, $\zeta_2'$, union a horizontal segment of length at most $\frac 1 2$, $\zeta_2$, which is either contained in $\gamma_2$ or disjoint from it. Call the resulting closed curve $\hat{\gamma}_2$.
Any vertical segment of length $i$ on $g_{\log(q_k)}\hat{X}$ is a translate of $\gamma_1$ and so we may close it up so that it is a translate of $\hat{\gamma}_1$. As in the proof of Lemma \[lemma:DK\], the intersection of any translate of $\hat{\gamma}_1$ with $\hat{\gamma}_2$ is constant (it is a topological invariant of these curves). So now we study the intersection of translates of $\hat{\gamma}_1$ and $\zeta_2\cup \zeta_2'$. $\gamma_1$ can intersect $\zeta_2$ between $0$ and $i$ times. $\gamma_1$ does not intersect $\zeta_2'$ and $\zeta_1$ does not intersect $\zeta_2$. Also $\zeta_1$ intersects $\zeta_2'$ at most once. To summarize the intersections with $\gamma_2$ of any two translates of $\gamma_1$ differ by at most $i+1$.
Observe that on every horizontal line, a segment of at least $\frac{1}{4}$ has that all the corresponding translates of $\hat{\gamma}_1$ for this line segment intersect $\zeta_2$ or all of them do not. (Indeed there is a segment of size at least $\frac 1 2$ so that a vertical segment of length 1 from any point on this segment misses $\zeta_2$ and $\{j\|q_k\alpha\|:0\leq j\leq i\}$ is contained in an interval of length at most $\frac 1 4 $. That is, there is a subinterval of size $\frac 1 4$ so that for each $x$ in this subinterval we have that $j\|q_k\alpha\|+x$ is in the subinterval of size $\frac 1 4 $ for all $0\leq j\leq i$.) So on a subset of this set of measure at least $\frac 1 8$ the translates of $\hat{\gamma}_2$ either all intersect $\zeta_2'$ or all miss $\zeta_2'$. This set satisfies the lemma and it is either within one of the maximal or within $1$ of the minimal.
\[lem:hit interval\] If $\hat{I}$ is an interval of size at least $\gamma \|q_k\alpha\|$ and $q_L>12 \gamma^{-1}\|q_k\alpha\|$ then for all $x$ we have $\frac 1 {q_L}\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [\frac 1 2 \lambda(\hat{I}),2 \lambda(\hat{I})]$. Also for all $\gamma>0$ there exists $u$ so that if $\hat{I}$ is an interval of size at least $\gamma \|q_k\alpha\|$ then $\frac 1 {q_L}\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [\frac 1 2 \lambda(\hat{I}),2 \lambda(\hat{I})]$ for all $L>k+u$ and if $t>q_{k+u}$ we have $\frac 1 t \sum_{j=0}^{t-1}\chi_{\hat{I}}(R^jx)\geq \frac 1 4 \lambda(\hat{I})$.
This follows by Lemma \[lemma:DK\]. Indeed $\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [q_L\lambda(\hat{I})-2,q_L\lambda(\hat{I})+2]$ so if the lemma follows if $q_L\lambda(\hat{I})>4$. Since $\|q_k\alpha\|<\frac 1 {3q_{k+1}}$ this is the case. To see the second claim first notice that $q_{k+2}=q_{k+1}+q_k>2q_k$. So if $2^b>12\gamma^{-1}$ then $q_{L}>12\gamma^{-1}\|q_k\alpha\|$ whenever $L\geq k+2b$, and the second claim follows from the first with $u=2b$. To see the last claim, notice $\sum_{j=0}^{t-1}\chi_{\hat{I}}(R^jx) \geq \sum_{i=0}^{\lfloor \frac t {q_{k+u}}\rfloor-1} \sum_{j=0}^{q_{k+u}-1}\chi_{\hat{I}}(R^{j}R^{iq_{k+u}}x)$ and apply the previous sentence to obtain that this is at least $\lfloor \frac t {q_{k+u}}\rfloor q_{k+u}\frac 1 2 \lambda(\hat{I})$. Since $t<2 \lfloor \frac t {q_{k+u}}\rfloor q_{k+u}$ we have the final claim.
\[lem:sum bound\] $\underset{k \to \infty}{\lim}\frac{\sum_{i=1}^ka_i}{q_k}=0.$
By Lemma \[lemma:q:k:plus:1:ak:qk\] we have that $q_\ell>a_{\ell}q_{\ell-1}$ and $q_{\ell+2}>2q_\ell.$ So by induction we have that $q^k>2^{\frac{j}2}\prod_{i=1}^{k}a_i$ where $j=|\{i<k:(a_i,a_{i-1})=(1,1)\}|$.
Applying the Criterion {#sec:work}
======================
In this section we show that (A0)-(A9) imply the assumptions of Proposition \[prop:fact:qfact\]. Proposition \[prop:technical:cond3\] connects between rotations and powers of 3-IETs. Lemma \[lemma:first:sum\] is an intermediate step in showing that (A0)-(A9) imply the assumptions of Proposition \[prop:technical:cond3\] and Section \[sec:subsec:proof:of:prop:technical\] completes the argument.
In view of Proposition \[prop:fact:qfact\] and Corollary \[cor:trivial:joining\], to prove Theorem \[theorem:disjoint:powers\] it is enough to prove the following:
\[prop:satisfy:2a:2b:2c\] Suppose assumptions (A0)-(A9) are satisfied. Then, there exists a constant $\epsilon_* > 0$ (depending only on the constants in (A0)-(A9)) such that the following holds: Suppose $n \in {{{\mathbb}N}}$, $m'<m<n$, and $$\label{eq:sublacunary}
\left|\frac{m'}{m} - 1\right| \le \epsilon_*.$$ Then the assumptions of Proposition \[prop:fact:qfact\] can be satisfied for $X_1 = X_2 = J$, $T_1 = S^n$ and $T_2$ either $S^m$ or $S^{m'}$. (Note that we view $T_1$ and $T_2$ as maps from $J$ to $J$).
[[**Notation.** ]{}]{} Before starting the proof of Proposition \[prop:satisfy:2a:2b:2c\] we introduce some notation. Let $$\psi_M(x) = \sum_{\ell=0}^{M -1} \chi_J(R^\ell x).$$ Then, for any $x \in J$ so that $R^Mx\in J$, $$\label{eq:time:change:general}
S^{\psi_M(x)} x = R^M x.$$
[[**Picking parameters.** ]{}]{} Let $$\label{eq:def:c}
c = \frac{m - m'}{n}.$$ Since we have $0 < m' < m \le n$, note that $$\frac{m}{m'} = 1 + \frac{c n}{m'} \ge 1 + c.$$ Thus, in order to prove Theorem \[theorem:disjoint:powers\], we may assume that $c$ is small.
Let $k$, $L$ be such that assumptions (A0)-(A9) are satisfied. Let $r_k= \lfloor \frac{\lambda(J)L}n \rfloor$. Let $$w_k = \lfloor \frac{r_k}{\lambda(J)}\rfloor.$$ Then, $$(m-m') w_k = \frac{(m-m')}{n } L + O(\frac{m-m'}{2 \lambda(J)}) = c L +
O(n).$$ Thus, in view of (A0) and (\[eq:def:c\]), we have $$\label{eq:m:mprime:wk:range}
(m-m') w_k \in [q_{k-1},q_k).$$
The proof of Proposition \[prop:satisfy:2a:2b:2c\] relies on the following technical result:
\[prop:technical:cond3\] There exists $c_0, \tilde{c}>0$, $\hat{C} > 0$ and $u \in {{{\mathbb}N}}$ depending only on the constants $C_1,...,C_4$ of the assumptions (A0)-(A4) so that if $c < c_0$ (where $c$ is as in (\[eq:def:c\])), then there exists $\hat{m}\in \{m,m'\}$ and $d \in \{-3,-2,-1,1,2,3\}$ so that, after passing to a subsequence, for all large enough $k$,
1. there exists $\tilde{A} \subset J$ with $\lambda(\tilde{A})>\tilde{c}$ so that for all $y \in \tilde{A}$ we have $$\label{eq:Rqk:S:hat:m:rk}
R^{q_k}S^{\hat{m}r_k}y = S^d S^{\hat{m}r_k} R^{q_k}y.$$
2. If $N>\frac{q_{k+u}}{2\hat{m}}$ then for any $y \in [0,1]$ we have $$\frac{1}{N} \sum_{i=0}^{N-1}\chi_{\tilde{A}}(S^{i\hat{m}}y) > \frac{\lambda(\tilde{A})}{\hat{C}} \text{ and } \frac{1}{N} \sum_{i=0}^{N-1}\chi_{R^{q_k}\tilde{A}}(S^{i\hat{m}}y) > \frac{\lambda(\tilde{A})}{\hat{C}}.$$
Proof of Proposition \[prop:satisfy:2a:2b:2c\] assuming Proposition \[prop:technical:cond3\]
--------------------------------------------------------------------------------------------
We now prove Proposition \[prop:satisfy:2a:2b:2c\] assuming Proposition \[prop:technical:cond3\]. We assume that the constant $c$ in Proposition \[prop:technical:cond3\] is small enough, and $d=1$. The case of $d\in\{3,2,-1,-2,-3\}$ is similar.
In view of assumption (A8) there exists an interval $K_1(L)\subset \mathbb{N}$ of size at most $k_c$ so that for any $x \in J$, $\psi_L(x) \in K_1(L)$. Since $R(J^c) \subset J$, there exists an interval $K_2(L)$ of size $(k_c+1)$ such that for all $x \in [0,1]$, $\psi_L(x) \in K_2(L)$.
We have $$\int_0^1 \psi_L(x) \, d\lambda(x) = \lambda(J) L.$$ Therefore, $\lfloor \lambda(J) L \rfloor \in K_2(L)$. Since $R(J^c) \subset J$, it follows that for all $x \in J$, $$\lfloor \lambda(J) L \rfloor = \psi_{L'(x)}(x) + \delta(x)
\qquad\text{ where $0 \le \delta(x) \le 1$, and $|L'(x) -
L| < (k_c+1)$. }$$ Write $$n r_k = \lfloor \lambda(J) L \rfloor + \epsilon_k, \qquad \text{ where
$|\epsilon_k| < n$.}$$ Now, for $x \in J$, by (\[eq:time:change:general\]), $$T_1^{r_k} x = S^{n r_k} x = S^{\epsilon_k} S^{\lfloor \lambda(J) L
\rfloor} x = S^{\epsilon_k} R^{L'(x)} x = S^{\epsilon_k} R^{L'(x) -
L} R^L x$$ Thus, by (A6) condition (1) of Proposition \[prop:fact:qfact\] follows, (with the size of the partition dependent on $n$).
Let $F_k$ be the first return map of $R^{q_k}$ to $J$. (Essentially we want $F_k$ to be $R^{q_k}$, but we want $F_k$ to be a map from $J$ to $J$). Since $R^{q_k}$ tends to the identity map as $k \to \infty$, condition (2a) of Proposition \[prop:fact:qfact\] follows.
For $x \in \hat{A}$, and since we are assuming $d=1$, (\[eq:Rqk:S:hat:m:rk\]) becomes $$R^{q_k} T_2^{r_k} x = S T_2^{r_k} R^{q_k} x.$$ Since $R^{q_k}$ tends to the identity as $k \to \infty$, there exists a subset $E \subset J$ of almost full measure such that for $x \in E$, $R^{q_k} x = F_k x$. Then, for $x \in E \cap \hat{A}$, $$R^{q_k} T_2^{r_k} x = S T_2^{r_k} F_k x.$$ Condition (2b) of Proposition \[prop:fact:qfact\] follows.
We now begin the proof of Condition (3) of Proposition \[prop:fact:qfact\]. In (A0)-(A9) we choose $\eta <
(96\cdot 25)^{-1}$. Let $\rho = \hat{c}_\eta$, and choose $\delta_0 <
\frac{1}{12} \eta \rho$. Then, by (A5), we can either choose $a$ such that for $i$ sufficiently large, $$\label{eq:choice:of:a}
\frac{1}{96\hat{C}} \lambda(U_a^{(i)}) > 14 \delta_0 + 27
\lambda(\bigcup_{\ell < a} U_\ell^{(i)}) \text{ and }
\lambda(U_a^{(i)}) > \rho,$$ or choose $a'$ so that for $i$ sufficiently large, $$\label{eq:choice:of:a:prime}
\frac{1}{96\hat{C}} \lambda(U_{a'}^{(i)}) > 14 \delta_0 + 27
\lambda(\bigcup_{\ell > a'} U_\ell^{(i)}) \text{ and }
\lambda(U_{a'}^{(i)}) > \rho,$$ We need a lemma to obtain Condition (3) of Proposition \[prop:fact:qfact\] from Proposition \[prop:technical:cond3\]:
\[lemma:strectch:in\] For every $\rho>0$ there exists $b \in {{{\mathbb}N}}$ so that if for some $s \in
{{{{\mathbb}Z}}}$, $\lambda(\psi_L^{-1}(s))>\rho$ where $L\geq q_\ell$ so that either $L=q_\ell$ or $a_{\ell+1}>4$ and $L=iq_\ell$ for $i\leq \lfloor \frac{a_{\ell+1}}4 \rfloor$ then there exists a measurable set $V$ with the following properties:
- We have $$\lambda\left(\bigcup_{j=0}^{q_{\ell-b}-1}R^j(V)\right)>\frac{1}{2} \lambda(\psi_L^{-1}(s)).$$
- We have $R^j(V) \subset \psi_L^{-1}(s)$ for all $0\leq
j<q_{\ell-b}$.
- The sets $R^j(V)$, $0\leq j<q_{\ell-b}$ are pairwise disjoint.
We claim that there exists $b' \in {{{\mathbb}N}}$ so that $$\label{eq:claim:stretch:in}
\lambda(\{x: R^jx\in \psi^{-1}_L(s) \text{ for all }0\leq
j<5q_{j-b'}\})> \frac 4 5\lambda(\psi_L^{-1}(s)).$$ To prove (\[eq:claim:stretch:in\]), note that for any $x \in [0,1]$ there exist at most $2$ different $0\leq j<q_\ell-1$ so that $\psi_L(R^{j+1}x)\neq
\psi_L(R^jx)$. (Indeed the set where $\psi_L(x) \neq \psi_L(Rx)$ is two intervals of length $i\|q_\ell \alpha\|<\frac i {a_{\ell+1}q_\ell }\leq \frac 1 {4q_\ell}<\|q_{\ell-1}\|$. Any orbit of length $q_\ell-1$ can hit each of these intervals at most once.) Thus for any $r \in
{{{\mathbb}N}}$, $$\label{eq:tmp:claim:stretch:in}
\lambda(\{x:\psi_L(x)\neq \psi_L(R^jx) \text{ for some }0<j<r\})\leq
\frac{3r}{q_\ell-1}.$$ (Indeed each orbit of length $q_\ell-1$ can have at most 3 consecutive stretches in this set. These stretches have length at most $r$.) Now (\[eq:tmp:claim:stretch:in\]) implies (\[eq:claim:stretch:in\]).
Now we build $V$. For each $x \in G= \{x: R^jx\in \psi^{-1}_L(s) \text{ for all }0\leq
j<5q_{\ell-b'}\}$ let $m_x=\max\{j:R^jx\notin \psi_L^{-1}(s)\}+1$ and $M_x= \min\{j:R^j\notin \psi_L^{-1}(s)\}-1$. Let $$V=\bigcup_{x\in G}\bigcup_{j=0}^{\lfloor \frac{M_x-m_x}{q_{\ell-b}}\rfloor-1} R^{m_x+jq_{\ell-b}}x.$$
We assume that (\[eq:choice:of:a\]) holds. (The proof in the case (\[eq:choice:of:a:prime\]) holds is virtually identical). Let $a$ and $\rho$ be as in (\[eq:choice:of:a\]). We then apply Lemma \[lemma:strectch:in\] with this $\rho$ and $s = a$. Then, let $V$, $\ell$ and $b$ be as in Lemma \[lemma:strectch:in\]. We assume $\epsilon_*$ (and thus $c$) is small enough so that (A7) implies that $$\label{eq:L:prime:big}
q_{\ell - b} >q_{k+u+4}.$$ Let $\sigma$ be any joining of $S^n \times S^{\hat{m}}$ and for each $x$ let $\Sigma_x$ denote the points $y$ so that $(x,y)$ is $\sigma$-generic. Let $$E_1 = \bigcup_{i=0}^{n-1} R^i V \cap J.$$ We are assuming that $x \in V \subset J$, and also we are assuming that $Ry \in J$ whenever $y \not\in J$. Then, for at least half of $0 \le i < n$ we have $R^i x \in J$, it follows that $$\lambda(E_1) > \frac{n}{2} \lambda(V) > \frac{n}{4 q_{\ell-b}}
\lambda(\psi_L^{-1}(a)).$$ We can choose $N \in {{{\mathbb}N}}$ so that $N \ge \frac{q_{\ell-b}}{12n}$ and also $$\label{eq:phi:L:Snj:x:a}
\psi_L(S^{nj} x ) = a \quad\text{ for all $x \in E_1$ and all $0 \le j
< 2N$.}$$ Let $$E_1' = \bigcup_{j=0}^{N-1} S^{nj} E_1.$$ Then, in view of (\[eq:phi:L:Snj:x:a\]), $$\label{eq:E1:subset:phi:L:inverse:a}
S^{nj} E_1' \subset \psi_L^{-1}(a) \qquad\text{for all $0 \le j < N$.}$$ Note that $R^ix$, $R^jx$ are in distinct $S^n$ orbits if $|i-j|<n$ and $R^ix,R^jx \in J$. This means that the above union is disjoint, and thus $$\lambda(E_1') = N \lambda(E_1) > \frac{N n}{4q_{\ell-b}}
\lambda(\psi_L^{-1}(a)) > \frac{1}{48} \lambda(\psi_L^{-1}(a)).$$ Since $\sigma$ is a self joining of $\lambda$, we can find $E \subset
E_1 {\times}[0,1]$ so that $$\label{eq:measure:sigma:E}
\sigma(E) > \frac{1}{2} \lambda(E_1) > \frac{1}{96} \lambda(\psi_L^{-1}(a)).$$ We have, by (\[eq:L:prime:big\]), $$\label{eq:N:big}
N \ge \frac{q_{\ell-b}}{12 n} > \frac{q_{k+u+4}}{12 n} > \frac{q_{k+u}}{\hat{m}}.$$ where $\hat{m} \in \{m,m'\}$ and $u$ is as in Proposition \[prop:technical:cond3\]. Let $\chi_{\tilde{A}}$ denote the characteristic function of $\tilde{A}$, we have, in view of (\[eq:N:big\]) and conclusion (2) of Proposition \[prop:technical:cond3\], for any $y \in [0,1)$, $$\label{eq:most:in:hatA}
\frac{1}{N} \sum_{j = 0}^{N - 1}
\chi_{\tilde{A}}(S^{\hat{m} j} y) \ge \frac{\lambda(\tilde{A})}{\hat{C}}.$$ Then, since $\sigma$ is $S^{n} {\times}S^{\hat{m}}$ invariant, $$\begin{aligned}
\sigma(\psi_L^{-1}(a) \times \tilde{A}) & = \frac{1}{N} \sum_{j=0}^{N-1}
\sigma(S^{-n j} \psi_L^{-1}(a) \times S^{-\hat{m} j } \tilde{A})
& \\
& \geq \frac{1}{N} \sum_{j=0}^{N-1}
\sigma((S^{-n j} \psi_L^{-1}(a) \times S^{-\hat{m} j } \tilde{A}) \cap
E) & \\
& = \int_{E} \left(\frac{1}{N} \sum_{j=0}^{N-1}
\chi_{\tilde{A}}(S^{\hat{m}j} y)\right) \, d\sigma(x,y) & \text{by (\ref{eq:E1:subset:phi:L:inverse:a})}
\\
& \ge \frac{\lambda(\tilde{A}) \sigma(E)}{\hat{C}} & \text{by (\ref{eq:most:in:hatA})}
\\
& > \frac{1}{96 \hat{C}} \lambda(\psi_L^{-1}(a)) \, \lambda(\tilde{A}), &\text{by (\ref{eq:measure:sigma:E}).} \end{aligned}$$ Condition (3) of Proposition \[prop:fact:qfact\] now follows immediately from (\[eq:choice:of:a\]).
The main lemma
--------------
The next lemma about rotations is the key step in the proof of Proposition \[prop:technical:cond3\].
\[lemma:first:sum\] Assume (A0)-(A4) are satisfied and also that $(m-m')w_k
\in [q_{k-1},q_k)$. Let $C_1, \dots, C_4$ be as in assumptions (A0)-(A4). Then there exist $c_2 > 0$ and $C' > 0$ depending only on $C_1, \dots,
C_4$ such that for all $k \in {{{\mathbb}N}}$ there exists an interval $I \subset [0,1)$ and a set of natural numbers $E=\{e,...,e+c_2q_k\}$ so that
1. $|I| \ge C'{\| q_{k}\alpha\|}$
2. For all $x \in \bigcup_{i \in E}R^iI$. $$\label{eq:E:property}
\sum_{\ell=0}^{mw_k-1}(\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x))-\sum_{\ell=0}^{m'w_k-1}(\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))\in \{-1,1\}.$$
In the rest of this subsection, we will prove Lemma \[lemma:first:sum\]. We will derive Proposition \[prop:technical:cond3\] from Lemma \[lemma:first:sum\] in §\[sec:subsec:proof:of:prop:technical\]. The proof of this lemma is complicated and so we provide a brief sketch: We use (\[eq:difference:non:zero\]) to have a criterion for $\sum_{\ell=0}^{mw_k-1}(\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x))-\sum_{\ell=0}^{m'w_k-1}(\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))\in \{-1,1\}.$ Claims \[claim:hit:both\], \[claim:choose:E0\] and \[claim:E0:works\] use this criterion to prove the claim. Claim \[claim:hit:both\] and subsequent comments identify $I$. Claim 4.6 identifies $E$. Claim 4.7 is used to show the critierion given by (\[eq:difference:non:zero\]) holds for these $I$ and $E$.
Let $$c_2 \leq \frac{1}{C_4} \qquad\text{ so that $c_2 < 1$.}$$ Recall that $J=[0,z]$. Assume $k$ is odd. (This is an assumption of convenience of exposition. If $k$ is odd then $R^{q_k} {0} =-\|q_k\alpha\|$, if $k$ is even it is $\|q_k\alpha\|$. Thus if $k$ is even all sets $[-\|q_k\alpha\|,0)$ should be $[0,\|q_k\alpha\|)$ and $[z-\|q_k\alpha\|,z)$ should be $[z,z+\|q_k\alpha\|)$).
Observe $$\begin{gathered}
\label{eq:sum:rearrangement}
\sum_{\ell=0}^{mw_k-1}\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x)-(\sum_{\ell=0}^{m'w_k-1}\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))=\\
\sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^\ell
R^{m'w_k}x)-\sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^{\ell+q_k}R^{m'w_k}x))
\equiv F(R^{m' w_k} x), \end{gathered}$$ where $$F(y) = \sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^\ell y)-
\sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^{\ell+q_k}y)).$$ Recall that $J=[0,z)$. We have $$\label{eq:F:alternative}
F(y) = \sum_{\ell=0}^{(m-m')w_k-1}\chi_{[-\|q_k \alpha\|,0)}(R^\ell y)
- \sum_{\ell=0}^{(m-m')w_k-1}\chi_{[z-\|q_k \alpha\|,z)}(R^\ell y).$$ Note that, since $(m-m')w_k<q_k$, by Lemma \[lemma:orbit:sep\], each of the sums in (\[eq:F:alternative\]) is at most $1$. Thus, $F(y) \in \{-1, 0, 1\}$, and $$\begin{gathered}
\label{eq:difference:non:zero}
F(y) \in \{ 1, -1 \} \\ \text{if and only if $\{R^\ell y \}_{\ell=0}^{(m-m')w_k}$
hits $[-{\| q_k\alpha\|},0) \cup [z-{\| q_k\alpha\|},z)$ exactly
once. }\end{gathered}$$
Consider $[-{\| q_k\alpha\|},0)$. By Lemma \[lemma:ret:time\], the function that assigns to a point in $[-\|q_k\alpha\|,0)$ its first return time takes two values, $q_{k+1}$ and $q_{k+1}+q_k$. The return time of $q_{k+1}+q_k$ occurs on $[-{\| q_{k+1}\alpha\|},0)$.
\[claim:hit:both\]
- For every $x \in [-\|q_k\alpha\|,0)$ there exists $j \in \{1,...,q_{k+1}+q_k\}$ so that $$R^j x \in [z-{\| q_k\alpha\|},z).$$
- We have $$|\{0\leq j \leq q_{k+1}+q_k:\exists x
\in[-\|q_k\alpha\|,0) \text{ with }R^j x \in
[z-\|q_k\alpha\|,z)\}|\leq 4.$$
- There exists $j \in
\{1,...,q_{k+1}+q_k\}$ such that $$\lambda(R^j([-\|{q_{k+1}\alpha}\|,0))\cap [z-{\| q_k\alpha\|},z)) >
\frac{1}{4}\|q_{k+1} \alpha\|$$
Since $R$ is minimal, if $J'$ is an interval then for every $x$, $R^ix\in J'$ for some $0\leq i< \underset{x \in J'}{\max}\min\{j>0:R^jx \in
J'\}$. For any interval of size $\|q_k\alpha\|$ this is $q_{k+1}+q_k$, see Lemma \[lemma:ret:time\]. This proves (a). By Lemma \[lemma:orbit:sep\], for any $\ell$ there exists at most two $j$ in $\{\ell,...,\ell+q_{k+1}-1\}$ so that there exists $x \in[-\|q_k\alpha\|,0) \text{ with }R^j x \in
[z-\|q_k\alpha\|,z)$. Since $q_{k+1}+q_k<2q_{k+1}$ this implies (b). The statement (c) follows from (a) and (b). Indeed, let $\Delta$ denote the set of $j$ in part (b). Then, in view of (b), $|\Delta| \le 4$. For each $j \in \Delta$, let $I_j$ denote the set of $x \in [-\|q_{k+1} \alpha\|, 0)$ such that $R^j x \in [z-{\| q_k\alpha\|},z)$. Then, by (a), $[-\|q_{k+1} \alpha\|,0) = \bigcup_{j \in \Delta} I_j$. Thus, there exists $j \in \Delta$ such that $\lambda(I_j) \ge \frac 1 4 \|q_{k+1} \alpha\|$.
We now continue the proof of Lemma \[lemma:first:sum\]. We choose $j$, $0 \le j \le q_{k+1} + q_k$ so that $$\lambda(R^j([-\|q_{k}\alpha\|,0))\cap [z-\|q_k\alpha\|,z)) \text{ is maximal.}$$ Let $$I=R^{-j}([z-\|q_k\alpha\|,z)) \cap [-\|q_{k+1}\alpha\|,0)).$$ Note that by Claim \[claim:hit:both\](c), $\lambda(I)>\frac{1}{4} \|q_{k+1}\alpha\|$.
\[claim:choose:E0\] Either $$\label{eq:j:small}
(j - c_2 q_k, j+q_k) \subset \{1, q_{k+1}+q_k - 1\}$$ or $$\label{eq:j:big}
(j - q_k, j+ c_2 q_k) \subset \{1, q_{k+1}+q_k - 1\}$$
Note that by (A4), for $0<i<q_k/C_4$, $$\label{eq:first:comment}
\text{ if $x \in [-\|q_k\alpha\|,0)$ then $R^i x \not\in [z-\|q_k\alpha\|,z)$.}$$ Therefore, $j > q_k/C_4 \ge c_2 q_k$. Similarly, for $0<i<q_k/C_4$, $$\label{eq:first:comment:prime}
\text{if
$x \in [z-\|q_k\alpha\|,z)$ then $R^i x \not\in [-\|q_k\alpha\|,0)$.}$$ Therefore, $j < q_{k+1} + q_k - \frac{q_k}{C_4} \le q_{k+1} + q_k -
c_2 q_k$.
Now if $j < q_{k+1}$ then (\[eq:j:small\]) holds, and if $j > q_k$ then (\[eq:j:big\]) holds. This completes the proof of Claim \[claim:choose:E0\].
\[claim:E0:works\] Suppose (\[eq:j:small\]) holds and $\ell \in (j - c_2 q_k, j+q_k)$ or (\[eq:j:big\]) holds and $\ell \in (j-q_k, j+c_2 q_k)$. Also assume that $\ell \ne j$. Then, $$R^\ell I \cap (([-\|q_k\alpha\|,0) \cup [z-\|q_k\alpha\|,z)) = \emptyset.$$
Recall that $I \subset [-\|q_{k+1} \alpha \|,0)$ and thus by Lemma \[lemma:ret:time\], $$R^\ell I \cap [-\|q_k\alpha\|,0) = \emptyset \qquad \text{ for $1 \le
\ell \le q_{k+1} + q_k - 1$.}$$ Also, by Lemma \[lemma:ret:time\], the return time of any point in the interval $[z-\|q_k\alpha\|,z)$ to itself is at least $q_{k+1} >
q_k$. Thus, for $\ell$ such that $|\ell - j| < q_k$, $$R^\ell I \cap [z-\|q_k\alpha\|,z) = \emptyset.$$ Claim \[claim:E0:works\] follows.
We now continue the proof of Lemma \[lemma:first:sum\]. Let $r =
\min(c_2 q_k, (m-m') w_k)$. Recall that $(m-m')w_k < q_k$. If (\[eq:j:small\]) holds, let $$E = (j-r, j-r+c_2 q_k), \qquad \text{ so that $E+[0,(m-m')w_k) \subset (j-c_2 q_k, j+q_k)$.}$$ If (\[eq:j:big\]) holds, let $$\begin{gathered}
E = (j-(m-m')w_k, j-(m-m')w_k + c_2 q_k) \\ \text{ so that
$E+[0,(m-m')w_k) \subset (j- q_k, j+c_2 q_k)$.}\end{gathered}$$ Then, for all $i \in E$, $$i \le j \le i+(m-m')w_k.$$ Hence, by Claim \[claim:E0:works\], for all $x \in I$ and for all $i
\in E$, $$\label{eq:tmp:Rellix}
|\{R^{i+\ell} x\}_{\ell=1}^{(m-m')w_k}\cap ([-\|q_k\alpha\|,0)\cup
[z-\|q_k\alpha\|,z))|=1.$$ (the only contribution is from the case where $i + \ell = j$). Therefore, in view of (\[eq:sum:rearrangement\]) and (\[eq:difference:non:zero\]), for $x \in R^{-m' w_k} I$ and $\ell \in E$, (\[eq:E:property\]) holds.
From the definition, $|E| \ge c_2 q_k$. We now estimate $\lambda(I)$. By Lemma \[lemma:q:k:plus:1:ak:qk\], $$q_{k+2} = a_{k+2} q_{k+1} + q_{k} < (a_{k+2}+1) q_{k+1}.$$ $$q_{k+1} = a_{k+1} q_k + q_{k-1} < (a_{k+1} + 1) q_{k}$$ By Lemma \[lemma:good:bound\], $$\begin{gathered}
\lambda(I) \ge \frac{1}{4} \|q_{k+1} \alpha\| \ge \frac{1}{4}
\frac{1}{q_{k+2} + q_{k+1}} \ge \frac{1}{4} \frac{1}{(a_{k+2} +
2)q_{k+1}} \ge \\ \ge \frac{1}{4} \frac{1}{(a_{k+2} + 2)(a_{k+1}+1) q_k}
\ge \frac{1}{4} \frac{a_{k+1}}{(a_{k+2} + 2)(a_{k+1}+1)}\|q_k \alpha\|.\end{gathered}$$ Thus, by (A3), $$\lambda(I) \ge \frac{1}{8(C_3+2)} \|q_k \alpha \|.$$ This completes the proof of Lemma \[lemma:first:sum\].
Proof of Proposition \[prop:technical:cond3\] from Lemma \[lemma:first:sum\] {#sec:subsec:proof:of:prop:technical}
----------------------------------------------------------------------------
Recall $w_k =\lfloor \frac{r_k}{\lambda(J)} \rfloor$ as above.
(Corollary to Lemma \[lemma:first:sum\]) Given $w_k$ so that $q_k <(m-m')w_k<q_{k+1}$ as before there exists $\hat{m}\in
\{m,m'\}$ and a set $A_k$ with $\lambda(A_k) \geq \tilde{c}$ (depending on our non-divergence condition, that is $C_1,...,C_4,C'$) so that for all $x \in A_k$ we have $$\label{eq:cor:first:sum}
\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^{i+q_k}x)=d\in \{-3,-2,-1,1,2,3\}.$$
Lemma \[lemma:first:sum\] establishes that there exists $\bar{c} > 0$ and $\bar{c}_1 > 0$, and for an infinite sequence of $k \in {{{\mathbb}N}}$ there exists an interval $I' \subset [0,1]$ with $\lambda(I')>\bar{c}\|q_k \alpha\|$ so that for any $x\in I'$ there exists $H_x \subset \{0,1, \dots ,q_k-1\}$ with $|H_x|>\bar{c}_1 q_k$ so that for any $x \in I'$ and any $\ell \in
H_x$, and any $w_k$ with $(m-m')w_k \in [q_{k-1},q_k)$ we have $$\begin{gathered}
\label{eq:the:main:sum}
\sum_{i=0}^{mw_k-1}(\chi_J(R^i R^\ell x)-\chi_J(R^{i+q_k}R^\ell
x)) \\ - \sum_{i=0}^{m'w_k-1}(\chi_J(R^i R^\ell
x)-\chi_J(R^{i+q_k}R^\ell x)) \in \{-1,1\}.\end{gathered}$$
Also note that by Lemma \[lemma:DK\], $\psi_{q_k}$ takes at most 5 values (which are also consecutive) and so for any $s \in {{{\mathbb}N}}$ and any $x$ we have $$\label{eq:rearrange:psi}
\psi_s(x)-\psi_s(R^{q_k}x)=\psi_{q_k}(x)-\psi_{q_k}(R^s x)\in \{-4,\dots, 4\}.$$ Note that the left-hand-side of (\[eq:the:main:sum\]) is $$(\psi_{mw_k}(R^\ell x) - \psi_{m w_k}(R^{q_k} R^\ell x)) -
(\psi_{m'w_k}(R^\ell x) - \psi_{m' w_k}(R^{q_k} R^\ell x)) \equiv
S_1(R^\ell x) - S_2(R^\ell x).$$ By (\[eq:the:main:sum\]), for all $x \in I'$ and for all $\ell \in
H_x$, $S_1(R^\ell x) - S_2(R^\ell x)
\in \{-1,1\}$, and by (\[eq:rearrange:psi\]), we have $|S_1(R^\ell x)| \le 4$, and $|S_2(R^\ell x)| \le 4$. It follows that for all $x \in I'$ and all $\ell \in H_x$, $$S_i(R^\ell x) \in \{-3,-2,-1,1,2,3\} \qquad\text{ for some $i \in \{1,2\}$.}$$ Thus, there exists $\hat{m}\in \{m,m'\}$ and $d \in \{-3,-2,-1,1,2,3\}$ and a set $A_k$ with $$\lambda(A_k) \ge \frac{1}{12} |H_x| \lambda(I') = \frac{\bar{c}\bar{c}_1 q_k
\|q_k\alpha\|}{12},$$ so that for for $x \in A_k$ (\[eq:cor:first:sum\]) holds.
We frequently use the following trivial result in this section.
\[lem:size bound\] If $\lambda(B)\geq \gamma$ and $B$ is the union of at most $\ell$ intervals then there exists $B'\subset B$ with $\lambda(B')\geq\frac 1 2 \lambda(B)$ and $B'$ is the union of intervals of size at least $\frac{\gamma}{2\ell}$.
There exists $A_k'\subset A_k$ with $\lambda(A_k')>\frac{1}{2} \lambda(A_k)$ and so that $A_k'$ is made of at most $4q_k+1$ intervals with length at least $\frac {\tilde{c}}2 \frac 1 {4q_k+1}$.
Recall that $A_k$ is a level set of $$\sum_{i=0}^{\hat{m}w_i-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_i-1}\chi_J(R^iR^{q_k}x)=\sum_{i=0}^{q_k-1}\chi_J(R^ix)-\sum_{i=\hat{m}w_i}^{\hat{m}w_i +q_k -1}\chi_J(R^ix),$$ a function which has at most $4q_k$ discontinuities. The lemma follows from Lemma \[lem:size bound\] since this implies that any level set is made of at most $4q_k+1$ intervals.
In the previous results we have proved properties of a level set of $\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^{i+q_k}x)$. In this lemma we relate that to proving nice properties about a set, $G_\ell=\{x:S^{\hat{m}n}x=R^\ell x\}$ for some $\ell$, to obtain the set $\tilde{A}$ in Proposition \[prop:technical:cond3\].
\[lemma:Ak:twoprime\] For all large enough $k$ there exists $\hat{m} \in \{m,m'\}, \, d\in \{-3,-2,-1,1,2,3\}$ and a set $\tilde{A}_k$ with $\lambda(\tilde{A}_k)> \frac{\tilde{c}}4$ and which is the union of at most $8q_k$ intervals of size at least $\frac {\tilde{c}}2 \frac 1 {8\cdot 4q_k}$ so that for $x \in \tilde{A}_k$, $$\label{eq:Ak:twoprime}
R^{q_k}S^{\hat{m}r_k}(x)=S^d S^{\hat{m}r_k}(R^{q_k}x).$$
By Lemma \[lemma:DK\], for all $h,j \in
{{{\mathbb}N}}$, and any $x \in [0,1]$, $$\label{eq:iterated:djk}
\left|-h q_j \lambda(J)+ \sum_{i=0}^{h q_j-1} \chi_J(R^i x)\right| \le 2h.$$ Let $0 < N < q_b$ be a positive integer, and write $$N = \sum_{i=0}^{b-1} h_i q_i, \qquad\text{ where $h_i \in {{{{\mathbb}Z}}}$, $0 \le h_i \le a_{i+1}$ and $h_{b-1}<\frac{N}{q_{b-1}}$.}$$ Let $D_0 = 0$, and for $0 < j \le b$, let $$D_j = \sum_{i=0}^{j} h_i q_i, \qquad\text{where $q_0 = 1$.}$$ Note that $D_b = N$. Then, by (\[eq:iterated:djk\]), for $x \in [0,1]$, $$\begin{aligned}
\label{eq:bounding}
\left| - N \lambda(J)+\sum_{i=0}^{N-1}\chi_J(R^i x)\right| & = \notag
\left| \sum_{j=0}^{b-1} \left( -(D_{j+1}-D_j) \lambda(J) + \sum_{i =
D_j}^{D_{j+1}-1} \chi_J(R^i x) \right) \right| \notag \\
& = \left| \sum_{j=0}^{b-1} \left( -h_j q_j \lambda(J) + \sum_{i
= 0}^{h_j q_j-1} \chi_J(R^i R^{D_j} x) \right) \right| \notag \\
& \le 2 \sum_{j=0}^{b-2} a_{j+1}+\frac{N}{q_{b-1}} \qquad\text{ by (\ref{eq:iterated:djk})} \notag \\
&=o(N) + \frac{N}{q_{b-1}}=o(N). \end{aligned}$$ For $x \in J$ define $N(x)$ so that $$\hat{m} r_k = \sum_{i=0}^{N(x)} \chi_J(R^i x).$$ We now apply (\[eq:bounding\]) with $N(x)$ instead of $N$. We obtain that for each $x \in J$ there exists $N(x) \in {{{\mathbb}N}}$ so that $$\label{eq:sum:is:hatm:rk}
\hat{m}r_k = \sum_{i=0}^{N(x)} \chi_J(R^i x), \qquad\text{ and $|N(x)
- \hat{m} w_k| \le 4\sum_{i=0}^{b-2} a_i+\frac{\hat{m}w_k}{q_{b-1}}$.}$$ Since $$\hat{m}w_k<q_{k+1}\frac{\max\{m,m'\}}{m-m'}<(C_2+1)q_{k}\frac{\max\{m,m'\}}{m-m'}$$ we have that there exists $D$ so that for all $k$, $\hat{m}w_k<Dq_k$. Also by Lemma \[lem:sum bound\] $$\sum_{j=0}^{b-2} a_{j+1} +\frac{N(x)}{q_{b-1}} = o(q_{b-1})+o(N(x))=o(N(x))$$ and so $N(x)=\hat{m}w_k+o(\hat{m}w_k)$. Therefore, for all large enough $k$ we have $|N(x) - \hat{m} w_k|<\frac{\tilde{c}}{32}q_k$.
Observe that if $$\label{eq:Nx:different:from:mwk}
\sum_{i=0}^{N(x)}\chi_J(R^ix)-\sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x)\neq
\sum_{i=0}^{\hat{m} w_k}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k}
\chi_J(R^iR^{q_k}x),$$ then $R^jx $ is in one of two intervals of size at most $\|q_k\alpha\|$ for some $$j \in [\min(N(x), \hat{m} w_k), \max(N(x), \hat{m} w_k)].$$ It follows that there exists $\tilde{A}_k \subset A_k'$ with $\lambda(\tilde{A}_k) >
\frac{\tilde{c}}4$ which is a union of intervals each of size at least $\frac {\tilde{c}}2 \frac 1 {8\cdot 4q_k}$, such that for $x
\in \tilde{A}_k$ (\[eq:Nx:different:from:mwk\]) does not hold. Indeed, we are removing at most $2\frac{\tilde{c}}{32}q_k$ intervals of size $\|q_k\alpha\|$ so we obtain a set of measure at least $\lambda(A_k')-\frac{\tilde{c}}{16}$ that is a union of at most $4q_k+1+4\frac{\tilde{c}}{32}q_k$ intervals and can invoke Lemma \[lem:size bound\].
Suppose $x \in \tilde{A}_k \subset A_k$. Then, in view of (\[eq:cor:first:sum\]), $$\sum_{i=0}^{N(x)}\chi_J(R^ix)-\sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x) = -d,$$ where $d \in \{-3,-2,-1,1,2,3\}$. In view of (\[eq:sum:is:hatm:rk\]), this can be rewritten as $$\label{eq:sum:is:d:plus:hatm:rk}
d+\hat{m} r_k= \sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x).$$ Now, in view of (\[eq:time:change:general\]), (\[eq:sum:is:hatm:rk\]) and (\[eq:sum:is:d:plus:hatm:rk\]), $$S^{\hat{m} r_k} x = R^{N(x)} x \qquad\text{and}\qquad
S^{d+\hat{m} r_k} R^{q_k} x = R^{N(x)}
R^{q_k} x.$$ The equation (\[eq:Ak:twoprime\]) follows.
To complete the proof we need to show $\{S^{i\hat{m}}x\}$ hits $\tilde{A}_k$ frequently enough. Lemma \[lem:hit interval\] lets us that show $R$ orbits hit $\tilde{A}_k$ frequently enough. The key observation we use is that if we define $j_i$ by $S^{\hat{m}i}x=R^{j_i}x$ then $j_{i+1}-j_i\leq 2 \hat{m}$. We call a set with this property *$2\hat{m}$ dense*. This motivates us to build an auxiliary set, $\hat{A}_k$, so that the hits of an $R$ orbit to $\hat{A}_k$ give a lower bound for the hits of an $S^{\hat{m}}$ orbit to $\tilde{A}_k$.
We assume $k$ is large enough so that Lemma \[lemma:Ak:twoprime\] holds, $q_k>16 \hat{m}$ and $4\hat{m}\|q_k\alpha\|<\frac{\tilde{c}}{16}$.
Consider $\tilde{A}_k$ and remove from it all $x$ so that $$\sum_{i=0}^{\hat{m}w_i}\chi_J(R^i(x))-\sum_{i=0}^{\hat{m}w_i}\chi_J(R^{i+q_k}x)\neq \sum_{i=0}^{\hat{m}w_i}\chi_J(R^iR^j(x))-\sum_{i=0}^{\hat{m}w_i}\chi_J(R^{i+q_k}R^jx)$$ for some $j<2\hat{m}$. This means that if $x$ remains in this set then $$\label{eq:stretch}
\exists\ j\leq 0\leq k \text{ so that } k-j\geq 2\hat{m}
\text{ and }R^\ell x\in \tilde{A}_k \text{ for all } \ell \in \{j,\dots, k\}.$$ The set remaining, $A'''_k$ has measure at least $\lambda(\tilde{A}_k)-4\hat{m}\|q_k\alpha\|$ and is made up of at most $8\hat{m}+8q_k$ intervals. (Indeed, we are removing at most $2\hat{m}$ pre-images of 2 intervals of size $\|q_k\alpha\|$.) Let $\hat{A}_k$ be a subset of $A'''_k$ of measure at least $\frac 1
2 \lambda(A'''_k)$ made of intervals of length at least $\frac{\tilde{c}}{32 \cdot 64(q_{k})}$. (Indeed, we invoke Lemma \[lem:size bound\] using that $A'''_k$ is a set of measure at least $\frac{\tilde{c}}8$ which is made up of at most $16q_k$ intervals.) By the estimate of the size of intervals in $\hat{A}_k$, Lemma \[lem:hit interval\] implies that if $\frac{q_{k+u}}{q_k}$ is large enough we have for $t > q_{k+u}$, $$\label{eq:sep:hit:bound}
\frac{1}{t}\sum_{i=0}^{t} \chi_{\hat{A}_k}(R^ix)>\frac 1 {4}
\lambda(\hat{A}_k).$$
Because $R^ix \in \hat{A}_k$, the equation (\[eq:stretch\]) implies there exists $j\leq i\leq k$ with $k-j\geq 2\hat{m}$ so that $R^\ell x\in \tilde{A}_k$ for all $\ell \in \{j,\dots,k\}$. Then, we have $$\frac 1 {2\hat{m}}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq |\{i \in \mathcal{C}:R^ix\in \tilde{A}_k\}|,$$ where $\mathcal{C}$ is any $2\hat{m}$ dense subset of $\{0,...,t+2\hat{m}\}$.
Observing that $\{j\in [0,k+2m]: \exists i \text{ with }S^{\hat{m}i}x=R^jx\}$ is $2\hat{m}$ dense this implies that $$\frac 1 {2\hat{m}}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq \sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x)$$ where $N_t(x)=\min\{j:S^{\hat{m}j}x=R^\ell x \text{ with }\ell\geq t\}$. We obtain the proposition with $\hat{C}=16$. Indeed, $\lambda(\hat{A})\geq \frac 1 4 \lambda(\tilde{A})$ and so by (\[eq:sep:hit:bound\]) we have that for $t>q_{k+u}$, we have that $$\label{eq:new:sep:hit:bound}
\frac{1}{t}\sum_{i=0}^t
\chi_{\hat{A}_k}(R^ix)>\frac{\lambda(\tilde{A})}{16}.$$ Lastly, $G_x:=\{j:\exists i \text{ with }S^{\hat{m}i}x=R^jx\}$ is at least $\hat{m}$ separated (that is if $j \in G_x$ and $|i-j|<\hat{m}$ then $i \notin G_x$) and so $N_t(x)\leq \frac{t}{\hat{m}}$, letting us obtain, using (\[eq:new:sep:hit:bound\]), $$\frac{1}{2\hat{m} \cdot 16}\lambda(\tilde{A})\leq \frac{1}{2\hat{m}t}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq \frac 1 {t}\sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x)\leq \frac {1} {\hat{m}N_t(x)}\sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x).$$ Multiplying the sequence of inequalities by $\hat{m}$ completes the estimate.
Renormalization {#sec:renorm}
===============
Recall that $X$ is a torus with two marked points related to a 3-IET, $T$ and $\hat{X}$ is the torus obtained by forgetting the two marked points.
[[**Divergence in the space of tori, $\mathcal{M}_1$:** ]{}]{} By Mahler’s compactness criterion the divergence of $g_t\hat{X}$ is controlled by the shortest (non-homotopicaly trivial) simple closed curve on $g_t\hat{X}$. This sequence is given by curves $\gamma_k$ with vertical holonomy $q_k$ and horizontal holonomy $\pm |q_k\alpha-p_k|=\pm {\| q_k\alpha\|}$. Coarsely, this curve is contracted from $t=0$ to $t= \log(q_k\sqrt{a_{k+1}})$ and then expanded. Additionally, there is a fixed compact set $\hat{K}$ so that $g_{\log(q_k)}\hat{X} \in \hat{K}$ for all $k$ (and in particular $|\gamma_k|$ is proportional to 1 at $g_{\log(q_k)}$).
\[lem:rot divergence\] For any $\hat{K} \subset \mathcal{M}_1$ there exists $\ell_{\hat{K}}$ so that for all $k$ we have $|\{t\in [\log(q_k),\log(q_{k+1})):g_t\hat{X} \in \hat{K}\}|<\ell_{\hat{K}}$.
For any $\hat{K}$ there exists $\delta$ so that if $
g_s\hat{X} \in \hat{K}$ then the shortest simple closed curve on $g_s\hat{X}$ is at least $ \delta$. As in the previous paragraph, consider the curve $\gamma_k$ on $\hat{X}$, with vertical holonomy $q_k$ and horizontal holonomy $\pm |q_k \alpha - p_k|$. On $g_s \hat{X}$ the curve $g_s \gamma_k$ has vertical holonomy $e^{-s} q_k$ and horizontal holonomy $\pm e^s |q_k \alpha - p_k|$. If $s\in [\log(q_k),\log(q_{k+1})]$ then, since we are assuming that the length of $g_s \gamma_k$ is at least $\delta$, we must have $e^s |q_k\alpha-p_k|\geq \frac \delta 2 $ or $e^{-s}q_k\geq \frac \delta 2 $. By Lemma \[lemma:good:bound\] the first condition can only hold if $e^s>\frac{\delta}2 {a_{k+1}q_k}$. Noticing that $a_{k+1}q_k>\frac 1 2 q_{k+1}$ this implies $s>\log(q_{k+1}) +2\log(2)+\log(\delta)$. The second condition can only hold if $s<2q_k \frac1{\delta}$. The lemma follows with $\ell_{\hat{K}}=-2\log(\delta)-3\log(2).$
We now assume the assumption of Proposition \[prop:good:assump\]. This means there exists a compact set $\mathcal{K}\subset \mathcal{M}_{1,2}$ so that $\underset{T \to \infty}{\limsup} \, \frac 1 T |\{0<t<T:g_tX \in \mathcal{K}\}|=c>0$. Let $D_1,...$ be a sequence chosen so that $$\frac 1 {D_i} |\{0<t<D_i:g_tX\in \mathcal{K}\}|>\frac {99c}{100}$$ and $$\underset{\zeta>D_1}{\sup}\, |\{0<t<\zeta: g_tX\in \mathcal{K}\}|-c<\frac c {100}.$$
The next lemma is used to obtain (A7).
\[lem:cont in cpct\] For all $r>0$ for all $i$ large enough we have $$|\{t<D_i:g_tX \in \mathcal{K}, |\{s\in [t,t+r]:g_sX \in \mathcal{K}\}|>\frac c {99} r\}|>\frac {8c} {9}.$$
This is a standard application of the Vitali covering lemma. Indeed let $$B=\{t<D_i:g_tX \in \mathcal{K}, \lambda(\{s\in [t,t+r]:g_sX \in \mathcal{K}\})>\frac c {99} r\}$$ and so for each $t \in B$ we have $\lambda(\{s\in [t,t+r]:g_sX \in \mathcal{K}\})<\frac c {99}r.$ By applying the Vitali covering lemma to the intervals $[t,t+r]$ where $t\in B$, we may take a disjoint subcollection of these intervals $I_1,\dots I_\ell$ so that $$\label{eq:vitali}\lambda(\cup_{i=1}^\ell \{s\in I_i:g_s{X}\in \mathcal{K})>\frac 1 3 \lambda( \cup_{t \in B}\{s\in[t,t+r]:g_sX\in \mathcal{K})\geq \frac{\lambda(B)}3.$$ Indeed let $U_1=\{[t,t+r]:t\in B\}$ and choose $I_1$ to be an interval $[t,t+r]$ in this set so that $\lambda(\{s\in [t,t+r]:g_sX\in \mathcal{K}\}$ is maximal. Let $U_2=\{[t,t+r]:t \in B \text{ and }[t,t+r]\cap I_1=\emptyset$ and let $I_2$ be an interval $[t,t+r]$ in this set so that $\lambda(\{s\in [t,t+r]:g_sX\in \mathcal{K}\}$ is maximal. Also observe $$\lambda(\{s: s\in [\tau,\tau+r] \text{ with }\tau \in B, \, [\tau,\tau+r]\cap I_1\neq \emptyset \text{ and } g_sX\in \mathcal{K}\})\leq 2\lambda(\{s\in I_1:g_sX\in \mathcal{K}\}).$$ Repeating this procedure we obtain our intervals $I_1,\dots, I_\ell$.
Having established (\[eq:vitali\]) we see at most $\frac c {99}r$ of the points in each interval are in $\mathcal{K}$ and the measure of the union of these intervals is at most $D_i$. This is a contradiction unless $\lambda(B)\leq \frac c {33}D_i+r$.
By the same proof we obtain:
\[lem:vit\]For all $\epsilon,\gamma>0$ there exists $\delta>0$ so that if $|A\cap[0,R]|>\gamma R$ then $$|\{t\in [0,R]\cap A:\lambda(\{t+s\in A\}_{t\in [0,T]})>\delta \gamma\}|>(1-\epsilon)\gamma R-2T.$$
Let $f(t)=\max\{j: q_j\leq e^{t}\}$. The next lemma is used to obtain (A8).
\[lem:in cpct\] For all $r,\epsilon>0$ there exists $\hat{K}\subset \mathcal{M}_{1}$ so that for all $i$ large enough we have $$|\{t<D_i:g_tX\in \mathcal{K}, g_\ell \hat{X} \in \hat{K} \text{ for all }\ell\in [\log(q_{f(t+r)}),\log(q_{f(t+r)+1})]\}|\geq (1-\epsilon) \frac {99c} {100} D_i.$$
We use the following straightforward consequence of Lemma \[lem:vit\].\
**Sublemma:** If $h:[0,\infty)\to \{0,1\}$ has $ \frac 1 R \int_0^R h(t)\geq \frac{99c}{100}$ then for all $\epsilon'>0,r, \ell$ there exists $L$ so that $$\begin{gathered}
|\{t<R:h(t)=1 \text { and there exists }0<\rho\leq r \text{ so that }\\ h(t+s)=0 \text{ for all but a set of measure $\ell$ of }
s \in [t+\rho,t+\rho+L] \} |<\epsilon' cR+2L.\end{gathered}$$
Apply Lemma \[lem:vit\] with $\epsilon=\epsilon'$ and $\gamma =\frac {99c}{100}$ to obtain $\delta$. Choose $L$ so that $\delta L>\ell+r$.
Let $\hat{C}$ be the compact set in $\mathcal{M}$ given by projecting $\mathcal{K}$ to $\mathcal{M}$ by forgetting the marked points. Let $\ell=\ell_{\hat{C}}$ as in Lemma \[lem:rot divergence\] and $\delta$ be the shortest simple closed curve on any surface in $\hat{C}$. Obtain $L$ from the sublemma with $r=r$, $\epsilon'=\frac{\epsilon}2$, $\ell=\ell$. Let $\hat{\mathcal{K}}$ denote the set of all tori whose shortest simple closed curve is at least $\delta e^{-L}$. The lemma holds for this $\hat{\mathcal{K}}$. Indeed, if $g_s\hat{X} \notin \hat{\mathcal{K}}$ then by examining the size of the shortest simple closed curve we see $g_{s+\tau}X\notin \mathcal{K}$ for all $-L<\tau<L$. That is, considering $A=\{s:g_sX \in \mathcal{K}\}$ and $\rho=\log(q_{f(t+r)})-t$ we are asking that $|\{s\in [t+\rho,t+\rho+L]:s\in A\}|<\ell$. So by the sublemma the set of such $t$ has small density and so we have the lemma.
We now begin the derivation of (A5), (A6) and (A8).
\[lemma:the:time\] For all $t$ there exists $-2\leq s\leq 2$ so that either
- there exists $k$ with $a_{k+1}>4$ and $i\leq \lfloor \frac{a_{k+1}}4 \rfloor$ so that $e^{t+s}=iq_k$
- or there exists $k$ so that $e^{t+s}=q_k$.
Let $j=f(t)$. If $a_{j+1}\leq4$ then since $e^2>5$ we may choose $s$ so that $e^{t+s}=q_j$ (and so $k=j$). If $a_{j+1}>4$ and $i>\frac {q_{j+1}}2$ choose $s$ so that $e^{t+s}=q_{j+1}$ (and so $k=j+1$). Otherwise choose $s$ so that $e^{s+t}=iq_j$ with $i\leq \lfloor \frac{a_{j+1}}4\rfloor$ (and so $k=j$).
The following is a corollary of Lemma \[lemma:the:time\] and Lemma \[lemma:happy:times\]:
\[cor:happy times\] For all $\eta>0$ there exists $\rho>0$ so that for all $t$, there exists $-2\leq s\leq 2$ with $\lambda(\psi_{e^{t+s}}(j))> \rho$ for some $j$ so that $$\begin{cases} \text{either }
j-\min\{\ell: \psi_{e^{s+t}}^{-1}(\ell) \neq \emptyset\}\leq 2 \text{ and } \lambda(\psi_{e^{s+t}}^{-1}((0,j))<\rho \eta\\ \text{or }
\max\{\ell:\psi_{e^{s+t}}^{-1}(\ell)\neq \emptyset\}-j\leq 2 \text{ and } \lambda(\psi_{e^{s+t}}^{-1}(j,\infty))<\rho\eta.
\end{cases}$$
For the proof we use the following trivial result:\
**Sublemma:** For all $\eta>0$ there exists $\rho>0$ so that if $x_0,x_1,x_2 >0$ and $\sum x_i>\frac 1 {12}$ then there exists $i$ so that $x_i>\rho$ and $\sum_{j=0}^{i-1}x_j<\eta \rho$. Also there exists $\ell$ so that $x_\ell>\rho$ and $\sum_{i=\ell+1}^2 x_i<\eta \rho$.
Applying Lemma \[lemma:the:time\] we obtain $e^{t+s}$. If $q_k=e^{t+s}$ then by Lemma \[lemma:DK\] we have $\psi_{q_k}$ takes at most 5 values that are consecutive. Letting $x_i$ be the measure of the $i^{th}$ level set and appying the sublemma implies the corollary. Otherwise by Lemma \[lemma:happy:times\] and the sublemma imply the corollary.
The next lemma is used to obtain (A0)-(A4). Its proof is similar to Lemma \[lem:in cpct\] and is omitted.
\[lem:A0 to A4\] Given any $\epsilon>0$ there exists $M$ so that $\lambda(E_2)=\lambda(\{t<D_i:g_tX\in \mathcal{K} \text{ and for all } s\in [-3,3] \text{ we have }a_{f(t+s)},a_{f(t+s)+1},a_{f(t+s)+2}<M \})>D_i(1-\epsilon)\frac{99c}{100}$.
By choosing $\epsilon=\frac 1 9$ in Lemmas \[lem:in cpct\], \[lem:cont in cpct\] and \[lem:A0 to A4\], for each $r$, we may choose a sequence of $t$ going to infinity which is simultaneously in the three sets whose measure is bounded from below in these Lemmas. For each $t$ there exists $s$ as in Lemma \[lemma:the:time\] and this choice verifies (A6) and (A9). Consider $L=e^{s+t}$ and $c=e^{-r}$. Since $|s|<3$, by Lemma \[lem:A0 to A4\] assumptions (A0-4) hold for this $L$ and $c$. Indeed, $C_1,C_2,C_3=M$ and $C_4$ is $e^{-3}$ times the minimum of the shortest distance between the marked points taken over surfaces in $\mathcal{K}$. By Lemma \[lem:cont in cpct\] (and the fact that the projection of $\mathcal{K}$ to $\mathcal{M}_1$ is compact) (A7) holds. Indeed if $\hat{C}$ is the projection of $\mathcal{K}$ to $\mathcal{M}$ and $\frac{c}{99}r>\ell_{\hat{C}}(u+2)$ (where $\ell_{\hat{C}}$ is as in Lemma \[lem:rot divergence\]) then $L>q_{k+u}$. Moreover, by Corollary \[cor:happy times\] for each $\eta>0$ there exists $\hat{c}_\eta$ so that (A5) holds. We now just need to show (A8) holds. If $e^{s+t}\in \{q_{f(t)},q_{f(t)+1}\}$ this is by Lemma \[lemma:DK\]. Otherwise note $\psi_{nq_i}$ is at most $5+2n$ valued by Lemma \[lemma:happy:times\]. By Lemma \[lem:in cpct\] there exist $N_{r,\epsilon}$ so that $a_{f(t)+1}<N_{r,\epsilon}$ and thus $e^{s+t}=\ell q_j$ for some $\ell<N_{r,\epsilon}$. We obtain (A8) with $k_{e^{-r}}=5+2N_{r,\epsilon}$.
The Sarnak conjecture and joinings of powers {#sec:appendix:A}
============================================
The following result is a trivial modification of a note [@Harper:note] of Harper, which is included for completeness. What is below is a lightly edited version of his note. See that note for connections with the work of other authors.
Let $(X, T)$ be a topological dynamical system. Assume that there exists $C>1$ so that for every $n$, the set $B_n=\{m<n:T^m \text{ is not disjoint from }T^n\}$ has the property that if $m>m'\in B_n$ then $\frac m {m'}>C$ then $T$ is disjoint from Möbius. Indeed for any continuous compactly supported function with integral 0, $F$, we have $\sum_{n=1}^M \mu(n)F(T^nx)=o(M)$.
Let $\mu$ denote the Möbius function.
To prove Theorem 1 we shall require a lemma concerning the additive function $$\omega_\tau (n) := \underset{
p|n,
p\leq e^{
\frac 1 \tau}}{\sum}1.$$
Define $\mu_\tau := \underset{
p\leq e
^{\frac 1 \tau}}{\sum}\frac 1 p$ and let $N$ be any natural number. Then we have the following variance estimate: $$\underset{
n\leq N}{\sum}
(\omega_\tau (n) - \mu_\tau )^
2 \leq N\mu_\tau + O(e^{
\frac 1 \tau }).$$
Lemma 1 is a special case of the Turán-Kubilius inequality, but since the proof is just a short calculation we shall give it in full. Expanding the sum in the statement we obtain $$\underset{
p,q\leq e
^{\frac 1 \tau}}
{\sum}
\underset{n\leq N}{\sum}
1_{p,q|n} - 2\mu_\tau
\underset{
p\leq e^{
\frac 1 \tau}}{\sum}
\lfloor \frac N p \rfloor + N\mu_\tau^ 2$$ and on removing the square brackets, and paying attention to the diagonal contribution in the double sum, we see that is at most $$\underset{
p,q\leq e^
{\frac 1 \tau}}{\sum}
[N/pq] - N\mu_\tau^2
+ N\mu_\tau + 2\mu_\tau \pi(e
^{\frac 1 \tau}),$$
Completion of proof
-------------------
Let $F(n)=F(T^nx)$ and in view of Lemma 1 and the Cauchy-Schwarz inequality, we have that $$\begin{gathered}
\label{eq:key estimate}|\sum_{n=1}^N \mu(n)F(n)|=\\ |\frac 1 {\mu_\tau} \sum_{n\leq N}\mu(n)F(n) \sum_{p |n, p\leq e^{\frac 1 \tau}}1 +\sum_{n=1}^N\mu(n)F(n) \frac{\mu_\tau-\omega_\tau(n)}{\mu_\tau}|\\\leq \frac 1 {\mu_\tau}|\sum_{n=1}^N \mu(n)F(n)\sum_{p|n,p\leq e^{\frac 1 \tau}}1|+\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}.\end{gathered}$$
Observe that for each $\tau$ there exists $N_0$ so that for all $N>N_0$ we have $\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}<\sqrt{N}\sqrt{\frac{2N}{\mu_\tau}}=N\sqrt{\frac{2}{\mu_\tau}}.$
So now we control
$$|\sum_{n\leq N} \mu(n)F(n) \sum_{p| n, p<e^{\frac 1 \tau}} 1|\leq |\sum_{n\leq N}\sum_{p|n,p<e^{\frac 1 \tau}}\mu(p)\mu(\frac n p)F(n)|+\sum_{n\leq N}2|\{p<e^{\frac 1 {\tau}}:p^2|n\}|\cdot \|F\|_{\sup} .$$ Because $\sum_{n\leq N}2|\{p<e^{\frac 1 {\tau}}:p^2|n\}|$ is $O(N)$ we focus on the other term, $$\label{eq:appendix bound}| \sum_{n\leq N}\sum_{p|n,p<e^{\frac 1 \tau}}\mu(p)\mu(\frac n p)F(n)|\leq \sum_{j\leq \log_2(N)} \sum_{2^j\leq k<2^{j+1}}| \mu(k)\sum_{p\leq \min\{e^{\frac 1 \tau},\frac Nk\}} \mu(p)F(pk)|.$$ We apply Cauchy-Schwartz to $ \sum_{2^j\leq k<2^{j+1}}| \mu(k)\sum_{p\leq \min\{e^{\frac 1 \tau},\frac Nk\}} \mu(p)F(pk)|$ and bound (\[eq:appendix bound\]) by $$\begin{gathered}
\sum_{j\leq\log_2(N)}\sqrt{2^j\sum_{2^j\leq m<2^{j+1}}|\sum_{p\leq \min\{e^{\frac 1 \tau},\frac N m\}}\mu(p)F(pm)|^2}\leq \\ \sum_{j\leq \log_2(N)} \sqrt{2^j \sum_{p_1,p_2\leq \min\{e^{\frac 1 \tau}, \frac N {2^j}\}} |\sum_{m=2^j}^{\min\{2^{j+1}, \frac N{p_1},\frac N {p_2}\}}F(p_1m)\overline{F(p_2m)} |}.\end{gathered}$$
The contribution of the diagonal terms ($p_1=p_2$) is at most $2^j \sqrt{\pi (\min \{e^{\frac 1 \tau}, \frac N {2^j})\}} \|F\|_{\sup} $ where $\pi(n)$ is the number of primes less than or equal to $n$. The contribution of the $p_1\neq p_2$ where $p_1$ and $p_2$ are not disjoint is at most $$2^j\sqrt{C \log (\min\{e^{\frac 1 \tau},\frac N {2^j}\})\pi (\min\{e^{\frac 1 \tau},\frac {N}{2^j})\}}\|F\|_{\sup}.$$ Summing over $j$ these terms give a contribution that is $O(N)$. Indeed, we estimate by $\sum_{j\leq \log_2(N)} 2^j\sqrt{C\log(\pi(\frac{N}{2^j}))\pi(\frac{N}{2^j})}\leq \sum_{j=1}^k C 2^j O((k-j)+1)2^{\frac 1 2 (k-j)}$ for $k=\lceil \log_2(N)\rceil$. This is clearly $O(N)$.
For $\tau$ fixed we choose $M_0$ large enough so that for any $M>M_0$, $p_1,p_2<e^{\frac 1 \tau}$ with $T^{p_1}$ disjoint from $T^{p_2}$, and $L\leq M$ we have $$|\sum_{n\leq L}F(p_1n)\overline{F(p_2n)}|<\tau M.$$ The contribution of the $p_1,p_2$ where $T^{p_1}$ and $T^{p_2}$ are disjoint and $2^j>M_0$ is at most $$\sqrt{2^j(\tau 2^j) \pi(\min \{e^{\frac 1 \tau},\frac N {2^j}\})^2}.$$ For fixed $\tau$, summing over $j$, this is also $O(N)$. Indeed we focus on $$\sum_{j:\frac{N}{2^j}<e^{\frac 1 \tau}}\sqrt{2^j(\tau 2^j) \pi(\min \{e^{\frac 1 \tau},\frac N {2^j}\})^2}$$ and observe that this is bounded by $O(N\tau \log(\frac 1 {\tau}))$. If $N$ is large enough the terms when $2^j<M_0$ are also $O(N)$. Since $\mu_n \to \infty$ plugging this into the last line of (\[eq:key estimate\]) and possibly choosing an even larger $N$ so that $\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}<N\sqrt{\frac{2}{\mu_\tau}}$ completes the proof.
Disjointness of powers for generic $3$-IET’s {#sec:appendix:B}
============================================
\[thm:disjoint powers\]For almost every $3$-IET, $T$ we have that $T^n$ is disjoint from $T^m$ for all $0<n<m$.
We prove this by the following straightforward disjointness criterion:
\[prop:criterion\] Let $T$ be an ergodic 3-IET, $R$ be an irrational rotation and $0<n<m$ be natural numbers. Assume there exists $c>0$, $r\in \mathbb{N}$, a sequence $k_1,...$ sets $F_i$, $G_i$ so that for all $i$
1. $\underset{i \to \infty}{\lim}\, \underset{x \in F_i}{\max} |T^{nk_i}x-x|=0$
2. $\underset{i \to \infty}{\lim}\, \underset{x \in G_i}{\max} |T^{mk_i}x-R^{-1}x|=0$
3. $1-\lambda(F_i)<\lambda(G_i)-c.$
Then $T^n$ and $T^m$ are disjoint.
Let $\sigma$ be an ergodic joining of $T^n \times T^m$ that is a probability measure. Because is $T$ is ergodic it suffices to show that $\sigma$ is $id \times T^{-1}$ invariant. By the fact that ergodic probability measures are mutually singular or the same it suffices to show that $(id \times R^{-1})_*\sigma$ is not singular with respect to $\sigma$. By our assumptions, for any $i$ we have $\sigma(F_i \times G_i)\geq c $. Similarly to Section 2, $\sigma$ is not singular with respect to $(id \times R^{-1})_* \sigma$.
For any $\alpha$ let ${\langle\langle q_j\alpha \rangle\rangle}=(-1)^j\|q_j\alpha\|$, the signed distance of $R^{q_j}x$ from $x$. If $x \in [0,1)$ there exists $b_1,...$ so that $b_i\leq a_i$, if $b_i=a_i$ then $b_{i+1}=0$ and $x=\sum_{i=1}^{\infty}b_i{\langle\langle q_{i-1}\alpha \rangle\rangle}$. Notice that for any fixed $\alpha$ the set of $x$ with (an allowable) Ostrowski expansion $b_1,...,b_k$ is an interval of size at least ${\| q_{k+1}\alpha\|}$.
\[lem:all ostrowski\]Given a 3-IET consider it as rotation by $\alpha$ induced on an interval $[0,x)$. Let $[a_1,\dots]$ be the continued fraction of $\alpha$ and $(b_1,...)$ be the $\alpha$-Ostrowski expansion of $x$. For $\lambda^2$ almost every $(\alpha,x)$ we have that for any ordered k-tuple of pairs $(c_1,d_1),...,(c_k,d_k)$ of natural numbers so that $d_i\leq c_i-1$ we have that there are infinitely many $i$ with $((a_i,b_i),...,(a_{i+k-1},b_{i+k-1}))=((c_1,d_1),...,(c_k,d_k))$.
For almost every $\alpha$ any $(k+1)$-tuple of natural numbers occurs infinitely often in its continued fraction expansion by the ergodicity of the Gauss map with respect to a fully supported finite invariant measure and the fact that having a fixed initial $(k+1)$-tuple $(c_1,\dots,c_{k+1})$ is a set of positive measure. For any $\alpha$ with this property, the set of $x$ so that the pair $(\alpha,x) $ satisfies the proposition is a set of full measure because the complement has no Lebesgue density points. Indeed let $\alpha$ have $a_{j+i}=c_i$ for $i\leq k+1$ and $y\in [0,1)$, then an interval of size at least $\|q_{j+k+1}\alpha\|$ in $B(y,\|q_j\alpha|)$ have that the $j+1$ through $j+k$ terms of their Ostrowski expansion are $d_1,\dots d_{k-1}$. Since $\frac{\|q_{j+k+1}\alpha\|}{\|q_j\alpha\|}>3^{-(j+1)}c_1\cdots c_{k+1}$ we have the claim.
By Lemma \[lem:all ostrowski\] it suffices to show that any 3-IET given by inducing rotation by $\alpha$ on $[0,x)$ where the sequence $(a_1,b_1),...$ contains all $k$-tuples $(c_1,d_1),...,(c_k,d_k)$ with the condition that $c_i-1\geq d_i$ infinitely often satisfies the assumptions of Proposition \[prop:criterion\] for some $c>0$. Let $(10m,0),(10m,0),(4m,1),(10m,0)$ be the pairs of $($continued fraction expansion, Ostrowski expansion$)$. Let $\ell$ be an index so that $(a_{\ell+1},b_{\ell+1}),(a_{\ell+2},b_{\ell+2}),(a_{\ell+3}, b_{\ell+3}),(a_{\ell+4},b_{\ell+4})=(10m,0),(10m,0),(4m,1),(10m,0)$ and $\ell+2$ is even (this can be done because the 8-tuple $$(10m,0),(10m,0),(4m,1),(10m,0),(10m,0),(4m,1),(10m,0),(10m,0)$$ occurs infinitely often). Let $x_{\ell-i}=\sum_{i=0}^{\ell-1}b_i{\langle\langle q_i\alpha \rangle\rangle}$. Note there exists $r<q_\ell$ so that $x_{\ell-1}=R^{r}0.$
**Sublemma:** There exists $j$ so that $\lambda(\{x:\sum_{i=0}^{q_{\ell+2}-1}\chi_{[0,x_{\ell-1})}(R^ix)\neq j\})<\frac 1 {100m^3}.$
Let $\phi_\ell(x)=\sum_{i=0}^{q_{\ell+2}-1}\chi_{[0,x_{\ell-1})}(R^ix)$
If $\phi_\ell(x)\neq \phi_\ell (Rx)$ then $\chi_{[0,q_{\ell-1}(x))}(x)-\chi_{[0,x_{\ell-1})}(R^{q_k}x)\neq 0$. Since $\ell$ is even this means that $x \in [-\|q_{\ell+2}\alpha\|,0)\cup [x_{\ell-1}-\|q_{\ell+2}\alpha\|,x_{\ell-1}\|)$. Observe that $R^{r}([-\|q_{\ell+2}\alpha\|,0))=[x_{\ell-1}-\|q_{\ell+2}\alpha\|,x_{\ell-1})$. It follows that $\phi_\ell$ has two level sets, one on $\cup_{i=1}^{r} R^i([-\|q_{\ell+2}\alpha\|,0))$ and the other on its complement. $r\lambda([-\|q_{\ell+2}\alpha\|,0))\leq q_\ell \|q_{\ell+2}\|<\frac 1 {100m^3}$.
Now consider $J=[0,x_{\ell-1}) \cup (J\setminus [0,x_{\ell-1})$. By the sublemma there exists $j$ so that $\lambda(\{x:\sum_{i=0}^{mq_{\ell+2}-1}\chi_{ [0,x_{\ell-1})}(R^ix)=mj\})>1-\frac m {100m^3}$ and $\lambda(\{x:\sum_{i=0}^{nq_{\ell+2}-1}\chi_{ [0,x_{\ell-1})}(R^ix)=nj\})>1-\frac n {100m^3}>1-\frac m {100m^3}$. Now since $b_{\ell+3}=0$ $$\begin{gathered}
{\| q_{\ell+2}\alpha\|}-\frac 1 {10m}{\| q_{\ell+2}\alpha\|} <{\| q_{\ell+2}\alpha\|}-{\| q_{\ell+4}\alpha\|}\leq|(J\setminus [0,x_{\ell-1})|<\\
{\| q_{\ell+2}\alpha\|}+{\| q_{\ell+3}\alpha\|}<{\| q_{\ell+2}\alpha\|}+\frac 1 {10m}{\| q_{\ell+2}\alpha\|}\end{gathered}$$ and $R^i(J \setminus [0,x_{\ell-1})$ are disjoint for all $j<q_{\ell+3}$ we have $$\lambda(\{x:\sum_{i=0}^{mq_{\ell+2}-1}\chi_J(R^ix)=mj+1\})>mq_{\ell+2}{\| q_{\ell+2}\alpha\|}-\frac 1 {100m^2}-mq_{\ell+2}\frac 1 {10m}{\| q_{\ell+2}\alpha\|}$$ and $$\lambda(\{x:\sum_{i=0}^{nq_{\ell+2}-1}\chi_J(R^ix)=nj\})\geq 1-nq_{\ell+2}{\| q_{\ell+2}\alpha\|}-\frac 1 {100m^2}-nq_{\ell+2}\frac 1 {10m}{\| q_{\ell+2}\alpha\|}.$$ We have verified the assumptions of Proposition \[prop:criterion\] with $c=(m-n-\frac{m+n}{10m})\|q_{\ell+2}\alpha\|q_{\ell+2}-\frac{2}{100m^2}$. Since $q_{\ell+2}\|q_{\ell+2}\alpha\|>\frac{1}{10m+3}$ this is positive.
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Ji Li, Dept. of Mathematics Thomas Roby, Dept. of Mathematics, University of Connecticut
For my friends, who keep me sane.
Without Ira’s brilliant and understanding mentorship, this research would not have been possible.
Without Susan’s friendship and guidance or the camaraderie of the members of my cohort, I never would have made it to the point of doing graduate research.
Without the encouragement David, Margaret, and Kim gave me, I might never have discovered mathematical research at all.
Without my parents’ patient and generous support or the forbearance of my many teachers from childhood into my graduate years, I would never even have discovered the academy.
Without Janet’s affection, fellowship, and tolerance of my eccentricities, my progress in these last years—and my spirits—would have been greatly diminished.
My life and this work are gift from and a testament to everyone whom I have known. Thank you all.
The theory of $\Gamma$-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case—bipartite blocks and general $k$-trees.
Historically, the algebra of generating functions has been a valuable tool in enumerative combinatorics. The theory of combinatorial species uses category theory to justify and systematize this practice, making clear the connections between structural manipulations of some objects of interest and algebraic manipulations of their associated generating functions. The notion of ‘quotient’ enumeration (that is, of counting orbits under some group action) has been applied in species-theoretic contexts, but methods for doing so have largely been ad-hoc. We will contribute a species-compatible way for keeping track of the way a group $\Gamma$ acts on structures of a species $F$, yielding what we term a $\Gamma$-species, which has the sort of synergy of algebraic and structural data that we expect from species. We will then show that it is possible to extract information about the $\Gamma$-orbits of such a $\Gamma$-species and harness this new method to attack several unsolved problems in graph enumeration—in particular, the isomorphism classes of nonseparable bipartite graphs and $k$-trees (that is, ‘unlabeled’ bipartite blocks and $k$-trees).
It is assumed that the reader of this thesis is familiar with the classical theory of groups and that he has encountered at least the basic vocabularies of category theory and graph theory. Results in these fields which are not original to this thesis will either be referenced from the literature or simply assumed, depending on the degree to which they are part of the standard body of knowledge one acquires when studying those disciplines.
In the first chapter, we outline the theory of species, develop several classical methods, and introduce the notion of a $\Gamma$-species. In the second chapter, we apply these techniques to the enumeration of unlabeled vertex-$2$-connected bipartite graphs, a historically open problem. In the third chapter, we apply these techniques to the more complex problem of the enumeration of unlabeled general $k$-trees, also historically unsolved. Finally, in an appendix we discuss algebraic and computational methods which allow species-theoretical insights to be translated into explicit algorithmic techniques for enumeration.
The theory of species {#c:species}
=====================
Introduction {#s:introspec}
------------
Many of the most important historical problems in enumerative combinatorics have concerned the difficulty of passing from ‘labeled’ to ‘unlabeled’ structures. In many cases, the algebra of generating functions has proved a powerful tool in analyzing such problems. However, the general theory of the association between natural operations on classes of such structures and the algebra of their generating functions has been largely ad-hoc. André Joyal’s introduction of the theory of combinatorial species in [@joy:species] provided the groundwork to formalize and understand this connection. A full, pedagogical exposition of the theory of species is available in [@bll:species], so we here present only an outline, largely tracking that text.
To begin, we wish to formalize the notion of a ‘construction’ of a structure of some given class from a set of ‘labels’, such as the construction of a graph from its vertex set or or that of a linear order from its elements. The language of category theory will allow us capture this behavior succinctly yet with full generality:
\[def:species\] Let $\catname{FinBij}$ be the category of finite sets with bijections and $\catname{FinSet}$ be the category of finite sets with set maps. Then a *species* is a functor $F: \catname{FinBij} \to \catname{FinSet}$. For a set $A$ and a species $F$, an element of $F \sbrac{A}$ is an *$F$-structure on $A$*. Moreover, for a species $F$ and a bijection $\phi: A \to B$, the bijection $F \sbrac{\phi}: F \sbrac{A} \to F \sbrac{B}$ is the *$F$-transport of $\phi$*.
A species functor $F$ simply associates to each set $A$ another set $F \sbrac{A}$ of its $F$-structures; for example, for $\specname{S}$ the species of permutations, we associate to some set $A$ the set $\specname{S} \sbrac{A} = \operatorname{Bij} \pbrac{A}$ of self-bijections (that is, permutations as maps) of $A$. This association of label set $A$ to the set $F \sbrac{A}$ of all $F$-structures over $A$ is fundamental throughout combinatorics, and functorality is simply the requirement that we may carry maps on the label set through the construction.
\[ex:graphspecies\] Let $\specname{G}$ denote the species of simple graphs labeled at vertices. Then, for any finite set $A$ of labels, $G \sbrac{A}$ is the set of simple graphs with $\abs{A}$ vertices labeled by the elements of $A$. For example, for label set $A = \sbrac{3} = \cbrac{1, 2, 3}$, there are eight graphs in $\specname{G} \sbrac{A}$, since there are $\binom{3}{2} = 3$ possible edges and thus $2^{3} = 8$ ways to choose a subset of those edges: $$\specname{G} \sbrac{\cbrac{1, 2, 3}} = \cbrac{
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The symmetric group $\symgp{3}$ acts on the set $\sbrac{3}$ as permutations. Consider the permutation $\pmt{(23)}$ that interchanges $2$ and $3$ in $\sbrac{3}$. Then $\specname{G} \sbrac{\pmt{(23)}}$ is a permutation on the set $\specname{G} \sbrac{\cbrac{1, 2, 3}}$; for example, $$\specname{G} \sbrac{\pmt{(23)}} \pbrac{
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Since the image of a bijection under such a functor is necessarily itself a bijection, many authors instead simply define a species as a functor $F: \catname{FinBij} \to \catname{FinBij}$. Our motivation for using this definition instead will become clear in \[s:quot\].
Note that, having defined the species $F$ to be a functor, we have the following properties:
- for any two bijections $\alpha: A \to B$ and $\beta: B \to C$, we have $F \sbrac{\alpha \circ \beta} = F \sbrac{\alpha} \circ F \sbrac{\beta}$, and
- for any set $A$, we have $F \sbrac{\Id_{A}} = \Id_{F \sbrac{A}}$.
Accordingly, we (generally) need not concern ourselves with the details of the set $A$ of labels we consider, so we will often restrict our attention to a canonical label set $\sbrac{n} := \cbrac{1, 2, \dots, n}$ for each cardinality $n$. Moreover, the permutation group $\symgp{A}$ on any given set $A$ acts by self-bijections of $A$ and induces *automorphisms* of $F$-structures for a given species $F$. The orbits of $F$-structures on $A$ under the induced action of $\symgp{A}$ are then exactly the ‘unlabeled’ structures of the class $F$, such as unlabeled graphs.
Finally, we note that it is often natural to speak of maps between classes of combinatorial structures, and that these maps are sometimes combinatorially ‘natural’. For example, we might wish to map the species of trees into the species of general graphs by embedding; to map the species of connected bicolored graphs to the species of connected bipartite graphs by forgetting some color information; or the species of graphs to the species of sets of connected graphs by identification. These maps are all ‘natural’ in the sense that they are explicitly structural and do not reference labels; thus, at least at a conceptual level, they are compatible with the motivating ideas of species. We can formalize this notion in the language of categories:
\[def:specmap\] Let $F$ and $G$ be species. A *species map* $\phi$ is a natural transformation $\phi: F \to G$ — that is, an association to each set $A \in \catname{FinBij}$ of a set map $\phi_{A} \in \catname{FinSet}$ such that the following diagram commutes: $$\begin{tikzpicture}[every node/.style={fill=white}]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=4em]
{
F \sbrac{A} & F \sbrac{B} \\
G \sbrac{A} & G \sbrac{B} \\
};
\path[->,font=\scriptsize]
(m-1-1) edge node {$\phi_{A}$} (m-2-1)
edge node {$F \sbrac{\sigma}$} (m-1-2)
(m-2-1) edge node {$G \sbrac{\sigma}$} (m-2-2)
(m-1-2) edge node {$\phi_{B}$} (m-2-2);
\end{tikzpicture}$$ We call the set map $\phi_{A}$ the *$A$ component of $\phi$* or the *component of $\phi$ at $A$*.
Such species maps may capture the idea that two species are essentially ‘the same’ or that one ‘contains’ or ‘sits inside’ another.
\[def:specmaptypes\] Let $F$ and $G$ be species and $\phi: F \to G$ a species map between them. In the case that the components $\phi_{A}$ are all bijections, we say that $\phi$ is a *species isomorphism* and that $F$ and $G$ are *isomorphic*. In the case that the components $\phi_{A}$ are all injections, we say that $\phi$ is a *species embedding* and that $F$ *embeds in* $G$ (denoted $\phi: F \hookrightarrow G$). Likewise, in the case that the components $\phi_{A}$ are all surjections, we say that $\phi$ is a *species covering* and that $F$ *covers* $G$ (denoted $\phi: F \twoheadrightarrow G$).
With the full power of the language of categories, we may make the following more general observation:
\[note:specmapcattheo\] Let $\catname{Spc}$ denote the functor category of species; that is, define $\catname{Spc} \defeq \catname{FinSet}^{\catname{FinBij}}$, the category of functors from $\catname{FinBij}$ to $\catname{FinSet}$. Species maps as defined in \[def:specmap\] are natural transformations of these functors and thus are exactly the morphisms of $\catname{Spc}$.
It is a classical theorem of category theory (cf. [@mac:cftwm]) that the epi- and monomorphisms of a functor category are exactly those whose components are epi- and monomorphisms in the target category if the target category has pullbacks and pushouts. Since $\catname{FinSet}$ is such a category, species embeddings and species coverings are precisely the epi- and monomorphisms of the functor category $\catname{Spc}$. Species isomorphisms are of course the categorical isomorphisms in $\catname{Spc}$.
In the case that $F$ and $G$ are isomorphic species, we will often simply write $F = G$, since they are combinatorially equivalent; some authors instead use $F \simeq G$, reserving the notation of equality for the much stricter case that additionally requires that $F \sbrac{A} = G \sbrac{A}$ as sets for all $A$. The notions of species embedding and species covering are original to this work.
In the motivating examples from above:
- The species $\mathfrak{a}$ of trees *embeds* in the species $\specname{G}$ of graphs by the map which identifies each tree with itself as a graph, since any two distinct trees are distinct as graphs.
- The species $\specname{BC}$ of bicolored graphs *covers* the species $\specname{BP}$ of bipartite graphs by the map which sends each bicolored graph to its underlying bipartite graph, since every bipartite graph has at least one bicoloring.
- The species $\specname{G}$ of graphs is *isomorphic* with the species $\specname{E} \pbrac{\specname{G}^{\specname{C}}}$ of sets of connected graphs by the map which identifies each graph with its set of connected components, since this decomposition exists uniquely.
Cycle indices and species enumeration {#s:cycind}
-------------------------------------
In classical enumerative combinatorics, formal power series known as ‘generating functions’ are used extensively for keeping track of enumerative data. In this spirit, we now associate to each species a formal power series which counts structures with respect to their automorphisms, which will prove to be significantly more powerful:
\[def:cycind\] For a species $F$, define its *cycle index series* to be the power series $$\label{eq:cycinddef}
\civars{F}{p_{1}, p_{2}, \dots} := \sum_{n \geq 0} \frac{1}{n!} \big( \sum_{\sigma \in \symgp{n}} \fix \pbrac{F \sbrac{\sigma}} p_{1}^{\sigma_{1}} p_{2}^{\sigma_{2}} \dots \big) = \sum_{n \geq 0} \frac{1}{n!} \big( \sum_{\sigma \in \symgp{n}} \fix \pbrac{F \sbrac{\sigma}} p_{\sigma} \big)$$ where $\fix \pbrac{F \sbrac{\sigma}} := \abs{\cbrac{s \in F \sbrac{A} : F \sbrac{\sigma} \pbrac{s} = s}}$, where $\sigma_{i}$ is the number of $i$-cycles of $\sigma$, and where $p_{i}$ are indeterminates. (That is, $\fix \pbrac{F \sbrac{\sigma}}$ is the *number* of $F$-structures fixed under the action of the transport of $\sigma$.) We will make extensive use of the compressed notation $p_{\sigma} = p_{1}^{\sigma_{1}} p_{2}^{\sigma_{2}} \dots$ hereafter.
In fact, by functorality, $\fix \pbrac{F \sbrac{\sigma}}$ is a class function[^1] on permutations $\sigma \in \symgp{n}$. Accordingly, we can instead consider all permutations of a given cycle type at once. It is a classical theorem that conjugacy classes of permutations in $\symgp{n}$ are indexed by partitions $\lambda \vdash n$, which are defined as multisets of natural numbers whose sum is $n$. In particular, conjugacy classes are determined by their cycle type, the multiset of the lengths of the cycles, which may clearly be identified bijectively with partitions of $n$. For a given partition $\lambda \vdash n$, there are $n! / z_{\lambda}$ permutations of cycle type $\lambda$, where $z_{\lambda} := \prod_{i} i^{\lambda_{i}} \lambda_{i}!$ where $\lambda_{i}$ denotes the multiplicity of $i$ in $\lambda$.. Thus, we can instead write $$\label{eq:cycinddefpart}
\civars{F}{p_{1}, p_{2}, \dots} := \sum_{n \geq 0} \sum_{\lambda \vdash n} \fix \pbrac{F \sbrac{\lambda}} \frac{p_{1}^{\lambda_{1}} p_{2}^{\lambda_{2}} \dots}{z_{\lambda}} = \sum_{n \geq 0} \sum_{\lambda \vdash n} \fix \pbrac{F \sbrac{\lambda}} \frac{p_{\lambda}}{z_{\lambda}}$$ for $\fix F \sbrac{\lambda} := \fix F \sbrac{\sigma}$ for some choice of a permutation $\sigma$ of cycle type $\lambda$. Again, we will make extensive use of the notation $p_{\lambda} = p_{\sigma}$ hereafter.
That the cycle index $\ci{F}$ usefully characterizes the enumerative structure of the species $F$ may not be clear. However, as the following theorems show, both labeled and unlabeled enumeration are immediately possible once the cycle index is in hand. Recall that, for a given sequence $a = \pbrac{a_{0}, a_{1}, a_{2}, \dots}$, the *ordinary generating function*[^2] of $a$ is the formal power series $\tilde{A} \pbrac{x} = \sum_{i = 0}^{\infty} a_{i} x^{i}$ and the *exponential generating function* is the formal power series $A \pbrac{x} = \sum_{i = 1}^{\infty} \frac{1}{i!} a_{i} x^{i}$. The scaling factor of $\frac{1}{n!}$ in the exponential generating function is convenient in many contexts; for example, it makes differentiation of the generating function a combinatorially-significant operation. The cycle index of a species is then directly related to two important generating functions:
\[thm:ciegf\] The exponential generating function $F \pbrac{x}$ of labeled $F$-structures is given by $$\label{eq:ciegf}
F \pbrac{x} = \civars{f}{x, 0, 0, \dots}.$$
\[thm:ciogf\] The ordinary generating function $\tilde{F} \pbrac{x}$ of unlabeled $F$-structures is given by $$\label{eq:ciogf}
\tilde{F} \pbrac{x} = \ci{F} \pbracs[big]{x, x^{2}, x^{3}, \dots}.$$
Proofs of both theorems are found in [@bll:species §1.2]. In essence, \[eq:ciegf\] counts each labeled structure exactly once (as a fixed point of the trivial automorphism on $\sbrac{n}$) with a factor of $1/n!$, while \[eq:ciogf\] simply counts orbits Burnside’s Lemma. In cases where the unlabeled enumeration problem is interesting, it is generally more challenging than the labeled enumeration of the same structures, since the characterization of isomorphism in a species may be nontrivial to capture in a generating function. If, however, we can calculate the complete cycle index for a species, both labeled and unlabeled enumerations immediately follow.
The use of $p_{i}$ for the variables instead of the more conventional $x_{i}$ alludes to the theory of symmetric functions, in which $p_{i}$ denotes the power-sum functions $p_{i} = \sum_{j} x_{j}^{i}$, which form an important basis for the ring $\Lambda$ of symmetric functions. When the $p_{i}$ are understood as symmetric functions rather than simply indeterminates, additional Pólya-theoretic enumerative information is exposed. In particular, the symmetric function in $x$-variables underlying a cycle index in $p$-variables may be said to count *partially*-labeled structures of a given species, where the coefficient on a monomial $\prod x_{i}^{\alpha_{i}}$ counts structures with $\alpha_{i}$ labels of each sort $i$. This serves to explain why the coefficients of powers of $p_{1} = \sum_{i} x_{i}$ counts labeled structures (where the labels must all be distinct) and why the automorphism types of structures are enumerated by $\civars{f}{x, x^{2}, x^{3}, \cdots}$, which allows clusters of labels to be the same. Another application of the theory of symmetric functions to the cycle indices of species may be found in [@gessel:laginvspec].
A more detailed exploration of the history of cycle index polynomials and their relationship to classical Pólya theory may be found in [@jili:pointdet].
Of course, it is not always obvious how to calculate the cycle index of a species directly. However, in cases where we can decompose a species as some combination of simpler ones, we can exploit these relationships algebraically to study the cycle indices, as we will see in the next section.
Algebra of species {#s:specalg}
------------------
It is often natural to describe a species in terms of combinations of other, simpler species—for example, ‘a permutation is a set of cycles’ or ‘a rooted tree is a single vertex together with a (possibly empty) set of rooted trees’. Several combinatorial operations on species of structures are commonly used to represent these kinds of combinations; that they have direct analogues in the algebra of cycle indices is in some sense the conceptual justification of the theory. In particular, for species $F$ and $G$, we will define species $F + G$, $F \cdot G$, $F \circ G$, $\pointed{F}$, and $F'$, and we will compute their cycle indices in terms of $\ci{F}$ and $\ci{G}$. In what follows, we will not say explicitly what the effects of a given species operation are on bijections when those effects are obvious (as is usually the case).
\[def:specsum\] For two species $F$ and $G$, define their *sum* to be the species $F + G$ given by $\pbracs[big]{F + G} \sbrac{A} = F \sbrac{A} \sqcup G \sbrac{A}$ (where $\sqcup$ denotes disjoint set union).
In other words, an $\pbrac{F + G}$-structure is an $F$-structure *or* a $G$-structure. We use the disjoint union to avoid the complexities of dealing with cases where $F \sbrac{A}$ and $G \sbrac{A}$ overlap as sets.
\[thm:specsumci\] For species $F$ and $G$, the cycle index of their sum is $$\label{eq:specsumci}
\ci{F + G} = \ci{F} + \ci{G}.$$
In the case that $F = G_{1} + G_{2}$, we can simply invert the equation and write $F - G_{2} = G_{1}$. However, we may instead wish to study the species $F - G$ without first writing $F$ as a sum. In the spirit of the definition of species addition, we wish to define the species subtraction $F - G$ as the species of $F$-structures that ‘are not’ $G$-structures. For slightly more generality, we may apply the notions of \[def:specmaptypes\]:
\[def:specdif\] For two species $F$ and $G$ with a species embedding $\phi: G \to F$, define their *difference with respect to $\phi$* to be the species $F \specsub{\phi} G$ given by $\pbracs[big]{F \specsub{\phi} G} \sbrac{A} \defeq F \sbrac{A} - \phi \pbrac{G \sbrac{A}}$. When there is no ambiguity about the choice of embedding $\phi$, especially in the case that $G$ has a combinatorially natural embedding in $F$, we may instead simply write $F - G$ and call this species their *difference*.
For example, for $\specname{G}$ the species of graphs and $\mathfrak{a}$ the species of trees with the natural embedding, we have that $\specname{G} - \mathfrak{a}$ is the species of graphs with cycles.
We note also that species addition is associative and commutative (up to species isomorphism), and furthermore the empty species $\numspecname{0}: A \mapsto \varnothing$ is an additive identity, so species with addition form an abelian monoid. This can be completed to create the abelian group of *virtual species*, in which the subtraction $F - G$ of arbitrary species is defined; the two definitions in fact agree where our definition applies. We will not delve into the details of virtual species theory here, directing the reader instead to [@bll:species §2.5].
\[thm:specprod\] For two species $F$ and $G$, define their *product* to be the species $F \cdot G$ given by $\pbrac{F \cdot G} \sbrac{A} = \sum_{A = B \sqcup C} F \sbrac{B} \times G \sbrac{C}$.
In other words, an $\pbrac{F \cdot G}$-structure is a partition of $A$ into two sets $B$ and $C$, an $F$-structure on $B$, and a $G$-structure on $C$. This definition is partially motivated by the following result on cycle indices:
\[thm:specprodci\] For species $F$ and $G$, the cycle index of their product is $$\label{eq:specprodci}
\ci{F \cdot G} = \ci{F} \cdot \ci{G}.$$
Conceptually, the species product can be used to describe species that decompose uniquely into substructures of two specified species. For example, a permutation on a set $A$ decomposes uniquely into a (possibly empty) set of fixed points and a derangement of their complement in $A$. Thus, $\specname{S} = \specname{E} \cdot \operatorname{Der}$ for $\specname{S}$ the species of permutations, $\specname{E}$ the species of sets, and $\operatorname{Der}$ the species of derangements.
We note also that species multiplication is commutative (up to species isomorphism) and distributes over addition, so the class of species with addition and multiplication forms a commutative semiring, with the species $\numspecname{1}: \begin{cases} \varnothing \mapsto \cbrac{\varnothing} \\ A \neq \varnothing \mapsto \varnothing \end{cases}$ as a multiplicative identity; if addition is completed as previously described, the class of virtual species with addition and multiplication forms a true commutative ring.
In addition, the question of which species can be decomposed as sums and products without resorting to virtual species is one of great interest; the notions of *molecular* and *atomic* species are directly derived from such decompositions, and represent the beginnings of the systematic study of the structure of the class of species as a whole. Further details on this topic are presented in [@bll:species §2.6].
\[def:speccomp\] For two species $F$ and $G$ with $G \sbrac{\varnothing} = \varnothing$, define their *composition* to be the species $F \circ G$ given by $\pbrac{F \circ G} \sbrac{A} = \prod_{\pi \in P \pbrac{A}} \pbrac{F \sbrac{\pi} \times \prod_{B \in \pi} G \sbrac{B}}$ where $P \pbrac{A}$ is the set of partitions of $A$.
In other words, the composition $F \circ G$ produces the species of $F$-structures of collections of $G$-structures. The definition is, again, motivated by a correspondence with a certain operation on cycle indices:
\[def:cipleth\] Let $f$ and $g$ be cycle indices. Then the *plethysm* $f \circ g$ is the cycle index $$\label{eq:cipleth}
f \circ g = f \pbrac{g \pbrac{p_{1}, p_{2}, p_{3}, \dots}, g \pbrac{p_{2}, p_{4}, p_{6}, \dots}, \dots},$$ where $f \pbrac{a, b, \dots}$ denotes the cycle index $f$ with $a$ substituted for $p_{1}$, $b$ substituted for $p_{2}$, and so on.
This definition is inherited directly from the theory of symmetric functions in infinitely many variables, where our $p_{i}$ are basis elements, as previously discussed. This operation on cycle indices then corresponds exactly to species composition:
\[thm:speccompci\] For species $F$ and $G$ with $G \sbrac{\varnothing} = \varnothing$, the cycle index of their plethysm is $$\label{eq:speccompci}
\ci{F \circ G} = \ci{F} \circ \ci{G}$$ where $\circ$ in the right-hand side is as in \[eq:cipleth\].
Many combinatorial structures admit natural descriptions as compositions of species. For example, every graph admits a unique decomposition as a (possibly empty) set of (nonempty) connected graphs, so we have the species identity $\specname{G} = \specname{E} \circ \specname{G}^{C}$ for $\specname{G}$ the species of graphs and $\specname{G}^{C}$ the species of nonempty connected graphs.
Diligent readers may observe that the requirement that $G \sbrac{\varnothing} = \varnothing$ in \[def:speccomp\] is in fact logically vacuous, since the given construction would simply ignore the $\varnothing$-structures. However, the formula in \[thm:speccompci\] fails to be well-defined for any $\ci{G}$ with non-zero constant term (corresponding to species $G$ with nonempty $G \sbrac{\varnothing}$) unless $\ci{F}$ has finite degree (corresponding to species $F$ with support only in finitely many degrees). Consider the following example:
Let $\specname{E}$ denote the species of sets, $\specname{E}_{3}$ its restriction to sets with three elements, $\numspecname{1}$ the species described above (which has one empty structure), and $X$ the species of singletons (which has one order-$1$ structure). If $\specname{E} \pbrac{\numspecname{1} + X}$ were well-defined, it would denote the species of ‘partially-labeled sets’. However, for fixed cardinality $n$, there is an $\specname{E} \pbrac{\numspecname{1} + X}$-structure on $n$ labels *for each nonnegative $k$*—specifically, the set $\sbrac{n}$ together with $k$ unlabeled elements. Thus, there would be infinitely many structures of each cardinality for this ‘species’, so it is not in fact a species at all.
However, the situation for $\specname{E}_{3} \pbrac{\numspecname{1} + X}$ is entirely different. A structure in this species is a $3$-set, some of whose elements are labeled. There are only four possible such structures: $\cbrac{*, *, *}$, $\cbrac{*, *, 1}$, $\cbrac{*, 1, 2}$, and $\cbrac{1, 2, 3}$, where $*$ denotes an unlabeled element and integers denote labeled elements. Moreover, by discarding the unlabeled elements, we can clearly see that $\specname{E}_{3} \pbrac{\numspecname{1} + X} = \sum_{i = 0}^{3} \specname{E}_{i}$.
In our setting, we will not use this alternative notion of composition, so we will not develop it formally here.
Several other binary operations on species are defined in the literature, including the Cartesian product $F \times G$, the functorial composition $F \square G$, and the inner plethysm $F \boxtimes G$ of [@travis:inpleth]. We will not use these here. However, we do introduce two unary operations: $\pointed{F}$ and $F'$.
\[def:specderiv\] For a species $F$, define its *species derivative* to be the species $\deriv{F}$ given by $\deriv{F} \sbrac{A} = F \sbrac{A \cup \cbrac{*}}$ for $*$ an element chosen not in $A$ (say, the set $A$ itself).
It is important to note that the label $*$ of an $\deriv{F}$-structure is *distinguished* from the other labels; the automorphisms of the species $\deriv{F}$ cannot interchange $*$ with another label. Thus, species differentiation is appropriate for cases where we want to remove one ‘position’ in a structure. For example, for $\specname{L}$ the species of linear orders and $\specname{C}$ the species of cyclic orders, we have $\specname{L} = \deriv{\specname{C}}$; a cyclic order on the set $A \cup \cbrac{*}$ is naturally associated with the linear order on the set $A$ produced by removing $*$. Terming this operation ‘differentiation’ is justified by its effect on cycle indices:
\[thm:specderivci\] For a species $F$, the cycle index of its derivative is given by $$\label{eq:specderivci}
\civars{\deriv{F}}{p_{1}, p_{2}, \dots} = \frac{\partial}{\partial p_{1}} \civars{F}{p_{1}, p_{2}, \dots}.$$
We note that we cannot in general recover $\ci{F}$ from $\ci{\deriv{F}}$, since there may be terms in $\ci{F}$ which have no $p_{1}$-component (corresponding to $F$-structures which have no automorphisms with fixed points).
Finally, we introduce a variant of the species derivative which allows us to *label* the distinguished element $*$:
\[def:specpoint\] For a species $F$, define its *pointed species* to be the species $\pointed{F}$ given by $\pointed{F} \sbrac{A} = F \sbrac{A} \times A$ (that is, pairs of the form $\pbrac{f, a}$ where $f$ is an $F$-structure on $A$ and $a \in A$) with transport $\pointed{F} \sbrac{\sigma} \pbrac{f, a} = \pbrac{F \sbrac{\sigma} \pbrac{f}, \sigma \pbrac{a}}$. We can also write $\pointed{F} \sbrac{A} = X \cdot \deriv{F}$ for $X$ the species of singletons.
In other words, an $\pointed{F} \sbrac{A}$-structure is an $F \sbrac{A}$-structure with a distinguished element taken from the set $A$ (as opposed to $\deriv{F}$, where the distinguished element is new). Thus, species pointing is appropriate for cases such as those of rooted trees: for $\mathfrak{a}$ the species of trees and $\specname{A}$ the species of rooted trees, we have $\specname{A} = \pointed{\mathfrak{a}}$. leads directly to the following:
\[thm:specpointci\] For a species $F$, the cycle index of its corresponding pointed species is given by $$\label{eq:specpointci}
\ci{\pointed{F}} = \ci{X} \cdot \ci{\deriv{F}}.$$
Note that, again, we cannot in general recover $\ci{F}$ from $\ci{\pointed{F}}$, for the same reasons as in the case of $\ci{\deriv{F}}$.
Multisort species {#s:mult}
-----------------
A species $F$ as defined in \[def:species\] is a functor $F: \catname{FinBij} \to \catname{FinSet}$; an $F$-structure in $F \sbrac{A}$ takes its labels from the set $A$. The tool-set so produced is adequate to describe many classes of combinatorial structures. However, there is one particular structure type which it cannot effectively capture: the notion of distinct *sorts* of elements within a structure. Perhaps the most natural example of this is the case of $k$-colored graphs, where every vertex has one of $k$ colors with the requirement that no pair of adjacent vertices shares a color. Automorphisms of such a graph must preserve the colorings of the vertices, which is not a natural restriction to impose in the calculation of the classical cycle index in \[eq:cycinddef\]. We thus incorporate the notion of sorts directly into a new definition:
\[def:ksortset\] For a fixed integer $k \geq 1$, define a *$k$-sort set* to be an ordered $k$-tuple of sets. Say that a $k$-sort set is *finite* if each component set is finite; in that case, its *$k$-sort cardinality* is the ordered tuple of its components’ set cardinalities. Further, define a *$k$-sort function* to be an ordered $k$-tuple of set functions which acts componentwise on $k$-sort sets. For two $k$-sort sets $U$ and $V$, a $k$-sort function $\sigma$ is a *$k$-sort bijection* if each component is a set bijection. For $k$-sort sets of cardinality $\pbrac{c_{1}, c_{2}, \dots, c_{k}}$, denote by $\symgp{c_{1}, c_{2}, \dots, c_{k}} = \symgp{c_{1}} \times \symgp{c_{2}} \times \dots \times \symgp{c_{k}}$ the *$k$-sort symmetric group*, the elements of which are in natural bijection with $k$-sort bijections from a $k$-sort set to itself. Finally, denote by $\catname{FinBij}^{k}$ the category of finite $k$-sort sets with $k$-sort bijections.
We can then define an extension of species to the context of $k$-sort sets:
\[def:multisort\] A *$k$-sort species* $F$ is a functor $F: \catname{FinBij}^{k} \to \catname{FinBij}$ which associates to each $k$-sort set $U$ a set $F \sbrac{U}$ of *$k$-sort $F$-structures* and to each $k$-sort bijection $\sigma: U \to V$ a bijection $F \sbrac{\sigma}: F \sbrac{U} \to F \sbrac{V}$.
Functorality once again imposes naturality conditions on these associations.
Just as in the theory of ordinary species, to each multisort species is associated a power series, its *cycle index*, which carries essential combinatorial data about the automorphism structure of the species. To keep track of the multiple sorts of labels, however, we require multiple sets of indeterminates. Where in ordinary cycle indices we simply used $p_{i}$ for the $i$th indeterminate, we now use $p_{i} \sbrac{j}$ for the $i$th indeterminate of the $j$th sort. In some contexts with small $k$, we will denote our sorts with letters (saying, for example, that we have ‘$X$ labels’ and ‘$Y$ labels’), in which case we will write $p_{i} \sbrac{x}$, $p_{i} \sbrac{y}$, and so forth. In natural analogy to \[def:cycind\], the formula for the cycle index of a $k$-sort species $F$ is given by $$\begin{gathered}
\label{eq:multcycinddef}
\civars{F}{p_{1} \sbrac{1}, p_{2} \sbrac{1}, \dots; p_{1} \sbrac{2}, p_{2} \sbrac{2}, \dots; \dots; p_{1} \sbrac{k}, p_{2} \sbrac{k}, \dots} = \\
\sum_{\substack{n \geq 0 \\ a_{1} + a_{2} + \dots + a_{k} = n}} \frac{1}{a_{1}! a_{2}! \dots a_{k}!} \sum_{\sigma \in \symgp{a_{1}, a_{2}, \dots, a_{k}}} \fix \pbrac{F \sbrac{\sigma}} p^{\sigma_{1}}_{\sbrac{1}} p^{\sigma_{2}}_{\sbrac{2}} \dots p^{\sigma_{k}}_{\sbrac{k}}.\end{gathered}$$ where by $p^{\sigma_{i}}_{\sbrac{i}}$ we denote the product $\prod_{j} \pbrac{p_{j} \sbrac{i}}^{\pbrac{\sigma_{i}}_{j}}$ where $\pbrac{\sigma_{i}}_{j}$ is the number of $j$-cycles of $\sigma_{i}$.
The operations of addition and multiplication extend to the multisort context naturally. To make sense of differentiation and pointing, we need only specify a sort from which to draw the element or label which is marked; we then write $\deriv[X]{F}$ and $\pointed[X]{F}$ for the derivative and pointing respectively of $F$ ‘in the sort $X$’, which is to say with its distinguished element drawn from that sort. When $F$ is a $1$-sort species and $G$ a $k$-sort species, the construction of the $k$-sort species $F \circ G$ is natural; in other settings, we will not define a general notion of composition of multisort species.
$\Gamma$-species and quotient species {#s:quot}
-------------------------------------
It is frequently the case that interesting combinatorial problems admit elegant descriptions in terms of quotients of a class of structures $F$ under the action of a group $\Gamma$. In some cases, this group action will be *structural* in the sense that it commutes with permutations of labels in the species $F$, or, informally, that it is independent of the choice of labelings on each $F$-structure. In such a case, we may also say that $\Gamma$ acts on ‘unlabeled structures’ of the class $F$.
\[ex:graphcomp\] Let $\specname{G}$ denote the species of simple graphs. Let the group $\symgp{2}$ act on such graphs by letting the identity act trivially and letting the non-trivial element $\pmt{(12)}$ send each graph to its complement (that is, by replacing each edge with a non-edge and each non-edge with an edge). This ‘complementation action’ is structural in the sense described previously.
We note that a group action is structural is exactly the condition that each $\gamma \in \Gamma$ acts by a species isomorphism $\gamma: F \to F$ in the sense of \[def:specmaptypes\].
We now incorporate such species-compatible actions into a new definition:
\[def:gspecies\] For $\Gamma$ a group, a *$\Gamma$-species* $F$ is a combinatorial species $F$ together with an action of $\Gamma$ on $F$-structures by species isomorphisms. Explicitly, for $F$ a $\Gamma$-species, the diagram $$\begin{tikzpicture}[every node/.style={fill=white}]
\matrix (m) [matrix of math nodes, row sep=4em, column sep=4em, text height=1.5ex, text depth=0.25ex]
{
A & F \sbrac{A} & F \sbrac{A} \\
B & F \sbrac{B} & F \sbrac{B} \\
};
\path[->,font=\scriptsize]
(m-1-1) edge node {$F$} (m-1-2)
edge node {$\sigma$} (m-2-1)
(m-1-2) edge node {$\gamma_{A}$} (m-1-3)
edge node {$F \sbrac{\sigma}$} (m-2-2)
(m-1-3) edge node {$F \sbrac{\sigma}$} (m-2-3)
(m-2-1) edge node {$F$} (m-2-2)
(m-2-2) edge node {$\gamma_{B}$} (m-2-3);
\end{tikzpicture}$$ commutes for every $\gamma \in \Gamma$ and every set bijection $\sigma: A \to B$. (Note that commutativity of the left square is required for $F$ to be a species at all.)
$\specname{G}$ is then a $\symgp{2}$-species with the action described in \[ex:graphcomp\].
For such a $\Gamma$-species, of course, it is then meaningful to pass to the quotient under the action by $\Gamma$:
\[def:qspecies\] For $F$ a $\Gamma$-species, define $\nicefrac{F}{\Gamma}$, the *quotient species* of $F$ under the action of $\Gamma$, to be the species of $\Gamma$-orbits of $F$-structures.
\[ex:graphcompquot\] Consider $\specname{G}$ as a $\symgp{2}$-species in light of the action defined in \[ex:graphcomp\]. The structures of the quotient species $\nicefrac{\specname{G}}{\symgp{2}}$ are then pairs of complementary graphs. We may choose to interpret each such pair as representing a $2$-partition of the set of vertex pairs of the complete graph (that is, of edges of the complete graph). More natural examples of quotient structures will present themselves in later chapters.
For each label set $A$, let $\quomap{\Gamma} \sbrac{A}: F \sbrac{A} \to \nicefrac{F}{\Gamma} \sbrac{A}$ denote the map sending each $F$-structure over $A$ to its quotient $\nicefrac{F}{\Gamma}$-structure over $A$. Then $\quomap{\Gamma} \sbrac{A}$ is an injection for each $A$, and the requirement that $\Gamma$ acts by natural transformations implies that the induced functor map $\quomap{\Gamma}: F \to \nicefrac{F}{\Gamma}$ is a natural transformation. Thus, the passage from $F$ to $\nicefrac{F}{\Gamma}$ is a species cover in the sense of \[def:specmaptypes\].
A brief exposition of the notion of quotient species may be found in [@bll:species §3.6], and a more thorough exposition (in French) in [@bous:species]. Our motivation, of course, is that combinatorial structures of a given class are often ‘naturally’ identified with orbits of structures of another, larger class under the action of some group. Our goal will be to compute the cycle index of the species $\nicefrac{F}{\Gamma}$ in terms of that of $F$ and information about the $\Gamma$-action, so that enumerative data about the quotient species can be extracted.
As an intermediate step to the computation of the cycle index associated to this quotient species, we associate a cycle index to a $\Gamma$-species $F$ that keeps track of the needed data about the $\Gamma$-action.
\[def:gcycind\] For a $\Gamma$-species $F$, define the $\Gamma$-cycle index $\gci{\Gamma}{F}$ as in [@hend:specfield]: for each $\gamma \in \Gamma$, let $$\gcivars{\Gamma}{F}{\gamma} = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \symgp{n}} \fix \pbrac{\gamma \cdot F \sbrac{\sigma}} p_{\sigma} \label{eq:gcycinddef}$$ with $p_{\sigma}$ as in \[eq:cycinddef\].
We will call such an object (formally a map from $\Gamma$ to the ring $\ringname{Q} \sbrac{\sbrac{p_{1}, p_{2}, \dots}}$ of symmetric functions with rational coefficients in the $p$-basis) a *$\Gamma$-cycle index* even when it is not explicitly the $\Gamma$-cycle index of a $\Gamma$-species, and we will sometimes call $\gcielt{\Gamma}{F}{\gamma}$ the “$\gamma$ term of $\gci{\Gamma}{F}$”. So the coefficients in the power series count the fixed points of the *combined* action of a permutation and the group element $\gamma$. Note that, in particular, the classical (‘ordinary’) cycle index may be recovered as $\ci{F} = \gcielt{\Gamma}{F}{e}$ for any $\Gamma$-species $F$.
The algebraic relationships between ordinary species and their cycle indices generally extend without modification to the $\Gamma$-species context, as long as appropriate allowances are made. The actions on cycle indices of $\Gamma$-species addition and multiplication are exactly as in the ordinary species case considered componentwise:
\[def:gspecsumprod\] For two $\Gamma$-species $F$ and $G$, the $\Gamma$-cycle index of their sum $F + G$ is given by $$\label{eq:gspecsum}
\gcielt{\Gamma}{F + G}{\gamma} = \gcielt{\Gamma}{F}{\gamma} + \gcielt{\Gamma}{G}{\gamma}$$ and the $\Gamma$-cycle index of their product $F \cdot G$ is given by $$\label{eq:gspecprod}
\gcielt{\Gamma}{F \cdot G}{\gamma} = \gcielt{\Gamma}{F}{\gamma} \cdot \gcielt{\Gamma}{G}{\gamma}$$
The action of composition, which in ordinary species corresponds to plethysm of cycle indices, can also be extended:
\[def:gspeccomp\] For two $\Gamma$-species $F$ and $G$, define their *composition* to be the $\Gamma$-species $F \circ G$ with structures given by $\pbrac{F \circ G} \sbrac{A} = \prod_{\pi \in P \pbrac{A}} \pbrac{F \sbrac{\pi} \times \prod_{B \in \pi} G \sbrac{B}}$ where $P \pbrac{A}$ is the set of partitions of $A$ and where $\gamma \in \Gamma$ acts on a $\pbrac{F \circ G}$-structure by acting on the $F$-structure and the $G$-structures independently.
The requirement in \[def:gspecies\] that the action of $\Gamma$ commutes with transport implies that this is well-defined. Informally, for $\Gamma$-species $F$ and $G$, we have defined the composition $F \circ G$ to be the $\Gamma$-species of $F$-structures of $G$-structures, where $\gamma \in \Gamma$ acts on an $\pbrac{F \circ G}$-structure by acting independently on the $F$-structure and each of its associated $G$-structures. A formula similar to that \[thm:speccompci\] requires a definition of the plethysm of $\Gamma$-symmetric functions, here taken from [@hend:specfield §3]:
\[def:gcipleth\] For two $\Gamma$-cycle indices $f$ and $g$, their *plethysm* $f \circ g$ is a $\Gamma$-cycle index defined by $$\pbrac{f \circ g} \pbrac{\gamma} = f \pbrac{\gamma} \pbrac{g \pbrac{\gamma} \pbrac{p_{1}, p_{2}, p_{3}, \dots}, g \pbracs[big]{\gamma^{2}} \pbrac{p_{2}, p_{4}, p_{6}, \dots}, \dots}.
\label{eq:gcipleth}$$
This definition of $\Gamma$-cycle index plethysm is then indeed the correct operation to pair with the composition of $\Gamma$-species:
\[thm:gspeccompci\] If $A$ and $B$ are $\Gamma$-species and $B \pbrac{\varnothing} = \varnothing$, then $$\label{eq:gspeccompci}
\gci{\Gamma}{A \circ B} = \gci{\Gamma}{A} \circ \gci{\Gamma}{B}.$$
Thus, $\Gamma$-species admit the same sorts of ‘nice’ correspondences between structural descriptions (in terms of functorial algebra) and enumerative characterizations (in terms of cycle indices) that ordinary species do.
However, to make use of this theory for enumerative purposes, we also need to be able to pass from the $\Gamma$-cycle index of a $\Gamma$-species to the ordinary cycle index of its associated quotient species under the action of $\Gamma$. This will allow us to adopt a useful strategy: if we can characterize some difficult-to-enumerate combinatorial structure as quotients of more accessible structures, we will be able to apply the full force of species theory to the enumeration of the prequotient structures, *then* pass to the quotient when it is convenient. Exactly this approach will serve as the core of both of the following chapters.
Since we intend to enumerate orbits under a group action, we apply a generalization of Burnside’s Lemma found in [@gessel:laginvspec Lemma 5]:
\[lem:grouporbits\] If $\Gamma$ and $\Delta$ are finite groups and $S$ a set with a $\pbrac{\Gamma \times \Delta}$-action, for any $\delta \in \Delta$ the number of $\Gamma$-orbits fixed by $\delta$ is $\frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} \fix \pbrac{\gamma, \delta}$.
Recall from \[eq:cycinddef\] that, to compute the cycle index of a species, we need to enumerate the fixed points of each $\sigma \in \symgp{n}$. However, to do this in the quotient species $\nicefrac{F}{\Gamma}$ is by definition to count the fixed $\Gamma$-orbits of $\sigma$ in $F$ under commuting actions of $\symgp{n}$ and $\Gamma$ (that is, under an $\pbrac{\symgp{n} \times \Gamma}$-action). Thus, \[lem:grouporbits\] implies the following:
\[thm:qsci\] For a $\Gamma$-species $F$, the ordinary cycle index of the quotient species $\nicefrac{F}{\Gamma}$ is given by $$\label{eq:quotcycind}
\ci{F / \Gamma} = \qgci{\Gamma}{F} \defeq \frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} \gcielt{\Gamma}{F}{\gamma} = \frac{1}{\abs{\Gamma}} \sum_{\substack{n \geq 0 \\ \sigma \in \symgp{n} \\ \gamma \in \Gamma}} \frac{1}{n!} \pbrac{\gamma \cdot F \sbrac{\sigma}} p_{\sigma}.$$ where we define $\qgci{\Gamma}{F} = \frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} \gcielt{\Gamma}{F}{\gamma}$ for future convenience.
Note that this same result on cycle indices is implicit in [@bous:species §2.2.3]. With it, we can compute explicit enumerative data for a quotient species using cycle-index information of the original $\Gamma$-species with respect to the group action, as desired.
Recall from \[thm:ciegf,thm:ciogf\] that the exponential generating function $F \pbrac{x}$ of labeled $F$-structures and the ordinary generating function $\tilde{F} \pbrac{x}$ of unlabeled $F$-structures may both be computed from the cycle index $\ci{F}$ of an ordinary species $F$ by simple substitutions. In the $\Gamma$-species context, we may perform similar substitutions to derive analogous generating functions.
\[thm:gciegf\] The exponential generating function $F_{\gamma} \pbrac{x}$ of labeled $\gamma$-invariant $F$-structures is $$\label{eq:gciegf}
F_{\gamma} \pbrac{x} = \gcieltvars{\Gamma}{F}{\gamma}{x, 0, 0, \dots}.$$
\[thm:gciogf\] The ordinary generating function $\tilde{F}_{\gamma} \pbrac{x}$ of unlabeled $\gamma$-invariant $F$-structures is $$\label{eq:gciogf}
\tilde{F}_{\gamma} \pbrac{x} = \gcieltvars{\Gamma}{F}{\gamma}{x, x^{2}, x^{3}, \dots}.$$
These theorems follow directly from \[eq:ciegf,eq:ciogf\], thinking of $F_{\gamma} \pbrac{x}$ and $\widetilde{F_{\gamma} \pbrac{x}}$ as enumerating the combinatorial class of $F$-structures which are invariant under $\gamma$.
Note that the notion of ‘unlabeled $\gamma$-invariant $F$-structures’ is always well-defined precisely because \[def:gspecies\] requires that the action of $\Gamma$ commutes with transport of structures.
From these results and \[thm:qsci\], we can the conclude:
\[qgciegf\] The exponential generating function $F \pbrac{x}$ of labeled $\nicefrac{F}{\Gamma}$-structures is $$\label{eq:qgciegf}
F \pbrac{x} = \frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} F_{\gamma} \pbrac{x}.$$
Similarly,
\[qgciogf\] The ordinary generating function $\tilde{F} \pbrac{x}$ of unlabeled $\nicefrac{F}{\Gamma}$-structures is $$\label{eq:qgcogf}
\tilde{F} \pbrac{x} = \frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} \tilde{F}_{\gamma} \pbrac{x}.$$
Note also that all of the above extends naturally into the multisort species context. We will use this extensively in \[c:ktrees\]. It also extends naturally to weighted contexts, but we will not apply this extension here.
The species of bipartite blocks {#c:bpblocks}
===============================
Introduction {#s:bpintro}
------------
We first apply the theory of quotient species to the enumeration of bipartite blocks.
\[def:bcgraph\] A *bicolored graph* is a graph $\Gamma$ each vertex of which has been assigned one of two colors (here, black and white) such that each edge connects vertices of different colors. A *bipartite graph* (sometimes called *bicolorable*) is a graph $\Gamma$ which admits such a coloring.
There is an extensive literature about bicolored and bipartite graphs, including enumerative results for bicolored graphs [@har:bicolored], bipartite graphs both allowing [@han:bipartite] and prohibiting [@harprins:bipartite] isolated points, and bipartite blocks [@harrob:bipblocks]. However, this final enumeration was previously completed only in the labeled case. By considering the problem in light of the theory of $\Gamma$-species, we develop a more systematic understanding of the structural relationships between these various classes of graphs, which allows us to enumerate all of them in both labeled and unlabeled settings.
Throughout this chapter, we denote by $\specname{BC}$ the species of bicolored graphs and by $\specname{BP}$ the species of bipartite graphs. The prefix $\specname{C}$ will indicate the connected analogue of such a species.
We are motivated by the graph-theoretic fact that each *connected* bipartite graph may be identified with exactly two bicolored graphs which are color-dual. In other words, a connected bipartite graph is (by definition or by easy exercise, depending on your approach) an orbit of connected bicolored graphs under the action of $\symgp{2}$ where the nontrivial element $\tau$ reverses all vertex colors. We will hereafter treat all the various species of bicolored graphs as $\symgp{2}$-species with respect to this action and use the theory developed in \[s:quot\] to pass to bipartite graphs.
Although the theory of multisort species presented in \[s:mult\] is in general well-suited to the study of colored graphs, we will not need it here. The restrictions that vertex colorings place on automorphisms of bicolored graphs are simple enough that we can deal with them directly.
Bicolored graphs {#s:bcgraph}
----------------
We begin our investigation by directly computing the $\symgp{2}$-cycle index for the species $\specname{BC}$ of bicolored graphs with the color-reversing $\symgp{2}$-action described previously. We will then use various methods from the species algebra of \[c:species\] to pass to various other species.
### Computing $\gcielt{\symgp{2}}{\specname{BC}}{e}$ {#ss:ecibc}
We construct the cycle index for the species $\specname{BC}$ of bicolored graphs in the classical way, which in light of our $\symgp{2}$-action will give $\gcielt{\symgp{2}}{\specname{BC}}{e}$.
Recall the formula for the cycle index of a $\Gamma$-species in \[eq:gcycinddef\]: $$\gcielt{\Gamma}{F}{\gamma} = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \symgp{n}} \fix \pbrac{\gamma \cdot F \sbrac{\sigma}} p_{\sigma}.$$ Thus, for each $n > 0$ and each permutation $\pi \in \symgp{n}$, we must count bicolored graphs on $\sbrac{n}$ for which $\pi$ is a color-preserving automorphism. To simplify some future calculations, we omit empty graphs and define $\specname{BC} \sbrac{\varnothing} = \varnothing$. We note that the *number* of such graphs in fact depends only on the cycle type $\lambda \vdash n$ of the permutation $\pi$, so we can use the cycle index formula in \[eq:cycinddefpart\] interpreted as a $\Gamma$-cycle index identity.
Fix some $n \geq 0$ and let $\lambda \vdash n$. We wish to count bicolored graphs for which a chosen permutation $\pi$ of cycle type $\lambda$ is a color-preserving automorphism. Each cycle of the permutation must correspond to a monochromatic subset of the vertices, so we may construct graphs by drawing bicolored edges into a given colored vertex set. If we draw some particular bicolored edge, we must also draw every other edge in its orbit under $\pi$ if $\pi$ is to be an automorphism of the graph. Moreover, every bicolored graph for which $\pi$ is an automorphism may be constructed in this way Therefore, we direct our attention first to counting these edge orbits for a fixed coloring; we will then count colorings with respect to these results to get our total cycle index.
Consider an edge connecting two cycles of lengths $m$ and $n$; the length of its orbit under the permutation is $\lcm \pbrac{m, n}$, so the number of such orbits of edges between these two cycles is $mn / \lcm \pbrac{m, n} = \gcd \pbrac{m, n}$. For an example in the case $m = 4, n = 2$, see \[fig:exbcecycle\]. The number of orbits for a fixed coloring is then $\sum \gcd \pbrac{m, n}$ where the sum is over the multiset of all cycle lengths $m$ of white cycles and $n$ of black cycles in the permutation $\pi$. We may then construct any possible graph fixed by our permutation by making a choice of a subset of these cycles to fill with edges, so the total number of such graphs is $\prod 2^{\gcd \pbrac{m, n}}$ for a fixed coloring.
(a1)(b1)
We now turn our attention to the possible colorings of the graph which are compatible with a permutation of specified cycle type $\lambda$. We split our partition into two subpartitions, writing $\lambda = \mu \cup \nu$, where partitions are treated as multisets and $\cup$ is the multiset union, and designate $\mu$ to represent the white cycles and $\nu$ the black. Then the total number of graphs fixed by such a permutation with a specified decomposition is $$\fix \pbrac{\mu, \nu} = \prod_{\substack{i \in \mu \\ j \in \nu}} 2^{\gcd \pbrac{i, j}}$$ where the product is over the elements of $\mu$ and $\lambda$ taken as multisets. However, since $\mu$ and $\nu$ represent white and black cycles respectively, it is important to distinguish *which* cycles of $\lambda$ are taken into each. The $\lambda_{i}$ $i$-cycles of $\lambda$ can be distributed into $\mu$ and $\nu$ in $\binom{\lambda_{i}}{\mu_{i}} = \lambda_{i}! / \pbrac{\mu_{i}! \nu_{i}!}$ ways, so in total there are $\prod_{i} \lambda_{i}! / \pbrac{\mu_{i}! \nu_{i}!} = z_{\lambda} / \pbrac{z_{\mu} z_{\nu}}$ decompositions. Thus, $$\fix \pbrac{\lambda} = \frac{z_{\lambda}}{z_{\mu} z_{\nu}} \fix \pbrac{\mu, \nu} = \sum_{\mu \cup \nu = \lambda} \frac{z_{\lambda}}{z_{\mu} z_{\nu}} \prod_{\substack{i \in \mu \\ j \in \nu}} 2^{\gcd \pbrac{i, j}}.$$ Therefore we conclude:
$$\label{eq:ecibc}
\gcielt{\symgp{2}}{\specname{BC}}{e} = \sum_{n > 0} \sum_{\substack{\mu, \nu \\ \mu \cup \nu \vdash n}} \frac{p_{\mu \cup \nu}}{z_{\mu} z_{\nu}} \prod_{i, j} 2^{\gcd \pbrac{\mu_{i}, \nu_{j}}}$$
Explicit formulas for the generating function for unlabeled bicolored graphs were obtained in [@har:bicolored] using conventional Pólya-theoretic methods. Conceptually, this enumeration in fact largely mirrors our own. Harary uses the algebra of the classical cycle index of the ‘line group[^3]’ of the complete bicolored graph of which any given bicolored graph is a spanning subgraph. He then enumerates orbits of edges under these groups using the Pólya enumeration theorem. This is clearly analogous to our procedure, which enumerates the orbits of edges under each specific permutation of vertices.
### Calculating $\gcielt{\symgp{2}}{\specname{BC}}{\tau}$ {#ss:tcibc}
Recall that the nontrivial element of $\tau \in \symgp{2}$ acts on bicolored graphs by reversing all colors.
We again consider the cycles in the vertex set $\sbrac{n}$ induced by a permutation $\pi \in \symgp{n}$ and use the partition $\lambda$ corresponding to the cycle type of $\pi$ for bookkeeping. We then wish to count bicolored graphs on $\sbrac{n}$ for which $\tau \cdot \pi$ is an automorphism, which is to say that $\pi$ itself is a color-*reversing* automorphism. Once again, the number of bicolored graphs for which $\pi$ is a color-reversing automorphism is in fact dependent only on the cycle type $\lambda$. Each cycle of vertices must be color-alternating and hence of even length, so our partition $\lambda$ must have only even parts. Once this condition is satisfied, edges may be drawn either within a single cycle or between two cycles, and as before if we draw in any edge we must draw in its entire orbit under $\pi$ (since $\pi$ is to be an automorphism of the underlying graph). Moreover, all graphs for which $\pi$ is a color-reversing automorphism and with a fixed coloring may be constructed in this way, so it suffices to count such edge orbits and then consider how colorings may be assigned.
Consider a cycle of length $2n$; we hereafter describe such a cycle as having *semilength* $n$. There are exactly $n^{2}$ possible white-black edges in such a cycle. If $n$ is odd, diametrically opposed vertices have opposite colors, so we can have an edge of length $l = n$ (in the sense of connecting two vertices which are $l$ steps apart in the cycle), and in such a case the orbit length is exactly $n$ and there is exactly one orbit. See \[fig:exbctincycd\] for an example of this case. However, if $n$ is odd but $l \neq n$, the orbit length is $2n$, so the number of such orbits is $\frac{n^{2} - n}{2n}$. Hence, the total number of orbits for $n$ odd is $\frac{n^2 + n}{2n} = \ceil{\frac{n}{2}}$. Similarly, if $n$ is even, all orbits are of length $2n$, so the total number of orbits is $\frac{n^{2}}{2n} = \frac{n}{2} = \ceil{\frac{n}{2}}$ also. See \[fig:exbctincyce\] for an example of each of these cases.
Now consider an edge to be drawn between two cycles of semilengths $m$ and $n$. The total number of possible white-black edges is $2mn$, each of which has an orbit length of $\lcm \pbrac{2m, 2n} = 2 \lcm \pbrac{m, n}$. Hence, the total number of orbits is $2mn / \pbrac{2 \lcm \pbrac{m, n}} = \gcd \pbrac{m, n}$.
(a0)(b1)
All together, then, the number of orbits for a fixed coloring of a permutation of cycle type $2 \lambda$ (denoting the partition obtained by doubling every part of $\lambda$) is $\sum_{i} \ceil{\frac{\lambda_{i}}{2}} + \sum_{i < j} \gcd \pbrac{\lambda_{i}, \lambda_{j}}$. All valid bicolored graphs for a fixed coloring for which $\pi$ is a color-preserving automorphism may be obtained uniquely by making some choice of a subset of this collection of orbits, just as in \[ss:ecibc\]. Thus, the total number of possible graphs for a given vertex coloring is $$\prod_{i} 2^{\ceil{\frac{\lambda_{i}}{2}}} \prod_{i < j} 2^{\gcd \pbrac{\lambda_{i}, \lambda_{j}}},$$ which we note is independent of the choice of coloring. For a partition $2\lambda$ with $l \pbrac{\lambda}$ cycles, there are then $2^{l \pbrac{\lambda}}$ colorings compatible with our requirement that each cycle is color-alternating, which we multiply by the previous to obtain the total number of graphs for all permutations $\pi$ with cycle type $2 \lambda$. Therefore we conclude:
$$\label{eq:tcibc}
\gcielt{\symgp{2}}{\specname{BC}}{\tau} = \sum_{\substack{n > 0 \\ \text{$n$ even}}} \sum_{\lambda \vdash \frac{n}{2}} 2^{l \pbrac{\lambda}} \frac{p_{2 \lambda}}{z_{2 \lambda}} \prod_{i} 2^{\ceil{\frac{\lambda_{i}}{2}}} \prod_{i < j} 2^{\gcd \pbrac{\lambda_{i}, \lambda_{j}}}$$
Connected bicolored graphs {#s:cbc}
--------------------------
As noted in the introduction of this section, we may pass from bicolored to bipartite graphs by taking a quotient under the color-reversing action of $\symgp{2}$ only in the connected case. Thus, we must pass from the species $\specname{BC}$ to the species $\specname{CBC}$ of connected bicolored graphs to continue. It is a standard principle of graph enumeration that a graph may be decomposed uniquely into (and thus species-theoretically identified with) the set of its connected components. We must, of course, require that the component structures are nonempty to ensure that the construction is well-defined, as discussed in \[s:specalg\]. This same relationship holds in the case of bicolored graphs. Thus, the species $\specname{BC}$ of nonempty bicolored graphs is the composition of the species $\specname{CBC}$ of nonempty connected bicolored graphs into the species $\specname{E}^{+} = \specname{E} - 1$ of nonempty sets: $$\specname{BC} = \specname{E}^{+} \circ \specname{CBC} \label{eq:bcdecomp}$$
Reversing the colors of a bicolored graph is done simply by reversing the colors of each of its connected components independently; thus, once we trivially extend the species $\specname{E}^{+}$ to an $\symgp{2}$-species by applying the trivial action, \[eq:bcdecomp\] holds as an identity of $\symgp{2}$-species for the color-reversing $\symgp{2}$-action described previously.
To use the decomposition in \[eq:bcdecomp\] to derive the $\symgp{2}$-cycle index for $\specname{CBC}$, we must invert the $\symgp{2}$-species composition into $\specname{E}^{+}$. In the context of the theory of virtual species, this is possible; we write $\con := \pbrac{\specname{E} - 1}^{\abrac{-1}}$ to denote this virtual species. We can derive from [@bll:species §2.5, eq. (58c)] that its cycle index is $$\label{eq:zgamma}
\ci{\con} = \sum_{k \geq 1} \frac{\mu \pbrac{k}}{k} \log \pbrac{1 + p_{k}}$$ where $\mu$ is the Möbius function. We can then rewrite \[eq:bcdecomp\] as $$\specname{CBC} = \con \circ \specname{BC}$$ It then follows immediately from \[thm:gspeccompci\] that
$$\gci{\symgp{2}}{\specname{CBC}} = \ci{\con} \circ \gci{\symgp{2}}{\specname{BC}} \label{eq:zcbcdecomp}$$
Bipartite graphs {#s:bp}
----------------
As we previously observed, connected bipartite graphs are naturally identified with orbits of connected bicolored graphs under the color-reversing action of $\symgp{2}$. Thus, $$\specname{CBP} = \faktor{\specname{CBC}}{\symgp{2}}.$$ By application of \[thm:qsci\], we can then directly compute the cycle index of $\specname{CBP}$ in terms of previous results:
$$\ci{\specname{CBP}} = \qgci{\symgp{2}}{\specname{CBC}} = \frac{1}{2} \pbrac{\gcielt{\symgp{2}}{\specname{CBC}}{e} + \gcielt{\symgp{2}}{\specname{CBC}}{\tau}}.$$
Finally, to reach a result for the general bipartite case, we return to the graph-theoretic composition relationship previously considered in \[s:cbc\]: $$\specname{BP} = \specname{E} \circ \specname{CBP}.$$
This time, we need not invert the composition, so the cycle-index calculation is simple:
$$\ci{\specname{BP}} = \ci{\specname{E}} \circ \ci{\specname{CBP}}.$$
A generating function for labeled bipartite graphs was obtained first in [@harprins:bipartite] and later in [@han:bipartite]; the latter uses Pólya-theoretic methods to calculate the cycle index of what in modern terminology would be the species of edge-labeled complete bipartite graphs.
Nonseparable graphs {#s:nbp}
-------------------
We now turn our attention to the notions of block decomposition and nonseparable graphs. A graph is said to be *nonseparable* if it is vertex-$2$-connected (that is, if there exists no vertex whose removal disconnects the graph); every connected graph then has a canonical ‘decomposition’[^4] into maximal nonseparable subgraphs, often shortened to *blocks*. In the spirit of our previous notation, we we will denote by $\specname{NBP}$ the species of nonseparable bipartite graphs, our object of study.
The basic principles of block enumeration in terms of automorphisms and cycle indices of permutation groups were first identified and exploited in [@rob:nonsep]. In [@bll:species §4.2], a theory relating a specified species $B$ of nonseparable graphs to the species $C_{B}$ of connected graphs whose blocks are in $B$ is developed using similar principles. It is apparent that the class of nonseparable bipartite graphs is itself exactly the class of blocks that occur in block decompositions of connected bipartite graphs; hence, we apply that theory here to study the species $\specname{NBP}$. From [@bll:species eq. 4.2.27] we obtain
\[eq:nbpexp\] $$\label{eq:nbpexpmain}
\specname{NBP} = \specname{CBP} \pbrac{\specname{CBP}^{\bullet \abrac{-1}}} + X \cdot \deriv{\specname{NBP}} - X,$$ where by [@bll:species 4.2.26(a)] we have $$\label{eq:nbpexpsub}
\deriv{\specname{NBP}} = \con \pbrac{\frac{X}{\specname{CBP}^{\bullet \abrac{-1}}}}.$$
We have already calculated the cycle index for the species $\specname{CBP}$, so the calculation of the cycle index of $\specname{NBP}$ is now simply a matter of algebraic expansion.
A generating function for labeled bipartite blocks was given in [@harrob:bipblocks], where their analogue of \[eq:nbpexp\] for the labeled exponential generating function for blocks comes from [@forduhl:combprob1]. However, we could locate no corresponding unlabeled enumeration in the literature. The numbers of labeled and unlabeled nonseparable bipartite graphs for $n \leq 10$ as calculated using our method are given in \[tab:bpblocks\].
The species of $k$-trees {#c:ktrees}
========================
Introduction {#s:intro}
------------
### $k$-trees {#ss:ktrees}
Trees and their generalizations have played an important role in the literature of combinatorial graph theory throughout its history. The multi-dimensional generalization to so-called ‘$k$-trees’ has proved to be particularly fertile ground for both research problems and applications.
The class $\kt{k}$ of $k$-trees (for $k \in \ringname{N}$) may be defined recursively:
\[def:ktree\] The complete graph on $k$ vertices ($K_{k}$) is a $k$-tree, and any graph formed by adding a single vertex to a $k$-tree and connecting that vertex by edges to some existing $k$-clique (that is, induced $k$-complete subgraph) of that $k$-tree is a $k$-tree.
The graph-theoretic notion of $k$-trees was first introduced in 1968 in [@harpalm:acycsimp]; vertex-labeled $k$-trees were quickly enumerated in the following year in both [@moon:lktrees] and [@beinpipp:lktrees]. The special case $k=2$ has been especially thoroughly studied; enumerations are available in the literature for edge- and triangle-labeled $2$-trees in [@palm:l2trees], for plane $2$-trees in [@palmread:p2trees], and for unlabeled $2$-trees in [@harpalm:acycsimp] and [@harpalm:graphenum]. In 2001, the theory of species was brought to bear on $2$-trees in [@gessel:spec2trees], resulting in more explicit formulas for the enumeration of unlabeled $2$-trees. An extensive literature on other properties of $k$-trees and their applications has also emerged; Beineke and Pippert claim in [@beinpipp:multidim] that “\[t\]here are now over 100 papers on various aspects of $k$-trees”. However, no general enumeration of unlabeled $k$-trees appears in the literature to date.
To begin, we establish two definitions for substructures of $k$-trees which we will use extensively in our analysis.
\[def:hedfront\] A *hedron* of a $k$-tree is a $\pbrac{k+1}$-clique and a *front* is a $k$-clique.
We will frequently describe $k$-trees as assemblages of hedra attached along their fronts rather than using explicit graph-theoretic descriptions in terms of edges and vertices, keeping in mind that the structure of interest is graph-theoretic and not geometric. The recursive addition of a single vertex and its connection by edges to an existing $k$-clique in \[def:ktree\] is then interpreted as the attachment of a hedron to an existing one along some front, identifying the $k$ vertices they have in common. The analogy to the recursive definition of conventional trees is clear, and in fact the class $\mathfrak{a}$ of trees may be recovered by setting $k = 1$. For higher $k$, the structures formed are still distinctively tree-like; for example, $2$-trees are formed by gluing triangles together along their edges without forming loops of triangles (see \[fig:ex2tree\]), while $3$-trees are formed by gluing tetrahedra together along their triangular faces without forming loops of tetrahedra.
(a)[d]{} (a)(d)
(d)[c]{} (a)(c) (d)(c)
(d)[b]{} (a)(b) (c)(b)
(a)[f]{} (a)(f) (c)(f)
(f)[e]{} (f)(e) (c)(e)
In graph-theoretic contexts, it is conventional to label graphs on their vertices and possibly their edges. However, for our purposes, it will be more convenient to label hedra and fronts. Throughout, we will treat the species $\kt{k}$ of $k$-trees as a two-sort species, with $X$-labels on the hedra and $Y$-labels on their fronts; in diagrams, we will generally use capital letters for the hedron-labels and positive integers for the front-labels (see \[fig:exlab2tree\]).
(a)[d]{} (a)(d)
(d)[c]{} (a)(c) (d)(c)
at (barycentric cs:a=1,d=1,c=1) [B]{};
(d)[b]{} (a)(b) (c)(b)
at (barycentric cs:b=1,d=1) [D]{};
(a)[f]{} (a)(f) (c)(f)
at (barycentric cs:a=1,c=1,f=1) [C]{};
(f)[e]{} (f)(e) (c)(e)
at (barycentric cs:f=1,c=1,e=1) [A]{};
The dissymmetry theorem for $k$-trees {#s:dissymk}
-------------------------------------
Studies of tree-like structures—especially those explicitly informed by the theory of species—often feature decompositions based on *dissymmetry*, which allow enumerations of unrooted structures to be recharacterized in terms of rooted structures. For example, as seen in [@bll:species §4.1], the species $\mathfrak{a}$ of trees and $\specname{A} = \pointed{\mathfrak{a}}$ of rooted trees are related by the equation $$\specname{A} + \specname{E}_{2} \pbrac{\specname{A}} = \mathfrak{a} + \specname{A}^{2}$$ where the proof hinges on a recursive structural decomposition of trees. In this case, the species $\specname{A}$ is relatively easy to characterize explicitly, so this equation serves to characterize the species $\mathfrak{a}$, which would be difficult to do directly.
A similar theorem holds for $k$-trees.
\[thm:dissymk\] The species $\ktx{k}$ and $\kty{k}$ of $k$-trees rooted at hedra and fronts respectively, $\ktxy{k}$ of $k$-trees rooted at a hedron with a designated front, and $\kt{k}$ of unrooted $k$-trees are related by the equation $$\label{eq:dissymk}
\ktx{k} + \kty{k} = \kt{k} + \ktxy{k}$$ as an isomorphism of species.
We give a bijective, natural map from $\pbrac{\ktx{k} + \kty{k}}$-structures on the left side to $\pbrac{\kt{k} + \ktxy{k}}$-structures on the right side. Define a *$k$-path* in a $k$-tree to be a non-self-intersecting sequence of consecutively adjacent hedra and fronts, and define the *length* of a $k$-path to be the total number of hedra and fronts along it. Note that the ends of every maximal $k$-path in a $k$-tree are fronts. It is easily verified, as in [@kob:ktlogspace], that every $k$-tree has a unique *center* clique (either a hedron or a front) which is the midpoint of every longest $k$-path (or, equivalently, has the greatest $k$-eccentricity, defined appropriately).
An $\pbrac{\ktx{k} + \kty{k}}$-structure on the left-hand side of the equation is a $k$-tree $T$ rooted at some clique $c$, which is either a hedron or a front. Suppose that $c$ is the center of $T$. We then map $T$ to its unrooted equivalent in $\kt{k}$ on the right-hand side. This map is a natural bijection from its preimage, the set of $k$-trees rooted at their centers, to $\kt{k}$, the set of unrooted $k$-trees.
Now suppose that the root clique $c$ of the $k$-tree $T$ is *not* the center, which we denote $C$. Identify the clique $c'$ which is adjacent to $c$ along the $k$-path from $c$ to $C$. We then map the $k$-tree $T$ rooted at the clique $c$ to the same tree $T$ rooted at *both* $c$ and its neighbor $c'$. This map is also a natural bijection, in this case from the set of $k$-trees rooted at vertices which are *not* their centers to the set $\ktxy{k}$ of $k$-trees rooted at an adjacent hedron-front pair.
The combination of these two maps then gives the desired isomorphism of species in \[eq:dissymk\].
In general we will reformulate the dissymmetry theorem as follows:
\[cor:dissymkreform\] For the various forms of the species $\kt{k}$ as above, we have $$\label{eq:dissymkreform}
\kt{k} = \ktx{k} + \kty{k} - \ktxy{k}.$$ as an isomorphism of ordinary species.
This species subtraction is well-defined in the sense of \[def:specdif\], since the species $\ktxy{k}$ embeds in the species $\ktx{k} + \kty{k}$ by the centering map described in the proof of \[thm:dissymk\]. Essentially, \[eq:dissymkreform\] identifies each unrooted $k$-tree with itself rooted at its center simplex.
and the consequent \[eq:dissymkreform\] allow us to reframe enumerative questions about generic $k$-trees in terms of questions about $k$-trees rooted in various ways. However, the rich internal symmetries of large cliques obstruct direct analysis of these rooted structures. We need to break these symmetries to proceed.
Coherently-oriented $k$-trees
-----------------------------
### Symmetry-breaking {#ss:symbreak}
In the case of the species $\specname{A} = \pointed{\kt{1}}$ of rooted trees, we may obtain a simple recursive functional equation [@bll:species §1, eq. (9)]: $$\label{eq:rtrees}
\specname{A} = X \cdot \specname{E} \pbrac{\specname{A}}.$$ This completely characterizes the combinatorial structure of the class of trees.
However, in the more general case of $k$-trees, no such simple relationship obtains; attached to a given hedron is a collection of sets of hedra (one such set per front), but simply specifying which fronts to attach to which does not fully specify the attachings, and the structure of that collection of sets is complex. We will break this symmetry by adding additional structure which we can later remove using the theory of quotient species.
\[def:mirrorfronts\] Let $h_{1}$ and $h_{2}$ be two hedra joined at a front $f$, hereafter said to be *adjacent*. Each other front of one of the hedra shares $k-1$ vertices with $f$; we say that two fronts $f_{1}$ of $h_{1}$ and $f_{2}$ of $h_{2}$ are *mirror with respect to $f$* if these shared vertices are the same, or equivalently if $f_{1} \cap f = f_{2} \cap f$.
\[obs:mirrorfronts\] Let $T$ be a coherently-oriented $k$-tree with two hedra $h_{1}$ and $h_{2}$ joined at a front $f$. Then there is exactly one front of $h_{2}$ mirror to each front of $h_{1}$ with respect to their shared front $f$.
\[def:coktree\] Define an *orientation* of a hedron to be a cyclic ordering of the set of its fronts and an *orientation* of a $k$-tree to be a choice of orientation for each of its hedra. If two oriented hedra share a front, their orientations are *compatible* if they correspond under the mirror bijection. Then an orientation of a $k$-tree is *coherent* if every pair of adjacent hedra is compatibly-oriented.
See \[fig:exco2tree\] for an example. Note that every $k$-tree admits many coherent orientations—any one hedron of the $k$-tree may be oriented freely, and a unique orientation of the whole $k$-tree will result from each choice of such an orientation of one hedron. We will denote by $\ktco{k}$ the species of coherently-oriented $k$-trees.
By shifting from the general $k$-tree setting to that of coherently-oriented $k$-trees, we break the symmetry described above. If we can now establish a group action on $\ktco{k}$ whose orbits are generic $k$-trees we can use the theory of quotient species to extract the generic species $\kt{k}$. First, however, we describe an encoding procedure which will make future work more convenient.
(a)[d]{} (a)(d)
(d)[c]{} (a)(c) (d)(c)
\(B) at (barycentric cs:a=1,d=1.25,c=1); at (B) [B]{}; ;
(d)[b]{} (a)(b) (c)(b)
\(D) at (barycentric cs:b=1,d=1.25); at (D) [D]{}; ;
(a)[f]{} (a)(f) (c)(f)
\(C) at (barycentric cs:a=1,c=1,f=1.25); at (C) [C]{}; ;
(f)[e]{} (f)(e) (c)(e)
\(A) at (barycentric cs:f=1,c=1,e=1.75); at (A) [A]{}; ;
### Bicolored tree encoding {#ss:bctree}
Although $k$-trees are graphs (and hence made up simply of edges and vertices), their structure is more conveniently described in terms of their simplicial structure of hedra and fronts. Indeed, if each hedron has an orientation of its faces and we choose in advance which hedra to attach to which by what fronts, the requirement that the resulting $k$-tree be coherently oriented is strong enough to characterize the attaching completely. We thus pass from coherently-oriented $k$-trees to a surrogate structure which exposes the salient features of this attaching structure more clearly—structured bicolored trees in the spirit of the $R, S$-enriched bicolored trees of [@bll:species §3.2].
A $\pbrac{\specname{C}_{k+1}, \specname{E}}$-enriched bicolored tree is a bicolored tree each black vertex of which carries a $\specname{C}_{k+1}$-structure (that is, a cyclic ordering on $k+1$ elements) on its white neighbors. (The $\specname{E}$-structure on the black neighbors of each white vertex is already implicit in the bicolored tree itself.) For later convenience, we will sometimes call such objects *$k$-coding trees*, and we will denote by $\ct{k}$ the species of such $k$-coding trees.
We now define a map $\beta: \ktco{k} \sbrac{n} \to \ct{k} \sbrac{n}$. For a given coherently-oriented $k$-tree $T$ with $n$ hedra:
- For every hedron of $T$ construct a black vertex and for every front a white vertex, assigning labels appropriately.
- For every black-white vertex pair, construct a connecting edge if the white vertex represents a front of the hedron represented by the black vertex.
- Finally, enrich the collection of neighbors of each black vertex with a $\specname{C}_{k+1}$-structure inherited directly from the orientation of the $k$-tree $T$.
The resulting object $\beta \pbrac{T}$ is clearly a $k$-coding tree with $n$ black vertices.
We can recover a $T$ from $\beta \pbrac{T}$ by following the reverse procedure. For an example, see \[fig:exbctree\], which shows the $2$-coding tree associated to the coherently-oriented $2$-tree of \[fig:exco2tree\]. Note that, for clarity, we have rendered the black vertices (corresponding to hedra) with squares.
(4)[B]{} ; (4)(B)
(B)[5]{} (B)(5)
(B)[2]{} (B)(2)
(4)[D]{} ; (4)(D)
(D)[1]{} (D)(1)
(D)[6]{} (D)(6)
(4)[C]{} ; (4)(C)
(C)[9]{} (C)(9)
(C)[3]{} (C)(3)
(3)[A]{} ; (3)(A)
(A)[8]{} (A)(8)
(A)[7]{} (A)(7)
\[thm:bctreeenc\] The map $\beta$ induces an isomorphism of species $\ktco{k} \simeq \ct{k}$.
It is clear that $\beta$ sends each coherently-oriented $k$-tree to a unique $k$-coding tree, and that this map commutes with permutations on the label sets (and thus is categorically natural). To show that $\beta$ induces a species isomorphism, then, we need only show that $\beta$ is a surjection onto $\ct{k} \sbrac{n}$ for each $n$. Throughout, we will say ‘$F$ and $G$ have contact of order $n$’ when the restrictions $F_{\leq n}$ and $G_{\leq n}$ of the species $F$ and $G$ to label sets of cardinality at most $n$ are naturally isomorphic.
First, we note that there are exactly $k!$ coherently-oriented $k$-trees with one hedron—one for each cyclic ordering of the $k+1$ front labels. There are also $k!$ coding trees with one black vertex, and the encoding $\beta$ is clearly a natural bijection between these two sets. Thus, the species $\ktco{k}$ of coherently-oriented $k$-trees and $\ct{k}$ of $k$-coding trees have contact of order $1$.
Now, by way of induction, suppose $\ktco{k}$ and $\ct{k}$ have contact of order $n \geq 1$. Let $C$ be a $k$-coding tree with $n+1$ black vertices. Then let $C_{1}$ and $C_{2}$ be two distinct sub-$k$-coding trees of $C$, each obtained from $C$ by removing one black node which has only one white neighbor which is not a leaf. Then, by hypothesis, there exist coherently-oriented $k$-trees $T_{1}$ and $T_{2}$ with $n$ hedra such that $\beta \pbrac{T_{1}} = C_{1}$ and $\beta \pbrac{T_{2}} = C_{2}$. Moreover, $\beta \pbrac{T_{1} \cap T_{2}} = \beta \pbrac{T_{1}} \cap \beta \pbrac{T_{2}}$, and this $k$-coding tree has $n-1$ black vertices, so $T_{1} \cap T_{2}$ has $n-1$ hedra. Thus, $T = T_{1} \cup T_{2}$ is a coherently-oriented $k$-tree with $n+1$ black hedra, and $\beta \pbrac{T} = C$ as desired. Thus, $\beta^{-1} \pbrac{\beta \pbrac{T_{1}} \cup \beta \pbrac{T_{2}}} = T_{1} \cup T_{2} = T$, and hence $\ktco{k}$ and $\ct{k}$ have contact of order $n+1$.
Thus, $\ktco{k}$ and $\ct{k}$ are isomorphic as species; however, $k$-coding trees are much simpler than coherently-oriented $k$-trees as graphs. Moreover, $k$-coding trees are doubly-enriched bicolored trees as in [@bll:species §3.2], for which the authors of that text develop a system of functional equations which fully characterizes the cycle index of such a species. We thus will proceed in the following sections with a study of the species $\ct{k}$, then lift our results to the $k$-tree context.
### Functional decomposition of $k$-coding trees {#ss:codecomp}
With the encoding $\beta: \ktco{k} \to \ct{k}$, we now have direct graph-theoretic access to the attaching structure of coherently-oriented $k$-trees. We therefore turn our attention to the $k$-coding trees themselves to produce a recursive decomposition. As with $k$-trees, we will study rooted versions of the species $\ct{k}$ of $k$-coding trees first, then use dissymmetry to apply the results to unrooted enumeration.
Let $\ctx{k}$ denote the species of $k$-coding trees rooted at black vertices, $\cty{k}$ denote the species of $k$-coding trees rooted at white vertices, and $\ctxy{k}$ denote the species of $k$-coding trees rooted at edges (that is, at adjacent black-white pairs). By construction, a $\ctx{k}$-structure consists of a single $X$-label and a cyclically-ordered $\pbrac{k+1}$-set of $\cty{k}$-structures. See \[fig:ctxconst\] for an example of this construction.
(root) at (0,0) [$X$]{}; ;
(180/:1) node [$\specname{C}_{\numfronts}$]{};
iin [0, ..., ]{}
(childi) at (:3) [$Y$]{}; (root) – (childi); (childi) ++(+90:1) node [$\cty{\kval}$]{};
(childi) ++(:1cm) ++(180+-:2cm) arc (180+-:180++:2cm);
(childi) – ++(:2) node \[rotate=,fill=white\] [$\cdots$]{}; (childi) – ++(+:2); (childi) – ++(-:2);
Similarly, a $\cty{k}$-structure essentially consists of a single $Y$-label and a (possibly empty) set of $\ctx{k}$-structures, but with some modification. Every white neighbor of the black root of a $\ctx{k}$-structure is labeled in the construction above, but the white parent of a $\ctx{k}$-structure in this recursive decomposition is already labeled. Thus, the structure around a black vertex which is a child of a white vertex consists of an $X$ label and a linearly-ordered $k$-set of $\cty{k}$-structures. Thus, a $\cty{k}$-structure consists of a $Y$-label and a set of pairs of an $X$ label and an $\specname{L}_{k}$-structure of $\cty{k}$-structures. We note here for conceptual consistency that in fact $\specname{L}_{k} = \deriv{\specname{C}}_{k+1}$ for $\specname{L}$ the species of linear orders and $\specname{C}$ the species of cyclic orders and that $\deriv{\specname{E}} = \specname{E}$ for $\specname{E}$ the species of sets; readers familiar with the $R, S$-enriched bicolored trees of [@bll:species §3.2] will recognize echoes of their decomposition in these facts.
Finally, a $\ctxy{k}$-structure is simply an $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$-structure as described above (corresponding to the black vertex) together with a $\cty{k}$-structure (corresponding to the white vertex). For reasons that will become clear later, we note that we can incorporate the root white vertex into the linear order by making it last, thus representing a $\ctxy{k}$-structure instead as an $X \cdot \specname{L}_{k+1} \pbracs[big]{\cty{k}}$-structure. See \[fig:ctxyconst\] for an example of this construction.
(root) at (0,0) [$X$]{}; (root) ++(2\*:1cm) arc (2\*:360:1cm); (root) – ++(0:3) \[ultra thick\] node \[fill=white\] ;
(180/:1) node [$\specname{L}_{\numfronts}$]{};
iin [0, ..., ]{}
(childi) at (:3) [$Y$]{}; (root) – (childi); (childi) ++(+90:1) node [$\cty{\kval}$]{};
(childi) ++(:1cm) ++(180+-:2cm) arc (180+-:180++:2cm);
(childi) – ++(:2) node \[rotate=,fill=white\] [$\cdots$]{}; (childi) – ++(+:2); (childi) – ++(-:2);
The various species of rooted $k$-coding trees are therefore related by a system of functional equations:
\[obs:funcdecompct\] For the (ordinary) species $\ctx{k}$ of $X$-rooted $k$-coding trees, $\cty{k}$ of $Y$-rooted $k$-coding trees, and $\ctxy{k}$ of edge-rooted $k$-coding trees, we have the functional relationships
\[eq:ctfunc\] $$\begin{aligned}
\ctx{k} &= X \cdot \specname{C}_{k+1} \pbracs[big]{\cty{k}} \label{eq:ctxfunc} \\
\cty{k} &= Y \cdot \specname{E} \pbrac{X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}} \label{eq:ctyfunc} \\
\ctxy{k} &= \cty{k} \cdot X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}} = X \cdot \specname{L}_{k+1} \pbracs[big]{\cty{k}} \label{eq:ctxyfunc}
\end{aligned}$$
as isomorphisms of ordinary species.
However, a recursive characterization of the various ordinary species of $k$-coding trees is insufficient to characterize the species of $k$-trees itself, since $k$-coding trees represent $k$-trees with coherent orientations.
Generic $k$-trees {#s:genkt}
-----------------
To remove the additional structure of coherent orientation imposed on $k$-trees before their conversion to $k$-coding trees, we now apply the theory of $\Gamma$-species developed in \[s:quot\]. In [@gessel:spec2trees], the orientation-reversing action of $\symgp{2}$ on $\cyc_{\sbrac{3}}$ is exploited to study $2$-trees species-theoretically. We might hope to develop an analogous group action under which general $k$-trees are naturally identified with orbits of coherently-oriented $k$-trees under an action of $\symgp{k}$. Unfortunately:
\[prop:notransac\] For $k \geq 3$, no transitive action of any group on the set $\cyc_{\sbrac{k+1}}$ of cyclic orders on $\sbrac{k+1}$ commutes with the action of $\symgp{k+1}$ that permutes labels.
We represent the elements of $\cyc_{\sbrac{k+1}}$ as cyclic permutations on the alphabet $\sbrac{k+1}$; then the action of $\symgp{k+1}$ that permutes labels is exactly the conjugation action on these permutations. Consider an action of a group $G$ on $\cyc_{\sbrac{k+1}}$ that commutes with this conjugation action. Then, for any $g \in G$ and any $c \in \cyc_{\sbrac{k+1}}$, we have that $$\label{eq:transaction}
g \cdot c = g \cdot c c c^{-1} = c \pbrac{g \cdot c} c^{-1}$$ and so $c$ and $g \cdot c$ commute. Thus, $c$ commutes with every element of its orbit under the action of $G$. But, for $k \geq 3$, not all elements of $\cyc_{\sbrac{k+1}}$ commute, so the action is not transitive.
We thus cannot hope to attack the coherent orientations of $k$-trees by acting directly on the cyclic orderings of fronts. Accordingly, we cannot simply apply the results of \[ss:codecomp\] to compute a $\Gamma$-species $\ct{k}$ with respect to some hypothetical action of a group $\Gamma$ whose orbits correspond to generic $k$-trees. Instead, we will use the additional structure on *rooted* coherently-oriented $k$-trees; with rooting, the cyclic orders around black vertices are converted into linear orders, for which there is a natural action of $\symgp{k+1}$.
### Group actions on $k$-coding trees {#ss:actct}
We have noted previously that every labeled $k$-tree admits exactly $k!$ coherent orientations. Thus, there are $k!$ distinct $k$-coding trees associated to each labeled $k$-tree, which differ only in the $\specname{C}_{k+1}$-structures on their black vertices. Consider a rooted $k$-coding tree $T$ and a black vertex $v$ which is not the root vertex. Then one white neighbor of $v$ is the ‘parent’ of $v$ (in the sense that it lies on the path from $v$ to the root). We thus can convert the cyclic order on the $k+1$ white neighbors of $v$ to a linear order by choosing the parent white neighbor to be last. There is a natural, transitive, label-independent action of $\symgp{k+1}$ on the set of such linear orders which induces an action on the cyclic orders from which the linear orders are derived. However, only elements of $\symgp{k+1}$ which fix $k+1$ will respect the structure around the black vertex we have chosen, since its parent white vertex must remain last.
In addition, if we simply apply the action of some $\sigma \in \symgp{k+1}$ to the order on white neighbors of $v$, we change the coherently-oriented $k$-tree $\beta^{-1} \pbrac{T}$ to which $T$ is associated in such a way that it no longer corresponds to the same unoriented $k$-tree. Let $t$ denote the unoriented $k$-tree associated to $\beta^{-1} \pbrac{T}$; then there exists a coherent orientation of $t$ which agrees with orientation around $v$ induced by $\sigma$. The $k$-coding tree $T'$ corresponding to this new coherent orientation has the same underlying bicolored tree as $T$ but possibly different orders around its black vertices. If we think of the $k$-coding tree $T'$ as the image of $T$ under a global action of $\sigma$, orbits under all of $\symgp{}$ will be precisely the classes of $k$-coding trees corresponding to all coherent orientations of specified $k$-trees, allowing us to study unoriented $k$-trees as quotients. The orientation of $T'$ will be that obtained by applying $\sigma$ at $v$ and then recursively adjusting the other cyclic orders so that fronts which were mirror are made mirror again. This will ensure that the combinatorial structure of the underlying $k$-tree $t$ is preserved.
Therefore, when we apply some permutation $\sigma \in \symgp{k+1}$ to the white neighbors of a black vertex $v$, we must also permute the cyclic orders of the descendant black vertices of $v$. In particular, the permutation $\sigma'$ which must be applied to some immediate black descendant $v'$ of $v$ is precisely the permutation on the linear order of white neighbors of $v'$ induced by passing over the mirror bijection from $v'$ to $v$, applying $\sigma$, and then passing back. We can express this procedure in formulaic terms:
\[thm:rhodef\] If a permutation $\sigma \in \symgp{k+1}$ is applied to linearized orientation of a black vertex $v$ in rooted $k$-coding tree, the permutation which must be applied to the linearized orientation a child black vertex $v'$ which was attached to the $i$th white child of $v$ (with respect to the linear ordering induced by the orientation) to preserve the mirror relation is $\rho_{i} \pbrac{\sigma}$, where $\rho_{i}$ is the map given by $$\label{eq:rhodef}
\rho_{i} \pbrac{\sigma}: a \mapsto \sigma \pbrac{i + a} - \sigma \pbrac{i}$$ in which all sums and differences are reduced to their representatives modulo $k+1$ in $\cbrac{1, 2, \dots, k+1}$.
Let $v'$ denote a black vertex which is attached to $v$ by the white vertex $1$, which we suppose to be in position $i$ in the linear order induced by the original orientation of $v$. Let $2$ denote the white child of $v'$ which is $a$th in the linear order induced by the original orientation around $v'$. It is mirror to the white child $3$ of $v$ which is $\pbrac{i+a}$th in the linear order induced by the original orientation around $v$. After the action of $\sigma$ is applied, vertex $3$ is $\sigma \pbrac{i+a}$th in the new linear order around $v$. We require that $2$ is still mirror to $3$, so we must move it to position $\sigma \pbrac{i + a} - \sigma \pbrac{i}$ when we create a new linear order around $v'$. This completes the proof.
This procedure is depicted in \[fig:rhoapp\].
;
(v)[3]{} (v) edge \[bend right=45, thick\] node \[below\](a3d)[$i+a$]{} (3); (v) edge \[bend left=45, thick\] node \[above\](a3u)[$\sigma \pbrac{i+a}$]{} (3); (a3d) edge \[->, dashed, thick\] node \[auto\][$\sigma$]{} (a3u);
(v)[1]{} (v) edge \[bend left=45, thick\] node \[below\](a1d)[$i$]{} (1); (v) edge \[bend right=45, thick\] node \[above\](a1u)[$\sigma \pbrac{i}$]{} (1); (a1d) edge \[->, dashed, thick\] node \[auto\][$\sigma$]{} (a1u);
(1)[v’]{} ; (v’) edge \[thick\] node \[auto\](b1)[$0$]{} (1);
(v’)[2]{} (v’) edge \[bend left=45, thick\] node \[below\](b2d)[$a$]{} (2); (v’) edge \[bend right=45, thick\] node \[above\](b2u)[$\sigma \pbrac{i+a} - \sigma \pbrac{i}$]{} (2); (b2d) edge \[->, dashed, thick\] node \[auto\][$\rho_{i} \pbrac{\sigma}$]{} (b2u);
(b2d) edge \[->, dotted, thick, bend right=15\] node \[auto\][$\mu$]{} (a3d); (b2u) edge \[->, dotted, thick, bend left=15\] node \[auto\][$\mu$]{} (a3u);
As an aside, we note that, although the construction $\rho$ depends on $k$, the value of $k$ will be fixed in any given context, so we suppress it in the notation.
Any $\sigma$ which is to be applied to a non-root black vertex $v$ must of course fix $k+1$. We let $\Delta: \symgp{k} \to \symgp{k+1}$ denote the obvious embedding; then the image of $\Delta$ is exactly the set of $\sigma \in \symgp{k+1}$ which fix $k+1$. We then have an action of $\symgp{k}$ on non-root black vertices induced by $\Delta$. (Equivalently, we can think of $\symgp{k}$ as the subgroup of $\symgp{k+1}$ of permutations fixing $k+1$, but the explicit notation $\Delta$ will be of use in later formulas.)
In light of \[obs:funcdecompct\], we now wish to adapt these ideas into explicit $\symgp{k}$- and $\symgp{k+1}$-actions on $\ctx{k}$, $\cty{k}$, and $\ctxy{k}$ whose orbits correspond to the various coherent orientations of single underlying rooted $k$-trees. In the case of a $Y$-rooted $k$-coding tree $T$, if we declare that $\sigma \in \symgp{k}$ acts on $T$ by acting directly (as $\Delta \pbrac{\sigma}$) on each of the black vertices immediately adjacent to the root and then applying $\rho$-derived permutations recursively to their descendants, orbits behave as expected. The same $\symgp{k}$-action serves equally well for edge-rooted $k$-coding trees, where (for purposes of applying the action of some $\sigma$) we can simply ignore the black vertex in the root.
However, if we begin with an $X$-rooted $k$-coding tree, the cyclic ordering of the white neighbors of the root black vertex has no canonical choice of linearization. If we make an arbitrary choice of one of the $k+1$ available linearizations, and thus convert to an edge-rooted $k$-coding tree, the full $\symgp{k+1}$-action defined previously can be applied directly to the root vertex. The orbit under this action of some edge-rooted $k$-coding tree $T$ with a choice of linearization at the root then includes all possible linearizations of the root orders of all possible $X$-rooted $k$-coding trees corresponding to the different coherent orientations of a single $k$-coding tree.
### $k$-trees as quotients {#ss:ktquot}
Since these actions are label-independent, we may now treat $\cty{k}$ and $\ctxy{k}$ as $\symgp{k}$-species and $\ctxy{k}$ as an $\symgp{k+1}$-species. The $\symgp{k}$- and $\symgp{k+1}$-actions on $\ctxy{k}$ are compatible, but we will make explicit reference to $\ctxy{k}$ as an $\symgp{k}$- or $\symgp{k+1}$-species whenever it is important and not completely clear from context which we mean. As a result of the above results, we can then relate the rooted $\Gamma$-species forms of $\ct{k}$ to the various ordinary species forms of generic rooted $k$-trees in \[thm:dissymk\]:
\[thm:arootquot\] For the various rooted forms of the ordinary species $\kt{k}$ as in \[thm:dissymk\] and the various rooted $\Gamma$-species forms of $\ct{k}$ as in \[obs:funcdecompct\] as $\symgp{k}$- and $\symgp{k+1}$-species, we have
\[eq:arootquot\] $$\begin{aligned}
\kty{k} &= \faktor{\cty{k}}{\symgp{k}} \label{eq:ayquot} \\
\ktxy{k} &= \faktor{\ctxy{k}}{\symgp{k}} \label{eq:axyquot} \\
\ktx{k} &= \faktor{\ctxy{k}}{\symgp{k+1}} \label{eq:axquot}
\end{aligned}$$
as isomorphisms of ordinary species, where $\ctxy{k}$ is an $\symgp{k}$-species in \[eq:axyquot\] and an $\symgp{k+1}$-species in \[eq:axquot\].
As a result, we have explicit characterizations of all the rooted components of the original dissymmetry theorem, \[thm:dissymk\]. To compute the cycle indices of these components (and thus the cycle index of $\kt{k}$ itself), we need only compute the cycle indices of the various rooted $\ct{k}$ species, which we will do using a combination of the functional equations in \[eq:ctfunc\] and explicit consideration of automorphisms.
Automorphisms and cycle indices {#s:ktcycind}
-------------------------------
### $k$-coding trees: $\cty{k}$ and $\ctxy{k}$ {#ss:ctcycind}
of the dissymmetry theorem for $k$-trees has a direct analogue in terms of cycle indices:
\[thm:dissymkci\] For the various forms of the species $\kt{k}$ as in \[s:dissymk\], we have $$\label{eq:dissymkci}
\ci{\kt{k}} = \ci{\ktx{k}} + \ci{\kty{k}} - \ci{\ctxy{k}}.$$
Thus, we need to calculate the cycle indices of the three rooted forms of $\kt{k}$. From \[thm:arootquot\] and by \[thm:qsci\] we obtain:
\[thm:aquotci\] For the various forms of the species $\kt{k}$ as in \[s:dissymk\] and the various $\symgp{k}$-species and $\symgp{k+1}$-species forms of $\ct{k}$ as in \[ss:actct\], we have
\[eq:aquotci\] $$\begin{aligned}
\ci{\kty{k}} &= \qgci{\symgp{k}}{\cty{k}} = \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \gcielt{\symgp{k}}{\cty{k}}{\sigma} \label{eq:ayquotci} \\
\ci{\ctxy{k}} &= \qgci{\symgp{k}}{\ctxy{k}} = \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \gcielt{\symgp{k}}{\ctxy{k}}{\sigma} \label{eq:axyquotci} \\
\ci{\ktx{k}} &= \qgci{\symgp{k+1}}{\ctxy{k}} = \frac{1}{\pbrac{k+1}!} \sum_{\sigma \in \symgp{k+1}} \gcielt{\symgp{k+1}}{\ctxy{k}}{\sigma} \label{eq:axquotci}
\end{aligned}$$
We thus need only calculate the various $\Gamma$-cycle indices for the $\symgp{k}$-species and $\symgp{k+1}$-species forms of $\cty{k}$ and $\ctxy{k}$ to complete our enumeration of general $k$-trees.
In \[obs:funcdecompct\], the functional equations for the ordinary species $\cty{k}$ and $\ctxy{k}$ both include terms of the form $\specname{L}_{k} \circ \cty{k}$. The plethysm of ordinary species does have a generalization to $\Gamma$-species, as given in \[def:gspeccomp\], but it does not correctly describe the manner in which $\symgp{k}$ acts on linear orders of $\cty{k}$-structures in these recursive decompositions. Recall from \[s:quot\] that, for two $\Gamma$-species $F$ and $G$, an element $\gamma \in \Gamma$ acts on an $\pbrac{F \circ G}$-structure (colloquially, ‘an $F$-structure of $G$-structures’) by acting on the $F$-structure and on each of the $G$-structures independently. In our action of $\symgp{k}$, however, the actions of $\sigma$ on the descendant $\cty{k}$-structures are *not* independent—they depend on the position of the structure in the linear ordering around the parent black vertex. In particular, if $\sigma$ acts on some non-root black vertex, then $\rho_{i} \pbrac{\sigma}$ acts on the white vertex in the $i$th place, where in general $\rho_{i} \pbrac{\sigma} \neq \sigma$.
Thus, we consider automorphisms of these $\symgp{k}$-structures directly. First, we consider the component species $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$.
\[lem:ctyinvar\] Let $B$ be a structure of the species $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$. Let $W_{i}$ be the $\cty{k}$-structure in the $i$th position in the linear order. Then some $\sigma \in \symgp{k}$ acts as an automorphism of $B$ if and only if, for each $i \in \sbrac{k+1}$, we have $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}} W_{i} \cong W_{\sigma \pbrac{i}}$.
Recall that the action of $\sigma \in \symgp{k}$ is in fact the action of $\Delta \sigma \in \symgp{k+1}$. The $X$-label on the black root of $B$ is not affected by the application of $\Delta \sigma$, so no conditions on $\sigma$ are necessary to accommodate it. However, the $\specname{L}_{k}$-structure on the white children of the root is permuted by $\Delta \sigma$, and we apply to each of the $W_{i}$’s the action of $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}}$. Thus, $\sigma$ is an automorphism of $B$ if and only if the combination of applying $\Delta \sigma$ to the linear order and $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}}$ to each $W_{i}$ is an automorphism. Since $\sigma$ ‘carries’ each $W_{i}$ onto $W_{\sigma \pbrac{i}}$, we must have that $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}} W_{i} \cong W_{\sigma \pbrac{i}}$, as claimed. That this suffices is clear.
Consider a structure $T$ of the $\symgp{k}$-species $\cty{k}$ and an element $\sigma \in \symgp{k}$. As discussed in \[ss:codecomp\], $T$ is composed of a $Y$-label and a set of $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$-structures. The permutation $\sigma$ acts trivially on $Y$ and $\specname{E}$ and acts on each of the component $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$-structures independently. For each of these component structures, by \[lem:ctyinvar\], we have that $\sigma$ is an automorphism if and only if $\Delta \sigma$ carries each $\cty{k}$-structure to its $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}}$-image. Thus, when constructing $\sigma$-invariant $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$-structures, we must construct for each cycle of $\sigma$ a $\cty{k}$-structure which is invariant under the application of *all* the permutations $\Delta^{-1} \pbrac{\rho_{i} \pbrac{\Delta \sigma}}$ which will be applied to it along the cycle. For $c$ the chosen cycle of $\sigma$, this permutation is $\Delta^{-1} \pbrac{\prod_{i \in c} \rho_{i} \pbrac{\Delta \sigma}}$, where the product is taken over any chosen linearization of the cyclic order of the terms in the cycle. Once a choice of such a $\cty{k}$-structure for each cycle of $\sigma$ is made, we can simply insert the structures into the $\specname{L}_{k}$-structure to build the desired $\sigma$-invariant $X \cdot \specname{L}_{k} \pbracs[big]{\cty{k}}$-structure. Accordingly:
\[thm:ctyfuncci\] The $\symgp{k}$-cycle index for the species $\cty{k}$ is characterized by the recursive functional equation $$\begin{gathered}
\label{eq:ctyfuncci}
\gcielt{\symgp{k}}{\cty{k}}{\sigma} = p_{1} \sbrac{y} \\
\times \ci{\specname{E}} \circ \pbracs[Big]{p_{1} \sbrac{x} \cdot \prod_{c \in C \pbrac{\sigma}} \gci{\symgp{k}}{\cty{k}} \pbracs[Big]{\Delta^{-1} \prod_{i \in c} \rho_{i} \pbrac{\Delta \sigma}} \pbrac{p_{\abs{c}} \sbrac{x}, p_{2 \abs{c}} \sbrac{x}, \dots; p_{\abs{c}} \sbrac{y}, p_{2 \abs{c}} \sbrac{y}, \dots}}.
\end{gathered}$$ where $C \pbrac{\sigma}$ denotes the set of cycles of $\sigma$ (as a $k$-permutation) and the inner product is taken with respect to any choice of linearization of the cyclic order of the elements of $c$.
The situation for the $\symgp{k+1}$-species $\ctxy{k}$ is almost identical. Recall from \[ss:actct\] that $\sigma \in \symgp{k+1}$ acts on a $\ctxy{k}$-structure $T$ by applying $\sigma$ directly to the linear order on the $k+1$ white neighbors of the root black vertex and applying $\rho$-variants of $\sigma$ recursively to their descendants. Thus, we once again need only require that, along each cycle of $\sigma$, the successive white-vertex structures are pairwise isomorphic under the action of the appropriate $\rho_{i} \pbrac{\sigma}$. Thus, we again need only choose for each cycle $c \in C \pbrac{\sigma}$ a $\cty{k}$-structure which is invariant under $\prod_{i \in c} \rho_{i} \pbrac{\sigma}$. Accordingly:
\[thm:ctxyfuncci\] The $\symgp{k+1}$-cycle index for the species $\ctxy{k}$ is given by $$\begin{gathered}
\label{eq:ctxyfuncci}
\gcielt{\symgp{k+1}}{\ctxy{k}}{\sigma} = p_{1} \sbrac{x} \\
\times \prod_{c \in C \pbrac{\sigma}} \gci{\symgp{k}}{\cty{k}} \pbracs[Big]{\prod_{i \in c} \rho_{i} \sbrac{\sigma}} \pbrac{p_{\abs{c}} \sbrac{x}, p_{2 \abs{c}} \sbrac{x}, \dots, p_{\abs{c}} \sbrac{y}, p_{2 \abs{c}} \sbrac{y}, \dots}.
\end{gathered}$$ under the same conditions as \[thm:ctyfuncci\].
Terms of the form $\prod_{i \in c} \rho_{i} \pbrac{\sigma}$ appear in \[eq:ctyfuncci,eq:ctxyfuncci\]. For the simplification of calculations, we note here a two useful results about these products.
First, we observe that certain $\rho$-maps preserve cycle structure:
\[lem:rhofp\] Let $\sigma \in \symgp{k}$ be a permutation of which $i \in \sbrac{k}$ is a fixed point and let $\lambda$ be the map sending each permutation in $\symgp{k}$ to its cycle type as a partition of $k$. Then $\lambda \pbrac{\rho_{i} \pbrac{\sigma}} = \lambda \pbrac{\sigma}$.
Suppose $i + a \in \sbrac{k}$ is in an $l$-cycle of $\sigma$. Then $$\begin{aligned}
\pbrac{\rho_{i} \pbrac{\sigma}}^{j} \pbrac{a} =& \pbrac{\rho_{i} \pbrac{\sigma}}^{j - 1} \pbrac{\sigma \pbrac{i + a} - \sigma \pbrac{i}} \\
=& \pbrac{\rho_{i} \pbrac{\sigma}}^{j - 2} \pbrac{\sigma \pbrac{i + \sigma \pbrac{i + a} - \sigma \pbrac{i}} - \sigma \pbrac{i}} \\
=& \pbrac{\rho_{i} \pbrac{\sigma}}^{j - 2} \pbrac{\sigma^{2} \pbrac{i + a} - \sigma^{2} \pbrac{i}} \\
&\vdots \\
=& \sigma^{j} \pbrac{i + a} - \sigma^{j} \pbrac{i}
\end{aligned}$$ But the values of $\pbrac{\rho_{i} \pbrac{\sigma}}^{j} \pbrac{a} = \sigma^{j} \pbrac{i + a} - \sigma^{j} \pbrac{i}$ are all distinct for $j \leq l$, since $i + a$ is in an $l$-cycle and $i$ is a fixed point of $\sigma$. Furthermore, $\pbrac{\rho_{i} \pbrac{\sigma}}^{l} \pbrac{a} = \sigma^{l} \pbrac{i + a} = i+a$. Thus, $a$ is in an $l$-cycle of $\rho_{i} \pbrac{\sigma}$. This establishes a length-preserving bijection between cycles of $\rho_{i} \pbrac{\sigma}$ and cycles of $\sigma$, so their cycle types are equal.
But then we note that the products in the above theorems are in fact permutations obtained by applying such $\rho$-maps:
\[lem:rhoprod\] Let $\sigma \in \symgp{k}$ be a permutation with a cycle $c$. Then $\lambda \pbrac{\prod_{i \in c} \rho_{i} \pbrac{\sigma}}$ is determined by $\lambda \pbrac{\sigma}$ and $\abs{c}$.
Let $c = \pbrac{c_{1}, c_{2}, \dots, c_{\abs{c}}}$. First, we calculate: $$\begin{aligned}
\prod_{i = 1}^{\abs{c}} \rho_{c_{i}} \pbrac{\sigma} =& \rho_{c_{\abs{c}}} \pbrac{\sigma} \circ \dots \circ \rho_{c_{2}} \pbrac{\sigma} \circ \rho_{c_{1}} \pbrac{\sigma} \\
=& \rho_{c_{\abs{c}}} \pbrac{\sigma} \circ \dots \circ \rho_{c_{2}} \pbrac{\sigma} \pbrac{a \mapsto \sigma \pbrac{c_{1} + a} - \sigma \pbrac{c_{1}}} \\
=& \rho_{c_{\abs{c}}} \pbrac{\sigma} \circ \dots \circ \rho_{c_{3}} \pbrac{\sigma} \pbrac{a \mapsto \sigma \pbrac{c_{2} + \sigma \pbrac{c_{1} + a} - \sigma \pbrac{c_{1}}} - \sigma \pbrac{c_{2}}} \\
=& \rho_{c_{\abs{c}}} \pbrac{\sigma} \circ \dots \circ \rho_{c_{3}} \pbrac{\sigma} \pbrac{a \mapsto \sigma^{2} \pbrac{c_{1} + a} - \sigma^{2} \pbrac{c_{1}}} \\
&\vdots \\
=& a \mapsto \sigma^{\abs{c}} \pbrac{c_{1} + a} - \sigma^{\abs{c}} \pbrac{c_{1}} \\
=& \rho_{c_{1}} \pbrac{\sigma^{\abs{c}}}.
\end{aligned}$$ But $c_{1}$ is a fixed point of $\sigma^{\abs{c}}$, so by the result of \[lem:rhofp\], this has the same cycle structure as $\sigma^{\abs{c}}$, which in turn is determined by $\lambda \pbrac{\sigma}$ and $\abs{c}$ as desired.
From this and the fact that the terms of $X$-degree $1$ in all $\gci{\cty{k}}{\symgp{k}}$ and $\gci{\ctxy{k}}{\symgp{k+1}}$ are equal (to $p_{1} \sbrac{x} p_{1} \sbrac{y}^{k+1}$), we can conclude that:
\[thm:ctciclassfunc\] $\gcielt{\cty{k}}{\symgp{k}}{\sigma}$ and $\gcielt{\ctxy{k}}{\symgp{k+1}}{\sigma}$ are class functions of $\sigma$ (that is, they are constant over permutations of fixed cycle type).
This will simplify computational enumeration of $k$-trees significantly, since the number of partitions of $k$ grows exponentially while the number of permutations of $\sbrac{k}$ grows factorially.
### $k$-trees: $\kt{k}$ {#ss:ktcycind}
We now have all the pieces in hand to apply \[thm:dissymkci\] to compute the cycle index of the species $\kt{k}$ of general $k$-trees. characterizes the cycle index of the generic $k$-tree species $\kt{k}$ in terms of the cycle indices of the rooted species $\ktx{k}$, $\kty{k}$, and $\ctxy{k}$; \[thm:arootquot\] gives the cycle indices of these three rooted species in terms of the $\Gamma$-cycle indices $\gci{\symgp{k}}{\cty{k}}$, $\gci{\symgp{k}}{\ctxy{k}}$, and $\gci{\symgp{k+1}}{\ctxy{k}}$; and, finally, \[thm:ctyfuncci,thm:ctxyfuncci\] give these $\Gamma$-cycle indices explicitly. By tracing the formulas in \[eq:ctyfuncci,eq:ctxyfuncci\] back through this sequence of functional relationships, we can conclude:
\[thm:akci\] For $\mathfrak{a}_{k}$ the species of general $k$-trees, $\gci{\symgp{k}}{\cty{k}}$ as in \[eq:ctyfuncci\], and $\gci{\symgp{k+1}}{\ctxy{k}}$ as in \[eq:ctxyfuncci\] we have: \[thm:ktreecyc\]
\[eq:akci\] $$\begin{aligned}
\ci{\kt{k}} &= \frac{1}{\pbrac{k+1}!} \sum_{\sigma \in \symgp{k+1}} \gcielt{\symgp{k+1}}{\ctxy{k}}{\sigma} + \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \gcielt{\symgp{k}}{\cty{k}}{\sigma} - \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \gcielt{\symgp{k}}{\ctxy{k}}{\sigma} \label{eq:akciexplicit} \\
&= \qgci{\symgp{k+1}}{\ctxy{k}} + \qgci{\symgp{k}}{\cty{k}} - \qgci{\symgp{k}}{\ctxy{k}}. \label{eq:akciquot}
\end{aligned}$$
in fact represents a recursive system of functional equations, since the formulas for the $\Gamma$-cycle indices of $\cty{k}$ and $\ctxy{k}$ are recursive. Computational methods can yield explicit enumerative results. However, a bit of care will allow us to reduce the computational complexity of this problem significantly.
Unlabeled enumeration and the generating function $\tilde{\mathfrak{a}}_{k} \pbrac{x}$ {#ss:ktunlenum}
--------------------------------------------------------------------------------------
in \[thm:akci\] gives a recursive formula for the cycle index of the ordinary species $\kt{k}$ of $k$-trees. The number of unlabeled $k$-trees with $n$ hedra is historically an open problem, but by application of \[thm:ciogf\] the ordinary generating function counting such structures can be extracted from the cycle index $\ci{\kt{k}}$. Actually computing terms of the cycle index in order to derive the coefficients of the generating function is, however, a computationally expensive process, since the cycle index is by construction a power series in two infinite sets of variables. The computational process can be simplified significantly by taking advantage of the relatively straightforward combinatorial structure of the structural decomposition used to derive the recursive formulas for the cycle index.
Recall from \[thm:gciogf\] that, for a $\Gamma$-species $F$, the ordinary generating function $\tilde{F}_{\gamma} \pbrac{x}$ counting unlabeled $\gamma$-invariant $F$-structures is given by $$\tilde{F} \pbracs[big]{\gamma} \pbrac{x} = \gcieltvars{\Gamma}{F}{\gamma}{x, x^2, x^3, \dots}$$ and that the ordinary generating function for counting unlabeled $\nicefrac{F}{\Gamma}$-structures is given by $$\tilde{F} \pbrac{x} = \frac{1}{\abs{\Gamma}} \sum_{\gamma \in \Gamma} \tilde{F} \pbracs[big]{\gamma} \pbrac{x}.$$ These formula admits an obvious multisort extension, but we in fact wish to count $k$-trees with respect to just one sort of label (the $X$-labels on hedra), so we will not deal with multisort here. Each of the two-sort cycle indices in this chapter can be converted to one-sort by substituting $y_{i} = 1$ for all $i$. For the rest of this section, we will deal directly with these one-sort versions of the cycle indices.
We begin by considering the explicit recursive functional equations in \[thm:ctyfuncci,thm:ctxyfuncci\]. In each case, by the above, the ordinary generating function is exactly the result of substituting $p_{i} \sbrac{x} = x^{i}$ into the given formula. Thus, we have:
\[thm:ctrhoogf\] For $\cty{k}$ the $\symgp{k}$-species of $Y$-rooted $k$-coding trees and $\ctxy{k}$ the $\symgp{k+1}$-species of edge-rooted $k$-coding trees, the corresponding $\Gamma$-ordinary generating functions are given by
\[eq:ctrhoogf\] $$\begin{aligned}
\widetilde{\cty{k}} \pbrac{\sigma} \pbrac{x} &= \exp \pbracs[Big]{ \sum_{n \geq 1} \frac{x^{n}}{n} \cdot \prod_{c \in C \pbrac{\sigma^{n}}} \widetilde{\cty{k}} \pbracs[Big]{\Delta^{-1} \prod_{i \in c} \rho_{i} \pbracs[big]{\Delta \sigma^{n}}} \pbrac{x^{\abs{c}}}} \label{eq:ctyrhoogf} \\
\intertext{and}
\widetilde{\ctxy{k}} \pbrac{\sigma} \pbrac{x} &= x \cdot \prod_{c \in C \pbrac{\sigma}} \widetilde{\cty{k}} \pbracs[Big]{\prod_{i \in c} \rho_{i} \pbrac{\sigma}} \pbracs[big]{x^{\abs{c}}}. \label{eq:ctxyrhoogf}
\end{aligned}$$
where $\widetilde{\cty{k}}$ is an $\symgp{k}$-generating function and $\widetilde{\ctxy{k}}$ is an $\symgp{k+1}$-generating function.
However, as a consequence of \[thm:ctciclassfunc\], we can simplify these expressions significantly:
\[cor:ctogf\] For $\cty{k}$ the $\symgp{k}$-species of $Y$-rooted $k$-coding trees and $\ctxy{k}$ the $\symgp{k+1}$-species of edge-rooted $k$-coding trees, the corresponding $\Gamma$-ordinary generating functions are given by
\[eq:ctogf\] $$\begin{aligned}
\widetilde{\cty{k}} \pbrac{\lambda} \pbrac{x} &= \exp \pbracs[Big]{\sum_{n \geq 1} \frac{x^{n}}{n} \cdot \prod_{i \in \lambda^{n}} \widetilde{\cty{k}} \pbracs[big]{\lambda^{i}} \pbracs[big]{x^{i}}} \label{eq:ctyogf} \\
\intertext{and}
\widetilde{\ctxy{k}} \pbrac{\lambda} \pbrac{x} &= x \cdot \prod_{i \in \lambda} \widetilde{\cty{k}} \pbracs[big]{\lambda^{i}} \pbracs[big]{x^{i}} \label{eq:ctxyogf}
\end{aligned}$$
where $\lambda^{i}$ denotes the $i$th ‘partition power’ of $\lambda$ — that is, if $\sigma$ is any permutation of cycle type $\lambda$, then $\lambda^{i}$ denotes the cycle type of $\sigma^{i}$ — and where $f \pbrac{\lambda} \pbrac{x}$ denotes the value of $f \pbrac{\sigma} \pbrac{x}$ for every $\sigma$ of cycle type $\lambda$.
As in \[thm:ctyfuncci\], we have recursively-defined functional equations, but these are recursions of power series in a single variable, so computing their terms is much less computationally expensive. Also, as an immediate consequence of \[thm:ctciclassfunc\], we have that $\widetilde{\cty{k}}$ and $\widetilde{\ctxy{k}}$ are class functions of $\sigma$, so we can restrict our computational attention to cycle-distinct permutations.
Moreover, the cycle index of the species $\kt{k}$, as seen in \[eq:akci\], is given simply in terms of quotients of the cycle indices of the two $\Gamma$-species $\cty{k}$ and $\ctxy{k}$. Thus, we also have:
\[thm:akrhoogf\] For $\kt{k}$ the species of $k$-trees and $\widetilde{\cty{k}}$ and $\widetilde{\ctxy{k}}$ as in \[thm:ctrhoogf\], we have $$\label{eq:akrhoogf}
\tilde{\mathfrak{a}}_{k} \pbrac{x} = \frac{1}{\pbrac{k+1}!} \sum_{\sigma \in \symgp{k+1}} \widetilde{\ctxy{k}} \pbrac{\sigma} \pbrac{x} + \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \widetilde{\cty{k}} \pbrac{x} \pbrac{\sigma} - \frac{1}{k!} \sum_{\sigma \in \symgp{k}} \widetilde{\ctxy{k}} \pbrac{\sigma} \pbrac{x}.$$
Again, as a consequence of \[thm:ctciclassfunc\] by way of \[cor:ctogf\], we can instead write
For $\kt{k}$ the species of $k$-trees and $\widetilde{\cty{k}}$ and $\widetilde{\ctxy{k}}$ as in \[cor:ctogf\], we have $$\label{eq:akogf}
\tilde{\mathfrak{a}}_{k} \pbrac{x} = \sum_{\lambda \vdash k+1} \frac{1}{z_{\lambda}} \widetilde{\ctxy{k}} \pbrac{\lambda} \pbrac{x} + \sum_{\lambda \vdash k} \frac{1}{z_{\lambda}} \widetilde{\cty{k}} \pbrac{\lambda} \pbrac{x} - \sum_{\lambda \vdash k} \frac{1}{z_{\lambda}} \widetilde{\ctxy{k}} \pbrac{\lambda \cup \cbrac{1}} \pbrac{x}.$$
This direct characterization of the ordinary generating function of unlabeled $k$-trees, while still recursive, is much simpler computationally than the characterization of the full cycle index in \[eq:akci\]. For computation of the number of unlabeled $k$-trees, it is therefore much preferred. Classical methods for working with recursively-defined power series suffice to extract the coefficients quickly and efficiently. The results of some such explicit calculations are presented in \[s:ktenum\].
Special-case behavior for small $k$
-----------------------------------
Many of the complexities of the preceding analysis apply only for $k$ sufficiently large. We note here some simplifications that are possible when $k$ is small.
### Ordinary trees ($k = 1$)
When $k = 1$, an $\kt{k}$-structure is merely an ordinary tree with $X$-labels on its edges and $Y$-labels on its vertices. There is no internal symmetry of the form that the actions of $\symgp{k}$ are intended to break. The actions of $\symgp{2}$ act on ordinary trees rooted at a *directed* edge, with the nontrivial element $\tau \in \symgp{2}$ acting by reversing this orientation. The resulting decomposition from the dissymmetry theorem in \[thm:dissymk\] and the recursive functional equations of \[obs:funcdecompct\] then clearly reduce to the classical dissymmetry analysis of ordinary trees.
### $2$-trees
When $k=2$, there is a nontrivial symmetry at fronts (edges); two triangles may be joined at an edge in two distinct ways. The imposition of a coherent orientation on a $2$-tree by directing one of its edges breaks this symmetry; the action of $\symgp{2}$ by reversal of these orientations gives unoriented $2$-trees as its orbits. The defined action of $\symgp{3}$ on edge-rooted oriented triangles is simply the classical action of the dihedral group $D_{6}$ on a triangle, and its orbits are unoriented, unrooted triangles. We further note that $\rho_{i}$ is the trivial map on $\symgp{2}$ and that $\rho_{i} \pbrac{\sigma} = \pbrac{1\ 2}$ for $\sigma \in \symgp{3}$ if and only if $\sigma$ is an odd permutation, both regardless of $i$. We then have that:
\[eq:rest2trees\] $$\begin{aligned}
\gci{\symgp{2}}{\cty{2}} &= p_{1} \sbrac{y} \cdot \ci{\specname{E}} \circ \pbracs[Big]{p_{1} \sbrac{x} \cdot \prod_{c \in C \pbrac{\sigma}} \gcieltvars{\symgp{2}}{\cty{2}}{e}{p_{\abs{c}} \sbrac{x}, p_{2 \abs{c}} \sbrac{x}, \dots; p_{\abs{c}} \sbrac{y}, p_{2 \abs{c}} \sbrac{y}, \dots}} \label{eq:ctyfuncci2} \\
\gci{\symgp{3}}{\ctxy{2}} &= p_{1} \sbrac{x} \cdot \prod_{c \in C \pbrac{\sigma}} \gci{\symgp{2}}{\cty{2}} \pbracs[big]{\rho \pbrac{\sigma}^{\abs{c}}} \pbrac{p_{\abs{c}} \sbrac{x}, p_{2 \abs{c}} \sbrac{x}, \dots; p_{\abs{c}} \sbrac{y}, p_{2 \abs{c}} \sbrac{y}, \dots}. \label{eq:ctxyfuncci2}
\end{aligned}$$
where, by abuse of notation, we let $\rho$ represent any $\rho_{i}$. By the previous, the argument $\rho \pbrac{\sigma}^{\abs{c}}$ in \[eq:ctxyfuncci2\] is $\tau$ if and only if $\sigma$ is an odd permutation and $c$ is of odd length. This analysis and the resulting formulas for the cycle index $\ci{\kt{2}}$ are essentially equivalent to those derived in [@gessel:spec2trees].
Computation in species theory {#c:comp}
=============================
Cycle indices of compositional inverse species {#s:compinv}
----------------------------------------------
In \[s:nbp\], our results included two references to the compositional inverse $\specname{CBP}^{\bullet \abrac{-1}}$ of the species $\specname{CBP}^{\bullet}$. Although we have not explored computational methods in depth here, the question of how to compute the cycle index of the compositional inverse of a specified species efficiently is worth some consideration. Several methods are available, including one developed in [@bll:species 4.2.19] as part of the proof that arbitrary species have compositional inverses, but our preferred method is one of iterated substitution.
Suppose that $\Psi$ is a species (with known cycle index) of the form $X + \Psi_{2} + \Psi_{3} + \dots$ where $\Psi_{i}$ is the restriction of $\Psi$ to structures on sets of cardinality $i$ and that $\Phi$ is the compositional inverse of $\Psi$. Then $\Psi \circ \Phi = X$ by definition, but by hypothesis $$X = \Psi \circ \Phi = \Phi + \Psi_{2} \pbrac{\Phi} + \Psi_{3} \pbrac{\Phi} + \dots$$ also. Thus $$\label{eq:compinv}
\Phi = X - \Psi_{2} \pbrac{\Phi} - \Psi_{3} \pbrac{\Phi} - \dots.$$ This recursive equation is the key to our computational method. To compute the cycle index of $\Phi$ to degree $2$, we begin with the approximation $\Phi \approx X$ and then substitute it into the first two terms of \[eq:compinv\]: $\Phi \approx X - \Psi_{2} \pbrac{X}$ and thus $\ci{\Phi} \approx \ci{X} - \ci{\Psi_{2}} \circ \ci{X}$. All terms of degree up to two in this approximation will be correct. To compute the cycle index of $\Phi$ to degree $3$, we then take this new approximation $\Phi \approx X - \Psi_{2} \pbrac{X}$ and substitute it into the first three terms of \[eq:compinv\]. This process can be iterated as many times as are needed; to determine all terms of degree up to $n$ correctly, we need only iterate $n$ times. With appropriate optimizations (in particular, truncations), this method can run very quickly on a personal computer to reasonably high degrees; we were able to compute $\ci{\specname{CBP}^{\bullet \abrac{-1}}}$ to degree sixteen in thirteen seconds.
Enumerative tables {#c:enum}
==================
Bipartite blocks {#s:bpenum}
----------------
With the tools developed in \[c:bpblocks\], we can calculate the cycle indices of the species $\mathcal{NBP}$ of nonseparable bipartite graphs to any finite degree we choose using computational methods. This result can then be used to enumerate unlabeled bipartite blocks. We have done so here using Sage 1.7.4 [@sage] and code listed in \[s:bpbcode\]. The resulting values appear in \[tab:bpblocks\].
$n$ Unlabeled
----- ----------- --
1 1
2 1
3 0
4 1
5 1
6 5
7 8
8 42
9 146
10 956
: Enumerative data for unlabeled bipartite blocks with $n$ hedra[]{data-label="tab:bpblocks"}
$k$-trees {#s:ktenum}
---------
With the recursive functional equations for cycle indices of \[s:ktcycind\], we can calculate the explicit cycle index for the species $\kt{k}$ to any finite degree we choose using computational methods; this cycle index can then be used to enumerate both unlabeled and labeled (at fronts, hedra, or both) $k$-trees up to a specified number $n$ of hedra (or, equivalently, $kn + 1$ fronts). We have done so here for $k \leq 7$ and $n \leq 30$ using Sage 1.7.4 [@sage] using code available in \[s:ktcode\]. The resulting values appear in \[tab:ktrees\].
We note that both unlabeled and hedron-labeled enumerations of $k$-trees stabilize:
\[thm:ktreestab\] For $k \geq n + 2$, the numbers of unlabeled and hedron-labeled $k$-trees are independent of $k$.
We show that the species $\kt{k}$ and $\kt{k+1}$ have contact up to order $k+2$ by explicitly constructing a natural bijection. We note that in a $\pbrac{k+1}$-tree with no more than $k+2$ hedra, there will exist at least one vertex which is common to *all* hedra. For any $k$-tree with no more than $k+2$ hedra, we can construct a $\pbrac{k+1}$-tree with the same number of hedra by adding a single vertex and connecting it by edges to every existing vertex; we can then pass labels up from the $\pbrac{k+1}$-cliques which are the hedra of the $k$-tree to the $\pbrac{k+2}$-cliques which now sit over them. The resulting graph will be a $\pbrac{k+1}$-tree whose $\pbrac{k+1}$-tree hedra are adjacent exactly when the $k$-tree hedra they came from were adjacent. Therefore, any two distinct $k$-trees will pass to distinct $\pbrac{k+1}$-trees. Similarly, for any $\pbrac{k+1}$-tree with no more than $k+2$ hedra, choose one of the vertices common to all the hedra and remove it, passing the labels of $\pbrac{k+1}$-tree hedra down to the $k$-tree hedra constructed from them; again, adjacency of hedra is preserved. This of course creates a $k$-tree, and for distinct $\pbrac{k+1}$-trees the resulting $k$-trees will be distinct. Moreover, by symmetry the result is independent of the choice of common vertex, in the case there is more than one.
However, thus far we have neither determined a direct method for computing these stabilization numbers nor identified a straightforward combinatorial characterization of the structures they represent.
Code listing {#c:code}
============
Our results in \[c:bpblocks,c:ktrees\] provide a framework for enumerating bipartite blocks and general $k$-trees. However, there is significant work to be done adapting the theory into practical algorithms for computing the actual numbers of such structures. Using the computer algebra system Sage 1.7.4 [@sage], we have done exactly this. In each case, the script listed may be run with Sage by invoking
> sage --python scriptname.py args
on a computer with a functioning Sage installation. Alternatively, each code snippet may be executed in the Sage ‘notebook’ interface starting at the comment “`MATH BEGINS HERE`”; in this case, the final `print…` invocation should be replaced with one specifying the desired parameters.
Bipartite blocks {#s:bpbcode}
----------------
The functional \[eq:nbpexp\] characterizes the cycle index of the species $\specname{NBP}$ of bipartite blocks. Python/Sage code to compute the coefficients of the ordinary generating function $\widetilde{\specname{NBP}} \pbrac{x}$ of unlabeled bipartite blocks explicitly follows in \[lst:bpcode\]. Specifically, the generating function may be computed to degree $n$ by invoking
> sage --python bpblocks.py n
on a computer with a functioning Sage installation.
$k$-trees {#s:ktcode}
---------
The recursive functional equations in \[eq:ctyogf,eq:ctxyogf,eq:akogf\] characterize the ordinary generating function $\tilde{\mathfrak{a}}_{k} \pbrac{x}$ for unlabeled general $k$-trees. Python/Sage code to compute the coefficients of this generating function explicitly follows in \[lst:ktcode\]. Specifically, the generating function for unlabeled $k$-trees may be computed to degree $n$ by invoking
> sage --python ktrees.py k n
on a computer with a functioning Sage installation.
This code uses the class-function optimization of \[thm:ctciclassfunc\] extensively; as a result, it is able to compute the number of $k$-trees on up to $n$ hedra quickly even for relatively large $k$ and $n$. For example, the first thirty terms of the generating function for $8$-trees in \[tab:8trees\] were computed on a modern desktop-class computer in approximately two minutes.
[^1]: That is, the value of $\fix \pbrac{F \sbrac{\sigma}}$ will be constant on conjugacy classes of permutations, which we note are exactly the sets of permutations of fixed cycle type.
[^2]: Although these are called ‘functions’ for historical reasons, convergence of these formal power series is often not of immediate interest.
[^3]: The *line group* of a graph is the group of permutations of edges induced by permutations of vertices.
[^4]: Note that this decomposition does not actually partition the vertices, since many blocks may share a single cut-point, a detail which significantly complicates but does not entirely preclude species-theoretic analysis.
| ArXiv |
---
abstract: 'We propose a universal gate set acting on a qubit formed by the degenerate ground states of a Coulomb-blockaded time-reversal invariant topological superconductor island with spatially separated Majorana Kramers pairs: the “Majorana Kramers Qubit". All gate operations are implemented by coupling the Majorana Kramers pairs to conventional superconducting leads. Interestingly, in such an all-superconducting device, the energy gap of the leads provides another layer of protection from quasiparticle poisoning independent of the island charging energy. Moreover, the absence of strong magnetic fields – which typically reduce the superconducting gap size of the island – suggests a unique robustness of our qubit to quasiparticle poisoning due to thermal excitations. Consequently, the Majorana Kramers Qubit should benefit from prolonged coherence times and may provide an alternative route to a Majorana-based quantum computer.'
author:
- Constantin Schrade and Liang Fu
title: Quantum Computing with Majorana Kramers Pairs
---
@twocolumnfalse
1.5truecm
In recent years an increasing number of platforms have been proposed for realizing time-reversal invariant topological superconductors (TRI TSCs) [@bib:Schnyder2008]. Among the most notable platforms are nanowires and topological insulators in contact to unconventional superconductors (SCs) [@bib:Wong2012; @bib:Nagaosa2013; @bib:Zhang2013; @bib:Dumitrescu2014] and conventional SCs [@bib:Klinovaja2014; @bib:Gaidamauskas2014; @bib:Schrade2017; @bib:Klinovaja20142; @bib:Yan2018; @bib:Hsu2018], proximity-induced Josephson $\pi$-junctions in nanowires and topological insulators [@bib:Keselman2013; @bib:Haim2014; @bib:Schrade2015; @bib:Borla2017] as well as TSCs with an emergent time-reversal symmetry (TRS) [@bib:Huang2017; @bib:Reeg2017; @bib:Hu2017; @bib:Maisberger2017].
A common feature of TRI TSCs is that they host spatially separated Majorana Kramers pairs (MKPs) which form robust, zero energy modes protected by TRS. In spite of much fundamental interest in the properties of MKPs [@bib:Chung2013; @bib:Li2016; @bib:Pikulin2016; @bib:Kim2016; @bib:Camjayi2017; @bib:Bao2017; @bib:Schrade2018], a yet unsolved question is if MKPs can be employed for applications in quantum computation. Here, we answer this question in the affirmative.
The purpose of this work is to introduce a qubit formed by the degenerate ground states of a Coulomb-blockaded TRI TSC island with spatially separated MKPs: the “Majorana Kramers Qubit" (MKQ). We depict the minimal experimental setup for a single MKQ in Fig. \[fig:1\]. It comprises two SC leads which separately couple to two distinct MKPs on a U-shaped TRI TSC island. The two SC leads are weakly coupled among themselves by spin-flip and normal tunnelling barriers. Within this setup, we will implement single-qubit Clifford gates by making use of a measurement-based approach to quantum computing [@bib:Bonderson2008; @bib:Litinski2017]. Moreover, to achieve universal quantum computation we will implement a $\pi/8$-gate as well as a two-MKQ entangling gate by pulsing of tunnel couplings.
![(Color online) Setup consisting of a U-shaped, mesoscopic TRI TSC island (gray) realizing a MKQ. Tunable tunnel couplings (white, dashed) connect SC leads $\ell=\text{L,R}$ (red) to the MKPs $\gamma_{\ell,s}$ (yellow) with $s={\uparrow},{\downarrow}$. The SC leads themselves are also connected by a spin-flip and a normal tunnelling barrier with lengths $d, d'$. To facilitate Cooper pair splitting between these two tunnelling barriers and the TRI TSC island we require that the separation of the tunnelling contacts is smaller than the coherence length $\xi_{\text{SC}}$ of the SC leads. Moreover, to avoid couplings of the MKPs to fermionic corner modes [@bib:Loss2015], the length of the vertical segments of the TRI TSC islands are much longer than the MKP localization length $\xi_{\text{MKP}}$. Lastly, a gate voltage $V$ tunes the charge on the TRI TSC island via a capacitor with capacitance $C$. []{data-label="fig:1"}](Fig1){width="0.75\linewidth"}
The main conceptual lesson we will learn is that Majorana-based quantum computing is possible without invoking the need for magnetic fields. Besides that, there two interesting, yet more practical, features of our setup which are noteworthy: (1) Within the single-MKQ setup of Fig. \[fig:1\], single-electron tunnelling from the SC leads does not only require overcoming the charging energy of the TRI TSC island but also the breaking of a Cooper pair in the leads. Consequently, the SC gap of the leads provides an additional layer of protection against quasiparticle poisoning, independent of the island charging energy. (2) Quasiparticle poisoning due to thermal excitations within the TRI TSC island is strongly suppressed the SC gap of the island itself. Critically, the energy gap of a TRI TSC island is conceivably larger than the energy gap of TRS-breaking Majorana islands [@bib:Fu2010; @bib:Vijay2015; @bib:Vijay2016; @bib:Landau2016; @bib:Plugge2016; @bib:Vijay2016_2; @bib:Aasen2016; @bib:Karzig2016; @bib:Plugge2017; @bib:Schrade2018_2; @bib:Gau2018] since there is no magnetic field that would reduce the SC gap size. As a consequence, the MKQ should benefit from improved coherence times and may be a viable route towards a robust quantum computer.
[*Setup.*]{} As shown in Fig. \[fig:1\], our setup comprises a U-shaped TRI TSC islands hosting MKPs $\gamma_{\ell,s}$ with $s={\uparrow},{\downarrow}$ at spatially well separated boundaries $\ell=\text{L,R}$. The two members of a MKP are related by TRS, $$\mathcal{T}\gamma_{\ell,{\uparrow}}\mathcal{T}^{-1}=\gamma_{\ell,{\downarrow}}, \; \mathcal{T}\gamma_{\ell,{\downarrow}}\mathcal{T}^{-1}=-\gamma_{\ell,{\uparrow}}.$$ We assume that the dimensions of the horizontal island segments exceed the localization lengths $\xi_{\text{MKP}}$ of the MKPs. This avoids couplings of the MKPs to fermionic modes that are potentially localized at the island corners [@bib:Loss2015] and, thereby, ensures that the MKPs are, in fact, robust zero-energy states protected by TRS.
Since the TRI TSC island is of mesoscopic size, it acquires a charging energy given by $$U_{C} = \left(ne-Q\right)^{2}/ 2C.$$ Here, $Q$ is the island gate charges that is continuously tunable with a voltage across a capacitor with capacitance $C$. We assume that the gate charge $Q/e$ is tuned close to an even or odd integer for both islands. A sufficiently large charging energy $e^{2}/2C$ then fixes the joint parity of the MKPs on the TRI TSC island to [@bib:Fu2010; @bib:Xu2010] $$\label{TotalParity}
\gamma_{\text{L},{\uparrow}}\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}\gamma_{\text{R},{\downarrow}} = (-1)^{n_0}.$$ This constraint reduces the four-fold degeneracy of the ground state at zero charging energy, to a two-fold degenerate ground state which forms the MKQ. The Pauli operators acting on each of the two MKQs can be written as bilinears in the Majorana operators, $$\begin{split}
\hat{x}&=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}, \quad
\hat{y}=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{R},{\downarrow}} , \quad
\hat{z}=i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\downarrow}}.
\end{split}$$ Under TRS, the Pauli operators transform as $\mathcal{T}\hat{x}\mathcal{T}^{-1}=(-1)^{n_0}\hat{x}$, $\mathcal{T}\hat{y}\mathcal{T}^{-1}=-\hat{y}$ and $\mathcal{T}\hat{z}\mathcal{T}^{-1}=(-1)^{n_0}\hat{z}$.
In our setup, we choose to address the MKQ by weakly coupling each MKP to a separate $s$-wave SC lead. The Hamiltonian for the two SC leads reads $$H_{SC}=\sum_{\ell=\text{L,R}}\sum_{{{{\bf{k}}}}} \Psi_{\ell,{{{\bf{k}}}}}^\dagger \left(
\xi_{{{{\bf{k}}}}}\eta_{z}+\Delta_{\ell}\eta_{x}e^{i\varphi_{\ell}\eta_{z}}
\right)\Psi_{\ell,{{{\bf{k}}}}},$$ where $\Psi_{\ell,{{{\bf{k}}}}}=(c_{\ell,{{{\bf{k}}}}{\uparrow}},c^{\dag}_{\ell,-{{{\bf{k}}}}{\downarrow}})^{T}$ is a Nambu spinor with $c_{\ell,{{{\bf{k}}}}s}$ the electron annihilation operator at momentum ${{{\bf{k}}}}$ and spin $s$ in lead $\ell$. The Pauli matrices $\eta_{x,y,z}$ are acting in Nambu-space. Furthermore, $\xi_{{{{\bf{k}}}}}$ is the normal state dispersion and $\Delta_{\ell},\varphi\equiv\varphi_{\text{L}}-\varphi_{\text{R}}$ denote the magnitude and the relative phase difference of the SC order parameters. We assume sufficiently low temperatures such that no quasiparticle states in the SC leads are occupied with notable probability and can couple to the MKPs.
The most general tunneling Hamiltonian between the MKPs on the islands and the fermions on the $\ell$-SC lead is given by, $$\begin{aligned}
\label{Eq4}
H_{T}
&=
\sum_{\ell=\text{L,R}}
\sum_{{{{\bf{k}}}},s}
\lambda_{\ell}
c^{\dag}_{\ell,{{{\bf{k}}}}s}
\gamma_{\ell,s}
e^{-i\phi/2}
+
\text{H.c.}, \end{aligned}$$ where we have diagonalized the tunnelling Hamiltonian in spin-space by an appropriate rotation of the lead fermions [@bib:Schrade2018]. In particular, this rotation also constraints the point-like tunnelling amplitudes $\lambda_{\ell}$ to be real numbers. Furthermore, we remark that the operators $e^{\pm i\phi/2}$ raise/lower the total island charges by one unit, $[n,e^{\pm i\phi/2}]=\pm e^{\pm i\phi/2}$, while the MBSs operators $\gamma_{\ell,s}$ flip the respective electron number parities.
As evident from Fig. \[fig:1\], there are two types of couplings between the SC leads: The first type is an *indirect coupling* via the TRI TSC islands which is induced by the tunnelling Hamiltonian of Eq. . The second type is a *direct coupling* via two additional tunnelling barriers. The first tunnelling barrier only allows for normal tunnelling described by the tunnelling Hamiltonian, $$H_{N}
=
t_{N}
\sum_{{{{\bf{k}}}}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}}
+
c^{\dag}_{\text{L},{{{\bf{k}}}}{\downarrow}} c_{\text{R},{{{\bf{k}}}}{\downarrow}}
+ \text{H.c.},$$ with $t_{N}$ a complex, point-like tunnelling amplitude and $|t_{N}|\ll\lambda_{\ell}$. The second barrier only allows for spin-flip tunnelling described by a tunnelling Hamiltonian. $$H_{S}
=
t_{S}
\sum_{{{{\bf{k}}}}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\downarrow}}
-
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{{\bf{k}}}}{\downarrow}}
+ \text{H.c.},$$ with $t_{S}\ll\lambda_{\ell}$ a complex, point-like tunnelling amplitude and $|t_{S}|\ll\lambda_{\ell}$. We propose two ways to engineer such tunnellling barriers: (1) We consider barriers with a finite *intrinsic* spin-orbit coupling with spin-orbit length $\lambda_{\text{SO}}$ as well as different barrier lengths $d,d'$. Tuning $\lambda_{\text{SO}}/d'$ ($\lambda_{\text{SO}}/d$) to a positive integer (positive half integer) realizes a barrier with pure normal (spin-flip) tunnelling [@bib:Bercioux2015]. (2) We consider barriers without intrinsic spin-orbit coupling but with an *engineered* spin-orbit coupling due to a local, rotating magnetic field induced by a series of nanomagnets [@bib:Karmakar2011; @bib:Klinovaja2013]. By adjusting either the period of the rotating field through the separation of the nanomagnets or again the length of the barriers, we can realize barriers with pure normal or spin-flip tunnelling.
In summary, we conclude that the full Hamiltonian of our setup is given by $H=U_{C}+H_{SC}+H_{T}+H_{N}+H_{S}$.
![(Color online) (a) To third order in the tunnelling amplitudes, a Cooper pair moves between the two SC lead by splitting up between the normal tunnelling barrier and the TRI TSC island. The spin-flip tunnelling barrier is fully depleted by a local gate. The resulting terms in the effective Hamiltonian are $\propto\hat{z}$. (b) Same as (a) but this time the Cooper pair splits up between the spin-flip tunnelling barrier and the TRI TSC island. The normal tunnelling barrier is fully depleted by a local gate. The resulting terms in the effective Hamiltonian are now $\propto\hat{x}$. []{data-label="fig:2"}](Fig2){width="\linewidth"}
[*Single-qubit Clifford gates.*]{} In this section, we will implement single-qubit Clifford gates by “Majorana tracking" [@bib:Litinski2017]. This means that for a given circuit of single-qubit Clifford gates we record all Pauli operator redefinitions on a classical computer and use the quantum hardware only to perform suitable measurements of the $\hat{x}, \hat{y}, \hat{z}$-Pauli operators at the end of the computation.
As a starting point, for measuring the $\hat{z}$-Pauli operator, we consider the situation when a local gate depletes the spin-flip tunnelling barrier between the two SC leads such that $\text{Im}\, t_{S}=0$ for $n_0$ even and $\text{Re}\, t_{S}=0$ for $n_{0}$ odd.
In this case, to second order in the tunnelling amplitudes $t_{N}$, Cooper pairs tunnel between the SC leads only via the normal tunnelling barrier inducing a finite Josephson coupling $J_{N}\sim |t_{N}|^{2}/\Delta$. In particular, a Josephson coupling due to Cooper tunnelling between each SC lead and the TRI TSC is highly unfavorable as a result of the substantial island charging energy [@bib:Schrade2018_2]. We, hence, recognize that the island charging energy plays two significant roles in our setup: First, it suppresses quasiparticle poisoning due to single-electron tunnelling from the environment. Second, it also suppresses local mixing terms $\propto\hat{y}$ due to Cooper pair tunnelling between each SC lead and the TRI TSC island. Such local mixing terms are – as noticed in the previous literature [@bib:Wolms2014] – of importance for TRI TSCs with zero charging energy and, as we will see, can be utilized for measuring the $\hat{y}$-Pauli operator.
As a next step, we note that to third order in the tunnelling amplitudes $t_{N},\lambda_{\text{L}},\lambda_{\text{R}}$ Cooper pair splitting sequences between the TRI TSC island and the normal tunnelling barrier induce additional Josephson couplings, $J_{z}$ for $n_{0}$ even and $J'_{z}$ for $n_{0}$ odd. We depict an example of such a Cooper pair splitting sequence in Fig. \[fig:2\](a). In a first process, a Cooper pair on the left SC lead breaks up and one of the electrons tunnels via the normal tunnelling barrier to the right SC lead. This leaves the left SC lead in an excited state with one quasiparticle above the SC gap. In a second process, the quasiparticle on the left SC tunnels to the TRI TSC island and increments its charge by one unit. While the left SC returns to its ground state in this way, the TRI TSC island is now in an excited state with one excess charge. It, therefore, requires a third process to remove the extra charge from the TRI TSC by recombining it to a Cooper pair on the right SC lead. Critically, the tunnelling events via both the normal tunnelling barrier and the TRI TSC island conserve the electron spin. For that reason, the just described third-order sequences contribute terms $\propto\hat{z}=i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\downarrow}}=(-1)^{n_{0}}i\gamma_{\text{L},{\uparrow}}\gamma_{\text{R},{\uparrow}}$.
As the last step, we point out that to fourth order in the tunnelling amplitudes, Cooper pairs tunnel between the two SC leads only via the TRI TSC island yielding a Josephson coupling $J$ with a sign determined by the joint parity of all four MBSs on the TRI TSC island [@bib:Schrade2018].
In the limit of weak tunnel couplings, $\pi\nu_{\ell}\lambda_{\ell}^{2}\ll\Delta,e^{2}/2C$ with $\nu_{\ell}$ the normal-state density of states per spin of the $\ell$-SC lead at the Fermi energy, we compute the amplitudes of all above-mentioned sequences perturbatively. Up to fourth order in the tunnelling amplitudes, we then summarize our results in an effective Hamiltonians acting on the ground states of the SC leads and the TRI TSC island. For $n_{0}$ even and $n_{0}$ odd, we find that, $$\begin{split}
&H_{z,\text{even}}=-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi,
\\
&H_{z,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{z} \, J'_{z} \sin\varphi, \label{Hz}
\end{split}$$ where the detailed microscopic forms of the Josephson couplings $J$ and $J_{z}, J'_{z}$ are given in [@bib:Schrade2018] and [@bib:supplemental], respectively. Here, it suffices to remark that $J_{z}\neq0$ $(J'_{z}\neq0)$ provided $\text{Im}\, t_{N}\neq0$ ($\text{Re} \, t_{N}\neq0$). Moreover, we point out that the both effective Hamiltonian exhibit TRS: For $H_{z,\text{even}}$ both $\hat{z}$ and $\cos\varphi$ are time-reversal even, while for $H_{z,\text{odd}}$ both $\hat{z}$ and $\sin\varphi$ are time-reversal odd.
To measure the $z$-eigenvalue of the $\hat{z}$-Pauli operator, we adopt a two-step protocol: (1) First, we separately measure the Josephson current through the normal tunnelling barrier and through the TRI TSC island to determine $J_{N}$ and $J$. (2) Second, we measure the Josephson current through the entire device. For $n_{0}$ even, the latter is given by $I=I_{c}\sin\varphi$ with the critical current $I_c =2e(J_{N}-J+z\, J_{z})/\hbar$ fixing the $z$-eigenvalue. For $n_0$ odd, the current phase relation is of the form $I=I_{c}\sin(\varphi+\varphi_0)$. This time it is not the critical current $I_{c}=2e\,\text{sgn}(J_{N}+J)\sqrt{(J_{N}+J)^{2}+(J'_z)^{2}}/\hbar$ but the anomalous phase shift $\varphi_{0}=z\,\arctan[J'_{z}/(J_{N}+J)]$ which fixes the $z$-eigenvalue. We note that the finite anomalous phase shift results from the $\hat{z}$-eigenstates breaking TRS when $n_{0}$ odd.
For the measurement of the $\hat{x}$-Pauli operator, we now shift our attention to a fully depleted normal tunnelling barrier such that $\text{Im}\, t_{N}=0$ for $n_0$ even and $\text{Re}\, t_{N}=0$ for $n_{0}$ odd. Similar to our earlier discussions, second order co-tunnelling events induce a Josephson coupling $J_{S}\sim|t_{S}|^{2}/\Delta$ as a result of Cooper pair tunnelling via the spin-flip tunnelling barrier whereas fourth order co-tunnelling events mediate a Josephson coupling $J$ via the TRI TSC island. However, a qualitative difference to the preceding considerations arises for the third order Cooper pair splitting sequences, see Fig. \[fig:2\](b). These sequences now demand two spin-flips, one for an electron to move through the spin-flip tunnelling barrier and one for an electron to move through the TRI TSC island. As a consequence, the third-order sequences now contribute terms $\propto\hat{x}=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}=(-1)^{n_0}i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\uparrow}}$. More explicitly, up to fourth order in the weak tunnel couplings, the effective Hamiltonians for $n_0$ even and $n_0$ odd are given by, $$\begin{split}
&H_{x,\text{even}}=-(J_{N}-J+\hat{x}\, J_{x})\cos\varphi,
\\
&H_{x,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{x} \, J'_{z} \sin\varphi. \label{Hx}
\end{split}$$ Here, $J_{x}, J'_{x}$ denote the Josephson couplings due to the Cooper pair splitting processes. For their microscopic form, see [@bib:supplemental]. Here, we only remark that $J_{x}\neq 0$ ($J'_{x}\neq 0$) granted that $\text{Im}\,t_{S}\neq0$ ($\text{Re}\,t_{S}\neq0$). We further point out that both effective Hamiltonian exhibit TRS: For $H_{x,\text{even}}$ both $\hat{x}$ and $\cos\varphi$ are time-reversal even, while for $H_{x,\text{odd}}$ both $\hat{x}$ and $\sin\varphi$ are time-reversal odd. To measure the $\hat{x}$-Pauli operator, we readily see that the effective Hamiltonians are of the same form as those in Eq. . Consequently, our previously introduced measurement protocol for the $\hat{z}$-Pauli operator immediately carries over to the measurement of the $\hat{x}$-Pauli operator.
At this point, it is worth mentioning that a potential error source for the $\hat{x}$, $\hat{z}$-measurements occurs when both $\text{Im}\, t_{N}\neq0$, $\text{Im}\, t_{S}\neq0$ for $n_0$ even or $\text{Re}\, t_{N}\neq0$, $\text{Re}\, t_{S}\neq0$ for $n_{0}$ odd. In practice, this happens either when one of the tunnelling barriers is not fully depleted, or the barrier lengths $d,d'$ are not appropriately adjusted to the spin-orbit length $\lambda_{\text{SO}}$. Fortunately, this constitutes a static hardware error which can be addressed prior to all experiments. In particular, the error can be made controllably small with a careful design of a *conventional* Josephson junction.
In the last part of this section, we address $\hat{y}$-measurements. These require the tuning the charging energy of the TRI TSC to zero which is attainable – on demand – by coupling the TRI TSC island to a bulk SC through a gate-tunable valve [@bib:Aasen2016]. Critically, even at zero charging energy the value of the joint fermion parity in Eq. remains protected as a result of the lead SC gap. However, unlike in the case of a substantial charging energy, Cooper pairs can now tunnel in a second order process between each SC lead and the TRI TSC island inducing a Josephson coupling $\propto\hat{y}$ [@bib:Chung2013]. Consequently, the resulting Josephson current provides a means for measuring the $\hat{y}$ eigenvalue. The details of this measurement scheme are discussed in [@bib:Chung2013].
[*Universal quantum computation.*]{} So far, we have discussed the implementation of single-qubit Clifford gates. However, for universal quantum computation, we need to supplement the single-qubit Clifford gates by a single-qubit $\pi/8$-gate and a two-qubit entangling gate [@bib:Brylinski2001]. Clearly, by pulsing the Josephson couplings in the effective Hamiltonians of Eq. and we can perform arbitrary rotations on the MKQ Bloch sphere and, therefore, in particular, a $\pi/8$-gate. For this procedure, phase-independent contributions – which were irrelevant for the Josephson current – should now be included in the effective Hamiltonians, see [@bib:supplemental].
For a two-qubit entangling gate, we consider the setup of Fig. \[fig:3\] which comprises two SC leads addressing two MKQs $a,b$. Here, a local gate fully depletes both the normal and the spin-flip tunnelling barrier, $t_{N}=t_{S}=0$. Provided that the width of the SC leads is much smaller than the SC coherence length $\xi_{\text{SC}}$, a Cooper pair can now split up between the two TRI TSC islands and generate entanglement between the MKQs. For symmetric couplings and a ground state charge $n_0$ for both TRI TSC islands, we have computed the amplitudes of these processes in the weak coupling limit. An effective anisotropic Heisenberg interaction summarizes the results, $$\begin{split}
\label{HEff_Two_Qubit}
H_{ab} &=J_{y}\hat{y}_{a}\hat{y}_{b}\\
&+[J_{xz}+(-1)^{n_{0}+1}J'_{xz}\cos\varphi] (\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}).
\end{split}$$ For the microscopic form of the coupling constants $J_{xz},J'_{xz},J_{y}$, see [@bib:supplemental]. We note that Heisenberg interaction can be made isotropic by choosing the SC phase difference such that $\tilde{J}\equiv J_{y}=J_{xz}+(-1)^{n_{0}+1}J'_{xz}\cos\varphi$. Pulsing the couplings for a duration $\tau$ defined by $
\int^{\tau}_{0} \tilde{J}(t')\ \mathrm{d}t'= \pi/2 \ (\text{mod}\ \pi)
$ then implements a $\sqrt{\text{SWAP}}$-gate via the unitary time evolution operator. The latter, in combination with single-qubit gates, is sufficient for universal quantum computing [@bib:Loss1998].
![(Color online) Setup of two MKQs $a,b$ coupled to two SC leads. The width of the leads is much smaller than their SC coherence length $\xi_{\text{SC}}$, thereby, permitting Cooper pair splitting between the two MKQs. The resulting anisotropic Heisenberg interaction between the two MKQs is used to construct a two-qubit entangling gate. []{data-label="fig:3"}](Fig3){width="0.8\linewidth"}
[*Conclusions.*]{} In this work, we have introduced a qubit formed by the degenerate ground states of a Coulomb-blockaded TRI TSC island with spatially well separated MKPs: the “Majorana Kramers Qubit". By coupling a single MKQ to SC leads, we have shown that in principle single-qubit Clifford gates can be realized though qubit measurements. Furthermore, we argued that a $\pi/8$-gate as well as a two-MKQ entangling gate can be realized by pulsing of Josephson couplings. Besides providing the conceptual insight that strong magnetic fields are not required for Majorana-based quantum computing, we hope that the MKQ will also provide an alternative route towards a robust quantum computer.
[*Acknowledgments.*]{} We would like to thank Jagadeesh S. Moodera for helpful discussions. C.S. was supported by the Swiss SNF under Project 174980. L.F. and C.S. were supported by DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award $\text{DE-SC0010526}$.
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In the Supplemental Material, we provide more details on the derivation of the effective Hamiltonians required for the implementation of the single- and two-qubit quantum gates.
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Constantin Schrade and Liang Fu\
[*Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139*]{}
In the Supplemental Material, we provide more details on the derivation of the effective Hamiltonians required for the implementation of the single- and two-qubit quantum gates.
Effective Hamiltonian for the single-qubit gates
================================================
In this first section of the Supplemental Material, we derive the effective Hamiltonians for the single-qubit gates as given in Eq. (9) and (10) of the main text. For simplicity, we adopt two assumptions: First, we assume that the lead SC gaps are of equal magnitude, $\Delta\equiv\Delta_{\ell}$. Second, we assume that the gate charge of the TRI TSC island $Q/e$ is tuned to an integer value $n_{0}$. In this, so-called, Coulomb-valley regime the ground state of the TRI TSC island consists of $n_{0}$ units of charge. At the same time, adding/removing a single unit of charge from the TRI TSC island comes at an equal energy cost of $U\equiv e^{2}/2C$.\
Initially, we derive the effective Hamiltonian given in Eq. (9) of the main text that is used for the measurement of the $\hat{z}$-Pauli operator. Our focus are the contributions to the effective Hamiltonian which are most important for the measurement protocol and which occur at third-order in the tunnelling amplitudes. They are given by, $$\begin{split}
H^{(3)}_{z}&=
- P_{n_0} H_{T,z} \left(\left[H_{0}+U_{C}\right]^{-1}\left[1-P_{n_0}\right]H_{T,z}\right)^{2}P_{n_0}.
\label{H3z}
\end{split}$$ Here, $H_{0}=U_{C}+H_{\text{SC}}$ denotes the Hamiltonian of the uncoupled system and $H_{T,z}=H_{T}+H_{N}$ denotes the total tunnelling Hamiltonian. For the moment, we have set $t_{S}=0$. However, after having derived both effective Hamiltonians given in Eqs. (9) and (10) of the main text, we will see that it is sufficient to require $\text{Im}(t_{S})=0$ for $n_0$ even and $\text{Re}(t_{S})=0$ for $n_{0}$ odd. Finally, we note that $P_{n_0}=\Pi_{n_0}\Pi_{\text{BCS}}$ where $\Pi_{n_0}$ is a projector on the ground state of the TRI TSC island with $n_0$ units of charge and $\Pi_{\text{BCS}}$ is a projector on the BCS (Bardeen-Cooper-Schrieffer) ground states of the SC leads.
As the first step in our derivation, we interpret Eq. as the weighted sum of all three-step sequences of intermediate states that map the ground state manifold of $H_{0}$ onto itself. For the moment, we will only discuss the three-step sequences which comprise the transport of a Cooper pair from the left to the right SC lead or vice versa, see Fig. \[fig:1\_SM\]. Two example sequences of the type shown in Fig. \[fig:1\_SM\](a) are, $$\begin{split}
&\quad\
P_{n_{0}}
(
\lambda
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},{\uparrow}}
e^{-i\phi/2}
)
(
\lambda
\gamma_{\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi/2}
)
(
t^{*}_{N}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&=
t^{*}_{N}\lambda^{2}
P_{n_{0}}
(
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},{\uparrow}}
\gamma_{\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&=
(-1)^{n_{0}+1}
t^{*}_{N}\lambda^{2}
(
\Pi_{n_0}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\downarrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}})
\\
&=
i(-1)^{n_{0}}
t^{*}_{N}\lambda^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}})
\\
&=
i(-1)^{n_{0}+1}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t^{*}_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}})
\\
&=
i
(-1)^{n_{0}}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t^{*}_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
\Pi_{\text{BCS}}
\\
\\
&\quad\
P_{n_{0}}
(
\lambda
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{R},{\downarrow}}
e^{-i\phi/2}
)
(
\lambda
\gamma_{\text{L},{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
e^{i\phi/2}
)
(
t_{N}c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
t_{N}\lambda^{2}
P_{n_{0}}
(
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
t_{N}\lambda^{2}
(
\Pi_{n_0}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\downarrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
-i
t_{N}\lambda^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
i
t_{N}\lambda^{2}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}}
)
\\
&=
-it_{N}\lambda^{2}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
\Pi_{\text{BCS}}
\end{split}$$ We remark that in the fifth line of both calculations, we have rewritten the electron operators of SC leads in terms of Bogoliubov quasiparticles, $c_{\ell,{{{\bf{k}}}}{\uparrow}}=e^{i\varphi_{\ell}/2}(u_{{{{\bf{k}}}}}\gamma_{\ell,{{{\bf{k}}}}{\uparrow}}+v_{{{{\bf{k}}}}}\gamma^{\dag}_{\ell,-{{{\bf{k}}}}{\downarrow}})$ and $c_{\ell,-{{{\bf{k}}}}{\downarrow}}=e^{i\varphi_{\ell}/2}(u_{{{{\bf{k}}}}}\gamma_{\ell,-{{{\bf{k}}}}{\downarrow}}-v_{{{{\bf{k}}}}}\gamma^{\dag}_{\ell,{{{\bf{k}}}}{\uparrow}})$. Adding these two sequences as well as their hermitian-conjugated counterparts, multiplying by the energy denominator $-1/(E_{{{{\bf{k}}}}}+U)(2E_{{{{\bf{k}}}}})$ and carrying out the summation over all momenta, yields the contribution, $$\begin{split}
&-\hat{z} J^{\text{(a)}}_{z} \cos\varphi \quad \text{with} \quad J^{\text{(a)}}_{z}=4\lambda^{2}\,\text{Im}(t_{N})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{even},
\\
&\quad\ \hat{z} J'^{\text{(a)}}_{z} \sin\varphi \quad \text{with} \quad J'^{\text{(a)}}_{z}=-4\lambda^{2}\,\text{Re}(t_{N})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{odd}.
\end{split}$$ Here, for notational brevity, we have dropped the projectors on the ground state manifold of $H_{0}$. As the second step, we evaluate the Cooper pair transport sequences corresponding to Figs. \[fig:1\_SM\](b) to (f) in a similar way. This gives, $$\begin{split}
&-\hat{z} J_{z} \cos\varphi \quad \text{with} \quad J_{z}=J^{\text{(a)}}_{z}+J^{\text{(b)}}_{z}+J^{\text{(c)}}_{z}+J^{\text{(d)}}_{z}+J^{\text{(e)}}_{z}+J^{\text{(f)}}_{z} \quad \text{for} \ n_{0} \ \text{even},
\\
&\hspace{12pt}\hat{z} J'_{z} \sin\varphi \quad \text{with} \quad J'_{z}=J'^{\text{(a)}}_{z}+J'^{\text{(b)}}_{z}+J'^{\text{(c)}}_{z}+J'^{\text{(d)}}_{z}+J'^{\text{(e)}}_{z}+J'^{\text{(f)}}_{z} \quad \text{for} \ n_{0} \ \text{odd},
\end{split}$$ where we have introduced the coupling constants, $$\begin{split}
&J^{\text{(a)}}_{z}=J^{\text{(b)}}_{z}=J^{\text{(e)}}_{z}=J^{\text{(f)}}_{z}
\quad , \quad
J^{\text{(c)}}_{z}=J^{\text{(d)}}_{z}=8\lambda^{2}\,\text{Im}(t_{N})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2},
\\
&J'^{\text{(a)}}_{z}=J'^{\text{(b)}}_{z}=J'^{\text{(e)}}_{z}=J'^{\text{(f)}}_{z}
\quad , \quad
J'^{\text{(c)}}_{z}=J'^{\text{(d)}}_{z}=-8\lambda^{2}\,\text{Re}(t_{N})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}.
\end{split}$$ Adding the second order contribution, which corresponds to a conventional Josephson effect through the normal tunnelling barrier, as well as the fourth order contribution, which corresponds to a parity-controlled $2\pi$-Josephson effect through the TRI TSC island [@bib:Schrade2018_SM], we arrive at the effective Hamiltonian given in Eq. (9) of the main text, $$\begin{split}
&H_{z,\text{even}}=-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi \quad, \quad H_{z,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{z} \, J'_{z} \sin\varphi.
\end{split}$$ Up to this point, we have only considered Cooper pair transport sequences which induce terms proportional to the SC phase difference in the effective Hamiltonian. Terms that are independent of the SC phase difference do not modify the Josephson current and, for that reason, have been omitted for the $\hat{z}$-measurement protocol. However, terms which are independent of the SC phase difference are of relevance when pulsing the tunnel couplings to obtain a $\pi/8$-gate. For that reason, we now provide a derivation of these contributions. First, we again examine two example sequence of the type shown in Fig. \[fig:1\_SM\](a), $$\begin{split}
&\quad\
P_{n_{0}}
(
\lambda
\gamma_{\text{R},{\downarrow}}
c_{\text{R},-{{{\bf{k}}}}{\downarrow}}
e^{i\phi/2}
)
(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{L},{\downarrow}}
e^{-i\phi/2}
)
(
t^{*}_{N}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&
=
t^{*}_{N}\lambda^{2}
P_{n_{0}}
(
\gamma_{\text{R},{\downarrow}}
c_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{L},{\downarrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&=
t^{*}_{N}\lambda^{2}
(\lambda
\Pi_{n_0}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\downarrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}})
\\
&=
-i
t^{*}_{N}\lambda^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}})
\\
&=
-i
t^{*}_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}})
\\
&=
i
t^{*}_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
\Pi_{\text{BCS}},
\\
\\
&\quad\
P_{n_{0}}
(
\lambda
\gamma_{\text{R},{\uparrow}}
c_{\text{R},{{{\bf{k}}}}{\uparrow}}
e^{i\phi/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{\uparrow}}
e^{-i\phi/2}
)
(
t_{N}c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
t_{N}\lambda^{2}
P_{n_{0}}
(
\gamma_{\text{R},{\uparrow}}
c_{\text{R},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{\uparrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
(-1)^{n_{0}+1}
t_{N}\lambda^{2}
(
\Pi_{n_0}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\downarrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c_{\text{R},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
i(-1)^{n_{0}}
t_{N}\lambda^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c_{\text{R},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
i(-1)^{n_{0}}
t_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}}
)
\\
&=
i(-1)^{n_{0}+1}
t_{N}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{z}
\Pi_{n_0}
)
\Pi_{\text{BCS}}
\end{split}$$
![(Color online) Third-order sequences of intermediate states (up to hermitian-conjugation) relevant for the single-qubit gates. The enumerated blue lines label the tunnel couplings which are turned on for a given intermediate step within a third-order sequence. []{data-label="fig:1_SM"}](Fig1_SM){width="\linewidth"}
When combining the two sequences with their hermitian-conjugated counterparts, their contributions cancel each other for $n_{0}$ odd but yield a finite contribution for $n_{0}$ even. Incorporating all possible contributions, as shown in Figs. \[fig:1\_SM\](a) to (f), we find that for $n_{0}$ even the effective Hamiltonian changes to, $$\begin{split}
&H_{z,\text{even}}\rightarrow\hat{z}\tilde{J}_{z}-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi \quad \text{with} \quad
\tilde{J}_{z}=\tilde{J}_{z,a}+\tilde{J}_{z,b}+\tilde{J}_{z,c}+\tilde{J}_{z,d}+\tilde{J}_{z,e}+\tilde{J}_{z,f}.
\end{split}$$ Here, we have defined additional coupling constants, $$\begin{split}
&\tilde{J}_{z,a}=\tilde{J}_{z,b}=\tilde{J}_{z,e}=\tilde{J}_{z,f}=-4\lambda^{2}\,\text{Im}(t_{N})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}}
\quad , \quad
\tilde{J}_{z,c}=\tilde{J}_{z,d}=8\lambda^{2}\,\text{Im}(t_{N})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}.
\end{split}$$ We again remark that the effective Hamiltonian for $n_{0}$ odd remains unchanged as a result of the cancellations discussed above.\
We now proceed by deriving the effective Hamiltonian given in Eq. (10) of the main text that is used for measuring the $\hat{x}$-Pauli operator. The all-important third-order contributions to the effective Hamiltonian are given by, $$\begin{split}
H^{(3)}_{x}&=
- P H_{T,x} \left(\left[H_{0}+U_{C}\right]^{-1}\left[1-P\right]H_{T,x}\right)^{2}P,
\label{H3x}
\end{split}$$ where $H_{T,x}=H_{T}+H_{S}$ denotes the total tunnelling Hamiltonian. For now, we have again set $t_{N}=0$. However, it will become clear after this derivation that it is sufficient to require $\text{Im}(t_{N})=0$ for $n_0$ even and $\text{Re}(t_{N})=0$ for $n_{0}$ odd. To begin, we consider two example sequences of the type shown in Fig. \[fig:1\_SM\](a), $$\begin{split}
&\quad\
P_{n_{0}}
(
\lambda
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{R},{\downarrow}}
e^{-i\phi/2}
)
(
\lambda
\gamma_{\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{-i\phi/2}
)
(
t_{S}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&=
t_{S}\lambda^{2}
P_{n_{0}}
(
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
)
P_{n_{0}}
\\
&=
t_{S}\lambda^{2}
(
\Pi_{n_0}
\gamma_{\text{R},{\downarrow}}
\gamma_{\text{L},{\uparrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}}
)
\\
&=
i(-1)^{n_{0}+1}t_{S}\lambda^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}}
)
\\
&=
i(-1)^{n_{0}+1}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t_{S}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}})
\\
&=
i(-1)^{n_{0}}
e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t_{S}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
\Pi_{\text{BCS}},
\\
\\
&\quad\
P_{n_{0}}
(
\lambda
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},{\uparrow}}
e^{-i\phi/2}
)
(
\lambda
\gamma_{\text{L},{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
e^{i\phi/2}
)
(
-
t^{*}_{S}c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
-
t^{*}_{S}\lambda^{2}
P_{n_{0}}
(
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},{\uparrow}}
\gamma_{\text{L},{\downarrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{n_{0}}
\\
&=
-
t^{*}_{S}\lambda^{2}
(
\Pi_{n_0}
\gamma_{\text{R},{\uparrow}}
\gamma_{\text{L},{\downarrow}}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
i
t^{*}_{S}\lambda^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
i
t^{*}_{S}\lambda^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
c_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\Pi_{\text{BCS}}
)
\\
&=
ie^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t^{*}_{S}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
(
\Pi_{\text{BCS}}
\gamma_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\Pi_{\text{BCS}}
)
\\
&=
-
ie^{i(\varphi_{\text{L}}-\varphi_{\text{R}})}
t^{*}_{S}\lambda^{2}
(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}
(
\Pi_{n_0}
\hat{x}
\Pi_{n_0}
)
\Pi_{\text{BCS}}
\end{split}$$ Here, we have made the inessential assumption that the normal state dispersion of the SC obeys $\xi_{{{{\bf{k}}}}}=\xi_{-{{{\bf{k}}}}}$ such that $u_{{{{\bf{k}}}}}=u_{-{{{\bf{k}}}}}$, $v_{{{{\bf{k}}}}}=v_{-{{{\bf{k}}}}}$ and $E_{{{{\bf{k}}}}}=E_{-{{{\bf{k}}}}}$. Adding the two sequencesas well as their hermitian-conjugated counterparts, multiplying by the energy denominator $-1/(E_{{{{\bf{k}}}}}+U)(2E_{{{{\bf{k}}}}})$ and carrying out the summation over all momenta, gives the contribution, $$\begin{split}
&-\hat{x} J^{(\text{a})}_{x} \cos\varphi \quad \text{with} \quad J^{(\text{a})}_{x}=-4\lambda^{2}\,\text{Im}(t_{S})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{even},
\\
&\quad\ \hat{x} J'^{(\text{a})}_{x} \sin\varphi \quad \text{with} \quad J'^{(\text{a})}_{x} =-4\lambda^{2}\,\text{Re}(t_{S})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{odd}.
\end{split}$$ For notational brevity, we have again omitted the projectors on the ground state manifold of $H_{0}$. In a similar way, we also evaluate the Cooper pair transport sequences given in Fig. \[fig:1\_SM\](b) to (f). This yields, $$\begin{split}
&-\hat{x} J_{x} \cos\varphi \quad \text{with} \quad J_{x}=J^{\text{(a)}}_{x}+J^{\text{(b)}}_{x}+J^{\text{(c)}}_{x}+J^{\text{(d)}}_{x}+J^{\text{(e)}}_{x}+J^{\text{(f)}}_{x} \quad \text{for} \ n_{0} \ \text{even},
\\
&\hspace{12pt}\hat{z} J'_{x} \sin\varphi \quad \text{with} \quad J'_{x}=J'^{\text{(a)}}_{x}+J'^{\text{(b)}}_{x}+J'^{\text{(c)}}_{x}+J'^{\text{(d)}}_{x}+J'^{\text{(e)}}_{x}+J'^{\text{(f)}}_{x} \quad \text{for} \ n_{0} \ \text{odd}.
\end{split}$$ Here, we have defined the coupling constants, $$\begin{split}
&J^{\text{(a)}}_{x}=J^{\text{(b)}}_{x}=J^{\text{(e)}}_{x}=J^{\text{(f)}}_{x}
\quad , \quad
J^{\text{(c)}}_{x}=J^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Im}(t_{S})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2},
\\
&J'^{\text{(a)}}_{x}=J'^{\text{(b)}}_{x}=J'^{\text{(e)}}_{x}=J'^{\text{(f)}}_{x}
\quad , \quad
J'^{\text{(c)}}_{x}=J'^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Re}(t_{S})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}.
\end{split}$$ Once we add the second order contribution, which corresponds to a conventional Josephson effect via the spin-flip tunnelling barrier, as well as the fourth order contribution, which corresponds to a parity-controlled $2\pi$-Josephson effect via the TRI TSC island [@bib:Schrade2018_SM], we arrive at the effective Hamiltonian of Eq. (10) in the main text, $$\begin{split}
&H_{x,\text{even}}=-(J_{S}-J+\hat{x}\, J_{x})\cos\varphi \quad, \quad H_{x,\text{odd}}=-(J_{S}+J)\cos\varphi + \hat{x} \, J'_{x} \sin\varphi.
\end{split}$$ So far, we have again only considered Cooper pair transport sequences leading to terms in the effective Hamiltonian that depend on the SC phase difference. Terms which are independent of the SC phase difference do not affect the Josephson current and, for that reason, have been omitted for the $\hat{x}$-measurement protocol. However, those terms are clearly of relevance when pulsing the tunnel couplings to obtain a $\pi/8$-gate. As for the effective Hamiltonian for the $\hat{z}$-measurement protocol, we find that contributions that are independent of the SC phase difference and $\propto\hat{x}$ only occur for $n_{0}$ even. More concretely, the effective Hamiltonian modifies to, $$\begin{split}
&H_{x,\text{even}}\rightarrow\hat{x}\tilde{J}_{x}-(J_{S}-J+\hat{x}\, J_{x})\cos\varphi \quad \text{with} \quad
\tilde{J}_{x}=\tilde{J}^{\text{(a)}}_{x}+\tilde{J}^{\text{(b)}}_{x}+\tilde{J}^{\text{(c)}}_{x}+\tilde{J}^{\text{(d)}}_{x}+\tilde{J}^{\text{(e)}}_{x}+\tilde{J}^{\text{(f)}}_{x},
\end{split}$$ where the additional coupling constants are given by, $$\begin{split}
&\tilde{J}^{\text{(a)}}_{x}=\tilde{J}^{\text{(b)}}_{x}=\tilde{J}^{\text{(e)}}_{x}=\tilde{J}^{\text{(f)}}_{x}=4\lambda^{2}\,\text{Im}(t_{S})
\sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}}
\quad , \quad
\tilde{J}^{\text{(c)}}_{x}=\tilde{J}^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Im}(t_{S})
\sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}.
\end{split}$$ Before closing this first section of the Supplemental Material, we note that the third-order contributions to $H_{z,\text{even}}$ and $H_{x,\text{even}}$ only depend on $\text{Im}(t_{N})$ and $\text{Im}(t_{S})$, respectively. In particular, spin-flip tunnelling contribution with $\text{Re}(t_{S})\neq0$ do not alter the form of $H_{z,\text{even}}$. Similarly, normal tunnelling contribution with $\text{Re}(t_{N})\neq0$ also do not alter the form of $H_{x,\text{even}}$. Hence, we are able to relax our initial assumptions from $t_{S}=0$ to $\text{Im}(t_{S})=0$ for $H_{z,\text{even}}$ and from $t_{N}=0$ to $\text{Im}(t_{N})=0$ for $H_{x,\text{even}}$. Similar arguments apply to $H_{z,\text{odd}}$ and $H_{x,\text{odd}}$. Here, our initial assumption $t_{S}=0$ relaxes to $\text{Re}(t_{S})=0$ for $H_{z,\text{odd}}$ and $t_{N}=0$ relaxes to $\text{Re}(t_{N})=0$ for $H_{x,\text{odd}}$.
Effective Hamiltonian for the two-qubit gates
=============================================
In this second section of the Supplemental Material, we sketch the derivation of the effective Hamiltonian for the two-qubit gates as given in Eq. (11) of the main text. Up to fourth order in the couplings, the general form of the effective Hamiltonian reads, $$\begin{split}
H_{ab}&=
- P_{ab} H_{T,ab} \left[H_{0,ab}^{-1}(1-P_{ab})H_{T,ab}\right]^{3}P_{ab}.
\end{split}$$
![(Color online) Fourth-order sequences of intermediate states (up to hermitian-conjugation and mirror operations $m_{x}$, $m_{y}$) relevant for the two-qubit gate. []{data-label="fig:2_SM"}](Fig2_SM){width="0.9\linewidth"}
Here, we have dropped the second order contribution as it only leads to a constant shift in energy and consequently contributes neither to the entanglement generation between the two TRI TS islands nor the Josephson current between the SC grains. Moreover, $H_{0,ab}$ is the Hamiltonian of the uncoupled system comprised of the two SC leads and the two TRI TSC islands, $H_{T,ab}$ is the tunnelling Hamiltonian between the two SC leads and the TRI TSC islands and $P_{ab}$ is the projector on the reduced Hilbert space consisting of the BCS ground states of the SC leads as well as the charge ground states of both TRI TS islands. For simplicity, we will assume that both the TRI TSC islands as well as all their tunnel couplings to the SC leads are identical. In particular, both TRI TSC islands are tuned to a Coulomb valley with $n_{0}$ units of charge in the ground state and both are coupled to the ground through capacitors of equal capacitance $C$. The tunnelling amplitude between the TRI TSC islands and the SC leads will be denoted by $\lambda$.
To evaluate the effective Hamiltonian, we first list the different types of sequences of intermediate states, see Fig. \[fig:2\_SM\]. Then we compute the effective Hamiltonian for each type of sequence separately. A summation over all the different types then produces the final expression for the effective Hamiltonian, $$\begin{split}
H_{ab}
&=
H^{\text{(a)}}_{ab}
+
H^{\text{(b)}}_{ab}
+
H^{\text{(c)}}_{ab}
+
H^{\text{(d)}}_{ab}
+
H^{\text{(e)}}_{ab}
+
H^{\text{(f)}}_{ab}
\\
&+
H^{\text{(g)}}_{ab}
+
H^{\text{(h)}}_{ab}
+
H^{\text{(i)}}_{ab}
+
H^{\text{(j)}}_{ab}
+
H^{\text{(k)}}_{ab}
+
H^{\text{($\ell$)}}_{ab}
\end{split}$$ Before going into the details of the derivation, we first display the full result for all types of contributions, $$\begin{split}
&H^{\text{(a)}}_{ab}
=
(-1)^{n_{0}+1}2J_{1}\cos\varphi \quad, \quad
H^{\text{(b)}}_{ab}
=
(-1)^{n_{0}+1}2J_{2}\cos\varphi \quad, \quad
H^{\text{(c)}}_{ab}
=
(-1)^{n_{0}+1}2J_{3}\cos\varphi
\\
&H^{\text{(d)}}_{ab}
=
[(-1)^{n_{0}+1}J_{1}\cos\varphi-J_{4}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \quad, \quad
H^{\text{(e)}}_{ab}
=
[(-1)^{n_{0}+1}J_{2}\cos\varphi-J_{5}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})
\\
&H^{\text{(f)}}_{ab}
=
[(-1)^{n_{0}+1}J_{7}\cos\varphi+J_{6}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \quad\ , \quad
H^{\text{(g)}}_{ab}
=
[(-1)^{n_{0}+1}J_{9}\cos\varphi+J_{8}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})
\\
&
H^{\text{(h)}}_{ab}=
[(-1)^{n_{0}+1}J_{11}\cos\varphi-J_{10}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})\hspace{5pt} , \quad
H^{\text{(i)}}_{ab}
=
[(-1)^{n_{0}+1}J_{11}\cos\varphi+J_{12}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})
\\
&H^{\text{(j)}}_{ab}
=
-J_{13}\hat{y}_{a}\hat{y}_{b}
+J_{14}\hat{y}_{a}\hat{y}_{b} \quad, \quad
H^{\text{(k)}}_{ab}
=
J_{15}\hat{y}_{a}\hat{y}_{b}
+
J_{16}\hat{y}_{a}\hat{y}_{b} \quad, \quad
H^{\text{($\ell$)}}_{ab}
=
-J_{17}\hat{y}_{a}\hat{y}_{b}
-J_{18}\hat{y}_{a}\hat{y}_{b}.
\end{split}$$ Here, we have introduced the coupling constants $$\begin{split}
&J_{1}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})}
\hspace{75pt}, \quad
J_{2}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{\bf{q}}}}+U)}
\\
&J_{3}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)(4U)(E_{{{\bf{q}}}}+U)}
\hspace{65pt}, \quad
J_{4}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})
}
\\
&J_{5}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2}
}
{
(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)
}
\hspace{44pt}, \quad
J_{6}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)
}
\\
&J_{7}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}}
}
{
(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)
}
\hspace{52pt},\quad
J_{8}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{\bf{q}}}}+U)
}
\\
&J_{9}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}}
}
{
(E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{\bf{q}}}}+U)
}
\hspace{17pt},\quad
J_{10}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U)
}
\\
&J_{11}=16\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}}
}
{
(E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U)
}
\hspace{64pt},\quad
J_{12}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U)
}
\\
&J_{13}=
8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U)
}
\hspace{68pt},\quad
J_{14}=
8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)
}
\\
&J_{15}=
8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)
}
\hspace{68pt},\quad
J_{16}=
8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{{\bf{k}}}}}+U)
}
\\
&J_{17}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)
}
\hspace{43pt},\quad
J_{18}=8\lambda^{4}
\sum_{{{{\bf{k}}}},{{\bf{q}}}}
\frac{
(u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2}
}
{
(E_{{{{\bf{k}}}}}+U)^{2}
(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})
}.
\end{split}$$ As a next step, we collect the different contributions. This simplifies the expression of the effective Hamiltonian to $$\begin{split}
H_{ab}
=
(-1)^{n_{0}+1}J_{0}\cos\varphi
+
[(-1)^{n_{0}+1}J'_{xz}\cos\varphi+J_{xz}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})
+
J_{y}\hat{y}_{a}\hat{y}_{b}
\end{split}$$ with the coupling constants $$\begin{split}
J_{0}&=J_{1}+J_{2}+J_{3}
\\
J'_{xz}&=J_{1}+J_{2}+J_{7}+J_{9}+2J_{11}
\\
J_{xz}&=-J_{4}-J_{5}+J_{6}+J_{8}-J_{10}+J_{12}
\\
J'_{y}&=J_{13}+J_{14}-J_{15}+J_{16}+J_{17}-J_{18}.
\end{split}$$ Now that we have presented the full expression for the effective Hamiltonian we will give an overview of the derivation of the individual contributions.
Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{1},J_{2},J_{7},J_{9},J_{11}}$
---------------------------------------------------------------------------------------------------------------------
In this first subsection, we discuss sequences of intermediate states which lead to contributions $\propto J_{1},J_{2},J_{7},J_{9},J_{11}$ in the effective Hamiltonian. Initially, we will present two examples which yield contributions $\propto J_{1}$ in the effective Hamiltonian. Subsequently, we will explain how additional examples for the contributions $\propto J_{2},J_{7},J_{9},J_{11}$ can be obtained from these findings. Our first example is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{b,\text{R},{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{b,\text{R},{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
\gamma_{a,\text{R},{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
\gamma_{b,\text{R},{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
)
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}+1}P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}}
e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})}
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{z}_{a}\hat{z}_{b}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{R},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}}
e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})}
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{z}_{a}\hat{z}_{b}
P_{ab}
\label{c1}
\end{split}$$ and thus leads to a term $\propto\hat{z}_{a}\hat{z}_{b}$. In the second example the coupling to the MKPs is different compared to first example,
$$\begin{split}
&\quad\
P_{ab}(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{b,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{b,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{b,\text{R},{\uparrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{R},{\downarrow}}
)
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}}
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c_{\text{R},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}}
e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})}
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{x}_{a}\hat{x}_{b}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
(-1)^{n_{0}}
e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})}
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{x}_{a}\hat{x}_{b}
P_{ab}
\label{c2}
\end{split}$$
which leads to a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. Hence, after combining the above results with the corresponding hermitian-conjugated sequences and summing over all momenta, we indeed find a contribution $\propto J_{1}$.
Finally, we point out that examples for contributions $\propto J_{2},J_{7},J_{9},J_{11}$ can be obtained by suitably commuting the terms in the first line of Eq. and Eq. . The different coupling constants arise because the energy denominators for the resulting processes will be different than the ones we used for the two examples given above.
Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{4},J_{5},J_{6},J_{8},J_{10},J_{12}}$
----------------------------------------------------------------------------------------------------------------------------
In this second subsection, we continue our overview on the sequences of intermediate states which contribute to the effective Hamiltonian. More specifically, we will examine sequences that give contributions $\propto J_{4},J_{5},J_{6},J_{8},J_{10},J_{12}$. As a first step, we introduce two examples which produce contributions $\propto J_{4}$. The first example is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
\gamma_{a,\text{R},{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
\gamma_{b,\text{L},{\uparrow}}
)
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{R},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
P_{ab}
\label{c3}
\end{split}$$ and it leads to a term $\propto\hat{z}_{a}\hat{z}_{b}$. The second example is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{R},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{R},{\downarrow}}
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{R},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{R},{\downarrow}}
)
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
u_{{{{\bf{k}}}}}
v_{{{\bf{q}}}}
)^{2}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{R},-{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{R},-{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
u_{{{{\bf{k}}}}}
v_{{{\bf{q}}}}
)^{2}
P_{ab},
\label{c4}
\end{split}$$ and gives a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. After carrying out the summation over all momenta, we see that both examples indeed contribute to the term $\propto J_{4}$ in the effective Hamiltonian. Moreover, we remark that examples for sequences of intermediate states that give contributions $\propto J_{5}$ and $\propto J_{10}$ can be obtained by appropriately commuting the terms in the round brackets in the first line of Eqs. and . The required energy denominator for the examples $\propto J_{5}$ is $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)]$ and for examples $\propto J_{10}$ it is $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$.
We now proceed by presenting two examples of sequences of intermediate states leading to contributions $\propto J_{6}$. The first example is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{R},{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{a,\text{R},{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
\gamma_{b,\text{R},{\uparrow}}
\gamma_{b,\text{L},{\uparrow}}
)
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
u_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
(
\gamma_{\text{L},{{\bf{q}}}{\uparrow}}
\gamma_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{z}_{a}\hat{z}_{b}
(
u_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
P_{ab}
\label{c5}
\end{split}$$ leading to a term $\propto\hat{z}_{a}\hat{z}_{b}$ and the second example is given $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{R},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{R},{\downarrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{R},{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{R},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{R},{\downarrow}}
)
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
u_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
)^{2}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{x}_{a}\hat{x}_{b}
(
u_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
)^{2}
P_{ab}
\label{c6}
\end{split}$$ producing a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)]$. Hence, once we have summed over all momenta, we conclude that both examples give contributions $\propto J_{6}$. Finally, we remark that examples for contributions $\propto J_{8}$ and $\propto J_{12}$ can be obtained by appropriately commuting the terms in the round brackets in the first line of Eqs. and . The only difference occurs in the energy denominator. The latter is given by $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{{\bf{k}}}}}+U)]$ for the contributions $\propto J_{8}$ and by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$ for the contributions $\propto J_{12}$.
Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{13},J_{15}}$
----------------------------------------------------------------------------------------------------
In this third subsection we discuss sequences of intermediate states that lead to contributions $\propto J_{13}, J_{14}$ in the effective Hamiltonian. An examples that lead to a contribution $\propto J_{13}$ is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{L},{\downarrow}}
)
(
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{y}_{a}\hat{y}_{b}
(
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
\hat{y}_{a}\hat{y}_{b}
(
\gamma_{\text{L},-{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},-{{\bf{q}}}{\uparrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
\hat{y}_{a}\hat{y}_{b}
P_{ab}.
\label{c9}
\end{split}$$ and gives a term $\propto\hat{y}_{a}\hat{y}_{b}$. The energy denominator for thhis sequences is given by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$. Hence, after summing over all momenta we verify that both sequences indeed contribute to the term $\propto J_{13}$ in the effective Hamiltonian.
Providing examples of sequences that give contributions $\propto J_{15}$ in the effective Hamiltonian requires us to swap the first two terms in the round brackets in the first line of Eq. . However, these terms do not commute. Hence, we need to re-evaluate the modified sequences. We find that, $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{L},{\downarrow}}
)
(
c_{\text{L},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{y}_{a}\hat{y}_{b}
(
c_{\text{L},{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
c_{\text{L},{{{\bf{k}}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
u_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
\hat{y}_{a}\hat{y}_{b}
(
\gamma_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}}
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
u_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
)^{2}
\hat{y}_{a}\hat{y}_{b}
P_{ab}.
\label{c11}
\end{split}$$ The energy denominator for this sequence is still given by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$. Consequently, after summing over all momenta, we recognize that the example given in Eq. contributes to the term $\propto J_{15}$ in the effective Hamiltonian.
Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{14},J_{16}}$
----------------------------------------------------------------------------------------------------
In this fourth subsection, we discuss sequences of intermediate states that contribute to the terms $\propto J_{14},J_{16}$ in the effective Hamiltonian. We begin by presenting an examples fo a sequence of intermediate states that yield contribution $\propto J_{14}$ in the effective Hamiltonian, $$\begin{split}
&\quad\
P_{ab}(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{L},{\downarrow}}
)
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
\hat{y}_{a}\hat{y}_{b}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{y}_{a}\hat{y}_{b}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},-{{\bf{q}}}{\downarrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{y}_{a}\hat{y}_{b}
P_{ab}
\label{c14}
\end{split}$$ The energy denominator for this example is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)]$. This means that after summing over all momenta, both sequences indeed contribute to the term $\propto J_{14}$ in the effective Hamiltonian.
Finally, we note that examples for sequences of intermediate states leading to contributions $\propto J_{16}$ can be obtained by commuting the first two terms in the round brackets in the first line of Eq. and adapting the energy denominators accordingly.
Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{17},J_{18}}$
----------------------------------------------------------------------------------------------------
In this final subsection, we give examples on sequences of intermediate states which lead to contributions $\propto J_{17}, J_{18}$ in the effective Hamiltonian. An example leading to a contribution $\propto J_{18}$ is given by $$\begin{split}
&\quad\
P_{ab}(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
)
P_{ab}
\\
&=
P_{ab}\lambda^{4}
(
\gamma_{a,\text{L},{\uparrow}}
\gamma_{a,\text{L},{\downarrow}}
\gamma_{b,\text{L},{\uparrow}}
\gamma_{b,\text{L},{\downarrow}}
)
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
\hat{y}_{a}\hat{y}_{b}
(
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{y}_{a}\hat{y}_{b}
(
\gamma_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{\text{L},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{\bf{q}}}{\uparrow}}
\gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{y}_{a}\hat{y}_{b}
P_{ab}.
\label{c16}
\end{split}$$ The energy denominator is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. Hence, after summing over all momenta, we find a contribution $\propto J_{18}$.
An example for a sequence of intermediate states that yields a contribution $\propto J_{17}$ can be obtained by swapping the first two terms in round brackets in the first line of Eq. . More concretely, $$\begin{split}
&\quad\
P_{ab}(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
P_{ab}(
\lambda
c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}}
\gamma_{b,\text{L},{\downarrow}}
e^{-i\phi_{2}/2}
)
(
\lambda
\gamma_{b,\text{L},{\uparrow}}
c_{\text{L},{{\bf{q}}}{\uparrow}}
e^{i\phi_{2}/2}
)
(
\lambda
\gamma_{a,\text{L},{\downarrow}}
c_{\text{L},-{{\bf{q}}}{\downarrow}}
e^{i\phi_{1}/2}
)
(
\lambda
c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}}
\gamma_{a,\text{L},{\uparrow}}
e^{-i\phi_{1}/2}
)
P_{ab}
\\
&=
-
P_{ab}\lambda^{4}
v_{{{{\bf{k}}}}}
u_{{{\bf{q}}}}
v_{{{\bf{q}}}}
u_{{{{\bf{k}}}}}
\hat{y}_{a}\hat{y}_{b}
P_{ab}.
\end{split}$$ The energy denominator is now given by $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)]$. Consequently, after summing over all momenta, we find a contribution $\propto J_{17}$ in the effective Hamiltonian.\
\
In summary, in this second section of the Supplemental Material, we have provided an extensive overview of the numerous contributions which make up the effective Hamiltonian for the two-qubit gate as given in Eq. (11) of the main text.
[99]{}
C. Schrade and L. Fu, Phys. Rev. Lett. **120**, 267002 (2018).
| ArXiv |
---
abstract: 'We unify two recent results concerning equilibration in quantum theory. We first generalise a proof of Reimann \[PRL 101,190403 (2008)\], that the expectation value of ‘realistic’ quantum observables will equilibrate under very general conditions, and discuss its implications for the equilibration of quantum systems. We then use this to re-derive an independent result of Linden et. al. \[PRE 79, 061103 (2009)\], showing that small subsystems generically evolve to an approximately static equilibrium state. Finally, we consider subspaces in which all initial states effectively equilibrate to the same state.'
author:
- 'Anthony J. Short'
title: Equilibration of quantum systems and subsystems
---
Introduction
============
Recently there has been significant progress in understanding the foundations of statistical mechanics, based on fundamentally quantum arguments [@Mahler; @Goldstein1; @Goldstein2; @PopescuShortWinter; @Tasaki; @reimann1; @reimann2; @us1; @us2; @gogolin1; @gogolin2]. In particular, Reimann [@reimann1; @reimann2] has shown that the expectation value of any ‘realistic’ quantum observable will equilibrate to an approximately static value, given very weak assumptions about the Hamiltonian and initial state. Interestingly, the same assumptions were used independently by Linden *et al.* [@us1; @us2], to prove that any small quantum subsystem will evolve to an approximately static equilibrium state (such that even ‘unrealistic’ observables on the subsystem equilibrate). In this paper we unify these two results, by deriving the central result of Linden *et al.*[@us1] from a generalisation of Reimann’s result. We also offer a further discussion and extension of Reimann’s results, showing that systems will appear to equilibrate with respect to all reasonable experimental capabilities. Finally, we identify subspaces of initial states which equilibrate to the same state.
Equilibration of expectation values.
====================================
We prove below a generalisation of Reimann’s result that the expectation value of any operator will almost always be close to that of the equilibrium state [@reimann1]. We extend his results to include non-Hermitian operators (which we will use later to prove equilibration of subsystems), correct a subtle mistake made in [@reimann2] when considering degenerate Hamiltonians, and improve the bound obtained by a factor of 4. As in [@reimann2; @us2], we make one assumption in the proof, which is that the Hamiltonian has *non-degenerate energy gaps*. This means that given any four energy eigenvalues $E_k, E_l, E_m$ and $E_n$, $$\label{eq:non-degen}
E_k - E_l = E_m - E_n \Rightarrow \begin{array}{c} (E_k = E_l \; \textrm{and}\; E_m = E_n) \\ \textrm{or} \\ (E_k = E_m \; \textrm{and}\; E_l = E_n). \end{array}$$ Note that this definition allows degenerate energy levels, which may arise due to symmetries. However, it ensures that all subsystems physically interact with each other. In particular, given any decomposition of the system into two subsystems ${\mathcal{H}}= {\mathcal{H}}_A \otimes{\mathcal{H}}_B$, equation (\[eq:non-degen\]) will not be satisfied by any Hamiltonian of the form $H=H_A \otimes I_B + I_A \otimes H_B$ (unless either $H_A$ or $H_B$ is proportional to the identity) [^1].
Consider a $d$-dimensional quantum system evolving under a Hamiltonian $H=\sum_n E_n P_n$, where $P_n$ is the projector onto the eigenspace with energy $E_n$. Denote the system’s density operator by $\rho(t)$, and its time-averaged state by $\omega \equiv {\left\langle \rho(t) \right\rangle_t}$. If $H$ has non-degenerate energy gaps, then for any operator $A$, $$\label{eq:theorem}
\sigma_A^2 \equiv {\left\langle \left| {\operatorname{tr}}\left(A \rho\left(t\right) \right) - {\operatorname{tr}}\left( A \omega\right) \right|^2 \right\rangle_t} \leq \frac{\Delta(A)^2 }{4 d_{{\rm eff}}} \leq \frac{\|A\|^2}{d_{{\rm eff}}}$$ where $\|A\|$ is the standard operator norm [^2], $$\Delta(A) \equiv 2 \min_{c \in \mathbb{C}} \| A- c I \|,$$ and $$d_{{\rm eff}} \equiv \frac{1}{\sum_n \big( {\operatorname{tr}}(P_n \rho(0)) \big)^2}.$$
This bound will be most significant when the number of different energies incorporated in the state, characterised by the effective dimension $ d_{{\rm eff}}$, is very large. Note that $1 \leq d_{{\rm eff}} \leq d$, and that $d_{{\rm eff}}=N$ when a measurement of $H$ would yield $N$ different energies with equal probability. For pure states $d_{{\rm eff}} = {\operatorname{tr}}(\omega^2)^{-1}$ as in [@us1; @us2], but it may be smaller for mixed states when the Hamiltonian is degenerate.
The quantity $\Delta(A) $ gives the range of eigenvalues when $A$ is Hermitian, and gives a slightly tighter bound than the operator norm. Following [@reimann2], we could improve the bound further by replacing $\Delta(A) $ with a state- and Hamiltonian-dependent term [^3], however we omit this step here for simplicity.
**Proof:** To avoid some difficulties which arise when considering degenerate Hamiltonians, we initially consider a pure state $\rho(t) = {{| \psi(t) \rangle}\!{\langle \psi(t) |}}$, then extend the results to mixed states via purification.
We can always choose an energy eigenbasis such that ${| \psi(t) \rangle}$ has non-zero overlap with only a single energy eigenstate ${| n \rangle}$ of each distinct energy, by including states ${| n \rangle} = P_n {| \psi(0) \rangle}/\sqrt{{\langle \psi(0) |} P_n {| \psi(0) \rangle}}$ whenever ${\langle \psi(0) |} P_n {| \psi(0) \rangle}>0$. The state at time $t$ is then given by $${| \psi(t) \rangle} = \sum_{n} c_n e^{-i E_n t/\hbar} {| n \rangle},$$ where $c_n = {\left\langle n| \psi(0) \right\rangle}$. This state will evolve in the subspace spanned by $\{{| n \rangle}\}$ as if it were acted on by the non-degenerate Hamiltonian $H'=\sum_n E_n {{| n \rangle}\!{\langle n |}}$. For any operator $A$, it follows that $$\begin{aligned}
\sigma_A^2\!\!\! &=& {\left\langle |{\operatorname{tr}}(A [\rho(t) - \omega] )|^2 \right\rangle_t} \nonumber \\
&=& {\left\langle \left|\sum_{n \neq m} c_n c_m^* e^{i(E_m-E_n)t/\hbar} {\langle m |} A {| n \rangle} \right|^2 \right\rangle_t} \nonumber \\
&=& \!\!\!\! \sum_{\scriptsize \begin{array}{c} n \neq m \\ k\neq l \end{array}} \!\!\! \! c_n c_m^* c_k c_l^*{\left\langle e^{i(E_m-E_n + E_l - E_k)t/\hbar} \right\rangle_t}{\langle m |} A { | n \rangle \! \langle l |} A^{\dag} {| k \rangle} \nonumber \\
&=& \sum_{n,m} |c_n|^2 |c_m|^2 {\langle m |} A { | n \rangle \! \langle n |} A^{\dag} {| m \rangle} - \sum_{n} |c_n|^4 |{\langle n |}A {| n \rangle}|^2 \nonumber \\
& \leq &{\operatorname{tr}}( A \omega A^{\dag} \omega ) \nonumber \\
& \leq & \sqrt{{\operatorname{tr}}(A^{\dag}\!A\, \omega^2) {\operatorname{tr}}(A A^{\dag} \omega^2)} \nonumber \\
&\leq& \| A \|^2 {\operatorname{tr}}(\omega^2) \label{eq:pure_theorem} \nonumber \\
&=& \| A \|^2{\operatorname{tr}}\left[ \left(\sum_n |c_n|^2 {{| n \rangle}\!{\langle n |}}\right)^2\right] \nonumber \\
&=& \| A \|^2 \sum_n \big( {\operatorname{tr}}(P_n \rho(0)) \big)^2 \nonumber \\
&=& \frac{\| A \|^2}{d_{{\rm eff}}}. \end{aligned}$$ In the fourth line, we have used the assumption that the Hamiltonian has non-degenerate energy gaps, in the sixth line we have used the Cauchy-Schwartz inequality for operators with scalar product ${\operatorname{tr}}(A^{\dag} B)$ and the cyclic symmetry of the trace, and in the seventh line we have used the fact that for positive operators $P$ and $Q$, ${\operatorname{tr}}(PQ) \leq \|P\| {\operatorname{tr}}(Q) $. This gives the weaker bound in the theorem.
To obtain the tighter bound, we note that $\sigma_A$ is invariant if $A$ is replaced by $\tilde{A} = A- c I$ for any complex $c$. Performing this substitution with $c$ chosen so as to minimize $\|\tilde{A}\|$ we can replace $\|A\|$ with $\|\tilde{A}\|=\Delta(A)/2$.
An extension to mixed states can be obtained via purification, following the approach discussed in [@us2]. Given any initial state $\rho(0)$ on ${\mathcal{H}}$, we can always define a pure state ${| \phi(0) \rangle}$ on ${\mathcal{H}}\otimes {\mathcal{H}}$ such that the reduced state of the first system is $\rho(0)$. By evolving ${| \phi(t) \rangle}$ under the joint Hamiltonian $H'=H \otimes I$, we will recover the correct evolution $\rho(t)$ of the first system, and $H'$ will have non-degenerate energy gaps whenever $H$ does. The expectation value of any operator $A$ for $\rho(t)$ will be the same as the expectation value of $A'=A \otimes I$ on the total system, and we also obtain $\Delta(A')=\Delta(A)$, $\|A\|=\|A'\|$, and $d_{{\rm eff}}'=d_{{\rm eff}}$. However, note that ${\operatorname{tr}}(\omega'^2)$ does not equal ${\operatorname{tr}}(\omega^2)$. Using the result for pure states, we can obtain (\[eq:theorem\]) in the mixed state case from $$\sigma_A^2 = \sigma_{A'}^2 \leq \frac{\Delta(A')^2 }{4 d_{{\rm eff}}'} =\frac{\Delta(A)^2 }{4 d_{{\rm eff}}}.$$ This completes the proof. $\square$
In [@reimann1], Reimann proves that $\sigma_A^2 \leq \Delta(A)^2 {\operatorname{tr}}(\omega^2)$ when $A$ is Hermitian and the Hamiltonian has non-degenerate levels as well as non-degenerate gaps. However, it appears that there is a subtle mistake in [@reimann2] when extending this proof to degenerate Hamiltonians. Specifically, the step from equation (D.11) to (D.12) in [@reimann2] does not follow if the state has support on more than one energy eigenstate in a degenerate subspace. A counterexample is provided by the mixed state $\rho(0) = \frac{1}{k} {{| 0 \rangle}\!{\langle 0 |}} \otimes {I}$, of a qubit and a $k$-dimensional system, with $H=({ | 0 \rangle \! \langle 1 |} + { | 1 \rangle \! \langle 0 |}) \otimes {I}$ and $A = ({{| 0 \rangle}\!{\langle 0 |}}-{{| 1 \rangle}\!{\langle 1 |}}) \otimes {I}$. In this case $\sigma^2_A = \frac{1}{2}$, $\Delta(A)=2$ and ${\operatorname{tr}}(\omega^2) = \frac{1}{2k}$, giving $\sigma_A^2 > \Delta(A)^2 {\operatorname{tr}}(\omega^2)$ when $k>4$. However, subsequently in [@reimann2], $ {\operatorname{tr}}(\omega^2)$ is replaced by an upper bound of $\max_n {\operatorname{tr}}(\rho(0) P_n)$, and this also upper bounds $d_{{\rm eff}}^{-1}$, so later results are unaffected. Note that the bound given by Theorem 1 for the same example is satisfied tightly for all $k$, as $ d_{{\rm eff}}=2$ and thus $\sigma_A^2 = \frac{1}{2} = \frac{\Delta(A)^2}{4 d_{{\rm eff}}}$.
Distinguishability
==================
When $A$ represents a physical observable and $\rho(0)$ a realistic initial state, it is argued in [@reimann1] that the difference between ${\operatorname{tr}}(A \rho(t))$ and ${\operatorname{tr}}(A \omega)$ will almost always be less than realistic experimental precision. This is then taken to imply that $\rho(t)$ will be indistinguishable from $\omega$ for the overwhelming majority of times.
However, the fact that two states yield the same expectation value for a measurement does not necessarily imply that they cannot be distinguished by it. For example, a measurement yielding an equal mixture of $+1$ and $-1$ outcomes for one state and always yielding $0$ for a second state clearly can distinguish the two states, despite the expectation values in the two cases being identical. Furthermore, even though any particular realistic measurement cannot distinguish $\rho(t)$ from $\omega$ for almost all times, this does not imply that for almost all times, no realistic measurement can distinguish $\rho(t)$ from $\omega$. This is because the optimal measurement to distinguish the two states may change over time. Finally, the measurement precision is not easy to define for measurements with discrete outcomes.
To address these issues, we first note that the most general quantum measurement is not described by a Hermitian operator, but by a positive operator valued measure (POVM). For simplicity, we consider POVMs with a finite set of outcomes, which is reasonable for realistic measurements, as even continuous outputs such as pointer position cannot be determined or recorded with infinite precision [^4]. A general measurement $M$ is described by giving a positive operator $M_r$ for each possible measurement result $r$, satisfying $\sum_r M_r= I$. The probability of obtaining result $r$ when measuring $M$ on $\rho$ is given by ${\operatorname{tr}}(M_r \rho)$.
Suppose you are given an unknown quantum state, which is either $\rho_1$ or $\rho_2$ with equal probability. Your maximum success probability in guessing which state you were given after performing the measurement $M$ is $$p^{\textrm{succ}}_{M} = \frac{1}{2} ( 1 + D_{M} (\rho_1, \rho_2))$$ where $$D_{M} (\rho_1, \rho_2) \equiv \frac{1}{2} \sum_{r} | {\operatorname{tr}}(M_r \rho_1) - {\operatorname{tr}}(M_r \rho_2) |.$$ We refer to $D_{M} (\rho_1, \rho_2)$ as the distinguishability of $\rho_1$ and $\rho_2$ using the measurement $M$. Similarly, the distinguishability of two states using any measurement from a set ${\mathcal{M}}$ is given by $$D_{{\mathcal{M}}} (\rho_1, \rho_2) \equiv \max_{M \in {\mathcal{M}}} D_{M} (\rho_1, \rho_2).$$ Note that $$0 \leq D_{{\mathcal{M}}} (\rho_1, \rho_2) \leq D(\rho_1, \rho_2) \leq 1,$$ where $D(\rho_1, \rho_2) = \frac{1}{2} {\operatorname{tr}}|\rho_1 - \rho_2|$ is the trace-distance, which is equal to $D_{{\mathcal{M}}} (\rho_1, \rho_2)$ when $\mathcal{M}$ includes all measurements.
Effective equilibration of large systems
========================================
For typical macroscopic systems, the dimension of ${\mathcal{H}}$ will be incredibly large (e.g. For Avagardo’s number $N_A$ of spin-$\frac{1}{2}$ particles, we would have $d >10^{10^{23}}$), and it is unrealistic to be able to perform any measurement with this many outcomes, let alone all such measurements. For practical purposes, we are therefore restricted to some set of realistic physical measurements ${\mathcal{M}}$. In this case, we would expect ${\mathcal{M}}$ to be a finite set, as all realistic experimental setups (including all settings of variable parameters) will be describable within a finite number of pages of text.
We say that a state *effectively equilibrates* if $${\left\langle D_{{\mathcal{M}}} (\rho(t), \omega) \right\rangle_t} \ll 1.$$ This means that for almost all times, it is almost impossible to distinguish the true state $\rho(t)$ from the equilibrium state $\omega$ using any achievable measurement.
We can obtain an upper bound on the average distinguishability as a corollary of theorem 1.
Consider a quantum system evolving under a Hamiltonian with non-degenerate energy gaps. The average distinguishability of the system’s state $\rho(t)$ from $\omega$, given a finite set of measurements ${\mathcal{M}}$, satisfies $${\left\langle D_{{\mathcal{M}}} (\rho(t), \omega) \right\rangle_t} \leq \frac{\sum_{M \in {\mathcal{M}}} \sum_{r } \Delta(M_r)}{4\sqrt{d_{{\rm eff}}}} \leq \frac{N({\mathcal{M}})}{4\sqrt{d_{{\rm eff}}}}, \label{eq:effective_equilibration}$$ where $N({\mathcal{M}})$ is the total number of outcomes for all measurements in ${\mathcal{M}}$.
The first bound will be tighter when measurements are imprecise, as each outcome is weighted by $\Delta(M_r) \in [0,1]$, reflecting its usefulness in distinguishing states [^5].
**Proof:** $$\begin{aligned}
{\left\langle D_{{\mathcal{M}}} (\rho(t), \omega) \right\rangle_t} &=& {\left\langle \max_{M(t) \in {\mathcal{M}}} D_{M(t)} (\rho(t), \omega) \right\rangle_t} \nonumber\\
&\leq& \sum_{M \in {\mathcal{M}}} {\left\langle D_M (\rho(t), \omega) \right\rangle_t} \nonumber\\
&=&\frac{1}{2} \sum_{M \in {\mathcal{M}}} \sum_{r } {\left\langle | {\operatorname{tr}}(M_r \rho(t) ) - {\operatorname{tr}}(M_r \omega) | \right\rangle_t} \nonumber\\
&\leq&\frac{1}{2} \sum_{M \in {\mathcal{M}}} \sum_{r } \sqrt{ \sigma_{M_r}^2 } \nonumber\\
&\leq & \frac{ \sum_{M \in {\mathcal{M}}} \sum_{r } \Delta(M_r)}{4\sqrt{d_{{\rm eff}}}} \nonumber\\
&\leq & \frac{N({\mathcal{M}})}{4\sqrt{d_{{\rm eff}}}}. \label{eq:distinguishability} \qquad \square\end{aligned}$$
In realistic experiments, we would expect the bound on the right of (\[eq:effective\_equilibration\]) to be much smaller than 1, implying that the state of the system effectively equilibrates to $\omega$. Consider again our system of $N_A$ spins. If $d_{{\rm eff}}\geq~d^{\,0.1} $, even if we take ${\mathcal{M}}$ to include any experiment whose description could be written in $10^{19}$ words, each of which generates up to $10^{21}$ bytes of data, we would still obtain ${\left\langle D_{{\mathcal{M}}} (\rho(t), \omega) \right\rangle_t} \leq 1/(10^{10^{22}})$.
Equilibration of small subsystems
=================================
Now consider that the system can be decomposed into two parts, a small subsystem of interest $S$, and the remainder of the system which we refer to as the bath $B$. Then ${\mathcal{H}}= {\mathcal{H}}_S \otimes {\mathcal{H}}_B$, where ${\mathcal{H}}_{S/B}$ has dimension $d_{S/B}$. It is helpful to define the reduced states of the subsystem $\rho_S(t) = {\operatorname{tr}}_B(\rho(t))$ and $\omega_S = {\operatorname{tr}}_B(\omega)$.
In such cases, it was shown in [@us1; @us2] that for sufficiently large $d_{{\rm eff}}$ the subsystem’s state fully equilibrates, such that for almost all times, no measurement on the subsystem (even ‘unrealistic’ ones) can distinguish $\rho(t)$ from $\omega$. In particular, when $\rho(t)$ is pure and the Hamiltonian has non-degenerate energy levels as well as non-degenerate energy gaps, it is proven in [@us1] that $$\label{eq:oureqn}
{\left\langle D(\rho_S(t), \omega_S) \right\rangle_t} \leq \frac{1}{2} \sqrt{\frac{d_S^2}{d_{{\rm eff}}}}.$$ Extending this result to degenerate Hamiltonians and initially mixed states is discussed in [@us2].
We cannot recover this bound directly from (\[eq:effective\_equilibration\]) by considering the set of all measurements on the subsystem, because this set contains an infinite number of measurements. However, we can derive (\[eq:oureqn\]) from Theorem 1 by considering an orthonormal operator basis for the subsystem, given by the $d_S^2$ operators [@schwinger] $$F_{(d_Sk_0 + k_1)} = \frac{1}{\sqrt{d_S}} \sum_{l} e^{\frac{2 \pi i l k_0}{d_S}} {| (l+k_1)\, \textrm{mod}\, d_S \rangle} {\langle l |}$$ where $k_0,k_1 \in \{0,1,\ldots d_S-1 \}$ and the states ${| l \rangle}$ are an arbitrary orthonormal basis for the subsystem. Then writing $(\rho_S(t) - \omega_S) = \sum_{k} \lambda_k(t) F_k$ we have $$\begin{aligned}
{\left\langle D(\rho_S(t), \omega_S) \right\rangle_t}\!\! &=&\! \frac{1}{2} {\left\langle {\operatorname{tr}}\big|\sum_k \lambda_k(t) F_k \big| \right\rangle_t} \nonumber \\
&\leq&\! \frac{1}{2} {\left\langle \sqrt{ d_S {\operatorname{tr}}\big(\sum_{kl} \lambda_k(t) \lambda^*_l(t) F_l^{\dag} F_k \big)} \right\rangle_t} \nonumber \\
&\leq&\! \frac{1}{2} \sqrt{ d_S \sum_{kl} {\left\langle \lambda_k(t) \lambda^*_l(t) \right\rangle_t} {\operatorname{tr}}(F_l^{\dag} F_k)} \nonumber \\
&=&\! \frac{1}{2} \sqrt{ d_S \sum_k {\left\langle |\lambda_k(t)|^2 \right\rangle_t}} \nonumber \\
&=&\! \frac{1}{2} \sqrt{ d_S \sum_k {\left\langle \big|{\operatorname{tr}}\big((\rho(t) - \omega)F^{\dag}_k\! \otimes I \big)\big|^2 \right\rangle_t}} \nonumber \\
&\leq&\! \frac{1}{2} \sqrt{ d_S \sum_k \frac{\|F^{\dag}_k\! \otimes I\|^2 }{ d_{{\rm eff}}}} \nonumber \\
&\leq&\! \frac{1}{2} \sqrt{ \frac{d_S^2}{d_{{\rm eff}}}}.\end{aligned}$$ In the second line we have used a standard relation between the 1- and 2-norm, and in the sixth line we have used Theorem 1 for the non-Hermitian operator $F^{\dag}_k\! \otimes I$. Note that $\sqrt{d_S} F_k$ is unitary, and thus $\|F_k^{\dag} \otimes I\| = \frac{1}{\sqrt{d_S}}$.
Universality of equilibrium states
==================================
We have so far been concerned with when states equilibrate, rather than the nature of their equilibrium state. However, one of the notable properties of equilibration is that many initial states effectively equilibrate to the same state, determined only by macroscopic properties such as temperature. Given a particular Hamiltonian and a set of realistic measurements ${\mathcal{M}}$, we can construct a partition of the Hilbert space into a direct sum of subspaces ${\mathcal{H}}= \bigoplus_k {\mathcal{H}}_k$, such that all states within ${\mathcal{H}}_k$ with large enough $d_{{\rm eff}}$ effectively equilibrate to the same state $\Omega_k$.
One way to achieve this is to choose the subspaces such that each projector $\Pi_k$ onto ${\mathcal{H}}_k$ commutes with the Hamiltonian, and such that any two energy eigenstates in ${\mathcal{H}}_k$ are hard to distinguish. i.e. For some fixed $\epsilon$ satisfying $0 < \epsilon \ll 1$, and all normalised energy eigenstates ${| i \rangle}, {| j \rangle} \in {\mathcal{H}}_k$ $$D_{{\mathcal{M}}} ({{| i \rangle}\!{\langle i |}}, {{| j \rangle}\!{\langle j |}}) \leq \epsilon.$$ When $d_{{\rm eff}}$ is sufficiently large, it follows that all states in ${\mathcal{H}}_k$ effectively equilibrate to $\Omega_k = \Pi_k/ {\operatorname{tr}}(\Pi_k)$, as $$\begin{aligned}
{\left\langle D_{{\mathcal{M}}}(\rho(t), \Omega_k) \right\rangle_t}\! &\leq& {\left\langle D_{{\mathcal{M}}}(\rho(t), \omega) \right\rangle_t} + {\left\langle D_{{\mathcal{M}}}(\omega, \Omega_k) \right\rangle_t} \nonumber \\
\! &\leq&\!\! \frac{N({\mathcal{M}})}{4\sqrt{d_{{\rm eff}}}} + \sum_{i,j} \frac{{\langle i |} \omega {| i \rangle}}{{\operatorname{tr}}(\Pi_k)} D_{{\mathcal{M}}}({{| i \rangle}\!{\langle i |}}, {{| j \rangle}\!{\langle j |}}) \nonumber \\
\! &\leq& \!\!\frac{N({\mathcal{M}})}{4\sqrt{d_{{\rm eff}}}} + \epsilon.\end{aligned}$$ where the sums in the second line are over an eigenbasis of $\omega$ (which is also a basis of ${\mathcal{H}}_k$), and we have used the fact that $D_{{\mathcal{M}}} (\rho, \sigma)$ satisfies the triangle inequality ($D_{{\mathcal{M}}} (\rho, \sigma) \leq D_{{\mathcal{M}}} (\rho, \tau) + D_{{\mathcal{M}}} (\tau, \sigma)$) and convexity, $$D_{{\mathcal{M}}}\left( \sum_i p_i \rho_i, \sigma \right) \leq \sum_i p_i D_{{\mathcal{M}}}(\rho_i, \sigma),$$ where $p_i \geq 0$ and $\sum_i p_i= 1$.
When ${\mathcal{H}}_k$ can be chosen to be a small band of energies, the equilibrium state $\Omega_k$ will be the usual microcanonical state.
Conclusions
===========
To summarise, we have shown that two key results of [@us1; @reimann1] about the equilibration of large systems can be derived from very weak assumptions (non-degenerate energy gaps, and sufficiently large $d_{{\rm eff}}$), and a single theorem (Theorem 1). In particular, for almost all times, the state of an isolated quantum system will be indistinguishable from its equilibrium state $\omega$ using any *realistic* experiment, and the state of a small subsystem will be indistinguishable from $\omega_S$ using any experiment.
Although the first result has a similar flavour to the classical equilibration of course-grained observables such as density and pressure, it is really much stronger, as it encompasses any measurement you could describe and record the data from in a reasonable length of text, including microscopic measurements. The second result has no classical analogue, as it yields an essentially static description of the true micro-state of a subsystem, rather than the rapidly fluctuating dynamical equilibrium of particles in classical statistical mechanics. Given the difficulty of proving similar results in the classical case, it seems that quantum theory offers a firmer foundation for statistical mechanics.
*Acknowledgments.* The author is supported by the Royal Society.
[10]{}
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[^1]: To see this, consider the four energy eigenstates ${| k \rangle}={| 0 \rangle}{| 0 \rangle}$, ${| l \rangle}={| 0 \rangle}{| 1 \rangle}, {| m \rangle}= {| 1 \rangle}{| 0 \rangle}, {| n \rangle}={| 1 \rangle}{| 1 \rangle}$ which are products of eigenstates of $H_A$ and $H_B$.
[^2]: $ \|A\|=\sup \{ \sqrt{{\langle v |} A^{\dag} A {| v \rangle}} : {| v \rangle} \in {\mathcal{H}}\,\textrm{with}\, {\left\langle v| v \right\rangle}=1\}$, or equivalently $\|A\|$ is the largest singular value of $A$.
[^3]: In particular we could replace $\Delta(A)$ with $\Delta''(A)=\min_{\tilde{A}} 2\|\tilde{A}\|$, where the operators $\tilde{A}$ are obtained by subtracting any function of $H$ from $A$ and projecting onto the support of $\omega$.
[^4]: However, our results could be extended to continuous output sets using measure theory if desired
[^5]: Note that $\Delta(M_r)$ is the maximum difference in probability of that result occurring for any two states
| ArXiv |
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abstract: |
ola
We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of [*cyclic local*]{} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.\
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PACS number(s) 03.67.Mn, 03.65.Aa, 03.65.Ud.
author:
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A. M. Martins\
Instituto Superior Técnico, 1049-001 Lisboa, Portugal
title: Invariance of reduced density matrices under Local Unitary operations
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Introduction
============
Measuring and classifying quantum entanglement has been the object of extensive research work. The motivations are related to applications in quantum information and computation tasks [@Bennett1992; @Bennett1993] as well as to the foundations of quantum physics [@Ariano2010; @Pusey2012]. An exhaustive bibliography about these different aspects can be found in a recent review article by Horodecki and al. [@Horodecki2009]
A very fruitful approach to understand entanglement, was launched by the seminal work of Linden and al. [@Popescu1997; @Popescu1998] who first used group-theoretic methods to classify entanglement in multi-qubit systems through their classes of local unitary (LU) equivalent states. Two quantum states that can be transformed into each other by LU operations, have the same amount of entanglement and are characterized by local polynomial invariants [@Grassl1998; @Sudbery2001].
Among all possible local unitary operations that can be applied to a subsystem of a quantum system, there are the [*cyclic operations*]{} [@Fu2005], that fix the corresponding reduced state. These operations originate nonlocal effects in the global quantum state of the system that may distinguish product states from classically correlated states. Based on these operations new entanglement measures have been proposed [@Gharibian2008; @Illuminati2011]. LU operations that fix reduced states, leave also invariant local measurements (LM). We say that two quantum states are LM-equivalent when they have the same set of 1-party reduced density matrices.
In this work we answer the following question: Given an $n$-qubit input state $\rho$, pure or mixed, what is the set of states to which it can be converted by LU operations that leave invariant the corresponding reduced states? This is, what is the set of states that are LU and LM equivalent to $\rho$?
We say that a state, $\rho_U$ is LU-equivalent to $\rho$ if $\rho_U =U \rho U^{\dag}$, with $U= \otimes_j^{n} U_j ( \in G )$, where $U_j \in SU(2)$, is the unitary operator acting in qubit $j$ and $G= SU(2)^{\otimes n}$, is the local unitary group. Each equivalence class of LU-equivalent states is an orbit of this group. We say that $\rho_U$ is LM-equivalent to $\rho$ when their set $S$, of reduced states, is the same, i.e, ${\bf T}_{(j)} (\rho_U) = {\bf T}_{(j)} (\rho) =\rho_j \,\ ( i=1,..,n)$, where ${\bf T}_{(j)}$ is the partial trace over all qubits except qubit $j$. The set of local operators $U $ that fix each of the $n$ reduced states $\rho_j$, is the centralizer subgroup of the set $S$.
We derive necessary and sufficient conditions for an LU operator to belong to the centralizer subgroup of the set $S$ and identify all possible types of centralizers subgroups. We show that their dimension is directly related with the number of maximally mixed 1-qubit states. We also prove that the operator $U_i$ that fixes any non-maximally mixed reduced state $\rho_i$, is a 1-parameter unitary operator completely determined by the Bloch vector of $\rho_i$.
The partial trace operator play a central role in the derivation of the above mentioned results and deserve a place of their own right in this work. We explore the isomorphism existent between the orthogonal complement of the kernel of ${\bf T}_{(j)}$ and the Hilbert space of qubit $j$, to identify the vectors representing the LM-equivalence classes.
The paper is organized as follows. In Section 2, we use the partial trace operator to decompose the Hilbert space of the whole system in pairs of complementary subspaces. In Section 3, we define an isomorphism between the reduced density matrices and vectors of the Hilbert space of the $n$-qubits and derive the necessary and sufficient conditions obeyed by a local unitary operator that fix the corresponding reduced state. In Section 4, we compute all possible centralizer subgroups of a set of reduced states and give the explicit form of the LU/LM operators. Finally we conclude in Section 5.
The partial trace
==================
A suitable choice of the basis set to develop the density matrices may simplify considerably solving specific physical problems, or may help to identify new properties of the system. In this work, where systems are formed by $n$ similar $2$-level constituents, and where the partial trace operators play a determinant role, the natural choice of basis set is the generalized Bloch vector basis.
Let ${\cal V}_j$ denote the $4$-dimensional Hilbert space of $2 \times 2$ Hermitian matrices. A convenient basis for ${\cal V}_j$ is ${\cal B}_j= \{\sigma_{\alpha_j} ; {\alpha_j}=0,1,2,3 \}$, where $ \sigma_{\alpha_j}( \alpha_j=1,2,3)$ represents the usual Pauli matrices, and $\sigma_0 = {\bf 1}$, is the $2 \times 2$ identity matrix. Using in ${\cal V}_j$ the Hilbert-Schmidt inner product $(\sigma_{\alpha_i }, \sigma_{\alpha_k}) = Tr\{ \sigma_{\alpha_i }\sigma_{\alpha_k } \} = 2 \delta_{ij} $, then ${\cal B}_j $ is an orthogonal basis set. We are going to consider the set $ {\cal B}_{{\cal V}^{\otimes n}} = \{ \sigma_{\vec \alpha} \}$, where $$\label{vector2}
\sigma_{\vec \alpha} = \otimes_{j=1}^{n} \sigma_{\alpha_j}$$ The vector index ${\vec \alpha} =(\alpha_1, \alpha_2,... ,\alpha_n)$ is a $n$-tuple containing the $n$ indices $\alpha_j$. There exist $4^{n}$ such matrices all being traceless, except for $\sigma_{\vec 0 }= {\otimes}_{j=1}^n {\bf 1}_j$, which corresponds to the $2^n \times 2^n $ identity matrix with trace $Tr \{ \sigma_{\vec 0 } \} = 2^{n}$.
$ {\cal B}_{{\cal V}^{\otimes n}} $ is an orthogonal basis set of the complex $4^{n}$-dimensional Hilbert-Schmidt vector space ${\cal V}^{\otimes n} =\otimes_{j=1}^n {\cal V}_j $. Every complex square matrix, $(2^{n} \times 2^{n})$, can be seen as a vector $\bf v$, uniquely written in the form $$\label{vector}
{\bf v} = \sum_{\vec \alpha} v_{\vec \alpha} \,\ \sigma_{\vec \alpha}$$ where the components $v_{\vec \alpha} $ are given by $$\label{components}
v_{\vec \alpha} = \frac{1}{2^n} Tr\{ \sigma_{\vec \alpha} \,\ {\vec v} \}$$
Any $n$-qubit quantum state $\rho=\sum_{\vec \alpha} r_{\vec \alpha} \sigma_{\vec \alpha} \in {{\cal V}^{\otimes n}} $, must be hermitian, $\rho = \rho^{\dag}$, definite positive $\rho \geq 0$, and normalized $Tr \{ \rho \} =1$. These requirements on $\rho$ impose certain constrains to the components $ r_{ \vec \alpha} $: (a) $\forall_{\vec \alpha}, r_{\vec \alpha} \in \Re$, (b) $r_{\vec 0} = \frac{1}{2^n} $, (c) $r_{\vec \alpha} = \frac{1}{2^n} Tr\{ \sigma_{\vec \alpha} \,\ \rho \}$ and (d) $\sum_{ \vec \alpha} r_{ \vec \alpha}^{2} \leq 1 $, the equality is attained for pure states.
The translated vector, ${\bar \rho }= \rho - {\bf 1}^{ \otimes n} /2^{n}$,( ${\bf 1}^{ \otimes n} = \otimes_{j=1}^{n} {\bf 1}_j$), characterizes completely the quantum state $\rho$ and is the well known [*generalized Bloch vector representation*]{} of dimension $(4^{n} - 1)$.
Let $D_n$ be the set of the $n$-qubit density matrices $\rho$ and let ${\bf T}_{(i)} : {\cal V}^{\otimes n} \rightarrow {\cal V}_i $ be the linear transformation defined by $$\label{partial}
{\bf T}_{(i)} (\rho) =Tr_{n/ \{i \}} \{ \rho \} = \rho_i \,\,\,\,\,\,\,\,\,\,\,\,\ ( i=1,...,n)$$ where $Tr_{n/ \{i \}} \{ . \}$ is the partial trace operator over $(n-1)$ qubits, except qubit $i$.
This is a surjective map of ${\cal V}^{\otimes n}$, (the vector space of the $n$ qubits), onto ${\cal V}_i \equiv Im({\bf T}_{(i)} ) $, (the vector space of qubit $i$), where $Im({\bf T}_{(i)} )$ is the image space of ${\bf T}_{(i)} $. Let $ {\cal K}_i $ be the kernel of ${\bf T}_{(i)} $ and let ${\cal Q}_i$ be the orthogonal complement of $ {\cal K}_i $, i.e., $$\label{partial}
{\cal V}^{\otimes n} = {\cal Q}_i \oplus {\cal K}_i \,\,\ ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ \,\,\ (i=1,...,n)$$ The subspace $ {\cal Q}_i $ is a $4$-dimensional space isomorphic to ${\cal V}_i $. Applying the map ${\bf T}_{(i)} $ to the vectors of the basis set $$\label{basis2}
{\cal B}_{{\cal Q}_i} = \{ {\bf b}_{\alpha_i}= \otimes_{k=1}^{i-1} {\bf 1}_k \otimes \sigma_{\alpha_i} \otimes_{k'=i+1}^n {\bf 1}_{k'} ; \,\ \alpha_i = 0,1,2,3\}$$ where $ {\cal B}_{{\cal Q}_i} \subset {\cal B}_{{\cal V}^{\otimes n}} $, we obtain $$\label{correspondance}
{\bf T}_{(i)} ({\bf b}_{\alpha_i} )=2^{n-1} \sigma_{\alpha_i}$$ Let ${\cal B}_{{\cal K}_i} = {\cal B}_{{\cal V}^{\otimes n}} \setminus {\cal B}_{{\cal Q}_i} $. The image of any vector of ${\cal B}_{{\cal K}_i} $ under ${\bf T}_{(i)}$ is the zero vector $0_i \in {\cal V}_i $. We conclude that ${\cal B}_{{\cal Q}_i}$ and ${\cal B}_{{\cal K}_i}$ are orthogonal basis sets for the subspaces ${\cal Q}_i$ and ${\cal K}_i$, such that, ${\cal B}_{{\cal V}^{\otimes n}} = {\cal B}_{{\cal Q}_i} \cup {\cal B}_{{\cal K}_i} $.
Any density matrix $\rho \in D_n$ can be written in a unique way as $$\label{direct}
\rho =\rho_{{\cal Q}_i} + \rho_{{\cal K}_i}$$ where $\rho_{{\cal Q}_i}$ and $ \rho_{{\cal K}_i} $ are the projections of $\rho$ on the subspaces ${\cal Q}_i $ and $ {\cal K}_i $.
The projection operator $\pi_{{\cal Q}_i} : {\cal V}^{\otimes n} \rightarrow {\cal Q}_i$ is defined by $$\label{rhoq1}
\pi_{{\cal Q}_i} (\rho) = \rho_{{\cal Q}_i} = \sum_{{\alpha_i}=0}^3 \frac{(\rho , {\bf b}_{\alpha_i})}{\parallel {\bf b}_{\alpha_i} \parallel^2 } {\bf b}_{\alpha_i}= \sum_{{\alpha_i}=0}^3 r_{0_1 ... 0_{i-1} \alpha_i 0_{i+1}...0_n}{\bf b}_{\alpha_i}$$ The action of ${\bf T}_{(i)}$ on both sides of eq.(\[direct\]), gives $$\label{rhoq2}
{\bf T}_{(i)}(\rho ) ={\bf T}_{(i)}(\rho_{{\cal Q}_i} )= \rho_i$$ Any 1-qubit density matrix $\rho_i \in {\cal V}_i$ can be written in the basis ${\cal B}_i $ in the form $$\label{rhoi}
\rho_i = \frac{1}{2} ({\bf 1}_i+ {\vec r}_i . {\vec \sigma}(i)) \,\,\,\,\,\,\,\,\,\,\,\,\ ( i=1,...,n)$$ where, $ {\vec r}_i . {\vec \sigma}(i) = 2^n \sum_{a_i =1}^3 r_{0_1 ... 0_{i-1} a_i 0_{i+1}...0_n} \sigma_{a_i}$, and ${\vec r}_i $ is the Bloch vector of qubit $i$, such that $\| {\vec r}_i \| \leq 1 $.
Substituting (\[rhoi\]) in the l.h.s. of eq.(\[rhoq1\]), we obtain the following explicit one-to-one correspondence between the vectors $\rho_i \in {\cal V}_i $ and the vectors $\rho_{{\cal Q}_i} \in {\cal Q}_i$, $$\label{rhoq5}
\rho_{{\cal Q}_i} = \frac{1}{2^{n-1}} \otimes_{k=1}^{i-1} {\bf 1}_k \otimes \rho_i \otimes_{k'=i+1}^n {\bf 1}_{k'}$$
The translated vector $$\label{rhoqj2}
{\bar \rho}_{{\cal Q}_i} = \rho_{{\cal Q}_i} - \frac{1}{ 2^n}{\bf 1}^{ \otimes n}= \otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ {\vec r}_i . {\vec \sigma}(i)] \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ contains the same quantum information as the reduced density matrix $\rho_i$, this is, the quantum state of qubit $i$, is fully represented by the vector ${\bar \rho}_{{\cal Q}_i} \in {\cal V}^{\otimes n}$. Moreover, $$\label{orto}
( {\bar \rho}_{{\cal Q}_i} ,{\bar \rho}_{{\cal Q}_j} ) = \| {\bar \rho}_{{\cal Q}_i} \|^2 \delta_{ij}$$ i.e., vectors ${\bar \rho}_{{\cal Q}_i} $ associated to different qubits are orthogonal to each other and to any other vectors of the basis set $ {\cal B}_{{\cal V}^{\otimes n}} $. The norm, $\| {\bar \rho}_{{\cal Q}_i} \| = [ \lambda_{-}^2 (i) +\lambda_{+}^2 (i) ]^{1/2} $, where, $\lambda_{\mp}$, are the eigenvalues of ${\bar \rho}_{{\cal Q}_i}$. It is now obvious that any quantum state $\rho \in D_n$ can be written in the form, $$\label{rho4}
\rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{n} {\bar \rho}_{{\cal Q}_i} + \Delta$$ where $\Delta$ refers to the terms of $\rho$ containing all possible $k$-partite correlations ($ 2 \leq k \leq n $) existing between the $n$-qubits.
Reduced states and equivalence classes
=======================================
The possible outcomes of the measurement of any local observable $ {\hat A}_j \in {\cal V}_j $, performed on qubit $j$, are given by the eigenvalues $a_k(j)$ of the operator ${\hat A}_j $. The expectation value of this measurement, when the system is in the state $\rho$, is given by $$\label{expectation}
\langle {\hat A}_j \rangle = Tr \{{\hat A}_j \rho \} = Tr_j \{ {\hat A}_j \rho_j \}$$ where $Tr_j \{ \} $ is the trace in qubit $j$ and $Tr \{ \}$ is the trace in all qubits. This equality shows that measurements performed on qubit $j$ give the same result as if it would be in the reduced state $\rho_j = {\bf T}_{ (j) } (\rho) $. A imediate consequence of eq.(\[expectation\]) is that different global quantum states $\rho$ with equal reduced states $ \rho_j $ have equal $1$-qubit expectation values $\langle {\hat A}_j \rangle $. When two states $\rho $ and $\rho^{'} $ have the same image $\rho_j$, under the map ${\bf T}_{ (j) }$, their difference belong to the kernel ${\cal K}_j$, i.e., they are congruent modulo ${\cal K}_j$. The set of all states with reduced state $\rho_j$ forms a LM$_j$-equivalence class $ C_j$, this is, $$\label{class}
C_j = \{ \rho \in D_n : {\bf T}_{ (j) } (\rho) =\rho_j \}$$ The set of all LM$_j$-equivalence classes is the quotient space ${\cal V}^{\otimes n}/ {\cal K}_j$.
We have shown that for any quantum state $\rho$, there is a one-to-one correspondence between its reduced state $\rho_j$ and its projection $\rho_{{\cal Q}_j }$. This enables us to define a linear map $\psi_j: {\cal V}^{\otimes n}/ {\cal K}_j \rightarrow {\cal Q}_j $, such that $$\label{class}
\psi (C_j )= \rho_{{\cal Q}_j }$$ assigns to each class $C_j \in {\cal V}^{\otimes n}/ {\cal K}_j$ the vector $\rho_{{\cal Q}_j } \in {\cal Q}_j$, we say that the vector $\rho_{{\cal Q}_j } \in {\cal Q}_j$, is the representative state of the class $C_j $ and we may write [@Martins2008] $$\label{class1}
C_j = \{ \rho \in D_n : \rho =\rho_{{\cal Q}_j } + {\cal K}_j \}$$ The set of all $n$-qubit density matrices, such that their reduced density matrices belong to $S= \{ \rho_i = {\bf T}_{ (i) } (\rho), \,\ ( i=1,...,n ) \}$, is given by the intersection of the equivalence classes $C_i$, i.e., $$\label{class2}
{\bar C} = \bigcap_{i=1}^{n} C_i = \{ \rho \in D_n : \rho =\rho_{{\cal Q}_i } + {\cal K}_i \,\ ; i=1,...,n \}$$ saying it in another way, quantum states in the set ${\bar C}$ have their $\rho_{{\cal Q}_i}$ projections in the set $$\label{set3}
{\bar S} = \{ \rho_{{\cal Q}_i}= \pi_{{\cal Q}_i}(\rho) \,\ ; i=1,...,n \}$$ The set ${\bar S}$ is isomorphic to the set $S$ therefore, they have the same content of quantum information. This isomorphism is particularly useful when we are studying local properties of the qubits because, instead of working with the $n$ Hilbert spaces ${\cal V}_i$, we can use the original Hilbert space ${\cal V}^{\otimes n}$ of the $n$-qubits.
The unitary transformation $U \in G$ acts on a $n$-qubit state $\rho$ via the adjoint action, $$\label{LU}
\rho_U = ad \,\ U [ \rho] = U \rho U^{\dag} = \left( \otimes_{j=1}^{n} U_j \right) \rho \left( \otimes_{j=1}^{n} U_j^{\dag} \right)$$ where $G= SU(2)^{\otimes n}$ is a $3n$-dimensional Lie group and ${\cal L} = su(2) \oplus su(2) \oplus ... \oplus su(2) $ is the corresponding Lie algebra. The set ${\cal B}_{\cal L} =\{ {\bf b}_{a_i} \in {\cal B}_{{\cal Q}_i}; \,\ a_i =1,2,3 $ and $ i=1,...,n \}$ is a basis set for ${\cal L}$ whose elements are the generators of $G$.
In this work we are looking for all quantum states $\rho_U $, LU equivalent to $\rho$, such that measurements of any local observable ${\hat A}_i$ are not able to distinguish between $\rho$ and $\rho_U$. Having in mind eq.(\[expectation\]), we are looking for states $\rho_U $ with the same set $S$ of 1-qubit reduced density matrices. This is, $$\label{trace2}
{\bf T}_{(i)} (\rho_U) = {\bf T}_{(i)} (\rho) =\rho_i ; \,\,\,\,\ i=1,...,n$$ or, given the one-to-one correspondence between $\rho_i$ and $\rho_{{\cal Q}_i}$, the LU equivalent states are such that $$\label{trace3}
\pi_{{\cal Q}_i} (\rho_U) = \pi_{{\cal Q}_i} (\rho) = \rho_{{\cal Q}_i} ; \,\,\,\,\ i=1,...,n$$ i.e., the states $ \rho_U $ belong to the set ${\bar S}$. Not all adjoint actions of local unitary operators $U$ on $\rho$ obey this condition, however all local unitary operators leave the subspaces ${\cal Q}_j$ and ${\cal K}_j$ invariant, as we prove in the next Theorem.
[**Theorem 1:**]{} [*The subspaces ${\cal Q}_j$ and ${\cal K}_j$ are invariant under LU transformations.*]{}
[**Proof:**]{} Any vector ${\bf v}_{{\cal Q}_j} \in {\cal Q}_j$ has the form ${\bf v}_{{\cal Q}_j} = \otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ \sum_{\alpha_i=0}^{3}v _{\alpha_i } \sigma_{\alpha_i} ] \otimes_{k'=i+1}^n {\bf 1}_{k'} $. The adjoint action of $U $ on $\rho$ is $$\label{invariant}
ad \,\ U [ {\bf v}_{{\cal Q}_j} ] = U {\bf v}_{{\cal Q}_j} U^{\dag} =\otimes_{k=1}^{i-1} {\bf 1}_k \otimes [ \sum_{\alpha_i=0}^{3}v _{\alpha_i } U_i \sigma_{\alpha_i} U_i^{\dag} ] \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ As $$\label{rotation}
\sum_{\alpha_i=0}^{3}v _{\alpha_i } U_i \sigma_{\alpha_i} U_i^{\dag} = \sum_{\alpha_i=0}^{3}v _{\alpha_i }^{'} \sigma_{\alpha_i}$$ then $ U {\bf v}_{{\cal Q}_j} U^{\dag} \in {\cal Q}_j$.
When ${\cal Q}_j$ (or ${\cal K}_j$ ) is invariant under a unitary transformation so is the complementary subspace ${\cal K}_j$ (or ${\cal Q}_j$) [@Halmos1987]. $\Box$
[**Corollary 1:**]{} [*The subspace ${\bar {\cal K}} = \cap_{j=1}^{n} {\cal K}_j$ is invariant under the local adjoint action.*]{}
[**Corollary 2:**]{} [*The projection operator $\pi_{{\cal Q}_i}$ commutes with any LU transformation, i.e.,*]{} $$\label{comutador2}
\pi_{{\cal Q}_i} (U \rho U^{\dag}) = U [ \pi_{{\cal Q}_i} ( \rho ) ] U^{\dag} = U \rho_{{\cal Q}_i} U^{\dag}$$ [**Proof:** ]{} If a subspace is invariant under a linear transformation $U$ then $U$ commutes with every projection operator on that subspace [@Halmos1987]. $\Box$
This corollary shows that $\pi_{{\cal Q}_i} ( \rho_U) =U \rho_{{\cal Q}_i} U^{\dag}$. Imposing now the constrain of eq.(\[trace3\]), i.e., that $ \rho_U $ has the same set of 1-qubit reduced density matrices as $\rho$, we conclude that the LU transformations we are looking for, are such that $$\label{inv}
U \rho_{{\cal Q}_i} U^{\dag} = \rho_{{\cal Q}_i} \,\,\ ; \,\,\,\ i=1,...,n$$ $\rho_{{\cal Q}_i}$ is invariant under LU transformations. Local unitary operators $U \in G$ obeying condition (\[inv\]), for all elements of the set ${\bar S} $, belong to the centralizer subgroup $C_G ({\bar S})$ of the set ${\bar S}$, i.e. $$\label{centralizergroup}
C_G ({\bar S}) = \{U \in G : U \rho_{{\cal Q}_i} U^{\dag} = \rho_{{\cal Q}_i}, \forall_{\rho_{{\cal Q}_i} \in {\bar S}} \}$$
Next theorem sets the conditions obeyed by the local unitary transformations $U_i $ in order that equality (\[inv\]) holds.
[**Theorem 2 :**]{} [*A state $\rho_U $, LU equivalent to $\rho$, has the same set $S$ of reduced density matrices as $\rho$, iff each local unitary operator $U_i \in SU(2)$ commutes with $\rho_i$, i.e.,*]{} $$\label{comutador}
[U_i , \rho_i]=0 \,\,\ , \,\,\ i=1,...,n$$
[**Proof:** ]{} By Corollary 1, $$\label{trans}
\pi_{{\cal Q}_i} (U \rho U^{\dag}) =U \rho_{{\cal Q}_i} U^{\dag} = \otimes_{k=1}^{i-1} {\bf 1}_k \otimes U_i \rho_i U_i^{\dag} \otimes_{k'=i+1}^n {\bf 1}_{k'}$$ The condition (\[inv\]) is verified when, $U_i \rho_i U_i^{\dag} = \rho_i $ for each qubit $i$. This is equivalent to equality (\[comutador\]). $ \Box$
Theorem 2 refers to these multi-qubit LU operations and proves that the cyclic property is a necessary and sufficient condition for invariance of any number of reduced states. Moreover, local unitary operators acting in different qubits $i$ and $j$, commute with each other, i.e., $ [ U_i , U_j ] = 0$.
In conclusion, the general form of any quantum state $\rho_U$, LU equivalent to $\rho$, and with the same set $S$ of 1-qubit reduced density matrices is given by $$\label{flu}
\rho_U = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{n} {\bar \rho}_{{\cal Q}_i} + U \Delta U^{\dag}$$ where the operators $U$ belong to the centralizer subgroup $C_G ({\bar S})$. The problem of finding $\rho_U$ in the last equation is solved when the centralizer subgroup of a state $\rho$ is known.
We call $\rho $-family and denote by ${\cal F}_{\rho}$, the set of states $\rho_U$ given by eq.(\[flu\]). The elements of this family have the same type of entanglement but are not distinguishable by local measurements.
Not all states in the LU-orbit of $\rho$ belong to ${\cal F}_{\rho}$. Next proposition is a criterium to decide wether a state $\rho^{'}$, is not in the family ${\cal F}_{\rho}$.
[**Proposition 1:**]{} A state $\rho^{'}$, LM-equivalent to the state $\rho$, does not belong to the family ${\cal F}_{\rho}$, if $$\label{cond}
Tr\{ \rho^{' 2} \} \neq Tr\{ \rho^{2} \} \,\,\,\,\ \mbox{or if} \,\,\,\,\ Tr\{ \Delta^{' 2} \} \neq Tr\{ \Delta^{2} \}$$
Centralizers subgroups and LU/LM-equivalence
============================================
In this section we show that the translated vectors $ {\bar \rho}_{{\cal Q}_i}$, present in eq.(\[rho4\]), determine the centralizer subgroup $C_G ({\bar S})$ and the set of states LU/LM equivalent to each quantum state $\rho$.
Any generic local unitary operator $U_j \in SU(2)$ is a three real continuous parameter operator and can be written in the form $$\label{uni}
U_j ( \phi_j , \theta_j , \omega_j )= e^{i {\vec s}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j +i \sin (\omega_j ) {\hat n}_{ {\vec s}_j} . {\vec \sigma} (j)$$ where ${\hat n}_{ {\vec s}_j} = {\vec s}_j / \parallel {\vec s}_j \parallel \equiv ( \cos \phi_j \sin \theta_j , \sin \phi_j \sin \theta_j, \cos \theta_j )$ is a unit vector in the 3-dimensional Euclidian space (Bloch space of qubit $j$), parametrized by the azimuthal angle, $0 \leq \phi_j \leq 2 \pi $, and the polar angle, $0 \leq \theta_j \leq \pi $. The third parameter is $\omega_j = \parallel {\vec s}_j \parallel$ ($ 0 \leq \omega_j \leq \pi /2$) is the length of the vector ${\vec s}_j$.
Any 1-qubit density matrix can be written in the form (\[rhoi\]). When ${\vec r}_j =0$, then the 1-qubit density matrix is maximally mixed, i.e., $\rho_j^{*} =\frac{1}{2}{\bf 1}_j $, and any local unitary operation $U_j = e^{i {\vec s}_j .{\vec \sigma} (j) }$ commutes with $\rho_j$.
When ${\vec r}_j \neq 0$, then condition (\[comutador\]) is verified when ${\vec s}_j = \xi_j {\vec r}_j $ (see Appendix), with $\xi_j \in \Re$. The corresponding [*local cyclic*]{} operator is $$\label{uni2}
U_j (\xi_j )= e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j + i \sin (\omega_j ) {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j)$$ a single parameter unitary operator, where $ \omega_j = \xi_j \parallel {\vec r}_j \parallel $ is the continuous parameter. The direction ${\hat n}_{ {\vec r}_j}$ is fixed by the cyclic condition (\[comutador\]). Varying continuously the parameter $\xi_j $, in eq.(\[uni2\]), between, $0$, and, $\pi /2 \parallel {\vec r}_j \parallel $, then $U_j $ varies between $ {\bf 1}_j $ and $ ( i {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j) )$.
Invoking the local isomorphism between SU(2) and SO(3) we see that the unitary operator $U_j$, of eq.(\[uni2\]), represents a rotation of an angle $\omega_j$ around the vector ${\vec r}_j $ of the Bloch sphere of qubit $j$, which leaves this vector and the corresponding $\rho_j$ invariants. In the generalized $(4^{n}-1)$ Bloch vector space, the vectors ${\bar \rho}_{{\cal Q}_j}$, of different qubits, are orthogonal to each other. Local unitary operations of $SU(2)^{\otimes n}$ of the type $$\label{uni3}
U = \otimes_{j=1}^{m} e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } \otimes_{l=m+1}^{n} {\bf 1}_l \,\,\,\, (m=1,...,n)$$ correspond to $m$ independent rotations of the the group $SO(3)$ around each vector ${\bar \rho}_{{\cal Q}_j}$.
These results reveal the intimate relation between the set $S$ of the 1-qubit reduced density matrices and the centralizer subgroup $C_G ({\bar S})$ of the set ${\bar S}$. Different cases are possible.
[**Case 1:**]{} When $ {\bar \rho}_{{\cal Q}_i} =0 $, for all $i=1,...,n$, then eq.(\[rho4\]) reduces to $$\label{rho5}
\rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \Delta$$ and $\rho$ have maximally mixed 1-qubit reduced states, i.e., $$\label{set1}
S= \{ \rho_i^{*}= \frac{1}{2} {\bf 1}_i \,\ ; \,\ i=1,...,n\}$$ The centralizer subgroup $C_G ({\bar S})$ is the entire $G$ whose dimension is $3n$. The states $\rho_U $, are given by $$\label{rho6}
\rho_U = \frac{1}{2^n} {\bf 1}^{ \otimes n} + U \Delta U^{\dag}$$ where $ U =\otimes_{j=1}^{n} e^{i{\vec s}_j .{\vec \sigma} (j) } $. As each $U_j$ only acts on qubit $j$ then $U \Delta U^{\dag}$ can be easily computed by replacing each Pauli matrix $\sigma_{\alpha_j}$ in $\Delta$ by $U_j \sigma_{\alpha_j} U_j^{\dag}$.
Any n-qubit Werner state $\rho^{\cal W}$, has maximally mixed 1-qubit reduced states. All states in the LU-orbit of a $\rho^{\cal W}$ are LM-equivalent. The maximally mixed state $\rho^{*}= \frac{1}{2^n} {\bf 1}^{ \otimes n} $ is a special type of Werner state. When all local unitary operators are equal, i.e., $U_j =e^{i{\vec s} .{\vec \sigma} (j) }$ (independent o $j$) the state $\rho^{\cal W}$ is transformed into itself.
The $n$-GHZ entanglement class has maximally mixed 1-qubit density matrices. All states in this class are LM-equivalent.
[**Case 2:**]{} When, in eq.(\[rho4\]), there are $m < n$ operators $ {\bar \rho}_{{\cal Q}_k} \neq 0, (k=1,...,m) $ and the remaining $(n-m)$ operators are $ {\bar \rho}_{{\cal Q}_l} =0, (l=m+1,...,n) $, then $$\label{rho7}
\rho = \frac{1}{2^n} {\bf 1}^{ \otimes n} + \sum_{i=1}^{m} {\bar \rho}_{{\cal Q}_i} + \Delta$$ The corresponding 1-qubit reduced density matrices belong to the set $$\label{set2}
S=\{ \rho_k =\frac{1}{2}{\bf 1}_k + {\vec r}_k . {\vec \sigma} (k) ; \,\ ( k=1,...,m) \wedge \rho_l^{*} =\frac{1}{2}{\bf 1}_l ; ( l=m+1,...,n) \}$$ The centralizer subgroup $C_G ({\bar S})$ is $$\label{centralizergroup2}
C_G ({\bar S}) = \{U \in G : U = \otimes_{k=1}^{m}e^{i\xi_l {\vec r}_l .{\vec \sigma} (l) } \otimes_{l=m+1}^{n} e^{i {\vec s}_k .{\vec \sigma} (k) } \}$$ Different values of $m$ give rise to different centralizer subgroups with dimension $dim[ C_G ({\bar S})] =3n -2m$, the same as the number of independent continuous parameters. When $m=n$, $dim[ C_G ({\bar S})] =n$, this number is minimum. When $m=0$, then $dim[ C_G ({\bar S})] =3n $ is maximum and the centralizer subgroup is the entire $G$ (Case 1).
The states $\rho_U $, of the family ${\cal F}_{\rho}$ are obtained by replacing $\Delta$ in eq.(\[rho7\]) by $U \Delta U^{\dag}$, where $U $ belong to the stabilizer subgroup of eq.(\[centralizergroup2\]).
Biseparable states $$\label{by}
\rho = \rho^{(m)} \otimes \rho^{(n-m)}$$ where $\rho^{(m)}$ is any state of $m$-qubits and the remaining $(n-m)$-qubits are in a state $\rho^{(n-m)}$, (for instance, $(n-m)$-GHZ or $(n-m)$-Werner states), have centralizers subgroups of the form (\[centralizergroup2\]). A product state $$\label{product}
\rho = \otimes_{k=1}^{m} \rho_k \otimes_{l=m+1}^{n} \rho_l^{*}$$ is a special case of biseparable state. The $\rho$-family of a product state is the product state itself.
Concluding remarks
==================
We have investigated a special type of local unitary operations that fix the set of reduced states of a pure or mixed multi-qubit state. We have shown that the possible forms of the [*cyclic local*]{} transformations is determined by the 1-qubit reduced density matrices. The dimension of the centralizer subgroups of the set reduced states is minimum when no 1-qubit reduced matrix is maximally mixed and it is maximum when all 1-qubit reduced states are maximally mixed.
We have shown that local [*cyclic*]{} unitary operations of multiqubit states, with non maximally mixed 1-qubit reductions, are a subgroup of $SU(2)^{\otimes n}$ whose elements are given by the tensorial product of $n$-single parameter unitary operators, this suggests that it is possible to analyze in a continuous way the measures of entanglement and of non-classicality [@Gharibian2012], by varying independently these parameters. Simultaneous application of these local [*cyclic*]{} operations to different qubits goes beyond the bipartite studies [@Fu2005; @Gharibian2008] and may reveal new nonlocal effects.
Appendix
========
[**Theorem 3**]{}: [*The commutation relation $ [U_j , \rho_j ]=0 $, where the 1-qubit reduced state is $\rho_j = \frac{1}{2} {\bf 1}_j+ {\vec r}_j . {\vec \sigma}(j)$, with ${\vec r}_j \neq 0$, is verified iff* ]{} $$\label{uni5}
U_j (\xi_j )= e^{i \xi_j {\vec r}_j .{\vec \sigma} (j) } = \cos ( \omega_j ) {\bf 1}_j + i \sin (\omega_j ) {\hat n}_{ {\vec r}_j} . {\vec \sigma} (j)$$
[**Proof**]{}: Any local unitary operator has the general form (\[uni\]). The cyclic condition is equivalent to $$\label{comutador2}
[ { {\vec s}_j} . {\vec \sigma} (j) , {\vec r}_j . {\vec \sigma}(j) ] = 0$$ Computing the above commutator, we obtain $$\label{comutador3}
[ {\vec s}_j . {\vec \sigma} (j) , {\vec r}_j . {\vec \sigma}(j) ] = \sum_{k=1}^{3} \sum_{l=1}^{3} s_{jk} r_{jl} [ \sigma_k (j) , \sigma_l (j) ] = 2 i \{ \sum_{k=1}^{3} \sum_{l=1}^{3} s_{jk} r_{jl} \epsilon_{klu} \} Ê\sigma_r (j)$$ where $\epsilon_{klu}$ is the Levi-Civita symbol. After some straightforward calculations we show that the commutator will be zero iff, ${\vec s}_j = \xi_j {\vec r}_j $.
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| ArXiv |
---
abstract: 'We discuss the soundness of the scaling functional (SF) approach proposed by Aubouy Guiselin and Raphaël (Macromolecules **29**, 7261 (1996)) to describe polymeric interfaces. In particular, we demonstrate that this approach is a variational theory. We emphasis the role of SF theory as an important link between ground-state theories suitable to describe adsorbed layers, and “classical” theories for polymer brushes.'
author:
- Manoel Manghi
- Miguel Aubouy
title: |
Validity of the scaling functional approach for polymer interfaces\
as a variational theory
---
Introduction
============
Polymer interfaces are layers made of polymeric chains in direct contact with a boundary which may be a solid/liquid, liquid/liquid interface or a more complex surface such as a membrane. Because they have applications in such diverse fields as colloid stabilization, coating, tribology, galenic, they have been the subject of active research since the 80’s both from a fundamental and applied point of view. At present, there are two well established self-consistent-field (SCF) theories to describe polymer layers. They both start from the partition function of an ensemble of chains in contact with the interface treated in mean-field, but they soon proceed in a marked different way. Eventually, they become very different type of theories, depending on whether the chains are reversibly adsorbed, and there is an adsorbed state which dominates the solution of the Schrödinger equation associated (ground state dominance (GSD) theories [@holbook; @Semenov-Joanny]), or they are end-tethered to a repulsive surface (so-called ”brushes”), and the path integral is dominated by the classical solution (classical theories [@Semenov; @MWC; @Zhulina]).
Because the two types of theories are very different in spirit, there is a conceptual gap for intermediate cases. In other words, there is no mean-field theory available to describe both adsorption and grafting of polymers within the same formalism. Such case arises, e.g., when chains are grafted onto an attractive surface. In principle, at least, one should be able to go in a continuous way from adsorbed-like to brush-like layers by tuning the amount of chains per unit surface.
A tentative to bridge such gap was proposed in a series of paper where the so-called Scaling Functional (SF) approach is developed [@AGR; @se]. This is an approach where the layer of monodisperse adsorbed chains ($N$ monomers of size $a$) is considered as a thermodynamic ensemble of interacting loops and tails. These loops are polydisperse in size, and the main tool is the “loop size profile”, $S$, such that $$S(n)=S_{0}\int_{n}^{N}P(u)du,
\label{defS}$$ where $P$ is the statistical distribution of loop sizes in monomer units, and $S_{0}$ is the total number (per $cm^{2}$) of loops. The free-energy (per $cm^2$) of the layer of chains is written as: $$\begin{aligned}
\mathcal{F}\{S\} &\cong& \frac{k_BT}{a^2} \int_0^N
\left\{k[a^2S(n)]^{\beta}\right. \nonumber \\ &+& \left.[-a^2S'(n)] \ln
\left[-\frac{S'(n)}{S_0}\right]\right\} dn, \label{fenergie}\end{aligned}$$ where $k\cong 1$ is a constant, $k_BT$ is the thermal energy and $S'(n)=dS/dn$. The first term in rhs of Eq. (\[fenergie\]) accounts for loop interactions (which depend on solvent conditions through the value of the exponent $\beta$, see Table \[table\]). The second term in the rhs of Eq. (\[fenergie\]) is the usual entropy associated with a set of polydisperse objects. Similarly, the extension of the layer is computed following $$L\{S\} \cong a\int_0^N[a^2S(n)]^{\alpha}dn ,$$ where the exponent $\alpha$ is given in Table \[table\]. In the SF approach, the layer of chains is actually described as a polydisperse polymer brush (the role of the chains being played here by the “pseudo-loops”, i.e. half loops) *plus* an entropic term which stems for the fact that the size distribution is not fixed by any external operator, but the system of loops is in thermodynamic equilibrium.
type of solvent good $\Theta$ melt “mean-field”
----------------- ------ ---------- ------ --------------
$\alpha$ 1/3 1/2 1 1/3
$\beta$ 11/6 2 3 5/3
: \[table\] Values of the scaling exponents for the layer thickness and the free energy.
If we impose monodisperse pseudo-loops ($P(u)=\delta (u-N)$) and $S_0=\sigma$, the grafting density, we immediately recover the standard results for polymer brushes. In good solvent conditions, these are: the extension $L\cong
aN(a^2\sigma)^{1/3}$, the free energy $\mathcal{F}\cong k_BTN(a^2\sigma)^{11/6}$ and the volume fraction of monomers $\Phi \cong (a^2\sigma)^{2/3}$. On the other hand, if we let the polydispersity free to minimize the thermodynamical potential (with $S_0=a^{-2}$ to account for attraction), we recover the results found for reversibly adsorbed chains. In good solvent conditions, we find that the volume fraction of monomer scales as $\Phi(z)\cong (a/z)^{4/3}$, and the extension as $L\cong aN^{3/5}$.
Such idea proved to be successful in describing many different kinds of polymer layers (grafted, reversibly adsorbed [@AGR], irreversibly adsorbed [@Guiselin]), whatever the solvent quality (good solvent, $\Theta$-solvent and melt, i.e. no solvent). The approach was further expanded to the cases of convex interfaces [@AGRspheres].
The success of this phenomenological approach lead us to address the status of Eq. (\[fenergie\]). The SF approach is so far an elegant model but not a theory because Eq. (\[fenergie\]) is not deduced from first principles, and the set of approximations involved is not explicited. Recently, the SF approach was applied to the issue of surface tension of polymeric liquids [@ManoPRL; @ManoMacromol; @ManoColloid]. Here again, the SF approach proved to be successful in reproducing the experimental features in great detail. However, because the results presented in Ref. [@ManoPRL] are different from the results of the self-consistent field theory on the same issue, it seems important to clarify the soundness of the SF approach. This question is addressed here in some detail.
The SF approach raises two questions essentially: *a)* is it sound ? *b)* is it valid ? The first question addresses the status of the SF approach, the second has to do with the validity of the results that we will find by using it. Obviously, these two issues are linked. Because “sound” is sometimes used for “crude” or “inaccurate”, it is useful to carefully explain what we mean by “sound” and “valid” before we start arguing.
As it stands, the SF approach is a phenomenological description. This is useful on issues where we do not have any theory available. On the other hand, suppose we are in a position to compare a phenomenological approach to a theory on the same issue. The theory will always prevail. If the two results are in agreement, this is fine, but then the phenomenology is a trick to understand qualitatively the issue, and essentially does not bring new features. If, on the contrary, the two results are different, there is always the suspicion that the phenomenological approach is a good idea extrapolated to an issue where this idea is too simple, and therefore, the result is wrong. We simply say “the approach is not sound”. Accuracy then is less relevant.
The debate is quite different when we have to compare two theories on the same issue. If somehow we were able to deduce the SF approach from first principles, and therefore prove that this is a theory, then the question of soundness is resolved. Of course this would be done within approximations, and the theory may be crude or inaccurate to treat the issue, but it is sound. Then the debate over accuracy is essential to evaluate the results.
We see that the status of the SF approach is the first question to be addressed, and depending on the answer, the debate over validity will be different. In Section \[status\], we deduce the effective free energy, Eq. (\[fenergie\]), from first principles. In doing so, we demonstrate that the SF approach is indeed a variational theory for polymer layers. Then we are lead to ask the second question: is it valid ? Such task involves comparing the results found with SF theory to SCF theories both at a formal level, and at the level of the results. In Section \[validity\], we address this question.
Status \[status\]
=================
Variational free energy
-----------------------
We consider a set of $N_C$ monodisperse, linear, neutral chains in contact with a solid plane (area $\Sigma $). We assume that the layer is uniform in the directions parallel to the surface. Our starting point is the partition function, $\mathcal{Z}$, of the chains, each characterized by the path $z_i(n)$, where $z_i$ is the position normal to the surface and $n$ is the curvilinear index ($1\leq i\leq N_C$): $$\mathcal {Z} =\prod_i^{N_C} \int_0^{\infty} dz_i(0)
\int_0^{\infty} dz_i(N)\int \mathcal{D}\{z_i\}\exp \left[ - \frac{{\mathcal
H}}{k_BT}\right]$$ where the effective Hamiltonian, $\mathcal{H}$, is the sum of an elastic (entropic) contribution, $$\mathcal{H}_{\mathrm{el}}=\frac32\frac{k_BT}{a^2}\sum_i^{N_C}\int_0^N
\left(\frac{dz_i}{dn}\right)^2 dn$$ and an excluded-volume (two-body) interaction with parameter $v$, $$\mathcal{H}_{\mathrm{ex}}=\frac{vk_{B}T}{2\Sigma}\sum_{i,j}^{N_C}
\int_0^N\int_0^N\delta \left(z_i(n)-z_j(n')\right) dndn'$$ where $\delta$ is the Dirac distribution. We limit ourselves to two-body interactions and thus neglect interactions of further order. This is not valid for a $\Theta$-solvent (where $v=0$), but as we shall see in Section \[solvent\], the correct free energy for this type of solvent is easily introduced afterwards. The volume fraction at distance $z$ writes $$\phi(z)=\frac{1}{\Sigma}\sum_{i=1}^{N_C}\int_0^N\delta(z-z_i(n))dn.
\label{philoc}$$
![\[figure\] As far as the Hamiltonian is concerned, each chain in contact with the boundary (a) with associated path $z(n)$ ($1<n<N$), is formally equivalent to the set of pseudo-loops (b) obtained by cutting the loops into two equal pieces with associated paths $\{z_{\alpha}(n),1<n<m_{\alpha}\}$.](figure1.eps)
Regardless of the particular microscopic situation that is realized, we can always decompose the chain into loops and tails, and rewrite $\mathcal{H}$ accordingly. This amounts to cut the integrals into smaller pieces, by identifying the monomers either in contact with the surface or at the top of the loops and that we note $n_{i,\alpha}$. Each piece corresponds to the complete path of a loop or a tail. The “cutting” scheme is described in Fig. \[figure\]. We implicitly assume that the loops are symmetric, which comes from the translation invariance parallel to the solid surface. Hence mathematically, these identified monomers have a “null velocity”: $dz_i/dn|_{n_{i,\alpha}}=0$ for each $i$ and $\alpha$. The chain $i$ is then cut in $N_i$ pieces of size $m_{i,\alpha}=n_{i,\alpha}-n_{i,\alpha-1}$ where $1\leq\alpha\leq N_i$ with $\sum_{\alpha }m_{i,\alpha}=N$. For tails, we consider the full path from the extreme monomer to the first monomer in direct contact with the surface, as expected. The loop are cut into two pieces of equal length, which we shall call pseudo-loops. Clearly, as far as mathematics is concerned, tails and pseudo-loops are similar objects : these are chain segments starting at the surface and ending somewhere in the solution with no velocity at these extreme monomers (cf. Fig. \[figure\]). For that reason, we shall not distinguish between tails and pseudo-loops in the rest of the letter, and refer to both of them as “pseudo-loops”. As is obvious, such decomposition *a)* is always possible, *b)* is unambiguous, *c)* let the partition function $\mathcal{Z}$ identical without any approximation, provided that we supplement the cutting procedure by the constraint (later referred to as $\mathcal{C}$), that the free extremities of the chain *segments* originating from the same loop should be at the same height $z$. Then $\left(m_{i,\alpha},
\left\{z_{i,\alpha}\right\}\right)_{\alpha=1,N_i}$ designates the set of sizes and paths of pseudo-loops for chain $i$, and we rewrite the Hamiltonian:
$$\mathcal{H} =
\frac{3}{2}\frac{k_{B}T}{a^{2}}\sum_{i=1}^{N_{C}}\sum_{\alpha
=1}^{N_{i}}\int_{0}^{m_{i,\alpha }}\left( \frac{dz_{i,\alpha }}{dn}\right)
^{2}dn + \frac{vk_{B}T}{2\Sigma
}\sum_{i,j=1}^{N_{C}}\sum_{\alpha ,\beta =1}^{N_{i}}\int_{0}^{m_{i,\alpha
}}\int_{0}^{m_{j,\beta }}\delta \left( z_{i,\alpha }(n)-z_{j,\beta }(n^{\prime
})\right) dndn^{\prime } .
\label{Hdécomposé2}$$
Computing exactly the partition function of the system with the Hamiltonian Eq. (\[Hdécomposé2\]) is clearly out of reach. Rather, we implement the variational principle which necessitates two steps [@variationnel]. First, we need to choose a trial probability such that $\mathcal{P}_{T}$ is a good approximation of the actual probability, $\mathcal{P}=
\mathcal{Z}^{-1} \exp [-\mathcal{H}/k_{B}T]$, but nevertheless allows for analytical calculations. Second, we approximate the exact free-energy, $\mathcal{F}$, of the system by the extremum of the functional $\mathcal{F}_{\mathrm{var}}\{\mathcal{P}_{T}\} = \left\langle
\mathcal{H}\right\rangle_{\mathcal{P}_{T}}+ k_BT\left\langle
\ln\mathcal{P}_{T}\right\rangle_{\mathcal{ P}_{T}}$. Of these two steps, the second one is the simplest because it is purely a matter of calculation. Only the first one is significant as regard to the physics, since the success of the variational theory lies in finding an appropriate trial function. The guess $\mathcal{P}_{T}$ is a functional form with free unspecified parameters. By minimizing $\mathcal{F}_{\mathrm{var}}\{\mathcal{P}_{T}\}$ with respect to these parameters, we will obtain $\mathcal{P}_{T}$ with the chosen functional form that best approximates $\mathcal{P}$. This is what ultimately controls the difference between $\mathcal{F}$ and the approximation $\mathcal{F}_{\mathrm{var}}$. Note that the choice of the ensemble of functions over which we shall perform the minimization is arbitrary. It is a guess, not an approximation which could be somehow quantified *a priori*.
Our guess for $\mathcal{P}_{T}$ is: $$\mathcal{P}_{T}\left(\left\{(m_{i,\alpha},
\{z_{i,\alpha}\})_{\alpha=1,N_i}\right\}_{i=1,N_C}\right)=
\prod_{i=1}^{N_C}\prod_{\alpha=1}^{N_i}P(m_{i,\alpha},\{z_{i,\alpha}\}),
\label{defP}$$ where $\int_0^N P(m_{i,\alpha},\{z_{i,\alpha}\})dm=1$. Equation (\[defP\]) is a mean-field type of approximation for the pseudo-loops since their probability distributions are decorellated (hypothesis *A*). Furthermore, we assume that *the path $\{z_{i,\alpha}\}$ is the same for all the pseudo-loops and is noted $\{z\}$* (hypothesis *B*). Because $P(m_{i,\alpha},\{z_{i,\alpha}\})$ does not depend on the particular pseudo-loop that is considered, we can drop the indexes and write $P(m,\{z\})$. Hence the probability distribution reads $\mathcal{P}_{T}=P(m,\{z\})^{B}$ where $B=\sum_{i=1}^{N_C}N_i$ is the number of pseudo-loops at the interface. The crucial point is that $\mathcal{P}_{T}$ does no more depend on the complete set of sizes and path, $\left\{ m_{i,\alpha },\left\{ z_{i,\alpha }\right\} \right\}
$, but only on *a)* the size of the pseudo-loop, $m$, and on *b)* the path $z$, chosen to be the same for all pseudo-loops. Importantly, the constraint $\mathcal{C}$ is automatically fulfilled with our approximation since two pseudo-loops originating from the same loop have the same size, $m$, and thus terminate at the same height $z(m)$. Then, the system is described by two functions: $P(m)$, the probability that we have a pseudo-loop of size $m$, and $z(n)$, the path of the chain segments. Hence, the trial free energy is obtained by minimizing $\mathcal{F}_{\mathrm{var}}$ with respect to changes in $z$ and $P$ (later, we will find it more convenient to work with $S$, rather than $P$).
With (\[defP\]), we find $$\left\langle\mathcal{H}_{\mathrm{ex}}\right\rangle_{\mathcal{P}_{T}} =
\frac12 \frac{v\Sigma}{a^6} k_BT \int
dz\Phi^2(z)
\label{Hex}$$ where $$\begin{aligned}
\Phi(z) &\equiv& \left\langle\phi(z)\right\rangle_{\mathcal{P}_{T}} \nonumber
\\ &=& a^3S_0\int_0^N dmP(m)\int_0^m\delta(z-z(n))dn
\label{phim}\end{aligned}$$ and $$\left\langle \mathcal{H}_{\mathrm{el}}\right\rangle
_{\mathcal{P}_{T}}=\frac32 \frac{k_BT}{a^2}\Sigma S_0
\int_0^Ndm P(m)\int_0^m\dot{z}^2(n)dn
\label{Hel}$$ where $B=\Sigma S_0$ (hence $S_0$ is the “grafting density” of pseudo-loops), and $\dot{z}=dz/dn$. Similarly, the entropic part of $\mathcal{F}_{\mathrm{var}}$ is found to be $$k_BT\left\langle \ln \mathcal{P}_{T}\right\rangle_{\mathcal{P}_{T}} = k_BT\Sigma
S_0\int_0^NdmP(m)\ln P(m).$$ Combining all these results and integrating by parts and using Eq. (\[defS\]), we find $$\begin{aligned}
\frac{{\mathcal F}_{\mathrm{var}}(\{S\},\{\dot{z}\})}{k_BT\Sigma} &=& \int_0^N
\left\{\frac{3}{2a^2}\dot{z}^2(n)S(n) +
\frac{v}{2}\frac{S^2(n)}{\dot{z}(n)} \right. \nonumber\\
&-& \left. S'(n)\ln\left(-\frac{S'(n)}{S_0}\right)
\right\}dn .
\label{centralresult}\end{aligned}$$ Note that $\Phi(z)=S(n(z))/\dot{z}$. Equation (\[centralresult\]) is the central result of this letter which we now discuss. To get the best approximation, we minimize Eq. (\[centralresult\]) with respect to $\dot{z}$ which yields $\dot{z}=\left(va^{2}/6\right)^{1/3} S^{1/3}$, and when this result is introduced back into Eq. (\[centralresult\]), we find Eq. (\[fenergie\]) with $\beta =5/3$ and $k=\frac{3.6^{1/3}}{4}\left(v/a^3\right)^{2/3}$. We thus find the mean-field version of our effective free energy Eq. (\[fenergie\]) with a numerical coefficient, $k$, of order 1.
The formal derivation presented here brings an interesting remark. In the early developpements of the SF theory, the entropic part in Eq. (\[fenergie\]) was introduced (and interpreted) as a contribution arising from combinatorial arrangements of pseudo-loops at the surface: the presence of the interface breaks down the symmetry of the solution and these monomers in contact with the surface become *distinguishable*. We see that the entropic term in Eq. (\[centralresult\]) is formally that contribution arising from the entropy of the trial probability.
Generalization to other solvent conditions \[solvent\]
------------------------------------------------------
The generalization to other solvent conditions, i.e. good solvent, $\Theta$-solvent and melt, has been done in other references [@AGR; @se] and deserves some comments.
In the case of a melt, the excluded volume interactions are screened at all scales, and our mean-field approximation for pseudo-loops is automatically verified. The probability distribution is then related to the Green function of a chain by $P(m)\propto G(0,z(m);m)$ where $z(m)$ is self-consistently determined *via* the constraint $\dot{z}(n)=S_0\int_n^NP(m)dm$ ($\phi(z)=1$ everywhere in an incompressible melt).
For a good solvent, the osmotic Eq. (\[Hel\]) and elastic term Eq. (\[Hex\]) are easily renormalized, following the des Cloiseaux law [@desCloiseauxlaw], and using semi-dilute blobs [@PGGbook]. However, the approximation which consists in neglecting correlations between pseudo-loops is *a priori* not verified. Thus, the transformation of $k_BT\langle\ln{\mathcal
P}_T\rangle_{{\mathcal P}_T}$ in $k_BT\Sigma S_0\int_0^Ndm P(m)\ln P(m)$ is not justified. However, correlations between monomers inside the same pseudo-loop are taken into account through the blob renormalization.
Hence we have demonstrated that the SF approach is a variational theory, and Eq. (\[fenergie\]) is sound.
Validity \[validity\]
=====================
Of course, that the SF theory is sound (in the sense that it is deduced from first principles) does not guarantee at all that it is accurate, or even simply valid to describe polymeric layers. This is because we have made approximations whose range of validity remains to be examined.
*A priori*, we could distinguish three different point of view to discuss the issue of accuracy: *a)* internal, *b)* external and *c)* experimental.
Internal estimate of accuracy
-----------------------------
Internal means that we are able to estimate the error that we have made in approximating the initial Hamiltonian, and thus propose an internal criterium of validity, very much like the Lifchitz criterium of validity for mean-field theories. This requires that we define a relevant parameter which would quantify the difference between the initial and the approximated Hamiltonian, i.e. the two assumption that we made.
Concerning hypothesis *A*, we know that the mean-field approximation for the loops is not valid in good solvent conditions. This implies that the last term of Eq. (\[centralresult\]) is wrong. However, the renormalization with semi-dilute blobs of the first two terms takes into account the swelling of the pseudo-loops (hence correlations between monomers) on scales smaller than the pseudo-loop sizes. Thus for loops at least larger than one blob size, the excluded volume interactions are screened and this loops are decorellated. Hence, the entropic term of Eq. (\[centralresult\]) is justified for a large number of the pseudo-loops and even if it is not fully satisfying, this is the best way we can take into account these correlations unless we are lead to use renormalization group theory, which has been done for one chain but not for many chains [@Eisenriegler].
The hypothesis *B* is the crudest assumption in our theory. We assume that all pseudo-loops have the same *mean* path $z(n)$. It is easy to show that for a melt, we find by minimization $z_{\mathrm{eq}}(n) \simeq
n^{1/2}$, which is the best variational approximation with our probability distribution Eq. (\[defP\]). This result is quite similar to the Flory theorem $R \simeq aN^{1/2}$ for the extension of a polymer chain in a melt. Of course this result is valid for large $n$, since for a random walk, fluctuations around this value is proportional to $n^{-1/2}$. This result may not be valid for small loops. However, with variational theories, the estimate of this error is impossible.
External estimate of accuracy
-----------------------------
External means that we compare the SF theory with another theory. For polymeric layers, the obvious candidate is SCF theories. *A priori*, there are two ways to do that: *a)* a formal comparison, *b)* a comparison of the results that we obtain on a given issue. A formal comparison is simple when the two theories have a common language. Unfortunately, this is not the case for SCF theories and the SF theory. The former is deduced from the initial Hamiltonian through a mean-field type of approximation for monomer-monomer correlations which is then applied to the problem of polymer at interfaces, whereas the latter proceeds in *first* rewriting the Hamiltonian for chains at interfaces and *then* using a mean-field approximation for pseudo-loops. Because of this different order for these two steps, we do not know the way to formally compare SCF and SF theories. Then we are left with comparing the results.
There are two issues where such comparison is possible: *a)* brushes in the infinite stretching limit, “mean-field” solvent conditions, and *b)* reversibly adsorbed layers, “mean-field” solvent conditions. This issues are conceptually important because we know exactly the solution of the SCF theory in the asymptotic limit $N\rightarrow\infty$.
### Brushes
As shown by Netz and Schick [@Netz] and Li and Witten [@Li], the theory of polymer brushes proposed simultaneously by Milner, Witten, and Cates (MWC) and Zhulina *et al.* in Refs. [@MWC; @Zhulina], which consists in keeping the classic path in the partition function, can also be considered as a variational approach. However, the trial probability is different, and the layer is described by two functions: $g$, such that $g(z_0)dz_0$ is the probability that the chain free extremity belongs to the interval $[z_0,z_0+dz_0]$, and $e$, such that $e(z,z_0)=|dz/dn|$ is the extension at position $z$ for a chain whose free-extremity is situated at $z_0$. Paths (described by $e$) are chosen such as polymers are grafted at one end (with grafting density $\sigma $), i.e. $\int_0^{\infty}\frac{dz_0}{e(z,z_0)}=N$ (which leads to the so-called equal time argument). The variational free energy (per $cm^2$) is [@Netz]: $$\begin{aligned}
\frac{\mathcal{F}_{\mathrm{MWC}}}{k_BT} &=&
\frac{v}{2}\int_0^{\infty}\Phi^2(z)dz \nonumber \\ &+& \sigma
\int_0^{\infty}dz_0 g(z_0)\int_0^{z_0}\frac{3}{2a^2}e(z,z_0)dz \nonumber \\ &+&
\sigma \int_0^{\infty}g(z_0)\ln [g(z_0)]dz_0, \label{NRJSchick}\end{aligned}$$ with $\Phi(z)=\sigma \int_z^{\infty}dz_0 \frac{g(z_0)}{e(z,z_0)}$. Note that in the context of brushes, the entropic contribution in Eq. (\[NRJSchick\]), which is similar to that in Eq. (\[centralresult\]), is the entropy of the chain end distribution [@Netz; @g1; @g2]. Simple arguments show that the first two terms in the rhs of Eq. (\[NRJSchick\]) scale as $N(a^2\sigma)^{5/3}$, whereas $\int g(z_0)\ln [g(z_0)]dz_0\sim 1$. Hence, in the strong stretching limit, $N(a^2\sigma)^{2/3}\gg 1$, the entropic contribution to $\mathcal{F}_{\mathrm{MWC}}$ is negligible [@Netz]. However, this term is conceptually important and has a physical signification since $e(z_0,z_0)$ is the tension sustained by the free chain-ends. Hence, we see that Eq. (\[centralresult\]) and Eq. (\[NRJSchick\]) are formally very close, but the choices for, respectively, $z(n)$ and $e(z,z_0)$ are different.
To compare the SF theory with the MWC theory, we concentrate on monodisperse brushes (hence the entropic contribution in Eq. (\[centralresult\]) disappears) in the strong stretching limit (hence, we neglect the entropic contribution in Eq. (\[NRJSchick\])). We find in equilibrium $\mathcal{F}_{\mathrm{MWC}}^*=0.892\,\mathcal{F}^*$. We see that the extremum of $\mathcal{F}_{\mathrm{MWC}}^*$ is lower and according to the variational criterium, the MWC theory is a better approximation of the exact free energy. See [@Li; @milner] for a thorough discussion of this difference. It is related to the different choices for the paths where the MWC choice (i.e. the equal time argument) is less restrictive. The reason is that in the SF theory for brushes, we impose an additional constraint: all chain free extremities are situated in the outer edge of the layer, in a fashion similar to the Flory approach (or the Alexander-de Gennes, which is similar in spirit but introduces the correct scaling exponents). Formally, this amounts to impose a delta type of function for $g$, a restriction motivated by our desire to keep the SF theory tractable in a wider range of situations. Eventually, we find the same results for $L$ and $\mathcal{F}^*$ at the scaling level, although the description of the volume fraction profile is more accurate in the MWC theory.
### Adsorbed layers
Presumably, the case of reversibly adsorbed polymers is more significant for our purpose since our variational approach is based on a “loop description”, which is justified for the homogeneous adsorption.
If we go to reversible adsorption, we have to turn our attention to GSD theory. Although desirable, it is not so simple to compare the SF theory with GSD theories. There are two reasons for this: *a)* the GSD theory uses the analogy between the partition function $\mathcal{Z}$ and the Green propagator in quantum mechanics, which does not allow a description in “polymer trajectories”; *b)* in this theory, the free energy is expressed in terms of the mean monomer concentration $\Phi(z)$, a quantity not simply related to our probability density $P(m).$ Indeed, the partition function of a chain having one end at $z$ and the other free, $\mathcal{Z}(N,z)$, in the SCF theory, is the solution of the Schrödinger equation : $\partial
\mathcal{Z}/\partial N=\frac{a^2}{6}\partial^2 \mathcal{Z}/\partial
z^2-U\mathcal{Z}$, where the external potential, $U$, is the sum of the attractive potential due to the surface, $U_{\mathrm{surf}}$, and the self-consistent potential, $U_{\mathrm{SCF}}$. For adsorbed chains, there is a ground state of negative energy $-\varepsilon Nk_BT$ which dominates the solution, and (in the limiting case where $\varepsilon N\gg 1$) the free energy approximates to: $$\mathcal{F}_{\mathrm{GSD}}=k_BT\int_0^{\infty}dz\left[\kappa(\Phi)\left(
\frac{d\Phi}{dz}\right)^2 + U(z)\Phi(z)\right] , \label{GSD}$$ where $\kappa(\Phi)=a^2/(24\Phi)$. As shown by Lifshitz and des Cloiseaux [@Lifshitz; @des; @Cloiseaux], the square gradient term in Eq. (\[GSD\]) has essentially an entropic origin [@note], whereas the polymeric nature of the liquid can be neglected in the molecular field $U_{\mathrm{SCF}}(z)$ (which is estimated for a monomeric liquid). Then we are lead to think that the elastic and entropic part in Eq. (\[centralresult\]) are related to the square gradient term, but we are not able to rewrite the former as the latter at the moment.
In the absence of any clue to formally compare Eqs. (\[centralresult\]) and (\[GSD\]), we shall compare their results for infinite chains and mean-field potential, a limit where the GSD theory happens to be exact. If we minimize the free energy Eq. (\[centralresult\]) with the boundary conditions $S(0)=a^{-2}$, $S(N\rightarrow\infty)=0$, we find $a^2
S_{\mathrm{eq}}(n)=k'^{3/2}/(n+k')^{3/2}$ where $k'=(3/(2k))^{4/9}$ which yields $\Phi(z)\sim z^{-2}$, essentially the solution found by minimizing Eq. (\[GSD\]). Similarly, we find that $\mathcal{F}^*\cong k_BT/a^2$ as with GSD theory. Hence, we find a very good agreement for infinite chains.
That the agreement should be better (in the sense that both the scaling and the concentration profile are identical) for adsorption than for brushes reflects the validity of our initial assumption that all pseudo-loops have the same path. As explained in Ref. [@MWCpoly], for very polydisperse layers, we expect a stratification of the locations of the free chain ends above the surface. This is because the free ends of a long chain locates further away from the surface than that of a short chain to take advantage of a lower osmotic pressure (the concentration decreases away from the interface). In the continuum limit, this argument suggests that every pseudo-loops are similarly extended, and therefore validates our guess.
Experimental estimate of accuracy
---------------------------------
To evaluate the accuracy of a variational theory, the ultimate and major argument is to compare the value of the free energy at its minimum to experiments. The good candidate is thus the surface tension of polymeric liquids, $\gamma$. We have shown in Refs. [@ManoPRL; @ManoMacromol; @ManoColloid], that the SF approach allows the calculation of the variations of $\gamma(N)$ in very good agreement with experimental data found in the literature. This is a good test for the theory which has been done both for melts and semi-dilute solutions (in good solvent).
It is important to note that the SCF theory in the GSD approximation leads to a different result for the melt surface tension. The finite chain correction in that case is proportional to $N^{-1}$. We found a larger correction in $\ln
N/N^{1/2}$. An explanation of this discrepancy is that the SCF description relates the surface tension to the gradients in volume fraction which are localized in a very thin layer of thickness $a$ (indeed this approach is not valid for large gradients). We argue that this dependence comes from the chains reorganization on a larger layer of thickness the radius of gyration of a chain. In this layer, the volume fraction is constant. Thus it cannot be described by the SCF approach whereas the SF approach uses different tools, namely $z(n)$ and $S(n)$, which allows such a description. Hence, for adsorbed layers from a melt and a semi-dilute solution, we see that these two approaches are quite different.
Concluding remarks
==================
This article aims at clarifying the debate concerning the “soundness” of the Scaling Functional approach. In view of this, the demonstration that the SF approach is a variational theory is certainly the essential and most significant result of this article (Section \[status\]). But we think we have made clear a certain number of points (Section \[validity\]). These are:
1. The SF approach is a variational theory and therefore has the same epistemological status as SCF theories for brushes (“classical” solution) and adsorbed chains (GSD). Of course, the approximations made are different and each of these theories has a different range of validity.
2. Because the SF theory is a variational theory, we are not able to properly quantify the approximations that are involved, and therefore we are unable to define the range of validity of this theory.
3. There is no way that we know to formally compare the two theories because the first step in approximating the initial Hamiltonian are different.
4. Each time a direct comparison with SCF theory is possible, we find *a)* always the same scaling results, and *b)* sometimes the same analytical result. Thus we conclude that the SCF theory does not provide any argument against the SF theory.
That the GSD approximation to the SCF theory is formally justified and quantifiable in “mean-field” solvent conditions does not guarantee that the result that we find is accurate for real solvent conditions and notably for the melt case. A description only in terms of volume fraction (see Eq. (\[GSD\])) comes also from a variational argument [@des; @Cloiseaux] and has not been quantitatively justified in real systems. In other words, the GSD in the limit $N\rightarrow\infty$, is the exact solution of the SCF theory, but still an approximate solution of the initial Hamiltonian.
The crucial point regarding our approximations (cutting loops into two tails and describing all the pseudo-loops with the same path) is whether the distinction between loops and tails is important enough to modify the conclusions of a simple theory in which it is neglected. When we are in a position to directly evaluate the consequences of these approximations, we find that this distinction does not affect the scaling results. It is interesting to note that the distinction between loops and tails has been done self-consistently for the SCF theory but is put *ad hoc* for other type of solvents [@Semenov-Joanny]. Therefore we conclude that there is no valid argument to support that these approximations are not sound, provided that we remain at a scaling level of description.
Finally, we assume in this approach that a large number of loops are formed at the interface. This impose both a sharp interface and the presence of many adsorbed chains. Therefore, this theory does not apply to single chain adsorption and to systems such as interfaces between incompatible polymers or diblock copolymers, for which other approaches based on the SCF theory have been developed [@Helfand; @Leibler].
As a conclusion, the SF theory proposes a compromise between a precise description of the polymeric layer, and a wide ranging scaling type of theory valid for arbitrary polymer layers, various solvent conditions and various geometries. Since it does not require a comparable amount of mathematics and has a wider range of applicability than both theories, it is very likely that the SF theory will become an important piece of our understanding of polymeric interfaces.
We are grateful to B. Fourcade for stimulating conversations. We also benefitted from discussions with A.N. Semenov and R.R. Netz.
[99]{}
G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove and B. Vincent, *Polymer at Interfaces* (Chapman et Hall, London, 1993).
A.N. Semenov, J. Bonet-Avalos, A. Johner and J.-F. Joanny, Macromolecules **29**, 2179 (1996).
A.N. Semenov, Sov. Phys. JETP **61**, 733 (1985).
S.T. Milner, T.A. Witten and M.E. Cates, Europhys. Lett. **5**, 413 (1988).
E.B. Zhulina, V.A. Priamitsyn O.V. Borisov, Polym. Sci. USSR **31**, 205 (1989); A.M. Skvortsov, A.A. Gorbunov, V.A. Palushkov, E.B. Zhulina, O.V. Borisov, V.A. Priamitsyn, Polym. Sci. USSR **30**, 1706 (1988); E.B. Zhulina, O.V. Borisov, V.A. Priamitsyn, T.M. Birshtein, Macromolecules **24**, 140 (1991).
M. Aubouy, O. Guiselin and E. Raphaël, Macromolecules **29**, 7261 (1996).
M. Aubouy, Phys. Rev. E **56**, 3370 (1997). Note that Sections II and IV proved to be incorrect see \[9,10\].
O. Guiselin, Europhys. Lett. **17**, 225 (1992).
M. Aubouy and E. Raphaël, Macromolecules **31**, 4357 (1998).
M. Aubouy, M. Manghi and E. Raphaël, Phys. Rev. Lett. **84**, 4858 (2000).
M. Manghi and M. Aubouy, Macromolecules **33**, 5721 (2000).
M. Manghi and M. Aubouy, Adv. Colloid. Interf. Sc. **94**, 21 (2001).
P.M. Chaikin and T.C. Lubensky, *Principles of Condensed Matter Physics* (Cambridge University Press, Cambridge, 1995).
J. des Cloiseaux, J.Phys. France **36**, 281 (1975).
P.-G. de Gennes, *Scaling Concepts in Polymer Physics* (Cornell University Press, Ithaca, 1985).
E. Eisenriegler, *Polymers near Surfaces* (World Scientific, Singapore, 1993).
R.R. Netz and M. Schick, Macromolecules **31**, 5105 (1998).
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The same expression holds for the interfacial energy of two incompatible polymers. In that context, the gradient term is found with the Random Phase Approximation. See for instance : R.A.L. Jones and R.W. Richards, *Polymers at Surfaces and Interfaces* (Cambridge University Press, Cambridge, 1999).
S.T. Milner, T.A. Witten and M.E. Cates, Macromolecules **22**, 853 (1989).
E. Helfand, Polymer Interfaces in *Polymer Compatibility and Incompatibility* (H. Solc. Chur, Harwood, 1982).
L. Leibler, Macromolecules **13**, 1602 (1980).
| ArXiv |
---
abstract: 'This review describes the multiboson algorithm for Monte Carlo simulations of lattice QCD, including its static and dynamical aspects, and presents a comparison with Hybrid Monte Carlo.'
address: 'ETH, CH-8092 Zürich, Switzerland'
author:
- Philippe de Forcrand
title: The MultiBoson method
---
Monte Carlo, fermions, algorithms
section1.tex section2.tex section3.tex section4.tex concl.tex
| ArXiv |
---
abstract: 'We investigate the field-angle-dependent zero-energy density of states for YNi$_2$B$_2$C with using realistic Fermi surfaces obtained by band calculations. Both the 17th and 18th bands are taken into account. For calculating the oscillating density of states, we adopt the Kramer-Pesch approximation, which is found to improve accuracy in the oscillation amplitude. We show that superconducting gap structure determined by analyzing STM experiments is consistent with thermal transport and heat capacity measurements.'
address:
- '$^{1}$Department of Physics, University of Tokyo, Tokyo 113-0033, Japan'
- '$^{2}$CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Tokyo 110-0015, Japan'
- '$^{3}$CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan'
- '$^{4}$Department of Basic Science, University of Tokyo, Tokyo 153-8902, Japan'
- '$^{5}$CNR-INFM, CASTI Regional Lab, I-67010 Coppito (L’Aqulia), Italy'
- '$^{6}$Department of Physics, Kobe University, Nada, Kobe 657-8501, Japan'
author:
- 'Yuki Nagai$^{1}$, Nobuhiko Hayashi$^{2,3}$, Yusuke Kato$^{1,4}$, Kunihiko Yamauchi$^{5}$ and Hisatomo Harima$^{6}$'
title: 'Field angle dependence of the zero-energy density of states in unconventional superconductors: analysis of the borocarbide superconductor YNi$_2$B$_2$C'
---
The discovery of the nonmagnetic borocarbide superconductor YNi$_2$B$_2$C [@Cava] has attracted considerable attention because of the growing evidence for highly anisotropic superconducting gap and high superconducting transition temperature 15.5K. In recent years, Maki [*et al*]{}. [@Maki] theoretically suggested that the gap symmetry of this material is $s$+$g$ wave and the gap function has zero points (point nodes) in the momentum space. Motivated by this prediction, field-angle dependence of the heat capacity [@Park] and the thermal conductivity [@Izawa] have been measured on YNi$_{2}$B$_{2}$C. The gap symmetry can be deduced from their oscillating behavior. Those experimental results [@Park; @Izawa] were considered to be consistent with the $s$+$g$-wave gap. However, the present authors [@NagaiJ] recently found that the local density of states (LDOS) around a vortex calculated for the $s$+$g$-wave gap on an isotropic Fermi surface (FS) is not consistent with measurements by scanning tunneling microscopy and spectroscopy (STM/STS) [@Nishimori]. Therefore, we calculated the LDOS around a vortex with the use of a realistic FS of the 17th band obtained by a band calculation [@NagaiY]. We also investigated the density of states (DOS) under zero field and the field-angle dependence of the zero-energy DOS (ZEDOS). Consequently, we proposed alternative gap structure for YNi$_{2}$B$_{2}$C and succeeded in reproducing those experimental observations consistently [@NagaiY]. In this paper, we investigate the field-angle-dependent ZEDOS taking the 18th band into account in addition to the 17th band considered previously. While the so-called Doppler-shift (DS) method was previously utilized in Ref. [@NagaiY], we adopt here a more reliable method [@NagaiLett] for calculating the ZEDOS.
The DOS is the basis for analyzing physical quantities such as the specific heat and the thermal conductivity [@Vorontsov]. For example, the specific heat $C/T$ is proportional to the ZEDOS in the zero temperature limit $T \rightarrow 0$. The DOS is obtained from the regular Green function $g$ within the quasiclassical theory of superconductivity, which is represented by a parametrization with $a$ and $b$ as $g = - (1-a b)/(1+ a b)$. They follow the Riccati equations ($\hbar=1$) [@Schopohl]: $$\begin{aligned}
\Vec{v}_{\rm F} \cdot \Vec{\nabla} a + 2 \tilde{\omega}_n a + a \Delta^{\ast} a - \Delta &=& 0,
\label{eq:ar}\\
\Vec{v}_{\rm F} \cdot \Vec{\nabla} b - 2 \tilde{\omega}_n b - b \Delta b + \Delta^{\ast} &=& 0.
\label{eq:br}\end{aligned}$$ Here, $\Vec{v}_{\rm F}$ is the Fermi velocity, and $i \tilde{\omega}_n = i \omega_n + (e/c) \Vec{v}_{\rm F} \cdot \Vec{A}$ with the Matsubara frequency $\omega_n$ and the vector potential $\Vec{A}$.
To analyze the field-angle-dependent experiments quantitatively, we have developed a method on the basis of the Kramer-Pesch approximation (KPA) [@NagaiLett]. This method enables us to take account of the vortex-core contribution which is neglected in the DS method. By virtue of it, one can achieve quantitative accuracy. We consider a single vortex situated at the origin of the coordinates. To take account of the contributions of a vortex core, in the KPA we expand the Riccati equations (\[eq:ar\]) and (\[eq:br\]) up to first order in the impact parameter $y$ around a vortex and the energy $\omega_n$ [@NagaiJ] ($i\omega_n \to E+i\eta$ and $y$ is the coordinate along the direction perpendicular to $\Vec{v}_{\rm F}$). By this expansion, one can obtain an analytic solution of the Riccati equations around a vortex. We then find the expression for the angular-resolved DOS [@NagaiLett] $$N(E, \alpha_{\rm M}, \theta_{\rm M})
=
\frac{v_{\rm F0} \eta}
{2 \pi^2 \xi_0}
\Biggl{\langle}
\int
\frac{ d S_{\rm F} }
{ |\Vec{v}_{\rm F}| }
\frac{
\lambda
\bigl[ \cosh
(x/\xi_0)
\bigr]^{\frac{-2 \lambda}{\pi h}}
}{(E-E_y)^2 + \eta^2}
\Biggl{\rangle}_{\rm SP}.
\label{eq:dos}$$ Here, the azimuthal (polar) angle of the magnetic field $\Vec{H}$ is $\alpha_{\rm M}$ ($\theta_{\rm M}$) in a spherical coordinate frame fixed to crystal axes, and $d S_{\rm F}$ is an area element on the FS \[e.g., $d S_{\rm F}=k_{\rm F}^2 \sin\theta d\phi d\theta$ for a spherical FS in the spherical coordinates $(k,\phi,\theta)$, and $d S_{\rm F}=k_{{\rm F}ab} d\phi dk_c$ for a cylindrical FS in the cylindrical coordinates $(k_{ab},\phi,k_c)$\]. In the cylindrical coordinate frame $(r,\alpha,z)$ with ${\hat z} \parallel \Vec{ H}$ in the real space, the pair potential is $\Delta \equiv \Delta_0 \Lambda(\kv_{\rm F}) \tanh(r/\xi_0) \exp(i\alpha)$ around a vortex, $\Delta_0$ is the maximum pair amplitude in the bulk, $\lambda = |\Lambda|$, and $\langle \cdots \rangle_{\rm SP}
\equiv \int_0^{r_a} r dr \int_0^{2 \pi} \cdots d \alpha/ (\pi r_a^2)$ is the real-space average around a vortex, where $r_a/\xi_0 = \sqrt{H_{c2}/H}$ \[$H_{c2} \equiv \Phi_0 / (\pi \xi_0)$, $\Phi_0 = \pi r_a^2 H$\]. $x=r\cos(\alpha -\theta_v)$, $y=r\sin(\alpha -\theta_v)$, and $E_y = \Delta_0 \lambda^2 y/(\xi_0 h)$. $\theta_v(\kv_{\rm F},\alpha_{\rm M}, \theta_{\rm M})$ is the angle of $\Vec{v}_{\rm F \perp}$ in the plane of $z=0$, where $\alpha$ and $\theta_v$ are measured from a common axis [@NagaiJ; @NagaiY]. $\Vec{v}_{\rm F \perp}$ is the vector component of $\Vec{v}_{\rm F} (\kv_{\rm F})$ projected onto the plane normal to ${\hat {\Vec{H}}}=(\alpha_{\rm M}, \theta_{\rm M})$. $|\Vec{v}_{\rm F \perp}(\kv_{\rm F},\alpha_{\rm M}, \theta_{\rm M})|
\equiv v_{\rm F0}(\alpha_{\rm M}, \theta_{\rm M}) h(\kv_{\rm F},\alpha_{\rm M}, \theta_{\rm M})$ and $v_{\rm F0}$ is the FS average of $|\Vec{v}_{\rm F \perp}|$ [@NagaiY]. $\xi_0$ is defined as $\xi_0 = v_{{\rm F}0}/(\pi \Delta_0)$. We consider here a clean SC in the type-II limit. The impurity effect can be incorporated through the smearing factor $\eta$.
We calculate the angular dependence of the ZEDOS ($E=0$ in Eq. (\[eq:dos\])) for YNi$_{2}$B$_{2}$C with using the band structure calculated by Yamauchi [*et al*]{}. [@Yamauchi]. When integrating Eq. (\[eq:dos\]), the band structure is reflected in $d S_{\rm F}$ and $\Vec{v}_{\rm F}$ ($\Vec{v}_{\rm F \perp}$, $v_{\rm F0}$, and $h$ are obtained from $\Vec{v}_{\rm F}$). In YNi$_{2}$B$_{2}$C there are three bands crossing the Fermi level, which are called the 17th, 18th, and 19th bands. The electrons of the 17th band predominantly contribute to the superconductivity, since the DOS at the Fermi level for the 17th, 18th, and 19th bands are 48.64, 7.88, and 0.38 states/Ry, respectively [@Yamauchi]. We consider here the FSs of the 17th and 18th bands and neglect the 19th band because the DOS is substantially small on the 19th FS. We use an anisotropic $s$-wave gap structure obtained in our preceding paper (Eq. (27) in Ref. [@NagaiY]) for the 17th FS. This gap structure was determined by analyzing STM observations [@Nishimori]. As for the 18th FS, we assume an isotropic gap with an amplitude equal to the maximum gap on the 17th FS, according to angular-resolved photo emission measurements [@Baba].
![\[fig:1\]Angular dependence of the ZEDOS for YNi$_{2}$B$_{2}$C (a):with use of the 17th band, (b): that of 18th band and (c) that of the 17th and 18th bands. The magnetic field tilts from the $c$ axis by polar angle $\theta_{M}=\pi/2$. $\eta = 0.05 \Delta_{0}$ and $r_{a} = 7 \xi_{0}$.](17thfi.eps){width="14pc"}
![\[fig:1\]Angular dependence of the ZEDOS for YNi$_{2}$B$_{2}$C (a):with use of the 17th band, (b): that of 18th band and (c) that of the 17th and 18th bands. The magnetic field tilts from the $c$ axis by polar angle $\theta_{M}=\pi/2$. $\eta = 0.05 \Delta_{0}$ and $r_{a} = 7 \xi_{0}$.](18thfi.eps){width="14pc"}
![\[fig:1\]Angular dependence of the ZEDOS for YNi$_{2}$B$_{2}$C (a):with use of the 17th band, (b): that of 18th band and (c) that of the 17th and 18th bands. The magnetic field tilts from the $c$ axis by polar angle $\theta_{M}=\pi/2$. $\eta = 0.05 \Delta_{0}$ and $r_{a} = 7 \xi_{0}$.](1718thfis.eps){width="14pc"}
We rotate the applied magnetic field in the basal plane perpendicular to the $c$ axis ($\theta_{\rm M}=\pi/2$) and investigate the azimuthal angle $\alpha_{\rm M}$ dependence of the ZEDOS. First, we show the partial ZEDOS on the 17th FS only. As seen in Fig. \[fig:1\](a), the oscillation amplitude is of the order 9 %. Previously we have calculated the ZEDOS by the DS method under the same condition and obtained the amplitude of the order 30 % [@NagaiY], which was too large in comparison to experimental results [@Park; @Izawa]. The KPA adopted in this paper yields substantial improvement in the amplitude.
Second, we show the partial ZEDOS on the 18th FS only. As seen in Fig. \[fig:1\](b), the cusp-like minima appear. The origin of the cusp-like structure is different from that of the 17th-FS case, since the gap is isotropic on the 18th FS while it is anisotropic on the 17th FS. The cusp-like minima are due to the FS anisotropy on the 18th FS. According to the band calculation (Fig. 2(b) in Ref. [@Yamauchi]), the shape of the 18th FS is like a square as viewed from the $k_{c}$ axis in the momentum space. The sides of this square are parallel to the $k_{a}$ or $k_{b}$ axis. Therefore, the region where the Fermi velocity is parallel to the $k_{a}$ or $k_{b}$ axis has a high proportion of the area on the 18th FS. Now, the quasiparticles with the Fermi velocity parallel to the magnetic field do not contribute to the ZEDOS (e.g., see Ref. [@NagaiLett]). As a result, when the magnetic field is rotated near the directions parallel to those axes, the change in the ZEDOS is drastic, leading to cusp-like minima. Udagawa [*et al*]{}. [@Udagawa] previously obtained cusp-like minima in the field-angle-dependent ZEDOS by using a square-like FS model. The origin of their cusp-like minima is the same as that of the present result for the 18th FS. It should be noted that the DOS ratio $r$ of the 18th FS to the 17th FS is $r = N_{\rm F 18}/N_{\rm F 17} = 7.88/48.64 \sim 0.16$ [@Yamauchi]. The contribution of the 18th FS is not dominant, and therefore the cusp-like minima observed in the experiments [@Park; @Izawa] probably cannot be attributed to the square shape of the FS only.
Finally, we show the total ZEDOS on both the 17th and 18th FSs. As seen in Fig. \[fig:1\](c), the oscillation amplitude is of the order 8 % and the overall behavior is almost similar to that for the 17th FS shown in Fig. \[fig:1\](a). In the thermal conductivity measurements in YNi$_{2}$B$_{2}$C, the oscillation amplitude in the field 1 T and 0.5 T is of the order 2 % [@Izawa] and 4.5 % [@Kamada], respectively, at the same temperature 0.43 K. In the heat capacity measurements [@Park], the oscillation amplitude in 1 T at 2 K is of the order 4.7 %. The KPA is an approximation appropriate in the limit of the low temperature and low magnetic field. Therefore, in terms of the order of the oscillation amplitude, our result could be considered to be consistent with that of those experiments. As for the directions of the minima, the result is also consistent with the experiments. Those mean that the assumed gap structure determined by analyzing STM experiments (Eq. (27) in Ref. [@NagaiY]) is consistent with the heat capacity [@Park] and the thermal transport [@Izawa] observations. So far we have found here that inclusion of the 18th-FS contribution does not drastically alter our previous conclusion [@NagaiY], and the use of the KPA improves the value of the amplitude compared to our result [@NagaiY] by the DS method. We should note, however, that the sharpness of the cusp is rather weak in Fig. \[fig:1\](c) in comparison to the experimental observations [@Park; @Izawa]. This point may be resolved by considering carefully the $k_{c}$ dependence of the gap structure. The gap structure used here was determined to explain the azimuthal dependence of the observations in the basal plane [@NagaiY]. A modification of the gap structure would be necessary to explain the observed polar-angle dependence [@Izawa]. This is left for future studies. Here, it should be noted that such a detailed discussion is almost impossible till the KPA [@NagaiLett] is developed as a new method with high precision.
In conclusion, we calculated the field-angle-dependent ZEDOS on the FSs from 17th and 18th band of YNi$_2$B$_2$C. We adopted the gap structure [@NagaiY] determined by comparison with STM observations [@Nishimori]. The oscillation amplitude was found to be of the order 8 %. This amplitude does not deviate so much from the heat capacity [@Park] and the thermal transport [@Izawa; @Kamada] measurements. As for the directions of the minima, the result is also consistent with those experiments. This suggests that the assumed gap structure on the 17th FS is essentially appropriate for the in-plane properties. Further modification of the gap structure is expected in terms of the $k_c$ dependence. The use of the KPA improved the value of the amplitude compared to our previous result [@NagaiY] by the DS method. The KPA [@NagaiLett] appears to be an efficient method with high precision for analyzing experimental data.
This research was supported by Grant-in-Aid for JSPS Fellows (204840), and was also partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research on Priority Areas, 20029007.
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==========
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| ArXiv |
---
abstract: 'The general solution of Einstein’s gravity equation in $D$ dimensions for an anisotropic and spherically symmetric matter distribution is calculated in a bulk with position dependent cosmological constant. Results for $n$ concentric $(D-2)-$branes with arbitrary mass, radius, and pressure with different cosmological constant between branes are found. It is shown how the different cosmological constants contribute to the effective mass of each brane. It is also shown that the equation of state for each brane influences the dynamics of branes, which can be divided into eras according to the dominant matter. This scenario can be used to model the universe in the $D=5$ case, which may presents a phenomenology richer than the current models. The evolution law of the branes is studied, and the anisotropic pressure that removes divergences is found. The Randall-Sundrum metric in an outside the region in the flat branes limit is also derived.'
author:
- 'I. C. Jardim'
- 'R. R. Landim'
- 'G. Alencar'
- 'R. N. Costa Filho'
title: 'Construction of multiple spherical branes cosmological scenario\'
---
Introduction
============
The general model for the cosmos is based on the description of the universe as a perfect fluid that admits a global cosmic time. This scenario has a space-time with constant curvature given by the Friedmann-Robertson-Walker metric, where its dynamics is determined by a cosmological scale factor that depends on the fluid state equation. Despite the success of that model in the description of the primordial nucleosintesys and the cosmic microwave background, it has failures that led to the emergence of new models. Among the major flaws of the current model are the problem of dark energy showing the accelerated expansion in the currently observed universe, and the dark matter which is the divergence between the rotation of the halo of some galaxies and the amount of matter contained in them according to gravitational dynamics [@weinberg:cosmology].
Cosmological models with extra dimensions appeared first in Kaluza-Klein models with extra dimensions and later in Randall-Sundrum scenarios [@Randall:1999vf; @Randall:1999ee]. These models describe the observed universe as a brane universe in a hyper-dimensional space-time. Despite the fact that the Friedmann-Robertson-Walker metric did not determine the geometry of the observed universe. The majority of studies focused on plane geometry. Because of its simplicity, this geometry is not able to change the dynamics of the universe and thus cannot solve the problem of dark matter or the initial singularity.
Although the first studies to describe the universe as a spherical shell back to the 80’s [@Rubakov:1983bb; @Visser:1985qm; @Squires:1985aq], the spherical brane-universe has shown very rich phenomenology in the past decade [@Gogberashvili:1998iu; @Boyarsky:2004bu]. Besides being compatible with the observational data [@Tonry:2003zg; @Luminet:2003dx; @Overduin:1998pn], the models provide an explanation for; the galaxy isotropic runaway (isotropic expansion), the existence of a preferred frame, and a cosmic time. They show how the introduction of different cosmological constants in each region of the bulk can change the dynamics of the cosmological scale factor so as to make it compatible with the observed dynamics [@Knop:2003iy; @Riess:2004nr] without the introduction of dark energy [@Gogberashvili:2005wy]. Similar to other models with extra dimensions, the spherical shell models open the possible to obtain an energy scale in order to solve the problem of hierarchy [@Gogberashvili:1998vx] and can be used as a basis for systems with varying speed of light in the observed universe [@Gogberashvili:2006dz].
The introduction of other branes and different cosmological constants can modify the overall dynamics of the observed universe. Local density fluctuations of density can change the local dynamics such as galactic dynamics (since the field of other branes interacts gravitationally with the matter of the brane-universe) without dark matter. Herein this piece of work we extend and generalize the scenario of the world as one expanding shell [@Gogberashvili:1998iu] to multiple concentric spherical $(D-2)$-branes in a $D$ dimensional space-time. For this, we solve the Einstein’s equation in $D$ dimensions to $n$ $(D-2)-$branes with different masses in a space with different cosmological constants between the branes. A previous study considered a continuous distributions of matter. However, only one cosmological constant was used [@Das:2001md]. We solve the $D-$dimensional case, but for a cosmological model we limited ourselves to the case $D=5$, since the observed universe has only three spatial dimensions.
This work is organized as follows: In the second section we review the Einstein’s equations in $D$ dimensions with a cosmological constant for spherically symmetric matter distribution. In the third section we solve this set of equations for $n$ shells with different cosmological constants $\Lambda$ between them. In Sec. 4, the energy-momentum tensor conservation law is used to determine the possible anisotropic pressure which removes the divergences in brane evolution equation. In the fifth section we particularize the solution found to take the flat brane limit in order to obtain the Randall-Sundrum metric in the exterior region. In the last section we discuss the conclusions and possible consequences.
Static and Spherically Symmetric Space-time in $D$ Dimensions
=============================================================
To learn about the gravitational effect of a distribution of matter we must determine the geometry of space-time. For this we need to know the $D(D+1)/2$ independent components of the metric solving the Einstein’s equation. However, it is possible to use the symmetry of the problem to reduce these components to just two, given by the invariant line element[@Gogberashvili:1998iu], $$ds^{2} = -A(r,t)dt^{2} +B(r,t)dr^{2} +r^{2}d\Omega^{2}_{D-2}$$ where $\Omega_{D-2}$ is the element of solid angle in $D$ dimensions, formed by $D-2$ angular variables.
Therefore we are left only with two functions, $A(r,t)$ and $B(r,t)$, to be determined by the Einstein’s equation in $D$ dimensions $$\label{einD}
R_{\mu}^{\nu} -\frac{1}{2}R\delta_{\mu}^{\nu} +\Lambda\delta_{\mu}^{\nu} = \kappa_{D}T_{\mu}^{\nu},$$ where $\Lambda$ is the cosmological constant, which depends on $r$ and possibly on $t$. Also $\kappa_{D}$ is the gravitational coupling constant in $D$ dimensions. Due to the symmetries of the problem we only have four non null independent components of the Einstein’s equation (\[einD\]), which are
$$\begin{aligned}
\kappa_{D}T_{0}^{0} &=&-\frac{D-2}{2r^{2}}\left[(D-3)\left(1 -B^{-1}\right) +\frac{rB'}{B^{2}}\right] +\Lambda \label{ein00}, \\
\kappa_{D}T_{1}^{1} &=& -\frac{D-2}{2r^{2}}\left[(D-3)\left(1 -B^{-1}\right) -\frac{rA'}{AB}\right] +\Lambda \label{ein11},\\
\kappa_{D}T^{1}_{0} &=& \frac{D-2}{2r}\frac{\dot{B}}{B^{2}}, \label{ein10}\\
\kappa_{D}T_{2}^{2} &=& \frac{1}{4A}\left[\frac{\dot{A}\dot{B}}{AB} +\frac{\dot{B}^{2}}{B^{2}} -\frac{2\ddot{B}}{B}\right] +\frac{(D-3)(D-4)}{2Br^{2}} - \nonumber
\\&&-\frac{2(D-3)(D-4)}{r^{2}} +\frac{(D-3)}{2Br}\left(\frac{A'}{A} -\frac{B'}{B}\right) + \nonumber
\\&& +\frac{1}{4B}\left[\frac{2A''}{A} -\frac{A'^{2}}{A^{2}} -\frac{A'B'}{AB}\right] +\Lambda \label{ein22},\end{aligned}$$
where the prime means derivation with respect to $r$ and the dot is the derivative with respect to $t$.
We can see that if we know $T^{0}_{0}$, $T^{1}_{1}$ and $\Lambda$ we can, from (\[ein00\]) and (\[ein11\]), completely determine the solutions with two boundary conditions. This comes from the fact that we have two first order differential equations. In this case the remaining equations determine the flow of energy $T^{1}_{0} $, and the tangential stresses $ T^{2}_{2}$. To find the exact solution we need to specify the form of matter $T^{\mu}_{\nu}$ which we use.
General Solution for Thin Spherical Branes
===========================================
The cosmological scenario we shall consider consists of $n$ concentric spherical delta type $(D-2)$-branes in a $D$ dimensional space with different cosmological constant between them. As said in the introduction this is a generalization of [@Gogberashvili:1998iu]. For this we fix the energy-momentum tensor and the cosmological constant to the form $$\label{fixT}
T_{0}^{0}(r,t) = -\sum_{i=1}^{n}\rho_{i}\delta(r-R_{i}), \quad
T_{1}^{1} = \sum_{i=1}^{n}P_{i}\delta(r-R_{i}),$$ and
$$\Lambda(r,t) = \frac{(D-1)(D-2)}{2}\sum_{i=0}^{n}\lambda_{i}\left[\theta(r-R_{i}) -\theta(r -R_{i+1})\right],$$
where the dependence on $t$ should be solely due to the branes radii ($ R_{i} = R_{i}(t) $). The $\theta$ function is defined in such way that is 1 when the argument vanish, it’s made to ensure that above expresion cover all space, including $r=0$ point.
The cosmological constant can be understood as a special fluid, so we can think that the difference between the cosmological constant is because each brane contains a fluid with different density. Fixing $T^{0}_{0}$ and $\Lambda$ we can find $B(r,t)$ using equation (\[ein00\]) in the form $$\kappa_{D}T^{0}_{0} = -\frac{D-2}{2r^{D-2}}\left[r^{D-3}\left(1 -B^{-1}\right)\right]' +\Lambda \label{ein002}$$ and according to the above equation, $B$ has a first order discontinuity in $R_{i}$ because $T^{0}_{0}$ has a second order one and $\Lambda$ has first order discontinuity only. Where we consider that in the region $R_{i} \leq r <R_{i +1}$, $B(r,t) = B_{i}(r)$, since in this region $B$ does not depend on $t$. The above-mentioned time dependence occurs in the region where this solution is valid.
The region between the branes has no matter. Therefore, the equation (\[ein10\]) assures us that the solution is static in this region. This information is contained in the Birkhoff’s theorem. Integrating (\[ein002\]) from $ R_{j} - \epsilon $ to $R_{j} + \epsilon$ and taking the limit $ \epsilon \to 0$, we obtain the discontinuity in the point $ r = R_ {j} $ $$\label{descont}
B_{j}^{-1}(R_{j}) -B_{j-1}^{-1}(R_{j}) = -\frac{2\kappa_{D}}{D-2}\rho_{j}R_{j}$$ The limit $\epsilon \to 0$ eliminate the $\Lambda$ term because its divergence is first order only. Integrating (\[ein002\]) from $R_{j} +\epsilon$ to $r< R_{j+1}$, where $B$ is continuous, and taking the limit $\epsilon \to 0$ we obtain $$\begin{aligned}
B_{j}^{-1}(r) &=& 1- \left(\frac{R_{j}}{r}\right)^{D-3}\left[1 -\lambda_{j}R_{j}^{2} -B_{j}^{-1}(R_{j})\right] -\lambda_{j}r^{2}
\\&=& 1- \left(\frac{R_{j}}{r}\right)^{D-3}\left[1 -\lambda_{j}R_{j}^{2} +\frac{2\kappa_{D}}{D-2}\rho_{j}R_{j} \right] + \\&& +\left(\frac{R_{j}}{r}\right)^{D-3}B_{j-1}^{-1}(R_{j})-\lambda_{j}r^{2}.\end{aligned}$$ By recurrence we find that $$\begin{aligned}
B_{j}^{-1}(r) &=& 1- \frac{1}{r^{D-3}}\sum_{i=1}^{j}\left[\frac{2\kappa_{D}}{D-2}\rho_{i}R_{i}^{D-2} -\Delta\lambda_{i}R^{D-1}_{i}\right] +
\\&&+ \left(\frac{R_{1}}{r}\right)^{D-3}\left(1-\lambda_{0}R_{1}^{2} -B_{0}^{-1}(R_{1})\right) -\lambda_{j}r^{2} ,\end{aligned}$$ where $\Delta\lambda_{i} =\lambda_{i} -\lambda_{i-1}$. Considering that inside all branes the solution is a de Sitter vacuum, i.e., $
B_{0}(r) =\left( 1-\lambda_{0}r^{2}\right)^{-1},
$ we get $$\begin{aligned}
B_{j}^{-1}(r) &=& 1- \frac{1}{r^{D-3}}\sum_{i=1}^{j}\left[\frac{2\kappa_{D}}{D-2}\rho_{i}R_{i}^{D-2} -\Delta\lambda_{i}R^{D-1}_{i}\right] -\nonumber
\\&& -\lambda_{j}r^{2}.\label{Bj}\end{aligned}$$ The above solution is valid only in the region $R_{j}\leq r<R_{j+1}$, but we can write the solution valid in any region in the form $$B^{-1}(r,t) = 1- \frac{2G_{D}M(r,t)}{r^{D-3}} -r^{2}\lambda(r,t) \label{B}$$ where $M(r,t)$ and $\lambda(r,t)$ are defined by
$$\begin{aligned}
M(r,t) &\equiv & \sum_{i=0}^{n}\left[\frac{\kappa_{D}}{(D-2)G_{D}}\rho_{i}R_{i}^{D-2} -\frac{\Delta\lambda_{i}}{2G_{D}}R^{D-1}_{i}\right]\theta(r-R_{i}),
\\ \lambda(r,t) &\equiv& \sum_{i=0}^{n}\lambda_{i}[\theta(r -R_{i}) - \theta(r -R_{i+1})],\end{aligned}$$
and the time dependence is implicit in $R_{i}$. It is important to note that $M(r,t)$ is not positive defined, in order to enable a repulsive gravitational situation. Using the above definition in (\[ein11\]) we find the equation which governs $A$ $$\frac{A'}{A} = \frac{2\kappa_{D}}{D-2}BrT^{1}_{1} +2B\left[(D-3)G_{D}\frac{M(r,t)}{r^{D-2}} -r\lambda(r,t) \right]. \nonumber$$ Taking the way $T^{1}_{1}$ was fixed at (\[fixT\]), $A$ has a second order discontinuity. Now, using the same procedure to find $B$ we can show that $$A_{j}(r) = B_{j}^{-1}(r)A_{0}(R_{1})B_{0}(R_{1})\prod_{i=1}^{j}\frac{B_{i}(R_{i})}{B_{i-1}(R_{i})}e^{\pi_{i}}.\nonumber$$ where $$\pi_{i} \equiv \frac{2\kappa_{D}}{D-2}R_{i}B_{i}(R_{i})P_{i}.$$ The asymptotic behavior of $B(r,t)$ is $\lim_{r\to\infty} B(r) = \left[ 1-\lambda(r)r^{2}\right]^{-1}$, which is the generalization of the de Sitter vacuum to a cosmological constant that is position dependent. Likewise we expect that $ A(r) $ behaves asymptotically as the vacuum, i.e., $\lim_{r\to\infty} A(r) = 1-\lambda(r)r^{2}$, so that we can use this to fix the multiplicative constants appearing in the temporal solution and write $$A_{j}(r) = B_{j}^{-1}(r)\prod_{i=j+1}^{n}\frac{B_{i-1}(R_{i})}{B_{i}(R_{i})}e^{-\pi_{i}}.\nonumber$$ In the same way we did for $B$, we can rewrite the above solution in order to be valid in all space as $$\begin{aligned}
A(r,t) &=& B^{-1}(r,t)\prod_{i=1}^{n}{e}^{-\pi_{i}\theta(R_{i} -r)}\times \nonumber
\\&& \times\left[1+\left(\frac{B_{i-1}(R_{i})}{B_{i}(R_{i})}-1\right)\theta(R_{i}-r)\right],\label{A}\end{aligned}$$ where $B_{j}$ is defined by (\[Bj\]).
The solutions (\[A\]) and (\[B\]) are generalizations of the Kottler solution [@Kottler-Ann.Phys.361] in $D$ dimensions with position dependent $\Lambda $. In the case where $\lambda$ is constant, these solutions agree with those found by Das [@Das:2001md]. However, these solutions only make sense if the branes are not in a time-like region. In order to avoid a singularity in the solutions, we impose that the radius of the brane relates to the masses so that they are beyond their respective generalized Kottler’s radii, i.e. $$\begin{aligned}
-\lambda(R_{i}) R_{i}^{D-1} +R_{i}^{D-3} -2GM(R_{i}) >0. \end{aligned}$$ The above solutions perfectly agree with the Birkhoff’s theorem, and despite a constant, it is the Schwarzschild solutions with a cosmological constant (Kottler Solution). The temporal dependence of the solutions is in $R_{i}$, so that in each region the solution is static. The multiplicative constants in the temporal part of the solutions indicates the gravitational redshift, even within the shells. Mathematically this makes the solution continuous in all regions.
The time dependence on $B$ is given exclusively by $R_{i}$. Therefore in a dynamical case ($R_{i} = R_{i}(t)$) we can obtain the energy flow from (\[ein10\]). It’s easy to show that $$T^{1}_{0} = -\sum_{i=1}^{n}\rho_{i}V_{i}\delta(r -R_{i}),\nonumber$$ where $V_{i} \equiv \dot{R}_{i}$. The tangential stresses can be obtained from (\[ein22\]), but it is easier to compute from the energy-momentum tensor conservation law.
Energy-Momentum Tensor Conservation Law
=======================================
The Einstein’s field equation relates the energy-momentum tensor and the metric tensor. But due the symmetry only two components of energy-momentum tensor are necessary to determine the metric. So the other terms of energy-momentum tensor are determined by Einstein’s equation or by a conservation law. Taking the covariant derivative in the equation (\[einD\]) we obtain the $D$ dimensional conservation law. $$\begin{aligned}
T_{\mu;\nu}^{\nu} = \frac{\Lambda_{,\mu}}{\kappa_{D}},\end{aligned}$$ the above equation states that the energy and momentum are not conserved inside the brane because the extra dimensional pressure given by the difference between cosmological constants. This difference can be used to model the dark energy, which makes the universe expand. But in our case no strange matter in needed inside the brane like the usual dark matter models. In terms of independent components the above conservation law can be written as $$\begin{aligned}
\frac{\dot{\Lambda}}{\kappa_{D}} &=&\dot{T}^{0}_{0} +T^{1\prime}_{0} +\frac{\dot{B}}{2B}\left[T^{0}_{0} -T^{1}_{1}\right]+ \nonumber
\\&&+\frac{T^{1}_{0}}{2}\left[\frac{A'}{A} +\frac{B'}{B} +\frac{2(D-2)}{r}\right]
\\\frac{\Lambda'}{\kappa_{D}} &=&\dot{T}^{0}_{1} +T^{1\prime}_{1} +\left[\frac{A'}{2A} +\frac{(D-2)}{r}\right]T^{1}_{1}+ \nonumber
\\&& +\frac{T^{0}_{1}}{2}\left[\frac{\dot{A}}{A} +\frac{\dot{B}}{B}\right] -\left[\frac{A'}{2A}T^{0}_{0} +\frac{(D-2)}{r}T^{2}_{2}\right].\end{aligned}$$ The first equation is trivially satisfied if we use the known solution (\[A\]) and (\[B\]) and energy-momentum tensor components. The second equation gives us the propagation speed of the brane as a function of the tangential stress, the masses, and the cosmological constant in the same way as in (\[ein22\]). Taking $$T^{2}_{2} = \sum_{i=1}^{n}T_{i}\delta(r-R_{i})\nonumber$$ we can integrate the unsolved component of the conservation law from $R_{i}-\epsilon$ to $R_{i}+\epsilon$ to obtain $$\begin{aligned}
\frac{\Delta\Lambda_{i}}{\kappa_{D}} &=& \frac{B_{i}(R_{i})}{A_{i}(R_{i})}[\dot{\rho}_{i}V_{i} +\rho_{i}\dot{V}_{i}] +\frac{D-2}{R_{i}}\left[P_{i} -T_{i}\right]+
\\&&+\left[\left.\left(P_{i} +\rho_{i}\right)\frac{A'}{2A} +\rho_{i}V_{i}^{2}\left(\frac{B'}{A} -\frac{BA'}{A^{2}}\right)-\right.\right.
\\&&-\left.\left.\rho_{i}V_{i}\left(\frac{B\dot{A}}{2A^{2}} -\frac{3\dot{B}}{2A}\right)\right]\right|_{r=R_{i}}\end{aligned}$$ as the functions $A$ and $B$ have second order divergences in $r = R_{i}$ the last term in above expression have the same divergence. Analyzing separately the divergent terms
$$\begin{aligned}
\mbox{div} &=& \overbrace{\frac{\kappa_{D}}{D-2}B_{i}(R_{i})R_{i}\left(P_{i} +\rho_{i}\right)\left[P_{i}-\frac{B_{i}(R_{i})}{A_{i}(R_{i})}\rho_{i}V_{i}^{2} \right]\left.\delta(r-R_{i})\right|_{r=R_{i}}}^{\mbox{real divergence}} +
\\&&+B_{i}(R_{i})\left[P_{i} +\rho_{i} -4\rho_{i}V_{i}^{2}\frac{B_{i}(R_{i})}{A_{i}(R_{i})}\right]\left[(D-3)\frac{G_{D}M(R_{i})}{R_{i}^{D-2}} -R_{i}\lambda_{i}\right] -\rho_{i}V_{i}\frac{B_{i}(R_{i})}{2A_{i}(R_{i})}\left[K_{0}(R_{i}) -\sum_{j=i}^{n}\dot{\pi}_{j}\right]\end{aligned}$$
where $$K_{0} = \sum_{j=1}^{n}\left[\frac{\dot{B}_{j-1}(R_{j})}{B_{j-1}(R_{j})} -\frac{\dot{B}_{j}(R_{j})}{B_{j}(R_{j})} \right]\theta(R_{j}-r).\nonumber$$ To avoid a real divergence we need to fix $$P_{i} = -\rho_{i} \;\;\;\;\;\mbox{or}\;\;\;\;\; P_{i} =\frac{B_{i}(R_{i})}{A_{i}(R_{i})}\rho_{i}V_{i}^{2}.\nonumber$$ The first case indicates a cosmological constant state equation type, this equation is the only state that is independent of motion, i.e. the properties of a fluid with this state equation is independent of its movement. Therefore, it was already expected that the divergences found in the dynamic case can be removed. The second case relates the normal pressure with the brane velocity. That indicates an increase in the pressure if the velocity increases to keep the spherical shape of the brane. This relationship ensures that $P$ could vanishes in the static case, as in the Randall-Sundrum scenario. Assuming a linear state equation relating the tangential stresses and the energy density, $
T_{i} = \gamma_{i}\rho_{i},
$ and defining, for the $i$-th brane, the time $$dt_{i} = \sqrt{A_{i}(R_{i})/B_{i}(R_{i})}dt \nonumber$$ the brane evolution is given by $$\begin{aligned}
\rho_{i}\frac{dU_{i}}{dt_{i}}&=& \frac{\Delta\Lambda_{i}}{\kappa_{D}}\left(1- U_{i}^{2}\right) -\frac{D-2}{R_{i}}\left[P_{i} -\rho_{i}\left(\gamma_{i} +U_{i}^{2}\right)\right] - \nonumber
\\&&-B_{i}(R_{i})\left[P_{i} +\rho_{i} -2\rho_{i}U_{i}^{2}\right]\times \nonumber
\\&&\times\left[(D-3)\frac{G_{D}M(R_{i})}{R_{i}^{D-2}} -R_{i}\lambda_{i}\right],\label{evolution}\end{aligned}$$ where $
U_{i} \equiv \frac{d R_{i}}{dt_{i}} .
$ This indicates a different dynamic for each cosmological eras driven by a different state equation , i.e., by the related tangential pressures.
The Randall-Sundrum Flat Brane Limit
====================================
In the previous sections we found the general solution to $n$ spherical branes in a $D$-dimensional space-time with different cosmological constant between them. To find a scenario similar to Randall-Sundrum we need to fix $D=5$, $n=1$ and $\lambda_{0}=\lambda_{1}$. In this case the exterior solution is $$ds^{2} = -f(r)dt^{2} +f^{-1}(r)dr^{2} +r^{2}d\Omega_{3}^{2},\nonumber$$ where $$f(r)= \left(1 -\frac{2G_{5}M}{r^{2}} -\lambda r^{2}\right).$$ In order to obtain the Randall-Sundrum metric we define $$dz \equiv \left(1 -\frac{2G_{5}M}{r^{2}} -\lambda r^{2}\right)^{-1/2}dr\nonumber$$ or, fixing that $z$ vanishes when $r =R$, $$z = \frac{1}{2k}\ln\left[\frac{2k\left(k^{2}r^{4} +r^{2} -2G_{5}M\right)^{1/2} +2k^{2}r^{2} +1}{2k\left(k^{2}R^{4} +R^{2} -2G_{5}M\right)^{1/2} +2k^{2}R^{2} +1}\right],\nonumber$$ where $\lambda =-k^{2}$ to avoid the de Sitter horizon. To obtain the flat brane limit we will consider that $R$ and consequently $r$ tends to infinity. In this limit the dominant term, regarding that $M$ grows as $R^{3}$, is $$z = \frac{1}{2k}\ln\left[\frac{r^{2}}{R^{2}}\right]\nonumber$$ writing $r$ as function of $z$, we obtain the line element $$ds^{2} = -k^{2}{e}^{2kz}R^{2}dt^{2} +dz^{2} +{e}^{2kz}R^{2}d\Omega_{3}^{2}.\nonumber$$ Putting the constants into coordinates we obtain the Randall-Sundrum metric $$ds^{2} = {e}^{2kz}\eta_{\mu\nu}dx^{\mu}dx^{\nu} +dz^{2}.\nonumber$$ The exponential in the warp factor is positive because the bulk is anti de Sitter, instead the original RS scenario.
Conclusions and Perspectives
============================
In this work we built a scenario of multiple concentric membranes through the solution of Einstein’s equation in $D$ dimensions with different cosmological constant in each region. The results we found may serve as a basis for more specific scenarios, through the fixation of radii, masses on each brane, and bulk cosmological constant. In a dynamical case, the solutions we found can be used to model the universe for $D=5$. The model used here is more accurate than the previous ones, and as a consequence we have a multiplicative constant that appears in the temporal solution, which is the redshift measured by observers in the region inside the brane.
Through the momentum-energy tensor conservation law we obtained two possible anisotropic pressures that remove the singularities in the branes dynamics. These two possible pressures give us two possible fixations leading to more freedom in the construction of a cosmological model that can better fits the observed data. The tangential pressure and the difference between the cosmological constants are responsible for the evolution of each brane according to Eq. (\[evolution\]). This tangential pressure is found from a state equation that is determined by the dominant matter in each cosmological era. We show that the difference between the cosmological constants modifies the effective mass of the matter distribution, and can be fixed in a way that the observed universe expansion rate is independent of the dark energy. Finally, we were able to arrive at the Randall-Sundrum metric to a anti-de Sitter bulk from calculating the external solution in the limit of plane branes. This metric was first introduced in the literature as an ansatz[@Randall:1999ee]. Now, it is derived from the Kottler anti-de Sitter solution. The main extension of the model developed here is the phenomenological study of cosmology generated by solving a dynamic Universe Brane equation.
We would like to thank the financial support provided by Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP), the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and FUNCAP/CNPq/PRONEX.
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| ArXiv |
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abstract: 'A coupling-constant definition is given based on the compositeness property of some particle states with respect to the elementary states of other particles. It is applied in the context of the vector-spin-1/2-particle interaction vertices of a field theory, and the standard model. The definition reproduces Weinberg’s angle in a grand-unified theory. One obtains coupling values close to the experimental ones for appropriate configurations of the standard-model vector particles, at the unification scale within grand-unified models, and at the electroweak breaking scale.'
author:
- 'J. Besprosvany'
date: 'Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México 01000, D. F., México '
title: 'Standard-model coupling constants from compositeness'
---
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The coupling constants are the dimensionless numbers that measure the strength of nature’s interactions. Their values are fixed by experiment in the standard model (SM) of elementary particles, and depend on the energy scale. Clues to the origin of their values are suggested from the relations among the quantum numbers of the SM particles.
In general, the realization of unity among physical variables, originally thought as disconnected, has led to a new understanding and connections among additional ones. For example, by linking electric and magnetic phenomena, Maxwell’s theory showed that light is a phenomenon of the kind, and predicted its velocity in terms of likewise parameters. Indeed, recently proposed SM extensions including a unifying principle are able to provide information on the coupling constant values. Thus, grand unification$\cite{unification}$ assumes that the gauge groups describing the interactions originate in a common group, and it predicts a single unified coupling, to which distinct couplings indeed appear to converge at high energy. It is also able to predict the coupling-constant ratios. In addition, compactification configurations of additional dimensions associated to interactions[@Weinbergcoup], and the dilaton-field ground state in string theory[@Green] predict their values, but, as yet, not uniquely. Information on the coupling constants may be also derived from extended-spin models[@Jaime]. Even if the underlying dynamics is not obvious, these connections may become manifest through symmetry arguments, which give additional information.
Composite models are another class of unifying theories that address the SM particle-multiplicity problem. Utilizing the connections among the quantum numbers of the 27 or so SM particles, these particles are constructed in terms of fewer elementary fields[@haplon]. The SM Poincaré symmetry and gauge-invariant interactions provide the link.
In general, these symmetries dictate the few quantum numbers that describe a particle state. These are the configuration or momentum coordinates, the spin, the gauge-group representation, and the flavor for quarks and leptons. Flavor characterizes only fermions. In the SM, fermions belong in the spin-1/2 Lorentz representation, and the gauge bosons are vectors. Similarly, fermions belong in the fundamental representation of the gauge group, while the vector bosons belong in the adjoint. This means that the gauge and spin quantum numbers of the latter can be constructed in terms of the former.
In the case of composite models, this facilitates their modelling in terms of simpler fields. However, it is difficult then to reproduce the SM dynamics without introducing additional fields and interactions, which, in turn, reduces the models’ predictability. Also, no additional substructure of the SM particles has been found. Another appealing idea is to assume that the vector bosons are composed of the SM fermions. A quantum electrodynamics model was proposed in which the photon is constructed from an electron and a positron[@Bjorken]. This model requires an unobservable space asymmetry, and its renormalizability rules are unclear.
In this paper, we use the experimentally derived compositeness property of the SM particles to get information on the SM coupling constants. We focus on those vector quantum numbers that can be constructed in terms of those of the fermions. This is a remarkable SM property; fermions could otherwise belong to other representations transforming according to the Lorentz and gauge groups, without satisfying this property. As with grand unification, which assumes a connection among the quantum numbers of the vector bosons, this paper assumes a connection among those of the spin-1/2 particles and vector bosons. The associated symmetry provides the coupling information. In particular, the application of quantum mechanical rules leads to normalization constants, and Clebsch-Gordan coefficients that relate both representations, and ultimately relate to the coupling constants. We will also find that the grand-unified coupling ratio prescription is reproduced.
In addition, we show that this assumption is consistent with the SM. Indeed, we apply an equivalent field-theory formulation that makes this kind of compositeness explicit, keeping the SM assumption that the fields are fundamental, unlike the composite-model case; all the SM predictions are therefore maintained. Thus, while composite models require additional fields in terms of which SM or new particles are constructed, this assumption is model independent. Hence, the putative problems associated with substructure compositeness are not encountered.
We first give a general coupling-constant definition based on the normalization and the compositeness property of some particle states with respect to other particle elementary states. Using the Wigner spinor classification of Lorentz representations, one may express SM fields in terms of their spinor components. It follows that the SM Lagrangian and its fields can be rewritten and reinterpreted in this way. Finally, we classify the configurations of the vector particles in relation to their SM and grand-unified theory content, calculate corresponding coupling values at the electroweak breaking and unification scales, and present final comments.
Quantum numbers characterize particles, and the normalized state $|w_i \rangle$ represents a particle with eigenvalue $w_i$ of the appropriate operator. The numbers $a_{ij}$ in the composite state $$\begin{aligned}
\label {composite}
| W \rangle =\frac{1}{\sqrt{N}}\sum_{i,j} a_{ij} |w_i \rangle|
w_j \rangle ,\end{aligned}$$ normalized with $$\begin{aligned}
\label {normalisation}
N=\sum_{i,j} a_{ij}^* a_{ij},\end{aligned}$$ fix $\langle w_i w_j| W \rangle$. The same amplitude is reproduced by the corresponding operator $\hat W=\frac{1}{\sqrt{N}}\sum_{i,j} a_{ij}
|w_i \rangle\langle w_j | $, satisfying $tr \hat W^\dagger \hat
W=1$, through $\langle w_i |\hat W |w_j\rangle$. Thus, both structures keep the same information, and the same normalization prescription may be applied.
$\hat W$ is also the most general operator acting on the $|w_i
\rangle$ states. Symmetry can determine the coefficients $a_{ij}^\lambda$, up to a constant, where $\lambda$ labels the representation components of such symmetry. For example, the only (non-axial) vector operator that can be constructed out of spin-1/2 particle states is the Dirac matrix $\gamma_0\gamma^\mu$[@Dirac]; $\partial^\mu$ stems from configuration space, and, when coupled to a vector field, it is not relevant in the SM vector-spin-1/2 interaction Lagrangian because it is neither renormalizable nor gauge invariant. For each $\mu$ (no sum) $tr \gamma_0\gamma^\mu
\gamma_0\gamma^\mu=4$ normalizes covariantly the operator, and fully determines it by providing the remaining constant; so is the case for the corresponding composite state $| W \rangle$. Hence, the matrix element between the spin states $|i \rangle$ and $|j
\rangle$ $$\begin{aligned}
\label{matrixelement}
\langle i |\hat W^\mu|j \rangle\end{aligned}$$ is determined with $\hat
W^\mu=\frac{1}{2}\gamma_0\gamma^\mu$. The four-entry $\hat W^\mu$ acts on the space spanned essentially by the spin-1/2 particle, its antiparticle, and their two spin polarizations.
This procedure can be generalized to the case of greater number of degrees of freedom, using the rules for the direct product of vector spaces and the generalized operator that acts on such a space. The normalization for $M$ such operators, $\hat W^T= \hat
W_1... \hat W_M$, is the product of the traces of each operator $\hat W_i$ in its space.
The vertex interaction Lagrangian $\int {\mathcal L}_{f}$ with density $ {\mathcal L}_{f}=-\frac{1}{2}gA^a_\mu{\Psi^\alpha}^\dagger\gamma_0
\gamma^\mu G^a\Psi^\alpha$ is determined from Poincaré and gauge invariance. In general, the latter determines the interactions of the vector bosons with the other particles, and among themselves, up to the coupling constant $g$. In particular, $ {\mathcal
L}_{f}$ is the only boson-spin-1/2 vertex. In the SM the fermions belong in the fundamental representation. The vertex can be consistently viewed as the expectation value of the tensor-product operator $\hat W^{\mu a}=g \gamma_0\gamma^\mu G^a1_x 1_\alpha, $ with vector components $A^a_\mu(x), $ acting upon the spin-1/2 particles $\Psi^\alpha(x)$; $\mu$ is the spin-1 index, $G^a$ the gauge-group representation matrix of the fermions, $a$ the group-representation index, $x$ the spacetime coordinate with the diagonal[^1] $1_x=|x\rangle \langle x |$, and $1_\alpha$ the unit matrix over the flavor $\alpha$. A composite state $A^a_\mu(x)|x\rangle |\mu
\rangle | a\rangle$, with $|\mu \rangle$, $| a\rangle$ elements as in Eq. \[composite\], underlies the operator association leading to $\hat W^{\mu a}$: $|\mu \rangle\rightarrow (\gamma_0\gamma^\mu)_{\sigma\eta},$ $|a
\rangle\rightarrow G^a_{bc}$, $|x \rangle\rightarrow|x\rangle
\langle x| $; the fermion state is $\Psi_{\eta c}^\alpha(x)$. All are written explicitly in terms of $\sigma$, $\eta$ spin-1/2 indices, $b,$ $c,$ gauge-group representation indices, and the flavor. $A_\mu^a\hat W^{\mu a}$ is also the expression for the vector field in spin space, treated, e. g., in Ref. [@Wald] (the same generalization is applied to the gauge degrees of freedom). In that reference, a spinor description of the Lorentz representations is given. At each spacetime point, tensor spinorial objects are defined. In particular, a real basis of (bi)spinorial objects is constructed that spans the Lorentz vector representation. The component elements of such a basis are essentially constructed out of the unit and the Pauli matrices. A map is defined between these bispinor objects and vectors. Their identification follows from the fact that they have the same transformation properties. In fact, Maxwell’s equations can be equivalently formulated in terms of such objects, as two Dirac equations[@Bargmann]. The other Lagrangian terms can also be reinterpreted and formulated in terms of spin-projected fields. Canonical quantization in quantum field theory normalizes $A^a_\mu$; the compositeness assumption further imposes such condition on the $\hat W_i$ operators, which fully normalizes $A^a_\mu\hat W^{\mu a} $. In general, $A^a_\mu $ can be understood as an element in a polarization or group basis $A^a_\mu
=tr
n_{\mu }^a A^b_\nu n^{\nu b}$, where in our case $n^{\nu b }=\hat
W^{\nu b}$, and it is assumed to be normalized. Indeed, we recognize in the vertex $$\begin{aligned}
\label {vertex}{\mathcal L}_{f}=-{\Psi_{\sigma
b}^\alpha(x)}^\dagger A^a_\mu(x)\Psi_{\eta c}^\alpha(x) \langle
\sigma |\gamma_0\gamma^\mu | \eta \rangle \frac{1}{2}g
\langle b | G^a | c \rangle\end{aligned}$$ the matrix elements in Eq. \[matrixelement\], and the gauge-group ones. Within the compositeness assumption, we equate each matrix element in Eq. \[vertex\] with that of the composite vector in Eq. \[matrixelement\], and similarly for the group-representation matrices, all of which contain operators acting upon the spin-1/2 particles, which leads to the identification $$\begin{aligned}
\label {identi}
g\rightarrow 2\sqrt{ \frac{1}{ N}}.\end{aligned}$$ The normalization $N$ is calculated as in Eq. \[normalisation\], with the convention for the $\gamma$-matrices $$\begin{aligned}
\label {groupnorgam}
tr \gamma_\mu\gamma_\nu=4 g_{\mu\nu} ,\end{aligned}$$ and irreducible representations $$\begin{aligned}
\label {groupnor}
trG_i G_j=2 \delta_{ij}.
\end{aligned}$$ Essentially, we are setting normalization constants for the matrix elements in Eq. \[vertex\], which connect representations, and can be viewed as Clebsch-Gordan coefficients. ${\mathcal L}_{f}$ contains sums over matrix elements for each $\mu$ and $a$, which determine the coupling constant; only two polarizations $\mu$ have a physical-state interpretation, while gauge and Lorentz invariance demand a unique value. Quantum field theory admits arbitrary coupling constants for a vertex, which are obtained experimentally. The theoretical assignment of $g$ complements this theory.
In comparing the fermion states with the vector ones, we find that the latter are composite only in the Lorentz and the gauge groups, whereas the configuration variable $x$ is elementary for both types of field. In general, an additional fermion index $\beta$ independent of $A^a_\mu$ corresponds to $\hat W_F=\sum |\beta \rangle\langle \beta| =1_F$, a unit operator present in the vertex, not contributing to the coupling constant. This is the flavor’s case. However, there are two consistent coupling definitions when such a kind of operator acts in a fermion subspace. Thus, e.g., $ SU(2)_L$ generators in a grand-unified theory such as $SU(5)$ are constructed with their lepton ($l$) and baryon ($b$) components as $G_{SU(2)_Ll}+(G_{SU(2)_Lb}\times 1_{SU(3)})$, with $1_{SU(3)}$ a projection operator in color space (leptons are color singlets); $1_{SU(3)}$ does not commute with some $SU(5)$ generators, and the associated vector-field components interact with the other unified-group ones. Physically, this [*full*]{} case corresponds to active degrees of freedom. In a lower energy regime, the symmetry is broken, and the interactions are truncated to the weak $SU(2)_L$ and the other SM interactions, while $1_{SU(3)}$ commutes with these generators. Then, in this [*reduced*]{} case, $1_{SU(3)}$ drops out of the calculation.
Grand-unified theory predicts coupling-constant ratios under the condition that the SM generators belong to the same unified-group representation, which determines Weinberg’s angle at the unification energy scale[@unification], and the running of the coupling of each interaction gives values at lower energies.
Similarly, the configuration of the fields’ group representations $G_i$ gives a clue to the energy scale. To obtain unified and SM couplings we specify the normalized vector-field polarizations and gauge-group generators. The couplings are calculated using the fermion quantum numbers, which are the generators’ eigenvalues, and make the generators themselves (the Cartan subset). A generation of SM left-handed \[quarks; leptons\] is classified by $[Q,u^c,d^c;$ $L,e^c]$, with $L=(e,\nu)$, $Q=(u,d)$ $SU(2)_L$ doublets, and $u^c$, $d^c$, $e^c$, charge-conjugate singlets, according to their color-weak-hypercharge $SU(3)\times SU(2)_L\times U(1)_Y$ groups; the latter can be viewed as subgroups of the $SU(5)$ grand-unified theory. The multiplets are $ [( 3,2,1/3), (\bar 3,1,-4/3),(\bar 3,1,2/3);$ $(1,2,-1),
(1,1,2)]$. The fermions fit neatly into the $\bf 5$ and $\bf 10$ representations of this group. The hypercharge $Y$ and the weak interaction have different $\sigma_{\mu\pm}=\frac{1}{2}(1\pm
\gamma_5)\gamma_0\gamma_\mu$ components, with the pseudoscalar $\gamma_5=-i\gamma_0\gamma_1\gamma_2\gamma_3$, which uses ${\mathcal L}_{f}$ with possibly different $\hat
W^a_{\mu\pm}=\sigma_{\mu\pm}G^a_\pm $ components, in an obvious notation. One gets for $Y$ in the [*full*]{} configuration, with the above quantum numbers, the rules in Eqs. \[normalisation\] and \[identi\], and conventions in Eqs. \[groupnorgam\] and \[groupnor\], $g^\prime=2/[{2
(2 + 2^2 + 6(\frac{1}{3})^2+ 3 (\frac{2}{3})^2 +3
(\frac{4}{3})^2)]^{1/2}}$ $=\frac{1}{2}\sqrt{\frac{3}{5}}$, where the 2 in the denominator normalizes each chiral component $\sigma_{\mu\pm}$, to which corresponds one massless fermion polarization. The first two terms in the parenthesis $2+2^2=1^2+1^2+2^2$ are the lepton hypercharges and, the last three are the quark hypercharges; their multiplicity is taken into account. $Y$ may be also viewed as a generator of the $SU(5)$ interaction.
Two coupling definitions apply for the weak $SU(2)_L$ interaction, one of whose generators has diagonal components $I_{(l,b)} =(1,-1)$. For the [*full*]{} configuration, $g^{uni}=$ $2/{[2(1+3)(1^2+1^2)]^{1/2}}$ $ =\frac{1}{2}$, where the second factor in the denominator counts the lepton and quark doublets, which in turn give the third factor. Non-supersymmetric unified models[@DGLee] give experimentally consistent unification couplings of $g_{ex}^{uni}\sim.52 -.56$, at $10^{14}-10^{16}$ GeV. From the SM[@Glashow]-[@Salam], $tan(\theta_W)=g^\prime/g^{uni}$, and one reproduces the $SU(5)$ unification result for Weinberg’s angle$\cite{quinn}$ $sin^2(\theta_W^{uni})=3/8$. In general, the coupling definition in Eq. \[identi\] is consistent with the grand-unified prescription for such a coupling ratio.
The [*reduced*]{} configuration of the normalized weak vector implies that the color components drop from the calculation. It gives the same weight to quarks as to leptons, as is necessary if one omits unification-group information. We should get information on the electroweak-breaking scale to the extent that these weak and hypercharge configurations describe on-shell Z and W vector bosons. We find $g^{le} =2 /[{2 (2)(1^2+1^2)
)]^{1/2}}=\frac{1}{\sqrt{2}}\approx
.707$, while at the $M_Z$ scale$\cite{tables}$, $g_{ex}=.649519(20)$, where one standard-deviation uncertainty for the last digits is given in parenthesis.
Each isospin doublet component corresponds to a different hypercharge isospin singlet; this suggests, extending the rule to color components, that only the [*full*]{} configuration need be considered for $Y.$ Thus, $g'\approx .387$ is between $g_{ex}^\prime(M_Z)=.35603(6)$ and the unified hypercharge values $\sqrt{\frac{3}{5}}g_{ex}^{uni}\sim .40-.43$; the relatively narrow range provides a test of the prediction. From $tan(\theta_W)=g^\prime/g^{le}$, we find $sin^2(\theta_W)=3/13\approx .23078,$ while at $M_Z$ $sin^2(\theta_{Wex})= .23113(15).$
One may also interpret the [*reduced*]{} weak configuration within the minimal supersymmetric model, with a unified[@Amaldi] $g^{Suni}_{ex}=.69(4)$ at $10^{15.8\pm .4}$ GeV. $g^{le}$ reproduces a value also within a narrow low and high energy range. For the gluons’ coupling in the $1_{SU(2)_L}$-[*reduced*]{} case we use the $\lambda_3$ Pauli-matrix fundamental component of the $SU(3)$ (any other generator would also do) with the convention of Eq. \[groupnor\] $g_s =2 /[{2
(2)(1^2+1^2) )]^{1/2}}=1/\sqrt{2}\approx
.707,$ or $\alpha_s=\frac{g_s^2}{4
\pi}\approx .040$, while $\alpha_{s(ex)}(M_Z)=.1172(20)$. Then $g_s$ provides a lower limit around the unification scale.
While only the fermion-vector vertex has been examined, the results are valid for a more general Lagrangian. The coupling constants in the other Lagrangian terms get a unique value, because gauge invariance demands it for each gauge group. All along, flavor is assumed to belong to the [*reduced*]{} configuration, for it does not influence interactions.
In a grand-unified theory and in the SM, the electroweak-field components at the unification scale, and at the symmetry-breaking scale, are determined, respectively, through the ratios of the electroweak couplings, namely, Weinberg’s angle. The SM fermions and bosons, and their simple interactions conform to a compositeness assumption. Under this assumption, the allowed vertices and the fields’ normalized polarization generate the coupling constants. Specifically, these are obtained by associating composite-field configurations both to the unification scale and the W and Z particle regime. Already at tree level, Weinberg’s angle is reproduced for the $SU(5)$ unified theory, and a value close to the experimental one is obtained at the $M_Z$ electroweak-breaking scale, which validates the ascribed configuration in each regime. This set of two coupling constant or Weinberg angle values provides a connection between the two energy scales through, e.g., the renormalization group equations, which have to be supplemented with boundary conditions. Although the low-energy ratio $tan (\theta_{W}) $ does not contain couplings at precisely the same energy, it contains information on the group-generator structure, stemming from the compositeness assumption; this is not in contradiction with the coupling running that should be applied, and whose corrections cancel among the two couplings. The couplings are also interpreted consistently within the minimal supersymmetric model. The calculation of $\theta_{W}$ can be viewed as complement to, or as alternative to, that of $\theta_{W}^{uni}$. In the first approach, the coupling constants relate energies in the unified and symmetry-breaking scales. In the second approach, one obtains information on the $M_Z$ scale, understood as fundamental[@Dim].
The paper’s approach may also be applied in other extensions, which require only the consideration of reducible representations. The compositeness hypothesis is supported with coupling constants obtained among a limited number of allowed configurations, and that reproduce experimental values, which are within a narrow range at different energy scales.
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[**Acknowledgement.**]{} The author acknowledges support from DGAPA-UNAM, project IN120602, and CONACYT, project 42026-F.
[^1]: $1_x$ only connects local fields, without compositeness. Formally, $a^x_{x^\prime x^{\prime
\prime}}=\delta_{x^\prime x^{\prime\prime}}\delta_{x x^\prime}.$ $A^a_\mu(x)$ normalizes in $x$ space for $tr1_x=1$.
| ArXiv |
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abstract: 'A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum of two squares. We investigate the growth of the number of such lattices with a fixed determinant as the determinant grows, exhibiting some determinant sequences on which it is particularly large. To this end, we also study the behavior of the associated zeta function, comparing it to the Dedekind zeta function of Gaussian integers and to the Solomon zeta function of ${\mathbb Z}^2$. Our results extend automatically to well-rounded sublattices of any lattice $A {\mathbb Z}^2$, where $A$ is an element of the real orthogonal group $O_2({\mathbb R})$.'
address: 'Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711-6420'
author:
- Lenny Fukshansky
bibliography:
- 'esm.bib'
title: 'On distribution of well-rounded sublattices of $\mathbb Z^2$'
---
Ł[[L]{}]{} [H]{}
Introduction and statement of results
=====================================
Let $N \geq 2$ be an integer, and let $\Lambda \subseteq \real^N$ be a lattice of full rank. Define the [*minimum*]{} of $\Lambda$ to be $$|\Lambda| = \min_{\bx \in \Lambda \setminus \{\bo\}} \|\bx\|,$$ where $\|\ \|$ stands for the usual Euclidean norm on $\real^N$. Let $$S(\Lambda) = \{ \bx \in \Lambda : \|\bx\| = |\Lambda| \}$$ be the set of [*minimal vectors*]{} of $\Lambda$. We say that $\Lambda$ is a [*well-rounded*]{} lattice (abbreviated WR) if $S(\Lambda)$ spans $\real^N$. WR lattices come up in a wide variety of different contexts, including sphere packing, covering, and kissing number problems, coding theory, and the linear Diophantine problem of Frobenius, just to name a few. Still, the WR condition is special enough so that one would expect WR lattices to be relatively sparce. However, in 2005 C. McMullen [@mcmullen] showed that in a certain sense [*unimodular*]{} WR lattices are “well distributed” among all [*unimodular*]{} lattices in $\real^N$, where a unimodular lattice is a lattice with determinant equal to 1. More specifically, he proved the following theorem, from which he derived the 6-dimensional case of the famous Minkowski’s conjecture for unimodular lattices.
\[[@mcmullen]\] \[mcmullen\] Let $A \subseteq SL_N(\real)$ be the subgroup of diagonal matrices with positive diagonal entries, and let $\Lambda$ be a full-rank unimodular lattice in $\real^N$. If the closure of the orbit $A \Lambda$ is compact in the space of all full-rank unimodular lattices in $\real^N$, then it contains a WR lattice.
Notice that in a certain sense this is a statement about distribution of WR lattices in the space of all unimodular lattices in a fixed dimension. Motivated by this beautiful theorem, we want to investigate the distribution of WR sublattices of $\zed^N$, which is a natural arithmetic problem. For instance, for a fixed positive integer $t$, does there necessarily exist a WR subllatice $\Lambda \subseteq \zed^N$ so that $\det(\Lambda) = t$? If so, how many different such sublattices are there? The first trivial observation is that if $t = d^N$ for some $d \in \zed_{>0}$ and $I_N$ is the $N \times N$ identity matrix, then the lattice $\Lambda = (d I_N) \zed^N$ is WR with $\det(\Lambda) = t$ and $|\Lambda| = d$. It seems however quite difficult to describe [*all*]{} WR sublattices of $\zed^N$ in an arbitrary dimension $N$. This paper is concerned with providing such a description in dimension two.
From now on we will write $\WR(\Omega)$ for the set of all full-rank WR sublattices of a lattice $\Omega$; in this paper we will concentrate on $\WR(\zed^2)$. In section 3 we develop a certain parametrization of lattices in $\WR(\zed^2)$, which we then use to investigate the determinant set $\D$ of such lattices and to count the number of them for a fixed value of determinant. Specifically, let $\D$ be the set of all possible determinant values of lattices in $\WR(\zed^2)$, and let $\Mm$ be the set of all possible values of squared minima of these lattices, i.e. $\Mm = \{ |\Lambda|^2 : \Lambda \in \WR(\zed^2) \}$. It is easy to see that $\Mm$ is precisely the set of all positive integers, which are representable as a sum of two squares. Then it is interesting to understand how dense are these sets in $\zed_{>0}$.
For any subset $\PP$ of $\zed$ and $M \in \zed_{>0}$, we write $$\PP(M) = \{ n \in \PP : n \leq M\}.$$ Define [*lower density*]{} of $\PP$ in $\zed$ to be $$\DL_{\PP} = \liminf_{M \rightarrow \infty} \frac{|\PP(M)|}{M},$$ and its [*upper density*]{} in $\zed$ to be $$\DU_{\PP} = \limsup_{M \rightarrow \infty} \frac{|\PP(M)|}{M}.$$ Clearly, $0 \leq \DL_{\PP} \leq \DU_{\PP} \leq 1$. If $0 < \DL_{\PP}$, we say that $\PP$ [*has density*]{}, and if $\DL_{\PP} = \DU_{\PP}$, i.e. if $\lim_{M \rightarrow \infty} \frac{|\PP(M)|}{M}$ exists, we say that $\PP$ [*has asymptotic density*]{} equal to the value of this limit, which could be 0.
With this notation, we will show that $\D$ has density. More specifically, we prove the following.
\[dense\] The determinant set $\D$ of lattices in $\WR(\zed^2)$ has representation $$\D = \left\{ (a^2+b^2)cd\ :\ a,b \in \zed_{\geq 0},\ \max\{a,b\} >0,\ c,d \in \zed_{>0},\ 1 \leq \frac{c}{d} \leq \sqrt{3} \right\},$$ and lower density $$\label{D_dens}
\DL_{\D} \geq \frac{3^{\frac{1}{4}}-1}{2 \cdot 3^{\frac{1}{4}}} \approx 0.12008216 \dots$$ The minima set $\Mm$ has asymptotic density 0.
We prove Theorem \[dense\] in section 4. Now, if $\Lambda \in \WR(\zed^2)$, let $\bx,\bwy$ be a minimal basis for $\Lambda$, and let $\theta$ be the angle between the vectors $\bx$ and $\bwy$; it is a well known fact that in dimensions $\leq 4$ a lattice is always generated by vectors corresponding to its successive minima, so such a basis certainly exists (see, for instance, [@pohst]). Then there is a simple connection between the minimum and the determinant of $\Lambda$: $$\det(\Lambda) = \|\bx\| \|\bwy\| \sin\theta = |\Lambda|^2 \sqrt{ 1 - \frac{\left( \bx^t \bwy \right)^2}{|\Lambda|^4}} = \sqrt{ |\Lambda|^4 - \left( \bx^t \bwy \right)^2 }.$$ Lemma \[gauss\] below implies that $0 \leq |\bx^t \bwy| \leq \frac{|\Lambda|^2}{2}$. Therefore we have $$\frac{\sqrt{3}\ |\Lambda|^2}{2} \leq \det(\Lambda) \leq |\Lambda|^2.$$ In view of this relation, it is especially interesting that the determinant set has positive density while the minima set has density 0.
Next, for each $u \in \D$ we want to count the number of $\Lambda \in \WR(\zed^2)$ such that $\det(\Lambda) = u$. We need some additional notation. Suppose $t \in \zed_{>0}$ has prime factorization of the form $$\label{prim_fact}
t = 2^w p_1^{2k_1} \dots p_s^{2k_s} q_1^{m_1} \dots q_r^{m_r},$$ where $p_i \equiv 3\ (\md 4)$, $q_j \equiv 1\ (\md 4)$, $w \in \zed_{\geq 0}$, $k_i \in \frac{1}{2} \zed_{>0}$, and $m_j \in \zed_{>0}$ for all $1 \leq i \leq s$, $1 \leq j \leq r$. Let $\alpha(t)$ be the number of representations of $t$ as a sum of two squares ignoring order and signs, that is $$\label{alpha}
\alpha(t) = \left| \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = t,\ a \leq b \right\} \right|.$$ Also define $$\label{alpha*}
\alpha_*(t) = \left| \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = t,\ a \leq b,\ \gcd(a,b)=1 \right\} \right|,$$ for all $t > 2$, and define $\alpha_*(1) = \alpha_*(2) = \frac{1}{2}$. It is a well-known fact that $\alpha(t)$ is given by $$\alpha(t) = \left\{ \begin{array}{ll}
0 & \mbox{if any $k_i$ is a half-integer} \\
\frac{1}{2} B & \mbox{if each $k_i$ is an integer and $B$ is even } \\
\frac{1}{2} \left( B - (-1)^w \right) & \mbox{if each $k_i$ is an integer and $B$ is odd,}
\end{array}
\right.$$ where $B=(m_1+1) \dots (m_r+1)$ (see, for instance [@silverman], [@weisstein]). Clearly, when $t$ is squarefree, $\alpha_*(t)=\alpha(t)$. It is also a well-known fact that for $t$ as in (\[prim\_fact\]) $$\alpha_*(t) = \left\{ \begin{array}{ll}
0 & \mbox{if $s \neq 0$ or $w > 1$} \\
2^{r-1} & \mbox{if $s=0$ and $w = 0$ or $1$.}
\end{array}
\right.$$ We also define the function $$\label{beta}
\beta_{\nu}(t) = \left| \left\{ d \in \zed_{>0} : d\ |\ t\ \text{and } \frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t} \right\} \right|,$$ for every $1 < \nu \leq 3^{1/4}$. The value of $\nu$ which will be particularly important to us is $\nu = 3^{1/4}$, therefore we define $$\beta(t) = \beta_{3^{1/4}}(t).$$ We discuss the function $\beta_{\nu}(t)$ in more detail in section 2; at least it is clear that for each given $t$, $\beta_{\nu}(t)$ is effectively computable for every $\nu$. Finally, for any $t \in \zed_{>0}$ define $$\delta_1(t) = \left\{ \begin{array}{ll}
1 & \mbox{if $t$ is a square} \\
2 & \mbox{if $t$ is not a square,}
\end{array}
\right.$$ and $$\delta_2(t) = \left\{ \begin{array}{ll}
0 & \mbox{if $t$ is odd} \\
1 & \mbox{if $t$ is even, $\frac{t}{2}$ is a square} \\
2 & \mbox{if $t$ is even, $\frac{t}{2}$ is not a square.}
\end{array}
\right.$$ With this notation, we can state our second main theorem.
\[count\] Let $u \in \zed_{>0}$, and let $\N(u)$ be the number of lattices in $\WR(\zed^2)$ with determinant equal to $u$. If $u=1$ or $2$, then $\N(u)=1$, the corresponding lattice being either $\zed^2$ or $\left( \begin{matrix} 1&-1 \\ 1&1 \end{matrix} \right) \zed^2$, respectively. Let $u>2$, and define $$t = t(u) = \left\{ \begin{array}{ll}
u & \mbox{if $u$ is odd} \\
\frac{u}{2} & \mbox{if $u$ is even.}
\end{array}
\right.$$ Then: $$\begin{aligned}
\label{N_formula}
\N(u) & = & \delta_1(t) \beta(t) + \delta_2(t) \beta \left(\frac{t}{2} \right) + 4 \mathop{\sum_{n|t, 1<n<t/2}}_{n\ \text{not a square}} \alpha_* \left(\frac{t}{n} \right) \beta(n) \nonumber \\
& + & 2 \mathop{\sum_{n|t, 1 \leq n<t/2}}_{n\ \text{a square}} \alpha_* \left(\frac{t}{n} \right) (2\beta(n)-1).\end{aligned}$$ In particular, if $u \notin \D$, then the right hand side of (\[N\_formula\]) is equal to zero.
Theorem \[count\] can also be easily extended to a more general class of lattices. Namely, write $O_2(\real)$ for the real orthogonal group, then for every $A \in O_2(\real)$ and every $\bx,\bwy \in \real^2$ we have $(A\bx)^t (A\bwy) = \bx^t \bwy$, i.e. $O_2(\real)$ is the isometry group of $\real^2$ with respect to the Euclidean norm. Therefore, if $A \in O_2(\real)$ then $\Lambda \in \WR(A\zed^2)$ if and only if $A^t \Lambda \in \WR(\zed^2)$. This immediately implies the following result.
\[gen\] Let $A \in O_2(\real)$. Then the determinant set and the minima set of lattices in $\WR(A\zed^2)$ are $\D$ and $\Mm$ respectively, as defined above. Moreover, for each $u \in \D$ the number of lattices in $\WR(A\zed^2)$ with determinant equal to $u$ is given by $\N(u)$ as in Theorem \[count\].
We prove Theorem \[count\] in section 5. In section 6 we use Theorem \[count\] to work out simple examples of our formula in the case of prime power and product of two primes determinants. We also describe the “orthogonal” elements of $\WR(\zed^2)$, which come from ideals in Gaussian integers; these are quite sparse among all lattices in $\WR(\zed^2)$. We then derive easy to use bounds on $\N(u)$ and on the normal order of $\N(u)$. We also demonstrate examples of “extremal” sequences of determinant values, for which $\N(u)$ is especially large; see Corollary \[size\_N\]. In section 7 we derive a formula for the number of lattices in $\WR(\zed^2)$ of fixed minimum.
In section 8 we study some basic properties of a zeta function, corresponding to the well-rounded sublattices of $\zed^2$. Namely, for $s \in \cee$ define $$\label{WR_zeta}
\zeta_{\WR(\zed^2)}(s) = \sum_{\Lambda \in \WR(\zed^2)} (\det(\Lambda))^{-s} = \sum_{u=1}^{\infty} \N(u) u^{-s},$$ where $\N(u)$ is as above. In particular, $\N(u) \neq 0$ if and only if $u \in \D$. For a Dirichlet series $\sum_{n=1}^{\infty} c_n n^{-s}$, we say that it has a [*pole of order*]{} $\mu$ at $s=s_0$, where $\mu$ and $s_0$ are positive real numbers, if $$\label{pole_def}
0 < \lim_{s \rightarrow s_0^+} |s-s_0|^{\mu} \sum_{n=1}^{\infty} |c_n n^{-s}| < \infty.$$ We will also say that such a Dirichlet series is [*bounded from above (or below)*]{} by a Dirichlet series $\sum_{n=1}^{\infty} b_n n^{-s}$, if $\sum_{n=1}^{\infty} |c_n n^{-s}| \leq \sum_{n=1}^{\infty} |b_n n^{-s}|$ (respectively, $\geq \sum_{n=1}^{\infty} |b_n n^{-s}|$). In section 8 we prove the following result.
\[zeta\] Let the notation be as above, then $\zeta_{\WR(\zed^2)}(s)$ is analytic for all $s \in \cee$ with $\Re(s) > 1$, and is bounded from below by a Dirichlet series that has a pole of order 2 at $s=1$. Moreover, for every real $\eps >0$ there exists a Dirichlet series with a pole of order $2+\eps$ at $s=1$, which bounds $\zeta_{\WR(\zed^2)}(s)$ from above.
Notice that Theorem \[zeta\] provides additional information about the growth of $\N(u)$. In section 8 we prove Theorem \[zeta\] by means of considering the behavior of some related Dirichlet series, namely the generating functions of $\alpha_*$ and $\beta_{\nu}$. We should remark that we are not using the notion of a pole here in a sense that would imply the existence of an analytic continuation, but only to reflect on the growth of the coefficients; in fact, the arithmetic function $\N(u)$ behaves sufficiently erraticaly that one would doubt $\zeta_{\WR(\zed^2)}(s)$ having an analytic continuation to the left of $s=1$. We are now ready to proceed.
A special divisor function
==========================
As above, let $1 < \nu \leq 3^{1/4}$. In this section we briefly discuss bounds on the divisor function $\beta_{\nu}(t)$. Let $$\M_{\nu}(t) = \left\{ d \in \zed_{>0} : d\ |\ t\ \text{and } \frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t} \right\}.$$
\[beta\_gcd\] If $d_1,d_2 \in \M_{\nu}(t)$, then $\gcd(d_1,d_2)>1$.
Suppose $d_1,d_2 \in \M_{\nu}(t)$, and $\gcd(d_1,d_2)=1$. Then $d_1d_2|t$, but $$\frac{t}{\nu^2} < d_1d_2 \leq t.$$ Notice that $d_1d_2 \neq t$, since this would imply $d_1=d_2=\sqrt{t}$. Then $$1 < \frac{t}{d_1d_2} < \nu^2 \leq \sqrt{3},$$ but $\frac{t}{d_1d_2} \in \zed$, which is a contradiction.
Lemma \[beta\_gcd\] implies in particular that $\M_{\nu}(t)$ can contain at most one prime $q$, and in this case every $d \in \M_{\nu}(t)$ must be divisible by $q$. Write $p(t)$ for the smallest prime divisor of $t$. Another immediate consequence of Lemma \[beta\_gcd\] is that gaps between two consecuitive elements of $\M_{\nu}(t)$ must be greater or equal than $p(t)$. Therefore, since $\beta_{\nu}(t)=|\M_{\nu}(t)|$, we obtain $$\label{beta_bound1}
\beta_{\nu}(t) \leq \left[ \left( \frac{\nu - 1}{\nu p(t)} \right) \sqrt{t} \right] + 1,$$ where $p(t) \geq 2$ for each $t \in \zed$, however for most $t$ better bounds are known.
Let us write $\tau(t)$ for the number of distinct divisors of $t$ and $\omega(t)$ for the number of distinct prime divisors of $t$ (see [@hall] for detailed information on $\tau(t)$ and $\omega(t)$). Hooley’s $\Delta$-function of $t$ is defined by $$\Delta(t) = \max_x \left| \left\{ d \in \zed_{>0} : d|t,\ e^x < d \leq e^{x+1} \right\} \right|.$$ If we take $x = \log \frac{\sqrt{t}}{\nu}$, then it is easy to see that $$\M_{\nu}(t) \subseteq \left\{ d \in \zed_{>0} : d|t,\ e^x < d \leq e^{x+1} \right\},$$ and hence $\beta_{\nu}(t) \leq \Delta(t)$. Now, as stated in [@ten1] (see also [@hall]) a consequence of Sperner’s theorem is that $$\label{beta_bound2}
\beta_{\nu}(t) \leq \Delta(t) \leq O \left( \frac{\tau(t)}{\sqrt{\omega(t)}} \right),$$ and if $t$ is squarefree the constant in $O$-notation is equal to 2. By Theorem 317 of [@hardy], for any $\eps > 0$ there exists an integer $t_0(\eps)$ such that for all $t > t_0(\eps)$, $$\label{tau_bound}
\tau(t) < 2^{(1+\eps) \frac{\log t}{\log \log t}}.$$ Then combining (\[beta\_bound2\]) with (\[tau\_bound\]), we see that for any $\eps > 0$ there exists an integer $t_0(\eps)$ such that for all $t > t_0(\eps)$,$$\label{beta_bound3}
\beta_{\nu}(t) \leq O \left( t^{\frac{(1+\eps) \log 2}{\log \log t}} \right),$$ which is much better than $O(\sqrt{t})$, the bound of (\[beta\_bound1\]), when $t$ is sufficiently large. In fact, for most $t$ we expect $\beta_{\nu}(t)$ to be even much smaller. A result of [@ten1] states that if $\psi(t)$ is any function such that $\psi(t) \rightarrow \infty$, as slowly as we wish, as $t \rightarrow \infty$, then $$\label{beta_av}
\beta_{\nu}(t) \leq \Delta(t) < \psi(t) \log \log t$$ for all positive integers $t$ in a sequence of asymptotic density 1.
The function $\beta_{\nu}(t)$ also has a geometric interpretation. Write the prime decomposition for $t$ as $$t = p_1^{e_1} \dots p_{\omega(t)}^{e_{\omega(t)}}$$ for corresponding distinct primes $p_1, \dots, p_{\omega(t)}$ and positive integers $e_1, \dots, e_{\omega(t)}$. If $d$ is a divisor of $t$, then $$d = p_1^{x_1} \dots p_{\omega(t)}^{x_{\omega(t)}}$$ for some integers $0 \leq x_i \leq e_i$ for each $1 \leq i \leq \omega(t)$. Then $\frac{\sqrt{t}}{\nu} \leq d \leq \sqrt{t}$ if and only if $$\log \frac{\sqrt{t}}{\nu} \leq \sum_{i=1}^{\omega(t)} x_i \log p_i \leq \log \sqrt{t}.$$ In other words, $\beta_{\nu}(t)$ is precisely the number of integer lattice points in the polytope $P_{\nu}(t)$ in $\real^{\omega(t)}$ bounded by the hyperplanes $$\label{poly_pt}
x_i = 0,\ x_i = e_i\ \forall\ 1 \leq i \leq \omega(t),\ \sum_{i=1}^{\omega(t)} x_i = \log \frac{\sqrt{t}}{\nu},\ \sum_{i=1}^{\omega(t)} x_i = \log \sqrt{t}.$$ In other words, $\beta_{\nu}(t) = \left|P_{\nu}(t) \cap \zed^{\omega(t)}\right|$, and $P_{\nu}(t)$ may be an irrational polytope. Counting integer lattice points in irrational polytopes is a very hard problem; a generating function for this problem is defined in [@barvinok], but almost nothing seems to be known about it.
Parametrization of well-rounded lattices
========================================
In this section we present an explicit description and a convenient parametrization of lattices in $\WR(\zed^2)$, which we later use to prove the main results of this paper.
First we introduce some additional notation, following [@near:ort]. An ordered collection of linearly independent vectors $\{ \bx_1, \dots, \bx_k \} \subset \real^N$, $2 \leq k \leq N$, is called [*nearly orthogonal*]{} if for each $1 < i \leq k$ the angle between $\bx_i$ and the subspace of $\real^N$ spanned by $\bx_1, \dots, \bx_{i-1}$ is in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$. In other words, this condition means that for each $1 < i \leq k$ $$\label{near}
\frac{| \bx_i^t \bwy |}{\|\bx_i\| \|\bwy\|} \leq \frac{1}{2},$$ for all non-zero vectors $\bwy \in \spn_{\real} \{ \bx_1, \dots, \bx_{i-1} \}$. The following result is Theorem 1 of [@near:ort]; in case $N=2$ this was proved by Gauss.
\[[@near:ort]\] \[no\] Suppose that an ordered basis $\{ \bx_1, \dots, \bx_k \}$ for a lattice $\Lambda$ in $\real^N$ of rank $1 < k \leq N$ is nearly orthogonal. Then it contains a minimal vector of $\Lambda$.
In particular, if all vectors $\bx_1, \dots, \bx_k$ of Theorem \[no\] have the same norm, then $\Lambda$ is a WR lattice; we will call such a basis $\bx_1, \dots, \bx_k$ minimal.
For the rest of this paper we will restrict to the case $N=2$. Here is a first characterization of WR sublattices of $\zed^2$.
\[gauss\] A sublattice $\Lambda \subseteq \zed^2$ of rank 2 is in $\WR(\zed^2)$ if and only if it has a basis $\bx,\bwy$ with $$\label{gauss_cond}
\|\bx\| = \|\bwy\|,\ |\cos \theta| = \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} \leq \frac{1}{2},$$ where $\theta$ is the angle between $\bx$ and $\bwy$. Moreover, if this is the case, then the set of minimal vectors $S(\Lambda) = \{ \pm \bx, \pm \bwy \}$. In particular, a minimal basis for $\Lambda$ is unique up to $\pm$ signs and reordering.
Suppose first that $\Lambda$ contains a basis $\bx,\bwy$ satisfying (\[gauss\_cond\]). By Theorem \[no\] this must be a minimal basis, meaning that $\Lambda$ is WR.
Next assume that $\Lambda$ is WR, and let $\bx,\bwy \in S(\Lambda)$ be linearly independent vectors. It is a well known fact that for lattices of rank $\leq 4$ linearly independent minimal vectors form a basis, hence $\bx,\bwy$ is a basis for $\Lambda$, and $|\Lambda| = \|\bx\| = \|\bwy\|$. Let $\theta$ be the angle between $\bx$ and $\bwy$. We can assume without loss of generality that $\cos \theta > 0$: if not, replace $\bx$ with $-\bx$ or $\bwy$ with $-\bwy$. Notice that $\bo \neq \bx-\bwy \in \Lambda$, and $$\|\bx - \bwy\| = \sqrt{ \|\bx\|^2 + \|\bwy\|^2 - 2 \bx^t \bwy } = |\Lambda| \sqrt{ 2 (1 - \cos \theta) }.$$ If $\cos \theta > \frac{1}{2}$, then $\|\bx - \bwy\| < |\Lambda|$, which is a contradiction. This proves (\[gauss\_cond\]), and also implies that the angle between two minimal linearly independent vectors in $\Lambda$ must lie in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$.
Now assume that $\Lambda \in \WR(\zed^2)$, and let $\bx,\bwy$ be a minimal basis for $\Lambda$, so $|\Lambda| = \|\bx\| = \|\bwy\|$. Let $$\theta_1,\theta_2,\theta_3,\theta_4$$ be angles between pairs of vectors $\{\bx,\bwy\}$, $\{\bwy,-\bx\}$, $\{-\bx,-\bwy\}$, and $\{-\bwy,\bx\}$ respectively. Since all of these vectors are in $\Lambda$ and have length $|\Lambda|$, it must be true that $$\theta_1,\theta_2,\theta_3,\theta_4 \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right].$$ On the other hand, $$\theta_2 = \theta_4 = \pi - \theta_1,\ \theta_3 = \theta_1.$$ Assume there exists a vector $\bz \in \Lambda$ of length $|\Lambda|$ which is not equal to $\pm \bx, \pm \bwy$. Then all the angles it makes with the vectors $\pm \bx, \pm \bwy$ must lie in the interval $\left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$. This means that at least one of these angles must be equal to $\frac{\pi}{3}$, assume without loss of generality that this is the angle $\bz$ makes with $\bx$. Then $$\bz = \left( \begin{matrix} z_1 \\ z_2 \end{matrix} \right) = \left( \begin{matrix} \cos \left( \frac{\pi}{3} \right)&-\sin \left( \frac{\pi}{3} \right) \\ \sin \left( \frac{\pi}{3} \right)&\cos \left( \frac{\pi}{3} \right) \end{matrix} \right) \left( \begin{matrix} x_1 \\ x_2 \end{matrix} \right) = \left( \begin{matrix} \frac{x_1}{2} - \frac{x_2\sqrt{3}}{2} \\ \frac{x_1\sqrt{3}}{2} + \frac{x_2}{2} \end{matrix} \right),$$ where $x_1,x_2 \in \zed$ are coordinates of $\bx$. Since $\bz \in \Lambda$, it must be true that $z_1,z_2 \in \zed$, but this is not possible. Hence a vector $\bz$ like this cannot exist, and this completes the proof.
Next we develop a certain convenient explicit parametrization of lattices in $\WR(\zed^2)$. We start with lemmas describing two different families of such lattices.
\[abcd1\] Let $a,b,c,d \in \zed$ be such that $$\label{cond_esm1}
0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0.$$ Then $$\label{esm1}
\Lambda = \left( \begin{matrix} ac+bd&ac-bd \\ bc-ad&bc+ad \end{matrix} \right) \zed^2$$ is in $\WR(\zed^2)$ with $$\label{det_esm1}
\det(\Lambda) = 2(a^2+b^2)|cd|.$$
Suppose $a,b,c,d \in \zed$ satisfy (\[cond\_esm1\]). Let $\bx = \left( \begin{matrix} ac+bd \\ bc-ad \end{matrix} \right)$ and $\bwy = \left( \begin{matrix} ac-bd \\ bc+ad \end{matrix} \right)$, then $$\begin{aligned}
\label{square}
\|\bx\|^2 & = & (ac+bd)^2 + (bc-ad)^2 \nonumber \\
& = & (a^2+b^2)(c^2+d^2) \nonumber \\
& = & (ac-bd)^2 + (bc+ad)^2 = \|\bwy\|^2.\end{aligned}$$ Let $\Lambda = \spn_{\zed} \{\bx, \bwy\}$, then $\rk \Lambda = 2$. Let $\theta$ be the angle between $\bx$ and $\bwy$, and let $c = \gamma d$, where by (\[cond\_esm1\]), $1 \leq |\gamma| \leq \sqrt{3}$. Then, by (\[square\]) and (\[cond\_esm1\]) $$\begin{aligned}
|\cos(\theta)| & = & \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} = \frac{|(a^2+b^2)(c^2-d^2)|}{(a^2+b^2)(c^2+d^2)} \\
& = & \frac{c^2-d^2}{c^2+d^2} = \frac{\gamma^2-1}{\gamma^2+1} \leq \frac{1}{2}.\end{aligned}$$ Therefore $\theta \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$, and so, by Lemma \[gauss\], $\Lambda$ is WR; (\[det\_esm1\]) follows. This completes the proof.
\[abcd2\] Let $a,b,c,d \in \zed$ be such that $$\label{cond_esm2}
c^2+d^2 \geq 4|cd|,\ \max \{|a|,|b|\} > 0.$$ Then $$\label{esm2}
\Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2$$ is in $\WR(\zed^2)$ with $$\label{det_esm2}
\det(\Lambda) = (a^2+b^2) |c^2-d^2|.$$
Suppose $a,b,c,d \in \zed$ satisfy (\[cond\_esm2\]). Let $\bx = \left( \begin{matrix} ac-bd \\ ad+bc \end{matrix} \right)$ and $\bwy = \left( \begin{matrix} ad-bc \\ ac+bd \end{matrix} \right)$, and define $\Lambda = \spn_{\zed} \{\bx, \bwy\}$. Then $\rk \Lambda = 2$, and $\|\bx\|, \|\bwy\|$ are the same as in (\[square\]). Let $\theta$ be the angle between $\bx$ and $\bwy$. Then, by (\[square\]) and (\[cond\_esm2\]) $$\begin{aligned}
|\cos(\theta)| & = & \frac{| \bx^t \bwy |}{\|\bx\| \|\bwy\|} = \frac{2|cd|(a^2+b^2)}{(a^2+b^2)(c^2+d^2)} \\
& = & \frac{2|cd|}{c^2+d^2} \leq \frac{1}{2}.\end{aligned}$$ Therefore $\theta \in \left[ \frac{\pi}{3}, \frac{2\pi}{3} \right]$, and so, by Lemma \[gauss\], $\Lambda$ is WR; (\[det\_esm2\]) follows. This completes the proof.
\[2-class\] Suppose $\Lambda \in \WR(\zed^2)$. Then $\Lambda$ is either of the form as described in Lemma \[abcd1\] or as in Lemma \[abcd2\].
Suppose $\Lambda \in \WR(\zed^2)$, and let $\bx,\bwy \in \Lambda$ be a minimal basis $$\label{c1}
\|\bx\|^2 = x_1^2 + x_2^2 = |\Lambda|^2 = y_1^2+y_2^2 = \|\bwy\|^2.$$ Notice that due to (\[c1\]) it must be true that either the pairs $x_1,y_1$ and $x_2,y_2$, or the pairs $x_1,y_2$ and $x_2,y_1$ are of the same parity. Indeed, suppose this is not true, then we can assume without loss of generality that $x_1,x_2$ are even and $y_1,y_2$ are odd. But then $$x_1^2 + x_2^2 \equiv 0\ (\md\ 4),\ y_1^2+y_2^2 \equiv 2\ (\md\ 4),$$ which contradicts (\[c1\]). Therefore, either $$\label{c2}
\frac{x_1-y_1}{2},\ \frac{x_1+y_1}{2},\ \frac{y_2-x_2}{2},\ \frac{x_2+y_2}{2} \in \zed,$$ or $$\label{c3}
\frac{x_1-y_2}{2},\ \frac{x_1+y_2}{2},\ \frac{y_1-x_2}{2},\ \frac{x_2+y_1}{2} \in \zed.$$ First assume (\[c2\]) is true. Then let $$\label{c4}
c = \gcd \left( \frac{x_1+y_1}{2}, \frac{x_2+y_2}{2} \right),\ a = \frac{x_1+y_1}{2c},\ b = \frac{x_2+y_2}{2c},\ d = \frac{(y_2-x_2)c}{x_1+y_1}.$$ Clearly $a,b,c \in \zed$. We will now show that $d \in \zed$. Indeed, $$d = \frac{(y_2-x_2)c}{x_1+y_1} = \frac{y^2_2-x^2_2}{(x_2+y_2)\left(\frac{x_1+y_1}{c}\right)},$$ and of course $(x_2+y_2)\ |\ (y^2_2-x^2_2)$. Also, by (\[c1\]) $$\left(\frac{x_1+y_1}{c}\right)\ |\ (x_1+y_1)\ |\ (x_1^2-y_1^2) = (y^2_2-x^2_2),$$ and by definition of $c$ in (\[c4\]), $$\gcd\left( x_2+y_2,\ \frac{x_1+y_1}{c} \right) = 1,$$ which implies that $$(x_2+y_2)\left(\frac{x_1+y_1}{c}\right)\ |\ (y^2_2-x^2_2),$$ and hence $d \in \zed$. With these definitions of $a,b,c,d$, it is easy to see that $$x_1 = ac+bd,\ x_2 = bc-ad,\ y_1 = ac-bd,\ y_2 = bc+ad,$$ and hence $\Lambda$ is precisely of the form (\[esm1\]). Moreover, since it is WR, Lemma \[gauss\] implies that it must satisfy condition (\[gauss\_cond\]), which implies (\[cond\_esm1\]). This finishes the proof in case (\[c2\]) is true. The proof in case (\[c3\]) is true is completely analogous, in which case $\Lambda$ is of type (\[esm2\]), and then (\[cond\_esm2\]) is satisfied.
Suppose now that a lattice $$\Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2$$ with $$c^2+d^2 \geq 4|cd|,\ \max \{|a|,|b|\} > 0$$ as in Lemma \[abcd2\], and $$\det(\Lambda) = (a^2+b^2) |c^2-d^2|$$ is even. We will show that in this case $\Lambda$ can be represented in the form as in Lemma \[abcd1\]. First assume that $a^2+b^2$ is even, then $a^2,b^2$, and hence $a,b$, must be of the same parity, meaning that $a+b$ and $a-b$ are even. Define $$a_1 = \frac{a-b}{2},\ b_1 = \frac{a+b}{2},$$ then $a^2+b^2 = 2(a_1^2+b_1^2)$. Let $c_1,d_1$ be such that $c_1 d_1 = c^2-d^2$, and $$|c_1| = \max\{ |c-d|, |c+d| \},\ |d_1| = \min\{ |c-d|, |c+d| \}.$$ Suppose for instance that $c_1 = c+d$ and $d_1 = c-d$ (the argument is completely analogous in case $c_1 = c-d$ and $d_1 = c+d$). Then $$c = \frac{c_1+d_1}{2},\ d = \frac{c_1-d_1}{2},$$ and so $4cd = c_1^2-d_1^2$. On the other hand $c^2+d^2 = \frac{c_1^2+d_1^2}{2}$. The fact that $c^2+d^2 \geq 4|cd|$ implies that $$\frac{c_1^2+d_1^2}{2} \geq |c_1^2 - d_1^2| = c_1^2 - d_1^2,$$ since $|c_1| \geq |d_1|$, and so $$|c_1| \leq \sqrt{3}\ |d_1|.$$ This choice of $a_1,b_1,c_1,d_1$ satisfies the conditions of (\[cond\_esm1\]), and it is easy to see that $$\label{lat_eq}
\Lambda = \left( \begin{matrix} ac-bd&ad-bc \\ ad+bc&ac+bd \end{matrix} \right) \zed^2 = \left( \begin{matrix} a_1c_1+b_1d_1&a_1c_1-b_1d_1 \\ b_1c_1-a_1d_1&b_1c_1+a_1d_1 \end{matrix} \right) \zed^2,$$ and $\det(\Lambda) = 2(a_1^2+b_1^2)|c_1d_1|$.
Next assume that $c^2-d^2$ is even. Then $(c+d)(c-d)$ is even and $(c+d)+(c-d) = 2c$ is even, which implies that $(c+d)$ and $(c-d)$ must both be even, in particular $c^2-d^2$ is divisible by $4$. Let $c_1,d_1$ be such that $4 c_1 d_1 = c^2-d^2$, and $$|c_1| = \frac{1}{2} \max\{ |c-d|, |c+d| \},\ |d_1| = \frac{1}{2} \min\{ |c-d|, |c+d| \}.$$ By an argument as above, we can easily deduce again that $$|d_1| \leq |c_1| \leq \sqrt{3}\ |d_1|.$$ Let $$a_1 = a-b,\ b_1 = a+b,$$ then $2(a^2+b^2) = a_1^2+b_1^2$, and so $$\det(\Lambda) = (a^2+b^2)|c^2-d^2| = 4(a^2+b^2)|c_1d_1| = 2(a_1^2+b_1^2)|c_1d_1|.$$ Once again, it is easy to check that with $a_1,b_1,c_1,d_1$ defined this way (\[lat\_eq\]) holds.
Let $$\label{E1}
\E' = \left\{ (a,b,c,d) \in \zed^4\ :\ 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0 \right\},$$ and $$\begin{aligned}
\label{O1}
\OO' = \{ (a,b,c,d) \in \zed^4 & : & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ \max \{|a|,|b|\} > 0, \nonumber \\
& & \text{so that } 2 \nmid (a^2+b^2) |cd| \}.\end{aligned}$$ Define two classes of integral lattices $$\label{E}
\E = \left\{ \Lambda(a,b,c,d) = \left( \begin{matrix} ac+bd&ac-bd \\ bc-ad&bc+ad \end{matrix} \right) \zed^2 : (a,b,c,d) \in \E' \right\},$$ and $$\label{O}
\OO = \left\{ \Lambda(a,b,c,d) = \left( \begin{matrix} \frac{ac+ad+bd-bc}{2}&\frac{ac-ad-bc-bd}{2} \\ \frac{ac+bc+bd-ad}{2}&\frac{ac+ad+bc-bd}{2} \end{matrix} \right) \zed^2 : (a,b,c,d) \in \OO' \right\}.$$ Then for every $\Lambda = \Lambda(a,b,c,d) \in \E$, $$\det(\Lambda) = 2(a^2+b^2) |cd|,$$ is even, and for every $\Lambda = \Lambda(a,b,c,d) \in \OO$, $$\det(\Lambda) = (a^2+b^2) |cd|$$ is odd. We proved the following theorem.
\[disjoint\] The set $\WR(\zed^2)$ can be represented as the disjoint union of $\E$ and $\OO$. Moreover, the set of all possible determinants of lattices in $\WR(\zed^2) = \E \cup \OO$ is $$\label{dets}
\D = \{ (a^2+b^2) |cd| : a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\ 0 < |d| \leq |c| \leq \sqrt{3} |d| \}.$$
\[min\_rem\] Notice also that $$|\Lambda|^2 = \left\{ \begin{array}{ll}
(a^2+b^2)(c^2+d^2) & \mbox{if $\Lambda \in \E$} \\
\frac{1}{2}(a^2+b^2)(c^2+d^2) & \mbox{if $\Lambda \in \OO$.}
\end{array}
\right.$$ Therefore the set of squared minima $\Mm(\E)$ of the lattices from $\E$ can be represented as $$\begin{aligned}
\label{min_rep1}
\Mm(\E) = \{ (a^2+b^2)(c^2+d^2) & : & a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\nonumber \\
& & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ 2 | (a^2+b^2)cd \},\end{aligned}$$ and the set of squared minima $\Mm(\OO)$ of the lattices from $\OO$ can be represented as $$\begin{aligned}
\label{min_rep2}
\Mm(\OO) = \Big\{ \frac{1}{2} (a^2+b^2)(c^2+d^2) & : & a,b,c,d \in \zed,\ 0 < \max \{|a|,|b|\},\nonumber \\
& & 0 < |d| \leq |c| \leq \sqrt{3} |d|,\ 2 \nmid (a^2+b^2)cd \Big\}.\end{aligned}$$ Then the set of squared minima $\Mm$ can be represented as $\Mm = \Mm(\E) \cup \Mm(\OO)$.
\[monoid\] The determinant set $\D$ in (\[dets\]) and the squared minima set $\Mm$ are commutative monoids under multiplication.
If $t_1 = (a_1^2+b_1^2)c_1d_1$ and $t_2 = (a_2^2+b_2^2)c_2d_2$ are in $\D$, then $t_1t_2 = (a_3^2+b_3^2)c_3d_3 \in \D$, where $a_3=a_1a_2+b_1b_2$, $b_3=b_1a_2-a_1b_2$, $c_3 = \pm \max\{|c_1d_2|,|d_1c_2|\}$, and $d_3 = \pm \min\{|c_1d_2|,|d_1c_2|\}$. It is also obvious that a product of two integers which are representable as sums of two squares is also representable as a sum of two squares.
Next we will use Theorem \[disjoint\] to investigate the structure of the set $\D$ and to count the number of lattices in $\WR(\zed^2)$ of a fixed determinant.
Proof of Theorem \[dense\]
==========================
The description of the set $\D$ in the statement of Theorem \[dense\] follows immediately from (\[dets\]). In this section we will mostly be concerned with deriving the estimate (\[D\_dens\]) for the lower density of $\D$.
For each real number $1 < \nu \leq 3^{1/4}$, define the set $$\label{prod_set}
\B_{\nu} = \left\{ n \in \zed_{>0} : \exists\ d \in \zed_{>0}\ \text{such that } d\ |\ n\ \text{and } \frac{\sqrt{n}}{\nu} \leq d \leq \sqrt{n} \right\}.$$ Then notice that another description of the set $\D$ in (\[dets\]) is $$\D = \Mm \B_{3^{1/4}} = \{ m n : m \in \Mm,\ n \in \B_{3^{1/4}} \},$$ where $\Mm$ is the set of squared minima of lattices in $\WR(\zed^2)$, as before, so $$\label{sum_set}
\Mm = \{ m \in \zed_{>0} : m = k^2+l^2\ \text{for some } k,l \in \zed \}.$$ By a well known theorem of Fermat, the set $\Mm$ consists precisely of those positive integers $m$ in whose prime factorization every prime of the form $(4k+3)$ occurs an even number of times. Since $1 \in \Mm \cap \B_{3^{1/4}}$, we have $\Mm, \B_{3^{1/4}} \subset \D$; on the other hand, $6 \in \B_{3^{1/4}} \setminus \Mm$ and $2 \in \Mm \setminus \B_{3^{1/4}}$, hence $\Mm \subsetneq \D$ and $\B_{3^{1/4}} \subsetneq \D$. Moreover, $\D \subsetneq \zed_{>0}$, since for instance $3 \notin \D$.
It is a well-known result of Landau (see, for instance [@motohashi]) that $\Mm$ has asymptotic density equal to 0, specifically $$\lim_{M \rightarrow \infty} \frac{1}{M} \left| \{ n \in \Mm : n \leq M \} \right| = \lim_{M \rightarrow \infty} \frac{1}{\sqrt{\log M}} = 0.$$
Let us investigate the density of the sets $\B_{\nu}$ for a fixed $\nu \in (1,3^{1/4}]$. As before, for each $M \in \zed_{>0}$ we write $$\B_{\nu}(M) = \{ n \in \B_{\nu} : n \leq M \}.$$ For each $n \in \zed_{>0}$ define $$I_{\nu}(n) = \left\{n^2, n(n-1), \dots, n \left(n-\left[\left(\frac{\nu-1}{\nu}\right)n\right]\right)\right\}.$$ Notice that every $k \in \B_{\nu}(M)$ is of the form $k = n(n-i)$ for some $n$ and $i \leq \left[\left(\frac{\nu-1}{\nu}\right)n\right]$, and so if $n \leq [\sqrt{M}]$, then $k \in \bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n)$. For each $n \in \zed_{>0}$, $$\label{I_card}
|I_{\nu}(n)| = \left[\left(\frac{\nu-1}{\nu}\right)n\right] + 1.$$ There may also be some $k = n(n-i) \in \B_{\nu}(M)$ with $n > [\sqrt{M}]$ for some $i \leq \left(\frac{\nu-1}{\nu}\right)n$. Then $k = n^2 - ni \leq M$, and so $i \geq n - \frac{M}{n}$. It is easy to see that this is only possible if $n \leq \left[ \sqrt{\nu M} \right]$, and so for each $[\sqrt{M}] < n \leq \left[ \sqrt{\nu M} \right]$ define $$J_{\nu,M}(n) = \left\{ n(n-i) : \left[ n - \frac{M}{n} \right] + 1 \leq i \leq \left[ \left(\frac{\nu-1}{\nu}\right)n \right] \right\}.$$ Clearly, $J_{\nu,M}(n) \subseteq I_{\nu}(n)$ for each such $n$, and $$\label{B_contain_J}
\B_{\nu}(M) = \left( \bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n) \right) \cup \left( \bigcup_{n=[\sqrt{M}]+1}^{\left[ \sqrt{\nu M} \right]} J_{\nu,M}(n)\right).$$ For simplicity of approximation notice that $$\label{B_contain}
\bigcup_{n=1}^{[\sqrt{M}]} I_{\nu}(n) \subseteq \B_{\nu}(M) \subseteq \bigcup_{n=1}^{\left[ \sqrt{\nu M} \right]} I_{\nu}(n).$$ We immediately obtain an upper bound on $|\B_{\nu}(M)|$.
\[I\_upp\] For all $M \in \zed_{>0}$, $$\label{I_upp1}
|\B_{\nu}(M)| \leq \frac{\nu-1}{2}\ M + \frac{\nu-1}{2 \sqrt{\nu}}\ \sqrt{M}.$$
Combining (\[B\_contain\]) and (\[I\_card\]), we obtain: $$|\B_{\nu}(M)| \leq \sum_{n=1}^{\left[ \sqrt{\nu M} \right]} |I_{\nu}(n)| \leq \frac{\nu-1}{\nu} \sum_{n=1}^{\left[ \sqrt{\nu M} \right]} n \leq \frac{\nu-1}{2 \nu} \sqrt{\nu M} \left( \sqrt{\nu M} + 1 \right).$$ The bound of (\[I\_upp1\]) follows.
Next we want to produce a lower bound on $|\B_{\nu}(M)|$. For this we first consider the pairwise intersections of the sets $I_{\nu}(n)$.
\[I\_int\] Let $m < n \leq [\sqrt{M}]$.
(1) If $n > m \sqrt{\nu}$, or if $n \geq m \sqrt{\nu}$ and $\sqrt{\nu}$ is irrational, then $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$.
(2) If $\gcd(m,n)=1$, then $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$.
(3) If $m < n < m \sqrt{\nu}$, then $$|I_{\nu}(n) \cap I_{\nu}(m)| \leq \left[ \frac{\nu-1}{\nu}\ \gcd(m,n) \right] + 1.$$
Define $Q=\frac{\nu-1}{\nu}$. Let $m < n \leq [\sqrt{M}]$, and suppose that $k \in I_{\nu}(n) \cap I_{\nu}(m)$. Then $$k = n(n-x) = m(m-y),$$ for some integers $0 \leq x \leq Qn$ and $0 \leq y \leq Qm$. Define a line $$L(n,m) = \{ (x,y) \in \real^2 : nx-my=n^2-m^2 \},$$ and a rectangular box $$R(n,m) = \{ (x,y) \in \real^2 : 0 \leq x \leq Qn,\ 0 \leq y \leq Qm \}.$$ It follows immediately that $$|I_{\nu}(n) \cap I_{\nu}(m)| = |L(n,m) \cap R(n,m) \cap \zed^2|.$$ The line $L(m,n)$ passes through the points $\left( \frac{n^2-m^2}{n}, 0 \right)$ and $\left( 0, -\frac{n^2-m^2}{n} \right)$, so in particular $L(n,m) \cap R(n,m) = \emptyset$ if $\frac{n^2-m^2}{n} > Qn$, i.e. if $n > m \sqrt{\nu}$. Also if $\sqrt{\nu}$ is irrational, then $m \sqrt{\nu}$ is never an integer, and so $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$ if $n \geq m \sqrt{\nu}$, proving (1).
Now suppose $m < n < m \sqrt{\nu}$, and let $(x,y) \in L(n,m) \cap R(n,m) \cap \zed^2$. Then $$y = -\frac{n(n-x)}{m} + m \in \zed_{>0},$$ hence $m\ |\ n(n-x)$. Clearly, $m \nmid n$, and so we must have $$\lcm(m,n) = \frac{mn}{\gcd(m,n)}\ |\ n(n-x),$$ meaning that $$\label{gcd1}
\frac{m}{\gcd(m,n)}\ |\ n-x.$$ In particular, if $\gcd(m,n)=1$, we must have $m\ |\ n-x$, but $n-x \leq n < 2m$, meaning that in order for $n-x$ to be divisible by $m$, it must be equal to $m$. This would imply that $x=n-m < \frac{n^2-m^2}{n}$, hence $y < 0$, meaning that $(x,y) \notin R(n,m)$, which is a contradiction. This proves (2).
Now assume that $(x_1,y_1),(x_1+t,y_2) \in L(n,m) \cap R(n,m) \cap \zed^2$, where $t$ is as small as possible. By (\[gcd1\]), $\frac{m}{\gcd(m,n)}\ |\ n-x_1$ and $\frac{m}{\gcd(m,n)}\ |\ n-x_1-t$, so $\frac{m}{\gcd(m,n)}\ |\ t$, and by minimality of $t$ we must have $t = \frac{m}{\gcd(m,n)}$. Therefore $x$-coordinates of points in $L(n,m) \cap R(n,m) \cap \zed^2$ must satisfy $$\frac{n^2-m^2}{n} \leq x < Qn,$$ and the distance between $x$-coordinates of any two such points must be at least $\frac{m}{\gcd(m,n)}$. Hence, $$\begin{aligned}
|I_{\nu}(n) \cap I_{\nu}(m)| & = & |L(n,m) \cap R(n,m) \cap \zed^2| \leq \left[ \frac{Qn - \frac{n^2-m^2}{n}}{\frac{m}{\gcd(m,n)}} \right] + 1 \\
& = & \left[ \frac{(\nu m^2 - n^2) \gcd(m,n)}{\nu mn} \right] + 1 \leq \left[ \frac{\nu-1}{\nu}\ \gcd(m,n) \right] + 1.\end{aligned}$$ This proves (3).
\[I\_low\] For all $M \in \zed_{>0}$, $$\label{I_low1}
|\B_{\nu}(M)| > \frac{\nu-1}{2 \nu}\ M \left( 1 - \frac{ \log \log \sqrt{M}}{\log \sqrt{M}} \right)^2.$$
Let $N=\pi(\sqrt{M})$, i.e. the number of primes up to $\sqrt{M}$. It is a well-known fact that for all $\sqrt{M} \geq 11$, $$\label{N_bound}
N \geq \frac{\sqrt{M}}{\log \sqrt{M}}.$$ Hence suppose that $M \geq 121$, and let $p_1, \dots, p_N$ be all the primes up to $\sqrt{M}$ in ascending order. By part (2) of Lemma \[I\_int\], $$I_{\nu}(p_i) \cap I_{\nu}(p_j) = \emptyset,$$ for all $1 \leq i \neq j \leq N$. As above, we let $Q=\frac{\nu-1}{\nu}$. Therefore, using (\[I\_card\]) we obtain: $$\label{low2}
|\B_{\nu}(M)| \geq \sum_{i=1}^N |I_{\nu}(p_i)| \geq Q \sum_{i=1}^N p_i.$$ A result of R. Jakimczuk [@jakimczuk] implies that $$\label{prime_bound}
\sum_{i=1}^N p_i > \frac{N^2}{2}\ \log^2 N.$$ The bound (\[I\_low1\]) follows upon combining (\[low2\]) with (\[prime\_bound\]) and (\[N\_bound\]).
Now assume that $M < 121$, so $\sqrt{M} < 11$. A direct verification shows that in this case $I_{\nu}(n) \cap I_{\nu}(m) = \emptyset$ for all $1 \leq n \neq m \leq [\sqrt{M}]$, and so $$|\B_{\nu}(M)| \geq \sum_{n=1}^{[\sqrt{M}]} |I_{\nu}(n)| \geq Q \sum_{n=1}^{[\sqrt{M}]} n \geq \frac{\nu-1}{2 \nu}\ M.$$ This completes the proof.
Combining Lemmas \[I\_upp\] and \[I\_low\], we obtain the following result.
\[B\_density\] $$\frac{\nu-1}{2 \nu} \leq \DL_{\B_{\nu}} \leq \DU_{\B_{\nu}} \leq \frac{\nu-1}{2}.$$
Using (\[I\_low1\]), we see that $$\DL_{\B_{\nu}} = \liminf_{M \rightarrow \infty} \frac{|\B_{\nu}(M)|}{M} \geq \frac{\nu-1}{2 \nu} \lim_{M \rightarrow \infty} \left( 1 - \frac{ \log \log \sqrt{M}}{\log \sqrt{M}} \right)^2 = \frac{\nu-1}{2 \nu},$$ and using (\[I\_upp1\]), we see that $$\DU_{\B_{\nu}} = \limsup_{M \rightarrow \infty} \frac{|\B_{\nu}(M)|}{M} \leq \frac{\nu-1}{2} + \frac{\nu-1}{2 \sqrt{\nu}} \lim_{M \rightarrow \infty} \frac{1}{\sqrt{M}} = \frac{\nu-1}{2}.$$ This completes the proof.
It is also possible to produce bounds on $|\B_{\nu}(M)|$ using decomposition (\[B\_contain\_J\]) instead of (\[B\_contain\]), and employing the fact that $$|J_{\nu,M}(n)| = \left[ \left(\frac{\nu-1}{\nu}\right)n \right] - \left[ n - \frac{M}{n} \right].$$ It is also possible to employ the full power of Lemma \[I\_int\], in particular part (3), to further refine the lower bound on $|\B_{\nu}(M)|$. These estimates however produce only marginally better constants, but much messier bounds in general.
We could also lift the restriction that $\nu \leq 3^{1/4}$ with essentially no changes to the arguments, but in any case the important situation is that with $\nu$ being close to 1, and we want to emphasize that the case of utmost importance to us is that with $\nu = 3^{1/4}$.
Now (\[D\_dens\]) of Theorem \[dense\] follows immediately by recalling that $\B_{3^{1/4}} \subseteq \D$, and applying Theorem \[B\_density\] with $\nu = 3^{1/4}$. The bounds on density of $\B_{\nu}$ are of independent interest, and will also be used in section 8 below to determine the order of the pole of the zeta function of well-rounded lattices.
Proof of Theorem \[count\]
==========================
We start with a lemma which identifies all the 4-tuples from $\E' \cup \OO'$ which parametrize the same lattices.
\[transform1\] Let $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E' \cup \OO'$. Then $$\label{gamma1}
\Lambda(a_1,b_1,c_1,d_1) = \Lambda(a_2,b_2,c_2,d_2)$$ if and only if there exists $0 \neq \gamma \in \que$ such that $(a_1,b_1,c_1,d_1)$ is equal to one of the following: $$\begin{aligned}
\label{gamma2}
& & \left( \frac{a_2}{\gamma}, \frac{b_2}{\gamma}, \gamma c_2, \gamma d_2 \right),\ \left( \frac{b_2}{\gamma}, -\frac{a_2}{\gamma}, -\gamma d_2, \gamma c_2 \right),\ \left( -\frac{b_2}{\gamma}, \frac{a_2}{\gamma}, -\gamma d_2, \gamma c_2 \right), \nonumber \\
& & \left( -\frac{a_2}{\gamma}, -\frac{b_2}{\gamma}, \gamma c_2, \gamma d_2 \right),\ \left( -\frac{a_2}{\gamma}, -\frac{b_2}{\gamma}, -\gamma c_2, \gamma d_2 \right),\ \left( \frac{b_2}{\gamma}, -\frac{a_2}{\gamma}, \gamma d_2, \gamma c_2 \right), \nonumber \\
& & \left( -\frac{b_2}{\gamma}, \frac{a_2}{\gamma}, \gamma d_2, \gamma c_2 \right), \left( \frac{a_2}{\gamma}, \frac{b_2}{\gamma}, -\gamma c_2, \gamma d_2 \right).\end{aligned}$$
If $(a_1,b_1,c_1,d_1)$ is equal to one of the 4-tuples as in (\[gamma2\]), then a direct verification shows that (\[gamma1\]) is true. Suppose, on the other hand, that (\[gamma1\]) is true. Let $$\bx_1 = \left( \begin{matrix} a_1c_1+b_1d_1 \\ b_1c_1-a_1d_1 \end{matrix} \right),\ \bwy_1 = \left( \begin{matrix} a_1c_1-b_1d_1 \\ b_1c_1+a_1d_1 \end{matrix} \right),$$ and $$\bx_2 = \left( \begin{matrix} a_2c_2+b_2d_2 \\ b_2c_2-a_2d_2 \end{matrix} \right),\ \bwy_2 = \left( \begin{matrix} a_2c_2-b_2d_2 \\ b_2c_2+a_2d_2 \end{matrix} \right),$$ if $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E'$, or $$\bx_1 = \left( \begin{matrix} \frac{a_1c_1+a_1d_1+b_1d_1-b_1c_1}{2} \\ \frac{a_1c_1+b_1c_1+b_1d_1-a_1d_1}{2} \end{matrix} \right),\ \bwy_1 = \left( \begin{matrix} \frac{a_1c_1-a_1d_1-b_1c_1-b_1d_1}{2} \\ \frac{a_1c_1+a_1d_1+b_1c_1-b_1d_1}{2} \end{matrix} \right),$$ and $$\bx_2 = \left( \begin{matrix} \frac{a_2c_2+a_2d_2+b_2d_2-b_2c_2}{2} \\ \frac{a_2c_2+b_2c_2+b_2d_2-a_2d_2}{2} \end{matrix} \right),\ \bwy_2 = \left( \begin{matrix} \frac{a_2c_2-a_2d_2-b_2c_2-b_2d_2}{2} \\ \frac{a_2c_2+a_2d_2+b_2c_2-b_2d_2}{2} \end{matrix} \right),$$ if $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \OO'$. By Lemma \[gauss\], this means that the basis matrix $(\bx_1\ \bwy_1)$ for $\Lambda(a_1,b_1,c_1,d_1)$ must be equal to one of the following basis matrices for $\Lambda(a_2,b_2,c_2,d_2)$: $$\begin{aligned}
& & (\bx_2\ \bwy_2),\ (-\bx_2\ \bwy_2),\ (\bx_2\ -\bwy_2),\ (-\bx_2\ -\bwy_2), \\
& & (\bwy_2\ \bx_2),\ (-\bwy_2\ \bx_2),\ (\bwy_2\ -\bx_2),\ (-\bwy_2\ -\bx_2).\end{aligned}$$ A direct verification shows that in each of these cases $(a_1,b_1,c_1,d_1)$ is equal to one of the 4-tuples as in (\[gamma2\]), in the same order. This completes the proof.
\[sqr\_free\] Notice that in Lemma \[transform1\] $(a_1,b_1,c_1,d_1)$ can be equal to the second, third, sixth, or seventh 4-tuple in (\[gamma2\]) only if $$|c_1|=|\gamma||d_2|,\ |d_1|=|\gamma||c_2|,$$ but on the other hand we know that $(a_1,b_1,c_1,d_1), (a_2,b_2,c_2,d_2) \in \E' \cup \OO'$. Combining these facts we obtain $$\label{cd_eq}
|c_1| = |\gamma||d_2| \leq |\gamma||c_2| = |d_1| \leq |c_1|,$$ which implies that there must be equality everywhere in (\[cd\_eq\]). In this case the determinant of the corresponding lattice $\Lambda$ is equal to $(a_1^2+b_1^2) c_1^2$ if $\Lambda \in \OO$ or to $2(a_1^2+b_1^2) c_1^2$ if $\Lambda \in \E$.
[*Proof of Theorem \[count\].*]{} Let $u \in \D$. If $u=1,2$, the proof is by direct verification. Assume from here on that $u>2$, and let $$t = t(u) = \left\{ \begin{array}{ll}
u & \mbox{if $u$ is odd} \\
\frac{u}{2} & \mbox{if $u$ is even.}
\end{array}
\right.$$ Define $$D(t) = \{ n \in \zed_{>0} : n | t \},$$ i.e. $D(t)$ is the set of positive divisors of $t$. Define $$D_1(t) = \{ (c,d) \in D(t) \times D(t) : d \leq c \leq \sqrt{3} d,\ cd | t \}.$$ For each $(c,d) \in D_1(t)$, define $$S_t(c,d) = \left\{ (a,b) \in \zed^2_{\geq 0} : a^2+b^2 = \frac{t}{cd},\ a \leq b \right\}.$$ Also let $$T(t) = \{ (a,b,c,d) \in \zed^4_{\geq 0} : (c,d) \in D_1(t),\ (a,b) \in S_t(c,d) \}.$$ Define an equivalence relation on $T(t)$ by writing $$(a_1,b_1,c_1,d_1) \sim (a_2,b_2,c_2,d_2)$$ if $(a_1,b_1,c_1,d_1) = \left( \frac{a_2}{\gamma},\frac{b_2}{\gamma},\gamma c_2,\gamma d_2 \right)$ for some $\gamma \in \que_{>0}$. Then let $T_1(t)$ be the set of all equivalence classes of elements of $T(t)$ under $\sim$, i.e. $T_1(t) = T(t)/\sim$. By abuse of notation, we will write $(a,b,c,d)$ for an element of $T_1(t)$. We first have the following lemma.
\[gcd\_ab\] For each equivalence class in $T_1(t)$ it is possible to select a unique representative $(a,b,c,d)$ with $\gcd(a,b)=1$.
Let $(a,b,c,d) \in T_1(t)$, and let $q=\gcd(a,b)$, then it is easy to see that $$(a,b,c,d) \sim \left( \frac{a}{q},\frac{b}{q},qc,qd \right).$$ Moreover, suppose that $(a_1,b_1,c_1,d_1),(a_2,b_2,c_2,d_2) \in T_1(t)$ are such that $$\gcd(a_1,b_1)=\gcd(a_2,b_2)=1,$$ and $$(a_1,b_1,c_1,d_1) \sim (a_2,b_2,c_2,d_2).$$ Then there exists $\gamma = \frac{s}{q} \in \que_{>0}$ with $\gcd(s,q)=1$ such that $$a_1=\frac{q}{s} a_2,\ b_1=\frac{q}{s} b_2,\ c_1=\frac{s}{q} c_2,\ d_1=\frac{s}{q} d_2.$$ Then $s|a_2$, $s|b_2$, and so $s|\gcd(a_2,b_2)=1$, hence $s=1$. Also $q|qa_2=a_1$, $q|qb_2=b_1$, and so $q|\gcd(a_1,b_1)=1$, hence $q=1$. Therefore $$(a_1,b_1,c_1,d_1) = (a_2,b_2,c_2,d_2).$$ This completes the proof.
Therefore, for each $t$ we only need to count the lattices produced by the 4-tuples $(a,b,c,d) \in T_1(t)$ with $\gcd(a,b)=1$. Let $(a,b,c,d)$ be such a 4-tuple, then either $(a,b) = (0,1)$ or $a,b,c,d \neq 0$, since $\gcd(0,b)=b$. Moreover, $a \neq b$ unless $a=b=1$.
First assume $c \neq d$. If $(a,b) \neq (0,1),(1,1)$. By Lemma \[transform1\] and Remark \[sqr\_free\] only the following 4-tuples produce the same lattice $\Lambda(a,b,c,d)$: $$\begin{aligned}
& & (a,b,c,d),\ (-a,-b,-c,-d),\ (-a,-b,c,d),\ (a,b,-c,-d),\\
& & (a,b,-c,d),\ (a,b,c,-d),\ (-a,-b,-c,d),\ (-a,-b,c,-d).\end{aligned}$$ Then each $(a,b,c,d) \in T_1(t)$ gives rise to the four distinct lattices: $$\label{sqrfr_lat1}
\Lambda(a,b,c,d),\ \Lambda(-a,b,c,d),\ \Lambda(b,a,c,d),\ \Lambda(-b,a,c,d),$$ since $\Lambda(a,b,d,c)=\Lambda(-b,a,c,d)$. Also, each of $(0,1,c,d),(1,1,c,d),(a,b,1,1) \in T_1(t)$ gives rise to the following pairs of distinct lattices, respectively: $$\begin{aligned}
\label{sqrfr_lat2}
& & \Lambda(0,1,c,d),\ \Lambda(1,0,c,d); \nonumber \\
& & \Lambda(1,1,c,d),\ \Lambda(-1,1,c,d); \nonumber \\
& & \Lambda(a,b,1,1),\ \Lambda(-a,b,1,1).\end{aligned}$$
Now suppose that $c=d$. Then $$\Lambda(a,b,c,c) = \Lambda(-b,a,c,c),\ \Lambda(-a,b,c,c) = \Lambda(b,a,c,c).$$ Hence, if $(a,b) \neq (0,1),(1,1)$, then each $(a,b,c,c) \in T_1(t)$ gives rise to two distinct lattices, $\Lambda(a,b,c,c)$ and $\Lambda(b,a,c,c)$. Notice also that $$\Lambda(0,1,c,c)=\Lambda(1,0,c,c),\ \Lambda(1,1,c,c)=\Lambda(-1,1,c,c).$$ Hence 4-tuples $(0,1,c,c), (1,1,c,c) \in T_1(t)$ give rise to only one lattice each.
The formula for $\N(u)$, $u \in \D$, of Theorem \[count\] follows. Also notice that if $u \in \zed_{>0} \setminus \D$, then for every divisor $n$ of $u$, either $\alpha_*(t/n)$ or $\beta(n)$ is equal to zero, and so the right hand side of (\[N\_formula\]) is equal to zero by construction. This completes the proof of the theorem.
\[points\] Our problem can be interpreted in terms of counting integral points on certain varieties. Let us say that two points $$\bx = (x_1,x_2,x_3,x_4)^t,\ \bwy = (y_1,y_2,y_3,y_4)^t \in \real^4$$ are equivalent if there exists $U \in GL_2(\zed)$ such that $$U \left( \begin{matrix} x_1&x_3 \\ x_2&x_4 \end{matrix} \right) = \left( \begin{matrix} y_1&y_3 \\ y_2&y_4 \end{matrix} \right).$$ Notice that the number of [*all*]{} full-rank sublattices of $\zed^2$ with determinant equal to $u$ is precisely the number of integral points on the hypersurface $$x_1x_4 - x_2x_3 = u,$$ modulo this equivalence. This number is well known: one formula, for instance, is given by (\[all\_sublattices\]) below. On the other hand, by Lemma \[gauss\], the number of [*well-rounded*]{} full-rank sublattices of $\zed^2$ with determinant equal to $u$ is the number of integral points on the subset of the variety $$x_1x_4 - x_2x_3 = u,\ x_1^2+x_2^2-x_3^2-x_4^2 = 0,$$ defined by the inequality $$2|x_1x_3+x_2x_4| \leq x_1^2+x_2^2,$$ modulo the same equivalence. This makes direct counting much harder, and so our parametrization is quite useful.
Corollaries
===========
The first immediate consequence of Theorem \[count\] is the following.
\[zero\_even\] If $u \in \zed_{>0}$ is odd, then $\N(u) = \N(2u)$.
To demonstrate some examples of our formulas at work, we derive the following simpler looking expressions for the case of prime-power determinants.
\[primep\] Let $p$ be a prime, $k \in \zed_{>0}$. Let $u = p^k$ or $2p^k$. Then $$\N(u) = \left\{ \begin{array}{ll}
0 & \mbox{if $p \equiv 3\ (\md 4)$ and $k$ is odd} \\
1 & \mbox{if $p \equiv 3\ (\md 4)$ and $k$ is even} \\
1 & \mbox{if $p=2$} \\
k+1 & \mbox{if $p \equiv 1\ (\md 4)$}
\end{array}
\right.$$
First assume that $p \neq 2$. Define $t$ as in the statement of Theorem \[count\], then $t=p^k$. If $k$ is even, then by Theorem \[count\] $$\begin{aligned}
\N(u) & = & \beta(p^k) + 4 \sum_{j=1}^{\frac{k}{2}} \alpha_*(p^{k+1-2j}) \beta(p^{2j-1}) + 2 \sum_{j=0}^{\frac{k}{2}-1} \alpha_*(p^{k-2j}) (2\beta(p^{2j})-1) \\
& = & 1 + 2 \sum_{j=0}^{\frac{k}{2}-1} \alpha_*(p^{k-2j}),\end{aligned}$$ since $\beta(p^{2j-1})=0$, and $\beta(p^{2j})=1$ for all $j$. If $p \equiv 3\ (\md 4)$, then $\alpha_*(p^{k-2j}) = 0$ for all $j$. If $p \equiv 1\ (\md 4)$, then $\alpha_*(p^{k-2j}) = 1$ for all $j$, in which case $$\N(u) = 1 + 2 \sum_{j=0}^{\frac{k}{2}-1} 1 = k+1.$$
Next assume $k>1$ is odd. Then, in the same manner as above, $$\begin{aligned}
\N(u) & = & 2 \beta(p^k) + 4 \sum_{j=1}^{\frac{k-1}{2}} \alpha_*(p^{k+1-2j}) \beta(p^{2j-1}) + 2 \sum_{j=0}^{\frac{k-1}{2}} \alpha_*(p^{k-2j}) (2\beta(p^{2j})-1) \\
& = & 2 \sum_{j=0}^{\frac{k-1}{2}} \alpha_*(p^{k-2j}),\end{aligned}$$ which is equal to 0 if $p \equiv 3\ (\md 4)$. If $p \equiv 1\ (\md 4)$, then $$\N(u) = 2 \sum_{j=0}^{\frac{k-1}{2}} 1 = k+1.$$
If $k=1$, then by Theorem \[count\] $$\N(u) = 2 \alpha_*(p) = \left\{ \begin{array}{ll}
0 & \mbox{if $p \equiv 3\ (\md 4)$} \\
2 & \mbox{if $p \equiv 1\ (\md 4)$}.
\end{array}
\right.$$
Now assume that $p=2$, $u=p^k$, and $k>1$: the case $k=1$, i.e. $u=2$ is considered separately in the statement of Theorem \[count\]. If $k$ is even, then $$\begin{aligned}
\N(u) & = & 2 \beta(2^{k-1}) + \beta(2^{k-2}) + 4 \sum_{j=1}^{\frac{k-2}{2}} \alpha_*(2^{k-2j}) \beta(2^{2j-1}) \\
& + & 2 \sum_{j=0}^{\frac{k-4}{2}} \alpha_*(2^{k-1-2j}) (2\beta(2^{2j})-1) = 1,\end{aligned}$$ since $\alpha_*(2^i)=0$ for all $i>1$, and $\beta(2^i)=0$ for all odd $i$. Now let $k$ be odd. Then $$\begin{aligned}
\N(u) & = & \beta(2^{k-1}) + 2 \beta(2^{k-2}) + 4 \sum_{j=1}^{\frac{k-3}{2}} \alpha_*(2^{k-2j}) \beta(2^{2j-1}) \\
& + & 2 \sum_{j=0}^{\frac{k-3}{2}} \alpha_*(2^{k-1-2j}) (2\beta(2^{2j})-1) = 1.\end{aligned}$$ This completes the proof.
In precisely the same manner, we obtain the following formulas for the case when determinant is a product of two odd primes.
\[two\_primes\] If $u=p_1p_2$, where $p_1 < p_2$ are odd primes, then $$\N(u) = \left\{ \begin{array}{ll}
0 & \mbox{if $p_1$ or $p_2 \equiv 3\ (\md 4)$ and $p_2>\sqrt{3}p_1$} \\
2 & \mbox{if $p_1$ or $p_2 \equiv 3\ (\md 4)$ and $p_2 \leq \sqrt{3}p_1$} \\
4 & \mbox{if $p_1$ and $p_2 \equiv 1\ (\md 4)$ and $p_2>\sqrt{3}p_1$} \\
6 & \mbox{if $p_1$ and $p_2 \equiv 1\ (\md 4)$ and $p_2 \leq \sqrt{3}p_1$.}
\end{array}
\right.$$
Direct verification.
The same way one can apply the formulas of Theorem \[count\] to obtain explicit expressions for $\N(u)$ for many other instances of $u$ as well.
Notice that some of the lattices in $\WR(\zed^2)$ come from ideals in $\zed[i]$. Namely, let $u = a^2+b^2 \in \D$ and consider the lattices $\Lambda_1(a,b) = \left( \begin{matrix} a&-b \\ b&a \end{matrix} \right) \zed^2$ and $\Lambda_2(a,b) = \left( \begin{matrix} a&b \\ -b&a \end{matrix} \right) \zed^2$ with $\det(\Lambda_1) = \det(\Lambda_2) = u$. Let $I_1(a,b)$ and $I_2(a,b)$ be the ideals in $\zed[i]$ generated by $a+bi$ and $a-bi$ respectively, then $-b+ai = i(a+bi) \in I_1(a,b)$ and $b+ai = i(a-bi) \in I_2(a,b)$. Hence $I_1(a,b)$ and $I_2(a,b)$ map bijectively onto $\Lambda_1(a,b)$ and $\Lambda_2(a,b)$ respectively under the canonical mapping $x+iy \rightarrow \left( \begin{matrix} x \\ y \end{matrix} \right)$, and $\Lambda_1(a,b) = \Lambda_2(a,b)$ if and only if $b=0$, which can only happen when $u$ is a square. Notice that such representation is only possible for the determinant values $u$ which are also in the minima set $\Mm$; in other words, a full-rank WR sublattice of $\zed^2$ comes from an ideal in $\zed[i]$ if and only if it has an orthogonal basis. It is easy to see that the number of such lattices of determinant $u \in \Mm$, which is precisely the number of ideals of norm $u$ in $\zed[i]$, is equal to $2\alpha(u)$ if $u$ is not a square, and $2\alpha(u) + 1$ if $u$ is a square. With this in mind, we can now state the following immediate consequence of Corollaries \[primep\] and \[two\_primes\].
\[ideals\] If $u \in \D$ is of the form $u = p^k, 2p^k$, where $p$ is a prime, or $u = p_1p_2$ where $\sqrt{3} p_1 < p_2$ are odd primes, then all lattices in $\WR(\zed^2)$ of determinant $u$ come from ideals of norm $u$ in $\zed[i]$.
This of course is not true in general, in fact the class of such lattices coming from ideals in $\zed[i]$ is quite thin. Notice in particular that in order for a lattice $\Lambda \in \WR(\zed^2)$ to come from an ideal of $\zed[i]$ it must first of all be true that $\det \Lambda \in \Mm$, which has density 0 versus the entire determinant set $\D$, which has positive density.
Let $\Pp = \{p_1,p_2,\dots\}$ be the collection of all primes in the arithmetic progression $4n+1$. By Dirichlet’s theorem on primes in arithmetic progressions, $\Pp$ is infinite. For each $p_i \in \Pp$ define $\PP_i = \{p_i^k,2p_i^k\}_{k=1}^{\infty}$. Then $\bigcup_{i=1}^{\infty} \PP_i \subset \D$, and Corollary \[primep\] implies that for each $i$, $$\N(u) = \frac{\log u}{\log p_i} + 1,$$ for each $u \in \PP_i$. In other words, there are infinite sequences in $\D$ on which $\N(u)$ grows at least logarithmically in $u$. For comparison, it is a well known fact (see for instance [@bgruber]) that for any positive integer $u$ with prime factorization $u=q_1^{c_1} \dots q_m^{c_m}$ the number of [*all*]{} full-rank sublattices of $\zed^2$ with determinant $u$ is $$\label{all_sublattices}
F(2,u) = \prod_{j=1}^m \frac{q_j^{c_j+1} - 1}{q_j - 1},$$ which grows linearly in $u$. It is therefore interesting to exhibit sequences of determinant values $u$ for which $\N(u)$ is especially large.
Recall that for an integer $u$, $\tau(u)$ and $\omega(u)$ are numbers of divisors and of prime divisors of $u$, respectively. We can report the following consequence of Theorem \[count\].
\[size\_N\] For each $u \in \zed_{>0}$, $$\label{size_O1}
\N(u) \leq O \left( \tau(u)^2 2^{\omega(u)} \right) \leq O \left( \left( \frac{\sqrt{2} \log u}{\omega(u)} \right)^{2 \omega(u)} \right).$$ Moreover, $$\label{size_O2}
\N(u) < O \left( (\log u)^{\log 8} \right),$$ for all $u \in \D$ outside of a subset of asymptotic density 0. However, there exist infinite sequences $\{ u_k \}_{k=1}^{\infty} \subset \D$ such that for every $k \geq 1$ $$\label{size_O3}
\N(u_k) \geq (\log u_k)^k.$$ For instance, there exists such a sequence with $u_k \leq \exp \left( O (k (\log k)^2) \right)$ and $\omega(u_k) = O(k \log k)$.
Notice that the right hand side of (\[N\_formula\]) is the sum of at most $\tau(u)$ nonzero terms. Combining (\[beta\_bound2\]) with the formula for $\alpha_*$ in section 1, it follows that each of these terms is at most $O \left( \tau(u) 2^{\omega(u)} \right)$. Let $u$ have a prime decomposition of the form $u = p_1^{e_1} \dots p_n^{e_n}$, so $\omega(u)=n$, then: $$\frac{\log u}{n} = \frac{1}{n} \sum_{i=1}^n e_i \log p_i \geq \left( \prod_{i=1}^n e_i \log p_i \right)^{\frac{1}{n}} \geq \left( O \left( \prod_{i=1}^n (e_i+1) \right) \right)^{\frac{1}{n}} = \left( O(\tau(u)) \right)^{\frac{1}{n}}.$$ This proves (\[size\_O1\]), and (\[size\_O2\]) follows from (\[size\_O1\]) combined with Theorems 431 and 432 of [@hardy], which state that the normal orders of $\omega(u)$ and $\tau(u)$ are $\log \log u$ and $2^{\log \log u}$, respectively.
Next, write $p_n$ for the $n$-th prime congruent to 1 mod 4. It is a well known fact that $$\label{n_prime}
p_n = O(n \log n).$$ For each $n \geq 1$, define $v_n = \prod_{i=1}^n p_i^2$. Write $\N_I(v_n)$ for the number of lattices in $\WR(\zed^2)$ with determinant $v_n$ that come from ideals in $\zed[i]$, then $$\label{NI}
\N(v_n) \geq \N_I(v_n) = 2\alpha(v_n)+1 = 3^n.$$
Let $k$ be a positive integer. We want to choose $n$ such that $$\label{nO_1}
\N(v_n) \geq 3^n \geq \left( \log \left( \prod_{i=1}^n p_i^2 \right) \right)^k = 2^k \left( \sum_{i=1}^n \log p_i \right)^k.$$ By (\[n\_prime\]), $$\label{nO_2}
\sum_{i=1}^n \log p_i = \sum_{i=1}^n \log \left( O(i \log i) \right) = \sum_{i=1}^n O \left( \log (i \log i) \right) = \sum_{i=1}^n O \left( \log i \right) \leq O(n \log n).$$ Combining (\[nO\_1\]) with (\[nO\_2\]) and taking logarithms, we see that it is sufficient to choose $n$ such that $$\frac{n}{\log n} \geq O(k),$$ hence we can take $n = O(k \log k)$. Then, by (\[nO\_2\]), for this choice of $n$ we have $$v_n = \exp \left( 2 \sum_{i=1}^n \log p_i \right) \leq \exp \left( O(n \log n) \right) = \exp \left( O (k (\log k)^2) \right).$$ Let $u_k = v_n$ for this choice of $n$, and so $n = \omega(u_k)$. This completes the proof.
Let $v_n$ be as in the proof of Corollary \[size\_N\] above, i.e. $v_n = \prod_{i=1}^n p_i^2$, where $p_1, p_2, \dots$ are primes congruent to 1 mod 4; for instance, the first 9 such primes are 5, 13, 17, 29, 37, 43, 47, 53, 61. For each $k$ choose the smallest $n$ so that $v_n > (\log v_n)^k$, and let $u_k = v_n$ for this choice of $n$. Here is the actual data table for the first few values of the sequence $\{u_k\}$ computed with Maple.
[*$k$*]{} [*$n$*]{} [*$u_k = v_n$*]{} [*$\N(u_k)$*]{} [*$(\log u_k)^k$*]{}
----------- ----------- ------------------------------ ----------------- ----------------------
1 2 $4225$ 9 8.34877454
2 4 $1026882025$ 518 430.5539044
3 7 $5741913252704971225$ 215002 80589.79464
4 9 $60016136730202390980384025$ 14324372 12413026.85
Notice that the choice of $n = O(k \log k)$ as in Corollary \[size\_N\] insures that not just $\N(u_k)$, but even the much smaller $\N_I(u_k)$ (compare for instance the values of $\N(u_k)$ in the table above to $\N_I(u_k) = 3^n$) is greater than $(\log u_k)^k$, and even with this stronger restriction $u_k$ and $\omega(u_k)$ grow relatively slow as functions of $k$.
Counting well-rounded lattices with fixed minimum
=================================================
Let $m \in \zed_{>0}$, then, as stated in [@martinet:venkov], there exist $\left[ \frac{m+1}{2} \right]$ WR lattices $\Lambda$, not necessarily integral, of rank 2 in $\real^2$ with $|\Lambda| = \sqrt{m}$, generated by a minimal basis $\bx,\bwy$ with $0 < \bx^t \bwy \leq \left[ \frac{m-1}{2} \right]$. This information, however, does not lead to an explicit formula for the number of WR sublattices of $\zed^2$ of prescribed minimum. We derive such a formula here.
Let $m \in \zed_{>0}$. Suppose that $\Lambda \in \WR(\zed^2)$ and $|\Lambda|^2=m$, then by Lemma \[gauss\] there exists a representation $\Lambda = \left( \begin{matrix} x_1&y_1 \\ x_2&y_2 \end{matrix} \right) \zed^2$ with $$\label{abcd_min}
\bx = \left(\begin{matrix} x_1 \\ x_2 \end{matrix} \right),\ \bwy = \left(\begin{matrix} y_1 \\ y_2 \end{matrix} \right) \in \zed^2,\ \|\bx\|=\|\bwy\|=m,\ \theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right],$$ where $\theta(\bx,\bwy)$ is the angle between vectors $\bx$ and $\bwy$, since if $\theta(\bx,\bwy) \in \left( \frac{\pi}{2}, \frac{2\pi}{3} \right]$ we can always replace $\bx$ with $-\bx$ or $\bwy$ with $-\bwy$ to ensure that $\theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right]$. Then define $$\C_m = \{ \bx \in \zed^2 : x_2 > 0,\ \|\bx\| = m \}.$$ For each $\bx \in \C_m$ let $$E_m(\bx) = \left\{ \bwy \in \zed^2 : y_2 \geq 0,\ \|\bwy\| = m,\ \theta(\bx,\bwy) \in \left[ \frac{\pi}{3}, \frac{\pi}{2} \right] \right\},$$ and define $\eta_m(\bx) = |E_m(\bx)|$. The following result follows immediately.
\[count\_min\] Let $m \in \Mm$. Let $\N'(m)$ be the number of lattices in $\WR(\zed^2)$ with minimum equal to $m$. Then $$\N'(m) = \sum_{\bx \in \C_m} \eta_m(\bx).$$
Notice that for each $\bx \in \C_m$, $\eta_m(\bx)$ is precisely the number of integer lattice points on the arc of the circle of radius $\sqrt{m}$, bounded by the points $\left( \begin{matrix} \frac{x_1}{2} - \frac{x_2\sqrt{3}}{2} \\ \frac{x_1\sqrt{3}}{2} + \frac{x_2}{2} \end{matrix} \right)$ and $\left( \begin{matrix} -x_2 \\ x_1 \end{matrix} \right)$. The angle corresponding to this arc is $\frac{\pi}{6}$. On the other hand, $\alpha(m)$ as defined by (\[alpha\]) is the number of integer lattice points on the quarter-circle of radius $\sqrt{m}$ centered at the origin and bounded by the points $(\sqrt{m},0)$, $(0, \sqrt{m})$. It is not difficult to see that for every $\bx \in \C_m$, $$\eta_m(\bx) \leq \left\{ \begin{array}{ll}
\alpha(m) & \mbox{if $m$ is not a square} \\
\alpha(m)+1 & \mbox{if $m$ is a square.}
\end{array}
\right.$$ Indeed, for each $\left( \begin{matrix} 0 \\ \sqrt{m} \end{matrix} \right) \neq \left( \begin{matrix} y_1 \\ y_2 \end{matrix} \right) \in E_m(\bx)$, either $\left( \begin{matrix} y_1 \\ y_2 \end{matrix} \right)$ or $\left( \begin{matrix} -y_2 \\ y_1 \end{matrix} \right)$ is contained in the first quadrant and so is counted by $\alpha(m)$; if $m$ is a square, we add one to account for the point $\left( \begin{matrix} 0 \\ \sqrt{m} \end{matrix} \right)$, which is not counted by $\alpha(m)$. In general, these bounds are sharp, for instance $\alpha(13)=\eta_{13}\left( \begin{matrix} 2 \\ 3 \end{matrix} \right)=1$. However, if $\alpha(m)$ is large and the integral points $\bx$ are well distributed on the quarter-circle, it is possible to do better. For this $m$ needs to satisfy certain special conditions. More precisely, write prime decomposition of $m$ as $$m = 2^w p_1^{2l_1} \dots p_s^{2l_s} q_1^{k_1} \dots q_r^{k_r},$$ where $p_i \equiv 3\ (\md 4)$, $q_j \equiv 1\ (\md 4)$, $w \in \zed_{\geq 0}$, $l_i \in \frac{1}{2} \zed_{>0}$, and $k_j \in \zed_{>0}$ for all $1 \leq i \leq s$, $1 \leq j \leq r$. If $l_i \notin \zed$ for any $1 \leq i \leq s$, then $\alpha(m) = 0$, so let us assume that $l_i \in \zed$ for all $1 \leq i \leq s$. Then $$\alpha(m) = \alpha \left( \frac{m}{2^w p_1^{2l_1} \dots p_s^{2l_s}} \right),$$ hence we can assume that $$\label{m_cong1}
m = q_1^{k_1} \dots q_r^{k_r},$$ where $q_j \equiv 1\ (\md 4)$ and $k_j \in \zed_{>0}$ for all $1 \leq j \leq r$. Define $$L(m) = \sqrt{ \frac{\log (q_1 \dots q_r)}{\log \alpha(m)} },$$ and let $$\Mm_1 = \left\{ m \in \Mm : m\ \text{as in (\ref{m_cong1}) and } L(m) \rightarrow 0 \text{ as } m \rightarrow \infty \right\}.$$ A result of Babaev [@babaev] implies that for $m \in \Mm_1$ $$\label{alphap_bound}
\eta_m(\bx) = \frac{\alpha(m)}{3} + O(L(m)\alpha(m)),$$ for each $\bx \in \C_m$. For $m \in \Mm \setminus \Mm_1$ I am not aware of upper bounds on $\eta_m(\bx)$ better than $\alpha(m)$; a classical result of Jarnik on the number of integral lattice points on convex curves [@jarnik] as well as more modern results, for instance of Bombieri and Pila [@pila], imply a general bound on $\eta_m(\bx)$ which is at best $O \left(m^{\frac{1}{4}+\eps} \right)$ for each $\bx \in \C_m$.
In precisely the same manner as Corollary \[gen\] follows from Theorem \[count\], the following is an immediate consequence of Theorem \[count\_min\].
\[gen\_min\] Let $A \in O_2(\real)$. Then for each $m \in \Mm$ the number of full-rank WR sublattices of $A\zed^2$ with squared minimum equal to $m$ is given by $\N'(m)$ as in Theorem \[count\_min\].
\[min\_det\] Notice that even fixing both, the minimum and the determinant, does not identify an element of $\WR(\zed^2)$ uniquely. For instance, if $u$ is representable as a sum of two squares, then the number of lattices $\Lambda \in \WR(\zed^2)$ with $|\Lambda|^2 = \det(\Lambda) = u$ is $$\N_I(u) = \left\{ \begin{array}{ll}
2\alpha(u) & \mbox{if $u$ is not a square} \\
2\alpha(u)+1 & \mbox{if $u$ is a square,}
\end{array}
\right.$$ i.e. precisely the number of lattices in $\WR(\zed^2)$ of determinant $u$ coming from ideals in $\zed[i]$, as defined in section 6. Hence even this number can tend to infinity with $u$.
Zeta function of well-rounded lattices
======================================
Given any finitely generated group $G$, it is possible to associate a zeta function $\zeta_G(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ to it, where the coefficients $a_n$ count the number of its subgroups of index $n$ and $s \in \cee$ (see [@lubot], Chapter 15 for details). Such zeta functions are extensively studied objects, since they encode important arithmetic information about the group in question and often have interesting properties. For example, by Theorem 15.1 of [@lubot] (see also (5) of [@reiner]) $$\label{all_zeta}
\zeta_{\zed^2}(s) = \sum_{\Lambda \subseteq \zed^2} (\det(\Lambda))^{-s} = \sum_{u=1}^{\infty} F(2,u) u^{-s} = \zeta(s) \zeta(s-1),$$ where the sum is taken over all sublattices $\Lambda$ of $\zed^2$ of finite index, $F(2,u)$ is given by (\[all\_sublattices\]), and $\zeta(s)$ is the Riemann zeta function. This $\zeta_{\zed^2}(s)$ is an example of Solomon’s zeta function (see [@reiner], [@solomon]). The identity (\[all\_zeta\]) holds in the half-plane $\Re(s) > 2$, where this series is absolutely convergent, and so the function $\zeta_{\zed^2}(s)$ is analytic (Proposition 1 of [@reiner]). Moreover, in this half-plane $\zeta_{\zed^2}(s) = \sum_{u=1}^{\infty} \sigma(u) u^{-s}$, where $\sigma(u) = \sum_{d|u} d$ (see Theorem 290 of [@hardy]); $\zeta_{\zed^2}(s)$ has a pole at $s=2$.
In this section we study the properties of the partial zeta function corresponding not to all, but only to the [*well-rounded*]{} sublattices of $\zed^2$ as defined by (\[WR\_zeta\]). We also define the Dedekind zeta function of Gaussian integers $\zed[i]$ $$\label{dedekind}
\zeta_{\zed[i]}(s) = \sum_{\Aa \subseteq \zed[i]} \Nn(\Aa)^{-s} = \mathop{\sum_{\Lambda \in \WR(\zed^2)}}_{|\Lambda|^2=\det(\Lambda)} (\det(\Lambda))^{-s} = \sum_{m=1}^{\infty} \N_I(m) m^{-s},$$ where $\N_I(m)$ is as above, and the first sum is taken over all the ideals $\Aa = (a+bi)\zed[i]$ for some $a,b \in \zed$, and $\Nn(\Aa) = a^2+b^2$ is the norm of such ideal. In other words, coefficients of $\zeta_{\zed[i]}(s)$ count the elements of $\WR(\zed^2)$ that come from ideals in $\zed[i]$ while coefficients of $\zeta_{\WR(\zed^2)}(s)$ count all elements of $\WR(\zed^2)$. We also note that $\zeta_{\zed[i]}(s)$ is analytic on $\Re(s) > 1/2$ except for a simple pole at $s=1$ (see Theorem 5 on p. 161 of [@lang]). It is clear that for all $u \in \zed_{>0}$ $$\label{zeta_c}
\N_I(u) \leq \N(u) \leq F(2,u),$$ in other words $\zeta_{\WR(\zed^2)}(s)$ is “squeezed” between $\zeta_{\zed[i]}(s)$ and $\zeta_{\zed^2}(s)$. Moreover, our estimates on coefficients in the previous sections suggest that $\zeta_{\WR(\zed^2)}(s)$ should be “closer” to $\zeta_{\zed[i]}(s)$ than to $\zeta_{\zed^2}(s)$. Theorem \[zeta\], which we will now prove, makes this statement more precise. We start by studying some related Dirichlet series.
\[Dir1\] Let $t = t(u)$ be as in Theorem \[count\]. The Dirichlet series $\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}}$ is absolutely convergent at least in the half-plane $\Re(s) > 1$ with a simple pole at $s=1$. Moreover, when $\Re(s) > 1$ it has an Euler product expansion: $$\label{Euler_prod}
\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}} = \left( 1 + \frac{1}{2^s} + \frac{1}{4^s} \right) \prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1}.$$
First of all, notice that for every $u \in \zed_{>0}$, $$2\alpha_*(t) \leq 2\alpha(t) \leq 2\alpha(u) + 1 \leq \N_I(u) + 1,$$ therefore $\sum_{u=1}^{\infty} 2\alpha_*(t) u^{-s}$ is absolutely convergent at least on the half-plane $\Re(s) > 1$ with at most a simple pole at $s=1$, since $\sum_{u=1}^{\infty} (\N_I(u)+1) u^{-s} = \zeta_{\zed[i]}(s) + \zeta(s)$ is. Next, let $$\alpha_*'(n) = \left\{ \begin{array}{ll}
\alpha_*(n) & \mbox{if $n$ is odd} \\
0 & \mbox{if $n$ is even.}
\end{array}
\right.$$ Notice that $2\alpha'_*$ is a multiplicative arithmetic function, specifically $2\alpha'_*(1)=1$ and $2\alpha'_*(mn) = 2\alpha'_*(m) 2\alpha'_*(n)$ for all $m,n \in \zed_{>0}$ with $\gcd(m,n)=1$. Therefore, by Theorem 286 of [@hardy] the series $\sum_{u=1}^{\infty} 2\alpha'_*(u) u^{-s}$ has the following Euler-type product representation, where $p$ is always a prime: $$\begin{aligned}
\sum_{u=1}^{\infty} 2\alpha'_*(u) u^{-s} & = & \prod_p \left( \sum_{k=0}^{\infty} 2\alpha'_*(p^k) p^{-ks} \right) = \prod_{p \equiv 1 (\md 4)} \left( 1 + 2 \sum_{k=1}^{\infty} p^{-ks} \right) \\
& = & \prod_{p \equiv 1 (\md 4)} \left( \frac{2}{1-p^{-s}} - 1 \right) = \prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1},\end{aligned}$$ whenever this product is convergent. Also notice that since $\alpha_*(2u) = 0$ if $2|u$ and $\alpha_*(2u) = \alpha_*(u)$ if $2 \nmid u$, we have $$\begin{aligned}
\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}} & = & \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \sum_{u=1}^{\infty} \frac{2\alpha_*(u)}{(2u)^{s}} \\
& = & \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \frac{1}{2^s} \left( \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} + \frac{1}{2^s} \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}} \right) \\
& = & \left( 1 + \frac{1}{2^s} + \frac{1}{4^s} \right) \sum_{u=1}^{\infty} \frac{2\alpha'_*(u)}{u^{s}},\end{aligned}$$ which proves (\[Euler\_prod\]) when $\prod_{p \equiv 1 (\md 4)} \frac{p^s+1}{p^s-1}$ is convergent. It is easy to notice that this happens when $\Re(s) > 1$, but $\prod_{p \equiv 1 (\md 4)} \frac{p+1}{p-1}$ diverges, meaning that $\sum_{u=1}^{\infty} \frac{2\alpha_*(t)}{u^{s}}$ must have a pole at $s=1$, and by our argument above we know that it must be a simple pole. This completes the proof.
For the next lemma, let $\B_{\nu}$ be as in (\[prod\_set\]) in section 4.
\[Dir2\] For each $1 < \nu \leq 3^{1/4}$, the Dirichlet series $\sum_{u \in \B_{\nu}} \frac{1}{u^{s}}$ is absolutely convergent in the half-plane $\Re(s) > 1$ with a simple pole at $s=1$ in the sense of (\[pole\_def\]).
First notice that $$\sum_{u \in \B_{\nu}} \left| \frac{1}{u^s} \right| \leq \sum_{n=1}^{\infty} \left| \frac{1}{n^s} \right|,$$ and so must be analytic when $\Re(s) > 1$ with at most a simple pole at $s=1$.
On the other hand, let the Dirichlet lower density of the set $\B_{\nu}$ be defined as $$\liminf_{s \rightarrow 1^+} \frac{\sum_{u \in \B_{\nu}} u^{-s}}{\sum_{u \in \zed_{>0}} u^{-s}} = \liminf_{s \rightarrow 1^+} \frac{1}{\zeta(s)} \sum_{u \in \B_{\nu}} u^{-s}.$$ It is a well known fact (see for instance equation (1.6) of [@ahlswede]) that the Dirichlet lower density of a set is greater or equal than its lower density. Hence, by Theorem \[B\_density\] $$0 < \frac{\nu-1}{2 \nu} \leq \DL_{\B_{\nu}} \leq \liminf_{s \rightarrow 1^+} \frac{1}{\zeta(s)} \sum_{u \in \B_{\nu}} u^{-s},$$ which implies that $\sum_{u \in \B_{\nu}} u^{-s}$ must have a pole of the same order as $\zeta(s)$ at $s=1$. This completes the proof.
\[Dir3\] For each $1 < \nu \leq 3^{1/4}$, Dirichlet series $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$ is absolutely convergent in the half-plane $\Re(s) > 1$, and is bounded below by a Dirichlet series with a pole of order 1 at $s=1$. Moreover, for every real $\eps >0$ there exists a Dirichlet series with a pole of order $1+\eps$ at $s=1$, which bounds $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$ from above.
For each $1 < \nu \leq 3^{1/4}$, define $\chi_{\nu}$ to be the characteristic function of the set $\B_{\nu}$, i.e. for each $u \in \zed_{>0}$, $$\chi_{\nu}(u) = \left\{ \begin{array}{ll}
1 & \mbox{if $u \in \B_{\nu}$} \\
0 & \mbox{if $u \notin \B_{\nu}$.}
\end{array}
\right.$$ Clearly, $\beta_{\nu}(u) \geq \chi_{\nu}(u)$, therefore $$\sum_{u=1}^{\infty} \left| \frac{\beta_{\nu}(u)}{u^{s}} \right| \geq \sum_{u=1}^{\infty} \left| \frac{\chi_{\nu}(u)}{u^{s}} \right| = \sum_{u \in \B_{\nu}} \left| \frac{1}{u^{s}} \right|,$$ which, combined with Lemma \[Dir2\], proves the lower bound of the lemma.
On the other hand, recall that $\beta_{\nu}(u) \leq \Delta(u)$ for all $u \in \zed_{>0}$, where $\Delta(u)$ is Hooley’s $\Delta$-function, as defined in section 2, hence $\sum_{u=1}^{\infty} \left| \beta_{\nu}(u) u^{-s} \right| \leq \sum_{u=1}^{\infty} \left| \Delta(u) u^{-s} \right|$. Hooley’s $\Delta$-function is known to satisfy $$\label{hooley_log}
\sum_{u=1}^{\infty} \left| \frac{\Delta(u)}{u^{s}} \right| \ll_{\eps} \sum_{u=1}^{\infty} \left| \frac{(\log u)^{\eps}}{u^{s}} \right|,$$ for every $\eps > 0$, which is a consequence of Tenenbaum’s bound on the average order of $\Delta(u)$ (see [@tenenbaum], also [@hall]), and so the upper bound of the lemma follows by observing that $\sum_{u=1}^{\infty} (\log u)^{\eps} u^{-s}$ has a pole of order $1+\eps$ at $s=1$. Since $\sum_{u=1}^{\infty} (\log u)^{\eps} u^{-s}$ is absolutely convergent in the half-plane $\Re(s) > 1$, (\[hooley\_log\]) also proves that so is $\sum_{u=1}^{\infty} \frac{\beta_{\nu}(u)}{u^{s}}$.
We are now ready to prove Theorem \[zeta\].
[*Proof of Theorem \[zeta\].*]{} First of all notice that (\[zeta\_c\]) combined with comparison test for series imply that $\sum_{u=1}^{\infty} \N(u) u^{-s}$ has a pole at $s=1$, since $\zeta_{\zed[i]}(s)$ has a pole at $s=1$, and is absolutely convergent, i.e. $\zeta_{\WR(\zed^2)}(s)$ is analytic, for $\Re(s) > 2$, since $\zeta_{\zed^2}(s)$ is analytic when $\Re(s) > 2$. In fact, we can do better. Let $$\beta'(n) = \left\{ \begin{array}{ll}
2\beta(n)-1 & \mbox{if $n$ is a square} \\
2\beta(n) & \mbox{if $n$ is not a square,}
\end{array}
\right.$$ for every $n \in \zed_{>0}$. Notice that for every $u \in \zed_{>0}$, $\N(u)$ can be expressed in terms of the Dirichlet convolution of arithmetic functions $2\alpha_*$ and $\beta'$: $$\N(u) = (2\alpha_* * \beta')(t) + \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) - 2\beta'(t) - \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) \right),$$ where $t=t(u)$, $\delta_1(t)$, and $\delta_2(t)$ are as in Theorem \[count\]. Therefore, by Theorem 284 of [@hardy] $$\begin{aligned}
\label{zeta_prod1}
\zeta_{\WR(\zed^2)}(s) & = & \sum_{u=1}^{\infty} (2\alpha_* * \beta')(t) u^{-s} \nonumber \\
& + & \sum_{u=1}^{\infty} \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) - 2\beta'(t) - \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) \right) u^{-s} \nonumber \\
& = & \left( \sum_{u=1}^{\infty} 2\alpha_*(t) u^{-s} - 2\right) \left( \sum_{u=1}^{\infty} \beta'(t) u^{-s} \right) \nonumber \\
& + & \sum_{u=1}^{\infty} \left( \delta_1(t)\beta(t) + \delta_2(t)\beta\left( \frac{t}{2} \right) \right) u^{-s} - \sum_{u=1}^{\infty} \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) u^{-s},\end{aligned}$$ whenever these three series are absolutely convergent. Now notice that $$\frac{1}{|2^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\delta_1(t)\beta(t)}{|u^s|} \leq 2 \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|},$$ and $$\frac{1}{|4^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\delta_2(t)\beta\left(\frac{t}{2}\right)}{|u^s|} = \frac{1}{|4^s|} \sum_{u=1}^{\infty} \frac{\delta_1(u)\beta(u)}{|u^s|} \leq \frac{2}{|4^s|} \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|},$$ as well as $$\frac{1}{|2^s|} O \left( \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \right) = \frac{1}{|2^s|} \sum_{u=1}^{\infty} \frac{\beta'(u)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\beta'(t)}{|u^s|} \leq \sum_{u=1}^{\infty} \frac{\beta'(u)}{|u^s|} = O \left( \sum_{u=1}^{\infty} \frac{\beta(u)}{|u^s|} \right),$$ whenever $\Re(s) > 0$. Also $$\sum_{u=1}^{\infty} \frac{1+(-1)^t}{2} \beta' \left( \frac{t}{2} \right) u^{-s} = 4^{-s} \sum_{u=1}^{\infty} \beta'(u) u^{-s}.$$ Now the conclusion of Theorem \[zeta\] follows by applying these observations along with Lemmas \[Dir1\], \[Dir3\] to (\[zeta\_prod1\]).
\[ded\_zeta\_sq\] Notice that one implication of Theorem \[zeta\] is that $\N(u)$, the coefficient of $\zeta_{\WR(\zed^2)}(s)$, grows, roughly speaking, like the coefficient of $\zeta_{\zed[i]}(s)^2$, which is $$\sum_{mn = u} \N_I(m) \N_I(n).$$
Finally, we mention that in the same manner one can define zeta functions of well-rounded sublattices of any lattice $\Omega$ in $\real^N$ for any $N$. Studying the properties of these functions may yield interesting arithmetic information about the distribution of such sublattices.
[**Acknowledgment.**]{} I would like to thank Kevin Ford, Preda Mihailescu, Baruch Moroz, Gabriele Nebe, Bogdan Petrenko, Sinai Robins, Eugenia Soboleva, Paula Tretkoff, Jeff Vaaler and Victor Vuletescu for their helpful comments on the subject of this paper. I would also like to acknowledge the wonderful hospitality of Max-Planck-Institut für Mathematik in Bonn, Germany, where a large part of this work has been done.
| ArXiv |
Introduction
============
The idea of current-carrying edge states[@Halperin-82] is one of the major paradigms in the theory of the quantum Hall (QH) effect. For simple filling fractions $\nu=(2m+1)^{-1}$, Wen has shown[@Wen-90; @Wen-91; @Wen-91A; @Wen-91B; @Wen-92rev] that edge modes can be represented as one-component chiral Luttinger liquids, with the universal coupling determined by $\nu$. Within this simple model, controlled calculations are possible. This lead to many beautiful results, including the universal inter-edge tunneling exponent[@Wen-91B; @Kane-Fisher-Tunnel], exact expressions for tunneling conductance, the non-linear tunneling $I$–$V$ curve[@Weiss-Exact; @Fendley-95B], and tunneling noise[@Kane-94A; @Fendley-95C; @QHpers-book].
Experimentally, however, there are more dimensions to this problem. The results of the first pinch-off tunneling experiment[@Milliken-96], where the scaling appeared to be in agreement with theory[@Wen-91B; @Kane-Fisher-Tunnel], have only recently received a partial confirmation[@Maasilta-97; @Turley-98]. Furthermore, in Ref. no scaling was observed at all, and in Ref. the measured tunneling exponent was off by a factor of two. Such discrepancies were attributed in part to edge reconstruction in samples with “soft” confinement[@softedge]. However, the tunneling measurements in cleaved-edge samples[@Chang-96; @Grayson-98], where the confining potential is expected to be sharp, yield tunneling exponents shifted off the predicted values even at the magic filling fractions $\nu=1$, $1/3$.
Previously, much effort[@mechanisms] was dedicated to identify mechanisms leading to (non-universal) corrections to tunneling exponents. In particular, the effect of the long-range Coulomb interaction was analyzed[@Zuelicke-96; @Moon-96; @Oreg-96; @Imura-97] in the geometry of two counterpropagating parallel edges ($\alpha\to0$ in Fig. \[fig:angles\]). In exact analogy with its effect in one-dimensional electron gas[@Emery-1DEG], repulsive Coulomb interaction renormalizes the Luttinger liquid coupling parameter. Thus, a weak impurity-associated inter-edge tunneling becomes a relevant perturbation, so that the current flow (from top to bottom in Fig. \[fig:angles\]) is [*enhanced*]{} at low temperature $T$ and applied voltage $V$. However, the same interaction [ *suppresses*]{}[@Imura-97] the tunneling in the dual configuration, of two semi-infinite non-chiral Luttinger liquids connected by a tunneling point ($\alpha\to\pi$ in Fig. \[fig:angles\]), and the system is pushed towards the insulating regime. This indicates that even the [*sign*]{} of the Coulomb interaction effect on the tunneling exponent is not the same in different geometries.
0.75 0.2pc
The purpose of this work is to analyze in detail the Coulomb interaction effect on the properties of QH tunneling junctions in different geometries. First, we demonstrate that the well-known duality relating weak tunneling and weak backscattering remains exact in the presence of long-range interactions. Then, we focus on scale-invariant [X]{}-shaped constrictions, and calculate the renormalized Luttinger coupling constant $g_\star^2$ (which, in particular, determines the power law dependence of the conductance on $T$ and $V$) as a function of the opening angle $\alpha$ (Fig. \[fig:angles\]). We show that the unscreened Coulomb interaction drives a zero temperature delocalization transition as a function of $\alpha$ in both integer and fractional QH constrictions. In the integer case the transition occurs precisely at the self-dual value $\alpha_c=\pi/2$, independent of the interaction strength. At the fractions $\nu=(2m+1)^{-1}$, the critical angle $\alpha_c$ is non-universal, but its value is always smaller than $\pi/2$. We also analyze the effect of Coulomb interactions in the geometry of cleaved-edge tunneling experiments.
The paper is organized as follows. In Sec. \[sec:effective-tunneling\] we introduce the tunneling action which accounts for the long-range interactions. A general proof of the duality between weak tunneling and weak backscattering is given in Sec. \[sec:duality\]. In Sec. \[sec:self-similar\], we present our results for the renormalized Luttinger coupling $g_\star^2$ in different geometries, and in Sec. \[sec:discussion\] we discuss the implications on tunneling experiments. Related analytic results are collected in Appendices: in App. \[sec:appendix-pi\], the case of $\alpha=\pi$ is solved; in App. \[sec:wiener-hopf\], the Wiener-Hopf technique is used to directly solve the self-dual case $\alpha=\pi/2$, and evaluate the lowest order correction for $|\cos\alpha|\ll1$.
The effective tunneling action {#sec:effective-tunneling}
==============================
Gapless edge excitations $u\equiv u(x,\tau)$ for Laughlin’s QH states with filling fractions $\nu\!=\!(2m+1)^{-1}$ can be described[@Wen-91A; @Wen-91B; @Wen-92rev] by the imaginary-time quadratic action $$\label{eq:edge-action}
{\cal S}_0=\frac{1}{4\pi}\int_0^\beta \!d\tau\int dx\,{\partial_x
u\,(i\partial_\tau{u}+ v \,\partial_x u)},$$ where $x$ is the coordinate along the edge, and $v\equiv v(x)$ is the edge wave velocity. The field $u$ is related to the linear charge density at the edge, $\rho=\sqrt\nu\,\partial_x u/(2\pi)$ (note the unconventional normalization).
Formally, gauge invariance requires that the field $u(x,\tau)$ be treated as a compact boson of radius $R=\sqrt\nu$, [*i.e.*]{}, the values $u$ and $u+2\pi\sqrt\nu$ must be identified. This, however, is [*not*]{} achieved within the usual path integral formalism[@Oreg-95] in a finite geometry if we assume the field $u(x,\tau)$ continuous everywhere along the circumference. Indeed, the equal-time commutation relationship $$[u(x),\,u(x')]=i\pi\sgn (x-x')$$ on the edge of length $L$ implies that the fields $u_0\equiv
u(0,\tau)$ and $u_L\equiv u(L,\tau)$ are canonically conjugated, which contradicts the continuity of the field along the circle. The difference $u_L-u_0$ (proportional to the topological charge associated with the zero mode) is also proportional to the total charge $Q=\sqrt\nu\,(u_L-u_0)/(2\pi)$ accumulated at the edge; only in the absence of tunneling into the edge this charge is a dynamically conserved quantized quantity. The correct zero-mode quantization spectrum can be obtained if we consider the variables $u_0$ and $u_L$ as independent, and write the bare edge action (\[eq:edge-action\]) more explicitly as[@Pryadko-98] $$\begin{aligned}
{\cal S}_0& =& \frac{1}{4\pi}\int_{0\strut}^\beta d\tau\int_0^L
dx\,{\partial_x u\,(i\partial_\tau{u}+ v
\,\partial_x u)}\nonumber \\
& & +\,{1\over8\pi}\int_0^{\beta\strut} d\tau \,(u_L-u_0)\,
i\partial_\tau (u_L+u_0).
\label{eq:finite-size-edge-action}\end{aligned}$$ The boundary term in the second line is added to fix the canonical quantization of the zero mode, and to decouple it from the edge modes with finite momenta.
Since the charge density $\rho$ is expressed linearly in terms of the field $u$, the action remains quadratic[@Wen-92rev; @Moon-96; @Hangmo-96] even in the presence of non-local Coulomb interaction $$\label{eq:coulomb-action}
{\cal S}_1= {\nu\,e^2\over 8\pi^2\varepsilon}\int_0^\beta
\!d\tau\int dx\,dy \,
u'(x)\,V\left(\left|{\bf r}_x-{\bf r}_y\right|\right)\,u'(y),$$ where ${\bf r}_x$ is the actual position of the point $x$ as measured along the edge, and $\varepsilon$ is the dielectric constant of the material. The problem is non-trivial because now both the distance $x$ measured along the edge, and the geometrical distance $\left|{\bf
r}_x-{\bf r}_y\right|$ are important.
The inter-edge tunneling is introduced by the non-linear term $$\label{eq:tunn-action}
{\cal S}_{\rm t}=\int_0^\beta \! d\tau \,
\re\,\lambda\,e^{i g \varphi}, \quad \varphi\equiv u(x_1)-u(x_2);$$ here $g\!=\!\sqrt\nu$ for the quasiparticles’ tunneling between the points $x_1$ and $x_2$ through the QH liquid with the filling fraction $\nu$, or $g\!\rightarrow \!\tilde g=1/\sqrt{\nu}$ for tunneling of electrons through the insulating region. The tunneling amplitude $\lambda$ is set by the details[@Jain-88] of the self-consistent potential near the tunneling point and considered as a phenomenological parameter.
The non-linear tunneling action (\[eq:tunn-action\]) depends on the values of the field $u(x,\tau)$ in the points $x_1$, $x_2$; the values of this field in all other points can be integrated out. Leaving the argument $\varphi$ of the tunneling term as the only independent variable, we can write the most general form of the effective action $$\label{eq:effective-model}
S={T\over4\pi}\sum_{n} %
|\omega_n|
\,{\cal K}(\omega_n)\,|\varphi_n|^2
+\int_0^\beta \! d\tau \,
\re\,\lambda\,e^{i g \varphi(\tau)},$$ where the harmonics $\varphi_n\equiv \int_0^\beta d\tau
\varphi(\tau)\,\exp(-i\omega_n\tau)$ and $\bar\varphi_n\equiv
\varphi_{-n}$ are evaluated at the Matsubara frequencies $\omega_n=2\pi\,n T$. This effective tunneling model is fully characterized by the frequency-dependent coupling ${\cal
K}(\omega_n)$, which contains all relevant information about the form of the interaction potential $V(r)$ and the geometry of the system. Formally, its functional form is defined by the correlator[@Pryadko-98] $$\label{eq:thermal-average}
{\cal K}^{-1}(\omega_n) ={|\omega_n|\over2\pi}
\bigl\langle\left|\varphi_n\right|^2\bigr\rangle_{\lambda=0}.$$
If the coupling ${\cal K}(\omega)$ is independent of the frequency, the effective action (\[eq:effective-model\]) can be visualized as describing an overdamped particle in a periodic (cosine) potential with Ohmic dissipation $\kappa={\cal K}/g^2$; the transport properties for this problem are known exactly[@Weiss-Exact; @Fendley-95B]. In general, however, the exact solution is not available, and we have to rely on the frequency-shell perturbative renormalization group (RG). The main idea is that the non-linear term is irrelevant for large-frequency modes $\varphi(\omega)$, as long as $|\omega|\gg
\lambda$. When such modes are integrated out, the tunneling constant for the remaining slow modes is reduced, $$\label{eq:eff-tunn}
\lambda(\Lambda) =\lambda(\Lambda_0)\left\langle
e^{ig\varphi}
\right\rangle_{\Lambda<\omega<\Lambda_0},$$ or, equivalently, $$-\ln{\lambda(\Lambda)\over \lambda(\Lambda_0)}=
{g^2}\int_{\Lambda_0}^\Lambda {d\omega\over2\pi} \left\langle
|\varphi(\omega)|^2\right\rangle_{\lambda=0}
={g^2}\int_{\Lambda_0}^\Lambda {d\omega\over\omega{\cal K}(\omega)},$$ where we used the definition (\[eq:thermal-average\]). After the frequencies are rescaled to restore the original upper cutoff, we arrive at the usual RG equation $${d\ln\lambda\over d\ln\Lambda} =1-g^2\,{\cal K}^{-1}(\Lambda) \equiv
1-g_\star^2(\Lambda).
\label{eq:define-gstar}$$ The renormalization stops at a lower cutoff scale determined either by the temperature or the applied voltage. Most importantly, for $g_\star^2 >1$, the tunneling amplitude flows to weak coupling as the temperature is lowered, so that the channel along the tunneling current becomes more insulating; for $g_\star^2 <1$ it flows to strong coupling.
It should be pointed out that in the case where ${\cal K}(\omega)$ is [*frequency-independent*]{}, the parameter $g_\star^2$ \[defined in Eq. (\[eq:define-gstar\])\] is a constant, and the effective Euclidean action describing the system can be recast in the simpler form $$S={T\over4\pi}\sum_{n} |\omega_n|\,|\varphi_n|^2
+\int_0^\beta \! d\tau \,
\re\,\lambda\,e^{i g_\star \varphi(\tau)}.
\label{eq:effective-action-disc}$$ Such is indeed the case (for sufficiently small $\omega$) for the scale-invariant models considered in detail in Sec. \[sec:self-similar\]. In this situation, the RG equation leads to the standard result[@Kane-Fisher-Tunnel; @QHpers-book] $$\lambda_{\rm eff}\sim \max(T,\,V)^{g_\star^2-1},
\label{eq:lameff}$$ which can be also obtained by expanding the exact solution[@Weiss-Exact; @Fendley-95B].
Duality between weak tunneling and weak bacscattering {#sec:duality}
=====================================================
The partition function corresponding to the effective action (\[eq:effective-model\]) \[which also describes an overdamped particle in a non-Ohmic dissipative environment, $\kappa(\omega)={\cal K}(\omega)/g^2$\] can be also rewritten[@Schmid-83; @Guinea-85] in terms of the dual variable $\Delta\theta$ with the identical action, up to a replacement ${\cal
K}(\omega_n)\to 1/{\cal K}(\omega_n)$, $g\to 1/g$, and the modified tunneling coefficient $\lambda\to \tilde\lambda$ (which has the meaning of fugacity for the instanton of the original field $\varphi$). In terms of edge modes, this duality[@Fendley-95B; @Weiss-Exact] represents a freedom to describe the same junction in terms of [*weak*]{} tunneling or [*strong*]{} backscattering, and vice versa. The main advantage of the duality is the ability to substitute a problem at [*strong*]{} tunneling with its dual, which can be then accessed perturbatively.
This argument relies heavily on the properties of the effective model (\[eq:effective-model\]), which, in principle, may or may not remain equivalent to the original edge model after the addition of the non-local coupling (\[eq:coulomb-action\]). To illustrate the mutual consistency of the two models, we derive the relationship between the coupling ${\cal K}(\omega)$ in the two tunneling geometries directly, using only the quadratic action ${\cal S}_{\rm q}\equiv {\cal S}_0+{\cal S}_1$.
Consider a field configuration with the boundary conditions fixed as in Fig. \[fig:dual-proof\]a, where $u_i=u_i(\tau)$ are given. Everywhere on the composite contour $C\equiv C_1+C_2$ the action is quadratic, and the corresponding Euler-Lagrange equation is linear, $$\partial_x\left[i\partial_\tau u+v(x)\,\partial_x u+
{\nu\,e^2\over 2\pi\varepsilon}\int_{C} dy\, V(|{\bf r}_x-{\bf
r}_y|)\,\partial_y u\right]=0.$$ The classical solution is uniquely determined by the given values $u_i(\tau)$ of the fields at the endpoints. The quadratic action (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]), evaluated along this classical solution, can be written as $$\begin{aligned}
{\cal S}_{\rm q}[u]&=&{\cal G}[u_1\!-\!u_0,\,u_3\!-\!u_2]
+\int d\tau{(u_1\!-\!u_0)\,i\partial_\tau (u_1\!+\!u_0)\over
4\pi}\nonumber\\
& &
+\int d\tau{(u_3\!-\!u_2)\,i\partial_\tau (u_3\!+\!u_2)\over 4\pi},
\label{eq:formally-evaluated-action}\end{aligned}$$ where ${\cal G}[a,b]$ is a quadratic, non-local in time, and generally very complicated functional of its arguments.
=0.9
The conservation of the total charge $$Q={\sqrt\nu \over 2\pi}(u_3-u_2+u_1-u_0)
\label{eq:charge-conserv}$$ requires that $\varphi\equiv{}u_1-u_0=u_2-u_3$, up to a time-independent constant. Setting the total charge to zero, we can write Eq. (\[eq:formally-evaluated-action\]) as $$\label{eq:formally-evaluated-action-two}
{\cal S}_{\rm q}[\varphi,\Delta\theta]=
{\cal G}[\varphi,-\varphi] -{1\over 2\pi}\! \int\!
d\tau\,{\varphi\,i\partial_\tau \Delta\theta}.$$ where $\Delta\theta\equiv (u_3\!+\!u_2\!-\!u_1\!-\!u_0)/2$. For the tunneling geometry in Fig. \[fig:dual-proof\]b, Eq. (\[eq:charge-conserv\]) implies that the field $u(x,\tau)$ can be chosen continuous everywhere along the combined edge $C_1+C_2$, $\Delta\theta=0$ and hence the effective quadratic action becomes $$S_{\rm q} ={\cal G}[\varphi,\,-\varphi]\equiv {T\over4\pi}\sum_{\omega_n=
2\pi\,nT} |\omega_n| \,{\cal K}(\omega_n)\,|\varphi_n|^2 ,$$ where we introduced the coupling ${\cal K}(\omega)$ as in Eq. (\[eq:effective-model\]).
For the tunneling geometry in Fig. \[fig:dual-proof\]c, the charges in upper and lower areas change with time as a result of the tunneling, and we must keep the field $u(x,\tau)$ discontinuous. The corresponding action becomes $$\tilde S_{\rm q} = {T\over4\pi}\sum_{n} |\omega_n| \,{\cal
K}(\omega_n)\,|\varphi_n|^2 +\omega_n
(\bar\varphi_{n}\Delta\theta_n-\Delta\bar\theta_{n}\varphi_n).$$ (Note that a different choice of $\Delta\theta$, [*e.g.*]{}, $\Delta\theta=u_3-u_0$ or $\Delta\theta=u_2-u_1$, only changes the Euclidean Lagrangian by a total time derivative, thus leaving the action $\tilde{S}_{\rm q}$ invariant.) The field $\varphi$ can be now trivially integrated out, and we arrive at the final form of quadratic action for this geometry, $$\label{eq:dual-effective-action}
\tilde S_{\rm q} = {T\over4\pi}\sum_{\omega_n}
|\omega_n| \,\tilde{\cal K}(\omega_n)\,|\Delta\theta_n|^2,
\quad \tilde{\cal K}(\omega_n) ={1\over{\cal K}(\omega_n)}.$$ This result can be generalized for systems with several junctions, where the coupling ${\cal K}(\omega)$ is replaced by a matrix, which is inverted when all junctions are replaced by their duals[@Pryadko-98].
This simple calculation shows that even in the presence of long-range interactions the duality between weak tunneling and weak backscattering for the model described by Eqns. (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]), (\[eq:tunn-action\]) coincides with the duality between weak and strong coupling for the effective tunneling model (\[eq:effective-model\]), independent of the actual geometry of the system. The only assumption we made is that the geometries in Fig. \[fig:dual-proof\]b and Fig. \[fig:dual-proof\]c should not differ “substantially”, that is, the size of a junction near a saddle point should be sufficiently small ([*e.g.*]{}, compared with a short-distance cut-off length, or, at small enough frequencies, with the wavelength $v/\omega$), so that the Coulomb potential would be the same in the points $u_0,\ldots,u_3$.
Scale-invariant models {#sec:self-similar}
======================
In the absence of long-range forces ($e^2=0$), the properties of any system are determined only by the relative location of the tunneling points along the edges. If such a system has only one tunneling point, in the limit where both contours $C_1$ and $C_2$ in Fig. \[fig:dual-proof\] become infinite, the system would not “know” the difference between the geometries in Fig. \[fig:dual-proof\]b and Fig. \[fig:dual-proof\]c, and the duality implies that the coupling has a universal self-dual value ${\cal K}(\omega)=1$, independent of the actual geometry of the edges. Of course, this statement requires that $\omega\,L/v\gg1$, otherwise one can obtain[@Pryadko-98] for Figs. \[fig:dual-proof\]b and \[fig:dual-proof\]c respectively $$%% \label{eq:K-matr-c1a}
{\cal K}^{(\ref{fig:dual-proof}b)}=
\left[{\cal K}^{(\ref{fig:dual-proof}c)}\right]^{-1}
=%{s_1 s_2-1\over(s_1-1)(s_2-1)}\equiv
%% {e^{\omega (L_1+L_2)}\!-\!1\over(e^{\omega L_1}\!-\!1) (e^{\omega
%% L_2}\!-\!1)}=
{1\over2}\left|\coth\Bigl({\omega L_1\over 2v}\Bigr)
+\coth\Bigl({\omega L_2\over 2v}\Bigr)\right|,$$ where $L_i$ is the length of the contour $C_i$, and a uniform edge velocity $v(x)=v$ is assumed for simplicity.
In the presence of Coulomb interactions, the functional form ${\cal
K}(\omega)$ has been previously found[@Moon-96; @Oreg-96; @Imura-97] only for two [*parallel*]{} edges ($\alpha\to 0$ or $\alpha=\pi$ in Fig. \[fig:angles\]), where the translational symmetry of the quadratic part of the action is restored. In any other geometry the distance $|x-y|$ measured along the edges, and the geometrical distance $R_{xy}\equiv |{\bf r}_x-{\bf r}_y|$ in Eq. (\[eq:coulomb-action\]) are no longer equivalent, and an analytic computation of the average (\[eq:thermal-average\]) with “non-interacting” quadratic action ${\cal S}_0+{\cal S}_1$ becomes virtually impossible.
Some simplification can be achieved for an idealized [X]{}-shaped geometry (see Fig. \[fig:angles\]), which can be also introduced as the zero-bias limit of the edges in a vicinity of a saddle point with the opening angle $\alpha$. For the special case of unscreened Coulomb potential, $$\label{eq:coulomb}
V(R)=\left({R^2+a^2}\right)^{-1/2},$$ the long-range interaction term (\[eq:coulomb-action\]) scales the same way as the local potential (velocity) term in Eq. (\[eq:edge-action\]). Then, if the edge velocity $v(x)=v$ is coordinate-independent the action becomes scale invariant for a sufficiently small short-distance cutoff $a$. This implies that the function ${\cal K}_\alpha(\omega)$, for a given opening angle $\alpha$, can depend on the frequency at most logarithmically. In this regime the geometry of the edges and the tunneling properties of the junction \[[*i.e.*]{}, the function ${\cal K}_\alpha(\omega)$\] are fully determined by the angle $\alpha$ and the dimensionless coupling constant[@Moon-96] $$\chi\equiv {\nu\,e^2/(\pi\hbar
v\varepsilon)}\label{eq:coupling-constant}.$$
The duality discussed in the previous section implies that ${\cal
K}_{\pi-\alpha}(\omega)={\cal K}^{-1}_{\alpha}(\omega)$, for given values of the coupling constant $\chi$ and the cut-off scale $a$. Therefore, in the self-dual geometry at $\alpha=\pi/2$, we expect $K_{\pi/2}=1$ exactly, independent of the form or the strength of the interaction potential $V(x)$.
To rewrite more explicitly the general Coulomb action (\[eq:coulomb-action\]) for the infinite geometry in Fig. \[fig:angles\], let us introduce the coordinate $x$ along each edge, with the origin at the tunneling point and positive direction to the right. Then the charge densities along the top and the bottom boundaries are respectively $\rho_1(x)=\sqrt\nu\partial_x\,u_1(x)/(2\pi)$ and $\rho_2(x)=-\sqrt\nu\partial_x\,u_2(x)/(2\pi)$ (the sign in the second expression differs because the coordinate is now chosen in the direction opposite to the edge velocity). The Coulomb part of the action becomes $${\cal S}_1= {\chi\over8\pi}\!\int\!
d\tau\!\int_{-\infty}^\infty\!\! dx\,dy \! \sum_{i,j=1,2}(-1)^{i+j}
\partial_x u_iV_{ij}(x,y)\partial_y u_j,$$ where the potential $V_{ij}(x,y)\equiv V\biglb(|{\bf r}_{i}(x)-{\bf
r}_{j}(y)|\bigrb)$ denotes the interaction energy between unit charges at the points $x$ and $y$ at the edges $i$ and $j$ respectively, and we changed the units of distance: from now on $v=1$. For symmetric geometries $V_{ij}(x,y)=V_{ij}(y,x)$, $V_{11}(x,y)=V_{22}(x,y)$, and the obtained expression can be diagonalized by introducing the symmetric and antisymmetric combinations $\varphi=u_1-u_2$, $\vartheta=u_1+u_2$. The quadratic part (\[eq:finite-size-edge-action\]), (\[eq:coulomb-action\]) of the Euclidean action becomes $$\begin{aligned}
\lefteqn{{\cal S}_{\rm q}={T\over8\pi}\sum_n\biggl\{\int dx\left[
2\omega_n\bar\varphi(x)\,\vartheta'_x+|\varphi'_x|^2
+|\vartheta'_x|^2\right]
\nonumber} & & \\
& &\;\, + {\chi\over2} \int \!dx\!\int
\!dy\,\left[\bar\varphi'_x\,V_+(x,y)\,\varphi'_y
+\bar\vartheta'_x\,V_-(x,y)\,\vartheta'_y\right]\biggr\},
\label{eq:symmetrized-action}\end{aligned}$$ where $V_\pm(x,y)\equiv V_{11}(x,y)\pm V_{12}(x,y)$ and the coordinate integrations are performed along the entire real axis. Note that the first term of the integrand is not written as $\omega_n(\bar\varphi\,\vartheta'_x-\bar\vartheta\,\varphi'_x)$ as would be expected from the action (\[eq:edge-action\]); the integrand in Eq. (\[eq:symmetrized-action\]) differs by a full spatial derivative, exactly equivalent to the surface term in the second line of Eq. (\[eq:finite-size-edge-action\]).
The interaction potential is always an even function with respect to simultaneous reflection of both coordinates, $V_\pm(x,y)=V_\pm(-x,-y)$, and the fields $\varphi=\varphi_s+\varphi_a$ and $\vartheta=\vartheta_s+\vartheta_a$ can be separated into symmetric ($s$) and antisymmetric ($a$) components. The first term of the action (\[eq:symmetrized-action\]) couples only the components of two fields with the opposite symmetry: $\varphi_s$ with $\vartheta_a$, and $\varphi_a$ with $\vartheta_s$. Since the tunneling term depends on the field $\varphi(0)=\varphi_s(x=0)$ only, the components $\varphi_a(x)$ and $\vartheta_s(x)$ decouple and can be integrated out independently of the value $\varphi(0)$. In the following, we shall presume that this symmetrization has been done, and use $$\varphi(x)=\varphi(-x),\quad \vartheta(x)=-\vartheta(-x),
\label{eq:field-symmetry}$$ with the indices “$s$” and “$a$” dropped for convenience.
Exactly-solvable example
------------------------
To illustrate the properties of the symmetrized action (\[eq:symmetrized-action\]), consider a model problem where the interaction happens only between the points at equal distance from the origin, $$\begin{aligned}
(\chi/2)\, V_{11}(x,y)&=&v_0\delta(x-y)+v_1\delta(x+y),\\
(\chi/2)\, V_{12}(x,y)&=&v_2\delta(x-y)+v_3\delta(x+y), \end{aligned}$$ where the velocity $v_0$ (measured in units of bare velocity $v$) denotes the strength of additional interaction at the same edge, $v_1$ and $v_2$ denote the interaction between the neighboring edges (left–right and top–bottom), while $v_3$ denotes the interaction between the points at the opposing edges. (Physically, this set of interactions corresponds to four locally-interacting chiral edges, running along the surface of a semi-infinite cylinder and meeting in the tunneling point at its near end).
With interaction of this simple form we can use the symmetry properties (\[eq:field-symmetry\]), and the quadratic action (\[eq:symmetrized-action\]) becomes entirely [*local*]{}, $${\cal S}_{\rm q}={T\over8\pi}\sum_n\int dx\left[
2\omega_n\bar\varphi(x)\,\partial_x\vartheta+ v_\varphi
|\partial_x\varphi|^2+ v_\vartheta |\partial_x\vartheta|^2 \right],$$ with $v_{\varphi,\vartheta}=1+v_0-v_3\pm (v_1-v_2)$. Now the field $\vartheta(x)$ can be trivially integrated out, and Eq. (\[eq:thermal-average\]) gives $${\cal K}^{-1}(\omega_n)={1+v_0-v_3+ (v_1-v_2)\over 1+v_0-v_3-
(v_1-v_2)}.$$ Clearly, under interchange $v_1\leftrightarrow v_2$ this expression goes to its inverse according to the duality relation derived in Sec. \[sec:duality\], and ${\cal K}(\omega_n)=1$ for the self-dual case $v_1=v_2$ where all edges are equivalent.
Coulomb interactions near a saddle point {#sec:coulomb-wedge}
----------------------------------------
Now let us consider more realistic long-distance interactions in the edge geometry shown in Fig. \[fig:angles\]. We write the intra- and inter-edge interaction potentials $$\begin{aligned}
V_{11}(x,y)&=&\tf(xy)V(x-y) +\tf(-xy)V(R_\alpha),\\
V_{12}(x,y)&=&\tf(xy)V(R_\alpha) +\tf(-xy)V(x-y),\end{aligned}$$ where the bulk distance $R_\alpha\equiv
({x^2+y^2-2xy\cos\alpha})^{1/2}$, $\tf(x)$ is the usual step function, $\tf(x)=1$ for $x>0$ and $\tf(x)=0$ otherwise, and $V(x)$ is, [ *e.g.*]{}, the Coulomb potential (\[eq:coulomb\]). The resulting effective action has the form (\[eq:symmetrized-action\]), with $$\begin{aligned}
\label{eq:v-plus}
V_+&=&V(x-y) +V(R_\alpha),\\
V_-&=&\left[V(x-y) -V(R_\alpha)\right]\sgn (xy).
\label{eq:v-minus}\end{aligned}$$
In the limit $\alpha=0$, $R_\alpha=|x-y|$, the antisymmetric part of the potential vanishes, $V_-(x,y)=0$, while $V_+(x,y)=2V(x-y)$, and we obtain the usual translationally-invariant action for two parallel edges. Integrating out the field $\vartheta$ and diagonalizing the remaining part of the action with the help of Fourrier transformation, we use Eq. (\[eq:thermal-average\]) to calculate the coupling, $$\label{eq:coupling-zero-angle}
{\cal K}_{\alpha=0}^{-1}(\bar\omega)
={2\bar\omega\over\pi}\int_0^\infty {d\zeta\over
\bar\omega^2+\zeta^2\,[1+2\chi\,K_0(\zeta)] },$$ where the Fourrier-transformed Coulomb potential (\[eq:coulomb\]), $V(\zeta)=2K_0(\zeta)$, is expressed in terms of the modified Bessel function $K_0$, and the reduced frequency $\bar\omega=a\omega/v$. Performing the integration with logarithmic accuracy, we obtain, in agreement with Refs. $${\cal K}_{\alpha=0}= \left[1+2\chi\ln\left({
2\sqrt{2\chi}\,e^{-\gamma}\over\bar\omega }\right)\right]^{1/2}
+{\cal O}\bigl(|\ln \bar\omega|^{-1/2}\bigr),
\label{eq:alpha0}$$ where $\gamma\approx 0.577$ is the Euler constant.
The case $\alpha=\pi$ corresponds to two semi-infinite non-chiral Luttinger liquids connected by a tunneling point ($\alpha\to\pi$ in Fig. \[fig:angles\]); by duality we expect[@Imura-97] ${\cal K}_{\alpha=\pi} =1/{\cal K}_{\alpha=0}$. This expression is proved again, specifically for this geometry, in Appendix \[sec:appendix-pi\].
We argued that in the self-dual case $\alpha=\pi/2$, ${\cal
K}(\omega)=1$ identically, independently of the properties of the potential $V(x)$, as long as it is appropriately regularized at short distances. We have also constructed a direct analytical solution for this case. The major simplification comes from an observation that the potential $V(R_{\pi/2})= V(\sqrt{x^2+y^2})$ is a symmetric function of $x$ and $y$ independently; the corresponding contribution vanishes from the action (\[eq:symmetrized-action\]) by the symmetry (\[eq:field-symmetry\]). As a result, only the potentials $V(x\pm y)$ with the distance measured along the edge enter the extremum equations, and these equations can be solved exactly using the Wiener-Hopf method, as detailed in Appendix \[sec:wiener-hopf\]. This direct solution confirms the universal result ${\cal
K}_{\alpha=\pi/2}=1$. In addition, the explicitly found extremum configuration of the fields $\varphi(x)$, $\vartheta(x)$ is used to get a perturbative expression for ${\cal K}_\alpha(\omega)$ near the self-dual point $\alpha_0=\pi/2$. This yields (see Appendix B) $$\label{eq:perturbation}
K_\alpha(\omega\to0)\approx 1+{\cal N}(\chi)\,\chi\cos\alpha,
\quad |\cos\alpha|\ll 1,$$ where ${\cal N}(\chi)$ is [*independent of*]{} $\omega$. In the limit of weak Coulomb interactions, ${\cal N}(\chi\to0)\approx1.51$, while ${\cal N}(\chi=1.0)\approx0.21$.
To get a handle on the dependence of the coupling ${\cal
K}_\alpha(\omega)$ on the parameters and the cut-off scales, we have also evaluated the average (\[eq:thermal-average\]) numerically for the quadratic action (\[eq:symmetrized-action\]) with the Coulomb potential (\[eq:coulomb\]) at different frequencies $\omega$, and for different values of the angle $\alpha$ and the dimensionless coupling constant $\chi$.
To perform this calculation we wrote a discretized version of the quadratic action (\[eq:symmetrized-action\]) in terms of lattice values $\varphi(x_n)$ and $\vartheta'(x_n)$, $0<n<N-1$, and then integrated out the values of the fields away from the origin, which only required inverting two $N\times N$ matrices. In addition to the cut-off distance $a$ in Eq. (\[eq:coulomb\]), the discretization involved two explicit cut-off scales: the total system size $L$ and the lattice grid size $h=L/N$. The calculations were performed in the regime $h\ll a\ll L$; the results are independent of these cut-off scales in the frequency range $h\ll v/\omega\ll L$. These inequalities substantially limited the dynamical range where the results are accurate.
Typical results of the calculations are illustrated in Fig. \[fig:K-frequency\] and Fig. \[fig:K-angle\]. The curves in Fig. \[fig:K-frequency\] with marked values of $\cos(\alpha)$ show superimposed values ${\cal K}_\alpha(\bar\omega)$, ${\cal
K}^{-1}_{\pi-\alpha}(\bar\omega)$ calculated with the lattice size $N=1600$, for cut-off parameters $a=0.05$, $0.1$. The deviation betwen the curves shows that our discretization violated the self-duality of the problem at both large and small cutoff scales. Nevertheless, as illustrated in Fig. \[fig:K-angle\], the self-duality holds with a very good numerical accuracy near the middle of the dynamical range, $a\,\omega/v\sim 0.1$.
As indicated by finite-size scaling analysis of our data (not shown), at small enough $\omega$, ${\cal K}_\alpha(\omega)$ saturates to a frequency-independent value in the range $0<\alpha<\pi$. This behavior is consistent with the small-angle expansion (\[eq:perturbation\]). In addition, Fig. \[fig:K-angle\] indicates that Eq. (\[eq:perturbation\]) provides a good approximation to ${\cal K}_\alpha(\omega)$ in a rather wide range of $\alpha$. For small $\alpha\ll1$, as the frequency is reduced, the numerical values ${\cal K}_\alpha(\omega)$ seem to closely follow the logarithmically divergent line (\[eq:alpha0\]), but eventually cross over to a constant value ${\cal
K}_\alpha(\omega=0,\chi)$, which (logarithmically) depends on the angle and the cut-off scale $a$.
=
=
Coulomb interactions in the cleaved edge geometry {#sec:Grayson}
-------------------------------------------------
Here we consider the effect of long-range interactions in the cleaved-edge geometry[@Chang-96; @Grayson-98], where the tunneling happens between a three-dimensional metal and the edge of a 2DEG, located in the plane perpendicular to the surface of the metal. It is believed that the tunneling in these experiments is dominated by localized “hot” spots or impurities. Chamon and Fradkin[@Chamon-97] demontstated that in the absence of interactions, a point contact between a 3D metal and a QH edge with the filling fraction $\nu$ is equivalent to a point tunneling junction between such an edge and an ideal non-interacting $\nu=1$ edge; furthermore, they mapped this latter problem to that of tunneling between two identical edges with filling fractions $\nu_*=2\nu/(1+\nu)$.
=
The effect of the Coulomb interaction in this setup is limited to the chiral Luttinger liquid, the “real” quantum Hall edge, while the Fermi-liquid nature of quasiparticles in the metal imply that they remain non-interacting for the purposes of tunneling measurements. The metallic surface only provides additional screening charges, which modify the form of the interaction potential $V\left(|{\bf r}_x-{\bf
r}_y|\right)$. Assuming characteristic frequencies at the edge are small compared with the plasma frequency of electrons in metal (which is always true for a good metal), the retardation can be neglected, and the modified interaction potential is obtained simply by adding the appropriate image charges.
The quadratic part of the action for the translationally-invariant geometry shown in the left part of Fig. \[fig:cleaved\] ([*i.e*]{}., the case $\alpha=0$) is obtained by combining Eq. (\[eq:finite-size-edge-action\]) with the Coulomb energy $$\label{eq:cleaved-action-coulomb}
{\cal S}_1=
{\chi\over 8\pi}\int_0^\beta d\tau\int_{-\infty}^\infty \!\!dx\,dy
\,\partial_x u\,\hat V(x-y)\, \partial_y u ,$$ where $\hat V(x)\equiv V(x)-V(\sqrt{x^2+4a^2})$ is corrected for the image potential, and the units of length are again chosen so that the edge velocity $v=1$. Because we work with the chiral field now, the surface term in the second line of Eq. (\[eq:finite-size-edge-action\]) is absolutely essential even in an infinite geometry. To properly account for this term, we formally separate the field $u=\phi+\theta$ into its symmetric $\phi(x,\tau)=\phi(-x,\tau)$ and antisymmetric $\theta(x,\tau)=-\theta(-x,\tau)$ compontents; then the surface term can be absorbed after an integration by parts, and the action (\[eq:finite-size-edge-action\]) becomes $$\label{eq:cleaved-symmetrized-quadratic}
{\cal S}_0\!=\!{1\over 4\pi}\! \int%_0^\beta
\! d\tau\!\int_{-\infty}^\infty\!\! dx \left[
2i\partial_\tau\phi\,\partial_x\theta+ (\partial_x\phi)^2+
(\partial_x\theta)^2\right].$$ This transformation is equivalent to “folding” the chiral edge in half, which produces two non-chiral fields defined on a semiaxis, and simultaneously eliminates the zero mode and associated subtleties. The translationally invariant action can be now diagonalized by a Fourrier transformation, and, after integrating out the fluctuations away from the origin, we obtain the single-edge contribution to the quadratic part of the effective action, $$\label{eq:half-action}
{\cal S}^{(1)}_{\rm q}={T\over 2\pi}\sum_n |\omega_n| \,\hat{\cal
K}(\omega_n)\,|\phi_1|^2,$$ where $\phi_1\equiv u(0)=\phi(0)$ by definition, and $$\begin{aligned}
\label{eq:cleaved-parallel-K}
\hat{\cal K}^{-1}(\omega)&=&{2|\omega|\over\pi}\!\int_0^\infty\!\!
{dk\,Z(k)\over \omega^2+k^2\,Z^2(k)}, \\ \nonumber
%% {\rm where}\;\;
Z(k)&=&1+{\chi\over2}\,\hat V(k).\end{aligned}$$ The argument[@Chamon-97] that a point contact with a metal is equivalent to that with a non-interacting $\nu=1$ edge holds independently of the interactions affecting the “real” edge. Therefore, the full effective action can be written as $$\label{eq:full-asymmetric-action}
{\cal S}={T\over2\pi}\sum_n |\omega_n|
\left({ \hat{\cal K}\,|\phi_1|^2
\! +|\phi_2|^2}\right)
\!+\!\int\! d\tau\,\re\,\lambda\,e^{i(g \phi_1-\phi_2)},$$ where we used $\hat{\cal K}=1$ for the auxiliary $\nu=1$ edge. The canonical form (\[eq:effective-model\]) of the tunneling action can be obtained by introducing the tunneling degree of freedom, $\varphi=g
\phi_1-\phi_2$, with the corresponding effective coupling ${\cal
K}_{\rm eff}$ calculated, [*e.g.*]{}, using the average as in Eq. (\[eq:thermal-average\]). As before, the resulting model describes an overdamped particle in a washboard potential; the corresponding non-Ohmic “friction” coefficient $$\kappa_{\rm eff}(\omega)\equiv{{\cal K}_{\rm eff}\over g_{\rm
eff}^2}={2\over g^2/\hat{\cal K}(\omega)+1}.
\label{eq:friction-cleaved}$$ In the non-interacting limit $\hat{\cal K}(\omega)=1$ this expression safely goes into the result[@Chamon-97] obtained by a different method.
Notice that the long-distance part of the Coulomb potential $\hat V$ in Eq. (\[eq:cleaved-action-coulomb\]) is screened by the metallic surface. Then, at sufficiently small frequencies, $a\omega\ll v_{\rm r}\equiv Z(0)\,v$, the momentum dependence of the coefficient $Z(k)$ can be ignored, and the integral (\[eq:cleaved-parallel-K\]) gives precisely the non-interacting coupling, $\hat{\cal K}=1$. This is not at all surprising, since the interaction happens within a single chiral edge, and its long-range part (most dangerous at small frequencies) is screened. As usual[@Volkov-88], the only effect of the additional interaction in this chiral system is the velocity renormalization, $v\to v_{\rm r}$.
The translational symmetry is lost for the “wedge” geometry shown in the right part of Fig. \[fig:cleaved\]. The Coulomb part of the corresponding action can be written in the form (\[eq:cleaved-action-coulomb\]) with the potential $\hat
V(x-y)\to%\hat V(x,y)=
V_-(x,y)$ given by Eq. (\[eq:v-minus\]). In the limit $\alpha\to0$, the potential $V_-(x,y)$ vanishes identically, and hence $\hat{\cal K}(\omega)=1$ in this case as well.
At general values of $\alpha$ we again use the “folding” trick by introducing symmetric and antisymmetric variables $\phi$, $\theta$. Up to an overall coefficient, the resulting action looks like Eq. (\[eq:symmetrized-action\]), with the exception that both components $\phi$ and $\theta$ couple with the [*same*]{} potential $V_-(x,y)$. The most prominent difference is that at $\alpha=\pi/2$ the symmetry no longer leads to a cancellation of the part $V(\sqrt{x^2+y^2})$ of the total potential, and the effect of the long-distance interactions is no longer trivial, $\hat{\cal
K}_{\pi/2}(\omega)\neq1$. Again, this comes as no surprise, since there is no self-duality in this geometry.
Finally, in the limiting case $\alpha=\pi$, the potential $V_-(x,y)$ becomes an even function of each argument; as a result, the coupling with the symmetric field $\phi$ (antisymmetric derivative $\partial_x\phi$) vanishes by symmetry. Up to an overall coefficient, the resulting action is identical to that considered in Appendix \[sec:appendix-pi\], and we obtain \[note that the extra coefficient was already accounted for in the corresponding effective action, [*cf*]{}. Eqns. (\[eq:half-action\]) and (\[eq:effective-model\])\], $$\hat{\cal K}_{\alpha=\pi}(\omega)={\cal K}_{\alpha=\pi}(\omega)
={2|\omega|\over \pi}\!\int_{0}^\infty\!\! {dk\over \omega^2+k^2
\biglb(1+\chi\,V(k)\bigrb)}.$$ This result is quite intuitive: metallic screening becomes non-effective in the case where a wire is perpendicular to the conducting surface.
Our calculations imply that the tunneling exponent is modified by the Coulomb interaction only if the edge is bent near the tunneling point. In an ideal sample, the edge runs along a straight line parallel to the surface of the metal, and long-range interactions do not modify the tunneling exponents. In any real sample, however, imperfections near the tunneling point always reduce the effective coupling ${\cal
K}(\omega)$, or, equivalently, systematically [*increase*]{} the tunneling exponent in Eq. (\[eq:lameff\]). Nevertheless, we do not believe this effect would be sufficient to explain a $10\%$ increase of the tunneling exponent observed[@Grayson-98] by Grayson [*et al*]{}. near $\nu=1$: cleaved-edge samples are characterized by sharp confinement and large drift velocities, meaning that the corresponding dimensionless coupling constant $\chi$ \[see Eq. (\[eq:coupling-constant\])\] is small.
Discussion {#sec:discussion}
==========
We have shown that the effect of long-range interactions on transport through a QH tunneling junction depends crucially on its geometry. In particular, in a self-similar [X]{}-shaped junction (see Fig. \[fig:angles\]) characterized by an opening angle $\alpha$, unscreened Coulomb interactions only renormalize the effective Luttinger-liquid exponent, $$g_\star^2=g^2/{\cal K}_\alpha(\omega=0,\,\chi),$$ where $g^2=1/\nu$ for electron tunneling between the edges of 2DEGs with Laughlin fractions $\nu$. Therefore, the renormalized exponent depends non-universally on the angle $\alpha$ and the dimensionless Coulomb interaction strength $\chi$.
This implies that the system should exhibit a zero-temperature delocalization transition at a critical angle characterized by $g_\star^2=1$. This is in contrast with the transport properties expected in the absence of long range interactions, which are exclusively determined by the filling fraction $\nu$. For integer QH systems with $\nu=1$, the transition always corresponds to a self-dual geometry, [*i.e.*]{}, $\alpha_c=\pi/2$, independently of the details of the interaction. In fractional QH constrictions, however, the transition (if any) occurs at a non-universal critical angle $\alpha_c<\pi/2$, such that ${\cal K}_{\alpha_c}(0,\chi)=\nu^{-1}$.
Properties of all charge transfer processes through the junction are defined by the parameter $g_\star$ in the effective action (\[eq:formally-evaluated-action-two\]), which determines the tunneling exponents[@Wen-91B; @Kane-Fisher-Tunnel] \[see Eq. (\[eq:lameff\])\], the form of the non-linear $I$–$V$ curve[@Weiss-Exact; @Fendley-95B], as well as the tunneling noise [@Kane-94A; @Fendley-95C; @QHpers-book]. In the limit of weak tunneling, the quantization of transferred charge is ultimately determined by gauge invariance, and a shot noise measurement would show current transferred by unit charges. However, the shot noise measured in the opposite, strongly coupled limit (reached, [*e.g.*]{}, by driving a large tunneling current through the junction), is set[@Sandler-noise] by the instanton charge for the effective tunneling action (\[eq:formally-evaluated-action-two\]). The value of this charge is determined solely by the value of $g_\star$. Hence, in this regime a noise measurement would show a non-universal charge $$e_\star/e=1/g^2_\star=\nu\,{\cal K}_\alpha(0,\,\chi),$$ clearly an interaction effect.
The described situation applies to ideal systems without screening. More realistically, Coulomb interactions are screened at some finite length $\xi$. Then, for a junction with finite opening angle, $|\cos\alpha|<1$, the correction to tunneling exponents always vanishes in the static limit, ${\cal K}^{\rm (scr)}_\alpha(0)=1$, even though it may be significant at larger frequencies, $\omega\gtrsim
v/\xi$ (this corresponds to a temperature $0.1$ K for $\xi=1\;\,\mu$m and $v=10^{-7}$ cm/s). Consequently, a system at a fractional $\nu$ with originally metallic behavior would eventually localize at small enough temperatures.[@Moon-96; @Imura-97] Contrarily, the interaction-induced flow in an integer junction would gradually stop without changing its direction.
For an [X]{}-shaped junction with a given opening angle $\alpha$, the magnitude of the renormalization parameter ${\cal
K}_\alpha(0,\chi)$ is determined by the value of the dimensionless Coulomb interaction constant (\[eq:coupling-constant\]), which, in turn, is defined by the edge wave (drift) velocity. For cleaved-edge samples, edge magnetoplasmon velocities have been measured[@velocity-cleaved] by Ashoori [*et al.*]{}, yielding $v\sim 10^8$ cm/s which corresponds to $\chi\sim 0.05$. On the other hand, edge electric fields equivalent to drift velocities as small as $v\sim 10^6$ cm/s have been measured by Maasilta and Goldman[@Maasilta-97], who analyzed discrete energy levels of a quantum antidot. This value of velocity results in a relatively large coupling constant value $\chi\sim5$.
We must point out, however, that our discussion of Coulomb interaction effects was based on a single-mode sharp edge, which implies large confining electric fields of order ${\cal E}\sim E_g/(e\ell)$, where $E_g$ is the energy gap associated with the incompressible QH state, and $\ell$ is the magnetic length. Using the drift velocity $v=c
{\cal E}/B$, we obtain $$\chi={\nu \,e^2\over \pi\varepsilon\hbar v}
\sim\left({\nu\,e^2\over \pi\varepsilon\ell}\right)\,E_g^{-1},$$ which, for a typical QH sample, leads to $\chi\lesssim 1$. Samples with much larger values of the Coulomb coupling are likely to have a tendency to edge reconstruction. This would lead to additional polarization at the edge due to neutral modes, and, consequently, a partial screening of Coulomb interaction.
Therefore, to observe the predicted effects, samples with well-defined, but not too sharp edges are necessary. This excludes the cleaved-edge samples (where the drift velocity $v$ is large), as well as the samples with electrostatically-defined geometry (where confinement tends to be soft). The best choice would therefore be a Hall bar with lithographically defined [X]{}-shaped constriction and a narrow local gate to fine-tune the tunneling. For a given base temperature $T$, the linear size of the constriction should be at least of order $\xi\sim\hbar v/T$, [*i.e.*]{}, approaching a millimeter scale for millikelvin temperature range. Tunneling junctions with small opening angles will give larger values of ${\cal
K}_\alpha$ \[in principle, limited only by the logarithm (\[eq:alpha0\]), divergent at small frequencies\]. However, as illustrated in Fig. \[fig:K-frequency\], for such junctions the renormalized Luttinger parameter $g_\star^2$ is more likely to retain some frequency (temperature) dependence, which would modify the measured exponents.
We gratefully acknowledge useful discussions with C. de C. Chamon, M. Fogler, C. Glattli, E. Gwinn, S. A. Kivelson, D.-H. Lee, Z. Nussinov, S. L. Sondhi, and X.-G. Wen.
A.A. acknowledges support of the Israeli Science Foundation and the fund for Promotion of Research in the Technion. E.S. acknowledges support by grant no. 96–00294 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. L.P.P. was supported in part by DOE Grant No. DE-FG02-90ER40542.
Coupling at $\alpha=\pi$. {#sec:appendix-pi}
=========================
Here we derive the form of the coupling ${\cal K}(\omega)$ for the saddle-point geometry shown in Fig. \[fig:angles\] in the special limit $\alpha=\pi$, which corresponds to two vertical semi-infinite wires connected by a single tunneling point. In this case the distance $R_\alpha=|x+y|$, and the contribution of the symmetric potential $V_+(x,y)=V(x-y)+V(x+y)$ to Eq. (\[eq:symmetrized-action\]) vanishes by symmetry (\[eq:field-symmetry\]), so that only the part $V_-(x,y)=[V(x-y)-V(x+y)]\sgn (xy)$ remains. The symmetry of the derivative $\partial_x\vartheta$ implies that both parts of the potential $V_-$ give identical contribution, and the quadratic part of the action (\[eq:symmetrized-action\]) can be written as $$\begin{aligned}
\lefteqn{{\cal S}_{\rm q}
={T\over8\pi}\sum_n\biggl\{\int_{-\infty}^\infty dx\left[
2\omega_n\bar\varphi(x)\,\partial_x\vartheta
+|\partial_x\varphi|^2
+|\partial_x\vartheta|^2\right]
\nonumber} & & \\
& &\quad + {\chi} \int_{-\infty}^\infty dx
\,dy\,\left[\partial_x\bar\vartheta\,V(x-y)\,\partial_y\vartheta\,{\rm
sgn}(xy)
\right]\biggr\}.
\label{eq:wedge-action-pi}\end{aligned}$$ Unlike the case $\alpha=0$, the non-local interaction in the second line cannot be diagonalized by a simple Fourrier transformation; we need to get rid of the sign function first. Naively, this could be done by multiplying both $\varphi(x)$ and $\vartheta(x)$ by ${\rm
sgn}(x)$. However, since $\varphi(0)\neq0$, the function $\varphi(x)\sgn (x)$ would not be continuous at the origin, so that spurious $\delta$-functions may be generated. Instead, we define auxiliary continuous functions $u(x)$, $g(x)$, so that $$\varphi(x)=\varphi(0)+\sgn (x)\,u(x),\quad
g(x)=\vartheta(\infty)-\sgn (x)\,\vartheta(x),$$ and $u(0)=g(\infty)=0$. After integrating out the field $u(x)$, the effective action becomes $$\begin{aligned}
{\cal S}_{\rm q}
&=&{T\over8\pi}\sum_n\biggl\{
-4\omega_n\bar\varphi(0)\,g(0)\\ & &
\quad+\int {dk\over2\pi} |g_k|^2
\left[ \omega_n^2+k^2\biglb(1+\chi
\,V(k) \bigrb)
\right]\biggr\}. \end{aligned}$$ In the first term here we substitute $g(0)=\int {dk\,g_k/(2\pi)}$ in terms of the Fourrier-transformed field $g_k$, integrate this field out, and obtain the effective action for the the field $\varphi(0)$ alone, $${\cal S}_{\rm q} ={T\over4\pi}\sum_n \omega_n \,|\varphi(0)|^2
\left[ {2\omega_n\over \pi}\!\int_{0}^\infty\!\! {dk\over
\omega_n^2+k^2 \biglb(1+\chi\,V(k)\bigrb)}\right];$$ comparing the result with the general form of the effective action (\[eq:effective-model\]), and the result (\[eq:coupling-zero-angle\]) for $\alpha=0$, we conclude that $${\cal K}_{\alpha=0}(\omega_n)\, {\cal K}_{\alpha=\pi}(\omega_n)=1$$ exactly, independent of the form of the potential $V(x)$.
Self-dual tunneling junction, $\alpha=\pi/2$ {#sec:wiener-hopf}
============================================
General Wiener-Hopf solution.
-----------------------------
Here we give a direct solution of the extremum equations for the self-dual case $\alpha=\pi/2$. This solution gives the coupling ${\cal K}_{\pi/2}=1$ directly, without utilizing the self-duality of the problem. In addition, it allows us to calculate $K_\alpha$ perturbatively for small values of $|\cos(\alpha)|\ll 1$.
Begin with the Euclidean action (\[eq:symmetrized-action\]) at $\alpha=\pi/2$, $$\begin{aligned}
\label{eq:wedge-action-half-pi}
{\cal S}_{\rm q}
& =& {T\over8\pi}\sum_n\int dx\biggl\{
2\omega_n\bar\varphi(x)\,\vartheta'_x
+\int dy\,\bar\varphi'_x\,Z(x-y)\,\varphi'_y
\nonumber \\
& & \qquad+\int dy\, \bar\vartheta'_x\,Z(x-y)\sgn (xy)\,\vartheta'_y
\biggr\},\end{aligned}$$ where the total potential $$Z(x-y)=\delta(x-y)+{\chi\over2}\,V(x-y); \label{eq:total-potential}$$ note that due to the symmetry (\[eq:field-symmetry\]), the contribution from the part of the potential with geometrical distance, $V(R_{\pi/2})=V(\sqrt{x^2+y^2})$, was cancelled. The Euler-Lagrange equations (valid at $x\neq0$, where the non-linear tunneling term gives no contribution) are $$\begin{aligned}
\label{eq:EL-phi}
\omega\partial_x\vartheta&-&\partial_x\int_{-\infty}^\infty \!dy \,
Z(x-y)\,\partial_y\varphi =0,\\
\label{eq:EL-theta}
\omega\partial_x\varphi&-&\partial_x \int_{-\infty}^\infty\! dy \,
Z(x-y)\sgn (xy)\,\partial_y\vartheta=0.\end{aligned}$$ We assume that both fields are continuous everywhere, and that $\varphi(x)$ and $\partial_x\vartheta(x)$ vanish at infinity. Multiplying the first of the obtained equations by $\bar\varphi(x)$, the second by $\bar\vartheta(x)$, and subtracting the results from the integrand in the action (\[eq:wedge-action-half-pi\]), with the help of the definition (\[eq:total-potential\]) we obtain $$\begin{aligned}
\lefteqn{{\cal S}_{\rm q}= {T\over8\pi}\sum_n\!\int\!\!
dx\,\partial_x\biggl[
\omega_n\bar\varphi\,\vartheta\!+\!\bar\varphi(x)\!\int\!\! dy
\,Z(x-y)\,\partial_y\varphi}\nonumber\\
& & \qquad+ \bar\vartheta(x) \int\! \!dy \,Z(x-y)
\sgn(xy)\,\partial_y\vartheta\biggr]
\nonumber\\
& =& - {T\over8\pi}\sum_n\bar\varphi(0)\,\Delta\varphi'_0, \quad
\Delta\varphi'_0\equiv \varphi'(0_+)-\varphi'(0_-),
\label{eq:effective-action-evaluated}\end{aligned}$$ where the integration was performed over the entire axis excluding the point $x=0$. The Euler-Lagrange equations (\[eq:EL-phi\]), (\[eq:EL-theta\]) can be simplified by defining linear combinations (symmetric with respect to $x$) $$A,B(x)=[\varphi(x)\pm\vartheta(x)\sgn(x)]/2,
\label{eq:a-b-defined}$$ then, multiplying Eq. (\[eq:EL-theta\]) by $\sgn(x)$ and taking symmetric and antisymmetric combinations of the result with Eq. (\[eq:EL-phi\]), we obtain at $x\neq0$ $$\label{eq:EL-A}
\omega\sgn(x)\partial_x A-\partial_x\int_{-\infty}^\infty \!dy \,
Z(x-y)\,\partial_y A =0,$$ and an identical equation (up to the substitution $\omega\to-\omega$) for $B(x)$. We integrate, keeping in mind that Eq. (\[eq:EL-A\]) is valid for $x\neq0$, $$\label{eq:integrated-EL-A}
\omega A\sgn(x)-\int_{-\infty}^\infty \!dy \, Z(x-y)\,\partial_y A
=C_a\sgn(x),$$ where the integration constants in the intervals $x<0$ and $x>0$ were related using the symmetry $A(x)=A(-x)$. The value of the constant $C_a$ is determined by the boundary conditions; using the definition (\[eq:a-b-defined\]) we obtain $$\label{eq:discontinuity-a}
2C_a= \omega\,\varphi(0)-\varphi'(0_+)-\vartheta'(0)
=\omega\,\vartheta(\infty).$$ Similarly, the integration of the corresponding equation for the function $B(x)$ yields $$\label{eq:discontinuity-b}
2C_b= -\omega\,\varphi(0)-\varphi'(0_+)+\vartheta'(0)
=\omega\,\vartheta(\infty).$$ Together, Eqns. (\[eq:discontinuity-a\]) and (\[eq:discontinuity-b\]) imply that $$\label{eq:CaCb}
C_a=C_b={-\varphi'(0_+)/ 2}.$$
Because of the sign function multiplying the first term in the l.h.s., Eq. (\[eq:integrated-EL-A\]) cannot be solved directly by a Fourrier transformation. It is, however, of the form solvable by the Wiener-Hoph technique[@wiener-hopf-book]. Following the standard prescription, we introduce the functions $A_\pm(x)=A(x)\,\tf(\pm x)$, so that, [*e.g.*]{}, $A(x)=A_+(x)+A_-(x)$, $A(x)\sgn(x)=A_+(x)-A_-(x)$. After this substitution we can Fourrier-transform Eq. (\[eq:integrated-EL-A\]), $$\label{eq:fourrier-transformed-A}
\omega [A_+-A_-]+ik\, Z(k)\, [A_++A_-] =2i\,C_a{\cal P}{1\over k},$$ where ${\cal P}$ denotes the principal value, and the Fourrier-transformed functions $A_\pm\equiv A_\pm(k)$ have no singularities above and below the real axis respectively (regularization at infinity ensures that they are also analytic everywhere along the real axis). The functions $A_\pm(x)$ are only discontinuous in the origin, and the asymptotic form of their Fourrier transformations at $|k|\to \infty$ is $$\label{eq:asymptotic-ug}
A_\pm(k)=\pm{ i\over k}\,A_\pm(0_\pm)+{\cal O}(|k|^{-2})
=\pm i\,{\varphi(0)\over2 k}+\ldots$$
The independent functions in Eq. (\[eq:fourrier-transformed-A\]) can be rearranged as follows, $$\begin{aligned}
A_+(k)&=&-{\cal R}(k)\, A_-(k)+{2C_a\over
k\,Z-i\omega}\,{\cal P}{1\over k}, \label{eq:function-R}\\
{\cal R}(k)&\equiv&
{k\,Z+i\omega\over k\,Z-i\omega}={{\cal
R}_-(k)\over{\cal R}_+(k)}\label{eq:fraction-decomposition}\end{aligned}$$ where the function ${\cal R}(k)$ was separated into the ratio of the function ${\cal R}_-(k)$ which has neither singularities nor zeros at and below the real axis, and ${\cal R}_+(k)$, which has the same properties at and above the real axis. This separation is possible because the function ${\cal R}(k)$ is analytic in a vicinity of the real axis (which is correct for any $\omega$, assuming that the interaction potential $V(x)$ is properly regularized at infinity). In the absence of the long-distance interactions, $\chi=0$, the decomposition is trivial, ${\cal R}_\pm^{0}=(k\pm i\omega)^{-1}$, where we assume $\omega>0$. At very large values of $k$ the long-distance part of the potential should not matter. Therefore, to ensure the regularity of the decomposition (\[eq:fraction-decomposition\]) at $\chi>0$, we can use the Cauchy formula $$\ln {r}_\pm(q)=\!\int_{-\infty}^\infty {dk\over2\pi i}\,
{\ln {r}(k)\over q-k\pm i0},\;\; {r}_\pm(q)\equiv{ {\cal
R}_\pm(q)\over {\cal R}^{0}_\pm(q)}
\label{eq:Cauchy-R}$$ for the ratio $r(k)={\cal R}(k)/{\cal R}^{0}(k)$. Since ${r}(k)\to1$ at large $k$, this expression implies that $r_\pm(k)\to1$ (and hence that ${\cal R}_\pm\sim 1/k$) as $|k|\to\infty$.
Multiplying Eq. (\[eq:function-R\]) by ${\cal
R}_+$, and separating the free term of the obtained expression into a sum of functions analytic above and below the real axis respectively, we obtain $$A_+(k)\,{\cal R}_+-2C_a\,h_+=-A_-(k)\,{\cal
R}_-+2C_a\,h_-.
\label{eq:eqn-A-decomposed}$$ Here the functions $h_\pm\equiv h_\pm(k)$, analytic in the upper (lower) complex half-plane, are defined so that $h_+(k)+h_-(k)=h(k)$, where $$\label{eq:function-h-decomposed}
h(k)\equiv {{\cal R}_+(k)\over k\,Z-i\omega} \,{\cal P} {1\over k}
={{\cal R}_-(k)-{\cal R}_+(k)\over 2i\,\omega}\,{\cal P}{1\over k};$$ these functions can be found using the Cauchy formula $$h_\pm(q)=\mp {1\over 2\pi\, i}\,\int_{-\infty}^\infty\!
{dk}\, {h(k)\over q-k\pm i0}.
\label{eq:Cauchy-h}$$ We assumed that ${\cal R}_\pm(k)$ are non-singular in the origin (and elsewhere along the real axis), therefore, using the identity ${\cal
R}_-(0)={\cal R}(0)\,{\cal R}_+(0)=-{\cal R}_+(0)$, we obtain $$\label{eq:h-plus-evaluated}
h_\pm(k)=\pm i\,{{\cal R}_\pm(k)\over
2\omega \,(k\pm i0)}.$$
By construction, the LHS of Eq. (\[eq:eqn-A-decomposed\]) has no singularities at and above the real axis, while its RHS has no singularities at and below the real axis. Therefore, the whole expression is analytic everywhere in the complex plane, and, as long as it is uniformly limited at infinity, it can only be a constant. Moreover, since both sides of Eq. (\[eq:eqn-A-decomposed\]) actually [*vanish*]{} at infinity \[as follows from Eq. (\[eq:asymptotic-ug\]) and the properties of the functions ${\cal R}_\pm$, $h_\pm$\], this implies that the whole expression can only be zero everywhere at the complex plane $k$. We obtain $$\label{eq:solution-trivial}
A_\pm(k)=2C_a\,{h_\pm(k)\over {\cal R}_\pm(k)}=\pm {i C_a\over
\omega\, (k\pm i0)},$$ and by matching with the asymptotic expansion (\[eq:asymptotic-ug\]), we get $$C_a={\omega\,\varphi(0)\over2}, \quad A_\pm(k)=\pm {i
\varphi(0)\over2\, (k\pm i0)}.\label{eq:A-found}$$ Comparing to Eq. (\[eq:CaCb\]), we obtain $$\Delta\varphi'_0=2\varphi'(0_+)=-2\omega\varphi(0)$$ and the contribution at the frequency $\omega>0$ to the effective action (\[eq:effective-action-evaluated\]) becomes $${\cal S}_{\rm q}(\omega)={T\over4\pi}
|\omega|\,|\varphi(0)|^2.$$ One can also obtain an identical contribution at $\omega<0$, so that $$\label{eq:K-half-pi-exact}
{\cal K}_{\alpha=\pi/2}(\omega)= 1,$$ as expected by the self-duality of the problem.
The analogue of Eq. (\[eq:EL-A\]) for the function $B(x)$ differs only by the sign of $\omega$, which leads to a replacement ${\cal
R}\to1/{\cal R}$, ${\cal R}_\pm\to 1/{\cal R}_\pm$. Instead of Eq. (\[eq:eqn-A-decomposed\]) we get $$\label{eq:eqn-B-decomposed}
B_+(k)\,{\cal R}^{-1}_+-2C_b\,f_+=-B_-(k)\,{\cal
R}^{-1}_-+2C_b\,f_-.$$ By analogy with Eq. (\[eq:h-plus-evaluated\]), we obtain $$\label{eq:f-plus-evaluated}
f_\pm(k)=\mp i\,{{\cal R}^{-1}_\pm(k)\over
2\omega \,(k\pm i0)}.$$ By the same analyticity argument, both sides of equation (\[eq:eqn-B-decomposed\]) are analytic everywhere in the complex plane; at $|k|\to\infty$ they asymptotically approach a constant value $i\varphi(0)$. Therefore, $$B_\pm(k)= \pm{i \varphi(0)}\,\left[{\cal R}_\pm(k)-{1\over 2(k\pm
i0)}\right] ,$$ and, combining with Eq. (\[eq:A-found\]), we can use the definition (\[eq:a-b-defined\]) to restore the original fields in the extremum, $$\begin{aligned}
\label{eq:final-phi-half-pi}
\varphi(x)&=&\varphi(0)\int {dk\over2\pi i}\left[ {\cal R}_-(k)-{\cal
R}_+(k)\right] e^{-ikx},\\
\vartheta(x)&=&\sgn(\omega\,x)\,[\varphi(0)-\varphi(x)],
\label{eq:final-theta-half-pi}\end{aligned}$$ where the $\sgn(\omega)$ in the second line is needed because the case $\omega<0$ is equivalent to the interchange of $A$ and $B$, which changes the sign of $\theta(x)$.
It is easy to verify that the obtained functions obey the boundary conditions assumed when deriving Eqns. (\[eq:effective-action-evaluated\]), (\[eq:discontinuity-a\]), (\[eq:discontinuity-b\]). This self-consistency check ensures that the obtained expressions give us the exact formal solution of the problem.
To understand the structure of this solution, let us introduce the expansion $$\chi V(x)=\sum_{l=1}^N {A_l\over a_l} \,e^{-a_l\,|x|}, \quad
\chi V(k)=\sum_{l=1}^N{2A_l\over
k^2+a_l^2}, \label{eq:exponent-expansion}$$ which, for sufficiently large $N$, gives an adequate regularized representation of any non-pathological even function $V(x)$. For example, the Coulomb potential $ V(x)=1/|x|$ can be rewritten as follows, $${1\over |x|}=\lim_{a\to0}{a\over 1-\exp({-a |x|})}
=\lim_{a\to0}a\sum_{l=0}^\infty e^{-a l\,|x|},$$ so that, given a finite $a$, any partial sum provides a regularization of the form (\[eq:exponent-expansion\]) with $a_l=a\,l$ and $A_l=\chi a^2l$.
We obtain $$\begin{aligned}
Z&=&1+{\chi\over2} V(k)=1+\sum_{l=1}^N {A_l\over k^2+a_l^2}, \\
k Z-i\omega&=&{P_{2N+1}(k) \prod_{l=1}^N (k^2+a_l^2)^{-1}},\end{aligned}$$ where the polynomial $$P_{2N+1}(k)= \prod_{s=1}^{2N+1}(k-i\kappa_s)$$ has precisely $(2N+1)$ purely imaginary distinct roots $k_s\equiv
i\kappa_s\neq0$. One can also show that for $\omega>0$ exactly $N$ of the roots lie below the imaginary axis; we shall assume $\kappa_s<0$ for $1<s<N$. The Cauchy integral (\[eq:Cauchy-R\]) is readily evaluated, and we obtain $$\begin{aligned}
{\cal R}_+={(k-i\kappa_1)\ldots (k-i\kappa_N)\over
(k+i\kappa_{N+1})\ldots (k+i\kappa_{2N+1})};
\label{eq:exponent-expansion-R}\end{aligned}$$ using the form similar to that in the first part of Eq. (\[eq:function-h-decomposed\]), the extremum solution (\[eq:final-phi-half-pi\]) can be explicitly rewritten as $$\varphi(x)=2|\omega|\,\varphi(0)\int {dk\over 2\pi}\,
{(k^2+a_1^2)\ldots (k^2+a_N^2)\,\cos(kx)\over (k^2+\kappa^2_{N+1})\ldots
(k^2+\kappa_{2N+1}^2)}.\label{eq:found-explicit-phi}$$
Expansion around the self-dual solution {#sec:perturbation-theory}
---------------------------------------
To get an approximate expression for ${\cal K}(\alpha)$ in a vicinity of $\alpha=\pi/2$, we expand $V_\pm(x,y)$ to first order in $\cos\alpha$, and employ perturbation theory. The solution of the extremum equations at $\alpha_0=\pi/2$ is unique, and the lowest order non-degenerate perturbation theory suffices. This amounts to evaluating the Euclidean action (\[eq:symmetrized-action\]) along the non-perturbed solution $\varphi(x)$, $\vartheta(x)$, $$\begin{aligned}
\delta{\cal S}_{\rm q}&\equiv&{T\over 4\pi}\sum_n|\omega_n|\,\delta {\cal
K}_\alpha\,|\varphi(0)|^2\\
&=&{T\over 4\pi}\sum_n
{\chi\over2} \int dx \,dy
\left[\bar\varphi'_x\,\delta V_+\,\varphi'_y
+\bar\vartheta'_x\,\delta V_-\,\vartheta'_y\right],\end{aligned}$$ where the integration is performed everywhere except the origin, and the potentials $$\begin{aligned}
\delta V_+&=&-{ xy\,\cos\alpha\over \sqrt{x^2+y^2}}V'(\sqrt{x^2+y^2}),\\
\delta V_-&=&-\delta V_+\,\sgn(xy). \end{aligned}$$ were found by expanding Eqns. (\[eq:v-plus\]), (\[eq:v-minus\]).
According to our solution (\[eq:final-theta-half-pi\]), the functions $\varphi'(x)$, $-\vartheta'(x)\,\sgn(\omega\,x)$ are identical, and the two terms give equal contributions, leading to $$\delta{\cal K}_\alpha=-{\chi\cos\alpha\over|\omega|\,|\varphi(0)|^2}
\int_{-\infty}^\infty\! dx\,dy\,\bar\varphi'_x\,\varphi'_y\,{y}\,
\partial_x\,V(\sqrt{x^2+y^2}).$$ For the Coulomb potential (\[eq:coulomb\]), this gives $$\delta{\cal K}_\alpha={4\chi\cos\alpha\over|\omega|\,|\varphi(0)|^2}
\int_0^\infty\! dx\int_0^\infty\!dy\,
{x \,y\,\bar\varphi'_x\,\varphi'_y\over (x^2+y^2+a^2)^{3/2}}.$$ This integral converges at small distances even if we set $a\to0$; in this scale-invariant limit the “wavefunctions” $\varphi(x)$ can depend only on the dimensionless quantities $|\omega|x$ and $\chi$, $\varphi(x)\equiv\varphi(0)\,\phi_\chi(|\omega|\,x)$. Scaling out the frequency leads to a [*frequency-independent*]{} correction, $$\begin{aligned}
\nonumber
\delta{\cal K}_\alpha(\omega,\chi)&=&{\chi\,{\cal
N}(\chi)\,\cos\alpha}+{\cal O}(\chi^2\cos^2\alpha),\quad
\omega\,a\ll1,\\
{\cal N}(\chi)&\equiv& 4\!\int_0^\infty\!\!
dx\int_0^\infty\!\!dy\, {x
\,y\,\bar\phi'_\chi(x)\,\phi'_\chi(y)\over
(x^2+y^2)^{3/2}}. \label{eq:app-linear-expansion}\end{aligned}$$ This result supports the numerical data, which indicates that ${\cal K}_\alpha(\omega)$ is [*independent*]{} of $\omega$ at small enough frequencies. This statement is true for all finite angles, $|\cos\alpha|<1$, while ${\cal K}_{\alpha=0}(\omega)$ diverges logarithmically according to Eq. (\[eq:alpha0\]).
The specific value of the correction depends on the coupling constant $\chi$. In the weak-coupling limit, $\chi\ll1$, the function $\phi_{\chi\to0}(x)=\exp(-|x|)$, and the integration produces $${\cal N}(\chi\to0)\approx 1.51.$$ For finite $\chi>0$, and any given $N$ in the expansion (\[eq:exponent-expansion\]), the explicit form of the integrand in Eq. (\[eq:app-linear-expansion\]) can be found with the help of Eq. (\[eq:found-explicit-phi\]), and the corresponding value ${\cal N}(\chi)$ can be evaluated numerically.
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| ArXiv |
---
abstract: 'Our recent discovery of magnetic fields in a small number of Herbig Ae/Be stars has required that we survey a much larger sample of stars. From our FORS1 and ESPaDOnS surveys, we have acquired about 125 observations of some $70$ stars in which no magnetic fields are detected. Using a Monte Carlo approach, we have performed statistical comparisons of the observed longitudinal fields and LSD Stokes $V$ profiles of these apparently non-magnetic stars with a variety of field models. This has allowed us to derive general upper limits on the presence of dipolar fields in the sample, and to place realistic upper limits on undetected dipole fields which may be present in individual stars. This paper briefly reports the results of the statistical modeling, as well as field upper limits for individual stars of particular interest.'
author:
- 'G.A. Wade'
- 'E. Alecian'
- 'C. Catala'
- 'S. Bagnulo'
- 'J.D. Landstreet'
- 'J. Flood'
- 'T. Böhm'
- 'J.-C. Bouret'
- 'J.-F. Donati'
- 'C.P. Folsom'
- 'J. Grunhut'
- 'J. Silvester'
date: 'March 8, 2003'
title: |
How non-magnetic are\
“non-magnetic” Herbig Ae/Be stars?
---
Introduction {#intr}
============
Observations of magnetic fields in pre-main sequence Herbig Ae/Be (HAeBe) stars can serve to address several important astrophysical problems: (1) The role of magnetic fields in mediating accretion, and the validity of models which propose that HAeBe stars are simply higher-mass analogues of the T Tau stars. (2) The origin of the magnetic fields of main sequence A and B type stars. (3) The development and evolution of chemical peculiarities and chemical abundance structures in the atmospheres of A and B type stars. (4) The loss of rotational angular momentum which leads to the slow rotation observed in some main sequence A and B type stars.
Since 2004, we have been engaged in a systematic assay of the magnetic properties of bright ($m_{\rm V}\ltsim 12$) HAeBe stars using the FORS1 spectropolarimeter at the ESO-VLT (Wade et al. 2007), and the ESPaDOnS spectropolarimeter at the CFHT (Wade et al., in preparation). We have acquired about 130 Stokes $V$ (circular polarisation) spectra of over 75 HAeBe stars, with the aim of measuring the longitudinal Zeeman effect in their spectra.
The ESPaDOnS observations are of high resolving power ($R\sim 65000$), and provide the capability to resolve the complex line profiles presented by many HAeBe stars. The longitudinal magnetic field was measured from each observation using the standard first-moment method applied to Least-Squares Deconvolved (LSD; Donati et al. 1997) profiles. The dependence of the field diagnosis on the LSD masks was explored in detail for each star, and “clean” photospheric masks were constructed by excluding lines in the spectrum that exhibited clear contamination by emission, or other significant departures from the predictions of an LTE synthetic spectrum. In most cases we found that global departures of the metallic line spectrum were relatively small, and the improvement achieved using tuned masks was minor. The magnetic field diagnosis obtained from the ESPaDOnS data is very sensitive to the projected rotational velocity $v\sin i$, and the formal uncertainties achieved consequently span a large range of precision, from very good (for most stars with $v\sin i\ltsim 80$ ) to essentially useless (for some stars with $v\sin i\gtsim 150$ ).
The FORS1 observations, on the other hand, are of low resolving power ($R\sim 1000-1500$). Although such spectra fail to resolve the complex profiles of most lines, they are relatively insensitive to rotational broadening. Consequently, the longitudinal magnetic fields derived from FORS1 spectra (using the linear regression method developed by Bagnulo et al. 2000) provide a relatively uniform diagnosis over a large range of $v\sin i$.
From these surveys magnetic fields have been detected in 6 stars[^1]. As the detailed properties of these magnetic HAeBe stars are discussed by Alecian et al. (these proceedings) and Folsom et al. (these proceedings), we will only review the general results here. The two stars modeled in great detail (HD 200775, HD 72106) show stable, oblique, dipolar magnetic fields with polar intensities of about 1 kG, low $v\sin i$, and rotation periods of several days. The two stars modeled in moderate detail (HD 190073, V380 Ori) show stable, organised magnetic fields, low $v\sin i$, and rotation periods of days, possibly years. Finally, the two stars with a small number of observations (HD 101412, HD 104237) show strong longitudinal fields and simple Stokes $V$ profiles, suggesting organised magnetic fields. Both of these stars have low $v\sin i$.
Based on these results, it is clear that some Herbig Ae/Be stars host strong, organised magnetic fields, qualitatively identical to those of the main sequence Ap/Bp stars. Here, we turn our eye to our much larger collection of $\sim 125$ observations of $\sim 70$ undetected HAeBe stars, in order to examine the extent to which our observations can constrain their magnetic properties. In other words, how non-magnetic are “non-magnetic” HAeBe stars?
Modeling and Results
====================
To explore the properties of the undetected sample of HAeBe stars, we have followed a Monte Carlo approach similar to that employed by Wade et al. (2007). We developed synthetic populations of magnetic stars where each star was characterised by the inclination of its rotation axis $i$, the obliquity of its dipolar magnetic field $\beta$, the rotation phase at which it was observed $\phi$, and the intensity of its dipolar magnetic field at the magnetic pole, $B_{\rm d}$. For the purposes of our simultations, the parameters $i$ and $\phi$ were selected randomly for each star ($i$ with a $\sin i$ PDF, $\phi$ with a uniform PDF), $\beta$ was randomly set equal to either $0\degr$ or $90\degr$ (with equal probability), while the magnetic intensity $B_{\rm d}$ was fixed for all stars in a given population (and therefore defined the characteristic field strength of that population). We then created synthetic distributions of longitudinal field measurements from each of these populations, assuming the same uncertainties characterising the real observations of undetected HAeBe stars. This procedure was repeated 100 times, using different realisations of the randomly-selected variables.
Fig. 1 compares the observed histogram of uncertainty-normalised longitudinal field measurements $z=\langle B_z\rangle/\sigma_B$ (including both FORS1 and ESPaDOnS data), with synthetic histograms compiled from the Monte Carlo simultations.
To quantitatively test whether the observed and computed distributions are representative of the same population (with the practical goal of testing if the observations imply fields which are weaker than those which characterise the models), we have performed a one-sided Kolmogorov-Smirnov (K-S) test (e.g. Conover 1971) on the cumulative distributions of longitudinal fields. The test statistic $D$ used in the K-S test is the maximum fractional difference betwen two cumulative distributions (i.e. the observed distribution and that compiled from a model). In this case, we find differences $D=0.058, 0.143, 0.228$ and 0.297 for models corresponding to dipole fields of polar intensity 0, 300, 450 and 600 G, respectively. For a sample size $N\sim 125$, a model distribution can be rejected at the 99% level if $D\geq 0.135$. Therefore the longitudinal field measurements allow us to rule out uniform populations of stars with fields above about 300 G. On the other hand, the observations are consistent with a uniform population of stars with fields of about 300 G or smaller[^2].
------------- -----
Model \#
$B_{\rm d}$ det
100 G 1
300 G 2
450 G 6
600 G 6
1000 G 6
2000 G 13
------------- -----
: Results of LSD profile modeling. See text for details.[]{data-label="t1"}
----------- ------ ----------------------- ------ ----- ------------
Target Spec $B_{\rm d}^{\rm max}$ P(%) \# $\sigma_B$
Type (G) obs (G)
HD 17081 B8 100 95 2 9
HD 139614 A6 300 98 3 14
HD 36112 A4 450 96 4 30
HD 142666 A6 450 98 6 35
HD 169142 A8 600 92 3 24
HD 31648 A3 2000 95 2 52
HD 144432 A9 2000 93 2 30
BF Ori A5 2000 86 1 32
----------- ------ ----------------------- ------ ----- ------------
: Results of LSD profile modeling. See text for details.[]{data-label="t1"}
We then set out to provide a clearer evaluation of the upper limits on dipole fields for individual stars. Unfortunately, such upper limits are nearly impossible to derive using small numbers of longitudinal field measurements because the longitudinal field for most stars becomes null at some point during the rotation cycle, even in the case of a strong surface field. We have therefore employed the individual LSD profiles obtained for those stars observed using ESPaDOnS. The velocity-resolved LSD profiles allow the detection of the magnetic field even when the longitudinal field is null, thanks to the spectral separation of polarised contributions from different parts of the stellar disc due to rotational Doppler effect. However, the interpretation of an LSD profile is more complicated than a longitudinal field measurement, requiring that we create synthetic LSD profiles corresponding to each observation (reproducing its associated LSD profile depth, $v\sin i$ and signal-to-noise ratio).
To model the LSD profiles, we first fit each LSD Stokes $I$ profile with a rotationally-broadened model, to determine $v\sin i$, line depth and radial velocity. We then used the model populations of magnetic stars to create synthetic Stokes V LSD profiles corresponding to each of our observations (using the profile synthesis procedure described by Alecian et al. 2007), and introduced synthetic Gaussian noise corresponding to the noise level in the real LSD $V$ profile. Finally, for each synthetic LSD profile we evaluated the probability that a Stokes $V$ signature was detected, using the same criteria that are applied in the real LSD procedure (see Donati et al. 1997). Again, this procedure was repeated 100 times for each observation and for each model, using different realisations of the randomly-selected variables. Examples of observed and synthetic LSD profiles are shown in Fig. 2.
The results of this procedure were twofold: first, a global comparison of the predictions of each of the population models with the observations, and secondly a quantitative evaluation of the compatibility of [*each*]{} LSD profile with the predictions of dipole surface field models of different intensities.
Table 1 summarises the results of this analysis. On the left, we show the number of detections of individual stars we would expect in the case of each population. Even for models as weak as 300 G, we obtain detections of small numbers of stars in over 90% of model realisations. This result is consistent with that derived from the longitudinal field measurements, and demonstrates that the LSD profiles strongly constrain models which propose the presence of weak, organised magnetic fields in all HAeBe stars. On the other hand, there may still exist a small number of magnetic stars, with magnetic properties similar to the detected magnetic HAeBe stars, present in our “non-magnetic” sample (but which remain undetected with the current observations). We note however that the dipole intensities derived by Alecian et al. and Folsom et al. (these proceedings) for the detected magnetic HAeBe stars, when evolved to the main sequence (Table 1 of Alecian et al.), are typical of those of the majority of Ap/Bp stars (Power et al., these proceedings). This indicates that the Herbig stars in which fields have been detected do not have unusually strong fields, and therefore that most of the magnetic stars in the sample are probably already detected.
On the right-hand side of Table 1, we show the inferred upper limits for dipole fields $B_{\rm d}^{\rm max}$ in individual sample stars obtained from fitting the LSD profiles. We also report the fraction of model realisations which generate a detection P(%) when $B_{\rm d}=B_{\rm d}^{\rm max}$, the number of observations, and the derived longitudinal field error bar $\sigma_B$. Of particular interest are the lack of detections for stars in which marginal magnetic field detections had previously been claimed (HD 139614, HD 144432, HD 31648, HD 36112 and BF Ori; Hubrig et al. 2004, 2006a; Wade et al. 2007) based on observations obtained with FORS1.
EA is funded by the Marie-Curie FP6 programme, while JDL and GAW acknowledge support from NSERC and DND-ARP (Canada).
Alecian E., Catala C., Wade G.A., Donati J.-F., et al., 2007, MNRAS, in press Bagnulo S., Landstreet J. D., Lo Curto G., Szeifert T., Wade G. A., 2003, A&A 403, 635 Conover W.J., 1971, [*Practical nonparametric statistics*]{}, 1st ed., p. 400, Wiley (New York) Donati, J.-F., Semel, M., Carter, B. D., Rees, D. E., Cameron, A. C., 1997, MNRAS 291, 658 Hubrig, S., Sch$\ddot{\rm{o}}$ller, M., Yudin, R. V., 2004, A&A 428, L1 Hubrig, S.; Yudin, R. V.; Schšller, M.; Pogodin, M. A., 2006, A&A 446, 1089 Wade G.A., Bagnulo S., Drouin D., Landstreet J.D. and Monin D., 2007, MNRAS 376, 1145
[^1]: One magnetic star (HD 101412) discovered using FORS1, 4 magnetic stars (V380 Ori, HD 72106, HD 190073, HD 200775) discovered using ESPaDOnS, and 1 magnetic star (HD 104237) discovered previously by Donati et al. (1997). We do not discuss here results from the survey of HAeBe stars in young open clusters, briefly introduced by Alecian et al. (these proceedings).
[^2]: We underscore that these field intensities refer to the dipole field polar strength at the stellar surface, and not to the mean longitudinal field.
| ArXiv |
---
abstract: 'A preliminary group classification of the class 2D nonlinear heat equations $u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})$, where $f$ is arbitrary smooth function of the variables $x,y,u,u_x$ and $u_y$ using Lie method, is given. The paper is one of the few applications of an algebraic approach to the problem of group classification: the method of preliminary group classification.'
address: 'School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran.'
author:
- 'M. Nadjafikhah'
- 'R. Bakhshandeh-Chamazkoti'
title: |
Preliminarily group classification of a class of\
2D nonlinear heat equations
---
,
,
$2$D Nonlinear heat equation, Optimal system, Preliminarily group classification.
Introduction
============
It is well known that the symmetry group method plays an important role in the analysis of differential equations. The history of group classification methods goes back to Sophus Lie. The first paper on this subject is [@[1]], where Lie proves that a linear two-dimensional second-order PDE may admit at most a three-parameter invariance group (apart from the trivial infinite parameter symmetry group, which is due to linearity). He computed the maximal invariance group of the one-dimensional heat conductivity equation and utilized this symmetry to construct its explicit solutions. Saying it the modern way, he performed symmetry reduction of the heat equation. Nowadays symmetry reduction is one of the most powerful tools for solving nonlinear partial differential equations (PDEs). Recently, there have been several generalizations of the classical Lie group method for symmetry reductions. Ovsiannikov [@[2]] developed the method of partially invariant solutions. His approach is based on the concept of an equivalence group, which is a Lie transformation group acting in the extended space of independent variables, functions and their derivatives, and preserving the class of partial differential equations under study. In an attempt to study nonlinear effects Saied and Hussain [@[3]] gave some new similarity solutions of the (1+1)-nonlinear heat equation. Later Clarkson and Mansfield [@[4]] studied classical and nonclassical symmetries of the (1+1)-heat equation and gave new reductions for the linear heat equation and a catalogue of closed-form solutions for a special choice of the function $f(x,y,u,u_x,u_y)$ that appears in their model. In higher dimensions Servo [@[5]] gave some conditional symmetries for a nonlinear heat equation while Goard et al. [@[6]] studied the nonlinear heat equation in the degenerate case. Nonlinear heat equations in one or higher dimensions are also studied in literature by using both symmetry as well as other methods [@[7]; @[8]].There are a number of papers to study (1+1)-nonlinear heat equations from the point of view of Lie symmetries method. The (2+1)-dimensional nonlinear heat equations $$\begin{aligned}
u_t=f(u)(u_{xx}+u_{yy}),\label{eq:1}\end{aligned}$$ are investigated in [@[9]] and in present paper we studied $$\begin{aligned}
u_t=f(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:2}\end{aligned}$$ Similarity techniques are applied in [@[10]; @[11]; @[12]; @[13]] for (2+1)-dimensional wave equations.
Symmetry Methods
================
Let a partial differential equation contains $p$ dependent variables and $q$ independent variables. The one-parameter Lie group of transformations $$\begin{aligned}
x_i\longmapsto x_i+\epsilon\xi^i(x,u)+O(\epsilon^2);\hspace{1cm}
u_{\alpha}\longmapsto
u_{\alpha}+\epsilon\varphi^{\alpha}(x,u)+O(\epsilon^2),\label{eq:3}\end{aligned}$$ where $i=1,\ldots,p$ and $\alpha=1,\ldots,q$. The action of the Lie group can be recovered from that of its associated infinitesimal generators. we consider general vector field $$\begin{aligned}
X=\sum_{i=1}^p\xi^i(x,u)\frac{{\rm \partial}}{{{\rm \partial}}x_i}+
\sum_{\alpha=1}^q\varphi^{\alpha}(x,u)\frac{{\rm \partial}}{{{\rm \partial}}u^{\alpha}}.\label{eq:4}\end{aligned}$$ on the space of independent and dependent variables. The symmetry generator associated with (\[eq:4\]) given by $$\begin{aligned}
X=\xi^1(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}x}+\xi^2(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}y}+
\xi^3(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}t}+\varphi(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}u}.\label{eq:6}\end{aligned}$$ The second prolongation of $X$ is the vector field $$\begin{aligned}
\nonumber
X^{(2)}=X+\varphi^x\frac{{\rm \partial}}{{{\rm \partial}}u_x}+\varphi^y\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\varphi^t\frac{{\rm \partial}}{{{\rm \partial}}u_t}+
\varphi^{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_{xx}}+\varphi^{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_{xy}}
+\varphi^{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_{xt}}
+\varphi^{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_{yy}}
+\varphi^{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_{yt}}+\varphi^{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_{tt}},\\\label{eq:7}\end{aligned}$$ that its coefficients are obtained with following formulas $$\begin{aligned}
\label{eq:8}
&&\varphi^x={D}_x\varphi-u_x{D}_x\xi^1-u_y{D}_x\xi^2-u_t{D}_x\xi^3,\hspace{2cm}
\varphi^y={D}_y\varphi-u_x{D}_y\xi^1-u_y{D}_y\xi^2-u_t{D}_y\xi^3,\\\nonumber
&&\varphi^t={D}_t\varphi-u_x{D}_t\xi^1-u_y{D}_t\xi^2-u_t{\rm
D}_t\xi^3,\hspace{2.1cm}
\varphi^{xx}={D}_x\varphi^x-u_{xx}{D}_x\xi^1-u_{xy}{D}_x\xi^2-u_{xt}{D}_x\xi^3\\\nonumber
&&\hspace{-2mm}\varphi^{yy}={D}_y\varphi^y-u_{xy}{D}_y\xi^1-u_{yy}{D}_y\xi^2-u_{yt}{D}_y\xi^3\hspace{1.58cm}
\varphi^{tt}={D}_t\varphi^t-u_{xt}{D}_t\xi^1-u_{yt}{D}_t\xi^2-u_{tt}{D}_t\xi^3\\\nonumber
&&\hspace{-2mm}\varphi^{yt}={D}_y\varphi^t-u_{xy}{D}_y\xi^1-u_{yy}{D}_y\xi^2-u_{yt}{D}_y\xi^3\hspace{1.58cm}
\varphi^{xt}={D}_t\varphi^t-u_{xt}{D}_t\xi^1-u_{yt}{D}_t\xi^2-u_{tt}{D}_t\xi^3\end{aligned}$$ where the operators $D_x$, $D_y$ and $D_t$ denote the total derivatives with respect to $x,y$ and $t$: $$\begin{aligned}
\nonumber
D_x&=&\frac{{\rm \partial}}{{{\rm \partial}}x}+u_x\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+u_{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+
u_{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+\ldots\\
D_y&=&\frac{{\rm \partial}}{{{\rm \partial}}y}+u_y\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+u_{yx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+
u_{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+\ldots\\\label{eq:9}
D_t&=&\frac{{\rm \partial}}{{{\rm \partial}}t}+u_t\frac{{\rm \partial}}{{{\rm \partial}}u}+u_{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_t}+u_{tx}\frac{{\rm \partial}}{{{\rm \partial}}u_x}+
u_{ty}\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\ldots\nonumber\end{aligned}$$ By theorem 6.5. in [@[14]], $X^{(2)}[u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})]|_{(6)}=0$ whenever $$\begin{aligned}
u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})=0.\label{eq:10}\end{aligned}$$ Since $$X^{(2)}[u_t-f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})]=\varphi^t-
(f_x\xi^1+f_y\xi^2+f_u\varphi+f_{u_x}\varphi^x+f_{u_y}\varphi^y)
(u_{xx}+u_{yy})-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy}),$$ therefore we obtain the following determining function: $$\begin{aligned}
\varphi^t-(f_x\xi^1+f_y\xi^2+f_u\varphi+f_{u_x}\varphi^x+f_{u_y}\varphi^y)
(u_{xx}+u_{yy})
-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy})=0.\label{eq:11}\end{aligned}$$ In the case of arbitrary $f$ it follows $$\begin{aligned}
\xi^1=\xi^2=\varphi=0,\label{eq:12}\end{aligned}$$ or $$\begin{aligned}
\xi^1=\xi^2=\varphi=0,\;\;\;\;\;\xi^3=C.\label{eq:13}\end{aligned}$$ Therefore, for arbitrary $f(x,y,u,u_x,u_y)$ Eq. (\[eq:1\]) admits the one-dimensional Lie algebra ${{\goth g}}_1$, with the basis $$\begin{aligned}
X_1=\frac{{\rm \partial}}{{{\rm \partial}}t}.\label{eq:14}\end{aligned}$$ ${{\goth g}}_1$ is called the principle Lie algebra for Eq. (\[eq:1\]). So, the remaining part of the group classification is to specify the coefficient $f$ such that Eq. (\[eq:1\]) admits an extension of the principal algebra ${{\goth g}}_1$. Usually, the group classification is obtained by inspecting the determining equation. But in our case the complete solution of the determining equation (\[eq:11\]) is a wasteful venture. Therefore, we don’t solve the determining equation but, instead we obtain a partial group classification of Eq. (\[eq:1\]) via the so-called method of preliminary group classification. This method was suggested in [@[10]] and applied when an equivalence group is generated by a finite-dimensional Lie algebra ${{\goth g}}_{{\mathscr E}}$. The essential part of the method is the classification of all nonsimilar subalgebras of ${{\goth g}}_{{\mathscr E}}$. Actually, the application of the method is simple and effective when the classification is based on finite-dimensional equivalence algebra ${{\goth g}}_{{\mathscr E}}$.
Equivalence transformations
===========================
An equivalence transformation is a nondegenerate change of the variables $t,x,y,u$ taking any equation of the form (\[eq:1\]) into an equation of the same form, generally speaking, with different $f(x,y,u,u_x,u_y)$. The set of all equivalence transformations forms an equivalence group ${{\mathscr E}}$. We shall find a continuous subgroup ${{\mathscr E}}_C$ of it making use of the infinitesimal method.
We consider an operator of the group ${{\mathscr E}}_C$ in the form $$\begin{aligned}
Y=\xi^1(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}x}+\xi^2(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}y}+
\xi^3(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}t}+\varphi(x,y,t,u)\frac{{\rm \partial}}{{{\rm \partial}}u}
+\mu(x,y,t,u,u_x,u_y,u_t,f)\frac{{\rm \partial}}{{{\rm \partial}}f},\label{eq:15}\end{aligned}$$ from the invariance conditions of Eq. (\[eq:1\]) written as the system: $$\begin{aligned}
\label{eq:3-16}
u_t&-&f(x,y,u,u_x,u_y)(u_{xx}+u_{yy})=0,\\\nonumber
f_t&=&f_{u_t}=0,\end{aligned}$$ where $u$ and $f$ are considered as differential variables: $u$ on the space $(x,y,t)$ and $f$ on the extended space $(x,y,t,u,u_x,u_y)$.
The invariance conditions of the system (\[eq:3-16\]) are $$\begin{aligned}
\label{eq:17}
Y^{(2)}(u_t&-&f(x,y,u,u_x,u_y)(u_{xx}+u_{yy}))=0,\\\nonumber
Y^{(2)}(f_t)&=&Y^{(2)}(f_{u_t})=0,\end{aligned}$$ where $Y^{(2)}$ is the prolongation of the operator (\[eq:15\]): $$\begin{aligned}
\label{eq:3-18}
Y^{(2)}=Y+\varphi^x\frac{{\rm \partial}}{{{\rm \partial}}u_x}+\varphi^y\frac{{\rm \partial}}{{{\rm \partial}}u_y}+\varphi^t\frac{{\rm \partial}}{{{\rm \partial}}u_t}+
\varphi^{xx}\frac{{\rm \partial}}{{{\rm \partial}}u_{xx}}+\varphi^{xy}\frac{{\rm \partial}}{{{\rm \partial}}u_{xy}}
+\varphi^{xt}\frac{{\rm \partial}}{{{\rm \partial}}u_{xt}}&+&\varphi^{yy}\frac{{\rm \partial}}{{{\rm \partial}}u_{yy}}\\\nonumber
&+&\varphi^{yt}\frac{{\rm \partial}}{{{\rm \partial}}u_{yt}}+\varphi^{tt}\frac{{\rm \partial}}{{{\rm \partial}}u_{tt}}+
\mu^t\frac{{\rm \partial}}{{{\rm \partial}}f_{t}}+\mu^{u_t}\frac{{\rm \partial}}{{{\rm \partial}}f_{u_t}}.\end{aligned}$$ The coefficients $\varphi^x, \varphi^y, \varphi^t, \varphi^{xx},
\varphi^{xy}, \varphi^{xt}, \varphi^{yy}, \varphi^{yt},
\varphi^{tt}$ are given in (\[eq:8\]) and the other coefficients of (\[eq:3-18\]) are obtained by applying the prolongation procedure to differential variables $f$ with independent variables $(x,y,t,u,u_x,u_y,u_t)$. we have $$\begin{aligned}
\mu^t&=&\widetilde{D}_t(\mu)-f_x\widetilde{D}_t(\xi^1)-f_y\widetilde{D}_t(\xi^2)
-f_u\widetilde{D}_t(\varphi)-f_{u_x}\widetilde{D}_t(\varphi^x)-f_{u_y}\widetilde{D}_t(\varphi^y),\label{eq:19}\\
\mu^{u_t}&=&\widetilde{D}_{u_t}(\mu)-f_x\widetilde{D}_{u_t}(\xi^1)-f_y\widetilde{D}_{u_t}(\xi^2)
-f_u\widetilde{D}_{u_t}(\varphi)-f_{u_x}\widetilde{D}_{u_t}(\varphi^x)-f_{u_y}\widetilde{D}_{u_t}(\varphi^y),\label{eq:20}\end{aligned}$$ where $$\begin{aligned}
\widetilde{D}_t=\frac{{\rm \partial}}{{{\rm \partial}}t},\hspace{1cm}\widetilde{D}_{u_t}=\frac{{\rm \partial}}{{{\rm \partial}}u_t}.\label{eq:21}\end{aligned}$$ So, we have the following prolongation formulas: $$\begin{aligned}
\label{eq:22}
\mu^t&=&\mu_t-f_x\xi_t^1-f_y\xi_t^2-f_u\varphi_t-f_{u_x}(\varphi^x)_t-f_{u_y}(\varphi^y)_t,\\\nonumber
\mu^{u_t}&=&\mu_{u_t}-f_{u_x}(\varphi^x)_{u_t}-f_{u_y}(\varphi^y)_{u_t},\end{aligned}$$ By the invariance conditions (\[eq:17\]) give rise to $$\begin{aligned}
\mu^t=\mu^{u_t}=0,\label{eq:23}\end{aligned}$$ that is hold for every $f$. Substituting (\[eq:23\]) into (\[eq:22\]), we obtain $$\begin{aligned}
\begin{array}{ll}
\mu_t=\mu_{u_t}=0\\
\xi^1_x=\xi^2_t=\varphi_t=0\\
(\varphi^x)_t=(\varphi^x)_{u_t}=(\varphi^y)_t=(\varphi^y)_{u_t}=0\label{eq:24}
\end{array}\end{aligned}$$ Moreover with substituting (\[eq:3-18\]) into (\[eq:17\]) we obtain $$\begin{aligned}
\varphi^t-f(x,y,u,u_x,u_y)(\varphi^{xx}+\varphi^{yy})-\mu(u_{xx}+u_{yy})=0.\label{eq:25}\end{aligned}$$ We are left with a polynomial equation involving the various derivatives of $u(x,y,t)$ whose coefficients are certain derivatives of $\xi^1,\xi^2,\xi^3$ and $\varphi$. Since $\xi^1,\xi^2,\xi^3,\varphi$ only depend on $x,y,t,u$ we can equate the individual coefficients to zero, leading to the complete set of determining equations: $$\begin{aligned}
\xi^1&=&\xi^1(x,y)\label{eq:26}\\
\xi^2&=&\xi^2(y)\label{eq:27}\\
\xi^3&=&\xi^3(t)\label{eq:28}\\
\varphi_{uu}&=&0\label{eq:29}\\
2\varphi_{xu}&=&\xi^1_{xx}+\xi^1_{yy}\label{eq:30}\\
\varphi_{yu}&=&\xi^2_{xx}+\xi^2_{yy}\label{eq:31}\\
\varphi_u&=&\xi_x^1=\xi_y^2\label{eq:32}\\
\mu&=&(\xi^1_x-\xi_t^3)f\label{eq:33}\\
\varphi_{tt}&=&f(\varphi_{xx}+\varphi_{yy})\label{eq:34}\end{aligned}$$ so, we find that $$\begin{aligned}
\nonumber
&&\xi^1(x)=c_1x+c_2y+c_3,\hspace{1cm}\xi^2(t)=c_1y+c_4,\hspace{1cm}\xi^3(t)=a(t),\\
&&\hspace{1cm}\varphi(x,y,u)=c_1u+\beta(x,y),\hspace{1cm}\mu=(c_1-a'(t))f,\label{eq:35}\end{aligned}$$ with constants $c_1, c_2, c_3$ and $c_4$, also we have $\beta_{xx}=-\beta_{yy}$.
$\;\;\;\;$We summarize: The class of Eq. (\[eq:2\]) has an infinite continuous group of equivalence transformations generated by the following infinitesimal operators: $$\begin{aligned}
Y=(c_1x+c_2y+c_3)\frac{{\rm \partial}}{{{\rm \partial}}x}+ (c_1y+c_4)\frac{{\rm \partial}}{{{\rm \partial}}y}+
a(t)\frac{{\rm \partial}}{{{\rm \partial}}t}+(c_1u+\beta(x,y))\frac{{\rm \partial}}{{{\rm \partial}}u}
+(c_1-a'(t))f\frac{{\rm \partial}}{{{\rm \partial}}f}.\label{eq:36}\end{aligned}$$ Therefore the symmetry algebra of the Burgers’ equation (\[eq:2\]) is spanned by the vector fields $$\begin{aligned}
&Y_1=x\frac{{\rm \partial}}{{{\rm \partial}}x}+y\frac{{\rm \partial}}{{{\rm \partial}}y}+t\frac{{\rm \partial}}{{{\rm \partial}}t}+u\frac{{\rm \partial}}{{{\rm \partial}}u}+f\frac{{\rm \partial}}{{{\rm \partial}}f},
\hspace{1cm} Y_2=y\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_3=\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}Y_4=\frac{{\rm \partial}}{{{\rm \partial}}y}&\\\label{eq:37}
&Y_5=a(t)\frac{{\rm \partial}}{{{\rm \partial}}t}-a'(t)f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}
Y_{\beta}=\beta(x,y)\frac{{\rm \partial}}{{{\rm \partial}}u}.&\nonumber\end{aligned}$$
Moreover, in the group of equivalence transformations there are included also discrete transformations, i.e., reflections $$\begin{aligned}
t\longrightarrow-t,\hspace{1.5cm}x\longrightarrow-x,\hspace{1.5cm}u\longrightarrow-u,\hspace{1.5cm}
f\longrightarrow-f.\label{eq:38}\end{aligned}$$
$$\begin{aligned}
\hspace{-0.75cm}\begin{array}{llllll}
\hline
[\,,\,]&\hspace{2cm}Y_1 &\hspace{2cm}Y_2 &\hspace{2cm}Y_3 &\hspace{2cm}Y_4 &\hspace{2cm}Y_5 \hspace{2cm}Y_6 \\ \hline
Y_1 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{1.8cm}-Y_3 &\hspace{1.8cm}-Y_4 &\hspace{2cm}0 \hspace{1.6cm}-Y_6 \\
Y_2 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{1.8cm}-Y_3 &\hspace{2cm}0 \hspace{2cm}0\\
Y_3 &\hspace{2cm}Y_3 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{2cm}0 &\hspace{2cm}0 \hspace{2cm}0\\
Y_4 &\hspace{2cm}Y_4 &\hspace{2cm} Y_3 &\hspace{2cm} 0 &\hspace{2cm}0 &\hspace{2cm}0 \hspace{2cm}0 \\
Y_5 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{2cm}0 &\hspace{2cm}0 \hspace{2cm}0\\
Y_6 &\hspace{2cm}Y_6 &\hspace{2cm} 0 &\hspace{2cm} 0 &\hspace{2cm}0 &\hspace{2cm}0 \hspace{2cm}0\\
\hline
\end{array}\end{aligned}$$
$$\begin{aligned}
\hspace{-0.75cm}\begin{array}{llllll}
\hline
[\,,\,]&\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6
\\ \hline
Y_1 &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}e^sY_3&\hspace{2cm}e^sY_4&\hspace{2cm}Y_5\hspace{2cm}e^sY_6 \\
Y_2 &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4+sY_3&\hspace{2cm}Y_5\hspace{2cm}Y_6 \\
Y_3 &\hspace{1.5cm}Y_1-sY_3&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4 &\hspace{2cm}Y_5\hspace{2cm}Y_6 \\
Y_4 &\hspace{1.5cm}Y_1-sY_4&\hspace{1.6cm}Y_2-sY_3&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6 \\
Y_5 &\hspace{2cm}Y_1&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6 \\
Y_6 &\hspace{1.5cm}Y_1-sY_6&\hspace{2cm}Y_2&\hspace{2cm}Y_3&\hspace{2cm}Y_4&\hspace{2cm}Y_5\hspace{2cm}Y_6 \\
\hline
\end{array}\end{aligned}$$
Preliminary group classification
================================
One can observe in many applications of group analysis that most of extensions of the principal Lie algebra admitted by the equation under consideration are taken from the equivalence algebra ${\goth g}_{{\mathscr E}}$. We call these extensions ${\mathscr E}$-extensions of the principal Lie algebra. The classification of all nonequivalent equations (with respect to a given equivalence group $G_{{\mathscr E}}$,) admitting ${\mathscr E}$-extensions of the principal Lie algebra is called a preliminary group classification. Here, $G_{{\mathscr E}}$ is not necessarily the largest equivalence group but, it can be any subgroup of the group of all equivalence transformations.So, we can take any finite-dimensional subalgebra (desirable as large as possible) of an infinite-dimensional algebra with basis (\[eq:31\]) and use it for a preliminary group classification. We select the subalgebra ${\goth g}_6$ spanned on the following operators: $$\begin{aligned}
&Y_1=x\frac{{\rm \partial}}{{{\rm \partial}}x}+y\frac{{\rm \partial}}{{{\rm \partial}}y}+t\frac{{\rm \partial}}{{{\rm \partial}}t}+u\frac{{\rm \partial}}{{{\rm \partial}}u}+f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}
Y_2=y\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_3=\frac{{\rm \partial}}{{{\rm \partial}}x},\hspace{1cm}
Y_4=\frac{{\rm \partial}}{{{\rm \partial}}y},&\nonumber\\
&Y_5=\frac{{\rm \partial}}{{{\rm \partial}}t}-f\frac{{\rm \partial}}{{{\rm \partial}}f},\hspace{1cm}Y_6=\frac{{\rm \partial}}{{{\rm \partial}}u}.&\label{eq:39}\end{aligned}$$ The communication relations between these vector fields is given in Table 1. To each $s$-parameter subgroup there corresponds a family of group invariant solutions. So, in general, it is quite impossible to determine all possible group-invariant solutions of a PDE. In order to minimize this search, it is useful to construct the optimal system of solutions. It is well known that the problem of the construction of the optimal system of solutions is equivalent to that of the construction of the optimal system of subalgebras [@[2]; @[12]]. Here, we will deal with the construction of the optimal system of subalgebras of ${\goth
g}_5$.Let $G$ be a Lie group, with ${\goth g}$ its Lie algebra. Each element $T\in G$ yields inner automorphism $T_a\longrightarrow
TT_aT^{-1}$ of the group $G$. Every automorphism of the group $G$ induces an automorphism of ${\goth g}$. The set of all these automorphism is a Lie group called [*the adjoint group $G^A$*]{}. The Lie algebra of $G^A$ is the adjoint algebra of ${\goth g}$, defined as follows. Let us have two infinitesimal generators $X,Y\in L$. The linear mapping ${\rm
Ad}X(Y):Y\longrightarrow[X,Y]$ is an automorphism of ${\goth g}$, called [*the inner derivation of ${\goth g}$*]{}. The set of all inner derivations ${\rm ad}X(Y)(X,Y\in{\goth g})$ together with the Lie bracket $[{\rm Ad}X,{\rm Ad}Y]={\rm Ad}[X,Y]$ is a Lie algebra ${\goth g}^A$ called the [*adjoint algebra of ${\goth
g}$*]{}. Clearly ${\goth g}^A$ is the Lie algebra of $G^A$. Two subalgebras in ${\goth g}$ are [*conjugate*]{} (or [*similar*]{}) if there is a transformation of $G^A$ which takes one subalgebra into the other. The collection of pairwise non-conjugate $s$-dimensional subalgebras is the optimal system of subalgebras of order $s$. The construction of the one-dimensional optimal system of subalgebras can be carried out by using a global matrix of the adjoint transformations as suggested by Ovsiannikov [@[2]]. The latter problem, tends to determine a list (that is called an [*optimal system*]{}) of conjugacy inequivalent subalgebras with the property that any other subalgebra is equivalent to a unique member of the list under some element of the adjoint representation i.e. $\overline{{\goth h}}\,{\rm
Ad(g)}\,{\goth h}$ for some ${\rm g}$ of a considered Lie group. Thus we will deal with the construction of the optimal system of subalgebras of ${\goth g}_6$. The adjoint action is given by the Lie series $$\begin{aligned}
{\rm Ad}(\exp(s\,Y_i))Y_j
=Y_j-s\,[Y_i,Y_j]+\frac{s^2}{2}\,[Y_i,[Y_i,Y_j]]-\cdots,\label{eq:40}\end{aligned}$$ where $s$ is a parameter and $i,j=1,\cdots,6$. The adjoint representations of ${\goth g}_6$ is listed in Table 2, it consists the separate adjoint actions of each element of ${\goth g}_6$ on all other elements. [**Theorem 4.1.**]{} [*An optimal system of one-dimensional Lie subalgebras of general Burgers’ equation (\[eq:2\]) is provided by those generated by*]{} $$\begin{aligned}
&1)&Y^1=Y_1=x{{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{2.2cm}2)~Y^2=Y_2=y{{\rm \partial}}_x,\\
&3)&Y^3=-Y_4=-{{\rm \partial}}_y,\hspace{5.7cm}4)~Y^4=Y_1+Y_5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\\
&5)&Y^5=Y_1-Y_2=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{0.5cm}6)~Y^6=Y_2-Y_4=y{{\rm \partial}}_x-{{\rm \partial}}_y,\\
&7)&Y^7=-Y_4+Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_u,\hspace{4.1cm}8)~Y^{8}=-Y_4-Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
&9)&Y^9=Y_2+Y_5=y{{\rm \partial}}_x+{{\rm \partial}}_t-f{{\rm \partial}}_f,\hspace{3.3cm}10)~Y^{10}=Y_2-Y_5=y{{\rm \partial}}_x-{{\rm \partial}}_t+f{{\rm \partial}}_f,\\
&11)&Y^{11}=Y_2+Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_u,\hspace{4.1cm}12)~Y^{12}=Y_2-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_u,\\
&13)&Y^{13}=Y_1+Y_2=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1mm}14)~Y^{14}=-Y_4+Y_5+Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&15)&Y^{15}=Y_2-Y_4-Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,16)~Y^{16}=Y_2-Y_4+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u,\\
&17)&Y^{17}=Y_2-Y_4+Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,18)~Y^{18}=Y_2-Y_4-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
&19)&Y^{19}=Y_1+Y_2+Y_5=(x+y){{\rm \partial}}_x+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\hspace{0.5cm}20)~Y^{20}=Y_2+Y_5+Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&21)&Y^{21}=Y_2+Y_5-Y_6=y{{\rm \partial}}_x+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,\hspace{1.6cm}22)~Y^{22}=Y_2-Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
&23)&Y^{23}=Y_2-Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1.6cm}24)~Y^{24}=-Y_4-Y_5-Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
&25)&Y^{25}=-Y_4-Y_5+Y_6=-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1.2cm}26)~Y^{26}=-Y_4+Y_5-Y_6=-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&27)&Y^{27}=Y_2-Y_4+Y_5+Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
&28)&Y^{28}=Y_1+Y_2-Y_5=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
&29)&Y^{29}=Y_1-Y_2-Y_5=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
&30)&Y^{31}=Y_1-Y_2+Y_5=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u,\\
&31)&Y^{31}=Y_1-Y_5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u+2f{{\rm \partial}}_f\\
&32)&Y^{32}=Y_2-Y_4-Y_5-Y_6=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u+f{{\rm \partial}}_f,\end{aligned}$$ [**Proof.**]{} Let ${\goth g}_6$ is the symmetry algebra of Eq. (\[eq:2\]) with adjoint representation determined in Table 2 and $$\begin{aligned}
Y=a_1Y_1+a_2Y_2+a_3Y_3+a_4Y_4+a_5Y_5+a_6Y_6,\end{aligned}$$ is a nonzero vector field of ${\goth g}_6$. We will simplify as many of the coefficients $a_i;i=1,\ldots,6$, as possible through proper adjoint applications on $Y$. We follow our aim in the below easy cases:[*Case 1:*]{} At first, assume that $a_1\neq 0$. Scaling $Y$ if necessary, we can assume that $a_1=1$ and so we get $$\begin{aligned}
Y=Y_1+a_2Y_2+a_3Y_3+a_4Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Using the table of adjoint (Table 2) , if we act on $Y$ with ${\rm
Ad}(\exp(a_3Y_3))$, the coefficient of $Y_3$ can be vanished: $$\begin{aligned}
Y'=Y_1+a_2Y_2+a_4Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Then we apply ${\rm Ad}(\exp(a_4Y_4))$ on $Y'$ to cancel the coefficient of $Y_4$: $$\begin{aligned}
Y''=Y_1+a_2Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ At last, we apply ${\rm Ad}(\exp(a_6Y_6))$ on $Y''$ to cancel the coefficient of $Y_6$: $$\begin{aligned}
Y'''=Y_1+a_2Y_2+a_5Y_5.\end{aligned}$$ [*Case 1a:*]{}\
If $a_2,a_5\neq 0$ then we can make the coefficient of $Y_2$ and $Y_5$ either $+1$ or $-1$. Thus any one-dimensional subalgebra generated by $Y$ with $a_2,a_5\neq 0$ is equivalent to one generated by $Y_1\pm Y_2\pm Y_5$ which introduce parts 19), 28), 29) and 30) of the theorem.[*Case 1b:*]{} For $a_2=0, a_5\neq0$ we can see that each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $
Y_1\pm Y_5$ which introduce parts 4) and 31) of the theorem.[*Case 1c:*]{} For $a_2\neq0, a_5=0$, each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_1\pm Y_2$ which introduce parts 5) and 13) of the theorem.[*Case 1d:*]{} For $a_2=0, a_5=0$, each one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_1$ which introduce parts 1) of the theorem.[*Case 2:*]{} The remaining one-dimensional subalgebras are spanned by vector fields of the form $Y$ with $a_1=0$. [*Case 2a:*]{} If $a_4\neq 0$ then by scaling $Y$, we can assume that $a_4=-1$. Now by the action of ${\rm Ad}(\exp a_3Y_3))$ on $Y$, we can cancel the coefficient of $Y_3$: $$\begin{aligned}
\overline{Y}=a_2Y_2-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ Let $a_2\neq0$ then by scaling $Y$, we can assume that $a_2=1$, and we have $$\begin{aligned}
\overline{Y}'=Y_2-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2a-1:*]{} Suppose $a_5=a_6=0$, then the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4$ which introduce parts 6). [*Case 2a-2:*]{} Suppose $a_5=0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4\pm
Y_6$ which introduce parts 16) and 18).[*Case 2a-3:*]{} Suppose $a_5\neq0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $Y_2-Y_4\pm Y_5\pm Y_6$ which introduce parts 15), 17), 27), and 32). Now if $a_2=0$, we have $$\begin{aligned}
\overline{Y}''=-Y_4+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2a-4:*]{} Suppose $a_5=a_6=0$, then the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4$ which introduce parts 3). [*Case 2a-5:*]{} Suppose $a_5=0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4\pm Y_6$ which introduce parts 7) and 8).[*Case 2a-6:*]{} Suppose $a_5\neq0, a_6\neq0$, all of the one-dimensional subalgebra generated by $Y$ is equivalent to one generated by $-Y_4\pm Y_5\pm Y_6$ which introduce parts 14), 24), 25) and 26).
[*Case 2b:*]{} $~~~~$ Let $a_4=0$ then $Y$ is in the form $$\begin{aligned}
\widehat{Y}=a_2Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ Suppose that $a_2\neq 0$ then if necessary we can let it equal to $1$ and we obtain $$\begin{aligned}
\widehat{Y}'=Y_2+a_5Y_5+a_6Y_6.\end{aligned}$$ [*Case 2b-1:*]{} Let $a_5=a_6=0$, then $Y_2$ is remained and find 2) section of the theorem.[*Case 2b-2:*]{} If $a_5\neq0, a_6\neq0$, then $\widehat{Y}'$ is equal to $Y_2\pm Y_5\pm Y_6$. Hence this case suggests part 20), 21), 22) and 23).[*Case 2b-3:*]{} If $a_5\neq0, a_6=0$, then $\widehat{Y}'=Y_2\pm Y_5$ . Hence this case suggests part 9) and 10).[*Case 2b-4:*]{} If $a_5=0, a_6\neq0$, then $Y_2\pm Y_6$ is obtained. So, this case suggests part 11) and 12). There is not any more possible case for studying and the proof is complete. $\Box$ The coefficients $f$ of Eq. (\[eq:2\]) depend on the variables $x,y,u,u_x,u_y$. Therefore, we take their optimal system’s projections on the space $(x,y,u,u_x,u_y,f)$. we have $$\begin{aligned}
\hspace{-0.7cm}
\begin{array}{rlrl}
1)&Z^1=Y^1=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f, \hspace{1cm}&17)&Z^{17}=Y^{17}=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\\
2)&Z^2=Y^2=y{{\rm \partial}}_x,\hspace{1cm}&18)&Z^{18}=Y^{18}=(x+y){{\rm \partial}}_x+u{{\rm \partial}}_u,\\
3)&Z^3=Y^3=-{{\rm \partial}}_y,\hspace{1cm}&19)&Z^{19}=Y^{19}=y{{\rm \partial}}_x+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
4)&Z^4=Y^4=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1cm}&20)&Z^{20}=Y^{20}=y{{\rm \partial}}_x-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
5)&Z^5=Y^5=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u,\hspace{1cm}&21)&Z^{21}=Y^{21}=y{{\rm \partial}}_x-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
6)&Z^6=Y^6=x{{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\hspace{1cm} &22)&Z^{22}=Y^{22}=y{{\rm \partial}}_x+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
7)&Z^7=Y^7=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+f{{\rm \partial}}_f,\hspace{1cm} &23)&Z^{23}=Y^{23}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u,\\
8)&Z^8=Y^8=y{{\rm \partial}}_x-{{\rm \partial}}_y,\hspace{1cm}&24)&Z^{24}=Y^{24}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u,\\
9)&Z^9=Y^9=-{{\rm \partial}}_y+{{\rm \partial}}_u,\hspace{1cm}&25)&Z^{25}=Y^{25}=-{{\rm \partial}}_y+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
10&Z^{10}=Y^{10}=-{{\rm \partial}}_y-{{\rm \partial}}_u,\hspace{1cm}&26)&Z^{26}=Y^{26}=-{{\rm \partial}}_y-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
11&Z^{11}=Y^{11}=y{{\rm \partial}}_x-f{{\rm \partial}}_f,\hspace{1cm}&27)&Z^{27}=Y^{27}=-{{\rm \partial}}_y+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
12)&Z^{12}=Y^{12}=y{{\rm \partial}}_x+f{{\rm \partial}}_f,\hspace{1cm}&28)&Z^{28}=Y^{28}=-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
\end{array}\end{aligned}$$ $$\begin{aligned}
\hspace{-0.7cm}
\begin{array}{rlrl}
13)&Z^{13}=Y^{13}=y{{\rm \partial}}_x+{{\rm \partial}}_u,\hspace{1cm}&29)&Z^{29}=Y^{29}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u-f{{\rm \partial}}_f,\\
14)&Z^{14}=Y^{14}=y{{\rm \partial}}_x-{{\rm \partial}}_u,\hspace{1cm}&30)&Z^{30}=Y^{30}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
15)&Z^{15}=Y^{15}=(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u,\hspace{1cm}&31)&Z^{31}=Y^{31}=y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u+f{{\rm \partial}}_f,\\
16)&Z^{16}=Y^{16}=(x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u+2f{{\rm \partial}}_f,\hspace{1cm}&32)&Z^{32}=Y^{32}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,
\end{array}\end{aligned}$$ [**Proposition 4.2.**]{} [*Let ${\goth g}_m:=\langle Y_1, \ldots,
Y_m\rangle$, be an $m$-dimensional algebra. Denote by $Y^i (i=1,
\ldots, r, 0<r\leq m, r\in{\Bbb N})$ an optimal system of one-dimensional subalgebras of ${\goth g}_m$ and by $Z^i\, (i =
1,\cdots, t, 0<t\leq r, t\in{\Bbb N})$ the projections of $Y^i$, i.e., $Z^i = {\rm pr}(Y^i)$. If equations $$\begin{aligned}
f = \Phi(x,y,u,u_x,u_y),\label{eq:18}\end{aligned}$$ are invariant with respect to the optimal system $Z^i$ then the equation $$\begin{aligned}
u_t = \Phi(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:19}\end{aligned}$$ admits the operators $X^i=$ projection of $Y^i$ on $(t,x,y,u,u_x,u_y)$.*]{} [**Proposition 4.3.**]{} [*Let Eq. (\[eq:19\]) and the equation $$\begin{aligned}
u_t = \Phi'(x,y,u,u_x,u_y)(u_{xx}+u_{yy}),\label{eq:20}\end{aligned}$$ be constructed according to Proposition 4.2. via optimal systems $Z^i$ and ${Z^i}'$, respectively. If the subalgebras spanned on the optimal systems $Z^i$ and ${Z^i}'$, respectively, are similar in ${\goth g}_m$, then Eqs. (\[eq:19\]) and (\[eq:20\]) are equivalent with respect to the equivalence group $G_m$, generated by ${\goth g}_m$.* ]{} Now we apply Proposition 4.2. and Proposition 4.3. to the optimal system (\[eq:17\]) and obtain all nonequivalent Eq. (\[eq:2\]) admitting ${\mathscr E}$-extensions of the principal Lie algebra ${\goth
g}$, by one dimension, i.e., equations of the form (\[eq:2\]) such that they admit, together with the one basic operators (\[eq:21\]) of ${\goth g}$, also a second operator $X^{(2)}$. For every case, when this extension occurs, we indicate the corresponding coefficients $f, g$ and the additional operator $X^{(2)}$.
We perform the algorithm passing from operators $Z^i\,(i=1,\cdots,32)$ to $f$ and $X^{(2)}$ via the following example. Let consider the vector field $$\begin{aligned}
Z^{32}=y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u-f{{\rm \partial}}_f,\label{eq:21}\end{aligned}$$ then the characteristic equation corresponding to $Z^6$ is $$\begin{aligned}
{dx\over y}={dy\over-1}=\frac{du}{-1}=\frac{df}{-f},\end{aligned}$$ and can be taken in the form $$\begin{aligned}
I_1=u+{x\over y},\hspace{5mm}I_2=e^{x\over y}f.\end{aligned}$$ From the invariance equations we can write $$\begin{aligned}
I_2=\Phi(I_1),\end{aligned}$$ it follows that $$\begin{aligned}
f=e^{-{x\over y}}\Phi(\lambda),\end{aligned}$$ where $\lambda=I_1$.
From Proposition 4.2. applied to the operator $Z^6$ we obtain the additional operator $X^{(2)}$ $$\begin{aligned}
y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u.\end{aligned}$$ After similar calculations applied to all operators (\[eq:17\]) we obtain the following result (Table 3) for the preliminary group classification of Eq. (\[eq:2\]) admitting an extension ${\goth
g}_3$ of the principal Lie algebra ${\goth g}_1$.
Conclusion
==========
In this paper, following the classical Lie method, the preliminary group classification for the class of heat equation (\[eq:2\]) and investigated the algebraic structure of the symmetry groups for this equation, is obtained. The classification is obtained by constructing an optimal system with the aid of Propositions 4.2. and 4.3.. The result of the work is summarized in Table 3. Of course it is also possible to obtain the corresponding reduced equations for all the cases in the classification reported in Table 3.
\[table:3\] $$\begin{aligned}
\hspace{-0.75cm}\begin{array}{l l l l l l} \hline
N &\hspace{1cm} Z &\hspace{1.1cm} \mbox{Invariant} &\hspace{1cm} \mbox{Equation}
&\hspace{1cm} \mbox{Additional operator}\,X^{(2)} \\ \hline
1 &\hspace{1cm} Z^1 &\hspace{1.1cm} {u\over x} &\hspace{1cm}u_t=x\Phi(u_{xx}+u_{yy})
&\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u \\
2 &\hspace{1cm} Z^2 &\hspace{1.1cm} u &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x \\
3 &\hspace{1cm} Z^3 &\hspace{1.1cm} u &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y\\
4 &\hspace{1cm} Z^4 &\hspace{1.1cm} {u\over x+y} &\hspace{1cm}u_t=y\Phi(u_{xx}+u_{yy})
&\hspace{1cm} (x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+u{{\rm \partial}}_u \\
5 &\hspace{1cm} Z^5 &\hspace{1.1cm} {u\over x}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
6 &\hspace{1cm} Z^6 &\hspace{1.1cm} {u\over x}&\hspace{1cm}u_t=x^2\Phi(u_{xx}+u_{yy})
&\hspace{1cm} x{{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
7 &\hspace{1cm} Z^7 &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=(x-y)\Phi(u_{xx}+u_{yy})
&\hspace{1cm}(x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+t{{\rm \partial}}_t+u{{\rm \partial}}_u \\
8 &\hspace{1cm} Z^8 &\hspace{1.1cm} u &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y \\
9 &\hspace{1cm} Z^9 &\hspace{1.1cm} u&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_u \\
10 &\hspace{1cm} Z^{10} &\hspace{1.1cm} x &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_u \\
11 &\hspace{1cm} Z^{11} &\hspace{1.1cm} u &\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t \\
12 &\hspace{1cm} Z^{12} &\hspace{1.1cm} u &\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t \\
13 &\hspace{1cm} Z^{13} &\hspace{1.1cm} u-{x\over y} &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_u \\
14 &\hspace{1cm} Z^{14} &\hspace{1.1cm} u+{x\over y} &\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_u \\
15 &\hspace{1cm} Z^{15} &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} (x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
16 &\hspace{1cm} Z^{16} &\hspace{1.1cm} {u\over x+y}&\hspace{1cm}u_t=(x+y)^2\Phi(u_{xx}+u_{yy})
&\hspace{1cm} (x+y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
17 &\hspace{1cm} Z^{17} &\hspace{1.1cm} {u\over x-y}&\hspace{1cm}u_t=(x-y)^2\Phi(u_{xx}+u_{yy})
&\hspace{1cm} (x-y){{\rm \partial}}_x+y{{\rm \partial}}_y+(t-1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
18 &\hspace{1cm} Z^{18} &\hspace{1.1cm} {u\over x+y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} (x+y){{\rm \partial}}_x+(t+1){{\rm \partial}}_t+u{{\rm \partial}}_u \\
19 &\hspace{1cm} Z^{19} &\hspace{1.1cm}u-{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t+{{\rm \partial}}_u\\
20 &\hspace{1cm} Z^{20} &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x+{{\rm \partial}}_t-{{\rm \partial}}_u \\
21 &\hspace{1cm} Z^{21} &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t-{{\rm \partial}}_u\\
22 &\hspace{1cm} Z^{22} &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_t+{{\rm \partial}}_u\\
23 &\hspace{1cm} Z^{23} &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_u\\
24 &\hspace{1cm} Z^{24} &\hspace{1.1cm}u+{x\over y}&\hspace{1cm}u_t=e^y\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_u \\
25 &\hspace{1cm} Z^{25} &\hspace{1.1cm} u+y&\hspace{1cm}u_t=\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u \\
26 &\hspace{1cm} Z^{26} &\hspace{1.1cm} u-y&\hspace{1cm}u_t=e^{-y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u \\
27 &\hspace{1cm} Z^{27} &\hspace{1.1cm} u+y&\hspace{1cm}u_t=e^{-y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u \\
28 &\hspace{1cm} Z^{28} &\hspace{1.1cm} u-y &\hspace{1cm}u_t=e^y\Phi(u_{xx}+u_{yy})
&\hspace{1cm} -{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u \\
29 &\hspace{1cm} Z^{29} &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t+{{\rm \partial}}_u \\
30 &\hspace{1cm} Z^{30} &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t-{{\rm \partial}}_u \\
31 &\hspace{1cm} Z^{31} &\hspace{1.1cm} u-{x\over y}&\hspace{1cm}u_t=e^{x\over y}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y-{{\rm \partial}}_t+{{\rm \partial}}_u \\
32 &\hspace{1cm} Z^{32} &\hspace{1.1cm} u+{x\over y}&\hspace{1cm}u_t=e^{-{x\over y}}\Phi(u_{xx}+u_{yy})
&\hspace{1cm} y{{\rm \partial}}_x-{{\rm \partial}}_y+{{\rm \partial}}_t-{{\rm \partial}}_u \\
\hline
\end{array}\end{aligned}$$
S. Lie, : Arch. for Math. 6, 328 (1881). L. V. Ovsiannikov, , Group Analysis of Differential Equations, Academic Press, New York, 1982. E.A. Saied, M.M. Hussain, Similarity solutions for a nonlinear model of the heat equation, J. Nonlinear Math. Phys. 3 (1–2) (1996) 219–225. P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys. D 70 (1993) 250–288. M.I. Servo, Conditional and nonlocal symmetry of nonlinear heat equation, J. Nonlinear Math. Phys. 3 (1–2) (1996) 63–67. J.M. Goard, P. Broadbridge, D.J. Arrigo, The integrable nonlinear degenerate diffusion equations, Z. Angew. Math. Phys. 47 (6) (1996) 926–942. P.G.Estevez, C. Qu, S.L. Zhang, Separation of variables of a generalized porous medium equation with nonlinear source, J. Math. Anal. Appl. 275 (2002) 44–59. P.W. Doyle, P.J. Vassiliou, Separation of variables in the 1-dimensional non-linear diffusion equation, Internat. J. Non-Linear Mech. 33 (2) (2002) 315–326. A. Ahmad, Ashfaque H. Bokhari, A. H. Kara, F. D. Zaman, Symmetry classifications and reductions of some classes of $(2+1)$-nonlinear heat equation, J. Math. Annal. Appl. 339 (2008) 175-181. M. Nadjafikhah, R. Bakhshandeh-Chamazkoti, and A. Mahdipour–Shirayeh, A symmetry classification for a class of $(2+1)$-nonlinear wave equation, Nonlinear Analysis,(2009), doi:10.1016/j.na.2009.03.087. R. Cimpoiasu, R. Constantinescu, Lie symmetries and invariants for $2$D nonlinear heat equation, Nonlinear Analysis 68 (2008) 2261-2268. N. H. Ibragimov, M. Tottisi, and A. Valenti, Preliminary group classification of equations $u_{tt}=f(x,u_x)u_{xx}+g(x,u_x)$, J. Math. phys, 32, No. 11:2988-2995, 1991. Lina song , and Hongqing zhang, Preliminary group classification for the nonlinear wave equation $u_{tt}=f(x,u)u_{xx}+g(x,u)$, Nonlinear Analysis, article in press. P.J. Olver , Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge, (1995).
| ArXiv |
---
abstract: 'The [*concordance*]{} cosmological model has been successfully tested over the last decades. Despite its successes, the fundamental nature of dark matter and dark energy is still unknown. Modifications of the gravitational action have been proposed as an alternative to these dark components. The straightforward modification of gravity is to generalize the action to a function, $f(R)$, of the scalar curvature. Thus one is able to describe the emergence and the evolution of the Large Scale Structure without any additional (unknown) dark component. In the weak field limit of the $f(R)$-gravity, a modified Newtonian gravitational potential arises. This gravitational potential accounts for an extra force, generally called fifth force, that produces a precession of the orbital motion even in the classic mechanical approach. We have shown that the orbits in the modified potential can be written as Keplerian orbits under some conditions on the strength and scale length of this extra force. Nevertheless, we have also shown that this extra term gives rise to the precession of the orbit. Thus, comparing our prediction with the measurements of the precession of some planetary motions, we have found that the strength of the fifth force must be in the range $[2.70-6.70]\times10^{-9}$ with the characteristic scale length to fix to the fiducial values of $\sim 5000$ AU.'
author:
- Ivan De Martino
- Ruth Lazkoz
- Mariafelicia De Laurentis
title: 'Analysis of the Yukawa gravitational potential in $f(R)$ gravity I: semiclassical periastron advance.'
---
Introduction {#uno}
============
Standard cosmology is entirely based on General Relativity. It is capable of explaining both the present period of accelerated expansion and the dynamics of self-gravitating systems resorting to Dark Energy and Dark Matter, respectively. The model has been confirmed by observations carried out over the last decades [@Planck16_13]. The need of having recourse to Dark Matter to explain the dynamics of stellar clusters, galaxies, groups and clusters of galaxies, among others astrophysical objects, has been well known for many decades now. Such systems show a deficit of mass when the photometric and spectroscopic estimates are compared with the dynamical one. Early astronomical candidates proposed to solve this problem of [*missing*]{} mass were MAssive Compact Halo Objects (MACHOs) and ReAlly Massive Baryon Objects (RAMBOs), sub-luminous compact objects (or clusters of objects) like Black Holes and Neutron Stars that could not have been observed due to several selection effects. Since the number of the observed sub-luminous objects was not enough to account for the [*missing matter*]{}, the idea that this matter was hidden in some exotic particles, weakly interacting with ordinary matter, emerged. Many candidates have been proposed such as Weakly Interacting Massive Particle (WIMP), axions, neutralino, Q-balls, gravitinos and Bose-Einstein condensate, among the others [@Bertone2005; @Capolupo2010; @Schive2014; @demartino2017b; @demartino2018; @Lopes2018; @Panotopoulos2018], but there are no experimental evidences of their existence so far [@Feng2010].
An alternative approach is to modify Newton’s law. Such a modification naturally arises in the weak field limit of some modified gravity models [@Moffat2006; @demartino2017; @PhysRept; @manos; @sergei] that attempt to explain the nature of Dark Matter and Dark Energy as an effect of the space-time curvature. These theories predict the existence of massive gravitons that may carry the gravitational interaction over a certain scale depending by the mass of these particles [@Bogdanos:2009tn; @felix; @Bellucci:2008jt; @graviton]. Thus, in their weak field limit, a Yukawa-like modification to Newton’s law emerges. One of those models is $f(R)$-gravity where the Einstein-Hilbert action, which is linear in the Ricci scalar $R$, is replaced with a more general function of the curvature, $f(R)$. In its the weak field limit, the modified Newtonian potential has the following functional form [@Annalen]: $$\label{eq:potyuk}
\Phi(r) = -\frac{G M}{(1+\delta)r}(1+\delta e^{-r/\lambda}),$$ where $M$ is the mass of the point-like source, $r$ the distance of a test particle ($m$) from the source, $G$ is Newton’s constant, $\delta$ is the strength of the Yukawa correction and the $\lambda$ represents the scale over which the Yukawa-force acts. Since $f(R)$-gravity is a fourth-order theory, the Yukawa scale length arises from the extra degrees of freedom (in the general paradigm, a $(2k+2)$-order theory of gravity gives rise to $k$ extra gravitational scales [@Quandt1991]). Both parameters are also related to the $f(R)$-Lagrangian as [@Annalen; @PhysRept]: $$\begin{aligned}
\delta = f'_0 - 1, \qquad \lambda = \sqrt{-\frac{6f''_0}{f'_0}},\end{aligned}$$ where $$f'_0=\frac{df(R)}{dR}\biggr|_{R=R_0}\,, \qquad f''_0=\frac{d^2f(R)}{dR^2}\biggr|_{R=R_0}.$$ Next, considering the field equations and trace of $f(R)$ gravity at the first order approximation in terms of the perturbations of the metric tensor, and choosing a suitable transformation and a gauge condition, one can relate the massive states of the graviton to the $f(R)$-Lagrangian and to the Yukawa-length: $$\begin{aligned}
m_g^2 \propto \frac{-f'_0}{f''_0} =\frac{2}{\lambda^2}.\end{aligned}$$ Therefore, it is customary to identify the Yukawa-length with the Compton wavelength of the massive graviton $\lambda_c = hc/m_g$. Thus, for example, we have $\lambda\sim10^{3}$ km with a mass of gravitons $m_g\sim 10^{-22}$ eV [@Lee2010; @Abbott2017]. Therefore, the effect of a modification of the Newtonian potential must naturally act at galactic and extragalactic scales, where $f(R)$-gravity has been successfully tested [@demartino2014; @demartino2015; @demartino2016]. Nevertheless, smaller effects could be detected at shorter scales [@Talmadge1988] where the strength of the Yukawa-correction has been bounded using the Pioneer anomaly [@Anderson1998; @Anderson2002] and S2 star orbits [@Borka20012; @Borka20013; @Zakharov2016; @Hees2017; @Zakharov2018; @Iorio2005]. Obviously, the most interesting systems to test gravitational theories are binary systems composed by coalescing compact stars, such as neutron stars, white dwarfs and/or black holes [@deLa_deMa2014; @deLa_deMa2015; @LeeS2017], but the study of stable orbits is equally important since it allows us to study possible variations of the gravitational interaction in the weak field limit. In the last decades, the orbital precession has been used to probe General Relativity [@Will2014; @Iorio2009], as well as to place bounds on “anti-gravity” due to the cosmological constant [@islam1983; @Iorio2006; @Sereno2006], on forces proposed as alternatives to dark matter [@Gron1996; @Khriplovich2006] and/or induced from extensions of General Relativity [@Capozziello2001; @Moffat2006; @Sanders2006; @Battat2008; @Nyambuya2010; @Ozer2017; @Liu2018].
In this paper we show, in a semi-classical approach, that the orbital motion under the modified gravitational potential in Eq. can be traced back to a Keplerian orbit with modified eccentricity, but with an orbital precession due to the Yukawa-term. We consider two point-like masses orbiting around each other and we use a Newtonian approach to compute the equation of the orbit, and a perturbative approach to compute the precession of the orbit. Finally, we use the current limits on the orbital precession of the planetary orbits to place a bound on the strength of the Yukawa-term. The paper is organized as follows: in Sec. \[due\] we introduce the equations of motion, in Sec. \[tre\] we compute the equation of the orbits, in Sec. \[quattro\] we compute analytically the precession effect due to the Yukawa potential, and we use current measurements of the orbital precession of Solar System’s planets to bound the parameter $\delta$ in Eq. . We consider, for each planet, a 3$\sigma$ interval around the best fit value of the precession, and we compute the lower and an upper limit on $\delta$ so that the predicted precession relies in the observed interval. In Sec. \[cinque\] we discuss some consequences of our results. Finally, in Sec. \[sei\] we give our conclusions.
Newtonian approach to two body problem in Yukawa potential {#due}
==========================================================
The starting point is the equation of motion of a massive point-like particle, $m$, in the gravitational potential well generated by the particle $M$, and given in Eq. . In polar coordinates $(r,\varphi)$ and with respect to the center of mass, the equations of motion read $$\begin{aligned}
\label{eq:1} & \ddot{r} = -\nabla\Phi(r)\,,\\
\label{eq:2} & \frac{d}{dt}(r^2 \dot{\varphi}) = 0 \,, \end{aligned}$$ and the total energy of the system can be written as [@deLa2011] $$\label{eq:energy}
E_T = \frac{1}{2}\mu(\dot{r}^2 + r^2\dot{\varphi}^2)-\frac{Gm M}{(1+\delta)r}(1+\delta e^{-r/\lambda}),$$ where $\displaystyle{\mu=\frac{mM}{m+M}}$ is the reduced mass, and $\Phi(r)$ is the modified gravitational potential of Eq. . Using the conservation of the angular momentum $L$ expressed in Eq. , it is straightforward to recast the total energy as a function of the radial coordinate: $$\label{eq:energy2}
E_T = \frac{1}{2}\mu\dot{r}^2 + \frac{L^2}{2\mu r^2} -\frac{Gm M}{(1+\delta)}\frac{(1+\delta e^{-r/\lambda})}{r}.$$
Eq. is the only one needed to compute the equation of motion for an unperturbed orbit. Nevertheless, we can learn more about the orbits by defining an effective potential as $$\label{eq:Veff}
V_{\rm eff}(r) = \frac{L^2}{2\mu r^2} - \frac{Gm M}{(1+\delta) r} - Gm M \frac{\delta}{(1+\delta) r} e^{-r/\lambda}.$$ Here, the first term accounts for the repulsive force associated to the angular momentum, the second term represents the gravitational attraction, and the third term can be interpreted as an additional force due to the Yukawa-like term in the gravitational potential acting on the particle. The effective potential demands other considerations: first, one needs $\delta \neq -1$ in order to avoid a singularity in the second and third terms; second, if $\delta$ assumes negative values, the second term stays attractive as far as the condition $ \delta > -1 $ is satisfied, and the last term becomes repulsive; third, the condition $\delta <-1$ makes the second term repulsive, rendering the third term attractive; fourth, if $\delta >0$ then both second and third terms are attractive.
For illustration, in Fig. \[fig1\](a) and (b) we plot the potential and the effective potential as a function of $r/\lambda$ showing their dependence on the strength of the Yukawa term. Notice that the minimum of the effective potential depends on the strength parameter $\delta$ of the Yukawa-term (Fig. \[fig1\](b)). As expected, a negative value of $\delta$ makes the potential well deeper as compared to the Newtonian case ($\delta=0$), while a positive value makes it flatter. This can be understood looking at Eq. , for $-1<\delta<0$ the effective mass $M'= M/(1+\delta)$ becomes larger, while for $\delta>0$ it becomes smaller than the “Newtonian mass" $M$.
{width="8.6cm"} {width="8.6cm"}
Differentiating with respect to the radial coordinate and looking for the minimum, one finds the condition: $$\frac{d V_{\rm eff}(r)}{dr} =0 \Rightarrow \frac{L^2}{\mu r} = \frac{G m M \left(\delta e^{-\frac{r}{\lambda}}+1\right)}{(\delta +1)}+\frac{\delta G m M e^{-\frac{r}{\lambda}}}{(\delta +1) \lambda}r\,.$$ The second derivative and the previous condition on the angular momentum leads to obtain the following expression $$\frac{d^2V_{\rm eff}(r)}{dr^2}= \frac{G m M e^{-\frac{r}{\lambda}}}{(\delta +1)r^3}\biggl[ \delta(-\lambda^{-2}r^2
+ \lambda^{-1}r +1) + e^{\frac{r}{\lambda}} \biggr].$$ A minimum in the effective potential exists if the following condition is satisfied $$\label{eq:condition1}
g(x)\equiv \delta(-x^2 + x +1) + e^x>0\,,$$ where we have defined $x\equiv r/\lambda$. Eq. is satisfied in the following cases: (i) $\delta>-1$ for $x\rightarrow0$, (ii) $\delta>-e$ for $x\rightarrow1$, and (iii) $\forall\,\delta$ in the limit $x\rightarrow\infty$. Let us notice that the first case, meaning $r\ll \lambda$, is the common configuration of an astrophysical system with its dynamics happening at scales much lower than the Compton wavelength of the massive graviton, such as planetary motion around the Sun. On the contrary, the second case ($r\sim\lambda$) represents systems such as S-Stars around the Galactic center, with their dynamics happening at scales of the order of parsecs. Finally, the last case ($r\gg \lambda$) can be associated to the extragalactic and cosmological scales. Since we are interested in studying systems on distance scales much smaller than the Compton scale of a massive graviton, the exponential term in previous equations Taylor expanded as $$\label{eq:approx_exp}
e^{\pm x} \approx 1 \pm x + \frac{x^2}{2} + \mathcal{O}\bigl(x^3\bigr).$$ When replacing Eq. in to Eq. , the first term gives the Newtonian force, the second term induces a shift in the energy of the system, and the third term gives rise to a constant radial acceleration (often called fifth force) that can be written as follows $$a_{\rm corr} = -\frac{a^*\delta}{2(1+\delta)}\frac{r^{*2}}{\lambda^2},$$ where $a^*$ is the Newtonian acceleration of an object at distance $r^*$. As an example, this correction can be applied to the Pioneer anomaly, thus obtaining a strength $|\delta|\leq 1.7\times10^{-4}$ at $\lambda\sim200$ AU [@Anderson1998; @Anderson2002]. It is important to remark that the approximation in Eq. is valid only for dynamics at the scale of planetary systems or stars with orbits having their semi-major axis much smaller than $\lambda$. In contrast, to study the dynamics of systems on larger scales, one cannot use the approximation in Eq. but rather, the equations of motion must be integrated numerically.
Let us analyze the condition for the existence a minimum in the effective potential at both $\mathcal{O}(x^{2})$ and $\mathcal{O}(x^{3})$ orders:
$\mathcal{O}(x^{2})$ order
: at this order of approximation, the effective potential becomes $$V_{eff}(r) = \frac{L^2}{2 \mu r^2} - \frac{G m M}{r}+\frac{\delta G m M}{(\delta +1) \lambda},$$ and we find the minimum at the radius $$\label{eq:rmin_order1}
r_{min} = \frac{L^2}{2\mu G m M},$$ which is the same as the one of Newtonian gravity (as expected), while the effective potential at the minimum is shifted with respect to the Newtonian one $$\begin{aligned}
\label{eq:veffmin_order1}
&&V_{eff, min}= - \frac{1}{2} G m M \left(\frac{G \mu m M}{L^2} -\frac{2 \delta }{(\delta +1)\lambda}\right)\,,\nonumber\\
\end{aligned}$$
$\mathcal{O}(x^{3})$ order
: the effective potential can be recast as $$\begin{aligned}
\label{eq:veff_order2}
&&V_{eff}(r) = \frac{L^2}{2 \mu r^2} - \frac{G m M}{r}+\frac{\delta G m M}{(\delta +1) \lambda} -\frac{\delta G m M r}{2 (\delta +1) \lambda^2}\,.\nonumber\\
\end{aligned}$$ Since we are looking for a strength force in the regime $\delta\ll1$, thus meaning a small deviation from Newtonian dynamic, the shift in $r_{min}$ is absolutely negligible. Thus, replacing Eq. in to Eq. we get $$\begin{aligned}
\label{eq:veffmin_order2}
V_{eff, min}& =& - \frac{G m M }{2} \left(\frac{G \mu m M}{L^2} -\frac{2 \delta }{(\delta+1)\lambda}\right) \nonumber\\
&-&\frac{L^2 \delta }{2 (1+\delta ) \lambda ^2 \mu }\,.
\end{aligned}$$
Therefore, at both $\mathcal{O}(\lambda^{-2})$ and $\mathcal{O}(\lambda^{-3})$ orders, the minimum of the effective potential always exists and it is located at the same radius ($r_{min}$) than in the Newtonian case. Finally, Eqs. and show that the minimum of the effective potential is shifted as qualitatively explained above and shown in Fig. \[fig1\]b.
Equation of the orbits {#tre}
======================
Hereby, we compute the equation of the closed orbit in both $\mathcal{O}(x^{2})$ and $\mathcal{O}(x^{3})$ approximations, and we show that, under some conditions on the eccentricity and the position of latus rectum, the orbit can be recast into the usual Keplerian form, where the correction due to the Yukawa-term getting hidden into the orbital parameters. We work in the regime $r\ll\lambda$ in order to replace the exponential term in Eq. with Eq. .
Approximation at $\mathcal{O}(x^{2})$-order {#treA}
-------------------------------------------
To compute the equation of the orbit we rewrite the radial component of the velocity as $$\dot{r}=-\dfrac{L}{\mu}\dfrac{d}{d\varphi}\dfrac{1}{r}\,,$$ then, at second order in the approximation of the Yukawa-term, the total energy of the system can be recast as $$\label{eq:energy3}
E_T = \frac{L^2}{2\mu}\left(\dfrac{d}{d\varphi}\dfrac{1}{r}\right)^2 + \frac{L^2}{2\mu r^2} -\frac{Gm M}{r} + \frac{Gm M \delta}{(1+\delta)\lambda}.$$ From the previous equation we can obtain the following differential equation $$\label{eq:energy4}
u'^2+u^2 - 2\beta_0 u = \beta_1,$$ where $u\equiv1/r$, $u'=du/d\varphi$ and $$\label{eq:betas01}
\gamma = G m M; \qquad \beta_0=\frac{\mu\gamma}{L^2}; \qquad \beta_1= \frac{2\mu E_T}{L^2}- \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}.$$ Differentiating Eq. , we get $$\label{eq:energy5}
u'\biggr(u''+u- \beta_0\biggr) = 0.$$ As we are looking for a Keplerian solution, we make the following [*ansatz*]{}: $$\label{eq:orbit0}
u\equiv\frac{1}{r}=\frac{1}{l}(1+\epsilon\cos\varphi),$$ where $l$ is the [*latus rectum*]{} and $\epsilon$ is the eccentricity. Therefore, inserting the Eq. in Eq. , we obtain the following condition for the [*latus rectum*]{}: $$\label{eq:orbit1}
l=\frac{1}{\beta_0}.$$ Then, we substitute Eqs. into Eq. thus obtaining the following expression for the eccentricity: $$\label{eq:orbit2}
\epsilon^2=1 + l^2 \beta_1,$$ that in terms of energy of the system is $$\label{eq:orbit3}
\epsilon^2 = 1 - \frac{2 L^2}{\mu\gamma} \frac{\delta}{(1+\delta)\lambda} + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2},$$ which for $\delta=0$ gets reduced to the Newtonian value: $$\label{eq:orbit3_1}
\epsilon^2 = 1 + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2}.$$ This shift is clearly not testable with observations given that we measure the orbital parameters, whereas the total energy is a theory dependent parameter. Nevertheless, looking at Eq. , it is straightforward to understand that, if the total energy and angular momentum are fixed then they correspond to an orbital motion with an eccentricity that would vary depending on the strength of the Yukawa correction, as shown in Figure \[fig2\].
![Illustration of the effect of the modified gravitational potential on the orbital parameters. Panel (a) shows the orbits for different values of $\delta$. The angular momentum and the total energy are set to those values that give rise to an elliptical orbit with eccentricity $\epsilon=0.5$ in Newtonian mechanics ($\delta=0$) showing that such an orbital solution would show a difference in the eccentricity when the Yukawa term is taken in to account. Panel (b) shows the relative difference with the Newtonian mechanics along the orbit.[]{data-label="fig2"}](fig2a.eps "fig:"){width="8.6cm"} ![Illustration of the effect of the modified gravitational potential on the orbital parameters. Panel (a) shows the orbits for different values of $\delta$. The angular momentum and the total energy are set to those values that give rise to an elliptical orbit with eccentricity $\epsilon=0.5$ in Newtonian mechanics ($\delta=0$) showing that such an orbital solution would show a difference in the eccentricity when the Yukawa term is taken in to account. Panel (b) shows the relative difference with the Newtonian mechanics along the orbit.[]{data-label="fig2"}](fig2b.eps "fig:"){width="8.6cm"}
Approximation at $\mathcal{O}(x^{3})$ order {#treB}
-------------------------------------------
Approximating the Yukawa-term at third order, the differential equation becomes $$\label{eq:energy6}
u'^2+u^2- 2\beta_0u - \beta_2\frac{1}{u} = \beta_1,$$ where $\beta_0$ and $\beta_1$ are given in Eq. , and $$\label{eq:beta2}
\beta_2=\frac{\mu\gamma\delta}{2 L^2\lambda(1+\delta)}.$$ By taking the derivative of Eq. we obtain $$\label{eq:energy7}
u'\left(u''+u +\frac{\beta_2}{u^2}-\beta_0\right)=0\,.$$
Let us introduce Eq. into Eq. and evaluate the expression at $\varphi=[0; \pi]$, which respectively correspond to the minimum and maximum distance between the two masses. Thus, we obtain two conditions: $$\begin{aligned}
& \label{eq:orbit4} (1 - l \beta_0) \epsilon^2 + 2 (1 - l \beta_0) \epsilon - l \beta_0 + l^3 \beta_2 +1 =0\,, \\
& \label{eq:orbit5} (1 - l \beta_0) \epsilon^2 - 2 (1 - l \beta_0) \epsilon - l \beta_0 + l^3 \beta_2 +1 =0\,.\end{aligned}$$ Subtracting Eqs. and we obtain the latus rectum which turns out to have the same expression as in Eq. . Finally, introducing Eq. in Eq. and evaluating it, once again, at $\varphi=[0; \pi]$ we obtain the following condition for the eccentricity $$\epsilon^2= 1+ l^2\beta_1-4 \beta_2.$$ Let us note that the previous expression reduces to Eq. when $\beta_2=0$, and thus to the Newtonian value when $\delta=0$. The previous equation can be straightforwardly recast in terms of energy of the system as $$\label{eq:orbit6}
\epsilon^2 = 1 + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2} - \frac{2 L^2}{\mu\gamma} \frac{\delta}{(1+\delta)\lambda} - \frac{2\mu\gamma\delta}{L^2\lambda(1+\delta)}.$$ The $\mathcal{O}(x^{3})$ order the shift is larger than at $\mathcal{O}(x^{2})$ order, and the difference due to the order of approximation is not negligible (see Fig. \[fig3\]).
![The plots and panels replicate the ones in Fig. \[fig2\] for the $\mathcal{O}(\lambda^{-3})$ approximation order.[]{data-label="fig3"}](fig3a.eps "fig:"){width="8.6cm"} ![The plots and panels replicate the ones in Fig. \[fig2\] for the $\mathcal{O}(\lambda^{-3})$ approximation order.[]{data-label="fig3"}](fig3b.eps "fig:"){width="8.6cm"}
Precession in Yukawa potential {#quattro}
==============================
To compute analytically the periastron advance due to the Yukawa-like term in the gravitational potential, we study small perturbations to the circular orbit. Thus, let us recast the total energy as $$\label{eq:prec1}
u'^2+u^2+ \frac{g(u)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta},$$ where $ g(u)$ account for the gravitational interaction. Let us impose a close orbit defined by a minimum and a maximum distance from the center: $r_-|_{\varphi=0} =a(1-\epsilon) $ and $r_+|_{\varphi=\pi} =a(1+\epsilon)$, respectively. Here $a$ is the semi-major axis of the orbit. Thus, those correspond to $u_0=1/r_-$ and $u_1=1/r_+$. Being $u'|_{u=u_0}=u'|_{u=u_1}=0$, the Eq. gives rise to the following two conditions $$\begin{aligned}
& u_0^2+ \frac{g(u_0)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}\,,\\
& u_1^2+ \frac{g(u_1)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}\,,\end{aligned}$$ from which one obtains $$\begin{aligned}
& L^2 = \frac{g(u_0) - g(u_1)}{u_1^2 - u_0^2}\,,\\
& E_T = \frac{u_1^2g(u_0) - u_0^2g(u_1)}{2\mu(u_1^2 - u_0^2)} + \frac{\mu\gamma\delta}{\mu(1+\delta)\lambda}\,.\end{aligned}$$ Then, the differential equation becomes $$\label{eq:prec2}
u' = \sqrt{G(u_0,u_1,u)}\,,$$ where $G(u_0,u_1,u)$ is
$$\begin{aligned}
G(u_0,u_1,u)=\frac{g(u_0)(u_1^2-u^2) + g(u_1)(u^2-u_0^2)-(b^2 - u_0^2)g(u)}{g(u_0) - g(u_1)}\, .\end{aligned}$$
We can find the amount of angle required to pass from $r_-$ to $r_+$ by integrating equation : $$\label{eq:prec_angle}
\varphi(r_+)-\varphi(r_-) = \int_{u_0}^{u_1} G(u_0,u_1,u)^{-1/2}du\,.$$ Hence the particle will move from $r_-$ to $r_+$ and back every time $\varphi\rightarrow\varphi+2\pi$, thus $r(\varphi)$ is periodic with period $2\pi$. Therefore, the precession for each revolution is $$\omega = 2|\varphi(r_+)-\varphi(r_-)| - 2\pi.$$
In the case of approximating the exponential term at $\mathcal{O}(x^{2})$ order, the function $g(u)$ only depends by a Newtonian term: $$\label{eq:prec3}
g(u) = -2\mu\gamma u = 2 \mu \Phi_N(1/u),$$ where $\Phi_N(1/u)$ is the classical Newtonian potential. Thus, the precession does not exist as expected for the Newtonian potential.
Nevertheless, when approximating the exponential term at $\mathcal{O}(\lambda^{-3})$ order we have $$\label{eq:prec4}
g(u) = 2 \mu \Phi_N(1/u) - \frac{\mu\gamma\delta}{\lambda(1+\delta)} \frac{1}{u}.$$ In order to solve the integral in Eq. we perform a change of variables $$u_1 = u_0+\eta; \qquad u= u_0+\eta\upsilon\,,$$ with $0<\upsilon<1$. Then, the Eq. can be recast as $$\label{eq:prec7}
\Delta \varphi \equiv \varphi(r_+)-\varphi(r_-) = \eta \int_0^1 g(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon)d\upsilon\,,$$ where $$g(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon) = \frac{1}{\sqrt{G(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon)}}.$$
Finally, defining the auxiliary variable $\xi\equiv(1+\delta)\lambda^2$, we find
$$\begin{aligned}
\label{eq:precYuk}
\Delta \varphi &= \pi\sqrt{1+\frac{2 \delta }{-3 \delta +2 u_0^2 \xi }} \biggl(1 -\frac{2 u_0 \delta \xi}{3 \delta ^2-8 u_0^2 \delta \xi +4 u_0^4 \xi^2} \eta
+ \frac{ \delta \left(-3 \delta ^3+16 u_0^2 \delta ^2 \xi -124 u_0^4 \delta \xi^2+144 u_0^6 \xi ^3\right)}{16 \left(3 u_0 \delta ^2-8 u_0^3 \delta \xi +4 u_0^5 \xi^2\right)^2}\eta ^2\biggr)\end{aligned}$$
To bound the strength $\delta$, we use the motion of the Solar system’s planets. Specifically, we use Mercury, Venus, Earth, Mars, Jupiter and Saturn for which the orbital precession has been measured [@Nyambuya2010]. We identify the allowed region of $\delta$ for which the predicted precession does not contradict the data. In Figure \[fig4\] we show the allowed zone of parameter space for each planet (light blue shades), and we also show that for $-1<\delta<0$ the precession in ongoing in the opposite direction with respect the observed one, while $\delta>0$ give rise to a precession in the right direction confirming the results found for $R^n$ gravity using the S-stars orbiting around the Galactic Center [@Borka20012; @Borka20013]. This results was rather expected since the effect produced by the modification of the gravitational potential must be greater or lower than the Newtonian one that is zero. The scale length has been fixed to the confidence value $\lambda=5000$ AU [@Zakharov2016], and a lower and upper limit on $\delta$ is inferred, and reported in Table \[tab:1\]. The tightest interval on $\delta$ is obtained with Saturn that is located at the highest distance from the Sun. This restricts $\delta$ to vary in the range from $2.70\times10^{-9}$ to $6.70\times10^{-9}$. With these values of the strength we have also predicted the precession for Uranus, Neptune and Pluto[^1], and we found a precession up to three order of magnitude larger than the one predicted by General Relativity meaning that the strength must be even smaller than $fews \times 10^{-9}$ to match the general relativistic constraints. All results are summarized in Table \[tab:2\].
{width="8.9cm"} {width="8.9cm"}\
{width="8.9cm"} {width="8.9cm"}\
{width="8.9cm"} {width="8.9cm"}\
[l c c c c c c c]{}\
&\
&\
**Planet** & & & $i$ & $\epsilon$ & $\dot{\omega}_{obs}$ & $\dot{\omega}_{GR}$ & $[\delta_{min}; \delta_{max}]$\
& ([AU]{}) & ([yrs]{}) & (degrees) & &\
\
[**Mercury**]{} & ${0.39}$ & ${0.24}$ & ${7.0}$ & ${0.206}$ & $43.1000\pm0.5000$ & $43.5$ & $[1.02;\, 1.09]\times10^{-2}$\
[**Venus**]{} & ${0.72}$ & ${0.62}$ & ${3.4}$ & ${0.007}$ & $8.0000\pm5.0000$ & $\,\,\,8.62$ & $[-0.76;\, 2.51]\times10^{-3}$\
[**Earth**]{} & ${1.00}$ & ${1.00}$ & ${0.0}$ & ${0.017}$ & $5.0000\pm1.0000$ & $\,\,\,3.87$ & $[1.45;\, 5.79]\times10^{-4}$\
[**Mars**]{} & ${1.52}$ & ${1.88}$ & ${1.9}$ & ${0.093}$ & $1.3624\pm0.0005$ & $\,\,\,1.36$ & $[5.90;\, 5.92]\times10^{-5}$\
[**Jupiter**]{} & ${5.20}$ & ${11.86}$ & ${1.3}$ & ${0.048}$ & $0.0700\pm0.0040$ & $\,\,\,0.0628$ & $[0.92;\, 1.30]\times10^{-7}$\
[**Saturn**]{} & ${9.54}$ & ${29.46}$ & ${2.5}$ & ${0.056}$ & $0.0140\pm0.0020$ & $\,\,\,0.0138$ & $[2.70;\, 6.70]\times10^{-9}$\
\
\
[l c c c c c c]{}\
&\
&\
**Planet** & & & $i$ & $\epsilon$ & $\dot{\omega}_{GR}$ & $[\dot{\omega}_{min}; \dot{\omega}_{max}]|$\
& ([AU]{}) & ([yrs]{}) & (degrees) &\
\
[**Uranus**]{} & ${19.2}$ & ${84.10}$ & ${0.8}$ & ${0.046}$ & $\,\,\,0.0024$ & $[0.05;\,0.12]$\
[**Neptune**]{} & ${30.1}$ & ${164.80}$ & ${1.8}$ & ${0.009}$ & $\,\,\,0.00078$ & $[0.18;\,0.45]$\
[**Pluto**]{} & ${39.4}$ & ${247.70}$ & ${17.2}$ & ${0.250}$ & $\,\,\,0.00042$ & $[0.11;\,0.30]$\
\
\
Implications for $f(R)$ gravity {#cinque}
===============================
To make compatible $f(R)$ models with local gravity constraints, these theories usually require a “screening mechanism”. When considering theories with a non-minimally coupled scalar field, one has to impose strong conditions on the effective mass of the scalar field that must depend on the space-time curvature or, alternatively, on the matter density distribution of the environment [@Khoury2004; @Khoury2009]. Thus, the scalar field can have a short range at Solar System scale escaping the experimental constraints, and have a long range at the cosmological scale, where it can propagate freely affecting the cosmological dynamics, and driving the accelerated expansion (see for details [@defelice2010]). With the same aim, similar mechanisms have been proposed for other models, such as the symmetron and the braneworld [@Dvali2000; @Nicolis2009; @Hinterbichler2010]. [ Nevertheless, these mechanisms are introduced [*ad hoc*]{} and particularized for each theory. In $f(R)$ gravity, the need of introducing a screening mechanism arises when, instead of working with higher order field equations, one performs a conformal transformation from the Jordan to the Einstein frame, where the field equations are of second order but a scalar field, related to the $f'(R)$ term, appears. Although it is simpler to work with second order field equations, and the two frames are mathematically equivalent, one should remember that the physical equivalence is not guaranteed in general [@Magnano; @Faraoni; @darkmetric]. Thus, one could prefer to work with high order field equations, staying in the Jordan frame, and handling the extra degrees of freedom as free parameters to be constrained by the data. In such a case, the scale dependence of these parameters plays the role of the screening mechanism. The screening mechanism is traced by the density of the self gravitating systems [@chameleon].]{}
Relatively, the results in Table \[tab:1\] can be straightforwardly interpreted as the fact that the Yukawa correction term to the Newtonian gravitational potential is screened at planetary scales. Indeed, the departure from Newtonian gravity is of the order of $10^{-9}$ in $\delta$. Finally, the values of the strength and the scale of the Yukawa potential highly degenerate at such small scales. To illustrate this degeneracy we have computed the lower and upper limit on $\delta$ varying $\lambda$ from 100 AU to $10^{4}$ AU. The results are shown in Fig. \[fig5\], where we have highlighted the parts of the parameter space that are (and are not) allowed. We show that a change of one order of magnitude in the scale length is reflected in change up to two order of magnitude in $\delta$. The plot is particularized for Saturn.
![Degeneracy between the strength and the scale length of the Yukawa gravitational potential. The plot is particularized for the case of Saturn.[]{data-label="fig5"}](fig5.eps){width="8.6cm"}
Conclusions and Remarks {#sei}
=======================
Measurements of the orbital precession of Solar System bodies can be used to compare observations with theoretical predictions arising from alternative theories of gravity. Specifically, $f(R)$ gravity models that, in their weak field limit, show a Yukawa-like correction to the Newtonian gravitational potential can be used to compute the orbital precession with a classical mechanics approach. We have computed an analytical expression for the orbital precession and compared its prediction with the values for the Solar System’s planets. We found that, fixing the characteristic scale length to $\lambda=5000$ AU [@Zakharov2016], the strength must rely in the range $[2.70; 6.70]\times10^{-9}$. Nevertheless, we must point out the presence of a degeneracy between the strength and the scale of the Yukawa potential. We find the direction of the orbital precession changing with the sign of the strength, confirming previous results [@Borka20012; @Borka20013]. If the change of the direction of the orbital precession can be used as an effective way to discriminate between General Relativity and its alternative, should be studied in a full relativistic approach where the motion happens along the geodesics [@pII].
Acknowledgements {#acknowledgements .unnumbered}
================
I.D.M acknowledge financial supports from University of the Basque Country UPV/EHU under the program “Convocatoria de contratación para la especialización de personal investigador doctor en la UPV/EHU 2015”, from the Spanish Ministerio de Economía y Competitividad through the research project FIS2017-85076-P (MINECO/AEI/FEDER, UE), and from the Basque Government through the research project IT-956-16. M.D.L. is supported by the ERC Synergy Grant “BlackHoleCam” – Imaging the Event Horizon of Black Holes (Grant No. 610058). M.D.L. acknowledge INFN Sez. di Napoli (Iniziative Specifiche QGSKY and TEONGRAV). This article is based upon work from COST Action CA1511 Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA), supported by COST (European Cooperation in Science and Technology).
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[^1]: Although the latter is not a planet, its large distance from the Sun and its small mass makes the object very useful to show the impact of the modified gravitational potential.
| ArXiv |
---
abstract: 'We study the effect of starlight from the first stars on the ability of other minihaloes in their neighbourhood to form additional stars. The first stars in the $\Lambda$CDM universe are believed to have formed in minihaloes of total mass $\sim 10^{5-6}\,M_\odot$ at redshifts $z\ga 20$, when molecular hydrogen ($\rm H_2$) formed and cooled the dense gas at their centres, leading to gravitational collapse. Simulations suggest that the Population III (Pop III) stars thus formed were massive ($\sim 100\,M_\odot$) and luminous enough in ionizing radiation to cause an ionization front (I-front) to sweep outward, through their host minihalo and beyond, into the intergalactic medium. Our previous work suggested that this I-front was trapped when it encountered other, nearby minihaloes, and that it failed to penetrate the dense gas at their centres within the lifetime of the Pop III stars ($\la 3\,\rm Myrs$). The question of what the dynamical consequences were for these target minihaloes, of their exposure to the ionizing and dissociating starlight from the Pop III star requires further study, however. Towards this end, we have performed a series of detailed, 1D, radiation-hydrodynamical simulations to answer the question of whether star formation in these surrounding minihaloes was triggered or suppressed by radiation from the first stars. We have varied the distance to the source (and, hence, the flux) and the mass and evolutionary stage of the target haloes to quantify this effect. We find: (1) trapping of the I-front and its transformation from R-type to D-type, preceded by a shock front; (2) photoevaporation of the ionized gas (i.e. all gas originally located outside the trapping radius); (3) formation of an $\rm H_2$ precursor shell which leads the I-front, stimulated by partial photoionization; and (4) the shock- induced formation of $\rm H_2$ in the minihalo neutral core when the shock speeds up and partially ionizes the gas. The fate of the neutral core is mostly determined by the response of the core to this shock front, which leads to molecular cooling and collapse that, when compared to the same halo without external radiation, is either: (a) expedited, (b) delayed, (c) unaltered, or (d) reversed or prevented, depending upon the flux (i.e. distance to the source) and the halo mass and evolutionary stage. When collapse is expedited, star formation in neighbouring minihaloes or in merging subhaloes within the host minihalo sometimes occurs [*within*]{} the lifetime of the first star. Roughly speaking, most haloes that were destined to cool, collapse, and form stars in the absence of external radiation are found to do so even when exposed to the first Pop III star in their neighbourhood, while those that would not have done so are still not able to. A widely held view that the first Pop III stars must exert either positive or negative feedback on the formation of the stars in neighbouring minihaloes should, therefore, be revisited.'
author:
- |
Kyungjin Ahn[^1] and Paul R. Shapiro[^2]\
Department of Astronomy, The University of Texas at Austin, 1 University Station C1400, Austin, TX 78712, USA
title: 'Does Radiative Feedback by the First Stars Promote or Prevent Second Generation Star Formation?'
---
cosmology: large-scale structure of universe – cosmology: theory – early universe – stars: formation – galaxies: formation
Introduction {#sec:Secondstar-Intro}
============
Cosmological minihaloes at high redshift – i.e. dark-matter dominated haloes with virial temperatures $T_{\rm vir} < 10^4 \,\rm K$, with masses above the Jeans mass in the intergalactic medium (IGM) before reionization ($10^4 \la M/M_\odot \la 10^8$) – are believed to have been the sites of the first star formation in the universe. To form a star, the gas inside these haloes must first have cooled radiatively and compressed, so that the baryonic component could become self-gravitating and gravitational collapse could ensue. For the neutral gas of H and He at $T < 10^4\,\rm K$ inside minihaloes, this requires that a sufficient trace abundance of $\rm H_2$ molecules formed to cool the gas by atomic collisional excitation of the rotational-vibrational lines of $\rm H_2$ . The formation of this trace abundance of $\rm H_2$ proceeds via the creation of intermediaries, $\rm H^-$ or $\rm H_{2}^{+}$, which act as catalysts, which in turn requires the presence of a trace ionized fraction, in the following two-step gas-phase reactions (see, e.g., @1968ApJ...154..891P [@1967Natur.216..976S; @1984ApJ...280..465L; @1987ApJ...318...32S]; @1994ApJ...427...25S, henceforth, “SGB94”; ): $$\begin{aligned}
&&{\rm H + e^- \rightarrow H^- + \gamma},\nonumber \\
&&{\rm H^- + H \rightarrow H_2 + e^-},
\label{eq:solomon}\end{aligned}$$ and $$\begin{aligned}
&&{\rm H + H^+ \rightarrow H_{2}^{+} + \gamma},\nonumber \\
&&{\rm H_{2}^{+} + H \rightarrow H_2 + H^+}.
\label{eq:solomon2}\end{aligned}$$ Unless there is a strong destruction mechanism for $\rm H^-$ (e.g. cosmic microwave background at $z\ga 100$), the former (equation \[eq:solomon\]) is generally the dominant process for $\rm H_2$ formation.
Gas-dynamical simulations of the Cold Dark Matter (CDM) universe suggest that the first stars formed in this way when the dense gas at the centres of minihaloes of mass $M \sim 10^{5 - 6}\, M_\odot$ cooled and collapsed gravitationally at redshifts $z \ga 20$ (e.g. @2000ApJ...540...39A [@2002Sci...295...93A]; @1999ApJ...527L...5B [@2002ApJ...564...23B]; @2003ApJ...592..645Y; @2001ApJ...548..509M [@2003MNRAS.338..273M]; @2006astro.ph..6106Y). This work and others further suggest that these stars were massive ($M_* \ga 100 \,M_\odot$), hot ($T_{\rm eff} \simeq 10^5 \,\rm K$), and short-lived ($t_* \la 3 \,\rm
Myrs$), thus copious emitters of ionizing and dissociating radiation.
These stars constitute the Population III (Pop III) stars, or zero metallicity stars, which are believed to have exerted a strong, radiative feedback on their environment. The details of this feedback and even the overall sign (i.e. negative or positive) are poorly understood. Once the ionizing radiation escaped from its halo of origin, it created H II regions in the IGM, beginning the process of cosmic reionization. The photoheating which accompanies this photoionization raises the gas pressure in the IGM, thereby preventing baryons from collapsing gravitationally out of the IGM into new minihaloes when they form inside the H II regions, an effect known as “Jeans-mass filtering” (SGB94; @1998MNRAS.296...44G; @2003MNRAS.346..456O). Inside the H II regions, whenever the I-fronts encounter pre-existing minihaloes, those minihaloes are subject to photoevaporation (@2004MNRAS.348..753S [henceforth, SIR]; @2005MNRAS.361..405I [henceforth, ISR]). A strong background of UV photons in the Lyman-Werner (LW) bands of $\rm H_2$ also builds up which can dissociate molecular hydrogen inside minihaloes even in the neutral regions of the IGM, thereby disabling further collapse and, thence, star formation (e.g. @1999ApJ...518...64O; @2000ApJ...534...11H; @2001ApJ...546..635O). This conclusion changes, however, if some additional sources of partial ionization existed to stimulate $\rm H_2$ formation without heating the gas to the usually high temperature of fully photoionized gas ($\sim 10^4
\,\rm K$) at which collisional dissociation occurs, such as X-rays from miniquasars [@1996ApJ...467..522H] or if stellar sources create a partially-ionized boundary layer outside of intergalactic H II regions [@2001ApJ...560..580R]. Such positive feedback effects, however, may have been only temporary, because photoheating would soon become effective as background flux builds up over time [@2006MNRAS.368.1301M].
The study of feedback effects has been limited mainly by technical difficulties. @2000ApJ...534...11H studied the feedback of LW, ultraviolet (UV), and X-ray backgrounds on minihaloes without allowing hydrodynamic evolution. @2001ApJ...560..580R studied the radiative feedback effect of stellar sources only on a static, uniform IGM. @2002ApJ...575...33R [@2002ApJ...575...49R] studied stellar feedback more self-consistently by performing cosmological hydrodynamic simulations with radiative transfer, but the resolution of these simulations is not adequate for resolving minihaloes. @2001ApJ...548..509M [@2003MNRAS.338..273M] also performed cosmological hydrodynamic simulations, with higher resolution, but radiative feedback was treated assuming the optically thin limit, which overestimates the ionization efficiency, especially in the high density regions which would initially be easily protected from ionizing radiation due to their high optical depth. The first self-consistent, radiation-hydrodynamical simulations of the feedback effect of external starlight on cosmological minihaloes were those of SIR and ISR, who studied the encounter between the intergalactic I-fronts that reionized the universe and individual minihaloes along their path. These simulations used Eulerian, grid-based hydrodynamics with radiative transfer and adaptive mesh refinement (AMR) to “zoom-in” with very high resolution, to demonstrate that the I-fronts from external ionizing sources are trapped when they encounter minihaloes, slowing down and transforming from weak, R-type to D-type, preceded by a shock. The gas on the ionized side of these I-fronts was found to be evaporated in a supersonic wind, and, if the radiative source continued to shine for a long enough time, the I-front eventually penetrated the minihaloes entirely and expelled all of the gas. These simulations elucidated the impact of the I-front and the physical effects of ionizing radiation on minihalo gas, quantifying the timescales and photon consumption required to complete the photoevaporation. They did not, however, address the aftermath of “interrupted” evaporation, when the source turns off before evaporation is finished.
Recent studies by @2005ApJ...628L...5O, @2006ApJ...639..621A, and @2006astro.ph..4148M addressed this question for minihaloes exposed to the radiation from the first Pop III star in their neighbourhood, instead of the effect of either a steadily-driven I-front during global reionization or a uniform global background. The results of @2005ApJ...628L...5O and @2006astro.ph..4148M are seriously misleading, however, since they did not account properly for the optical depth to hydrogen ionizing photons.
@2005ApJ...628L...5O assumed that the UV radiation from the first Pop III star that formed inside a minihalo in some region would fully ionize the gas in the neighbouring minihaloes. Using 3D hydrodynamics simulations, they found that, when the star turned off, $\rm H_2$ molecules formed in the dense gas that remained at the centre of the neighbouring minihalo, fast enough to cool the gas radiatively and cause gravitational collapse leading to more star formation. The $\rm H_2$ formation mechanism was the same as that described by @1987ApJ...318...32S, in which ionized gas of primordial composition at a temperature $T\ga 10^4 \,\rm K$ cools radiatively and recombines out of ionization equilibrium, enabling an enhanced residual ionized fraction to drive reaction (1) (and \[2\], as well) as the temperature falls below the level at which collisional dissociation suppresses molecule formation. As a result, @2005ApJ...628L...5O concluded that the radiative feedback of the first Pop III stars was positive, triggering a second generation of star formation in the minihaloes surrounding the one that hosted the first star.
@2006astro.ph..4148M also used 3D hydrodynamics simulations to consider the fate of the gas in the relic H II regions created by the first Pop III stars. they concluded that the radiative feedback of the first stars could be either negative or positive and estimated a critical UV intensity which would mark the transition from negative to positive feedback. @2006astro.ph..4148M, however, studied this effect only in the optically thin limit, as had also been done by @2001ApJ...548..509M [@2003MNRAS.338..273M]. The main mechanisms of the positive feedback effect in @2005ApJ...628L...5O and @2006astro.ph..4148M are, therefore, identical.
@2006ApJ...639..621A, on the other hand, performed a high-resolution ray-tracing calculation to track the position of the I-front created by the first Pop III star as it swept outward in the density field of a 3D cosmological SPH simulation of primordial star formation in the $\Lambda$CDM universe over the lifetime of the star. When this I-front encountered the minihaloes in the neighbourhood of the one which hosted the first Pop III star, it was trapped by the minihalo gas before it could reach the high-density region (core), due to the minihalo’s high column density of neutral hydrogen. This is consistent with the results of SIR and ISR mentioned above. According to @2006ApJ...639..621A, in fact, the lifetime of the Pop III star is less than the evaporation times determined by SIR and ISR for the relevant minihalo masses and flux levels in this case, so the neutral gas in the core is never ionized by the I-front. It seems that the initial assumption of full ionization of nearby haloes by @2005ApJ...628L...5O and the optically thin limit assumed by @2006astro.ph..4148M are invalid.
The final fate of this protected neutral core, however, is still unclear, because the I-front tracking calculations by @2006ApJ...639..621A did not include the hydrodynamical response of the minihalo gas to its ionization, a full treatment of radiative transfer or the primordial chemistry involving $\rm H_2$. One might naively expect that the nett effect would be negative, because heating from photoionization would ultimately expel most of gas from minihaloes, although the results of SIR and ISR, again, show that this minihalo evaporation would not be complete within the lifetime of the Pop III star. On the other hand, partial ionization beyond the I-front by hard photons from a Pop III star might be able to [*promote*]{} ${\rm H_2}$ formation, once the dissociating UV radiation from the star is turned off, which would then lead to a cooling and collapsing core. This issue can be addressed only by a fully coupled calculation of radiative transfer, chemistry, and hydrodynamics, which will be the focus of this paper.
We shall attempt to answer the following questions: Does the light from the first Pop III star in some neighbourhood promote or prevent the formation of more Pop III stars in the surrounding minihaloes? More specifically, do the neutral cores of these nearby minihaloes, which are shielded from the ionizing radiation from the external Pop III star, subsequently cool and collapse gravitationally, as they must in order to form stars, or are they prevented from doing so? Towards this end, we simulate the evolution of these target haloes under the influence of an external Pop III star using the 1-D spherical, Lagrangian, radiation-hydrodynamics code we have developed. We adopt a $120\,M_\odot$ Pop III star as a source, and place different mass haloes at different distances to explore a wide range of the parameter space for this problem. Masses of target haloes are chosen to span the range from those too low for haloes to cool and collapse by ${\rm H_2}$ cooling without external radiation to those massive enough to do so on their own.
Our calculation is the first self-consistent gas-dynamical calculation of the feedback effects of a single Pop III star on nearby haloes. A similar approach by 1-D radiation-hydrodynamics calculation has been performed by @2001MNRAS.326.1353K. Their work, however, focuses on the effect of a steady global background from quasars and from stars with surface temperatures $T_{*} \sim 10^{4} \,{\rm K}$, rather than a single, short-lived Pop III star with $T_{*} \sim 10^{5} \,{\rm K}$. In addition, while we were preparing this manuscript, a study which is similar to our work was reported by @2006ApJ...645L..93S, where a 3D radiation-hydrodynamics calculation with SPH particles was performed[^3]. A major difference of their work from ours is that they focus on the subclumps of the halo which hosts the first Pop III star, while we focus on external minihaloes in the neighbourhood of such a host halo. We also apply a more accurate treatment of ${\rm H_2}$ self-shielding, as well as a more complete chemistry network of neutral and ionic species of H, He, and ${\rm H_2}$. A more fundamental difference from these previous studies is our finding of a novel ${\rm
H_2}$ formation mechanism: [*collisional ionization of pre-I-front gas by a shock detached from a D-type I-front*]{}. This mechanism occurs at the centre of target haloes, which would otherwise remain very neutral. This mechanism creates new electrons abundant enough to promote further ${\rm H_2}$ formation, which can even expedite the core collapse.
In Section \[sec:code\] we describe the details of the 1-D spherical radiation-hydrodynamics code we have developed. Some details left out in Section \[sec:code\] will be described in Appendices. In Section \[sec:Initial-Setup\], we describe the initial setup of our problem. We briefly describe a test case in Section \[sec:opt-thin\], where we let a minihalo evolve from an initially ionized state, to show that our code reproduces the result of @2005ApJ...628L...5O in that case. In Section \[sec:mcm\] and Section \[sec:2star-Result\], we present the main results of our full radiation-hydrodynamics calculation. We summarize our results in Section \[sec:2star-Discussion\]. Throughout this paper, we use the $\Lambda$CDM cosmological parameters, ($\Omega_\Lambda$, $\Omega_0$, $\Omega_b$, $h$) = ($0.73$, $0.27$, $0.043$, $0.7$), consistent with the [*WMAP*]{} first-year data [@2003ApJS..148..175S][^4].
Numerical Method: 1-D spherical, radiation-hydrodynamics with primordial chemistry network {#sec:code}
==========================================================================================
In this section, we describe in detail the 1-D spherical, Lagrangian, radiation-hydrodynamics code we have developed for both dark and baryonic matter. We describe how hydrodynamics, dark matter dynamics, radiative transfer, radiative heating and cooling, and finally the nonequilibrium chemistry are handled. The finite differencing scheme, reaction rates, and certain other details not treated in this section will be described in Appendices. We include the neutral and ionic species of H, He and ${\rm H_{2}}$, namely H, $\rm H^+$, He, ${\rm He^{+}}$, ${\rm He^{++}}$, $\rm H^-$, $\rm H_2$, $\rm H_{2}^+$ and $e^-$, in order to treat the primordial chemistry fully. As deuterium and lithium exist in a negligible amount, we neglect ${\rm
D}$ and ${\rm Li}$ species[^5].
Hydrodynamic Conservation Equations {#sub:2star-Hydrodynamics}
-----------------------------------
The baryonic gas obeys inviscid fluid conservation equations, $$\frac{\partial\rho}{\partial t}+\frac{\partial}{r^{2}\partial
r}(r^{2}(\rho u))=0,
\label{eq:realhydro_mass}$$ $$\frac{\partial}{\partial t}(\rho u)+\frac{\partial}{\partial r}(p+\rho
u^{2})+\frac{2}{r}\rho u^{2}=-\rho\frac{Gm}{r^{2}},
\label{eq:realhydro_momentum}$$ $$\frac{De}{Dt}=-\frac{p}{\rho}\frac{\partial}{r^{2}\partial
r}(r^{2}u)+\frac{\Gamma-\Lambda}{\rho},
\label{eq:realhydro_energy}$$ where $e\equiv(3p)/(2\rho)$ is the internal energy per unit baryon mass, $\Gamma$ is the external heating rate, and $\Lambda$ is the radiative cooling rate. Note that all the variables in equations (\[eq:realhydro\_mass\]) - (\[eq:realhydro\_energy\]) denote baryonic properties, except for $m$, the mass enclosed by a radius $r$, which is composed of both dark and baryonic matter.
We do not change the adiabatic index $\gamma$ throughout the simulation. As long as monatomic species, H and He, dominate the abundance, $\gamma=5/3$ is the right value to use. This ratio of specific heats, $\gamma$, can change significantly, however, if a large fraction of H is converted into molecules. For example, the three-body ${\rm H_2}$ formation process, $${\rm H+H+H \rightarrow H_2 + H},
\label{threebody}$$ will occur vigorously when $n_{\rm H}\ga 10^8 \,{\rm cm\, s^{-1}}$ and $T\la
10^{3} \,{\rm K}$, which will invalidate the use of a constant $\gamma$. To circumvent such a problem, when such high density occurs, we simply stop the simulation. This process is, nevertheless, important in forming the protostellar molecular cloud (e.g. @2002Sci...295...93A). This issue will be further discussed in Section \[sec:2star-Result\], when we define the criterion for the collapse of cooling regions.
The shock is treated using the usual artificial viscosity technique (e.g. @artvis). The pressure $p$ in equations (\[eq:realhydro\_momentum\]) and (\[eq:realhydro\_energy\]) contains the artificial viscosity term. The details of this implementation are described in Appendix A.
Dark Matter Dynamics {#sub:fluid-approx}
--------------------
Gravity is contributed both by the dark matter and the baryonic components. Let us first focus on the dark matter component. In order to treat the dark matter gravity under spherical symmetry, almost all previous studies have used either a frozen dark matter potential or a set of self-gravitating dark matter shells in radial motion only (e.g. @1995ApJ...442..480T). Both methods have their own limitations. The frozen potential approximation cannot address the effect of a possible evolution of the gravitational potential. The radial-only dark matter approximation suffers from the lack of any tangential motion, producing a virialized structure whose central density profile is much steeper ($\rho\propto r^{-\beta}$ with $\beta \ge 2$; see e.g. ) than that of haloes in cosmological, 3-D N-body simulations ($\beta \approx 1$, as found in @1997ApJ...490..493N).
In order to treat the dynamics of dark matter more accurately than these previous treatments, we use the the fluid approximation we have developed and reported elsewhere [@2005MNRAS.363.1092A]. We briefly summarize its derivation here; for a detailed description, see @2005MNRAS.363.1092A. Collisionless CDM particles are described by the collisionless Boltzmann equation. When integrated, it yields an infinite set of conservation equations, which is called the BBGKY hierarchy (e.g. @1987gady.book.....B). However, CDM N-body simulations show that virialized haloes are well approximated by spherical symmetry. These simulations also show that the velocity dispersions are highly isotropic: radial dispersion is almost the same as the tangential dispersion. These two conditions make it possible to truncate the hierarchy of equations to a good approximation, which then yields only three sets of conservation equations. Amazingly enough, these equations are identical to the normal fluid conservation equations for the adiabatic index $\gamma=5/3$ gas: $$\frac{\partial\rho_d}{\partial t}+\frac{\partial}{r^{2}\partial
r}(r^{2}(\rho_d u_d))=0,
\label{eq:DM_mass}$$ $$\frac{\partial}{\partial t}(\rho_d u_d)+\frac{\partial}{\partial r}(p_d+\rho_d
u_{d}^{2})+\frac{2}{r}\rho_d u_{d}^{2}=-\rho_d \frac{Gm}{r^{2}},
\label{eq:DM_momentum}$$ $$\frac{De_d}{Dt}=-\frac{p_d}{\rho_d}\frac{\partial}{r^{2}\partial
r}(r^{2}u_d),
\label{eq:DM_energy}$$ where the subscript $d$ represents dark matter, the effective pressure $p_d \equiv \rho_d \left\langle u_d - \left\langle u_d\right\rangle
\right\rangle^2$ is the product of the dark matter density and the velocity dispersion at a given radius, and the effective internal energy per dark matter mass $e_d \equiv 3 p_d/ 2 \rho_d$. We use these effective fluid conservation equations (equation \[eq:DM\_mass\], \[eq:DM\_momentum\], \[eq:DM\_energy\]) to handle the motion of dark matter particles.
Note that dark matter shells in this code represent a collection of dark matter particles in spherical bins, in order to describe “coarse-grained” properties such as density ($\rho_d$) and the effective pressure ($p_d$). As these coarse-grained variables follow the usual fluid conservation equations, the hyperbolicity of these equations leads to the formation of an effective “shock.” The location of this shock will determine the effective “post-shock” region. This post-shock region corresponds to the dark matter shell-crossing region. Because of the presence of this effective shock, we also use the artificial viscosity technique. This collisional behaviour of our coarse-grained dark matter shells originates from our choice of physical variable. For further details, the reader is referred to @2005MNRAS.363.1092A and @2003RMxAC..18....4A for description and application of our fluid approximation.
The mass enclosed by a dark matter shell of radius $r$, $$m(<r)=m_{{\rm DM}}(<r)+m_{{\rm bary}}(<r),
\label{eq:totalmass}$$ enters equations (\[eq:realhydro\_momentum\]) and (\[eq:DM\_momentum\]). When computing $m(<r)$, we properly take account of the mismatch of the location of dark matter shells and baryon shells.
Radiative transfer
------------------
A full, multi-frequency, radiative transfer calculation is performed in the code. Since ${\rm H_2}$ cooling is of prime importance here, we first pay special attention to calculating the optical depth to UV dissociating photons in the LW bands and the corresponding ${\rm H_2}$ self-shielding function. We then describe how we calculate the optical depth associated with any other species depending upon the location of the radiation source. The finite difference scheme for the calculation of radiative rates is described in the Appendix A.
### Photodissociation of $\rm H_2$ and Self-Shielding {#sub:ss}
Hydrogen molecules are photodissociated when a UV photon in the LW bands between $11\,\rm eV$ and $13.6\,\rm
eV$ excites $\rm H_2$ to an excited electronic state from which dissociation sometimes occurs. When the column density of ${\rm H_2}$ becomes high enough ($N_{\rm H_2}\ga 10^{14} \,\rm cm^{-2}$), the optical depth to photons in these Lyman-Werner bands can be high, so ${\rm H_2}$ can “self-shield” from dissociating photons. Exact calculation of this self-shielding requires a full treatment of all 76 Lyman-Werner lines, even when only the lowest energy level transitions are included. Such a calculation is feasible under simplified conditions such as a radiative transfer problem through a static medium (e.g. @2000ApJ...534...11H; @2001ApJ...560..580R). Unfortunately, for combined calculations of radiative transfer and hydrodynamics, such a full treatment is computationally very expensive.
Under certain circumstances, however, one can use a pre-computed self-shielding function expressed in terms of the molecular column density $N_{{\rm
H_{2}}}$ and the temperature $T$ of gas, which saves a great amount of computation time. In a *cold, static* medium, for instance, one can use a self-shielding function provided by @1996ApJ...468..269D: $$F_{{\rm shield}}={\rm min}\left[1,\left(\frac{N_{{\rm
H_{2}}}}{10^{14}{\rm
cm^{-2}}}\right)^{-3/4}\right].
\label{eq:DB_shield_factor_cold}$$ The photodissociation rate is then given by $$k_{{\rm H_{2}}}=1.38\times10^{9}\left(J_{\nu}\right)_{h\nu=12.87{\rm
eV}}F_{{\rm shield}},
\label{eq:DB_rate}$$ where $\left(J_{\nu}\right)_{h\nu=12.87{\rm eV}}$ ($\rm
erg\,s^{-1}\,cm^{-2}\,Hz^{-1}\,sr^{-1}$) is the mean intensity in the spectral region of the LW bands. This approximation has been widely used in the study of high redshift structure formation (e.g. @2001MNRAS.326.1353K [@2001MNRAS.321..385G; @2003ApJ...592..645Y; @2004ApJ...613..631K]).
The problem with equation (\[eq:DB\_shield\_factor\_cold\]) is that when the gas temperature is high or gas has motion along the line of sight to the source, the thermal and velocity broadening of the LW bands caused by the Doppler effect can significantly reduce the optical depth. A better treatment for thermal broadening is also given by @1996ApJ...468..269D, now in terms of the molecular column density $N_{\rm H_{2}}$ and the velocity-spread parameter $b$ of the gas: $$\begin{aligned}
F_{{\rm shield}}= \frac{0.965}{(1+x/b_5)^2} +
\frac{0.035}{(1+x)^{0.5}} \nonumber \\
\times \exp[-8.5\times 10^{-4}(1+x)^{0.5}],
\label{eq:DB_shield_factor}\end{aligned}$$ where $x\equiv N_{\rm H_2}/5\times 10^{14}\, {\rm cm}^{-2}$, $b_5\equiv
b/10^5 {\rm cm \, s^{-1}} $, and $b=1.29 \times 10^4
\left(T_K/A\right)^{1/2} {\rm cm \, s^{-1}},$ where $A$ is the atomic weight [@1978ppim.book.....S]. For ${\rm H_2}$, $b=9.12 \,{\rm
km \, s^{-1}} \left(T/10^4 \, {\rm K}\right)^{1/2}$.
In the problem treated in this paper, we frequently find $T\approx
10^3 - 5\times 10^3 \, {\rm K}$ in the gas parcel (shell) which contributes most of the ${\rm H_2}$ column density. We also find that this gas parcel usually moves at $v \approx 2-5 \, {\rm km \,
s^{-1}}$ (see Section \[sub:H2shell\]). The combined effect of the thermal broadening and the Doppler shift on the shielding function, then, may be well approximated by a thermally broadened shielding function with $T\approx 10^4\,{\rm
K}$. Throughout this paper, therefore, we use equation (\[eq:DB\_shield\_factor\]) with $T= 10^4\,{\rm K}$ to calculate the self-shielding. For the photo-dissociation rate, we use equation (\[eq:DB\_rate\]).
We show in Fig. \[fig-DB\] how much the static, cold shielding function (equation \[eq:DB\_shield\_factor\_cold\]) may overestimate the self-shielding in our problem, by comparing this to the thermally-broadened shielding function (equation \[eq:DB\_shield\_factor\]) at $T= 10^4\,{\rm K}$. The biggest discrepancy between these two shielding functions exists for $N_{\rm
H_2} \approx 10^{14} - 10^{16} \, {\rm cm}^{-2}$. Interestingly enough, the ${\rm H_2}$ column density in our problem usually resides in this regime. It is crucial, therefore, to take into account the effects of thermal broadening and Doppler shift carefully, as we do in this paper.
### External Source {#sub:External-source}
Since our calculations are 1-D, spherically-symmetric, we have assumed the external radiation source contributes a radial flux $F_{\nu}^{{\rm ext}}(r)$ at frequency $\nu$ and radius $r$, measured from the minihalo centre, given by $$%
%
F_{\nu}^{{\rm ext}}(r)=
\frac{L_{\nu}^{{\rm ext}}}{4\pi D^{2}}e^{-\tau_{\nu}(>r)},
\label{eq:fnu_ext}$$ where $L_{\nu}^{{\rm ext}}$ is the source luminosity, and $\tau_{\nu}(>r)$ is the optical depth along the radial direction from radius $r$ to the source located at a distance $r=D$.
The radiative rate of species $i$ at radius $r$ is then given by $$k_{i}(r)=\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}4\pi J_{\nu}(r)}{h\nu}
=\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}F_{\nu}^{{\rm
ext}}(r)}{h\nu},
\label{eq:rad_rate_ext_body}$$ where we have used the fact that $4\pi J_{\nu} = F_{\nu}^{\rm ext}$, as long as the external radiation can be approximated as a 1D planar flux. In practice, one calculates this rate in a given grid-cell – i.e. spherical shell – with finite thickness. If such a grid-cell has a small optical depth, $F_{\nu}^{{\rm ext}}$ is almost constant across the grid, so one could take the grid-centred value of $F_{\nu}^{{\rm
ext}}$ to calculate $k_{i}(r)$. This naive scheme, however, does not yield an accurate result when a grid-cell is optically thick, where $F_{\nu}^{{\rm ext}}$ may vary significantly over the cell width. This problem occurs frequently for solving radiative transfer through optically thick media, where individual cells have large optical depth. In order to resolve this problem, we use a “photon-conserving” scheme like that described by @1999MNRAS.309..287R and @1999ApJ...523...66A. The details of our implementation of this scheme are described in the Appendix A.
Heating and Cooling {#sub:heating-cooling}
-------------------
### Photoheating {#sub:photo-heating}
Photoheating results from thermalization of the residual kinetic energy of electrons after they are photoionized. In general, the photoheating function is described by $$\begin{aligned}
\Gamma=\sum_{i}\Gamma_{i}&=&\sum_{i}n_{i}\int_{0}^{\infty}d\nu\frac{4\pi
J_{\nu}\sigma_{\nu}}{h\nu}(h\nu-h\nu_{i,{\rm th}}) \nonumber \\
&=&\sum_{i}n_{i}\int_{0}^{\infty}d\nu\frac{
F_{\nu}^{\rm ext}\sigma_{\nu}}{h\nu}(h\nu-h\nu_{i,{\rm th}}),
\label{eq:generic_photo_heat}\end{aligned}$$ where $h \nu_{i,{\rm th}}$ is the threshold energy over which the residual photon energy is converted into the kinetic energy of electrons, and the nett heating function $\Gamma$ is the sum of individual heating functions ({$\Gamma_{i}$}). In finite-differencing equation (\[eq:generic\_photo\_heat\]), we also use the photon-conserving scheme as we do for equation (\[eq:rad\_rate\_ext\_body\]). This prevents cells with large optical depth from obtaining unphysically high heating rates. See Appendix A for details.
### Radiative cooling
Cooling occurs through various processes. For atomic species, it comes from collisional excitation, collisional ionization, recombination, free-free emission, and CMB photons scattering off free electrons (Compton cooling/heating). For atomic H and He, cooling is dominated by collisional excitation (for $T\la 2\times 10^{5}{\rm K}$) and free-free emission (for $T\ga 2\times 10^{5}{\rm K}$). The atomic cooling rate decreases rapidly at $T\la 10^{4}{\rm K}$, as there are no collisions energetic enough to cause excitation. It is difficult, therefore, to cool gas below $T\approx10^{4}{\rm K}$ solely by atomic cooling of primordial gas.
Molecular hydrogen (${\rm H_{2}}$), however, is able to cool gas below $T\approx10^{4}{\rm K}$, down to $T\approx100{\rm K}$, by collisional excitation of rotational-vibrational lines by H atoms. An important question to address is how much ${\rm H_{2}}$ is created, maintained, or destroyed under the influence of an ionizing and dissociating radiation field. Even a small fraction, $n_{\rm H_{2}}/n_{\rm H}\ga 10^{-4}$, is sometimes enough to cool gas below $10^{4}{\rm K}$ (e.g. see @1987ApJ...318...32S).
We use cooling rates in the parametrized forms given by @1997NewA....2..209A, except for the hydrogen molecular cooling. For ${\rm H_2}$ cooling, we use the fit given by , where the low density cooling rate has been updated significantly from the previously used rate by @1984ApJ...280..465L, which suffers from the uncertainties associated with the only collisional coefficients available at that time. At low densities, $n_{\rm H} \la 10^2 {\rm cm^{-3}}$, the cooling rate of @1984ApJ...280..465L is bigger by an order of magnitude than that of at $T\approx 1000 K$.
Nonequilibrium chemistry {#sub:noneq_chem}
------------------------
The general rate equation for the abundance of species $i$ is given by $$\frac{\partial n_{i}}{\partial t}=C_{i}(T,\{ n_{j}\})-D_{i}(T,\{
n_{j}\})n_{i},
\label{eq:verygeneric_rate_eq}$$ where $C_{i}$ is the collective source term for the creation of species $i$, and the second term is the collective “sink” term for the destruction of species $i$. The processes included and adopted are shown in Table \[table:rates\] in Appendix B. Most of the rate coefficients are those from the fits by @1987ApJ...318...32S, with a few updates.
We also adopt the rate solving scheme proposed by @1997NewA....2..181A. It is well known that coupled rate equations in the form of equation (\[eq:verygeneric\_rate\_eq\]) are “stiff” differential equations, whose numerical solution suffers from instability if explicit ODE solvers are used. @1997NewA....2..181A show that their implicit, backward difference scheme provides enough stability. Accuracy of the solution is achieved by updating each species in some specific order, rather than updating all species simultaneously from their values at the last time step. In addition, the abundance of the relatively fast reactions of ${\rm H^{-}}$ and ${\rm H_{2}^{+}}$ are approximated by their equilibrium values, which are expressed by simple algebraic equations. See the Appendix A for the corresponding finite-differencing scheme.
We will frequently quote our results in terms of the fractional number density of species $i$, $y_{i}\equiv\frac{n_{i}}{n_{{\rm H}}}$, where $n_{{\rm H}}$ is the number density of the total atomic hydrogen atoms. We use $x$, however, to denote the fractional electron number density, $y_{e}$, which is a measure of the ionized fraction.
Code tests {#sec:codetest}
----------
We tested our code against the following problems which have analytic solutions:
\(A) the self-similar, spherical, cosmological infall and accretion shock resulting from a point-mass perturbation in an Einstein-de Sitter universe of gas and collisionless dark matter [@1985ApJS...58...39B];
\(B) the self-similar blast wave which results from a strong, adiabatic point explosion in a uniform gas – the Sedov solution (@1959sdmm.book.....S)
\(C) the propagation of an I-front from a steady point-source in a uniform, static medium
\(D) the gas-dynamical expansion of an H II region from a point source in a uniform gas [@1966ApJ...143..700L]
\(E) the gas-dynamical expansion-phase of the H II region from a point-source in a nonuniform gas whose density varies with distance $r$ from the source as $r^{-w}$, $w=3/2$ [@1990ApJ...349..126F].
Our code passed all the tests described above with an acceptable accuracy. Test results are described in Appendix C.
The Simulations
===============
Initial Setup {#sec:Initial-Setup}
-------------
We now describe the initial setup for the problem of radiative feedback effects of Pop III stars on nearby haloes at $z\approx20$. The first stars form inside rare, high density peaks at high redshift. We place target haloes of different mass $M=[2.5\times 10^{4}, \,
5\times 10^{4},\,
10^{5},\,
2\times 10^{5},\,
4\times 10^{5},\,
8\times 10^{5}]\,
M_{\odot}$ at different locations from the source, with proper distance $D=\{180,\,\,360,\,\,540,\,\,1000\}\,{\rm pc}$, which are all assumed to be affected directly by the radiation field from the source Pop III star of mass $M_{*}=120\,
M_{\odot}$[^6]. We expose the target halo to this radiation field for the lifetime of the star, $t_{*}(120\, M_{\odot})\simeq2.5\,{\rm Myrs}$ (). The source Pop III star is assumed to be located in a halo of mass $M\simeq10^{6}M_{\odot}.$ Time is measured from the arrival of the stellar radiation at the location of the target minihalo.
This setup is well justified by the cosmological simulations by @2006ApJ...639..621A. A cosmological gas and N-body simulation of structure formation in the $\Lambda$CDM universe on small scales by a GADGET/SPH code was used to identify the site at which the first Pop III star would form. This occurred at $z=20$, at the location of the highest density SPH particle in the simulation box, located within a halo of mass $M\simeq10^{6}M_{\odot}.$ This provided the initial density field for the I-front tracking calculations in @2006ApJ...639..621A. The I-front from this first star escaped from the host halo quickly with high escape fraction, traveling as a supersonic, weak R-type front. By the end of the lifetime of the star ($\sim[3-2]\,{\rm Myrs}$) for stellar masses in the range $M_* \sim[80-200]M_{\odot}$, the star’s H II region had reached a maximum radius of about $3\,{\rm kpc}$. We approximate the spectral energy distribution (SED) of the source star by a blackbody spectrum. A Pop III star of mass $M_{*}\approx120\, M_{\odot}$, according to , has the time-average effective temperature $T_{{\rm eff}}\approx10^{5}{\rm K}$ and luminosity $L=\int_{0}^{\infty}d\nu L_{\nu}\approx10^{6.243} L_{\odot}$. The corresponding ionizing photon luminosity with this blackbody spectrum is $Q_* \equiv \int_{\nu_{\rm H}}^{\infty}d\nu L_{\nu}/h \nu
=1.5\times
10^{50} \rm s^{-1}$, where $h\nu_{\rm H}\equiv 13.6\,{\rm eV}$ is the hydrogen ionization threshold energy. We assume that the source radiates with these time-averaged values throughout its lifetime, then stops. As the photons escape in a time scale short compared to the lifetime of the star and the escape fraction is high, we simply ignore the effect of the intervening gas (e.g. optical depth from the host halo and the IGM) and assume that the bare radiation field hits the edge of target haloes directly.
As we fix the luminosity of the source, different distances correspond to different fluxes. We express the frequency-integrated ionizing photon flux, $F$ in units of $\rm 10^{50}\, s^{-1}\, kpc^{-2}$, to give the dimensionless flux, $F_0 \equiv N_{\rm
ph,50}/D_{\rm kpc}^2 = N_{\rm ph,56}/D_{\rm Mpc}^2$, where $N_{\rm ph,50}$ is the ionizing photon luminosity (in units of $\rm
10^{50}\,s^{-1}$) and $D_{\rm kpc}$ ($D_{\rm Mpc}$) is the distance in units of kpc (Mpc), respectively. The value $F_0\approx 1$ is typical for minihaloes encountered by intergalactic I-fronts during global reionization (e.g. see @2004MNRAS.348..753S). Interestingly enough, $F_0$ for our “small-scale” problem has a similar value. The Pop III star in our problem has $N_{\rm ph,50}\equiv
Q_*/10^{50}\,s^{-1}=1.5$. For distances $180\,{\rm pc}$, $360\,{\rm pc}$, $540\,{\rm pc}$ and $1000\,{\rm pc}$, $F_0$ corresponds to 46.3, 11.6, 5.14 and 1.5, respectively.
Initial Halo Structure {#sub:IHS}
----------------------
For the initial halo structure, we adopt the minimum-energy truncated isothermal sphere (TIS) model (@1999MNRAS.307..203S; @2001MNRAS.325..468I), which will be described further in Section \[sub:phase1\]. The thermodynamic properties and chemical abundances of the gas in these target haloes, however, is somewhat ambiguous. The density and virial temperature of these haloes are higher than those of the IGM in general, which drives their chemical abundances to change from the IGM equilibrium state to a new equilibrium state. The most notable feature is the change of $y_{\rm
H_2}$ and $x$. The IGM equilibrium value of the electron abundance, $x\approx 10^{-4}$, is high enough to promote ${\rm H_2}$ formation inside minihaloes to yield a high molecule fraction, $y_{\rm H_2}\approx
10^{-4} - 10^{-3}$. At the density of gas in the halo core, this newly created ${\rm H_2}$ is capable of cooling the minihalo gas to $T\approx 100\, {\rm K}$, and depending on the virial temperature, the minihalo may, therefore, undergo a runaway collapse.
The time for this evolution of the target halo gas is short compared to the age of the universe when the first star forms in their neighbourhood. As a result, it is likely that the target haloes are exposed to the ionizing and dissociating radiation from that first star as they are in the midst of evolving, with fine-tuning required to catch all of them in a particular stage of this evolution. As the evolutionary “phase” of our target haloes is uncertain, we adopt two different phases as our representative initial conditions. In Phase I, chemical abundances have not yet evolved away from their IGM equilibrium values. This stage is characterized by low $\rm H_2$ fraction, $y_{\rm
H_2}\sim 2\times 10^{-6}$ and high electron fraction, $x\sim
10^{-4}$. Phase II is the state which is reached, after allowing the Phase I minihalo to evolve chemically, thermodynamically and hydrodynamically for a few million years (a small fraction of a Hubble time, $t_{\rm H}=186\,\rm Myrs$ at $z=20$), until the electron fraction has decreased to $x\sim 10^{-5}$. Phase II is characterized by high $\rm H_2$ fraction, $y_{\rm
H_2}\sim 10^{-4} - 10^{-3}$, and cooling-induced compression of the core relative to Phase I, by a factor between 1 and 20, higher for higher minihalo mass.
### Phase I: Unevolved Halo with IGM chemical abundance in hydrostatic equilibrium {#sub:phase1}
The first phase we choose is the initial state we assumed above, namely the nonsingular TIS structure with IGM chemical abundances. This phase is characterized by gas in hydrostatic equilibrium, with the truncation radius (outer boundary of the halo) $$\begin{aligned}
r_{t}&=&102.3
\left(\frac{\Omega_{0}}{0.27}\right)^{-1/3}
\left(\frac{h}{0.7}\right)^{-2/3} \nonumber \\
&&\times \left(\frac{M}{2\cdot10^{5}M_{\odot}}\right)^{1/3}
\left(\frac{1+z}{1+20}\right)^{-1}\,{\rm pc},\end{aligned}$$ the virial temperature $$\begin{aligned}
T&=&593.5
\left(\frac{\mu}{1.22}\right)
\left(\frac{\Omega_{0}}{0.27}\right)^{1/3}
\left(\frac{h}{0.7}\right)^{2/3} \nonumber \\
&&\times \left(\frac{M}{2\cdot10^{5}M_{\odot}}\right)^{2/3}
\left(\frac{1+z}{1+20}\right)\,{\rm K},
\label{eq:tvir}\end{aligned}$$ where $\mu$ is the mean molecular weight (1.22 for neutral gas and 0.59 for ionized gas) and the central density $$\rho_{0}=4.144\times10^{-22}
\left(\frac{\Omega_{0}}{0.27}\right)
\left(\frac{h}{0.7}\right)^{2}
\left(\frac{1+z}{1+20}\right)^{3}\,
{\rm g\,\, cm^{-3}},$$ which can also be expressed in terms of the hydrogen number density by $$\begin{aligned}
n_{{\rm H,0}}&=&\frac{X(\Omega_{b}/\Omega_{0})\rho_{0}}{m_{{\rm
H}}} \nonumber \\
&=&30\left(\frac{X}{0.76}\right)\left(\frac{\Omega_{b}}{0.043}\right)\left(\frac{h}{0.7}\right)^{2}\left(\frac{1+z}{1+20}\right)^{3}{\rm \, cm^{-3}},\end{aligned}$$ where $X$ is the hydrogen mass fraction in the baryon component. This central density is about $1.8\times 10^4 \,\overline{\rho}(z),$ where $\overline{\rho}(z)$ is the mean matter density at redshift $z$, while at $r=r_{\rm tr}$, $\rho=35\,\overline{\rho}(z)$. For more details, see @1999MNRAS.307..203S and @2001MNRAS.325..468I.
We assign chemical abundances that reflect the IGM equilibrium state, which is characterized by high electron fraction – high enough to promote ${\rm H_2}$ formation under the right conditions – and low ${\rm H_2}$ fraction – low enough to contribute negligible molecular cooling. We adopt $y_{\rm H}=1$, $y_{{\rm He}}=0.0789$, $x\simeq y_{{\rm H^{+}}}=10^{-4}$, $y_{{\rm H_{2}}}=2\times10^{-6}$, and $\{y_{i}\}=0$ for other species (see, e.g. SGB94; @2001ApJ...560..580R).
### Phase II: Evolved Halo with Recombining and Cooling Core {#sub:phase2}
The second initial condition we choose is the evolved state (Phase II) reached by allowing the system to evolve from Phase I initial conditions before the arrival of radiation from the Pop III star. In particular, we follow this evolution until the central electron fraction has dropped to $10^{-5}$ by recombination from Phase I. We choose this condition because it is now characterized by high molecule and low electron fraction, contrary to Phase I. The fate of this halo will then mainly be determined by how easily this abundant ${\rm H_2}$ is protected against dissociating radiation after the star turns on. The answer will also depend upon how much change has occurred hydrodynamically, because in some cases the halo core may have cooled and collapsed significantly enough to be unaffected by the feedback from [*late*]{} irradiation.
The time to reach Phase II is different for different mass haloes because of different gas properties. Initially, as we start from the TIS density profile whose central density is independent of the halo mass, the recombination rate is higher for smaller mass haloes, because hydrogen recombines according to the following: $$\label{eq:k2_dep}
\frac{dx}{dt} \propto n_{\rm H} n_{e^-} T^{-0.7}.$$ The situation becomes complicated, however, once evolution begins and density changes. The ${\rm H_2}$ cooling and collapse in the central region of the haloes is increasingly effective as halo mass increases, because of the increasingly large difference between the virial temperature and the ${\rm H_2}$ cooling temperature plateau, $\sim 100\,{\rm
K}$. The corresponding rapid collapse and cooling in massive haloes can easily offset the initial temperature dependence by obtaining high density and low temperature, as is seen in equation (\[eq:k2\_dep\]). Phase II for large mass haloes represents haloes that have already started their cooling and collapse.
In Fig. \[fig-init\], we show halo profiles in Phase I and Phase II for different halo masses. We also show how much time it takes for the haloes to evolve from Phase I to Phase II. The times for gas at the halo centre to recombine to $x=10^{-5}$ are in the range $7 \le \Delta t_{\rm I,\,II}(\rm Myrs)
\le 24$ for halo masses $0.25 \le M/(10^5\,M_\odot) \le 8$, peaked at $\Delta t_{\rm I,\,II}=24 \,\rm Myrs$ for $M=5\times 10^4 \, M_\odot$. In all cases, $\Delta t_{\rm I,\,II}
\ll t_{\rm H}=186\,\rm Myrs$, the age of the universe at $z=20$.
Halo Evolution from Fully-Ionized Initial Conditions: The Consequences of Irradiation Without Optical Depth {#sec:opt-thin}
===========================================================================================================
Before describing the results of our full radiative transfer, hydrodynamics calculation, we describe an experiment designed to show the effect of neglecting the optical depth of the minihalo to ionizing radiation from the external star during the star’s lifetime on the minihalo’s evolution after the star shuts off. For this purpose, we assume the target minihalo is initially fully-ionized and heated to the temperature of a photoionized gas as it would be if it were instantaneously flash-ionized by starlight in the optically-thin limit. Such a setup is equivalent to that used by @2005ApJ...628L...5O, where they find that second-generation star formation is triggered when the ionization of the minihalo caused by the nearby Pop III star leads to cooling by ${\rm
H_{2}}$. The high initial electron fraction is present because of the assumption of full ionization allows quick formation of ${\rm
H_{2}}$, which then cools the central region before it reaches the escape velocity.
For this experiment, we initialized ionized fractions as following: $y_{\rm H I}=6.4\times 10^{-4}$, $x=1.15$, $y_{\rm H II} = 1$, $y_{\rm He I}=6.8\times 10^{-6}$, $y_{\rm He II}=8.9\times 10^{-3}$, $y_{\rm He III}=7\times 10^{-2}$, $y_i =0$ for other species. Without disturbing the halo density profile – we use the TIS halo model, which is described in Section \[sub:phase1\] –, we also assigned a high initial temperature appropriate for photoionized gas, $T=2\times 10^{4} {\rm K}$. These abundance and temperature values roughly mimic the condition found in typical H II regions.
We find that such an initial condition leads to the collapse of the core region, when the formation of ${\rm H_{2}}$ stimulated by the high initial electron fraction enables $\rm H_2$ cooling. Gas in the outskirts evaporates from the halo, however, because pressure forces accelerate the gas to escape velocity before it can form ${\rm H_{2}}$ and cool. The ${\rm H_{2}}$ cooling and adiabatic cooling which happen later in this outflowing gas do not reverse the evaporation (Fig. \[fig-zap\]).
Our results for this case agree with the outcome of @2005ApJ...628L...5O. This led those authors to suggest that the first stars exerted a positive feedback effect on their surroundings, triggering a second generation of star formation. A question arises, however, as to whether this fully-ionized initial condition of nearby minihaloes is actually achieved by the first Pop III star to form in their neighbourhood. As already mentioned in Section \[sec:Secondstar-Intro\], @2006ApJ...639..621A found that the I-front from this Pop III star gets trapped in those minihaloes and cannot reach the central region before the star dies. In this paper, we will confirm that the fully-ionized initial condition of @2005ApJ...628L...5O is never achieved when one considers the coupled radiative and hydrodynamic processes more fully. We will also show that, if any protostellar region is to form in the target halo, it does so in the neutral core region which the ionizing photons do not penetrate.
Minimum Halo Mass for Collapse: the case without radiative feedback {#sec:mcm}
===================================================================
When a minihalo forms as a nonlinear, virialized, gravitationally-bound structure out of the linearly perturbed IGM, a change of chemical abundance occurs due to the change of gas properties. Most importantly, the hydrogen molecule fraction changes from the IGM equilibrium value, $y_{\rm H_2}\sim 2\times 10^{-6}$, to a new equilibrium value, $y_{\rm H_2}\ga 10^{-4}$. Even with such a small fraction, ${\rm H_2}$ can cool gas to $T_{\rm H_2}\simeq 100\,{\rm
K}$, where $T_{\rm H_2}$ represents the temperature “plateau” that gas in primordial composition can reach by $\rm H_2$ cooling.
There exists a minimum collapse mass of minihaloes, $M_{\rm c,min}$, above which haloes, in the absence of external radiation, can form cooling and collapsing cores within the Hubble time at a given redshift. The gap between the $\rm H_2$ cooling plateau temperature, $T_{\rm
H_2}$, and the minihalo virial temperature, $T_{\rm vir}$, given by equation (\[eq:tvir\]) is a useful indicator of the success or failure of collapse. For instance, at $z\approx
20$, $T_{\rm vir}\sim 160\,{\rm K}$ for $M=2.5\times
10^4\,M_\odot$. As $T_{\rm vir}\simeq T_{\rm H_2}$ , even after gas cools to $T_{\rm H_2}$, it cannot collapse fast enough to serve as a site for star formation. On the other hand, $T_{\rm
vir}\sim 10^3 \,\rm K $ for $M=4\times 10^5\,M_\odot$, and the temperature’s cooling down to $T_{\rm H_2}\approx 100\,\rm K$ will make the gas gravitationally unstable, which will lead to runaway collapse. This argument is supported by the results of @1996ApJ...464..523H, for example, that collapse can occur only in haloes with $T_{\rm vir}\ga 100\,\rm K$.
We model the initial minihalo structure by the TIS model as described in Section \[sub:phase1\] and let it evolve in the absence of radiation, starting from the IGM chemical abundance and minihalo virial temperature (Phase I). We determine $M_{\rm c,min}$ by the criterion $$t_{\rm coll}=t_{\rm H},$$ where $t_{\rm coll}$ is the time at which the central density reaches $n_{\rm H}=10^8\,\rm cm^{-3}$ (the density suitable for initiating three-body $\rm H_2$ formation; see e.g. @2000ApJ...540...39A), and $t_{\rm H}$ is the Hubble time at a given redshift.
We find that $M_{\rm c,min}\simeq 7\times 10^4\,M_\odot$ at $z=20$ (see Fig. \[fig-freeevol\]). We have plotted the evolution of minihalo centres in the absence of radiation, where each run starts from Phase I. This is in rough agreement with $M_{\rm c,min}\simeq 1.25\times 10^5\,M_\odot$, the value found by @2001ApJ...548..509M. The discrepancy is larger with results by @2000ApJ...544....6F and @2003ApJ...592..645Y, where they obtain $M_{\rm c,min}\simeq
7\times 10^5\,M_\odot$. The biggest contrast exists with @1997ApJ...474....1T, where they find $M_{\rm c,min}\simeq 2\times
10^6\,M_\odot$ at $z\approx 20$, almost 30 times as large as our findings.
We argue that this discrepancy in minimum collapse mass results primarily from how well the minihalo structure is resolved. Unless the centre, which gains the highest molecule formation rate due to the highest density, is fully resolved, one could be misled by a poor numerical resolution such that certain low-mass haloes, which can cool and collapse in reality, are in hydrostatic equilibrium in the simulation. The resolution becomes poorer in the following sequence: @2001ApJ...548..509M, which gives the best agreement with our result, used an adaptive mesh refinement (AMR) scheme, resolving baryonic mass down to $M_b\sim 5\,M_\odot$. Such high resolution is suitable to resolve even the central part of the smallest minihaloes whose total baryonic mass content is roughly $ 2-3\times 10^3\,\rm M_\odot$. @2000ApJ...544....6F and @2003ApJ...592..645Y, on the other hand, used the smoothed particle hydrodynamics (SPH) scheme, using SPH particles of mass $M_b \sim
40-140\times 10^2\,\rm M_\odot$. Finally, @1997ApJ...474....1T used a uniform top-hat model, where there is no radial variation in gas properties such as density and temperature, thus the central region is, in effect, completely unresolved. In addition, some of the rates used in @1997ApJ...474....1T were not accurate [@2000ApJ...544....6F].
We believe that $M_{\rm c,min}\simeq 7\times 10^4\,M_\odot$ at $z=20$ is close to reality, because our 1-D spherical setup is based upon the TIS model which is a highly concentrated structure, and the resolution of our code is superior to previous calculations[^7]. It is not our objective, however, to settle the exact value of $M_{\rm c,min}$. This estimate is based upon our specific criterion described in this section, and is subject to change under different criteria. This may also change if one adopts a more realistic halo formation history to account, for instance, for dynamical heating by accretion (see @2003ApJ...592..645Y). As the haloes we choose are rather conservatively divided into successful collapse (for $M\ge 10^5\,M_\odot$) and failure (for $M < 10^5\,M_\odot$), agreeing with AMR simulation result by @2001ApJ...548..509M, we shall proceed with our choice of parameter space and [*see how this fate of minihaloes changes as a result of external radiation from a Pop III star.*]{}
Results: Radiative Feedback on Nearby Minihaloes by an External Pop III Star {#sec:2star-Result}
============================================================================
As described in Section \[sec:Initial-Setup\], we expose target haloes of different mass to the radiation from a Pop III star whose spectrum is approximated as a $10^5\,\rm K$ blackbody radiation field and whose flux is attenuated by the geometrical factor $\left(\frac{D}{R_{*}}\right)^{-2}$ for different values of $D$. In this section, we summarize the simulation results for both the Phase I (early irradiation) and the Phase II (late irradiation) initial conditions.
I-front trapping and photo-evaporation
--------------------------------------
In all cases, even in the presence of evaporation, we find no evidence of penetration of ionizing radiation into the halo core. This is consistent with the results of @2006ApJ...639..621A for the H II regions of the first Pop III stars and of @2004MNRAS.348..753S and @2005MNRAS.361..405I for the encounters between intergalactic I-fronts and minihaloes during reionization. There are two main reasons for this behaviour. First, the total intervening hydrogen column density is initially high enough to trap the I-front outside the core. Second, the lifetime of the source is short compared to the evaporation time. If the source lived longer than the evaporation time, the I-front would eventually have reached the centre of the halo. In that case, @2004MNRAS.348..753S find that the minihalo gas is completely evaporated. In our problem, however, the slow evaporation does not allow the I-front to reach the centre within the lifetime of a Pop III star.
The I-front entering the minihaloes propagates as a weak R-type front in the beginning. The I-front then makes the transition to the D-type, after reaching the R-critical state. This R-critical state is reached when the I-front velocity $v_{\rm I}$ satisfies the following condition: $$v_{\rm I}=c_{\rm I,2}+(c_{\rm I,2}^2 - c_{\rm I,1}^2)^{0.5},$$ where $c_{\rm I}$ is the isothermal sound speed, $c_{\rm I}\equiv
\sqrt{p/\rho}$, and subscripts 1 and 2 represent pre-front and post-front, respectively. When the I-front propagates into a cold region ($T\ll
10^{4} \, {\rm K}$), as in our problem, this condition is approximately $v_{\rm I}\approx 2\,c_{\rm I,2} \approx 20\,{\rm km\,s^{-1}}.$ In all cases, we find that this transition occurs in times less than the lifetime of the source star, 2.5 Myrs. After reaching the R-critical state, gas in front of the I-front forms a shock, which then detaches from the slowed I-front. As an example, we plot in Fig. \[fig-Rcrit\] the profiles of Phase I, $4\times 10^5\,M_\odot$ halo at $t=t_{\rm R-crit}$ under different fluxes.
All of the post-front (ionized) gas, initially undisturbed, eventually evaporates away, accelerated outward by a large pressure gradient. As the line-of-sight is cleared by this evaporation, ionizing radiation penetrates deeper, until the source turns off. See Figs \[fig-Rcrit\], \[fig-midpoint\] and \[fig-endpoint\] for the evolution of the I-front.
This result invalidates the initial condition adopted by @2005ApJ...628L...5O and @2006astro.ph..4148M which led them to find that $\rm H_2$ formed in the core region after it was ionized and then cooled while recombining, once the source turned off. As we show, the core remains neutral before and after the source is turned off, so the mechanism explored by @2005ApJ...628L...5O does not work. This neutral core, therefore, must find a different way to cool and collapse if star formation is to happen in the target minihalo.
What happens to the initially ionized gas after the star turns off? This gas recombines as it cools radiatively and by adiabatic expansion, even forming $\rm H_2$ molecules. We find that this cooling cannot reverse the evaporation, however. Gas is simply carried away with the initial momentum given to it when it was in an ionized state. In Table \[table:ionized\], we list the fraction of the baryonic halo mass which is ionized during the lifetime of the star. This mass serves as a crude estimate of the mass lost from these haloes by evaporation. We found no major difference between Phase I and Phase II in this matter, so we provide only one table.
-------------------- ------------ --------------------- ----------- ----------- ----------- ------------
$D$ (pc) \[$F_0$\] $0.25 $ $0.5 \cdot 10^{4} $ $1 $ $2 $ $4 $ $8 $
$(0.043 )$ $(0.086 )$ $(0.17 )$ $(0.34 )$ $(0.69 )$ $(1.371 )$
180 \[46.3\] 0.95 0.92 0.88 0.84 0.82 0.79
360 \[11.6\] 0.85 0.81 0.77 0.74 0.70 0.67
540 \[5.14\] 0.78 0.74 0.70 0.66 0.62 0.59
1000 \[1.5\] 0.66 0.60 0.55 0.50 0.47 0.43
-------------------- ------------ --------------------- ----------- ----------- ----------- ------------
Formation of ${\rm H_2}$ precursor shell in Front of the I-Front {#sub:H2shell}
----------------------------------------------------------------
We find that a thin shell of ${\rm H_{2}}$ is formed just ahead of the I-front, with peak abundance $y_{{\rm H_{2}}}\approx10^{-4}$. It happens mainly because the increased electron fraction across the I-front promotes the formation of $\rm H_2$. More precisely, the gas ahead of the I-front is ionized to the extent that the electron abundance is large enough to form ${\rm H_{2}}$, but at the same time too low to drive significant collisional dissociation of $\rm H_2$. The width of this ${\rm H_{2}}$ shell and the amount of $\rm H_2$ in this region is determined by the hardness of the energy spectrum of the source: the width of the I-front is of the order of the mean free path of the ionizing photons. Pop III stars, in general, produce a large number of hard photons due to their high temperature, which can penetrate deeper into the neutral region than soft photons.
This precursor ${\rm H_2}$ shell feature is evident in Figs \[fig-Rcrit\], \[fig-midpoint\], and \[fig-endpoint\]. We show the detailed structure of these ${\rm H_{2}}$ shells in Fig. \[fig-midtrap\], where we plot the radial profile of the abundance of different species for the case of $M=4\times 10^5
\,M_\odot$, Phase I, $D=540\,\rm pc$ ($F_0=5.14$) at $t=0.5\, t_{*}$. We note the similarity between our results and those of @2001ApJ...560..580R for an I-front in a uniform, static IGM at the mean density (see Fig. 3 in @2001ApJ...560..580R) which also show a precursor $\rm H_2$ shell. A similar effect was reported by @2006ApJ...645L..93S, as well.
What is the importance of this $\rm H_2$ shell in protecting the central region of haloes from dissociating radiation? The molecular column density obtained by this $\rm H_2$ shell sometimes reaches $\sim 10^{16}\,\rm cm^{-2}$, which provides an appreciable amount of self-shielding. The self-shielding due to the $\rm H_2$ shell, however, is not the major factor that determines whether or not the $\rm H2$ in the core region is protected. A more important factor is which evolutionary phase the target halo is in when it is irradiated. Roughly speaking, when a target halo is irradiated early in its evolution (Phase I), the precursor $\rm H_2$ shell dominates the total $\rm H_2$ column density available to shield the central region, but this shielding is not sufficient to prevent photodissociation there anyway. On the other hand, if the halo is irradiated later in its evolution (Phase II), the $\rm H_2$ column density of the shell is only a small part of the total $\rm H_2$ column density, so shielding is successful independent of the precursor shell. We describe this in more detail as follows.
In order to understand quantitatively the importance of the $\rm
H_2$ shell in protecting the central $\rm H_2$ fraction, we have performed simulations with a source SED that is identical to the Pop III SED below 13.6 eV, but zero above 13.6 eV. As the radiation is now incapable of ionizing the halo gas, the $\rm H_2$ shell formation by partial ionization will not occur. This enables us to compare our results where the $\rm H_2$ shell is present to those cases without an $\rm H_2$ shell. We describe a specific case of $M=2\times 10^{5} \,M_\odot$ as an illustration. Roughly speaking, the ${\rm H_2}$ shell which forms only in the presence of ionizing radiation compensates for the amount by which the initial molecular column density, $N_{\rm H_2}$, is reduced when molecules in the ionized region are destroyed by collisional dissociation. The nett column density in the case where the $\rm H_2$ shell is present even exceeds that in the case without the $\rm H_2$ shell (Figs \[fig-nh2-C1\] and \[fig-nh2-C4\]). The nett effect is the increase of the self-shielding. Such an increase of the self-shielding, however, is not too dramatic. In the case of $M=2\times 10^{5} \,M_\odot$ with Phase I initial conditions, $y_{\rm
H_2}\approx 10^{-5.3}$ at the centre, about an order of magnitude higher than the central $y_{\rm H_2}$ of the case without ionizing photons (Fig. \[fig-nh2-C1\]). This molecule fraction is still too low, however, to cool the gas. On the other hand, in the case of $M=2\times 10^{5} \,M_\odot$ with Phase II initial conditions, $y_{\rm H_2}\approx 10^{-3.5}$ at the centre throughout the lifetime of the Pop III source, whether or not the ${\rm H_2}$ shell is formed. The depth (radius) of penetration of dissociating photons differs by a factor of 2 if the shell is included, but the central ${\rm H_2}$ is still protected because of the high ${\rm H_2}$ column density [*apart*]{} from the precursor shell (Fig. \[fig-nh2-C4\]). The major factor that determines the fate of the central $\rm H_2$ fraction is instead the evolutionary phase of a target halo when it is irradiated. The short lifetime of a Pop III star plays an important role of either reconstituting or protecting molecules in the core, depending upon the evolutionary phase of the halo, as will be described in Section \[sub:Collapse\].
Note that in all cases, we use equation (\[eq:DB\_shield\_factor\]), the shielding function for thermally-broadened lines with $T=10^{4}\,{\rm K}$. This is justified by the fact that the ${\rm H_2}$ shell moves inward with $v\approx 2-5 \,{\rm km\,s^{-1}}$ and the shell achieves $T=T_{\rm sh}\approx 10^{3}-5\times
10^{3}\,{\rm K}$, where $T_{\rm sh}$ denotes the temperature of the shell. If we take this peculiar velocity as sound speed, $v\approx 2-5 \,{\rm
km\,s^{-1}}$ corresponds to $T=T_p\equiv v^2 \mu m_{\rm H}/k=6\times 10^2
- 3.7\times 10^3\,{\rm K}$, where the subscript $p$ denotes the peculiar velocity. A crude way to imitate both effects by thermal broadening is to use the sum of these two temperatures ($T_{\rm sh}$ and $T_p$). We take the most conservative stand – the least self-shielding effect – in order not to overestimate the self-shielding, and use $T=10^{4}\,{\rm K}$ as the temperature responsible for the nett thermal broadening of the molecular LW bands.
Formation of shock and Evolution of core {#sub:shock}
----------------------------------------
After the I-front decelerates as it enters the target halo, transforming from R-type to D-type, a shock front forms to lead the D-type front. The neutral gas in the core is strongly affected by this shock front as it propagates. This shock plays an important role in providing both positive and negative feedback effects. By identifying successive evolutionary stages of the shock, we now describe how the core responds to the shock and evolves accordingly.
### Stage I: Formation and acceleration of Shock {#sub:shockstageI}
A shock starts to form as the I-front, initially moving supersonically as an R-type, slows down and turns into a D-type. The pre-front gas – neutral gas ahead of the I-front – can respond to the I-front before it is swept by the I-front, because the D-type front moves subsonically into the neutral gas. It is easier to understand the formation of the shock by using the I-front jump conditions: the pre-front gas speed in the rest frame of the I-front, $v_1$, derived from the I-front jump conditions, should satisfy either $v_1\ge v_R \equiv c_{\rm I,2}+(c_{\rm
I,2}^2 - c_{\rm I,1}^2)^{0.5},$ or $v_1\le v_D \equiv c_{\rm I,2}-(c_{\rm
I,2}^2 - c_{\rm I,1}^2)^{0.5}$, where $c_{\rm I,1}$ and $c_{\rm I,2}$ are the isothermal sound speeds of the pre-front and post-front gas, respectively. $v_R$ and $v_D$ have a gap of $2(c_{\rm
I,2}^2 - c_{\rm I,1}^2)^{0.5}$, which is nonzero in general. As the I-front slows down and $v_1$ starts to cross $v_R$, $v_1$ encounters a value which is not allowed mathematically. This paradox is resolved, however, because the pre-front gas now “prepares” a new hydrodynamic condition by forming a shock. The shock wave increases $\rho_1$ and thereby reduces $v_1$ and increases $v_D$, making it possible to satisfy the D-type condition, $v_1\le v_D$.
This shock-front then propagates inward, separating from the I-front, due to the discrepancy between the speed of the shock-front and the speed of the I-front. As the shock-front enters the flat-density core, the shock front starts to accelerate, leaving behind the post-shock gas with ever increasing temperature (e.g. see time steps 4 and 5 in Fig. \[fig-evol\], where the post-shock temperature increases as the radius $r$ decreases).
As the shock boosts the density and temperature in the neutral, post-shock gas, the ${\rm
H_2}$ formation rate there increases, boosting the $\rm H_2$ column density even further. We can understand the evolution of $y_{\rm H_2}$ in the presence of this shock quantitatively by using its equilibrium value, $y_{\rm
H_2,eq}$. The increase of density and temperature due to this shock promotes $\rm H_2$ formation, as follows. When there is no significant $\rm H^-$ destruction mechanism, the dominant $\rm H_2$ formation mechanism is through $\rm H^-$ (equation \[eq:solomon\]), and the $\rm H_2$ formation rate becomes equivalent to the $\rm H^-$ formation rate. Photo-dissociation dominates over collisional dissociation in destroying $\rm H_2$, which occurs when $x\la 4\times 10^{-3}\,T_{\rm
K}^{1/2}$ and $n_{\rm H}\ga 0.045 \times (F_{\rm LW}/10^{-21}\,\rm
erg\,s^{-1}\,cm^{-2}\,Hz^{-1})$ (e.g. @2001MNRAS.321..385G). Using the $\rm H^-$ formation rate coefficient [@1972AA....20..263D] $$k_{\rm H^-}=10^{-18}\,T_{\rm K}\,{\rm cm}^{3}\,{\rm s}^{-1},$$ and the photo-dissociation rate coefficient $k_{\rm H_2}$ given by equation (\[eq:DB\_rate\]), we obtain $$\begin{aligned}
y_{\rm H_2, eq}&=&4.1\times 10^{-5} \left( \frac{T}{5000\,{\rm K}}\right)\left(
\frac{x}{10^{-4}}\right)\nonumber \\
&& \times \left(
\frac{n_{\rm H}}{30\,{\rm cm}^{-3}}\right) \left( F_0\cdot F_{\rm
shield}\right)^{-1},
\label{eq:yh2}\end{aligned}$$ where we have used the fact that one can scale $F_{\rm LW}$ by $F_0$ according to the following: $$F_{\rm LW}\approx 3.25\times 10^{-21} \,{\rm erg
\,s^{-1}\,cm^{-2}\,Hz^{-1}}\, F_0,
\label{eq:flw}$$ if one adopts a black-body spectrum with $T=10^{5}\,\rm K$. As seen in equation (\[eq:yh2\]), both the high temperature ($\sim
1000-5000\,\rm K$) and increased density ($\times 4$ in the case of strong shock) of the post-shock gas contributes to boosting the $\rm H_2$ fraction. As $y_{\rm H_2}\propto F_{\rm shield}^{-1}$, molecular self-shielding also plays an important role in determining $y_{\rm H_2}$. If the shock boosts the formation rate of $\rm H_2$ and $y_{\rm H_2}$ increases, so will $N_{\rm H_2}$, and with it the shielding. These two effects, therefore, amplify each other.
There is an additional mechanism to create molecules: the shock-induced molecule formation (SIMF). The acceleration of the shock-front accompanied by an increasing post-shock temperature, leads to a partial ionization of the post-shock gas in many cases, when the right condition ($T\ga 10^4\,\rm K$) is met to trigger collisional ionization – see, for example, step 5 in Fig. \[fig-evol\]: the centre is shock-heated above $10^4\,\rm K$, with a boost in $x$. The electron fraction $x$ now reaches $\sim 10^{-4} - 10^{-2}$, which promotes further ${\rm H_2}$ formation. This mechanism is indeed identical to the $\rm H_2$ formation mechanism in a gas that has been shock-heated to temperatures above $10^4\,\rm K$ [@1987ApJ...318...32S; @1992ApJ...386..432K]. When a gas cools radiatively from a temperature well above $10^4\,\rm K$, it cools faster than it recombines. As a result, the recombination is out of equilibrium, and an enhanced electron fraction exists at temperatures even below $10^4\,\rm K$ compared to the equilibrium value. This electron fraction triggers the formation of $\rm H_2$ through the gas-phase reactions (equations \[eq:solomon\] and \[eq:solomon2\]).
SIMF does not always occur, however. The shock-front can accelerate when the pre-shock density remains almost constant (e.g. Fig. \[fig-evol\]). If the density increases faster than the shock propagates, on the other hand, the shock-front will encounter an ever increasing density “hill” and it will never accelerate to generate post-shock temperature above $10^4\,\rm K$ (e.g. Fig. \[fig-evol2e5C4\]). The dependence of SIMF on the halo mass, source flux, and the initial phase will be described in Section \[sub:Collapse\].
### Stage II: Cooling and Compression of Core
As the shock-front approaches the centre of the halo, the post-shock gas there becomes more concentrated and denser than the pre-shock gas. This shock-induced compression leads to a very fast molecular cooling in the core and further compression in almost a runaway fashion, as follows.
Molecular cooling occurs very rapidly at a high density and temperature condition. Assuming that the pre-shock gas of the halo core remains unchanged before the shock-front arrives – as is usually the case in Phase I – and the shock is strong, the post-shock density of the core becomes 4 times higher than that of the pre-shock, namely $n_{\rm HI}\approx
4\times 30\,\rm cm^{-3}=120\,\rm cm^{-3}$ in a TIS halo core at $z=20$. At the same time, post-shock temperature can be as high as $10^4\,\rm K$. The molecular cooling time, $t_{\rm cool,H_2}\equiv
T/(dT/dt)$, is $$t_{\rm cool,H_2}=\frac{kT}{X\mu(\gamma-1)y_{\rm H_2}n_{\rm
HI}\Lambda_{\rm H_2}},
\label{eq:tcool}$$ where $X=0.75$ is the hydrogen mass fraction, and $\Lambda_{\rm H_2}$ is the molecular cooling rate. For a gas with $n_{\rm HI}=120\,\rm
cm^{-3}$ and $T=10^4\,\rm K$, $\Lambda_{\rm H_2}\approx 3.4\times
10^{-22} \,\rm erg\,cm^{-3}\,s^{-1}$, and thus $$t_{\rm
cool,H_2}\approx 1.8\times 10^3\,{\rm yr}\,
\left(\frac{y_{\rm H_2}}{10^{-3}}\right)^{-1}.
\label{eq:tool-num}$$ With such a rapid cooling, the isothermal shock jump condition ($T_2=T_1$) is a good approximation, and the post-shock density becomes even higher than that of the adiabatic strong shock, because $\rho_{b,2}/\rho_{b_1}\approx M_{I,1}^2$ now. Such a strong compression of the core is observed very frequently in our parameter space of different halo masses and source fluxes. For example, Fig. \[fig-M2e5-ing\] shows how the centre of a halo with $M=2\times
10^4\,M_\odot$ evolves in response to the shock. As the shock hits the centre, density increases by many orders of magnitude.
Does this compression eventually lead to the core collapse? As the shock carries the kinetic energy as well as the thermal energy, the shock will bounce off the centre after it hits the centre. In the following section, we describe this final stage of the shock propagation and show how it will affect the core collapse.
### Stage III: Bounce of Shock and Collapse of Core
After the shock hits the centre, the shock wave will be reflected and propagate outward. In our 1D calculation, this reflection will mimic the transmission of the shock wave through the centre. This bouncing shock will try to disrupt the gas. The core that is undergoing cooling and compression due to the positive feedback effects mentioned so far will be affected by this negative feedback effect, as well.
The final fate of the core depends on how well the core endures such a disruption. As the shock bounces off the centre, density starts to decrease. If this bounce is weak, the core quickly reassembles, cools, and finally collapses. If this bounce is strong, the core will take a longer time to collapse and, in some cases, the core will never collapse within the Hubble time. Haloes of smaller mass seem to be more susceptible to this shock-bounce than those of larger mass (see Figs \[fig-M1e5-ing\] and \[fig-M8e5-ing\] for comparison).
If the core finally takes the collapse route, the central hydrogen number density increases to $\sim 10^{4}\,\rm cm^{-3}$, at which point the ro-vibrational levels of $\rm H_2$ are populated at their equilibrium values and the molecular cooling time becomes independent of density (e.g. @2002Sci...295...93A). Since then, adiabatic heating dominates over the molecular cooling, and the temperature increases as collapse proceeds. Finally, when $n_{\rm HI}$ reaches $\sim 10^{8}\,\rm
cm^{-3}$, the three-body hydrogen reaction ensues and converts most hydrogen atoms into the molecules, which will undergo a further collapse and form a proto-star.
Feedback of Pop III starlight on Nearby Minihaloes: parameter dependence of core collapse {#sub:Collapse}
-----------------------------------------------------------------------------------------
We now summarize the outcome of our full parameter study of radiative feedback effects of Pop III starlight on nearby minihaloes. As we have described in the previous section, positive and negative feedback effects of the shock compete and produce a nett effect which can be either 1) an expedited collapse, 2) delayed collapse, 3) neutral (unaffected) collapse, or 4) a disruption.
Overall, the radiative feedback effect of a Pop III star is not as destructive as naively expected. Minihaloes with $M\ga [1-2]\times 10^5\,M_\odot$, which can cool and collapse without radiation, are still able to form cooling and collapsing clouds at their centre even in the presence of Pop III starlight. The quantitative results are summarized in Tables \[table:case1\], \[table:case2\] and Fig. \[fig-coll\].
The relatively short lifetime of a Pop III star, compared to the recombination timescale in the core, is a key to understanding this behaviour. One of the necessary conditions for the core collapse is that $\rm H_2$ molecular cooling should occur in the core. As this requires a sufficient molecular fraction, namely $y_{\rm H_2}\ga
10^{-4}$, it is crucial to understand how molecules are created at such a level. In Phase I (low $y_{\rm H_2}$ and high $x$), radiation can easily dissociate $\rm H_2$ while the source is on, but after the source dies, the high electron fraction stimulates $\rm H_2$ formation. This is possible because the recombination time in the TIS core is longer than the lifetime of the source Pop III star. On the contrary, in Phase II (high $y_{\rm H_2}$ and low $x$), $\rm H_2$ is more easily protected against the dissociating radiation because the higher $\rm H_2$ column density provides self-shielding and compression increases the formation rate. Because the source irradiates these haloes for a short period of time, the [ *dissociation front*]{} does not reach the centre, and its high molecule fraction is preserved throughout the Pop III stellar lifetime.
### Phase I
When haloes start their evolution from Phase I – IGM chemical abundance and the TIS structure –, other than the change of collapse times, there is no reversal of collapse. In other words, haloes that were destined to cool and collapse would do so even when exposed to the first Pop III star in the neighbourhood. Minihaloes with $M\ga 10^5 \,M_\odot$ are able to collapse without radiation, while those with $M < 10^5 \,M_\odot$ are not. In the presence of radiation, haloes with $M\ga 10^5 \,M_\odot$ are still able to collapse, while those with $M < 10^5 \,M_\odot$ are still unable to do so, even with the help of shock-induced molecule formation (Fig. \[fig-coll\]; Table \[table:case1\]).
The core collapse in Phase I occurs mostly as an expedited collapse (Table \[table:case1\]). The shock plays a major role in driving such an expedited collapse: the $\rm H_2$ fraction becomes boosted by the higher density and high temperature delivered by the shock. Whether or not SIMF has occurred, such a boost in $y_{\rm H_2}$ is sufficient to expedite the core collapse.
There is one delayed collapse case at the low mass and the high flux end. For $M=10^5\,\rm M_\odot$ at $F_0=46.3$, the boosted molecule formation is not sufficient to bring the core to an immediate collapse. As the shock bounces, the momentum carries gas away from the centre until it cools and recollapses.
The unchanged collapses occur at the high mass and the low flux end. For $M=8\times
10^5\,M_\odot$ at $F_0=[1.5,\,5.14]$, the shock propagates into the already collapsing core. The shock energy delivered in these cases is not significant enough to change the course of collapse.
-------------------- ----------- ----------- ----------------------- ----------------------- ----------------------- -----------------------
$D$ (pc) \[$F_0$\] $0.25 $ $0.5 $ $1 $ $2 $ $4 $ $8 $
$(\cdot)$ $(\cdot)$ $(88.82)$ $(31.02)$ $(14.61)$ $(8.66)$
180 pc \[46.3\] $\cdot$ $\cdot$ $1.455$ $7.288 \cdot 10^{-2}$ $1.838 \cdot 10^{-1}$ $4.712 \cdot 10^{-1}$
360 pc \[11.6\] $\cdot$ $\cdot$ $1.935 \cdot 10^{-1}$ $1.308 \cdot 10^{-1}$ $3.597 \cdot 10^{-1}$ $8.177 \cdot 10^{-1}$
540 pc \[5.14\] $\cdot$ $\cdot$ $3.427 \cdot 10^{-1}$ $2.093 \cdot 10^{-1}$ $4.919 \cdot 10^{-1}$ $1.000$
1000 pc \[1.5\] $\cdot$ $\cdot$ $9.497 \cdot 10^{-1}$ $4.525 \cdot 10^{-1}$ $7.144 \cdot 10^{-1}$ $1.241$
-------------------- ----------- ----------- ----------------------- ----------------------- ----------------------- -----------------------
### Phase II
The overall effect of radiation from a Pop III star on neighbouring minihaloes in Phase II is similar to the effect on the minihaloes in Phase I: haloes that were destined to cool and collapse would do so even when exposed to the first Pop III star in the neighbourhood. A slight shift of the trend exists, however, in Phase II (Fig. \[fig-coll\]; Table \[table:case2\]). When haloes start their evolution from Phase II, those with $M\ga 10^5 \,M_\odot$ are able to collapse without radiation, while those with $M \la 2 \times 10^5 \,M_\odot$ are not. The collapse in Phase II is reversed (halted) for the low mass end: for $M=10^5\,M_\odot$, the shock disrupts the core and it never recollapses. SIMF occurs at $F_0>1.5$ for $M=10^5\,M_\odot$, but this does not prevent such a destructive process from happening.
As haloes start their evolution from Phase II, in which the halo cores are already cooling and collapsing, the neutral (unaffected) collapse cases occur more frequently than in Phase I. At high and intermediate masses, the collapse time hardly changes from the case without radiation. Haloes with $M=8\times 10^5 \,M_\odot$ collapse [*before*]{} the source dies, as they do without radiation, simply because the shock wave does not affect the core. In this case, shock propagates into the centre after collapse has advanced significantly.
There is one delayed collapse case: compared to the delayed collapse in Phase I, which occurred at low mass/high flux end ($M=10^5\,M_\odot$ at $F_0=46.3$), this now occurs at an intermediate mass/high flux end ($M=2\times 10^5\,M_\odot$ at $F_0=46.3$). Otherwise, for intermediate mass, collapse is either neutral or expedited.
-------------------- ----------- ----------- ----------- ----------------------- ----------------------- ------------------------
$D$ (pc) \[$F_0$\] $0.25 $ $0.5 $ $1 $ $2 $ $4 $ $8 $
$(\cdot)$ $(\cdot)$ $(65.66)$ $(14.49)$ $(4.23)$ $(1.65)$
180 pc \[46.3\] $\cdot$ $\cdot$ $\cdot$ $4.269 $ $7.151 \cdot 10^{-1}$ $9.541 \cdot 10^{-1}$
360 pc \[11.6\] $\cdot$ $\cdot$ $\cdot$ $4.997 \cdot 10^{-1}$ $1.155 $ $1.002 $
540 pc \[5.14\] $\cdot$ $\cdot$ $\cdot$ $6.740 \cdot 10^{-1}$ $9.794 \cdot 10^{-1}$ $9.964 \cdot 10^{-1}$
1000 pc \[1.5\] $\cdot$ $\cdot$ $\cdot$ $5.794 \cdot 10^{-1}$ $9.926 \cdot 10^{-1}$ $9.994 \cdot 10^{-1} $
-------------------- ----------- ----------- ----------- ----------------------- ----------------------- ------------------------
The structure of haloes at the moment of collapse {#sub:onset}
-------------------------------------------------
The structure of halo at collapse determines how a protostar evolves into a star and how the starlight will later propagate through the host halo. We first show how halo profiles at collapse vary for different mass without radiation. We then describe how halo structure is affected by the Pop III starlight.
We note that halo structure shows a strong dependence on the halo mass. For radius $r\ga 10^{-2}\,\rm pc$, density profiles of haloes without radiation are well fit by a power law, $\rho \propto r^{-w}$. The value of $w$, however, is dependent upon the mass of the halo. We find that $w=2.5$, 2.4, 2.3, and 2.2 for haloes of mass $M=10^5$, $M=2\times 10^5$, $M=4\times 10^5$, and $M=8\times 10^5\,M_\odot$, respectively. In all cases, the temperature is somewhat flat with $T\sim 10^{2.5} - 10^3\,\rm K$. The temperature at $r\approx 10^{-2}\,\rm pc$, where $\rho\approx 3\times 10^{-16} \rm g\,cm^{-3}$ (or $n_{\rm H}\approx 10^8\,\rm cm^{-3}$), is about $800 \,\rm K$ in all cases. The universality of these core properties seems to originate from the fact that the dominant process, $\rm H_2$ cooling, causes loss memory of the initial condition (e.g. different virial temperatures for different virial masses). The outer part of these haloes, however, still retain the memory virial equilibrium because radiative cooling is negligible. Overall, as mass decreases, density slope increases (see Fig. \[fig-onset\]).
The radiative feedback effect of the starlight on final halo profiles is found to be negligible in most cases. The region that has been photo-ionized during the stellar lifetime is obviously strongly affected. The neutral region, however, is almost indistinguishable from the case without radiation in most cases. The variance of temperature profile exists only at the low-mass end, $M=10^5\,M_\odot$, or the high-flux end, $F_0=46.3$ ($D=180\,\rm pc$). Such variance completely disappears at the high-mass end, $M=8\times 10^5\,M_\odot$, because collapse is mostly unaffected (Fig. \[fig-onset\]).
This result indicates that the mass of secondary Pop III stars would be almost identical to that of the Pop III stars which form without radiative feedback effect. A more fundamental variance may exist, however, due to the environmental variance of star forming regions: @2006astro.ph..7013O show that temperature variance of different regions result in the variance of protostellar masses, due to the corresponding variance of mass infall rate. As our simulation does not advance beyond $n_{\rm H}=10^8\,\rm cm^{-3}$, where three-body collision can produce copious amount of $\rm H_2$ molecules and change the adiabatic index of the gas, we are unable to quantify the final mass of the protostar at this stage.
Feedback of Pop III Starlight on Merging Haloes and Subclumps {#sub:abel}
-------------------------------------------------------------
While we were preparing this manuscript, two preprints were posted describing simulations of the radiative feedback of the first Pop III star on dense gas clumps even closer to the star than the external minihaloes we have considered so far, for the case of subclumps [@2006ApJ...645L..93S] and the case of a second minihalo undergoing a major merger with the minihalo that hosts the first star [@2006astro.ph..6019A]. The centre of the target halo or clumps in this case is well within the virial radius of the halo which hosts the first star, and, these authors find that secondary star formation occurs in these subhaloes. @2006astro.ph..6019A, for instance, report that the first star forms inside a minihalo of mass $M=4\times
10^5\,M_\odot$ as it merges with a second minihalo of mass $M=5.5\times
10^5\,M_\odot$ (the target halo). The centre of this target halo is at a distance of only 50 parsecs from the first star. Cooling and collapse leading to the formation of a protostar is found to occur inside the target halo about 6 Myrs [*after*]{} the first star has died.
We ask the same question that whether or not a halo would collapse to form a secondary Pop III star if a nearby Pop III star irradiates the halo at a distance of 50 pc. Note that the target halo we consider now would collapse anyway if there were no radiation, in $\sim 11$ Myrs for Phase I and $\sim 3$ Myrs for Phase II (see Table \[table:collsub\]). This problem requires us to extend our parameter space beyond what has been considered so far, because of the short distance (high flux) between the source and the target.
We have attempted to reproduce the result of @2006astro.ph..6019A using our code for a target halo of mass $M=5.5\times 10^5\,M_\odot$ and $D=50\,\rm pc$, corresponding to the ionizing flux $F_0=600$. Note that the LW band flux is very high: $F_{\rm LW}\sim
2000\times 10^{-21}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$ (equation \[eq:flw\]). As $D$ is smaller than the virial radius of the target halo, we truncated the halo profile at 50 pc. To be consistent with our previous calculations, we neglect the geometrical variation of the flux with position inside the target halo.
Surprisingly enough, contrary to the outcome of @2006astro.ph..6019A, we find that collapse is expedited, occurring [*within the lifetime of the first star*]{}, for both Phase I and Phase II initial conditions. The main mechanism was SIMF: initially, $\rm H_2$ is completely wiped out by a strong dissociating radiation, but as the SIMF occurs, newly created molecules lead to cooling and collapsing. This result is in disagreement with the result of @2006astro.ph..6019A, which shows that the second star forms [*after the star has died*]{}.
This puzzling result shows the importance of $\rm H_2$ self-shielding. @2006astro.ph..6019A performed an optically-thin calculation for Lyman-Werner bands, neglecting the $\rm H_2$ self-shielding, while our calculation took the self-shielding into account. In order to mimic their calculation more consistently, we artificially performed an optically-thin calculation for Lyman-Werner bands. We found that, if the target halo is irradiated without $\rm H_2$ self-shielding, the core collapse is delayed and occurs [*after the star dies*]{} both in Phase I and Phase II. In our simulations without $\rm
H2$ self-shielding, the core bounced and recollapsed in $\sim$44 Myrs and $\sim$111 Myrs after the star has turned off in Phase I and Phase II, respectively (Table \[table:collsub\]).
Qualitatively, our calculation without $\rm H_2$ self-shielding agrees with the result of @2006astro.ph..6019A, that collapse in the target halo occurs after the source dies. We find that SIMF is the main mechanism for the formation of $\rm H_2$. Initially, the strong LW band photons destroy molecules in the core. As the shock propagates inward, however, boosted density and temperature of the post-shock gas enhances the molecule fraction (equation \[eq:yh2\]), and increases the $\rm H_2$ column density. As the shock front accelerates, SIMF occurs, and newly created $\rm H_2$ is protected from the LW band photons because of increased self-shielding. If self-shielding is not accounted for, however, this $\rm H_2$ is destroyed and never restored, so collapse does not proceed during the lifetime of the source.
We conclude, therefore, that neglecting $\rm H_2$ self-shielding in calculation explains why @2006astro.ph..6019A observes a delayed collapse. The quantitative disagreement between our collapse times (when we neglect self-shielding) and theirs may originate from the difference in the structure and chemical abundances of the target halo when the source irradiates it. How do our results compare with those of @2006ApJ...645L..93S? A fundamental difference exists other than the fact that their work is limited to subclumps of a halo that hosts a Pop III star. They interpret the shock only as a carrier of negative feedback effect, while the shock, in our case, delivers both the positive and negative feedback effects. In their shock-driven evaporation (Model C) case, the collapsing core eventually fails to collapse, because the shock heats the core before it finishes collapse. Their successful collapse case (Model B) is simply an unaltered collapse: an already collapsing core finishes collapse before the shock front reaches the centre. On the other hand, we have observed expedited collapses as well as delayed or failed collapse. Such expedited collapses we observe are truly positive feedback effects. Quantitatively, because of their limited interpretation of the role of the shock, they argue that only regions with hydrogen number density $n_{\rm H}\ga 10^{2-3} \, \rm
cm^{-3}$, high enough to finish collapse before the shock front reaches the centre, can collapse under the influence of Pop III starlight. On the contrary, we find, for instance, that regions with $n_{\rm H}\sim 30 \, \rm cm^{-3}$ – core density of TIS haloes in Phase I – can cool and collapse even after the shock front has reached the centre. As the shock-front accelerates and delivers strong positive feedback effects in the small core region, high resolution is required to produce this mechanism in simulations. The relatively poor resolution of SPH simulations by @2006ApJ...645L..93S might have prevented them from fully resolving the shock structure in the core, and potentially producing the positive feedback effects.
Our result indicates that secondary star formation may occur even in subclumps of the host halo, which are subject to much stronger radiative feedback than isolated, nearby minihaloes. We have shown in this section that $\rm H_2$ self-shielding is important even at this high level of ionizing ($F_0=600$) and dissociating ($F_{\rm LW}=
2\times 10^{-18}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$) fluxes. It is even more surprising because the collapse is expedited and [*coeval formation*]{} of Pop III stars in the same neighbourhood is possible. The naive expectation of negative feedback effect of a Pop III star in its neighbourhood, therefore, should be revisited.
no radiation self-shielding no self-shielding
---------- -------------- ---------------- -------------------
Phase I 11.2 1.1 47
Phase II 2.7 1.3 114
: Collapse time (in units of Myrs) of a subclump with $M=5.5\times 10^5\,M_\odot$ irradiated by a Pop III star at distance $D=50\,\rm pc$ ($F_0=600$). For both Phase I and Phase II, we show how a case with a proper treatment of $\rm H_2$ self-shielding (2nd column) differs in collapse time from a case without self-shielding (3rd column) and a case without radiation. When $\rm H_2$ self-shielding is properly treated, collapse occurs in $\sim 1\,\rm Myr$, [*before*]{} the neighbouring Pop III turns off, while when $\rm H_2$ self-shielding is neglected, collapse occurs [*after*]{} the star turns off, which is qualitatively consistent with the simulation results by @2006astro.ph..6019A. []{data-label="table:collsub"}
Summary/Discussion {#sec:2star-Discussion}
==================
We have studied the radiative feedback effects of the first stars (i.e. Pop III stars) on their nearby minihaloes, by solving radiative transfer and hydrodynamics self-consistently using the 1-D spherical, radiation-hydrodynamics code we have developed. The results can be summarized as follows:
- We identified the minimum collapse mass, namely the mass of minihaloes which are able to have a core which cools and collapses in the absence of external radiation. We find that $M_{\rm c,min}\sim 7\times
10^4\,M_\odot$ at $z=20$. In determining $M_{\rm c,min}$, we applied two criteria. First, the collapsing region should reach $n_{\rm
H}=10^8\,\rm cm^{-3}$ to be considered as a collapse. Second, this should occur within the Hubble time. The minimum collapse mass we find roughly agrees with that of @2001ApJ...548..509M, where the AMR scheme they used seems to have resolved the inner structure of minihaloes.
- Minihaloes could have been in very different stages of their evolution when they were irradiated by a Pop III star. We used two different initial conditions to represent such phase differences. In Phase I, chemical abundances have not yet evolved away from their IGM equilibrium values. This stage is characterized by low $\rm H_2$ fraction, $y_{\rm
H_2}\sim 2\times 10^{-6}$ and high electron fraction, $x\sim
10^{-4}$ at the centre. Haloes can be irradiated in Phase II, which is the state of these haloes evolved from Phase I, where $x$ has dropped to $10^{-5}$ by recombination. Phase II is characterized by high $\rm H_2$ fraction $y_{\rm
H_2}\sim 10^{-4} - 10^{-3}$, low electron fraction $x=
10^{-5}$, and core density higher than that of Phase I.
- Within our parameter space, the I-front is trapped before reaching the core in all cases. Ionized gas evaporates, and a shock-front develops ahead of the I-front and travels into the core. The shock front leads to both positive and negative feedback effects. A boost in density and temperature by a shock increases the $\rm
H_2$ formation rate. In some cases, the shock accelerates and obtains a temperature above $10^4\,\rm K$, which is high enough to drive collisional ionization, which then leads to a further boost in $\rm H_2$ fraction. The high temperature and kinetic energy delivered by the shock, on the other hand, tries to disrupt the gas. The nett effect is either 1) an expedited collapse, 2) delayed collapse, 3) neutral (unaffected) collapse, or 4) a disruption, depending upon the flux, halo mass, and the initial condition when irradiated.
- At the moment of collapse, halo profiles under radiation are almost identical to those without radiation. Density profiles of different mass haloes are well fit by different power-law profiles, $\rho\propto r^{-w}$, where $w=2.5$, 2.4, 2.3, and 2.2 for $M=10^5$, $2\times 10^5$, $4\times 10^5$, and $8\times 10^5\,M_\odot$, respectively. Some variation in temperature profile exists at the low-mass end, $M=10^5\,M_\odot$, and the high-flux end $F_0=46.3$ ($D=180\,\rm pc$).
- Overall, the radiative feedback effect of Pop III stars is not as destructive as naively expected. Minihaloes with $M\ga [1-2]\times 10^5\,M_\odot$ are still able to form cooling and collapsing clouds at their centres even in the presence of radiation. A simple explanation is possible for such behaviour. In Phase I (low $y_{\rm H_2}$ and high $x$), radiation can easily dissociate $\rm H_2$ while the source is on, but after the source dies, high electron fraction allows $\rm H_2$ formation. On the contrary, in Phase II (high $y_{\rm H_2}$ and low $x$), $\rm H_2$ is more easily protected against the dissociating radiation because the higher $\rm H_2$ column density provides self-shielding and compression increases the formation rate. The situation becomes more complicated, however, by other feedback effects which will be described in the following bullets.
- Within our parameter space, haloes that are irradiated at Phase I experience expedited collapse predominantly for $10^5 \la
M/M_\odot \la 8\times 10^5$, except for the delayed or neutral collapses occurring at the low mass/high flux and the high mass/low flux extremes (e.g. for $M=10^5\,M_\odot$ at $F_0=46.3$ and for $M=8\times
10^5\,M_\odot$ at $F_0=[1.5,\,5.14]$).
- Haloes that are irradiated at Phase II show a more complicated behaviour. In this case, unaffected collapse is more frequent, in general, at high and intermediate masses, while for $M=10^5\,M_\odot$, core collapse is now reversed at any $F_0$. Delayed collapse occurs for $M=2\times 10^5\,M_\odot$ at $F_0=46.3$. Unaffected collapse occurs for $M=8\times 10^5\,M_\odot$ for any $F_0$, and for $M=4\times 10^5\,M_\odot$ at $F_0\la 11.6$. Otherwise, for intermediate mass, collapse is either neutral or expedited.
- We first find in this paper that coeval formation of Pop III stars is possible even under the influence of ionizing and dissociating radiation from a first star. This occurs either as an expedited collapse or an unaffected collapse. Among those parameters explored in this paper, expedited collapse occurs during the lifetime of the source star when a halo of mass $M=2\times 10^5 \,M_\odot$ in Phase I is irradiated by a Pop III star at a distance $D=180 \,\rm pc$ ($F_0=46.3$). Unaffected collapse occurs for haloes of mass $M=8\times 10^5$ in Phase II during the lifetime of the source star for all different distances (fluxes).
- Extending our parameter space to include a specific case studied by @2006astro.ph..6019A, a minihalo merging with a halo hosting a Pop III star, we find that the coeval formation of Pop III stars is possible even in this high ionizing ($F_0\approx 600$) and dissociating ($F_{\rm LW}\sim
2\times 10^{-18}\,\rm erg\,s^{-1}\,cm^{-2}\,Hz^{-1}$) flux case. While @2006astro.ph..6019A find that the secondary star formation in this target halo occurs after the first star dies because of $\rm H_2$ destruction by photodissociation, we find that the minihalo core collapse is expedited to form a star in $\sim 1\,\rm
Myr$, long before the first star dies, due to the SIMF and $\rm H_2$ self-shielding. This discrepancy comes from the fact that we account for the effect of $\rm H_2$ self-shielding, while they do not. A proper treatment of $\rm H_2$ self-shielding is important even for such a high flux regime, because the central $\rm H_2$ fraction can reach $y_{\rm H_2}\ga
10^-3$ due to the SIMF and strong $\rm H_2$ self-shielding is possible due to newly created $\rm H_2$.
We find the minimum collapse mass $M_{\rm c,min}\sim 7\times
10^4\,M_\odot$ at $z=20$ without radiation. While our result agrees roughly with that of the 3D AMR simulation by @2001ApJ...548..509M, discrepancy becomes larger with those of 3D SPH simulation results (e.g. @2000ApJ...544....6F; @2003ApJ...592..645Y) and a semi-analytical calculation using a uniform-sphere model [@1997ApJ...474....1T]. This implies that the central region of haloes should be resolved well in order to quantify the minimum collapse mass exactly.
What does the result of our paper imply for the “first” H II region created by Pop III stars? Because a significant fraction of nearby minihaloes can host second generation stars within the first H II region, it is possible that such a subsequent star formation may at least keep the first H II regions ionized. It may even be possible that individual H II regions grow and overlap, thus finishing the first cosmological reionization. A semi-analytic calculation of minihalo clustering around high density peaks, for example, might allow us to quantify how fast and how big such bubbles can grow. Without secondary star formation, this would simply be a relic H II region in which gas recombines and cools after the source star dies, possibly with metal enrichment from supernova explosion (e.g. @2003ApJ...596L.135B).
We found that the minimum collapse mass is $\sim
1-2\times 10^5 \,M_\odot$ even in the presence of Pop III starlight. Such a low value may affect the reionization history significantly. @2006ApJ...639..621A estimates that the instantaneous ionized mass fraction at $z=20$ is $\sim 0.1$, if individual $\sim 10^6
\,M_\odot$ haloes host one $\sim 100 \,M_\odot$ Pop III star each. If the typical mass scale of host haloes is $\sim 10^5 \,M_\odot$ instead, as the number density of haloes would be roughly 10 times as big as that for $M\sim 10^6 \,M_\odot$, Pop III stars alone would be able to finish cosmological reionization at $z \sim 20$[^8]. New reionization sources will form later in more massive haloes with $T_{\rm vir}\ga 10^4 \,\rm K$, which will host a region cooling by the hydrogen atomic cooling. Depending upon how fast such transition occurs, the global reionization history will have different characteristics (e.g. monotonic growth of ionization fraction vs. double reionization).
In this paper, we have considered only the radiative feedback effect. Pop III stars, however, may exert additional feedback effects. The H II region developed by a Pop III star inside the host halo breaks out as a “champagne flow” inside the host minihalo, where the I-front separates from the shock-front and runs ahead, transforming from D-type to R-type. The shock front left behind also expands into the IGM and nearby minihaloes would be encountered by this shock-front ultimately. Other feedback effects will come from supernova explosions. If the first star dies and explodes as a supernova, both dynamical and chemical feedback effects would alter the fate of nearby minihaloes, as well. How would the additional presence of $\rm H_2$ dissociating background radiation affect our results? In this paper, we have considered the effect of the radiation from an individual nearby Pop III star, whose SED takes a black body form for a short lifetime ($\sim 2.5$ Myrs). This is the case appropriate to the earliest star formation. It is valid whenever a minihalo resides in a place and time where the background from other, more distant stars is negligible. On average, however, the mean free path to $\rm H_2$ dissociating radiation is greater than that for ionizing radiation prior to reionization, so the situation can arise in which the ionizing radiation from distant sources is filtered out but the UV radiation in the LW bands is not. Suppose a minihalo is under the influence of both Pop III starlight from a nearby star and a persistent background radiation field in the LW bands. In the absence of the nearby star, the dissociating background can only hinder the formation of $\rm H_2$ and its cooling. As such, the $\rm H_2$ fraction inside the minihalo when the nearby Pop III star starts to irradiate it would be lower than it would have been without the background. In this case, even if the background were intense enough on its own to prevent the minihalo from cooling and collapsing, the minihalo could still host a cooling core if $\rm H_2$ formed by the positive feedback from the Pop III star, despite the presence of the background. Indeed, this could occur frequently, because we find that a high electron fraction – and, thus a high $\rm H_2$ fraction – can be achieved by collisional ionization in the postshock region in many cases (SIMF; see Section \[sub:shockstageI\]). This newly created $\rm H_2$ will then be easily protected from the dissociating background by self-shielding, since our simulation results show that this SIMF $\rm H_2$ survives even the much larger – albeit short-lived – flux of $\rm H_2$ dissociating radiation from a nearby star in our most extreme case, $F_{\rm
LW}\approx 2000 \times
10^{-21} \rm erg \, s^{-1}\,cm^{-2}\, Hz^{-1}$, as has been shown in Section \[sub:abel\]. Thus, the background would then only prevent those haloes that cannot “host” this SIMF mechanism from cooling and forming stars. We will address this issue further in the future.
As the focus of our paper is the fate of neutral cores of target haloes, in which the ionized fraction never exceeds $\sim
10^{-2}$, we neglected processes which are relevant only when gas achieves high ionized fraction, such as HD cooling and charge exchange between $\rm He^+$ ($\rm He$) and $\rm H$ ($\rm H^+$) (see e.g. @2006astro.ph..6106Y). These processes may be important, however, in the relic H II region outside the target minihalos. For instance, HD cooling may cool gas down below the $\rm H_2$ cooling temperature plateau, $T_{\rm H_2}\sim 100\,\rm K$, if $\rm H_2$ formation and cooling start from a highly ionized initial state (e.g. @2006MNRAS.366..247J).
We chose two different evolutionary phases of nearby minihaloes as our initial conditions. A more natural way to address this problem is to use the structure and chemical composition of minihaloes and IGM from 3-D, chemistry-hydrodynamics calculation. We intend to extend our study in a more consistent manner by combining a 3-D, chemistry-hydrodynamics simulation and the 1-D, radiation-hydrodynamics simulation in the future. In this paper, we simply adopted a model for virialized haloes (TIS profile). In the future, we will also implement a more realistic growth history of haloes (e.g. @2002ApJ...568...52W) to account for the dynamical effect of mass accretion.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank M. Alvarez, T. Abel, S. Glover, I. Iliev, B. O’shea, H. Susa, and D. Whalen for helpful discussions. We also acknowledge the Institute for Nuclear Theory at the University of Washington for their support and hospitality. This work was supported by NASA Astrophysical Theory Program grants NAG5-10825, NAG5-10826, NNG04G177G.
Numerical Method and Code Tests {#finite_appendix}
===============================
Here we describe the finite-difference scheme used for our 1-D spherical, radiation-hydrodynamics code. The subscript, unless noted otherwise, denotes the position of a shell. The superscript denotes the time. For instance, $\rho_{j+1/2}^{n+1}$ is the zone-centred density of shell $j+1$ at time $t^{n+1}$, and $r_{j}^{n}$ is the zone-edge-centred radius of shell $j$ at time $t^{n}$.
The Gas Dynamical Conservation Equations
----------------------------------------
Hydrodynamic conservation equations for the baryonic component (eqs. \[\[eq:realhydro\_mass\]\] - \[\[eq:realhydro\_energy\]\]) are solved following the finite-difference scheme by @1995ApJ...442..480T. We first update the velocity and position using the so-called “leap-frog” scheme, so that the velocity and the position are staggered in time: $$v_{j}^{n+1/2} = v_{j}^{n-1/2} -\left[
4\pi(r_{j}^{n})^{2}\frac{p_{j+1/2}^{n} -p_{j-1/2}^{n}}{dm_{j}}
+\frac{m_{j}^{n}}{(r_{j}^{n})^{2}}\right] dt^{n},
\label{eq:leapfrog1}$$ and $$r_{j}^{n+1} = r_{j}^{n} + v_{j}^{n+1/2}dt^{n+1/2},
\label{eq:leapfrog2}$$ which are second-order accurate. As the mass of each shell is conserved for such a Lagrangian scheme, density is updated following $$\rho_{j+1/2}^{n+1} = \frac{dm_{j+1/2}}{(4/3) \pi[(r_{j+1}^{n+1})^{3}
-(r_{j}^{n+1})^{3}]}.$$ In these equations, $$dt^{n}=\frac{1}{2}(dt^{n-1/2}+dt^{n+1/2}),$$ and $$dm_{j}=\frac{1}{2}(dm_{j-1/2}+dm_{j+1/2}).$$ We then advance the energy by $$\begin{aligned}
e_{i+1/2}^{n+1} &=& e_{i+1/2}^{n} -
p_{i+1/2}^{n}\left(\frac{1}{\rho_{i+1/2}^{n+1}} -
\frac{1}{\rho_{i+1/2}^{n}}\right) \nonumber \\
&& +\frac{(\Gamma-\Lambda)_{i+1/2}^{n}}{\rho_{i+1/2}^{n+1}}dt^{n+1/2}.
\label{eq:energy_tw}\end{aligned}$$
Shocks are treated with the usual artificial viscosity technique. The pressure in the momentum and energy conservation equations is replaced by $P=p+q$, where $$\begin{aligned}
q_{i+1/2}^{n+1} = -c_{q}\frac{2}{1/\rho_{i+1/2}^{n+1}
-1/\rho_{i+1/2}^{n}} \left|v_{i+1}^{n+1/2}-v_{i}^{n+1/2} \right|
\nonumber \\
\times (v_{i+1}^{n+1/2}-v_{i}^{n+1/2}),
\label{eq:arti_viscos}\end{aligned}$$ if $v_{i+1}^{n+1/2}-v_{i}^{n+1/2}<0$, and $q=0$ otherwise. We use $c_{q}=4$, which spreads the shock fronts over four or five cells.
Dark matter shells are also updated according to equations (\[eq:leapfrog1\]) - (\[eq:arti\_viscos\]) – note that we use fluid approximation as described in Section \[sub:fluid-approx\] –, except that the heating/cooling term is zero in equation (\[eq:energy\_tw\]). Note that the dark matter shells are allowed to have effective shock in our fluid approximation, and therefore we need to compute the artificial viscosity when dark matter shells are converging (equation \[eq:arti\_viscos\]), as in the case of the baryonic gas component.
Time Steps
----------
Time step for the finite-differencing is chosen such that important fluid variables do not change abruptly. The relevant time scales are the dynamical, sound-crossing (Courant), cooling(heating), and species-change time scales. In addition, to ensure that the fluid shells do not cross, we also adopt a shell-crossing time. $$dt=\min\{ dt_{{\rm dyn}},\, dt_{{\rm Cour}},\, dt_{{\rm cool}},\,
dt_{{\rm spec}},\, dt_{vel}\}$$ $$dt_{{\rm dyn}}=\min\left\{
c_{d}\sqrt{\frac{\pi^{2}r_{j}^{3}}{4m_{j}}}\right\} ,$$ $$dt_{{\rm Cour}}=\min\left\{ c_{{\rm
C}}\left|\frac{r_{j}-r_{j-1}}{\sqrt{\gamma(\gamma-1)u_{j}}}\right|\right\} ,$$ $$dt_{{\rm cool}}=\min\left\{
c_{c}\left|\frac{u_{j}\rho_{j}}{(\Gamma-\Lambda)_{j}}\right|\right\} ,$$ $$dt_{{\rm spec}}=\min\left\{ c_{{\rm sp}}\left|\frac{x_{j}}{dx_{j}/dt}\right|,\, c_{{\rm sp}}\left|\frac{y_{{\rm
H I},\, j}}{dy_{{\rm H I,}\, j}/dt}\right|\right\}$$ $$dt_{{\rm vel}}=\min\left\{
c_{v}\left|\frac{r_{j}-r_{j-1}}{v_{j}-v_{j-1}}\right|\right\} ,$$ where $c_{d}$, $c_{\rm C}$, $c_{c}$, $c_{\rm sp}$, and $c_{v}$ are coefficients that ensure accurate calculation of the finite difference equations. We use $c_{d}=0.1$, $c_{\rm C}=0.1$, $c_{c}=0.1$, $c_{\rm sp}=0.1$, and $c_{v}=0.05$.
In practice, we frequently find that $dt_{\rm dyn}$ can be very small compared to other time scales. We sometimes disregard $dt_{\rm dyn}$ in order to achieve computational efficiency. We confirmed, especially in our problem, that such a treatment does not produce any significant discrepancy from a calculation with $dt_{\rm dyn}$ considered. When the virial temperature of a halo is close to the cooling temperature plateau, for instance, $dt_{\rm dyn}$ must be irrelevant because gas would be almost hydrostatic.
Radiative Transfer
------------------
For the radiation field generated from a point source at the centre, the radiative rate coefficient of species $i$ at radius $r$ is given by equation (\[eq:rad\_rate\_ext\_body\]). Finite-differencing this rate coefficient, however, requires some caution. For the baryonic shell at position $j$ (smaller $j$ means closer to the centre) whose inner edge and outer edge have radii $r_{j-1/2}$ and $r_{j+1/2}$, respectively, the incident differential flux at the outer edge is $F_{\nu}^{{\rm int}}(r_{j+1/2})$, and one could naively calculate the rate coefficient of species $i$ by $$k_{i}(r_{j})=\int_{0}^{\infty}d\nu\frac{\sigma_{i,\nu}F_{\nu}^{{\rm
ext}}(r_{j+1/2})}{h\nu}.
\label{nonconserving_k_ext2}$$
As mentioned already in Section \[sub:External-source\] and Section \[sub:photo-heating\], however, this expression may not yield an accurate result when the shell $k$ is optically thick. In this case, $F_{\nu}$ may change substantially over the shell width, and equation (\[nonconserving\_k\_ext2\]) might overpredict the ionization rate by applying a constant flux over the shell width ($\Delta r_{j}\equiv r_{j+1/2}-r_{j-1/2}$). One may, in principle, choose to set up the initial condition such that all shells are optically thin. However, such a scheme can be very expensive computationally, especially when collapsed haloes are treated. In order to resolve this problem, we use the “photon-conserving scheme” by @1999MNRAS.309..287R and @1999ApJ...523...66A. In this treatment, the number of photons that are absorbed in a shell is the same as the number of ionization events. Equation (\[nonconserving\_k\_ext2\]) can then be re-written as $$\begin{aligned}
k_{i}(r_{j}) &=& \int_{0}^{\infty}d\nu\frac{L_{\nu}^{{\rm
ext}}(r_{j+1/2})-L_{\nu}^{{\rm
ext}}(r_{j-1/2})}{h\nu}\cdot\frac{1}{n_{i}V_{{\rm shell},j}}
\nonumber \\
&\simeq& \int_{0}^{\infty}d\nu\frac{F_{\nu}^{{\rm
ext}}(r_{j+1/2})}{h\nu}
\cdot\frac{1-e^{-\Delta\tau_{i,\nu}(r_{j})}}{n_{i}\Delta r_{j}},
\label{eq:conserving_k_ext2}\end{aligned}$$ where $L_{\nu}^{{\rm ext}}(r)=4\pi r^{2}F_{\nu}^{{\rm ext}}(r)$, $\Delta\tau_{i,\nu}(r_{j})\equiv n_{i}\Delta r_{j}\sigma_{i,\nu}$ is the optical depth of a shell $k$ on a species $i$, and $V_{{\rm
shell},j}\simeq4\pi r_{j}^{2}\Delta r_{j}$ is the volume of the shell. Note that when $\Delta\tau_{\nu}\ll1$, equation (\[eq:conserving\_k\_ext2\]) becomes equivalent to equation (\[nonconserving\_k\_ext2\]). For each species, the corresponding radiative reaction rate is calculated by quadrature, by summing the integrand in equation (\[eq:conserving\_k\_ext2\]), then summing over the frequency to obtain the nett radiative reaction rate.
Nonequilibrium Chemistry {#nonequilibrium-chemistry}
------------------------
As described in Section \[sub:noneq\_chem\], in order to update the abundance of species $i$, we adopt the finite difference scheme by @1997NewA....2..181A. Based upon equation (\[eq:verygeneric\_rate\_eq\]), each species $i$ is updated by $$n_{i}^{n+1}=\frac{C_{i}^{n+1}(T,\{ n_{j}\}) dt^{n+1/2} +
n_{i}^{n}}{1+D_{i}^{n+1}(T,\{ n_{j}\})dt^{n+1/2}},
\label{eq:backward-diff}$$ where the species $\{ n_{j}\}$ is the previously updated value in the order given by @1997NewA....2..181A (note that the letter $n$ ($n+1/2$, $n+1$) in superscript denotes the time $t^{n}$ ($t^{n+1/2}$, $t^{n+1}$). The order they find to be optimal is H, ${\rm H^{+}}$, He, ${\rm He^{+}}$, ${\rm He^{++}}$ and ${\rm e^{-}}$, followed by the algebraic equilibrium expressions for ${\rm H^{-}}$ and ${\rm H^{+}}$, and finally ${\rm H_{2}}$, again by equation (\[eq:backward-diff\]).
Numerical resolution {#sub:resolution}
--------------------
In practice, we use $500$ dark matter and $1000$ fluid shells sampled uniformly (in radius) from the centre to the truncation radius $r_{{\rm tr}}$. We put a small reflecting core at the centre with negligible size, namely $r_{{\rm core}}=10^{-4}r_{{\rm tr}}$. Such a core is found to be useful in reducing undesirable numerical instability at the centre. Our choice is conservative enough not to affect the overall answer.
A wide range of radiation frequency (energy), $h\nu\sim[0.7\,-\,7000]\,{\rm eV}$, is covered by $100$, logarithmically spaced bins, $\Delta E/E \approx 0.04$, together with additional, linearly-spaced bins where radiative cross sections change rapidly as frequency changes. About a dozen linearly spaced bins at each of those rapidly changing points turned out to produce reliable results.
Rate coefficients {#sub:rates}
=================
In Table \[table:rates\], we list the chemical reaction rates we implemented in our code and the corresponding references. The rate coefficients (1-19) and radiative cross sections (20-26) are mostly from the fit by @1987ApJ...318...32S, except for a few updates.
Reactions Reference
---- ------------------------------------------------------ --------------------------------------------
1 $\rm H + e^- \rightarrow H^+ + 2e^- $ @1987ephh.book.....J
2 $\rm H^+ + e^- \rightarrow H + \gamma $ Case B; @1989agna.book.....O
3 $\rm He + e^- \rightarrow He^+ + 2e^- $ @1987ephh.book.....J
4 $\rm He^+ + e^- \rightarrow He + \gamma $ @1973AA....25..137A
5 $\rm He^+ + e^- \rightarrow He^{++} + 2e^- $ AMDIS Database; @1997NewA....2..181A
6 $\rm He^{++} + e^- \rightarrow He^+ + \gamma $ @1978ppim.book.....S
7 $\rm H + e^- \rightarrow H^- + \gamma $ @1972AA....20..263D [@1987ApJ...318...32S]
8 $\rm H^- + H \rightarrow H_2 + e^- $ @bieniek
9 $\rm H + H^+ \rightarrow H_{2}^{+} + \gamma $ @1976PhRvA..13...58R
10 $\rm H_{2}^{+} + H \rightarrow H_2 + H^+ $ @1979JChPh..70.2877K
11 $\rm H_2 + H \rightarrow 3H $ @1986ApJ...311L..93D
12 $\rm H_2 + H^+ \rightarrow H_{2}^{+} + H $ @2004ApJ...606L.167S
13 $\rm H_2 + e^- \rightarrow 2H + e^- $ @1983ApJ...266..646M
14 $\rm H^- + e^- \rightarrow H + 2e^- $ @1987ephh.book.....J
15 $\rm H^- + H \rightarrow 2H + e^- $ @1984SvA....28...15I
16 $\rm H^- + H^+ \rightarrow 2H $ @1984inch.book.....D
17 $\rm H^- + H^+ \rightarrow H_{2}^{+} + e^- $ @1978JPhB...11L.671P
18 $\rm H_{2}^{+} + e^- \rightarrow 2H $ @1994ApJ...424..983S
19 $\rm H_{2}^{+} + H^- \rightarrow H + H_2 $ @1987IAUS..120..109D
20 $\rm H + \gamma \rightarrow H^+ + e^- $ @1989agna.book.....O
21 $\rm He^+ + \gamma \rightarrow He^{++} + e^- $ @1989agna.book.....O
22 $\rm He + \gamma \rightarrow He^{+ } + e^- $ @1989agna.book.....O
23 $\rm H^- + \gamma \rightarrow H + e^- $ @1972AA....20..263D [@1987ApJ...318...32S]
24 $\rm H_{2}^{+} + \gamma \rightarrow H + H^+ $ @1968PhRv..172....1D
25 $\rm H_2 + \gamma \rightarrow H_{2}^{+} + e^- $ @1978JChPh..69.2126O
26 $\rm H_{2}^{+} + \gamma \rightarrow 2H^{+} + e^- $ @1968JPhB....1..543B
27 $\rm H_2 + \gamma \rightarrow 2H $ Section \[sub:ss\]; @1996ApJ...468..269D
Code tests {#code-tests}
==========
We now extend the description of our code test problems in Section \[sec:codetest\] and show the results.
\(A) The self-similar, spherical, cosmological infall problem [@1985ApJS...58...39B]: A point mass, if placed in an unperturbed Einstein-de Sitter universe, will make all particles around it to be gravitationally bound, leading to a successive turnaround and collapse of spherically shells. Infalling matter will be shocked and form a virialized structure, whose profiles are well described by a self-similar solution. We restrict ourselves to purely baryonic fluid with the ratio of specific heats $\gamma = 5/3$.
The turnaround radius $r_{\rm ta}$, at which the Lagrangian proper velocity of a shell is zero, evolves as $$r_{\rm ta}(t)=\left( \frac{3\pi}{4}\right)^{-8/9} \left({\delta_{i}
R_{i}^{3}}\right)^{1/3} (t/t_{i})^{8/9},
\label{eq:bert-rta}$$ where $\delta_i R_{i}^{3}$ defines the seed mass $\delta m$ added to the Einstein-de Sitter universe, $$\delta m = \frac{4}{3}\pi \rho_{{\rm H},i} \delta_i R_{i}^{3},
\label{eq:bert-seed}$$ where the initial cosmic mean density $\rho_{{\rm H},i}=1/(6\pi G
t_{i}^{2})$ at $t=t_{i}$. The shock radius $r_s$ is a constant fraction of $r_{\rm ta}$: $r_s (t)=
0.338976 \,r_{\rm ta} (t)$ for $\gamma = 5/3$. The dimensionless radius $\lambda \equiv r/r_{\rm ta}$ and the dimensionless density $D\equiv \rho / \rho_{\rm H}$, where the cosmic mean density $\rho_{\rm H}=1/(6\pi G t^2)$, satisfy the unique Bertschinger solution. In Fig. \[fig-test\], we show the density profiles and $r_{\rm s}(t)$, obtained from the simulation with $\delta_i R_{i}^{3} = 1.84\times 10^{71} \,\rm cm^3$, $t_{i}=5.572\times
10^{14} \,\rm s$.
\(B) The self-similar blast wave from a strong, adiabatic point explosion in a uniform gas [@1959sdmm.book.....S]: A point explosion drives a self-similar blast wave through the initially static, uniform medium. A strong shock is generated, and $r_{\rm s} (t) = \xi_0 \left(
\frac{E}{\rho_0} \right)^{1/5} t^{2/5}$, where $E$ is the thermal energy of explosion, $\rho_0$ is the initial density, and $\xi_0$ is a dimensionless constant determined by $\gamma$. For $\gamma=5/3$, $\xi_0
= 1.152$. We use $E=1.053\times 10^{61} \,\rm erg$, $\gamma=5/3$, and $\rho_0 =
2.5626\times 10^{-24} \,\rm cm^{-3}$ for simulation results displayed in Fig. \[fig-test\].
\(C) The propagation of an I-front from a steady point-source in a uniform, static medium: This is the case where the classical description of the Strömgren radius is plausible, since gas is forced to remain static, and photoionization and recombination are the only physical processes determining the ionized fraction. The I-front from a point source with $N_*$ number of ionizing photons evolves as $$r_{\rm I} (t) = R_{\rm S} \left(1-\exp(-t/t_{\rm rec}) \right)^{1/3},
\label{eq:pure-ift}$$ where $R_{\rm S}\equiv \left[ 3N_{*}/(4\pi n_{\rm H}^2 \alpha) \right]^{1/3}$ is the Strömgren radius, $t_{\rm rec}\equiv 1/(n_{\rm H} \alpha)$ is the recombination time, and $\alpha$ is the recombination rate coefficient. We adopt $N_* = 10^{47} \,\rm s^{-1}$, $n_{\rm H}=10 \,\rm cm^{-3}$, and $\alpha = 1.05\times 10^{-13} \,\rm cm^{3} \,s^{-1}$. For this test, we use a monochromatic light whose frequency is slightly above the hydrogen ionization threshold.
\(D) the gas-dynamical expansion of an H II region from a point source in a uniform gas [@1966ApJ...143..700L]: The I-front, initially propagating as a weak R-type front into a uniform medium, slows down and travels as a D-type front, developing a shock front ahead of it. The I-front evolves as $$r_{\rm I} (\tilde{t}) = R_{\rm S,I} \left(1+\frac{7}{4}
\frac{\tilde{t}}{t_{\rm sc}}\right)^{4/7},
\label{eq:lasker}$$ where $R_{\rm S,I}$ is the initial Strömgren radius, $\tilde{t}\equiv t-t_c$ is the time measured from the moment $t_c$ when $dr_{\rm
I}/dt = c_{\rm I}$, and $c_{\rm I}\equiv (p/\rho)^{1/2}$ is the isothermal sound speed of the ionized gas [@1978ppim.book.....S]. We adopt $N_* = 2.45\times 10^{48} \,\rm s^{-1}$ and $n_{\rm H}=6.4\,\rm
cm^{-3}$. Following @1966ApJ...143..700L, we force temperature of the ionized gas to be $10^4 \,\rm K$, which gives $c_{\rm I}=12.86 \,\rm km/s$.
\(E) The gas-dynamical expansion-phase of the H II region from a point-source in a nonuniform gas whose density varies with distance $r$ from the source as $r^{-w}$, $w=3/2$ [@1990ApJ...349..126F]: This case is similar to the case (D), except that the density follows a power law, $n_{\rm H} \propto
r^{-w}$. Inside the core radius $r_c$, the density is constant at $n_{{\rm H},c}$. The I-front evolves as $$r_{\rm I} (t) = R_{w}
\left[1+\frac{7-2w}{4}\left(\frac{12}{9-4w}\right)^{1/2}\frac{c_{\rm I}
t}{R_w} \right],
\label{eq:franco}$$ where $R_w$ is the size of the initial H II region obtained by equating the ionization rate and the recombination rate. For instance, when $w=3/2$, $$R_{3/2} = r_c \exp
\left\{\frac{1}{3} \left[
\left( \frac{R_{\rm S}}{r_c} \right)^3 -1
\right] \right\},$$ where $R_{\rm S}\equiv \left[ 3N_{*}/(4\pi n_{{\rm H},c}^2 \alpha)
\right]^{1/3}$.
If $w\le 3/2$, the shock front always travels ahead of the I-front. If $w> 3/2$, however, the shock front is overtaken by the I-front, which soon runs to infinity in this “champagne” phase. We restrict ourselves to this critical exponent $w=3/2$. From equation (\[eq:franco\]), we obtain $r_{\rm I} (t) = R_{3/2}\left( 1+2c_{\rm I}/
R_{3/2}\right)$. In our simulation, we use $n_{{\rm
H},c}=2\times 10^6 \,\rm cm^{-3}$, $r_c = 2.1\times 10^{16} \,\rm
cm$, $N_* = 5\times 10^{49}\,\rm s^{-1}$, and $\alpha = 2.6\times
10^{-13}\,\rm cm^3 s^{-1}$. Temperature of the ionized gas is set at $T=8000\,\rm K$, such that $c_{\rm I} = 11.5\,\rm km/s$.
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[^1]: Email: [email protected]
[^2]: Email: [email protected]
[^3]: A new preprint by @2006astro.ph..6019A has also appeared which addresses this issue. We will discuss this further in Section \[sub:abel\]
[^4]: As we do not perform a statistical study, our result is independent of the cosmic density power spectrum. The three-year [*WMAP*]{} data does not show a big discrepancy in the set of cosmological parameters of the interest in this paper [@2006astro.ph..3449S]. The change in $\sigma_8$ and the index of the primordial power spectrum $n$ would translate to $\sim 1.4$ redshift delay of structure formation and reionization [@2006ApJ...644L.101A]
[^5]: D and Li components have usually been neglected due to their relatively low abundance, hence the negligible contribution to cooling (e.g. @1984ApJ...280..465L [@1987ApJ...318...32S]). Recent studies by @2005MNRAS.364.1378N and @2006MNRAS.366..247J, however, show that enough HD is generated in strongly-shocked, ionized primordial gas which then can cool below the temperature of $\sim 100 \,\rm K$ already achieved by $\rm
H_2$ cooling alone, down to the temperature of the CMB. As the HD cooling process is negligible if gas remains neutral (e.g. @2006MNRAS.366..247J), however, we may neglect the HD cooling process in our calculation as long as we are interested in the centre of target haloes which remains mostly neutral at any time. We will discuss this issue further in Section \[sec:2star-Discussion\].
[^6]: The additional case of $D=50\,\rm pc$, $F_0=600$, $M=5.5\times 10^5 \,M_\odot$, will be discussed separately in Section \[sub:abel\] with regard to the case in which the target minihalo is merging with the minihalo which hosts the star, separated by less than its virial radius from the star
[^7]: After this paper was written a new preprint was posted which is consistent with our description here, finding $M_{\rm c,min}\approx 10^5\, M_\odot$ [@2006astro.ph..7013O].
[^8]: This argument is based upon the fact that the comoving number density of haloes, $M\,dn/dM$, is roughly proportional to $M^{-1}$. The minihalo population, however, might have been severely reduced by the “Jeans-mass filtering” inside ionized bubbles created around rare, but more massive objects (e.g. @2006astro.ph..7517I), in which case sources hosted by minihaloes would make negligible contribution to cosmic reionization.
| ArXiv |
\[section\] \[lemma\][Theorem]{} \[lemma\][Definition]{} \[lemma\][Corollary]{} \[lemma\][Problem]{} \[lemma\][Proposition]{}
Homotopy types of strict $3$-groupoids {#homotopy-types-of-strict-3-groupoids .unnumbered}
======================================
Carlos SimpsonCNRS, UMR 5580, Université de Toulouse 3
It has been difficult to see precisely the role played by [*strict*]{} $n$-categories in the nascent theory of $n$-categories, particularly as related to $n$-truncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all $3$-types by any reasonable realization functor [^1] from strict $3$-groupoids (i.e. groupoids in the sense of [@KV]). More precisely we show that one does not obtain the $3$-type of $S^2$. The basic reason is that the Whitehead bracket is nonzero. This phenomenon is actually well-known, but in order to take into account the possibility of an arbitrary reasonable realization functor we have to write the argument in a particular way.
We start by recalling the notion of strict $n$-category. Then we look at the notion of strict $n$-groupoid as defined by Kapranov and Voevodsky [@KV]. We show that their definition is equivalent to a couple of other natural-looking definitions (one of these equivalences was left as an exercise in [@KV]). At the end of these first sections, we have a picture of strict $3$-groupoids having only one object and one $1$-morphism, as being equivalent to abelian monoidal objects $(G,+)$ in the category of groupoids, such that $(\pi _0(G),+)$ is a group. In the case in question, this group will be $\pi
_2(S^2)={{\bf Z}}$. Then comes the main part of the argument. We show that, up to inverting a few equivalences, such an object has a morphism giving a splitting of the Postnikov tower (Proposition \[diagramme\]. It follows that for any realization functor respecting homotopy groups, the Postnikov tower of the realization (which has two stages corresponding to $\pi _2$ and $\pi _3$) splits. This implies that the $3$-type of $S^2$ cannot occur as a realization.
The fact that strict $n$-groupoids are not appropriate for modelling all homotopy types has in principle been known for some time. There are several papers by R. Brown and coauthors on this subject, see [@RBrown1], [@BrownGilbert], [@BrownHiggins], [@BrownHiggins2]; a recent paper by C. Berger [@Berger]; and also a discussion of this in various places in Grothendieck [@Grothendieck]. Other related examples are given in Gordon-Power-Street [@Gordon-Power-Street]. The novelty of our present treatment is that we have written the argument in such a way that it applies to a wide class of possible realization functors, and in particular it applies to the realization functor of Kapranov-Voevodsky (1991) [@KV].
This problem with strict $n$-groupoids can be summed up by saying in R. Brown’s terminology, that they correspond to [*crossed complexes*]{}. While a nontrivial action of $\pi _1$ on the $\pi _i$ can occur in a crossed complex, the higher Whitehead operations such as $\pi _2\otimes \pi _2\rightarrow
\pi _3$ must vanish. This in turn is due to the fundamental “interchange rule” (or “Godement relation” or “Eckmann-Hilton argument”). This effect occurs when one takes two $2$-morphisms $a$ and $b$ both with source and target a $1$-identity $1_x$. There are various ways of composing $a$ and $b$ in this situation, and comparison of these compositions leads to the conclusion that all of the compositions are commutative. In a weak $n$-category, this commutativity would only hold up to higher homotopy, which leads to the notion of “braiding”; and in fact it is exactly the braiding which leads to the Whitehead operation. However, in a strict $n$-category, the commutativity is exact, so the Whitehead operation is trivial.
One can observe that one of the reasons why this problem occurs is that we have the exact $1$-identity $1_x$. This leads to wondering if one could get a better theory by getting rid of the exact identities. We speculate in this direction at the end of the paper by proposing a notion of [*$n$-snucategory*]{}, which would be an $n$-category with strictly associative composition, but without units; we would only require existence of weak units. The details of the notion of weak unit are not worked out.
A preliminary version of this note was circulated in a limited way in the summer of 1997.
I would like to thank: R. Brown, A. Bruguières, A. Hirschowitz, G. Maltsiniotis, and Z. Tamsamani.
[**.Strict $n$-categories**]{}
In what follows [*all $n$-categories are meant to be strict $n$-categories*]{}. For this reason we try to put in the adjective “strict” as much as possible when $n>1$; but in any case, the very few times that we speak of weak $n$-categories, this will be explicitly stated. We mostly restrict our attention to $n\leq 3$.
In case that isn’t already clear, it should be stressed that everything we do in this section (as well as most of the next and even the subsequent one as well) is very well known and classical, so much so that I don’t know what are the original references.
To start with, a [*strict $2$-category*]{} $A$ is a collection of objects $A_0$ plus, for each pair of objects $x,y\in A_0$ a category $Hom _A(x,y)$ together with a morphism $$Hom_A(x,y)\times Hom _A(y,z)\rightarrow Hom _A(x,z)$$ which is strictly associative in the obvious way; and such that a unit exists, that is an element $1_x\in Ob\, Hom _A(x,x)$ with the property that multiplication by $1_x$ acts trivially on objects of $Hom _A(x,y)$ or $Hom_A(y,x)$ and multiplication by $1_{1_x}$ acts trivially on morphisms of these categories.
A [*strict $3$-category*]{} $C$ is the same as above but where $Hom _C(x,y)$ are supposed to be strict $2$-categories. There is an obvious notion of direct product of strict $2$-categories, so the above definition applies [*mutatis mutandis*]{}.
For general $n$, the well-known definition is most easily presented by induction on $n$. We assume known the definition of strict $n-1$-category for $n-1$, and we assume known that the category of strict $n-1$-categories is closed under direct product. A [*strict $n$-category*]{} $C$ is then a category enriched [@Kelly] over the category of strict $n-1$-categories. This means that $C$ is composed of a [*set of objects*]{} $Ob(C)$ together with, for each pair $x,y\in Ob(C)$, a [*morphism-object*]{} $Hom _C(x,y)$ which is a strict $n-1$-category; together with a strictly associative composition law $$Hom _C(x,y)\times Hom _C(y,z) \rightarrow Hom _C(x,z)$$ and a morphism $1_x: \ast \rightarrow Hom _C(x,x)$ (where $\ast$ denotes the final object cf below) acting as the identity for the composition law. The [*category of strict $n$-categories*]{} denoted $nStrCat$ is the category whose objects are as above and whose morphisms are the transformations strictly perserving all of the structures. Note that $nStrCat$ admits a direct product: if $C$ and $C'$ are two strict $n$-categories then $C\times C'$ is the strict $n$-category with $$Ob(C\times C'):= Ob(C) \times Ob(C')$$ and for $(x,x'), \; (y,y') \in Ob(C\times C')$, $$Hom _{C\times C'}((x,x'), (y,y')):= Hom _C(x,y)\times Hom _{C'}(x',y')$$ where the direct product on the right is that of $(n-1)StrCat$. Note that the final object of $nStrCat$ is the strict $n$-category $\ast$ with exactly one object $x$ and with $Hom _{\ast}(x,x)= \ast$ being the final object of $(n-1)StrCat$.
The induction inherent in this definition may be worked out explicitly to give the definition as it is presented in [@KV] for example. In doing this one finds that underlying a strict $n$-category $C$ are the sets $Mor ^i(C)$ of [*$i$-morphisms*]{} or [*$i$-arrows*]{}, for $0\leq i\leq n$. The $0$-morphisms are by definition the objects, and $Mor ^i(C)$ is the disjoint union over all pairs $x,y$ of the $Mor ^{i-1}(Hom
_C(x,y))$. The composition laws at each stage lead to various compositions for $i$-morphisms, denoted in [@KV] by $\ast _j$ for $0\leq j < i$. These are partially defined depending on the [*source*]{} and [*target*]{} maps. For a more detailed explanation, refer to the standard references [@BrownHiggins] [@Street] [@KV] (and I am probably missing many older references which could date back even before [@Benabou] [@GabrielZisman]).
One of the most important of the axioms satisfied by the various compositions in a strict $n$-category is variously known under the name of “Eckmann-Hilton argument”, “Godement relations”, “interchange rules” etc. The following discussion of this axiom owes a lot to discussions I had with Z. Tamsamani during his thesis work. This axiom comes from the fact that the composition law $$Hom _C(x,y)\times Hom _C(y,z)\rightarrow Hom _C(x,z)$$ is a morphism with domain the direct product of the two morphism $n-1$-categories from $x$ to $y$ and from $y$ to $z$. In a direct product, compositions in the two factors by definition are independent (commute). Thus, for $1$-morphisms in $Hom _C(x,y)\times Hom _C(y,z)$ (where the composition $\ast _0$ for these $n-1$-categories is actually the composition $\ast _1$ for $C$ and we adopt the latter notation), we have $$(a,b) \ast _1 (c,d) = (a\ast _1c, b \ast _1d).$$ This leads to the formula $$(a\ast _0b) \ast _1 (c\ast _0d) = (a\ast _1c) \ast _0 (b \ast _1d).$$ This seemingly innocuous formula takes on a special meaning when we start inserting identity maps. Suppose $x=y=z$ and let $1_x$ be the identity of $x$ which may be thought of as an object of $Hom _C(x,x)$. Let $e$ denote the $2$-morphism of $C$, identity of $1_x$; which may be thought of as a $1$-morphism of $Hom _C(x,x)$. It acts as the identity for both compositions $\ast _0$ and $\ast _1$ (the reader may check that this follows from the part of the axioms for an $n$-category saying that the morphism $1_x: \ast
\rightarrow Hom _C(x,x)$ is an identity for the composition).
If $a, b$ are also endomorphisms of $1_x$, then the above rule specializes to: $$a\ast _1b =
(a\ast
_0e) \ast _1 (e\ast _0b) = (a\ast _1e) \ast _0 (e \ast _1b) = a \ast _0b.$$ Thus in this case the compositions $\ast _0$ and $\ast _1$ are the same. A different ordering gives the formula $$a\ast _1b =
(e\ast
_0a) \ast _1 (b\ast _0e) = (e\ast _1b) \ast _0 (a \ast _1e) = b \ast _0a.$$ Therefore we have $$a\ast _1b= b\ast _1a = a \ast _0b = b\ast _0a.$$ This argument says, then, that $Ob (Hom _{Hom _C(x,x)}(1_x, 1_x))$ is a commutative monoid and the two natural multiplications are the same.
The same argument extends to the whole monoid structure on the $n-2$-category $Hom _{Hom _C(x,x)}(1_x, 1_x)$:
\[godement\] The two composition laws on the strict $n-2$-category $Hom _{Hom _C(x,x)}(1_x, 1_x)$ are equal, and this law is commutative. In other words, $Hom _{Hom _C(x,x)}(1_x, 1_x)$ is an abelian monoid-object in the category $(n-2)StrCat$.
[$/$$/$$/$]{}
There is a partial converse to the above observation: if the only object is $x$ and the only $1$-morphism is $1_x$ then nothing else can happen and we get the following equivalence of categories.
\[scholium\] Suppose $G$ is an abelian monoid-object in the category $(n-2)StrCat$. Then there is a unique strict $n$-category $C$ such that $$Ob(C)= \{ x\}\;\;\; \mbox{and}\;\;\; Mor ^1(C)=Ob(Hom _C(x,x))=\{
1_x\}$$ and such that $Hom _{Hom _C(x,x)}(1_x, 1_x)=G$ as an abelian monoid-object. This construction establishes an equivalence between the categories of abelian monoid-objects in $(n-2)StrCat$, and the strict $n$-categories having only one object and one $1$-morphism.
[*Proof:*]{} Define the strict $n-1$-category $U$ with $Ob(U)= \{ u\}$ and $Hom _U(u,u)=G$ with its monoid structure as composition law. The fact that the composition law is commutative allows it to be used to define an associative and commutative multiplication $$U\times U \rightarrow U.$$ Now let $C$ be the strict $n$-category with $Ob(C)=\{ x\}$ and $Hom _C(x,x)=U$ with the above multiplication. It is clear that this construction is inverse to the previous one. [$/$$/$$/$]{}
It is clear from the construction (the fact that the multiplication on $U$ is again commutative) that the construction can be iterated any number of times. We obtain the following corollary.
\[iterate\] Suppose $C$ is a strict $n$-category with only one object and only one $1$-morphism. Then there exists a strict $n+1$-category $B$ with only one object $b$ and with $Hom _B(b,b)\cong C$.
[*Proof:*]{} By the previous lemmas, $C$ corresponds to an abelian monoid-object $G$ in $(n-2)StrCat$. Construct $U$ as in the proof of \[scholium\], and note that $U$ is an abelian monoid-object in $(n-1)StrCat$. Now apply the result of \[scholium\] directly to $U$ to obtain $B\in (n+1)StrCat$, which will have the desired property. [$/$$/$$/$]{}
[**.The groupoid condition**]{}
Recall that a [*groupoid*]{} is a category where all morphisms are invertible. This definition generalizes to strict $n$-categories in the following way [@KV]. We give a theorem stating that three versions of this definition are equivalent.
Note that, following [@KV], we [*do not*]{} require strict invertibility of morphisms, thus the notion of strict $n$-groupoid is more general than the notion employed by Brown and Higgins [@BrownHiggins].
Our discussion is in many ways parallel to the treatment of the groupoid condition for weak $n$-categories in [@Tamsamani] and our treatment in this section comes in large part from discussions with Z. Tamsamani about this.
The statement of the theorem-definition is recursive on $n$.
\[thmdef\] Fix $n<\infty$.
[**I. Groupoids**]{} Suppose $A$ is a strict $n$-category. The following three conditions are equivalent (and in this case we say that $A$ is a [*strict $n$-groupoid*]{}). (1) $A$ is an $n$-groupoid in the sense of Kapranov-Voevodsky [@KV]; (2) for all $x,y\in A$, $Hom _A(x,y)$ is a strict $n-1$-groupoid, and for any $1$-morphism $f:x\rightarrow y$ in $A$, the two morphisms of composition with $f$ $$Hom _A(y,z)\rightarrow Hom _A(x,z),\;\;\;\;
Hom _A(w,x)\rightarrow Hom_A(w,y)$$ are equivalences of strict $n-1$-groupoids (see below); (3) for all $x,y\in A$, $Hom _A(x,y)$ is a strict $n-1$-groupoid, and $\tau _{\leq
1}A$ (defined below) is a $1$-groupoid.
[**II. Truncation**]{} If $A$ is a strict $n$-groupoid, then define $\tau _{\leq k}A$ to be the strict $k$-category whose $i$-morphisms are those of $A$ for $i<k$ and whose $k$-morphisms are the equivalence classes of $k$-morphisms of $A$ under the equivalence relation that two are equivalent if there is a $k+1$-morphism joining them. The fact that this is an equivalence relation is a statement about $n-k$-groupoids. The set $\tau _{\leq 0}A$ will also be denoted $\pi _0A$. The truncation is again a $k$-groupoid, and for $n$-groupoids $A$ the truncation coincides with the operation defined in [@KV].
[**III. Equivalence**]{} A morphism $f:A\rightarrow B$ of strict $n$-groupoids is said to be an [*equivalence*]{} if the following equivalent conditions are satisfied: (a) (this is the definition in [@KV]) $f$ induces an isomorphism $\pi _0A\rightarrow
\pi _0B$, and for every object $a\in A$ $f$ induces an isomorphism $\pi _i(A,a)\stackrel{\cong}{\rightarrow} \pi _i(B, f(a))$ where these homotopy groups are as defined in [@KV]; (b) $f$ induces a surjection $\pi _0A\rightarrow \pi _0B$ and for every pair of objects $x,y\in A$ $f$ induces an equivalence of $n-1$-groupoids $Hom_A(x,y)\rightarrow Hom _B(f(x), f(y))$; (c) if $u,v$ are $i$-morphisms in $A$ sharing the same source and target, and if $r$ is an $i+1$-morphism in $B$ going from $f(u)$ to $f(v)$ then there exists an $i+1$-morphism $t$ in $A$ going from $u$ to $v$ and an $i+2$-morphism in $B$ going from $f(t)$ to $r$ (this includes the limiting cases $i=-1$ where $u$ and $v$ are not specified, and $i=n-1, n$ where “$n+1$-morphisms” mean equalities between $n$-morphisms and “$n+2$-morphisms” are not specified).
[**IV. Sub-lemma**]{} If $f: A\rightarrow B$ and $g: B\rightarrow C$ are morphisms of strict $n$-groupoids and if any two of $f$, $g$ and $gf$ are equivalences, then so is the third.
[**V. Second sub-lemma**]{} If $$A\stackrel{f}{\rightarrow}B\stackrel{g}{\rightarrow}C\stackrel{h}{\rightarrow}D$$ are morphisms of strict $n$-groupoids and if $hg$ and $gf$ are equivalences, then $g$ is an equivalence.
[*Proof:*]{} It is clear for $n=0$, so we assume $n\geq 1$ and proceed by induction on $n$: we assume that the theorem is true (and all definitions are known) for strict $n-1$-categories.
We first discuss the existence of truncation (part II), for $k\geq 1$. Note that in this case $\tau _{\leq k}A$ may be defined as the strict $k$-category with the same objects as $A$ and with $$Hom _{\tau _{\leq k}A}(x,y):= \tau _{\leq k-1}Hom _A(x,y).$$ Thus the fact that the relation in question is an equivalence relation, is a statement about $n-1$-categories and known by induction. Note that the truncation operation clearly preserves any one of the three groupoid conditions (1), (2), (3). Thus we may affirm in a strong sense that $\tau _{\leq k}(A)$ is a $k$-groupoid without knowing the equivalence of the conditions (1)-(3).
Note also that the truncation operation for $n$-groupoids is the same as that defined in [@KV] (they define truncation for general strict $n$-categories but for $n$-categories which are not groupoids, their definition is different from that of [@Tamsamani] and not all that useful).
For $0\leq k\leq k'\leq n$ we have $$\tau _{\leq k}(\tau _{\leq k'}(A)) = \tau _{\leq k}(A).$$ To see this note that the equivalence relation used to define the $k$-arrows of $\tau _{\leq k}(A)$ is the same if taken in $A$ or in $\tau _{\leq
k+1}(A)$—the existence of a $k+1$-arrow going between two $k$-arrows is equivalent to the existence of an equivalence class of $k+1$-arrows going between the two $k$-arrows.
Finally using the above remark we obtain the existence of the truncation $\tau
_{\leq 0}(A)$: the relation is the same as for the truncation $\tau _{\leq
0}(\tau _{\leq 1}(A))$, and $\tau _{\leq 1}(A)$ is a strict $1$-groupoid in the usual sense so the arrows are invertible, which shows that the relation used to define the $0$-arrows (i.e. objects) in $\tau _{\leq 0}(A)$ is in fact an equivalence relation.
We complete our discussion of truncation by noting that there is a natural morphism of strict $n$-categories $A\rightarrow \tau _{\leq k}(A)$, where the right hand side ([*a priori*]{} a strict $k$-category) is considered as a strict $n$-category in the obvious way.
We turn next to the notion of equivalence (part III), and prove that conditions (a) and (b) are equivalent. This notion for $n$-groupoids will not enter into the subsequent treatment of part (I)—what does enter is the notion of equivalence for $n-1$-groupoids, which is known by induction—so we may assume the equivalence of definitions (1)-(3) for our discussion of part (III).
Recall first of all the definition of the homotopy groups. Let $1^i_a$ denote the $i$-fold iterated identity of an object $a$; it is an $i$-morphism, the identity of $1^{i-1}_a$ (starting with $1^0_a=a$). Then $$\pi _i(A,a):= Hom _{\tau _{\leq i}(A)}(1^{i-1}_a, 1^{i-1}_a).$$ This definition is completed by setting $\pi _0(A):= \tau _{\leq 0}(A)$. These definitions are the same as in [@KV]. Note directly from the definition that for $i\leq k$ the truncation morphism induces isomorphisms $$\pi _i(A,a)\stackrel{\cong}{\rightarrow}\pi _i(\tau _{\leq k}(A), a).$$ Also for $i\geq 1$ we have $$\pi _i(A,a)= \pi _{i-1}(Hom _A(a,a), 1_a).$$ One shows that the $\pi _i$ are abelian for $i\geq 2$. This is part of a more general principle, the “interchange rule” or “Godement relations” refered to in §1.
Suppose $f:A\rightarrow B$ is a morphism of strict $n$-groupoids satisfying condition (b). From the immediately preceding formula and the inductive statement for $n-1$-groupoids, we get that $f$ induces isomorphisms on the $\pi _i$ for $i\geq 1$. On the other hand, the truncation $\tau _{\leq 1}(f)$ satisfies condition (b) for a morphism of $1$-groupoids, and this is readily seen to imply that $\pi _0(f)$ is an isomorphism. Thus $f$ satisfies condition (a).
Suppose on the other hand that $f:A\rightarrow B$ is a morphism of strict $n$-groupoids satisfying condition (a). Then of course $\pi _0(f)$ is surjective. Consider two objects $x,y\in A$ and look at the induced morphism $$f^{x,y}: Hom _A(x,y)\rightarrow Hom _B(f(x), f(y)).$$ We claim that $f^{x,y}$ satisfies condition (a) for a morphism of $n-1$-groupoids. For this, consider a $1$-morphism from $x$ to $y$, i.e. an object $r\in Hom _A(x,y)$. By version (2) of the groupoid condition for $A$, multiplication by $r$ induces an equivalence of $n-1$-groupoids $$m(r): Hom _A(x,x)\rightarrow Hom _A(x,y),$$ and furthermore $m(r)(1_x)=r$. The same is true in $B$: multiplication by $f(r)$ induces an equivalence $$m(f(r)): Hom _B(f(x), f(x))\rightarrow Hom _B(f(x), f(y)).$$ The fact that $f$ is a morphism implies that these fit into a commutative square $$\begin{array}{ccc}
Hom _A(x,x)&\rightarrow &Hom _A(x,y)\\
\downarrow && \downarrow \\
Hom _B(f(x), f(x))&\rightarrow &Hom _B(f(x), f(y)).
\end{array}$$ The equivalence condition (a) for $f$ implies that the left vertical morphism induces isomorphisms $$\pi _i(Hom _A(x,x), 1_x)\stackrel{\cong}{\rightarrow}
\pi _i(Hom _B(f(x), f(x)), 1_{f(x)}).$$ Therefore the right vertical morphism (i.e. $f_{x,y}$) induces isomorphisms $$\pi _i(Hom _A(x,y), r)\stackrel{\cong}{\rightarrow}
\pi _i(Hom _B(f(x), f(y)), f(r)),$$ this for all $i\geq 1$. We have now verified these isomorphisms for any base-object $r$. A similar argument implies that $f^{x,y}$ induces an injection on $\pi _0$. On the other hand, the fact that $f$ induces an isomorphism on $\pi _0$ implies that $f^{x,y}$ induces a surjection on $\pi _0$ (note that these last two statements are reduced to statements about $1$-groupoids by applying $\tau _{\leq 1}$ so we don’t give further details). All of these statements taken together imply that $f^{x,y}$ satisfies condition (a), and by the inductive statement of the theorem for $n-1$-groupoids this implies that $f^{x,y}$ is an equivalence. Thus $f$ satisfies condition (b).
We now remark that condition (b) is equivalent to condition (c) for a morphism $f:A\rightarrow B$. Indeed, the part of condition (c) for $i=-1$ is, by the definition of $\pi _0$, identical to the condition that $f$ induces a surjection $\pi _0(A)\rightarrow
\pi _0(B)$. And the remaining conditions for $i=0,\ldots , n+1$ are identical to the conditions of (c) corresponding to $j=i-1=-1,\ldots , (n-1)+1$ for all the morphisms of $n-1$-groupoids $Hom _A(x,y)\rightarrow Hom _B(f(x), f(y))$. (In terms of $u$ and $v$ appearing in the condition in question, take $x$ to be the source of the source of the source …, and take $y$ to be the target of the target of the target …). Thus by induction on $n$ (i.e. by the equivalence $(b)\Leftrightarrow (c)$ for $n-1$-groupoids), the conditions (c) for $f$ for $i=0,\ldots , n+1$, are equivalent to the conditions that $Hom _A(x,y)\rightarrow Hom _B(f(x), f(y))$ be equivalences of $n-1$-groupoids. Thus condition (c) for $f$ is equivalent to condition (b) for $f$, which completes the proof of part (III) of the theorem.
We now proceed with the proof of part (I) of Theorem \[thmdef\]. Note first of all that the implications $(1)\Rightarrow (2)$ and $(2)\Rightarrow (3)$ are easy. We give a short discussion of $(1)\Rightarrow (3)$ anyway, and then we prove $(3)\Rightarrow (2)$ and $(2)\Rightarrow (1)$.
Note also that the equivalence $(1)\Leftrightarrow (2)$ is the content of Proposition 1.6 of [@KV]; we give a proof here because the proof of Proposition 1.6 was “left to the reader” in [@KV].
[**$(1)\Rightarrow (3)$:**]{} Suppose $A$ is a strict $n$-category satisfying condition $(1)$. This condition (from [@KV]) is compatible with truncation, so $\tau _{\leq 1} (A)$ satisfies condition $(1)$ for $1$-categories; which in turn is equivalent to the standard condition of being a $1$-groupoid, so we get that $\tau _{\leq 1}(A)$ is a $1$-groupoid. On the other hand, the conditions $(1)$ from [@KV] for $i$-arrows, $1\leq i \leq n$, include the same conditions for the $i-1$-arrows of $Hom _A(x,y)$ for any $x,y\in Ob(A)$ (the reader has to verify this by looking at the definition in [@KV]). Thus by the inductive statement of the present theorem for strict $n-1$-categories, $Hom _A(x,y)$ is a strict $n-1$-groupoid. This shows that $A$ satisfies condition $(3)$.
[**$(3)\Rightarrow (2)$:**]{} Suppose $A$ is a strict $n$-category satisfying condition $(3)$. It already satisfies the first part of condition $(2)$, by hypothesis. Thus we have to show the second part, for example that for $f:
x\rightarrow y$ in $Ob(Hom _A(x,y))$, composition with $f$ induces an equivalence $$Hom _A(y,z)\rightarrow Hom _A(x,z)$$ (the other part is dual and has the same proof which we won’t repeat here).
In order to prove this, we need to make a digression about the effect of composition with $2$-morphisms. Suppose $f,g\in Ob (Hom _A(x,y))$ and suppose that $u$ is a $2$-morphism from $f$ to $g$—this last supposition may be rewritten $$u\in Ob(Hom _{Hom _A(x,y)}(f,g)).$$ [*Claim:*]{} Suppose $z$ is another object; we claim that if composition with $f$ induces an equivalence $Hom _A(y,z)\rightarrow Hom _A(x,z)$, then composition with $g$ also induces an equivalence $Hom _A(y,z)\rightarrow Hom _A(x,z)$.
To prove the claim, suppose that $h,k$ are two $1$-morphisms from $y$ to $z$. We now obtain a diagram $$\begin{array}{ccc}
Hom _{Hom _A(y,z)}(h,k) & \rightarrow & Hom _{Hom _A(x,z)}(hf, kf)\\
\downarrow && \downarrow \\
Hom _{Hom _A(x,z)}(hg, kg) & \rightarrow & Hom _{Hom _A(x,z)}(hf, kg),
\end{array}$$ where the top arrow is given by composition $\ast _0$ with $1_f$; the left arrow by composition $\ast _0$ with $1_g$; the bottom arrow by composition $\ast _1$ with the $2$-morphism $h\ast _0u$; and the right morphism is given by composition with $k\ast _0u$. This diagram commutes (that is the “Godement rule” or “interchange rule” cf [@KV] p. 32). By the inductive statement of the present theorem (version (2) of the groupoid condition) for the $n-1$-groupoid $Hom _A(x,z)$, the morphisms on the bottom and on the right in the above diagram are equivalences. The hypothesis in the claim that $f$ is an equivalence means that the morphism along the top of the diagram is an equivalence; thus by the sub-lemma (part (IV) of the present theorem) applied to the $n-2$-groupoids in the diagram, we get that the morphism on the left of the diagram is an equivalence. This provides the second half of the criterion (b) of part (III) for showing that the morphism of composition with $g$, $Hom _A(y,z)\rightarrow Hom _A(x,z)$, is an equivalence of $n-1$-groupoids.
To finish the proof of the claim, we now verify the first half of criterion (b) for the morphism of composition with $g$ (in this part we use directly the condition (3) for $A$ and don’t use either $f$ or $u$). Note that $\tau _{\leq
1}(A)$ is a $1$-groupoid, by the condition (3) which we are assuming. Note also that (by definition) $$\pi _0Hom _A(y,z)= Hom _{\tau _{\leq 1}A}(y,z) \;\;\; \mbox{and}\;\;\;
\pi _0Hom _A(x,z)= Hom _{\tau _{\leq 1}A}(x,z),$$ and the morphism in question here is just the morphism of composition by the image of $g$ in $\tau _{\leq 1}(A)$. Invertibility of this morphism in $\tau _{\leq 1}(A)$ implies that the composition morphism $$Hom _{\tau _{\leq 1}A}(y,z)\rightarrow Hom _{\tau _{\leq 1}A}(x,z)$$ is an isomorphism. This completes verification of the first half of criterion (b), so we get that composition with $g$ is an equivalence. This completes the proof of the claim.
We now return to the proof of the composition condition for (2). The fact that $\tau _{\leq 1}(A)$ is a $1$-groupoid implies that given $f$ there is another morphism $h$ from $y$ to $x$ such that the class of $fh$ is equal to the class of $1_y$ in $\pi _0Hom _A(y,y)$, and the class of $hf$ is equal to the class of $1_x$ in $\pi _0Hom _A(x,x)$. This means that there exist $2$-morphisms $u$ from $1_y$ to $fh$, and $v$ from $1_x$ to $hf$. By the above claim (and the fact that the compositions with $1_x$ and $1_y$ act as the identity and in particular are equivalences), we get that composition with $fh$ is an equivalence $$\{ fh\} \times Hom _A(y,z) \rightarrow Hom _A(y,z),$$ and that composition with $hf$ is an equivalence $$\{ hf\} \times Hom _A(x,z)\rightarrow Hom _A(x,z).$$ Let $$\psi _f: Hom _A(y,z)\rightarrow Hom _A(x,z)$$ be the morphism of composition with $f$, and let $$\psi _h: Hom _A(x,z)\rightarrow Hom _A(y,z)$$ be the morphism of composition with $h$. We have seen that $\psi _h\psi _f$ and $\psi _f\psi _h$ are equivalences. By the second sub-lemma (part (V) of the theorem) applied to $n-1$-groupoids, these imply that $\psi _f$ is an equivalence.
The proof for composition in the other direction is the same; thus we have obtained condition (2) for $A$.
[**$(2)\Rightarrow (1)$:**]{} Look at the condition (1) by refering to [@KV]: in question are the conditions $GR'_{i,k}$ and $GR''_{i,k}$ ($i<k\leq
n$) of Definition 1.1, p. 33 of [@KV]. By the inductive version of the present equivalence for $n-1$-groupoids and by the part of condition (2) which says that the $Hom _A(x,y)$ are $n-1$-groupoids, we obtain the conditions $GR'_{i,k}$ and $GR''_{i,k}$ for $i\geq 1$. Thus we may now restrict our attention to the condition $GR'_{0,k}$ and $GR''_{0,k}$. For a $1$-morphism $a$ from $x$ to $y$, the conditions $GR'_{0,k}$ for all $k$ with respect to $a$, are the same as the condition that for all $w$, the morphism of pre-multiplication by $a$ $$Hom _A(w,x)\times \{ a\} \rightarrow Hom _A(w,y)$$ is an equivalence according to the version (c) of the notion of equivalence (cf Part (III) of this theorem). Thus, condition $GR'_{0,k}$ follows from the second part of condition (2) (for pre-multiplication). Similarly condition $GR''_{0,k}$ follows from the second part of condition (2) for post-multiplication by every $1$-morphism $a$. Thus condition (2) implies condition (1). This completes the proof of Part (I) of the theorem.
For the sub-lemma (part (IV) of the theorem), using the fact that isomorphisms of sets satisfy the same “three for two” property, and using the characterization of equivalences in terms of homotopy groups (condition (a)) we immediately get two of the three statements: that if $f$ and $g$ are equivalences then $gf$ is an equivalence; and that if $gf$ and $g$ are equivalences then $f$ is an equivalence. Suppose now that $gf$ and $f$ are equivalences; we would like to show that $g$ is an equivalence. First of all it is clear that if $x\in Ob(A)$ then $g$ induces an isomorphism $\pi _i(B, f(x))\cong \pi _0(C, gf(x))$ (resp. $\pi _0(B)\cong \pi _0(C)$). Suppose now that $y\in Ob(B)$, and choose a $1$-morphism $u$ going from $y$ to $f(x)$ for some $x\in Ob(A)$ (this is possible because $f$ is surjective on $\pi
_0$). By condition (2) for being a groupoid, composition with $u$ induces equivalences along the top row of the diagram $$\begin{array}{ccccc}
Hom _B(y,y) &\rightarrow &Hom _B(y, f(x))&\leftarrow &Hom _B(f(x), f(x))\\
\downarrow && \downarrow && \downarrow \\
Hom _C(g(y),g(y)) &\rightarrow &Hom _C(g(y), gf(x))&\leftarrow &
Hom _C(gf(x), gf(x)).
\end{array}$$ Similarly composition with $g(u)$ induces equivalences along the bottom row. The sub-lemma for $n-1$-groupoids applied to the sequence $$Hom _A(x,x)\rightarrow Hom _B(f(x), f(x))\rightarrow Hom _C(gf(x), gf(x))$$ as well as the hypothesis that $f$ is an equivalence, imply that the rightmost vertical arrow in the above diagram is an equivalence. Again applying the sub-lemma to these $n-1$-groupoids yields that the leftmost vertical arrow is an equivalence. In particular $g$ induces isomorphisms $$\pi _i(B,y) = \pi _{i-1}(Hom _B(y,y), 1_y) \stackrel{\cong}{\rightarrow}
\pi _{i-1}(Hom _C(g(y), g(y)),1_{g(y)}) = \pi _i(C, g(y)).$$ This completes the verification of condition (a) for the morphism $g$, completing the proof of part (IV) of the theorem.
Finally we prove the second sub-lemma, part (V) of the theorem (from which we now adopt the notations $A,B,C,D,f,g,h$). Note first of all that applying $\pi _0$ gives the same situation for maps of sets, so $\pi _0(g)$ is an isomorphism. Next, suppose $x\in Ob(A)$. Then we obtain a sequence $$\pi _i(A,x)\rightarrow \pi _i(B,f(x))
\rightarrow \pi _i(C, gf(x))\rightarrow \pi _i(D, hgf(x)),$$ such that the composition of the first pair and also of the last pair are isomorphisms; thus $g$ induces an isomorphism $\pi _i(B , f(x))\cong \pi _i(C, gf(x))$. Now, by the same argument as for Part (IV) above, (using the hypothesis that $f$ induces a surjection $\pi
_0(A)\rightarrow \pi _0(B)$) we get that for any object $y\in Ob(B)$, $g$ induces an isomorphism $\pi _i(B , y)\cong \pi _i(C, g(y))$. By definition (a) of Part (III) we have now shown that $g$ is an equivalence. This completes the proof of the theorem. [$/$$/$$/$]{}
Let $nStrGpd$ be the category of strict $n$-groupoids.
We close out this section by looking at how the groupoid condition fits in with the discussion of \[scholium\] and \[iterate\]. Let $C$ be a strict $n$-category with only one object $x$. Then $C$ is an $n$-groupoid if and only if $Hom _C(x,x)$ is an $n-1$-groupoid and $\pi _0Hom _C(x,x)$ (which has a structure of monoid) is a group. This is version (3) of the definition of groupoid in \[thmdef\]. Iterating this remark one more time we get the following statement.
\[scholiumgpd\] The construction of \[scholium\] establishes an equivalence of categories between the strict $n$-groupoids having only one object and only one $1$-morphism, and the abelian monoid-objects $G$ in $(n-2)StrGpd$ such that the monoid $\pi _0(G)$ is a group.
[$/$$/$$/$]{}
\[iterategpd\] Suppose $C$ is a strict $n$-category having only one object and only one $1$-morphism, and let $B$ be the strict $n+1$-category of \[iterate\] with one object $b$ and $Hom _B(b,b)= C$. Then $B$ is a strict $n+1$-groupoid if and only if $C$ is a strict $n$-groupoid.
[*Proof:*]{} Keep the notations of the proof of \[iterate\]. If $C$ is a groupoid this means that $G$ satisfies the condition that $\pi
_0(G)$ be a group, which in turn implies that $U$ is a groupoid. Note that $\pi
_0(U)=\ast$ is automatically a group; so applying the observation \[scholiumgpd\] once again, we get that $B$ is a groupoid. In the other direction, if $B$ is a groupoid then $C=Hom _B(b,b)$ is a groupoid by versions (2) and (3) of the definition of groupoid. [$/$$/$$/$]{}
[**.Realization functors**]{}
Recall that $nStrGpd$ is the category of strict $n$-groupoids as defined above \[thmdef\]. Let $Top$ be the category of topological spaces. The following definition encodes the minimum of what one would expect for a reasonable realization functor from strict $n$-groupoids to spaces.
\[realizationdef\] A [*realization functor for strict $n$-groupoids*]{} is a functor $$\Re : nStrGpd \rightarrow Top$$ together with the following natural transformations: $$r:Ob (A) \rightarrow \Re (A);$$ $$\zeta _i(A,x): \pi _i (A, x) \rightarrow \pi _i (\Re (A), r(x)),$$ the latter including $\zeta _0(A): \pi _0(A)\rightarrow \pi _0(\Re (A))$; such that the $\zeta _i(A,x)$ and $\zeta _0(A)$ are isomorphisms for $0\leq i \leq n$, and such that the $\pi _i(\Re (A), y)$ vanish for $i>n$.
\[realization\] [([@KV])]{} There exists a realization functor $\Re$ for strict $n$-groupoids.
Kapranov and Voevodsky [@KV] construct such a functor. Their construction proceeds by first defining a notion of “diagrammatic set”; they define a realization functor from $n$-groupoids to diagrammatic sets (denoted $Nerv$), and then define the topological realization of a diagrammatic set (denoted $|
\cdot |$). The composition of these two constructions gives a realization functor $$G \mapsto \Re _{KV}(G):= | Nerv(G)|$$ from strict $n$-groupoids to spaces. Note that this functor $\Re_{KV}$ satisfies the axioms of \[realizationdef\] as a consequence of Propositions 2.7 and 3.5 of [@KV].
One obtains a different construction by considering strict $n$-groupoids as weak $n$-groupoids in the sense of [@Tamsamani] (multisimplicial sets) and then taking the realization of [@Tamsamani]. This construction is actually probably due to someone from the Australian school many years beforehand and we call it the [*standard realization*]{} $\Re _{\rm std}$. The properties of \[realizationdef\] can be extracted from [@Tamsamani] (although again they are probably classical results).
We don’t claim here that any two realization functors must be the same, and in particular the realization $\Re _{KV}$ could [*a priori*]{} be different from the standard one. This is why we shall work, in what follows, with an arbitrary realization functor satisfying the axioms of \[realizationdef\].
Here are some consequences of the axioms for a realization functor. If $C\rightarrow C'$ is a morphism of strict $n$-groupoids inducing isomorphisms on the $\pi _i$ then $\Re
(C)\rightarrow \Re (C')$ is a weak homotopy equivalence. Conversely if $f:C\rightarrow C'$ is a morphism of strict $n$-groupoids which induces a weak equivalence of realizations then $f$ was an equivalence.
[**.The case of the standard realization**]{}
Before getting to our main result which concerns an arbitrary realization functor satisfying \[realizationdef\], we take note of an easier argument which shows that the standard realization functor cannot give rise to arbitrary homotopy types.
\[compatiblelooping\] A collection of realization functors $\Re ^n$ for $n$-groupoids ($0\leq n <
\infty$) satisfying \[realizationdef\] is said to be [*compatible with looping*]{} if there exist transformations natural in an $n$-groupoid $A$ and an object $x\in Ob(A)$, $$\varphi (A, x): \Re ^{n-1}(Hom _A(x,x))\rightarrow \Omega ^{r(x)}\Re ^n(A)$$ (where $\Omega ^{r(x)}$ means the space of loops based at $r(x)$), such that for $i\geq 1$ the following diagram commutes: $$\begin{array}{ccc}
\pi _i(A, x) & = \pi _{i-1}(Hom _A(x,x), 1_x) \rightarrow &
\pi _{i-1}(\Re ^{n-1}(Hom _A(x,x)), r(1_x))\\
\downarrow &&\downarrow \\
\pi _i(\Re ^n(A), r(x)) & \leftarrow & \pi _{i-1}( \Omega ^{r(x)}\Re ^n(A),
cst(r(x)))
\end{array}$$ where the top arrow is $\zeta _{i-1}(Hom _A(x,x), 1_x)$, the left arrow is $\zeta _{i}(A,x)$, the right arrow is induced by $\varphi (A, x)$, and the bottom arrow is the canonical arrow from topology. (When $i=1$, suppress the basepoints in the $\pi _{i-1}$ in the diagram.)
[*Remark:*]{} The arrows on the top, the bottom and the left are isomorphisms in the above diagram, so the arrow on the right is an isomorphism and we obtain as a corollary of the definition that the $\varphi (A,x)$ are actually weak equivalences.
[*Remark:*]{} The collection of standard realizations $\Re ^n_{\rm std}$ for $n$-groupoids, is compatible with looping. We leave this as an exercise for the reader.
Recall the statements of \[iterate\] and \[iterategpd\]: if $A$ is a strict $n$-category with only one object $x$ and only one $1$-morphism $1_x$, then there exists a strict $n+1$-category $B$ with one object $y$, and with $Hom _B(y,y)=A$; and $A$ is a strict $n$-groupoid if and only if $B$ is a strict $n+1$-groupoid.
\[forstandard\] Suppose $\{ \Re ^n\}$ is a collection of realization functors \[realizationdef\] compatible with looping \[compatiblelooping\]. Then if $A$ is a $1$-connected strict $n$-groupoid (i.e. $\pi _0(A)=\ast$ and $\pi _1(A,x)=\{ 1\}$), the space $\Re ^n(A)$ is weak-equivalent to a loop space.
[*Proof:*]{} Let $A'\subset A$ be the sub-$n$-category having one object $x$ and one $1$-morphism $1_x$. For $i\geq 2$ the inclusion induces isomorphisms $$\pi _i(A', x) \cong \pi _i(A,x),$$ and in view of the $1$-connectedness of $A$ this means (according to the definition of \[thmdef\] III (a)) that the morphism $A'\rightarrow A$ is an equivalence. It follows (by definition \[realizationdef\]) that $\Re
^n(A')\rightarrow \Re ^n(A)$ is a weak equivalence. Now $A'$ satisfies the hypothesis of \[iterate\], \[iterategpd\] as recalled above, so there is an $n+1$-groupoid $B$ having one object $y$ such that $A'=Hom _B(y,y)$. By the definition of “compatible with looping” and the subsequent remark that the morphism $\varphi (B,y)$ is a weak equivalence, we get that $\varphi (B,y)$ induces a weak equivalence $$\Re ^n(A') \rightarrow \Omega ^{r(y)}\Re ^{n+1}(B).$$ Thus $\Re ^n(A)$ is weak-equivalent to the loop-space of $\Re ^{n+1}(B)$. [$/$$/$$/$]{}
The following corollary is a statement which seems to be due to C. Berger [@Berger] (although the statement appears without proof in Grothendieck [@Grothendieck]). See also R. Brown and coauthors [@RBrown1] [@BrownGilbert] [@BrownHiggins] [@BrownHiggins2].
\[berger\] [(C. Berger [@Berger])]{} There is no strict $3$-groupoid $A$ such that the standard realization $\Re _{\rm std} (A)$ is weak-equivalent to the $3$-type of $S^2$.
[*Proof:*]{} The $3$-type of $S^2$ is not a loop-space. By the previous corollary (and the fact that the standard realizations are compatible with looping, which we have above left as an exercise for the reader), it is impossible for $\Re _{\rm std}(A)$ to be the $3$-type of $S^2$. [$/$$/$$/$]{}
[**.Nonexistence of strict $3$-groupoids giving rise to the $3$-type of $S^2$**]{}
It is not completely clear whether Kapranov and Voevodsky claim that their realization functors are compatible with looping in the sense of \[compatiblelooping\], so Berger’s negative result (Corollary \[berger\] above) might not apply. The main work of the present paper is to extend this negative result to [*any*]{} realization functor satisfying the minimal definition \[realizationdef\], in particular getting a result which applies to the realization functor of [@KV].
\[noS2\] Let $\Re$ be any realization functor satisfying the properties of Definition \[realizationdef\]. Then there does not exist a strict $3$-groupoid $C$ such that $\Re (C)$ is weak-equivalent to the $3$-truncation of the homotopy type of $S^2$.
Let $\Re _{KV}$ be the realization functor of Kapranov and Voevodsky [@KV] cf the discussion above. If we assume that Propositions 2.7 and 3.5 of [@KV] (stating that $\Re _{KV}$ satisfies the axioms \[realizationdef\]) are true, then Corollary 3.8 of [@KV] is not true, i.e. $\Re_{KV}$ does not induce an equivalence between the homotopy categories of strict $3$-groupoids and $3$-truncated topological spaces.
[*Proof:*]{} According to Proposition \[noS2\], for any realization functor satisfying \[realizationdef\], the induced functor on the homotopy categories is not essentially surjective: its essential image doesn’t contain the $3$-type of $S^2$. [$/$$/$$/$]{}
Proposition \[noS2\] is very similar to the result of Brown and Higgins [@BrownHiggins] and also the recent result of C. Berger [@Berger] (cf \[berger\] above). As was noted in [@KV], the result of Brown and Higgins concerns the more restrictive notion of groupoid where one requires that all morphisms have strict inverses (however, see also [@RBrown1], [@BrownHiggins2]). As in [@KV], that restriction is not included in the definition \[thmdef\]. Berger considers strict $n$-groupoids according to the definition \[thmdef\] (i.e. with inverses non-strict) as well, but his negative result applies only to a standard realization functor and as such, doesn’t [*a priori*]{} directly contradict [@KV].
The basic difference in the present approach is that we make no reference to any particular construction of $\Re$ but show that the proposition holds for any realization construction having the properties of Definition \[realizationdef\].
The fact that strict $n$-groupoids don’t model all homotopy types is also mentionned in Grothendieck [@Grothendieck]. The basic idea in the setting of $3$-categories not necessarily groupoids, is contained in some examples which G. Maltsiniotis pointed out to me, in Gordon-Power-Street [@Gordon-Power-Street] where there are given examples of weak $3$-categories not equivalent to strict ones. This in turn is related to the difference between braided monoidal categories and symmetric monoidal categories, see for example the nice discussion in Baez-Dolan [@BaezDolan].
In order to prove \[noS2\], we will prove the following statement (which contains the main part of the argument). It basically says that the Postnikov tower of a simply connected strict $3$-groupoid $C$, splits.
\[diagramme\] Suppose $C$ is a strict $3$-groupoid with an object $c$ such that $\pi
_0(C)=\ast$, $\pi _1(C,c)=\{ 1\}$, $\pi _2(C,c)
= {{\bf Z}}$ and $\pi _3(C,c)=H$ for an abelian group $H$. Then there exists a diagram of strict $3$-groupoids $$C \stackrel{g}{\leftarrow} B \stackrel{f}{\leftarrow} A
\stackrel{h}{\rightarrow} D$$ with objects $b\in Ob(B)$, $a\in Ob(A)$, $d\in Ob(D)$ such that $f(a)=b$, $g(b)=c$, $h(a)=d$. The diagram is such that $g$ and $f$ are equivalences of strict $3$-groupoids, and such that $\pi _0(D)=\ast$, $\pi _1(D,d)=\{ 1\}$, $\pi _2(D,d)=\{ 0\}$, and such that $h$ induces an isomorphism $$\pi _3(h): \pi _3(A,a)=H \stackrel{\cong}{\rightarrow} \pi _3(D,d).$$
[*Proof of Proposition \[noS2\] using Proposition \[diagramme\]*]{}
Suppose for the moment that we know Proposition \[diagramme\]; with this we will prove \[noS2\]. Fix a realization functor $\Re$ for strict $3$-groupoids satisfying the axioms \[realizationdef\], and assume that $C$ is a strict $3$-groupoid such that $\Re (C)$ is weak homotopy-equivalent to the $3$-type of $S^2$. We shall derive a contradiction.
In [*résumé*]{} the argument is this: that applying the realization functor to the diagram given by \[diagramme\] and inverting the first two maps which are weak homotopy equivalences, we would get a map $$\tau _{\leq 3}(S^2)= \Re (C) \rightarrow \Re (D) = K(H, 3)$$ (with $H={{\bf Z}}$). This is a class in $H^3(S^2, H)$. The hypothesis that $\Re (h)$ is an isomorphism on $\pi _3$ means that this class is nonzero when applied to $\pi _3(S^2)$ via the Hurewicz homomorphism; but $H^3(S^2, {{\bf Z}})= 0$, a contradiction.
Here is a full description of the argument. Apply Proposition \[diagramme\] to $C$. Choose an object $c\in Ob(C)$. Note that, because of the isomorphisms between homotopy sets or groups \[realizationdef\], we have $\pi _0(C)=\ast$, $\pi _1(C,c)=\{ 1\}$, $\pi _2(C,c)
= {{\bf Z}}$ and $\pi _3(C,c)={{\bf Z}}$, so \[diagramme\] applies with $H={{\bf Z}}$. We obtain a sequence of strict $3$-groupoids $$C \stackrel{g}{\leftarrow} B \stackrel{f}{\leftarrow} A
\stackrel{h}{\rightarrow} D.$$ This gives the diagram of spaces $$\Re (C) \stackrel{\Re (g)}{\leftarrow} \Re (B) \stackrel{\Re
(f)}{\leftarrow} \Re
(A) \stackrel{\Re (h)}{\rightarrow} \Re (D).$$ The axioms \[realizationdef\] for $\Re$ imply that $\Re$ transforms equivalences of strict $3$-groupoids into weak homotopy equivalences of spaces. Thus $\Re (f)$ and $\Re (g)$ are weak homotopy equivalences and we get that $\Re (A)$ is weak homotopy equivalent to the $3$-type of $S^2$.
On the other hand, again by the axioms \[realizationdef\], we have that $\Re (D)$ is $2$-connected, and $\pi
_3(\Re (D), r(d))=H$ (via the isomorphism $\pi _3(D,d)\cong H$ induced by $h$, $f$ and $g$). By the Hurewicz theorem there is a class $\eta \in H^3(\Re (D),
H)$ which induces an isomorphism $${\bf Hur}(\eta ): \pi _3(\Re (D), r(d))\stackrel{\cong}{\rightarrow} H.$$ Here $${\bf Hur} : H^3(X , H)\rightarrow Hom (\pi _3(X,x), H)$$ is the Hurewicz map for any pointed space $(X,x)$; and the cohomology is singular cohomology (in particular it only depends on the weak homotopy type of the space).
Now look at the pullback of this class $$\Re (h)^{\ast}(\eta )\in H^3(\Re (A), H).$$ The hypothesis that $\Re (u)$ induces an isomorphism on $\pi _3$ implies that $${\bf Hur}(\Re (h)^{\ast}(\eta )): \pi _3(\Re (A),
r(a))\stackrel{\cong}{\rightarrow} H.$$ In particular, ${\bf Hur}(\Re (h)^{\ast}(\eta ))$ is nonzero so $\Re (h)^{\ast}(\eta )$ is nonzero in $H^3(\Re (A), H)$. This is a contradiction because $\Re (A)$ is weak homotopy-equivalent to the $3$-type of $S^2$, and $H={{\bf Z}}$, but $H^3(S^2 , {{\bf Z}})=\{ 0 \}$.
This contradiction completes the proof of Proposition \[noS2\], assuming Proposition \[diagramme\]. [$/$$/$$/$]{}
[*Proof of Proposition \[diagramme\]*]{}
This is the main part of the argument. We start with a strict groupoid $C$ and object $c$, satisfying the hypotheses of \[diagramme\].
The first step is to construct $(B,b)$. We let $B\subset C$ be the sub-$3$-category having only one object $b=c$, and only one $1$-morphism $1_b=1_c$. We set $$Hom _{Hom _B(b,b)}(1_b, 1_b):=Hom _{Hom _C(c,c)}(1_c, 1_c) ,$$ with the same composition law. The map $g: B\rightarrow C$ is the inclusion.
Note first of all that $B$ is a strict $3$-groupoid. This is easily seen using version (1) of the definition \[thmdef\] (but one has to look at the conditions in [@KV]). We can also verify it using condition (3). Of course $\tau _{\leq 1}(B)$ is the $1$-category with only one object and only one morphism, so it is a groupoid. We have to verify that $Hom _B(b,b)$ is a strict $2$-groupoid. For this, we again apply condition (3) of \[thmdef\]. Here we note that $$Hom _B(b,b)\subset Hom _C(c,c)$$ is the full sub-$2$-category with only one object $1_b=1_c$. Therefore, in view of the definition of $\tau _{\leq 1}$, we have that $$\tau _{\leq 1}Hom _B(b,b)\subset \tau _{\leq 1}Hom _C(c,c)$$ is a full subcategory. A full subcategory of a $1$-groupoid is again a $1$-groupoid, so $\tau _{\leq 1}Hom _B(b,b)$ is a $1$-groupoid. Finally, $Hom _{Hom _B(b,b)}(1_b, 1_b)$ is a $1$-groupoid since by construction it is the same as $Hom _{Hom _C(c,c)}(1_c, 1_c)$ (which is a groupoid by condition (3) applied to the strict $2$-groupoid $Hom _C(c,c)$). This shows that $Hom _B(b,b)$ is a strict $2$-groupoid an hence that $B$ is a strict $3$-groupoid.
Next, note that $\pi _0(B)=\ast$ and $\pi _1(B,b)=\{ 1\}$. On the other hand, for $i=2,3$ we have $$\pi _i(B,b)= \pi _{i-2}(Hom _{Hom _B(b,b)}(1_b, 1_b), 1^2_b)$$ and similarly $$\pi _i(C,c)= \pi _{i-2}(Hom _{Hom _C(c,c)}(1_c, 1_c), 1^2_c),$$ so the inclusion $g$ induces an equality $\pi _i(B,b) \stackrel{=}{\rightarrow}
\pi _i(C,c)$. Therefore, by definition (a) of equivalence \[thmdef\], $g$ is an equivalence of strict $3$-groupoids. This completes the construction and verification for $B$ and $g$.
Before getting to the construction of $A$ and $f$, we analyze the strict $3$-groupoid $B$ in terms of the discussion of \[scholium\] and \[scholiumgpd\]. Let $$G:= Hom _{Hom _B(b,b)}(1_b, 1_b).$$ It is an abelian monoid-object in the category of $1$-groupoids, with abelian operation denoted by $+: G\times G\rightarrow G$ and unit element denoted $0\in
G$ which is the same as $1_b$. The operation $+$ corresponds to both of the compositions $\ast _0$ and $\ast _1$ in $B$.
The hypotheses on the homotopy groups of $C$ also hold for $B$ (since $g$ was an equivalence). These translate to the statements that $(\pi _0(G), +) = {{\bf Z}}$ and $Hom _G(0,0)=H$.
We now construct $A$ and $f$ via \[scholium\] and \[scholiumgpd\], by constructing a morphism $(G',+)\rightarrow (G,+)$ of abelian monoid-objects in the category of $1$-groupoids. We do this by a type of “base-change” on the monoid of objects, i.e. we will first define a morphism $Ob(G')\rightarrow Ob(G)$ and then define $G'$ to be the groupoid with object set $Ob(G')$ but with morphisms corresponding to those of $G$.
To accomplish the “base-change”, start with the following construction. If $S$ is a set, let ${\bf E}(S)$ denote the groupoid with $S$ as set of objects, and with exactly one morphism between each pair of objects. If $S$ has an abelian monoid structure then ${\bf E}(S)$ is an abelian monoid object in the category of groupoids.
Note that for any groupoid $U$ there is a morphism of groupoids $$U\rightarrow {\bf E}(Ob(U)),$$ and by “base change” we mean the following operation: take a set $S$ with a map $p:S\rightarrow Ob(U)$ and look at $$V:= {\bf E}(S)\times _{{\bf E}(Ob(U))}U.$$ This is a groupoid with $S$ as set of objects, and with $$Hom _V(s,t)= Hom _U(p(s), p(t)).$$ If $U$ is an abelian monoid object in the category of groupoids, if $S$ is an abelian monoid and if $p$ is a map of monoids then $V$ is again an abelian monoid object in the category of groupoids.
Apply this as follows. Starting with $(G,+)$ corresponding to $B$ via \[scholium\] and \[scholiumgpd\] as above, choose objects $a,b \in Ob(G)$ such that the image of $a$ in $\pi
_0(G)\cong {{\bf Z}}$ corresponds to $1\in {{\bf Z}}$, and such that the image of $b$ in $\pi _0(G)$ corresponds to $-1\in {{\bf Z}}$. Let $N$ denote the abelian monoid, product of two copies of the natural numbers, with objects denoted $(m,n)$ for nonnegative integers $m,n$. Define a map of abelian monoids $$p:N \rightarrow Ob(G)$$ by $$p(m,n):= m\cdot a + n\cdot b := a+a+\ldots +a \, + \, b+b+\ldots +b.$$ Note that this induces the surjection $N\rightarrow \pi _0(G)={{\bf Z}}$ given by $(m,n)\mapsto m-n$.
Define $(G',+)$ as the base-change $$G':= {\bf E}(N) \times _{{\bf E}(Ob(G))} G,$$ with its induced abelian monoid operation $+$. We have $$Ob (G')= N,$$ and the second projection $p_2: G'\rightarrow G$ (which induces $p$ on object sets) is fully faithful i.e. $$Hom _{G'}((m,n), (m',n'))= Hom _G(p(m,n), p(m',n')).$$ Note that $\pi _0(G')={{\bf Z}}$ via the map induced by $p$ or equivalently $p_2$. To prove this, say that: (i) $N$ surjects onto ${{\bf Z}}$ so the map induced by $p$ is surjective; and (ii) the fact that $p_2$ is fully faithful implies that the induced map $\pi _0(G')\rightarrow \pi _0(G)={{\bf Z}}$ is injective.
We let $A$ be the strict $3$-groupoid corresponding to $(G',+)$ via \[scholium\], and let $f: A\rightarrow B$ be the map corresponding to $p_2: G'\rightarrow G$ again via \[scholium\]. Let $a$ be the unique object of $A$ (it is mapped by $f$ to the unique object $b\in Ob(B)$).
The fact that $(\pi _0(G'),+)={{\bf Z}}$ is a group implies that $A$ is a strict $3$-groupoid (\[scholiumgpd\]). We have $\pi _0(A)=\ast$ and $\pi
_1(A,a)=\{ 1\}$. Also, $$\pi _2(A,a)= (\pi _0(G'), +) = {{\bf Z}}$$ and $f$ induces an isomorphism from here to $\pi _2(B,b)=(\pi _0(G), +)={{\bf Z}}$. Finally (using the notation $(0,0)$ for the unit object of $(N,+)$ and the notation $0$ for the unit object of $Ob(G)$), $$\pi _3(A,a)= Hom _{G'}((0,0),(0,0)),$$ and similarly $$\pi _3(B,b)=Hom _G(0,0)=H;$$ the map $\pi _3(f): \pi _3(A,a)\rightarrow \pi _3(B,b)$ is an isomorphism because it is the same as the map $$Hom _{G'}((0,0),(0,0))\rightarrow Hom _G(0,0)$$ induced by $p_2: G'\rightarrow G$, and $p_2$ is fully faithful. We have now completed the verification that $f$ induces isomorphisms on the homotopy groups, so by version (a) of the definition of equivalence \[thmdef\], $f$ is an equivalence of strict $3$-groupoids.
We now construct $D$ and define the map $h$ by an explicit calculation in $(G',+)$. First of all, let $[H]$ denote the $1$-groupoid with one object denoted $0$, and with $H$ as group of endomorphisms: $$Hom _{[H]}(0,0):= H.$$ This has a structure of abelian monoid-object in the category of groupoids, denoted $([H], +)$, because $H$ is an abelian group. Let $D$ be the strict $3$-groupoid corresponding to $([H], +)$ via \[scholium\] and \[scholiumgpd\]. We will construct a morphism $h: A\rightarrow D$ via \[scholium\] by constructing a morphism of abelian monoid objects in the category of groupoids, $$h:(G', +)\rightarrow ([H], +).$$ We will construct this morphism so that it induces the identity morphism $$Hom _{G'}((0,0), (0,0))=H \rightarrow Hom _{[H]}(0,0)=H.$$ This will insure that the morphism $h$ has the property required for \[diagramme\].
The object $(1,1)\in N$ goes to $0\in \pi _0(G')\cong {{\bf Z}}$. Thus we may choose an isomorphism $\varphi : (0,0)\cong (1,1)$ in $G'$. For any $k$ let $k\varphi$ denote the isomorphism $\varphi + \ldots +\varphi$ ($k$ times) going from $(0,0)$ to $(k,k)$. On the other hand, $H$ is the automorphism group of $(0,0)$ in $G'$. The operations $+$ and composition coincide on $H$. Finally, for any $(m,n)\in N$ let $1_{m,n}$ denote the identity automorphism of the object $(m,n)$. Then any arrow $\alpha$ in $G$ may be uniquely written in the form $$\alpha = 1_{m,n} + k\varphi + u$$ with $(m,n)$ the source of $\alpha$, the target being $(m+k, n+k)$, and where $u\in H$.
We have the following formulae for the composition $\circ$ of arrows in $G'$. They all come from the basic rule $$(\alpha \circ \beta ) + (\alpha ' \circ \beta ')=
(\alpha + \alpha ') \circ (\beta + \beta ')$$ which in turn comes simply from the fact that $+$ is a morphism of groupoids $G'\times G'\rightarrow G'$ defined on the cartesian product of two copies of $G$. Note in a similar vein that $1_{0,0}$ acts as the identity for the operation $+$ on arrows, and also that $$1_{m,n} + 1_{m',n'} = 1_{m+m', n+n'}.$$
Our first equation is $$(1_{l,l} +k\varphi )\circ l\varphi = (k+l)\varphi .$$ To prove this note that $l\varphi + 1_{0,0}= l\varphi$ and our basic formula says $$(1_{l,l}\circ l_{\varphi} ) + (k\varphi \circ 1_{0,0})
=
(1_{l,l} +k\varphi )\circ (l\varphi + 1_{0,0} )$$ but the left side is just $l\varphi + k\varphi = (k+l)\varphi$.
Now our basic formula, for a composition starting with $(m,n)$, going first to $(m+l,n+l)$, then going to $(m+l+k, n+l+k)$, gives $$(1_{m+l,n+l} + k\varphi + u)\circ (1_{m,n} + l\varphi + v)$$ $$= (1_{m,n} + 1_{l,l} + k\varphi + u)\circ (1_{m,n} + l\varphi + v)$$ $$= 1_{m,n}\circ 1_{m,n} + (1_{l,l} +k\varphi )\circ l\varphi
+ u\circ v$$ $$= 1_{m,n} + (k+l)\varphi + (u\circ v)$$ where of course $u\circ v=u+v$.
This formula shows that the morphism $h$ from arrows of $G'$ to the group $H$, defined by $$h(1_{m,n} + k\varphi + u):= u$$ is compatible with composition. This implies that it provides a morphism of groupoids $h:G\rightarrow [H]$ (recall from above that $[H]$ is defined to be the groupoid with one object whose automorphism group is $H$). Furthermore the morphism $h$ is obviously compatible with the operation $+$ since $$(1_{m,n} + k\varphi + u)+ (1_{m',n'} + k'\varphi + u')=$$ $$(1_{m+m',n+n'} + (k+k')\varphi + (u+u'))$$ and once again $u+u'=u\circ u'$ (the operation $+$ on $[H]$ being given by the commutative operation $\circ$ on $H$).
This completes the construction of a morphism $h: (G, +)\rightarrow ([H], +)$ which induces the identity on $Hom (0,0)$. This corresponds to a morphism of strict $3$-groupoids $h: A\rightarrow D$ as required to complete the proof of Proposition \[diagramme\]. [$/$$/$$/$]{}
[**.A remark on strict $\infty$-groupoids**]{}
The nonexistence result of \[noS2\] holds also for strict $\infty$-groupoids as defined in [@KV]. Recall that Kapranov-Voevodsky [@KV] extend the notion of strict $n$-category and strict $n$-groupoid to the case $n=\infty$. The definition is made using condition (1), and the notion of equivalence is defined using (a) in \[thmdef\]. Note that the other characterizations of \[thmdef\] don’t actually make sense in the case $n=\infty$ because they are inductive on $n$.
The only thing we need to know about the case $n=\infty$ is that there are homotopy groups $\pi _i(A,a)$ of a strict $\infty$-groupoid $A$, and there are truncation operations on strict $\infty$-groupoids such that $\tau _{\leq n}(A)$ is a strict $n$-groupoid with a natural morphism $$A\rightarrow \tau _{\leq n}(A)$$ inducing isomorphisms on homotopy groups for $i\leq n$. (Here the $n$-groupoid $\tau _{\leq n}(A)$ is considered as an $\infty$-groupoid in the obvious way.) The homotopy groups and truncation are defined as in [@KV]—again, one has to avoid those versions of the definitions \[thmdef\] which are recursive on $n$.
We can extend the definition of \[realizationdef\] to the case $n=\infty$. It is immediate that for any realization functor $\Re$ satisfying the axioms \[realizationdef\] for $n=\infty$, the morphism $$\Re (A)\rightarrow \Re (\tau _{\leq n}A)$$ is the Postnikov truncation of $\Re (A)$. Applying \[noS2\], we obtain the following result.
\[noInfiniteS2\] For any realization functor $\Re$ satisfying the axioms \[realizationdef\] for $n=\infty$, there does not exist a strict $\infty$-groupoid $A$ (as defined by Kapranov-Voevodsky [@KV]) such that $\Re (A)$ is weak homotopy-equivalent to the $2$-sphere $S^2$.
[*Proof:*]{} Note that if $\Re$ is a realization functor satisfying \[realizationdef\] for $n=\infty$, then composing with the inclusion $i_3^{\infty}$ from the category of strict $3$-groupoids to the category of strict $\infty$-groupoids we obtain a realization functor $\Re i_3^{\infty}$ for strict $3$-groupoids, again satisfying \[realizationdef\]. If $A$ is a strict $\infty$-groupoid then the above truncation morphism, written more precisely, is $$A\rightarrow i_3^{\infty} \tau _{\leq 3}(A).$$ This induces isomorphisms on the $\pi _i$ for $i\leq 3$. Applying $\Re$ we get $$\Re (A) \rightarrow \Re i_3^{\infty} \tau _{\leq 3}(A),$$ inducing an isomorphism on homotopy groups for $i\leq 3$. In particular, if $\Re (A)$ were weak homotopy-equivalent to $S^2$ then this would imply that $\Re i_3^{\infty} \tau _{\leq 3}(A)$ is the $3$-type of $S^2$. In view of the fact that $\Re i_3^{\infty}$ is a realization functor according to \[realizationdef\] for strict $3$-groupoids, this would contradict \[noS2\]. Thus we conclude that there is no strict $\infty$-groupoid $A$ with $\Re (A)$ weak homotopy-equivalent to $S^2$. [$/$$/$$/$]{}
[**.Conclusion**]{}
One really needs to look at some type of weak $3$-categories in order to get a hold of $3$-truncated homotopy types. O. Leroy [@Leroy] and apparently, independantly, Joyal and Tierney [@JoyalTierney] were the first to do this. See also Gordon, Power, Street [@Gordon-Power-Street] and Berger [@Berger] for weak $3$-categories and $3$-types. Baues [@Baues] showed that $3$-types correspond to [*quadratic modules*]{} (a generalization of the notion of crossed complex) [@Baues]. Tamsamani [@Tamsamani] was the first to relate weak $n$-groupoids and homotopy $n$-types. For other notions of weak $n$-category, see [@BaezDolanLetter] [@BaezDolanIII] [@Batanin], [@Batanin2].
From homotopy theory (cf [@Lewis]) the following type of yoga seems to come out: that it suffices to weaken any one of the principal structures involved. Most weak notions of $n$-category involve a weakening of the associativity, or eventually of the Godement (commutativity) conditions.
It seems likely that the arguments of [@KV] would show that one could instead weaken the condition of being [*unary*]{} (i.e. having identities for the operations) and keep associativity and Godement. We give a proposed definition of what this would mean and then state two conjectures.
[*Motivation*]{}
Before giving the definition, we motivate these remarks by looking at the [*Moore loop space*]{} $\Omega ^x_M(X)$ of a space $X$ based at $x\in X$ (the Moore loop space is referred to in [@KV] as a motivation for their construction). Recall that $\Omega ^x_M(X)$ is the space of [*pairs*]{} $(r,
\gamma )$ where $r$ is a real number $r\geq 0$ and $\gamma = [0,r]\rightarrow
X$ is a path starting and ending at $x$. This has the advantage of being a strictly associative monoid. On the other side of the coin, the “length” function $$\ell : \Omega ^x_M(X)\rightarrow [0,\infty )\subset {{\bf R}}$$ has a special behavoir over $r=0$. Note that over the open half-line $(0,\infty )$ the length function $\ell$ is a fibration (even a fiber-space) with fiber homeomorphic to the usual loop space. However, the fiber over $r=0$ consists of a single point, the constant path $[0,0]\rightarrow X$ based at $x$. This additional point (which is the unit element of the monoid $\Omega ^x_M(X)$) doesn’t affect the topology of $\Omega ^x_M$ (at least if $X$ is locally contractible at $x$) because it is glued in as a limit of paths which are more and more concentrated in a neighborhood of $x$. However, the map $\ell$ is no longer a fibration over a neighborhood of $r=0$. This is a bit of a problem because $\Omega ^x_M$ is not compatible with direct products of the space $X$; in order to obtain a compatibility one has to take the fiber product over ${{\bf R}}$ via the length function: $$\Omega ^{(x,y)}_M(X\times Y)= \Omega ^x_M(X) \times _{{{\bf R}}} \Omega ^y_M(Y),$$ and the fact that $\ell$ is not a fibration could end up causing a problem in an attempt to iteratively apply a construction like the Moore loop-space.
Things seem to get better if we restrict to $$\Omega ^x_{M'}(X):=\ell ^{-1}((0,\infty ))\subset \Omega ^x_M(X) ,$$ but this associative monoid no longer has a strict unit. Even so, the constant path of any positive length gives a weak unit.
A motivation coming from a different direction was an observation made by Z. Tamsamani early in the course of doing his thesis. He was trying to define a strict $3$-category $2Cat$ whose objects would be the strict $2$-categories and whose morphisms would be the weak $2$-functors between $2$-categories (plus notions of weak natural transformations and $2$-natural transformations). At some point he came to the conclusion that one could adequately define $2Cat$ as a strict $3$-category except that he couldn’t get strict identities. Because of this problem we abandonned the idea and looked toward weakly associative $n$-categories. In retrospect it would be interesting to pursue Tamsamani’s construction of a strict $2Cat$ but with only weak identities.
[*Snucategories*]{}
Now we get back to looking at what it could mean to weaken the unit property for strict $n$-categories or strict $n$-groupoids. We will define a notion of [*$n$-snucategory*]{} (the initial ‘s’ stands for strict, ‘nu’ stands for non-unary) by induction on $n$. There will be a notion of direct product of $n$-snucategories. Suppose we know what these mean for $n-1$. Then an $n$-snucategory $C$ consists of a set $C_0$ of objects together with, for every pair of objects $x,y\in C_0$ an $n-1$-snucategory $Hom _C(x,y)$ and composition morphisms $$Hom _C(x,y)\times Hom _C(y,z) \rightarrow Hom _C(x,z)$$ which are strictly associative, such that the [*weak unary condition*]{} is satisfied. We now explain this condition. An element $e_x\in Hom _C(x,x)$ is called a weak identity if: —composition with $e$ induces equivalences of $n-1$-snucategories $$Hom _C(x,y)\rightarrow Hom_C(x,y) , \;\;\;
Hom _C(y,x)\rightarrow Hom_C(y,x);$$ —and if $e\cdot e$ is equivalent to $e$.
In order to complete the recursive definition we must define the notion of when a morphism of $n$-snucategories is an equivalence, and we must define what it means for two objects to be equivalent. A morphism is said to be an equivalence if the induced morphisms on $Hom$ are equivalences of $n-1$-snucategories and if it is essentially surjective on objects: each object in the target is equivalent to the image of an object. It thus remains just to be seen what equivalence of objects means. For this we introduce the [*truncations*]{} $\tau _{\leq i}C$ of an $n$-snucategory $C$. Again this is done in the same way as usual: $\tau _{\leq i}C$ is the $i$-snucategory with the same objects as $C$ and whose $Hom$’s are the truncations $$Hom_{\tau _{\leq i}C}(x,y):=\tau _{\leq i-1}Hom _C(x,y).$$ This works for $i\geq 1$ by recurrence, and for $i=0$ we define the truncation to be the set of isomorphism classes in $\tau _{\leq 1}C$. Note that truncation is compatible with direct product (direct products are defined in the obvious way) and takes equivalences to equivalences. These statements used recursively allow us to show that the truncations themselves satisfy the weak unary condition. Finally, we say that two objects are equivalent if they map to the same thing in $\tau _{\leq 0}C$.
Proceeding in the same way as in §2 above, we can define the notion of $n$-snugroupoid.
There are functors $\Pi _n$ and $\Re$ between the categories of $n$-snugroupoids and $n$-truncated spaces (going in the usual directions) together with adjunction morphisms inducing an equivalence between the localization of $n$-snugroupoids by equivalences, and $n$-truncated spaces by weak equivalences.
I think that the argument of [@KV] (which is unclear on the question of identity elements) actually serves to prove the above statement. I have called the above statement a “conjecture” because I haven’t checked this.
One might go out on a limb a bit more and make the following
The localization of the category of $n$-snucategories by equivalences is equivalent to the localizations of the categories of weak $n$-categories of Tamsamani and/or Baez-Dolan and/or Batanin by equivalences.
This of course is of a considerably more speculative nature.
[**Caveat**]{}: the above definition of “snucategory” is invented in an [*ad hoc*]{} way, and in particular one naturally wonders whether or not the equivalences $e\cdot e \sim e$ and higher homotopical data going along with that, would need to be specified in order to get a good definition. I have no opinion about this (the above definition being just the easiest thing to say which gives some idea of what needs to be done). Thus it is not completely clear that the above definition of $n$-snucategory is the “right” one to fit into the conjectures.
[MM2]{}
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M. Batanin. On the definition of weak $\omega$-category. Macquarie mathematics report number 96/207, Macquarie University, NSW Australia.
M. Batanin. Monoidal globular categories as a natural environment for the theory of weak $n$-categories. To appear, [*Adv. Math.*]{} and available at
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C. Berger. Double loop spaces, braided monoidal categories and algebraic $3$-type of space. Preprint (Univ. of Nice).
R. Brown. Computing homotopy types using crossed $n$-cubes of groups. [*Adams Memorial Symposium on Algebraic Topology*]{}, Vol 1, eds. N. Ray, G Walker. Cambridge University Press, Cambridge (1992) 187-210.
R. Brown, N.D. Gilbert. Algebraic models of 3-types and automorphism structures for crossed modules. [*Proc. London Math. Soc.*]{} (3) [**59**]{} (1989), 51-73.
R. Brown, P. Higgins. The equivalence of $\infty$-groupoids and crossed complexes. [*Cah. Top. Geom. Diff.*]{} [**22**]{} (1981), 371-386.
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W. Dwyer, D. Kan. Calculating simplicial localizations. [*J. Pure and Appl. Algebra*]{} [**18**]{} (1980), 17-35.
W. Dwyer, D. Kan. Function complexes in homotopical algebra. [*Topology*]{} [**19**]{} (1980), 427-440.
P. Gabriel, M. Zisman. [*Calculus of fractions and homotopy theory*]{}, Ergebnisse der Math. und ihrer Grenzgebiete [**35**]{}, Springer-Verlag, New York (1967).
R. Gordon, A.J. Power, R. Street. Coherence for tricategories [*Memoirs A.M.S.*]{} [**117**]{} (1995).
A. Grothendieck. [*Pursuing Stacks*]{} available from Université de Montpellier 2 or the University of Bangor.
A. Joyal, M. Tierney. Algebraic homotopy types. Occurs as an entry in the bibliography of [@BaezDolan].
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[^1]: Our notion of “reasonable realization functor” (Definition \[realizationdef\]) is any functor $\Re$ from the category of strict $n$-groupoids to $Top$, provided with a natural transformation $r$ from the set of objects of $G$ to the points of $\Re (G)$, and natural isomorphisms $\pi
_0(G)\cong \pi _0(\Re (G))$ and $\pi _i(G,x) \cong \pi _i(\Re (G), r(x))$. This axiom is fundamental to the question of whether one can realize homotopy types by strict $n$-groupoids, because one wants to read off the homotopy groups of the space from the strict $n$-groupoid. The standard realization functors satisfy this property, and the somewhat different realization construction of [@KV] is claimed there to have this property.
| ArXiv |
ArXiv |
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---
abstract: |
We consider the problem of maintaining a dynamic set of integers and answering queries of the form: report a point (equivalently, all points) in a given interval. Range searching is a natural and fundamental variant of integer search, and can be solved using predecessor search. However, for a RAM with $w$-bit words, we show how to perform updates in $O(\lg w)$ time and answer queries in $O(\lg\lg w)$ time. The update time is identical to the van Emde Boas structure, but the query time is exponentially faster. Existing lower bounds show that achieving our query time for predecessor search requires doubly-exponentially slower updates. We present some arguments supporting the conjecture that our solution is optimal.
Our solution is based on a new and interesting recursion idea which is “more extreme” that the van Emde Boas recursion. Whereas van Emde Boas uses a simple recursion (repeated halving) on each path in a trie, we use a nontrivial, van Emde Boas-like recursion on every such path. Despite this, our algorithm is quite clean when seen from the right angle. To achieve linear space for our data structure, we solve a problem which is of independent interest. We develop the first scheme for dynamic perfect hashing requiring sublinear space. This gives a dynamic Bloomier filter (an approximate storage scheme for sparse vectors) which uses low space. We strengthen previous lower bounds to show that these results are optimal.
author:
- |
Christian Worm Mortensen[^1]\
IT U. Copenhagen\
`[email protected]`
- |
Rasmus Pagh\
IT U. Copenhagen\
`[email protected]`
- |
Mihai Pǎtraşcu\
MIT\
`[email protected]`
bibliography:
- '../general.bib'
title: On Dynamic Range Reporting in One Dimension
---
Introduction
============
Our problem is to maintain a set $S$ under insertions and deletions of values, and a range reporting query. The query ${\texttt{findany}}(a,b)$ should return an arbitrary value in $S \cap [a,b]$, or report that $S
\cap [a,b] = \emptyset$. This is a form of existential range query. In fact, since we only consider update times above the predecessor bound, updates can maintain a linked list of the values in $S$ in increasing order. Given a value $x \in S \cap [a,b]$, one can traverse this list in both directions starting from $x$ and list all values in the interval $[a,b]$ in constant time per value. Thus, the ${\texttt{findany}}$ query is equivalent to one-dimensional range reporting.
The model in which we study this problem is the word RAM. We assume the elements of $S$ are integers that fit in a word, and let $w$ be the number of bits in a word (thus, the “universe size” is $u =
2^w$). We let $n = |S|$. Our data structure will use Las Vegas randomization (through hashing), and the bounds stated will hold with high probability in $n$.
Range reporting is a very natural problem, and its higher-dimensional versions have been studied for decades. In one dimension, the problem is easily solved using predecessor search. The predecessor problem has also been studied intensively, and the known bounds are now tight in almost all cases [@beame02predecessor]. Another well-studied problem related to ours is the lookup problem (usually solved by hashing), which asks to find a key in a set of values. Our problem is more general than the lookup problem, and less general than the predecessor problem. While these two problems are often dubbed “the integer search problems”, we feel range reporting is an equally natural and fundamental incarnation of this idea, and deserves similar attention.
The first to ask whether or not range reporting is as hard as finding predecessors were Miltersen et al in STOC’95 [@miltersen99asymmetric]. For the static case, they gave a data structure with space $O(nw)$ and constant query time, which cannot be achieved for the predecessor problem with polynomial space. An even more surprising result from STOC’01 is due to Alstrup, Brodal and Rauhe [@alstrup01range], who gave an optimal solution for the static case, achieving linear space and constant query time. In the dynamic case, however, no solution better than the predecessor problem was known. For this problem, the fastest known solution in terms of $w$ is the classic van Emde Boas structure [@veb77predecessor], which achieves $O(\lg w)$ time per operation.
For the range reporting problem, we show how to perform updates in $O(\lg w)$ time, while supporting queries in $O(\lg\lg w)$ time. The space usage is optimal, i.e. $O(n)$ words. The update time is identical to the one given by the van Emde Boas structure, but the query time is exponentially faster. In contrast, Beame and Fich [@beame02predecessor Theorem 3.7] show that achieving any query time that is $o(\lg w / \lg\lg w)$ for the predecessor problem requires update time $\Omega(2^{w^{1 - \epsilon}})$, which is doubly-exponentially slower than our update time. We also give an interesting tradeoff between update and query times; see theorem \[thm:range\] below.
Our solution incorporates some basic ideas from the previous solutions to static range reporting in one dimension [@miltersen99asymmetric; @alstrup01range]. However, it brings two important technical contributions. First, we develop a new and interesting recursion idea which is more advanced than van Emde Boas recursion (but, nonetheless, not technically involved). We describe this idea by first considering a simpler problem, the bit-probe complexity of the greater-than function. Then, the solution for dynamic range reporting is obtained by using the recursion for this simpler problem, on *every path* of a binary trie of depth $w$. This should be contrasted to the van Emde Boas structure, which uses a very simple recursion idea (repeated halving) on every root-to-leaf path of the trie. The van Emde Boas recursion is fundamental in the modern world of data structures, and has found many unrelated applications (e.g. exponential trees, integer sorting, cache-oblivious layouts, interpolation search trees). It will be interesting to see if our recursion scheme has a similar impact.
The second important contribution of this paper is needed to achieve linear space for our data structure. We develop a scheme for dynamic perfect hashing, which requires sublinear space. This can be used to store a sparse vector in small space, if we are only interested in obtaining correct results when querying non-null positions (the Bloomier filter problem). We also prove that our solution is optimal. To our knowledge, this solves the last important theoretical problem connected to Bloom filters. The stringent space requirements that our data structure can meet are important in data-stream algorithms and database systems. We mention one application below, but believe others exist as well.
Data-Stream Perfect Hashing and Bloomier Filters
------------------------------------------------
The Bloom filter is a classic data structure for testing membership in a set. If a constant rate of false-positives is allowed, the space *in bits* can be made essentially linear in the size of the set. Optimal bounds for this problem are obtained in [@pagh05bloom]. Bloomier filters, an extension of the classical Bloom filter with a catchy name, were defined and analyzed in the static case by Chazelle et al [@chazelle04bloom]. The problem is to represent a vector $V[0..u-1]$ with elements from $\{ 0, \dots, 2^r
- 1\}$ which is nonzero in only $n$ places (assume $n \ll u$, so the vector is sparse). Thus, we have a sparse set as before, but with values associated to the elements. The information theoretic lower bound for representing such a vector is $\Omega(n\cdot r + \lg
\binom{u}{n}) \approx \Omega(n (r + \lg u))$ bits. However, if we only want correct answers when $V[x] \ne 0$, we can obtain a space usage of roughly $O(nr)$ bits in the static case.
For the dynamic problem, where the values of $V$ can change arbitrarily at any point, achieving such low space is impossible regardless of the query and update times. Chazelle et al. [@chazelle04bloom] proved that $\Omega(n(r + \min(\lg\lg
\frac{u}{n^3}, \lg n)))$ bits are needed. No non-trivial upper bound was known. We give matching lower and upper bounds:
\[thm:bloomlb\] The randomized space complexity of maintaining a dynamic Bloomier filter for $r\geq 2$ is $\Theta(n(r + \lg\lg \frac{u}{n}))$ bits in expectation. The upper bound is achieved by a RAM data structure that allows access to elements of the vector in worst-case constant time, and supports updates in amortized expected $O(1)$ time.
To detect whether $V[x] = 0$ with probability of correctness at least $1-\epsilon$, one can use a Bloom filter on top. This requires space $\Theta(n\lg( 1/\epsilon ))$, and also works in the dynamic case [@pagh05bloom]. Note that even for $\epsilon = 1$, randomization is essential, since any deterministic solution must use $\Omega(n
\lg(u/n))$ bits of space, i.e. it must essentially store the set of nonzero entries in the vector.
With marginally more space, $O(n(r + \lg\lg u))$, we can make the space and update bounds hold with high probability. To do that, we analyze a harder problem, namely maintaining a perfect hash function dynamically using low space. The problem is to maintain a set $S$ of keys from $\{0, \dots, u-1\}$ under insertions and deletions, and be able to evaluate a perfect hash function (i.e. a one-to-one function) from $S$ to a small range. An element needs to maintain the same hash value while it is in $S$. However, if an element is deleted and subsequently reinserted, its hash value may change.
\[thm:hash\] We can maintain a perfect hash function from a set $S \subset \{ 0,
\dots, u-1 \}$ with $|S| \leq n$ to a range of size $n + o(n)$, under $n^{O(1)}$ insertions and deletions, using $O(n\lg\lg u)$ bits of space w.h.p., plus a constant number of machine words. The function can be evaluated in worst-case constant time, and updates take constant time w.h.p.
This is the first dynamic perfect hash function that uses less space than needed to store $S$ ($\lg \binom{u}{n}$ bits). Our space usage is close to optimal, since the problem is harder than dynamic Bloomier filtering. These operating conditions are typical of data-stream computation, where one needs to support a stream of updates and queries, but does not have space to hold the entire state of the data structure. Quite remarkably, our solution can achieve this goal without introducing errors (we use only Las Vegas randomization).
We mention an independent application of Theorem \[thm:hash\]. In a database we can maintain an index of a relation under insertions of tuples, using internal memory per tuple which is logarithmic in the length of the key for the tuple. If tuples have fixed length, they can be placed directly in the hash table, and need only be moved if the capacity of the hash table is exceeded.
Tradeoffs and the scheme of things {#scheme}
----------------------------------
We begin with a discussion of the greater-than problem. Consider an infinite memory of bits, initialized to zero. Our problem has two stages. In the update stage, the algorithm is given a number $a \in
[0..n-1]$. After seeing $a$, the algorithm is allowed to flip $O(T_u)$ bits in the memory. In the query stage, the algorithm is given a number $b \in [0..n-1]$. Now the algorithm may inspect $O(T_q)$ bits, and must decide whether or not $b > a$. The problem was previously studied by Fredman [@fredman82sums], who showed that $\max(T_u,
T_q) = \Omega(\lg n / \lg\lg n)$. It is quite tempting to believe that one cannot improve past the trivial upper bound $T_u = T_q = O(\lg
n)$, since, in some sense, this is the complexity of “writing down” $a$. However, as we show in this paper, Fredman’s bound is optimal, in the sense that it is a point on our tradeoff curve. We give upper and lower bounds that completely characterize the possible asymptotic tradeoffs:
\[thm:bitgt\] The bit-probe complexity of the greater-than function satisfies the tight tradeoffs:
$$\begin{aligned}
T_q \geq \lg\lg n,\ T_u \leq \lg n &:& T_u = \Theta(\lg_{T_q} n) \\
T_q \leq \lg\lg n,\ T_u \geq \lg n &:& 2^{T_q} = \Theta(\lg_{T_u} n) \\
\end{aligned}$$
As mentioned already, we use the same recursion idea as in the previous algorithm for dynamic range reporting, except that we apply this recursion to every root-to-leaf path of a binary trie of depth $w$. Quite remarkably, these structures can be made to overlap in-as-much as the paths overlap, so only one update suffices for all paths going through a node. Due to this close relation, we view the lower bounds for the greater-than function as giving an indication that our range reporting data structure is likewise optimal. In any case, the lower bounds show that markedly different ideas would be necessary to improve our solution for range reporting.
Let $T_{pred}$ be the time needed by one update and one query in the dynamic predecessor problem. The following theorem summarizes our results for dynamic range reporting:
\[thm:range\] There is a data structure for the dynamic range reporting problem, which uses $O(n)$ space and supports updates in time $O(T_u)$, and queries in time $O(T_q)$, $(\forall) T_u, T_q$ satisfying:
$$\begin{aligned}
T_q \geq \lg\lg w,\ \frac{\lg w}{\lg\lg w} \leq T_u \leq \lg w
&:& T_u = O(\lg_{T_q} w) + T_{pred} \\
T_q \leq \lg\lg w,\phantom{\ \ \frac{\lg w}{\lg\lg w} \leq}
T_u \geq \lg w &:& 2^{T_q} = O(\lg_{T_u} w) \\
\end{aligned}$$
Notice that the most appealing point of the tradeoff is the cross-over of the two curves: $T_u = O(\lg w)$ and $T_q = O(\lg\lg w)$ (and indeed, this has been the focus of our discussion). Another interesting point is at constant query time. In this case, our data structure needs $O(w^{\epsilon})$ update time. Thus, our data structure can be used as an optimal static data structure, which is constructed in time $O(n w^{\epsilon})$, improving on the construction time of $O(n \sqrt{w})$ given by Alstrup et al [@alstrup01range].
The first branch of our tradeoff is not interesting with $T_{pred} =
\Theta(\lg w)$. However, it is generally believed that one can achieve $T_{pred} = \Theta( \lg w / \lg\lg w)$, matching the optimal bound for the static case. If this is true, the $T_{pred}$ term can be ignored. In this case, we can remark a very interesting relation between our problem and the predecessor problem. When $T_u = T_q$, the bounds we achieve are identical to the ones for the predecessor problem, i.e. $T_u = T_q = O(\lg w / \lg\lg w)$. However, if we are interested in the possible tradeoffs, the gap between range reporting and the predecessor problem quickly becomes huge. The same situation appears to be true for deterministic dictionaries with linear space, though the known tradeoffs are not as general as ours. We set forth the bold conjecture (the proof of which requires many missing pieces) that all three search problems are united by an optimal time of $\Theta(\lg w /
\lg\lg w)$ in this point of their tradeoff curves.
We can achieve bounds in terms of $n$, rather than $w$, by the classic trick of using our structure for small $w$ and a fusion tree structure [@fredman93fusion] for large $w$. In particular, we can achieve $T_q = O(\lg\lg n)$ and $T_u = O\left( \frac{\lg n}{\lg\lg n}
\right)$. Compared with the optimal bound for the predecessor problem of $\Theta\left( \sqrt{\frac{\lg n}{\lg\lg n}} \right)$, our data structure improves the query time exponentially by sacrificing the update time quadratically.
Data-Stream Perfect Hashing
===========================
We denote by $S$ be the set of values that we need to hash at present time. Our data structure has the following parts:
- A hash function $\rho: \{0,\dots,u-1 \} \rightarrow
\{0,1\}^{v}$, where $v = O(\lg n)$, from a family of universal hash functions with small representations (for example, the one from [@dietzfel96universal]).
- A hash function $\phi: \{0,1\}^{v} \rightarrow \{1,\dots,r\}$, where $r=\lceil n/\lg^2 n \rceil$, taken from Siegel’s class of highly independent hash functions [@siegel04hash].
- An array of hash functions $h_1,\dots,h_r: \{0,1\}^v \rightarrow
\{0,1\}^s$, where $s=\lceil (6+2c)\lg\lg u \rceil$, chosen independently from a family of universal hash functions; $c$ is a constant specified below.
- A high performance dictionary [@dietzfel90highperf] for a subset $S'$ of the keys in $S$. The dictionary should have a capacity of $O(\lceil n/\lg u \rceil)$ keys (but might expand further). Along with the dictionary we store a linked list of length $O(\lceil n/\lg u \rceil)$, specifying certain vacant positions in the hash table.
- An array of dictionaries $D_1,\dots,D_r$, where $D_i$ is a dictionary that holds $h_i(\rho(k))$ for each key $k\in S \setminus
S'$ with $\phi(\rho(k))=i$. A unique value in $\{0,\dots,j-1\}$, where $j=(1+o(1))\lg^2 n$, is associated with each key in $D_i$. A bit vector of $j$ bits and an additional string of $\lg n$ bits is used to keep track of which associated values are in use. We will return to the exact choice of $j$ and the implementation of the dictionaries.
The main idea is that all dictionaries in the construction assign to each of their keys a unique value within a subinterval of $[1 .. m]$. Each of the dictionaries $D_1, \dots, D_r$ is responsible for an interval of size $j$, and the high performance dictionary is responsible for an interval of size $O(n/\lg u) = o(n)$.
The hash function $\rho$ is used to reduce the key length to $v$. The constant in $v = O(\lg n)$ can be chosen such that with high probability, over a polynomially bounded sequence of updates, $\rho$ will never map two elements of $S$ to the same value (the conflicts, if they occur, end up in $S'$ and are handled by the high performance dictionary).
When inserting a new value $k$, the new key is included in $S'$ if either:
- There are $j$ keys in $D_i$, where $i=\phi(\rho(k))$, or
- There exists a key $k'\in S$ where $\phi(\rho(k))=\phi(\rho(k'))=i$ and $h_i(\rho(k))=h_i(\rho(k'))$.
Otherwise $k$ is associated with the key $h_i(\rho(k))$ in $D_i$. Deletion of a key $k$ is done in $S'$ if $k\in S'$, and otherwise the associated key in the appropriate $D_i$ is deleted.
To evaluate the perfect hash function on a key $k$ we first see whether $k$ is in the high performance dictionary. If so, we return the value associated with $k$. Otherwise we compute $i=\phi(\rho(k))$ and look up the value $\Delta$ associated with the key $h_i(\rho(k))$ in $D_i$. Then we return $(i-1)j+\Delta$, i.e., position $\Delta$ within the $i$-th interval.
Since $D_1,\dots,D_r$ store keys and associated values of $O(\lg\lg
u)$ bits, they can be efficiently implemented as constant depth search trees of degree $w^{\Omega(1)}$, where each internal node resides in a single machine word. This yields constant time for dictionary insertions and lookups, with an optimal space usage of $O(\lg^2
n\lg\lg u)$ bits for each dictionary. We do not go into details of the implementation as they are standard; refer to [@hagerup98ram] for explanation of the required word-level parallelism techniques.
What remains to describe is how the dictionaries keep track of vacant positions in the hash table in constant time per insertion and deletion. The high performance dictionary simply keeps a linked list of all vacant positions in its interval. Each of $D_1,\dots,D_r$ maintain a bit vector indicating vacant positions, and additional $O(\lg n)$ summary bits, each taking the or of an interval of size $O(\lg n)$. This can be maintained in constant time per operation, employing standard techniques.
Only $o(n)$ preprocessing is necessary for the data structure (essentially to build tables needed for the word-level parallelism). The major part of the data structure is initialized lazily.
Analysis
--------
Since evaluation of all involved hash functions and lookup in the dictionaries takes constant time, evaluation of the perfect hash function is done in constant time. As we will see below, the high performance dictionary is empty with high probability unless $n/\lg u
> \sqrt{n}$. This means that it always uses constant time per update with high probability in $n$. All other operations done for update are easily seen to require constant time w.h.p.
We now consider the space usage of our scheme. The function $\rho$ can be represented in $O(w)$ bits. Siegel’s highly independent hash function uses $o(n)$ bits of space. The hash functions $h_1,\dots,h_r$ use $O(\lg n + \lg\lg u)$ bits each, and $o(n\lg\lg u)$ bits in total. The main space bottleneck is the space for $D_1,\dots,D_r$, which sums to $O(n\lg\lg u)$.
Finally, we show that the space used by the high performance dictionary is $O(n)$ bits w.h.p. This is done by showing that each of the following hold with high probability throughout a polynomial sequence of operations:
The function $\rho$ is one-to-one on $S$.
There is no $i$ such that $S_i = \{ k \in S \mid
\phi(\rho(k))=i \}$ has more than $j$ elements.
The set $S'$ has $O(\lceil n/\lg u \rceil)$ elements.
That 1. holds with high probability is well known. To show 2. we use the fact that, with high probability, Siegel’s hash function is independent on every set of $n^{\Omega(1)}$ keys. We may thus employ Chernoff bounds for random variables with limited independence to bound the probability that any $i$ has $|S_i| > j$, conditioned on the fact that 1. holds. Specifically, we can use [@schmidt95chernoff Theorem 5.I.b] to argue that for any $l$, the probability that $|S_{i}| > j$ for $j = \lceil \lg^2 n + \lg^{5/3} n \rceil$ is $n^{-\omega(1)}$, which is negligible. On the assumption that 1. and 2. hold, we finally consider 3. We note that every key $k'\in S'$ is involved in an $h_i$-collision in $S_i$ for $i=\phi(\rho(k'))$, i.e. there exists $k''\in S_i \setminus \{k'\}$ where $h_i(k')=h_i(k'')$. By universality, for any $i$ the expected number of $h_i$-collisions in $S_i$ is $O(\lg^4 n / (\lg u)^{6+2c}) = O((\lg
u)^{-(2+2c)})$. Thus the probability of one or more collisions is $O((\lg u)^{-(2+2c)})$. For $\lg u \geq \sqrt{n}$ this means that there are no keys in $S'$ with high probability. Specifically, $c$ may be chosen as the sum of the constants in the exponents of the length of the operation sequence and the desired high probability bound. For the case $\lg u < \sqrt{n}$ we note that the expected number of elements in $S'$ is certainly $O(n/\lg u)$. To see than this also holds with high probability, note that the event that one or more keys from $S_i$ end up in $S'$ is independent among the $i$’s. Thus we can use Chernoff bounds to get that the deviation from the expectation is small with high probability.
Lower Bound for Bloomier Filters
================================
For the purpose of the lower bound, we consider the following two-set distinction problem, following [@chazelle04bloom]. The problem has the following stages:
1. a random string $R$ is drawn, which will be available to the data structure throughout its operation. This is equivalent to drawing a deterministic algorithm from a given distribution, and is more general than assuming each stage has its own random coins (we are giving the data structure free storage for its random bits).
2. the data structure is given $A \subset [u], |A| \le n$. It must produce a representation $f_R(A)$, which for any $A$ has size at most $S$ bits, in expectation over all choices of $R$. Here $S$ is a function of $n$ and $u$, which is the target of our lower bound.
3. the data structure is given $B \subset [u]$, such that $|B| \le
n, A \cap B = \emptyset$. Based on the old state $f_R(A)$, it must produce $g_R(B, f_R(A))$ with expected size at most $S$ bits.
4. the data structure is given $x \in [u]$ and its previously generated state, i.e. $f_R(A)$ and $g_R(B, f_R(A))$. Now it must answer as follows with no error allowed: if $x \in A$, it must answer “A”; if $x \in B$, it must answer “B”; if $x \notin A
\cup B$, it can answer either “A” or “B”. Let $h_R(x,f,g)$ be the answer computed by the data structure, when the previous state is $f$ and $g$.
It is easy to see that a solution for dynamic Bloomier filters supporting ternary associated data, using expected space $o(n\lg\lg
\frac{u}{n})$, yields a solution to the two-set distinction problem with $S = o(n\lg\lg \frac{u}{n})$. We will prove such a solution does not exist.
Since a solution to the distinction problem is not allowed to make an error we can assume w.l.o.g. that step 3 is implemented as follows. If there exist appropriate $A, B \subset [u]$, with $x \in A$ such that $f_R(A) = f_0$ and $g_R(B, f_0) = g_0$, then $h_R(x, f_0, g_0)$ must be “A”. Similarly, if there exists a plausible scenario with $x \in
B$, the answer must be “B”. Otherwise, the answer can be arbitrary.
Assume that the inputs $A \times B$ are drawn from a given distribution. We argue that if the expected sizes of $f$ and $g$ are allowed to be at most $2S$, the data structure need not be randomized. This uses a bicriteria minimax principle. We have $E_{R,A,B}\left[ \frac{|f|}{S} + \frac{|g|}{S} \right] \leq 2$, where $|f|, |g|$ denote the length of the representations. Then, there exists a random string $R_0$ such that $E_{A,B} \left[ \frac{|f|}{S} +
\frac{|g|}{S} \right] \leq 2$. Since $|f|, |g| \geq 0$, this implies $E_{A,B}[|f|] \leq 2S, E_{A,B}[|g|] \leq S$. The data structure can simply use the deterministic sequence $R_0$ as its random bits; we drop the subscript from $f_R, g_R$ when talking about this deterministic data structure.
Lower Bound for Two-Set Distinction
-----------------------------------
Assume $u = \omega(n)$, since a lower bound of $\Omega(n)$ is trivial for universe $u \ge 2n$. Break the universe into $n$ equal parts $U_1,
\dots, U_n$; w.l.o.g. assume $n$ divides $u$, so $|U_i| =
\frac{n}{u}$. The hard input distribution chooses $A$ uniformly at random from $U_1 \times \dots \times U_n$. We write $A = \{ a_1,
\dots, a_n \}$, where $a_i$ is a random variable drawn from $U_i$. Then, $B'$ is chosen uniformly at random from the same product space; again $B' = \{b_1, \dots, b_n\}, b_i \gets U_i$. We let $B = B'
\setminus A$. Note that $E[|B|] = n \cdot \Pr[A_1 \ne B_1] = (1 -
\frac{n}{u}) \cdot n = (1 - o(1)) \cdot n$.
Let $A_i^p$ be the plausible values of $A_i$ after we see $f(A)$; that is, $A_i^p$ contains all $a \in U_i$ for which there exists a valid $A'$ with $a \in A'$ and $f(A') = f(A)$. Intuitively speaking, if $f(A)$ has expected size $o(n \lg\lg \frac{u}{n})$, it contains on average $o(\lg\lg \frac{u}{n})$ bits of information about each $a_i$. This is much smaller than the range of $a_i$, which is $\frac{u}{n}$, so we would expect that the average $|A_i^p|$ is quite large, around $\frac{u}{n} / (\lg \frac{u}{n})^{o(1)}$. This intuition is formalized in the following lemma:
With probability at least a half over a uniform choice of $A$ and $i$, we have $|A_i^p| \geq \frac{u/n}{2^{O(S/n)}}$.
The Kolmogorov complexity of $A$ is $n\lg \frac{u}{n} - O(1)$; no encoding for $A$ can have an expected size less than this quantity. We propose an encoding for $A$ consisting of two parts: first, we include $f(A)$; second, for each $i$ we include the index of $a_i$ in $|A_i^p|$, using $\lceil \lg|A_i^p| \rceil$ bits. This is easily decodable. We first generate all possible $A'$ with $f(A') = f(A)$, and thus obtain the sets $A_i^p$. Then, we extract from each plausible set the element with the given index. The expected size of the encoding is $2S + \sum_i E_{A}[\lg |A_i^p|] + O(n)$, which must be $\ge n\lg \frac{u}{n} - O(1)$. This implies $\lg \frac{u}{n} -
E_{i,A}[\lg |A_i^p|] \le \frac{2S}{n} + O(1)$. By Markov’s inequality, with probability at least a half over $i$ and $A$, $\lg \frac{u}{n} -
\lg |A_i^p| \le \frac{4S}{n} + O(1)$, so $\lg |A_i^p| \ge \lg
\frac{u}{n} - O(\frac{S}{n})$.
We now make a crucial observation which justifies our interest in $A_i^p$. Assume that $b_i \in A_i^p$. In this case, the data structure must be able to determine $b_i$ from $f(A)$ and $g(B,f(A))$. Indeed, suppose we compute $h(x,f,g)$ for all $x \in |A_i^p|$. If that data strucuture does not answer “B” when $x = b_i$, it is obviously incorrent. On the other hand, if it answers “B” for both $x = b_i$ and some other $x' \in A_i^p$, it also makes an error. Since $x'$ is plausible, there exist $A'$ with $x' \in A'$ such that $f(A') =
f(A)$. Then, we can run the data structure with $A'$ as the first set and $B$ as the second set. Since $f(A') = f(A)$, the data structure will behave exactly the same, and will incorrectly answer “B” for $x'$.
To draw our conclusion, we consider another encoding argument, this time in connection to the set $B'$. The Kolmogorov complexity of $B'$ is $n \lg \frac{u}{n} - O(1)$. Consider a randomized encoding, depending on a set $A$ drawn at random. First, we encode an $n$-bit vector specifying which indices $i$ have $a_i = b_i$. It remains to encode $B' \setminus A = B$. We encode another $n$-bit vector, specifying for which positions $i$ we have $b_i \in A_i^p$. For each $b_i \notin A_i^p$, we simply encode $B_i$ using $\lceil \lg
\frac{u}{n} \rceil$ bits. Finally, we include in the encoding $g(B,
f(A))$. As explained already, this is enough to recover all $b_i$ which are in $A_i^p$. Note that we do not need to encode $f(A)$, since this depends only on our random coins, and the decoding algorithm can reconstruct it.
The expected size of this encoding will be $O(n + S) + n\cdot
\Pr_{A,B',i} [b_i \notin A_i^p] \cdot \lg \frac{u}{n}$. We know that with probability a half over $A$ and $i$, we have $|A_i^p| \geq
\frac{u/n}{2^{O(S/n)}}$. Thus, $\Pr_{A,B',i} [b_i \in A_i^p] \geq
\frac{1}{2} \cdot 2^{-O(S/n)}$. Thus, the expected size of the encoding is at most $O(n + S) + (1 - 2^{-O(S/n)}) \cdot n \lg
\frac{u}{n}$. Note that by the minimax principle, randomness in the encoding is unessential and we can always fix $A$ guaranteeing the same encoding size, in expectation over $B$. We now get the bound:
$$\begin{aligned}
& & O(n + S) + (1 - 2^{-O(S/n)}) \cdot n \lg \frac{u}{n} \geq n \lg
\frac{u}{n} - O(1) \\
& \Rightarrow & O\left( \frac{S}{n} \right) \geq 2^{-O(S/n)} \lg
\frac{u}{n} - O(1) \Rightarrow 2^{O(S/n)} O(S / n) \geq \lg
\frac{u}{n} \Rightarrow \frac{S}{n} = \Omega \left( \lg\lg \frac{u}{n}
\right)\end{aligned}$$
A Space-Optimal Bloomier Filter
===============================
It was shown in [@carter78bloom] that the approximate membership problem (i.e., the problem solved by Bloom filters) can be solved optimally using a reduction to the exact membership problem. The reduction uses universal hashing. In this section we extend this idea to achieve optimal dynamic Bloomier filters.
Recall that Bloomier filters encode sparse vectors with entries from $\{0,\dots,2^r - 1\}$. Let $S\subseteq [u]$ be the set of at most $n$ indexes of nonzero entries in the vector $V$. The data structure must encode a vector $V'$ that agrees with $V$ on indexes in $S$, and such that for any $x\not\in S$, $\Pr[V'[x]\neq 0]\leq \epsilon$, where $\epsilon > 0$ is the error probability of the Bloomier filter. Updates to $V$ are done using the following operations:
- [Insert($x$, $a$)]{}. Set $V[x]:=a$, where $a\neq 0$.
- [Delete($x$)]{}. Set $V[x]:=0$.
The data structure assumes that only valid updates are performed, i.e. that inserts are done only in situations where $V[x]=0$ and deletions are done only when $V[x]\neq 0$.
\[thm:filter\] Let positive integers $n$ and $r$, and $\epsilon > 0$ be given. On a RAM with word length $w$ we can maintain a Bloomier filter $V'$ for a vector $V$ of length $u=2^{O(w)}$ with at most $n$ nonzero entries from $\{0,\dots,2^r - 1\}$, such that:
- [Insert]{} and [Delete]{} can be done in amortized expected constant time. The data structure assumes all updates are valid.
- Computing $V'[x]$ on input $x$ takes worst case constant time. If $V[x]\neq 0$ the answer is always ’V\[x\]’. If $V[x]=0$ the answer is ’0’ with probability at least $1-\epsilon$.
- The expected space usage is $O(n(\lg\lg(u/n) + \lg(1/\epsilon) +
r))$ bits.
The Data Structure
------------------
Assume without loss of generality that $u\geq 2n$ and that $\epsilon\geq u/n$. Let $v=\max(n \log(u/n), n/\epsilon)$, and choose $h: \{0,\dots,u-1\} \rightarrow \{0,\dots,v-1\}$ as a random function from a universal class of hash functions. The data structure maintains information about a minimal set $S'$ such that $h$ is 1-1 on $S
\setminus S'$. Specifically, it consists of two parts:
1. A dictionary for the set $S'$, with corresponding values of $V$ as associated information.
2. A dictionary for the set $h(S\backslash S')$, where the element $h(x)$, $x\in S\backslash S'$, has $V[x]$ as associated information.
Both dictionaries should succinct, i.e., use space close to the information theoretic lower bound. Raman and Rao [@raman03succinct] have described such a dictionary using space that is $1+o(1)$ times the minimum possible, while supporting lookups in $O(1)$ time and updates in expected amortized $O(1)$ time.
To compute $V'[x]$ we first check whether $x\in S'$, in which case $V'[x]$ is stored in the first dictionary. If this is not the case, we check whether $h(x)\in h(S\backslash S')$. If this is the case we return the information associated with $h(x)$ in the second dictionary. Otherwise, we return ’0’.
[Insert($x$, $a$)]{}. First determine whether $h(x)\in h(S\backslash
S')$, in which case we add $x$ to the set $S'$, inserting $x$ in the first dictionary. Otherwise we add $h(x)$ to the second dictionary. In both cases, we associate $a$ with the inserted element.
[Delete($x$)]{} proceeds by deleting $x$ from the first dictionary if $x\in S'$, and otherwise deleting $h(x)$ from the second dictionary.
Analysis
--------
It is easy to see that the data structure always return correct function values on elements in $S$, given that all updates are valid. When computing $V'[x]$ for $x\not\in S$ we get a nonzero result if and only if there exists $x'\in S$ such that $h(x)=h(x')$. Since $h$ was chosen from a universal family, this happens with probability at most $n/v \leq \epsilon$.
It remains to analyze the space usage. Using once again that $h$ was chosen from a universal family, the expected size of $S'$ is $O(n/\log(u/n))$. This implies that the expected number of bits necessary to store the set $S'$ is $\log\binom{u}{O(n/\log(u/n))} =
O(n)$, using convexity of the function $x\mapsto \binom{u}{x}$ in the interval $0\dots u/2$. In particular, the first dictionary achieves an expected space usage of $O(n)$ bits. The information theoretical minimum space for the set $h(S\backslash S')$ is bounded by $\log\binom{r}{n} = O(n \log(r/n)) = O(n \log\log(u/n) +
n\log(1/\epsilon))$ bits, matching the lower bound. We disregarded is the space for the universal hash function, which is $O(\log u)$ bits. However, this can be reduced to $O(\log n + \log\log u)$ bits, which is vanishing, by using slightly weaker universal functions and doubling the size $r$ of the range. Specifically, $2$-universal functions suffice; see [@pagh00dispers] for a construction. Using such a family requires preprocessing time $(\log u)^{O(1)}$, expected.
Upper Bounds for the Greater-Than Problem
=========================================
We start with a simple upper bound of $T_u = O(\lg n), T_q = O(\lg\lg
n)$. Our upper bound uses a trie structure. We consider a balanced tree with branching factor 2, and with $n$ leaves. Every possible value of the update parameter $a$ is represented by a root-to-leaf path. In the update stage, we mark this root-to-leaf path, taking time $O(\lg n)$. In the query stage, we want to find the point where $b$’s path in the trie would diverge from $a$’s path. This uses binary search on the $\lg n$ levels, as follows. To test if the paths diverge on a level, we examine the node on that level on $b$’s path. If the node is marked, the paths diverge below; otherwise they diverge above. Once we have found the divergence point, we know that the larger of $a$ and $b$ is the one following the right child of the lowest common ancestor.
For the full tradeoff, we consider a balanced tree with branching factor $B$. In the update stage, we need to mark a root-to-leaf path, taking time $\lg_B n$. In the query stage, we first find the point where $b$’s path in the trie would diverge from $a$’s path. This uses binary search on the $\lg_B n$ levels, so it takes time $O(\lg\lg_B
n)$. Now we know the level where the paths of $a$ and $b$ diverge. The nodes on that level from the paths of $a$ and $b$ must be siblings in the tree. To test whether $b > a$, we must find the relative order of the two sibling nodes. There are two strategies for this, giving the two branches of the tradeoff curve. To achieve small update time, we can do all work at query time. We simply test all siblings to the left of $b$’s path on the level of divergence. If we find a marked one, then $a$’s path goes to the left of $b$’s path, so $a < b$; otherwise $a > b$. This stragegy gives $T_u = O(\lg_B n)$ and $T_q = O(\lg(\lg_B
n) + B)$, for any $B \geq 2$. For $T_q > \Omega(\lg\lg n)$, we have $T_q = \Theta(B)$, so we have achieved the tradeoff $T_u = O(\lg_{T_q}
n)$.
The second strategy is to do all work at update time. For every node on $a$’s path, we mark all left siblings of the node as such. Then to determine if $b$’s path is to the left or to the right of $a$’s path, we can simply query the node on $b$’s path just below the divergence point, and see if it is marked as a left sibling. This strategy gives $T_u = O(B \lg_B n)$ and $T_q = O(\lg(\lg_B n))$. For small enough $B$ (say $B = O(\lg n)$), this strategy gives $T_q = O(\lg\lg n)$ regardless of $B$ and $T_u$. For $B = \Omega(\lg n)$, we have $\lg B =
\Theta(\lg T_u)$. Therefore, we can express our tradeoff as: $2^{T_q}
= O(\lg_{T_u} n)$.
Dynamic Range Reporting
=======================
We begin with the case $T_u = O(\lg w), T_q = O(\lg\lg w)$. Let $S$ be the current set of values stored by the data structure. Without loss of generality, assume $w$ is a power of two. For an arbitrary $t \in
[0, \lg w]$, we define the trie of order $t$, denoted $T_t$, to be the trie of depth $w / 2^t$ and alphabet of $2^t$ *bits*, which represents all numbers in $S$. We call $T_0$ the *primary trie* (this is the classic binary trie with elements from $S$). Observe that we can assign distinct names of $O(w)$ bits to all nodes in all tries. We call *active paths* the paths in the tries which correspond to elements of $S$. A node $v$ from $T_t$ corresponds to a subtree of depth $2^t$ in the primary trie; we denote the root of this subtree by $r_0(v)$. A node from $T_t$ corresponds to a 2-level subtree in $T_{t-1}$; we call such a subtree a *natural subtree*. Alternatively, a 2-level subtree of any trie is natural iff its root is at an even depth.
A root-to-leaf path in the primary trie is seen as the leaves of the tree used for the greater-than problem. The paths from the primary trie are broken into chunks of length $2^t$ in the trie of order $t$. So $T_t$ is similar to the $t$-th level (counted bottom-up) of the greater-than tree. Indeed, every node on the $t$-th level of that tree held information about a subtree with $2^t$ leaves; here one edge in $T_t$ summarizes a segment of length $2^t$ bits. Also, a natural subtree corresponds to two siblings in the greater-than structure. On the next level, the two siblings are contracted into a node; in the trie of higher order, a natural subtree is also contracted into a node. It will be very useful for the reader to hold these parallels in mind, and realize that the data structure from this section is implementing the old recursion idea *on every path*.
The root-to-leaf paths corresponding to the values in $S$ determine at most $n-1$ branching nodes in any trie. By convention, we always consider roots to be branching nodes. For every branching node from $T_0$, we consider the extreme points of the interval spanned by the node’s subtree. By doubling the universe size, we can assume these are never elements of $S$ (alternatively, such extreme points are formal rationals like $x + \frac{1}{2}$). We define $\overline{S}$ to be the union of $S$ and the two special values for each branching node in the primary trie; observe that $|\overline{S}| = O(n)$. We are interested in holding $\overline{S}$ for navigation purposes: it gives a way to find in constant time the maximum and minimum element from $S$ that fits under a branching node (because these two values should be the elements from $S$ closest to the special values for the branching node).
Our data structure has the following components:
- a linked list with all elements of $S$ in increasing order, and a predecessor structure for $S$.
- a linked list with all elements of $\overline{S}$ in increasing order, accompanied by a navigation structure which enables us to find in constant time the largest value from $S$ smaller than a given value from $\overline{S} \setminus S$. We also hold a predecessor structure for $\overline{S}$.
- every branching node from the primary trie holds pointers to its lowest branching ancestor, and the two branching descendants (the highest branching nodes from the left and right subtrees; we consider leaves associated with elements from $S$ as branching descendants). We also hold pointers to the two extreme values associated with the node in the list in item 2. Finally, we hold a hash table with these branching nodes.
- for each $t$, and every node $v$ in $T_t$, which is either a branching node or a child of a branching node on an active path, we hold the depth of the lowest branching ancestor of $r_0(v)$, using a Bloomier filter.
We begin by showing that this data structure takes linear space. Items 1-3 handle $O(n)$ elements, and have constant overhead per element. We show below that the navigation structure from 2. can be implemented in linear space. The predecessor structure should also use linear space; for van Emde Boas, this can be achieved through hashing [@willard83predecessor].
In item 4., there are $O(n)$ branching nodes per trie. In addition, there are $O(n)$ children of branching nodes which are on active paths. Thus, we consider $O(n\lg w)$ nodes in total, and hold $O(\lg
w)$ bits of information for each (a depth). Using our solution for the Bloomier filter, this takes $O(n(\lg w)^2 + w)$ bits, which is $o(n)$ words. Note that storing the depth of the branching ancestor is just a trick to reduce space. Once we have a node in $T_0$ and we know the depth of its branching ancestor, we can calculate the ancestor in $O(1)$ time (just ignore the bits below the depth of the ancestor). So in essence these are “compressed pointers” to the ancestors.
We now sketch the navigation structure from item 2. Observe that the longest run in the list of elements from $\overline{S} \setminus S$ can have length at most $2w$. Indeed, the leftmost and rightmost extreme values for the branching nodes form a parenthesis structure; the maximum depth is $w$, corresponding to the maximum depth in the trie. Between an open and a closed parenthesis, there must be at least one element from $S$, so the longest uninterrupted sequence of parenthesis can be $w$ closed parenthesis and $w$ open parenthesis.
The implementation of the navigation structure uses classic ideas. We bucket $\Theta(\sqrt{w})$ consecutive elements from the list, and then we bucket $\Theta(\sqrt{w})$ buckets. Each bucket holds a summary word, with a bit for each element indicating whether it is in $S$ or not; second-order buckets hold bits saying whether first order buckets contain at least one element from $S$ or not. There is also an array with pointers to the elements or first order buckets. By shifting, we can always insert another summary bit in constant time when something is added. However, we cannot insert something in the array in constant time; to fix that, we insert elements in the array on the next available position, and hold the correct permutation packed in a word (using $O(\sqrt{w} \lg w)$ bits). To find an element from $S$, we need to walk $O(1)$ buckets. The time is $O(1)$ per traversed bucket, since we can use the classic constant-time subroutine for finding the most significant bit [@fredman93fusion].
We also describe a useful subroutine, ${\texttt{test-branching}}(v)$, which tests whether a node $v$ from some $T_t$ is a branching node. To do that, we query the structure in item 4. to find the lowest branching ancestor of $r_0(v)$. This value is defined if $v$ is a branching node, but the Bloomier filter may return an arbitrary result otherwise. We look up the purported ancestor in the structure of item 3. If the node is not a branching node, the value in the Bloom filter for $v$ was bogus, so $v$ is not a branching node. Otherwise, we inspect the two branching descendants of this node. If $v$ is a branching node, one of these two descendants must be mapped to $v$ in the trie of order $t$, which can be tested easily.
Implementation of Updates
-------------------------
We only discuss insertions; deletions follow parallel steps uneventfully. We first insert the new element in $S$ and $\overline{S}$ using the predecessor structures. Inserting a new element creates exactly one branching node $v$ in the primary trie. This node can be determined by examining the predecessor and successor in $S$. Indeed, the lowest common ancestor in the primary trie can be determined by taking an xor of the two values, finding the most significant bit, and them masking everything below that bit from the original values [@alstrup01range].
We calculate the extreme values for the new branching node $v$, and insert them in $\overline{S}$ using the predecessor structure. Finding the branching ancestor of $v$ is equivalent to finding the enclosing parentheses for the pair of parentheses which was just inserted. But $\overline{S}$ has a special structure: a pair of parentheses always encloses two subexpressions, which are either values from $S$, or a parenthesized expression (i.e., the branching nodes from $T_0$ form a binary tree structure). So one of the enclosing parentheses is either immediately to the left, or immediately to the right of the new pair. We can traverse a link from there to find the branching ancestor. Once we have this ancestor, it is easy to update the local structure of the branching nodes from item 3. Until now, the time is dominated by the predecessor structure.
It remains to update the structure in item 4. For each $t > 0$, we can either create a new branching node in $T_t$, or the branching node existed already (this is possible for $t > 0$ because nodes have many children). We first test whether the branching node existed or not (using the ${\texttt{test-branching}}$ subroutine). If we need to introduce a branching node, we simply add a new new entry in the Bloomier filter with the depth of the branching ancestor of $v$. It remains to consider active children of branching nodes, for which we must store the depth of $v$. If we have just introduced a branching node, it has exactly two active children (if there exist more than two children on active paths, the node was a branching node before). These children are determined by looking at the branching descendants of $v$; these give the two active paths going into $v$. Both descendants are mapped to active children of the new branching node from $T_t$. If the branching node already existed, we must add one active child, which is simply the child that the path to the newly inserted value follows. Thus, to update item 4., we spend constant time per $T_t$. In total, the running time of an update is $T_{pred} + O(\lg w) = O(\lg
w)$.
Implementation of Queries
-------------------------
Remember that a query receives an interval $[a,b]$ and must return a value in $S \cap [a,b]$, if one exists. We begin by finding the node $v$ which is the lowest common ancestor of $a$ and $b$ in the primary trie; this takes constant time [@alstrup01range]. Note that $v$ spans an interval which includes $[a,b]$. The easiest case is when $v$ is a branching node; this can be recognized by a lookup in the hash table from item 3. If so, we find the two branching descendants of $v$; call the left one $v_L$ and the right one $v_R$. Then, if $S \cap
[a,b] \ne \emptyset$, either the rightmost value from $S$ that fits under $v_L$ or the leftmost value from $S$ that in fits under $v_R$ must be in the interval $[a,b]$. This is so because $[a,b]$ straddles the middle point of the interval spanned by $v$. The two values mentioned above are the two values from $S$ closest (on both sides) to this middle point, so if $[a,b]$ is non-empty, it must contain one of these two. To find these two values, we follow a pointer from $v_L$ to its left extreme point in $\overline{S}$. Then, we use the navigation structure from item 2., and find the predecessor from $S$ of this value in constant time. The rightmost value under $v_R$ is the next element from $S$. Altogether, the case when $v$ is a branching node takes constant time.
Now we must handle the case when $v$ is not a branching node. If $S
\cap [a,b] \ne \emptyset$, it must be the case that $v$ is on an active path. Below we describe how to find the lowest branching ancestor of $v$, *assuming that $v$ is on an active path*. If this assumption is violated, the value returned can be arbitrary. Once we have the branching ancestor of $v$, we find the branching descendant $w$ which is in $v$’s subtree. Now it is easy to see, by the same reasoning as above, that if $[a,b] \cap S \ne \emptyset$ either the leftmost or the rightmost value from $S$ which is under $w$ must be in $[a,b]$. These two values are found in constant time using the navigation structure from item 2., as described above. So if $[a,b] \cap S \ne \emptyset$, we can find an element inside $[a,b]$. If none of these two elements were in $[a,b]$ it must be the case that $[a,b]$ was empty, because the algorithm works correctly when $[a,b]
\cap S \ne \emptyset$.
It remains to show how to find $v$’s branching ancestor, assuming $v$ is on an active path, but is not a branching node. If for some $t >
0$, $v$ is mapped to a branching node in $T_t$, it will also be mapped to a branching node in tries of higher order. We are interested in the smallest $t$ for which this happens. We find this $t$ by binary search, taking time $O(\lg\lg w)$. For some proposed $t$, we check whether the node to which $v$ is mapped in $T_t$ is a branching node (using the ${\texttt{test-branching}}$ subroutine). If it is, we continue searching below; otherwise, we continue above.
Suppose we found the smallest $t$ for which $v$ is mapped to a branching node. In $T_{t-1}$, $v$ is mapped to some $z$ which is *not* a branching node. Finding the lowest branching ancestor of $v$ is identical to finding the lowest branching ancestor of $r_0(z)$ in the primary trie (since $z$ is a not a branching node, there is no branching node in the primary trie in the subtree corresponding to $z$). Since in $T_t$ $z$ gets mapped to a branching node, its natural subtree in $T_{t-1}$ must contain at least one branching node. We have two cases: either $z$ is the root or a leaf of the natural subtree (remember that a natural subtree has two levels). These can be distinguished based on the parity of $z$’s depth. If $z$ is a leaf, the root must be a branching node (because there is at least another active leaf). But then $z$ is an active child of a branching node, so item 4. tells us the branching ancestor of $r_0(z)$. Now consider the case when $z$ is the root of the natural subtree. Then $z$ is above any branching node in its natural subtree, so to find the branching ancestor of $r_0(z)$ we can find the branching ancestor of the node from $T_t$ to which the natural subtree is mapped. But this is a branching node, so the structure in item 4. gives the desired branching ancestor. To summarize, the only super-constant cost is the binary search for $t$, which takes $O(\lg\lg w)$ time.
Tradeoffs from Dynamic Range Reporting
======================================
Fix a value $B \in [2,\sqrt{w}]$; varying $B$ will give our tradeoff curve. For an arbitrary $t \in [0, \lg_B w]$, we define the trie of order $t$ to be the trie of depth $w / B^t$ and alphabet of $B^t$ bits, which represents all numbers in $S$. We call the trie for $t =
0$ the primary trie. A node $v$ in a trie of order $t$ is represented by a subtree of depth $B^t$ in the primary trie; we say that the root of this subtree “corresponds to” the node $v$. A node from a trie of order $t$ is represented by a subtree of depth $B$ in the trie of order $t-1$; we call such a subtree a “natural depth-$B$ subtree”. Alternatively, a depth-$B$ subtree is natural if it starts at a depth divisible by $B$.
The root-to-leaf paths from the primary trie are boken into chunks of length $B^t$ in the trie of order $t$. A trie of order $t$ is similar to the $t$-th level (counted bottom-up) of the tree used for the greater-than problem, since a path in the primary trie is seen as the leaves of that tree. Indeed, every node on the $t$-th level of that tree held information about a subtree with $B^t$ leaves; here one edge in a trie of order $t$ summarizes a segment of length $B^t$ bits. Also, a natural depth-$B$ subtree corresponds to $B$ siblings in the old structure. On the next level, the $B$ siblings are contracted into a node; in the trie of higher order, a natural depth-$B$ subtree is also contracted into a node.
Our data structure has the following new components:
- choose this for the first branch of the tradeoff (faster updates, slower queries): hold the same information as in 4. for each $t$, and every node $v$ in the trie of order $t$ which is not a branching node, is on an active path, and is the child of a branching node in the trie of order $t$.
- choose this for the second branch of the tradeoff: hold the same information as above for each $t$, and every node $v$ which is not a branching node, is on an active path, and has a branching ancestor in the same natural depth-$B$ subtree.
In item 5A., notice that for every $t$ there are at most $2n - 2$ children of branching nodes which are on active paths. We store $O(\lg
w)$ bits for each, and there are $O(\lg_B w)$ values of $t$, so we can store this in a Bloomier filter with $o(n)$ words of space. In item 5B., the number of interesting nodes blows up by at most $B$ compared to 5A., and since $B \leq \sqrt{w}$, we are still using $o(n)$ words of space.
#### Updates.
For each $t > 0$, we can either create a new branching node in the trie of order $t$, or the branching node existed already. We first test whether the branching node existed or not. If we just introduced a branching node, it has at most two children which are not branching nodes and are on active paths (if more than two such children exist, the node was a branching node before). If the branching node was old, we might have added one such child. These children are determined by looking at the branching descendents of $v$ (these give the two active paths going into $v$, one or both of which are new active paths going into the node in the subtrie of order $t$). For such children, we add the depth of $v$ in the structure from item 5A. If we are in case 5B, we follow both paths either until we find a branching node, or the border of the natural depth-$B$ subtree. For of these $O(B)$ positions, we add the depth of $v$ in item $5B$. To summarize, the running time is $O(T_{pred} + \lg_B w)$ if we need to update 5A., and $O(T_{pred} + B \lg_B w)$ is we need to update 5B.
#### Queries.
We need to show how to find $v$’s branching ancestor, assuming $v$ is on an active path, but is not a branching node. For some $t > 0$, and all $t$’s above that value, $v$ will be mapped in the trie of order $t$ to some branching node. That is the smallest $t$ such that the depth-$B^t$ natural subtree containing $v$ contains some branching node. We find this $t$ by binary search, taking time $O(\lg(\lg_B
w))$. For some proposed $t$, we check if the node to which $v$ is mapped is a branching node in the trie of order $t$ (using the subroutine described above). If it is, we continue searching below; otherwise, we continue above.
Say we found the smallest $t$ for which $v$ is mapped to a branching node. In the trie of order $t-1$, $v$ is mapped to some $w$ which is not a branching node. Finding the lowest branching ancestor of $v$ is identical to finding the lowest branching ancestor of the node corresponding to $w$ in the primary trie (since $w$ is a not a branching node, there is no branching node in the primary trie in the subtree represented by $w$). In the trie of order $t$, $w$ gets mapped to a branching node, so the natural depth-$B$ subtree of $w$ contains at least one branching node. The either: (1) there is some branching node above $w$ in its natural depth-$B$ subtree, or (2) $w$ is on the active path going to the root of this natural subtree (it is above any branching node).
We first deal with case (2). If $w$ is above any branching node in its natural subtree, to find $w$’s branching ancestor we can find the branching ancestor of the node from the trie of order $t$, to which this subtree is mapped. But this is a branching node, so the structure in item 4. gives the branching ancestor $z$. We can test that we are indeed in case (2), and not case (1), by looking at the two branching descendents of $z$, and checking that one of them is strictly under $v$.
Now we deal with case (1). If we have the structure 5B., this is trivial. Because $w$ is on an active path and has a branching ancestor in its natural depth-$B$ subtree, it records the depth of the branching ancestor of the node corresponding to $w$ in the primary trie. So in this case, the only super-constant cost is the binary search for $t$, which is $O(\lg(\lg_B w))$. If we only have the structure 5A., we need to walk up the trie of order $t-1$ starting from $w$. When we reach the child of the branching node above $w$, the branching node from the primary trie is recorded in item 5A. Since the branching node is in the same natural depth-$B$ subtree as $w$, we reach this point after $O(B)$ steps. One last detail is that we do not actually know when we have reached the child of a branching node (because the Bloomier filter from item 5A. can return arbitrary results for nodes not satisfying this property). To cope with this, at each level we hope that we have reached the destination, we query the structure in item 5A., we find the purported branching ancestor, and check that it really is the lowest branching acestor of $v$. This takes constant time; if the result is wrong, we continue walking up the trie. Overall, with the structure of 5A. we need query time $O(\lg(\lg_B w) + B)$.
We have shown how to achieve the same running times (as functions of $B$) as in the case of the greater-than function. The same calculation establishes our tradeoff curve.
Lower Bounds for the Greater-Than Problem
=========================================
A lower bound for the first branch of the tradeoff can be obtained based on Fredman’s proof idea [@fredman82sums]. We ommit the details for now. To get a lower bound for the second case ($T_q <
O(\lg\lg n)$), we use the sunflower lemma of Erdős and Rado. A sunflower is collection of sets (called petals) such that the intersection of any two of the sets is equal to the intersection of all the sets.
Consider a collection of $n$ sets, of cardinalities at most $s$. If $n > (p-1)^{s+1} s!$, the collection contains as a subcollection a sunflower with $p$ petals.
For every query parameter in $[0,n-1]$, the algorithm performs at most $T_q$ probes to the memory. Thus, there are $2^{T_q}$ possible execution paths, and at most $2^{T_q} - 1$ bit cells are probed on at least some execution path. This gives $n$ sets of cells of sizes at most $s = O(2^{T_q})$; we call these sets query schemes. By the sunflower lemma, we can find a sunflower with $p$ petals, if $p$ satisfies: $n > (p-1)^{s+1} s! \Rightarrow \lg n > \Theta(s (\lg p +
\lg s))$. If $T_q < (1-\epsilon) \lg\lg n$, we have $s\lg s = o(\lg
n)$, so our condition becomes $\lg n > \Theta(s \lg p)$. So we can find a sunflower with $p$ petals such that $\lg p = \Omega((\lg
n)/s)$. Let $P$ be the set of query parameters whose query schemes are these $p$ petals.
The center of the sunflower (the intersection of all sets) obviously has size at most $s$. Now consider the update schemes for the numbers in $P$. We can always find $T \subset P$ such that $|T| \geq |P| /
2^s$ and the update schemes for all numbers in $T$ look identical if we only inspect the center of the sunflower. Thus $\lg |T| = \lg |P| -
s = \Omega(\frac{\lg n}{s} - s)$. If $T_u < (\frac{1}{2} - \epsilon)
\lg\lg n$, we have $s = o(\frac{\lg n}{s})$, so we obtain $\lg |T| =
\Omega(\frac{\lg n}{s})$.
Now we restrict our attention to numbers in $T$ for both the update and query value. The cells in the center of the sunflower are thus fixed. Define the natural result of a certain query to be the result (greater than vs. not greater than) of the query if all bit cells read by the query outside the center of the sunflower are zero. Now pick a random $x \in T$. For some $y$ in the middle third of $T$ (when considering the elements of $T$ in increasing order), we have $\Pr[y
\leq x] \geq \frac{1}{3}, \Pr[y > x] \geq \frac{1}{3}$, so no matter what the natural result of querying $y$ is, it is wrong with probability at least $\frac{1}{3}$. So for a random $x$, at least a fraction of $\frac{1}{9}$ of the natural results are wrong. Consider an explicit $x$ with this property. The update scheme for $x$ must set sufficiently many cells to change these natural results. But these cells can only be in the petals of the queries whose natural results are wrong, and the petals are disjoint except for the center, which is fixed. So the update scheme must set at least one cell for every natural result which is wrong. Hence $T_u \geq |T|/9 \Rightarrow \lg
T_u = \Omega(\lg |T|) = \Omega(\frac{\lg n}{s}) = \Omega(\frac{\lg
n}{2^{T_q}}) \Rightarrow 2^{T_q} = \Omega(\lg_{T_u} n)$.
#### Acknowledgement. {#acknowledgement. .unnumbered}
The authors would like to thank Gerth Brodal for discussions in the early stages of this work, in particular on how the results could be extended to dynamic range counting.
[^1]: Part of this work was done while the author was visiting the Max-Planck-Institut für Informatik, Saarbrücken, as a Marie Curie doctoral fellow.
| ArXiv |
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Feza Gürsey Institute P.O.Box 6 Çengelköy, Istanbul 81220 Turkey\
May 7, 2001
PACS numbers: 11.10.Ef 02.30.Wd 02.30.Jr 03.40.Gc
[**Abstract**]{}
We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi-Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map.
This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3 N -2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a [*local*]{} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel’s recursion operator.
Introduction
============
In this paper we shall point out a general technique for the construction of inequivalent solutions to the inverse problem in the calculus of variations. We shall show that completely integrable partial differential equations in $1+1$ dimensions that admit multi-Hamiltonian structure can be cast into variational form with multiple Lagrangians. It is remarkable that all these new Lagrangians can be obtained directly from our present knowledge of complete integrability of the evolutionary system without doing any new calculations!
One of the important properties we expect from a completely integrable system is multi-Hamiltonian structure. A vector evolutionary system can then be cast into Hamiltonian form in more than one way $$u^i_{t_{\aleph+\alpha-1}} = \{u^i, H_\alpha \}_\aleph =
J^{ik}_{\aleph} \delta_k H_\alpha \qquad
\left\{ \begin{array}{l} i=1,2,...,n \\ \aleph=1,2,... N. \\
\alpha=-1,0,1,..., \infty
\end{array} \right. \label{hameq}$$ where the variational derivative is denoted by $\delta_k\equiv
\delta /\delta u^k$ and $J$ is a matrix of differential operators satisfying the properties of a Poisson tensor, namely skew-symmetry and Jacobi identity. For integrable systems there exists more than one such Hamiltonian operator and Hamiltonian function as the respective Hebrew and Greek indices indicate. Then, by the theorem of Magri [@magri] completely integrable systems admit infinitely many conserved Hamiltonian functions which are in involution with respect to Poisson brackets defined by compatible Hamiltonian operators.
The essential element in the multi-Hamiltonian approach to integrability is the construction of the Hamiltonian operators themselves. Fortunately this is a rich subject [@dorfman] that can be put to good use. We shall be interested in the consequences of multi-Hamiltonian structure on the Lagrangian formulation of completely integrable evolutionary equations. We shall work in the opposite direction to the traditional approach of deriving Hamiltonian structure from a Lagrangian. The crucial fact that we shall exploit is the relationship between Hamiltonian operators and Dirac brackets [@dirac] for degenerate Lagrangian systems which was first pointed out by Macfarlane [@mac]. In the case of completely integrable systems we have much more information on Hamiltonian structure than Lagrangian and it became clear only recently [@pavlov], [@nhepth], [@pavlov2] how we can construct multiple Lagrangians for systems that admit multi-Hamiltonian structure. We shall now present the general and most simple technique for generating these new Lagrangians.
Multi-Lagrangians {#sec-main}
=================
Evolutionary systems (\[hameq\]) cannot be cast into variational form with a local expression for the Lagrangian using the velocity fields $u^i$ alone but require the introduction of Clebsch potentials. In $1+1$-dimensions the general expression for Clebsch potentials is given by $$u^i = \phi^i_x
\label{clebsch}$$ and in this paper we shall only consider Lagrangians that are local functionals of these potentials. In the time-honored way we shall split the Lagrangian density for eqs.(\[hameq\]) into two $${\cal L} = {\cal T} - {\cal V} \label{lagtot}$$ that consist of the kinetic and potential pieces respectively. For the first Lagrangian density, an enumeration which will become clear presently, the kinetic term is always given by $${\cal T}_{1} = g_{ik} \; \phi^i_t \, \phi^k_x \label{kinetic}$$ where $g_{ik}$ are constants with $\det g_{ik} \ne 0$ and $${\cal V}_{1} = 2 {\cal H}_1 \label{potential}$$ is the Hamiltonian density. We note that the Hamiltonian function that appears in (\[hameq\]) is the space integral of the density. We shall number the conserved Hamiltonians by reserving the subscript $1$ to the “usual" Hamiltonian function but of course there exists conserved quantities such as Casimirs and the momentum which are of lower order. In fact, for complete integrability, an $n$-component vector evolutionary system (\[hameq\]) must admit $n$ infinite series of conserved Hamiltonians. We shall denote their densities by $${\cal H}_{\alpha ; [i]} \qquad \qquad i=1,2,..n; \quad
\alpha=-1,...,\infty \label{series}$$ and recall that each series starts with a Casimir $${\cal H}_{-1 ; [i]} = g_{ik} u^k \label{casimir}$$ which will carry the label minus one. One of these series is distinguished in that it contains the “usual" Hamiltonian function which is the one that appears in eq.(\[hameq\]). For the $2$-component systems that we shall discuss in this paper these are the Eulerian and Lagrangian series. We note also that the two series may coincide up to a relabelling dictated by the recursion operator. This is in fact the case for the $\gamma=2$ case of gas dynamics and in most examples of completely integrable dispersive equations except the Boussinesq equation.
The potential part of the Lagrangian does not depend on the velocities and from eq.(\[kinetic\]) it follows that the Hessian $$\det \left| \frac{\partial^2 {\cal L}_1}{\partial \phi^i_t \,
\partial \phi^k_t} \right| = 0$$ vanishes identically. We have therefore a degenerate Lagrangian system and in order to cast it into Hamiltonian form we must use Dirac’s theory of constraints [@dirac], or the covariant Witten-Zuckerman theory [@witten; @zuck] of symplectic structure. In particular, the first Hamiltonian operator obtained from the first Lagrangian is given by $$J^{ik}_{1} = g^{ik} \, D \qquad D \equiv \frac{d}{d x}
\label{j0}$$ where $g^{ik}$ is the inverse of the coefficients in the kinetic part of the first Lagrangian (\[kinetic\]) which is non-degenerate.
The construction of multiple Lagrangians relies on the use of the Lenard recursion relation which is implicit in eqs.(\[hameq\]) that in the Greek and Hebrew indices we have a symmetric matrix $$J^{ik}_{[\aleph} \delta_{|k|} H_{\alpha]} =0 \label{lenard}$$ where square brackets denote complete skew-symmetrization and bars enclose indices which are excluded in this process. Provided we can invert these Hamiltonian operators, we can construct recursion operators $$R_{\;\aleph_2 \; \;k}^{\aleph_1 \;\; i} = J_{\aleph_2}^{im}
(J^{mk}_{\aleph_1})^{-1}
\label{recop}$$ that map gradients of conserved Hamiltonians into each other (\[lenard\]).
For the construction of Lagrangians we start with the crucial observation that the first Lagrangian is of the form $${\cal L}_1 = {\cal H}_{-1 [i]} \, \phi^i_t - 2 {\cal H}_1
\label{observe}$$ which is manifest from (\[kinetic\]). The original fields that enter into the evolutionary system (\[hameq\]) are Casimirs which is evident from the subscript minus one. The second Lagrangian will be of the same general structure as (\[observe\]) if we further suppose that eqs.(\[hameq\]) can be written in bi-Hamiltonian form. Thus there will exist $H_2$ which is the next conserved Hamiltonian function in the hierarchy and the momentum $H_0$ which comes after Casimirs. The higher Lagrangian should simply be $${\cal L}_2 = {\cal H}_{0 [i]} \,
\phi^i_t - 2 {\cal H}_2$$ but there is an important refinement that we need to insert here. It is not the conserved density but rather the momentum map that enters into the kinetic part of the Lagrangian. The two differ only by total derivatives which is irrelevant in the context of conservation laws and therefore generally skipped over. However, these divergence terms are of crucial interest as the momentum map in the theory of symplectic structure. We shall show that given $\alpha^{th}$ local Hamiltonian structure, the full new Lagrangian is simply given by $${\cal L}_\alpha = \{ {\cal H}_{\alpha-2 [i]} + ({\cal G}_{\alpha
-2 [i]} )_x \} \phi^i_t - 2 {\cal H}_{\alpha} \label{genexp}$$ where ${\cal G}_{\alpha [i]}$ is a functional of the potentials. The coefficient of $\phi^i_t$ above is the momentum map and this is the only calculation necessary to find the new Lagrangian.
The fact that it is the momentum map rather than the conserved density that plays an important role in the Lagrangian can be seen at the level of the first Lagrangian. Now the Casimirs play the role of the momentum map and they are used to construct the next higher conserved quantity according to the construction of the canonical energy-momentum tensor $${\cal H}_0 = \frac{\partial {\cal L}_1}{\partial \phi^i_t}
\phi^i_x = {\cal H}_{-1 [i]} u^i = \frac{1}{2} g^{ik} {\cal H}_{-1
[i]}{\cal H}_{-1 [k]}$$ which is the momentum. This classical result for Lagrangians linear in the velocity can be generalized at each level we have a higher Lagrangian. We have $$2 {\cal H}_{\alpha-1} = g^{ik} [ {\cal H}_{\alpha-2 [i]} + ({\cal
G}_{\alpha-2 [i] } )_x ] {\cal H}_{-1 [k]} \label{check}$$ ending at the level where a local Lagrangian is no longer possible. In fact the validity of this equation is directly related to the existence of the Lagrangian. If a check of (\[check\]) fails for some $\alpha$, then there exists no local Lagrangian at $\alpha^{th}$ level.
Now we come to an important reservation that our new Lagrangians will necessarily carry. The Euler equations that follow from the variation of the action with the second Lagrangian will be $$R_{\;2 \; \;k}^{1 \;\; i} \,
\left[ u^k_{t} - J^{km}_{1} \delta_m H_1 \right]=0
\label{crucial}$$ so that the first variation of the second action will certainly be an extremum for the original equations of motion (\[hameq\]) but the Euler equations (\[crucial\]) require something weaker, namely linear combinations of functionals in the kernel of the recursion operator can be added to the right hand side of the equations of motion and the new action will still be an extremum.
From this construction it is manifest that for every Hamiltonian function in the infinite hierarchy of conserved Hamiltonians that we have for completely integrable systems, there exists a degenerate Lagrangian (\[genexp\]) that yields the equations of motion as its Euler equation up to functionals in the kernel of the recursion operator. The number of Lagrangians that can be constructed in this way is therefore infinite in number. Given bi-Hamiltonian structure we have two local Hamiltonian operators but the Lenard recursion operator (\[recop\]) is non-local. However, the special form of the first Hamiltonian operator (\[j0\]) leads to a local expression for the second Lagrangian in terms of Clebsch potentials. But it is clear that the repeated application of the recursion operator will require the introduction of non-local terms in higher Lagrangians. Strictly speaking, this is not a problem because the original Lagrangian is itself non-local in terms of the velocity fields $u^i$ which are the original variables. We swept this problem under the rug by introducing Clebsch potentials. Higher Lagrangians for evolutionary equations (\[hameq\]) can be written in local form by introducing potentials for the Clebsch potentials themselves! If, however, the equations of motion admit $N$ local Hamiltonian operators, then our construction guarantees the existence of $N$ Lagrangians which are local functionals of the Clebsch potentials. Thus we have
[**Theorem 1**]{} [*A completely integrable system that admits $N$-fold local first order Hamiltonian structure can be given $N$ different variational formulations with degenerate Lagrangians that are local functionals of the Clebsch potentials*]{}.
By a convenient abuse of language we claim that we have a Lagrangian for an equation that involves fields when the Lagrangian is in fact only a functional of the Clebsch potentials for these fields. Then we have the audacity to put in by hand the expression for the fields in terms of potentials after the variation! This can be at best only a shorthand for the real variational principle where we must impose the relationship between the fields and their potentials through Lagrange multipliers.
So far we have been guilty of this abuse ourselves. But now we must say that the first Lagrangian is actually $${\cal L}_{\ge 1}^{full} = {\cal L}_1(\phi^i, \phi^i_x,
\phi^i_{xx}...) + \lambda_i ( u^i - \phi^i_x ) \label{reallag}$$ so that upon variation with respect to all the variables $\phi^i,
u^i, \lambda_i$ we get (\[clebsch\]), $\lambda_i=0$ and we arrive at the equations of motion (\[hameq\]) expressed in terms of the original fields $u^i$ without fudging.
Now this obvious observation may seem correct but naive, however, we shall now find that it dramatically increases the number of new Lagrangians we can construct for integrable systems.
For every evolutionary equation that admits, say for simplicity, bi-Hamiltonian structure there exists a differential substitution $$u^i=M^i(r^k, r^k_x, ...) \label{m}$$ that brings the second Hamiltonian operator to the canonical form (\[j0\]) of Darboux. This differential substitution is a Miura transformation. Strictly speaking the theorem of Darboux remains unproved in field theory where the number of degrees of freedom is infinite but we shall assume it. Miura transformation works in a direction opposite to the usual action of the recursion operator. It leads to Hamiltonian equations $$r^i_{t} = \{r^i, H_0\}_1 = g^{ik} D \, \delta_{r^k}
H_0\Big|_{u^m=M^m(r^n)} \label{hameqmod}$$ where $H_0$ is the momentum for eqs.(\[hameq\]) expressed through (\[m\]). These are modified equations, different from the original equations, but the two sets are related by $$u^i_{t} - J_2^{ik} \delta_{u^k} H_0 (u) = {\cal O}^i_j (r^l) \,
\Big[ r^j_{t} - J_1^{jk} \delta_{r^k} H_0 \Big|_{u^m=M^m(r^n)}
\Big] \label{mm3}$$ up to functions in the kernel of some matrix differential operator ${\cal O}^i_j$.
A comparison of eqs.(\[crucial\]) and Miura’s relation (\[mm3\]) shows us that using the differential substitution of Miura we can obtain new Lagrangians for nonlinear evolution equations that admit multi-Hamiltonian structure in the opposite direction to our earlier construction. Transforming to the variables $r^i$ and using Clebsch potentials $$r^i = \psi_x^i \label{newclebsch}$$ we can write the classical Lagrangian for the modified system (\[hameqmod\]) $${\cal L}_1^{modified} = g_{ik} \psi_x^i \psi_t^k - 2 {\cal H}_0
\Big|_{u^m=M^m(\psi_x,\psi_{xx}...) } \label{lagmod}$$ where the labelling of ${\cal H}_0$ refers to its expression in the original variables $u^i$ but these need to be substituted for in terms of $r^i$ according to (\[m\]) and expressed through the potentials (\[newclebsch\]). We note that the Casimirs $ r^i =
\psi^i_x$ for the modified system are absent in the polynomial ${\cal H}_\alpha(u)$ hierarchy. We would expect naively that the Euler equations resulting from the first variation of the action with the Lagrangian (\[lagmod\]) would result in the modified equations (\[hameqmod\]). This would indeed be the case if we were to impose the constraint between the fields $r^i$ and their potentials $\psi^i$ as in (\[reallag\]) but now using (\[newclebsch\]). However, by imposing the constraint through Miura’s differential substitution $${\cal L}_0^{full} = {\cal L}_1^{modified}( \psi^i_x,
\psi^i_{xx}...) + \lambda_i [ u^i - M^i(\psi^i_x,\psi^i_{xx}...) ]
\label{reallag2}$$ we obtain a new Lagrangian for the original equations (\[hameq\]) in the original variables $u^i$. We shall use this construction to derive new Lagrangians, in particular for KdV in section \[sec-kdv\]. It is evident that this construction can be extended when there exists multi-Hamiltonian structure but, as we shall find in the example of KdV, sometimes it is possible to arrive at local Lagrangians using non-local Hamiltonian operators as well. Now we conclude
[**Theorem 2**]{} [*The first Lagrangian of every modified equation obtained through a Miura transformation will serve as a new zeroth Lagrangian for the original equations of motion provided the constraint between the fields and their potentials is imposed through a Miura-type differential substitution. For $N$-fold Hamiltonian structure there exists $N-1$ such new Lagrangians*]{}.
Miura transformation is in general not invertible because it is a differential substitution. But there exists interesting examples where it reduces to a point transformation which is invertible. In that case we can construct $N-1$ further Lagrangians.
We conclude that for an evolutionary system that admits $N$ fold first order Hamiltonian structure, the number of different variational principles where the first variation will be an extremum by virtue of the original equations of motion is $2N-1$ and in the case Miura transformation is invertible $3N-2$. We illustrate this situation for the case of bi-Hamiltonian structure in tables \[table1a\] and \[table1b\]. The general situation is much more complicated than what these tables would lead us to expect. Starting with tri-Hamiltonian structure the individual entries in each one of these tables will need to be table by itself because there are inequivalent Hamiltonian operators that yield the same equations of motion with the same Hamiltonian function. We shall discuss this interesting situation in a future publication on the Chaplygin-Born-Infeld equation.
[c||c|c|c]{} equations of motion & $u_t = J_2 \delta H_0 $ & $u_t = J_1
\delta H_1$ & $
J_2 J_1^{-1} u_t = - \delta_\phi H_2 $\
\
local Hamiltonian op. & $J_2$ & $J_1$ & no\
\
local Lagrangian & no & ${\cal L}_1$ & ${\cal L}_2$\
\
modified equations & $r_t = \tilde{J}_2 \delta H_0$ & $r_t = \tilde{J}_1 \delta H_1$ &\
\
\[-4.5mm\] local Hamiltonian op. & $\tilde{J}_2 = J_1$ & $\tilde{J}_1$ &\
\
\[-4.5mm\] local Lagrangian & ${\cal L}_0$ & no &
[c||c|c|c]{} equations of motion & & $u_t = J_2 \delta H_0 $ & ...\
\
local Hamiltonian op. & & $J_2$ & ...\
\
local Lagrangian & & no & ...\
\
modified equations & $\tilde{J}_1 \tilde{J}_2^{-1} r_t =
- \delta_\psi H_{-1} $& $r_t = \tilde{J}_2 \delta H_0$ & ...\
\
\[-4.5mm\] local Hamiltonian op. & no & $\tilde{J}_2 = J_1$ & ...\
\
\[-4.5mm\] local Lagrangian & ${\cal L}_{-1}$ & ${\cal L}_0$ & ...
KdV {#sec-kdv}
===
KdV stands as the symbol of completely integrable systems. We think we know it, but it turns out to be so rich that there is still new information to be learned about it. We recall that KdV $$u_t + 6 u \, u_x - u_{xxx} =0 \label{kdv}$$ admits the Kruskal sequence of conserved Hamiltonian densities $$\begin{aligned}
{\cal H}_{-1}^{KdV} &=& u \label{kdvcasimir1} \\
{\cal H}_{0}^{KdV} &=& \frac{1}{2} u^2 \label{kdvcasimir2} \\
{\cal H}_{1}^{KdV} &=& u^3 + \frac{1}{2} u_x^{2} \label{kdvh1} \\
{\cal H}_{2}^{KdV} &=& \frac{5}{2} \, u^{4} + 5 \, u \,
u_{x}^{2} + \frac{1}{2} \, u_{xx}^{\;\;2} \label{kdvh3}\\
& ... & \nonumber\end{aligned}$$ which are in involution with respect to Poisson brackets defined by two Hamiltonian operators $$J_1 = D, \qquad J_2 = - D^3 + 2 u D + 2 D u \label{kdvops}$$ that form a Poisson pencil. By introducing the potential $$u = \phi_x$$ KdV can be cast into variational form with two Lagrangians $$\begin{aligned}
{\cal L}_1^{KdV} & = & {\cal H}_{-1}^{KdV} \, \phi_t - 2 {\cal
H}_1^{KdV} \label{lagkdv}\\
{\cal L}_2^{KdV} & = & ( {\cal H}_{0}^{KdV} + \phi_{xx} ) \phi_t
- 2 {\cal H}_2^{KdV} \label{pavlov}\end{aligned}$$ which consist of the classical Lagrangian and the second Lagrangian [@pavlov] respectively.[^1] Here we observe that both (\[lagkdv\]) and (\[pavlov\]) are examples of our general expression (\[genexp\]) for higher Lagrangians.
The second application of Lenard’s recursion operator to $J_1$ results in a third Hamiltonian operator which is non-local so we cannot continue to generate higher Lagrangians. But we can use Theorem 2 to generate new lower Lagrangians for KdV. For this purpose we note that in both Lagrangians (\[lagkdv\]) and (\[pavlov\]) we should have added the constraint $ \lambda ( u -
\phi_x )$ and written the full Lagrangian. But following the convenient abuse of language we did not do so because it was manifest. It is, however, necessary to write the full Lagrangian in the case of lower Lagrangians.
According to our general construction of lower Lagrangians we first recall the original Miura transformation $$u = r^2 + r_{x} \label{miura}$$ that brings $J_2$ to the canonical form of $J_1$ in the variable $r$. The equation of motion for $r$ is mKdV which is different from (\[kdv\]) but under the substitution (\[miura\]) we have Miura’s result $$u_t + 6 u \, u_x - u_{xxx} = (D + 2 r ) \left( r_t + 6 r^2 r_x -
r_{xxx} \right) =0 \label{miura2}$$ so that, on shell, if mKdV is satisfied then so is KdV. Now we can introduce the Clebsch potential for the modified field variable $$r = \psi_x$$ and write the first Lagrangian for mKdV $${\cal L}_1^{mKdV} = \psi_x \psi_t + {\cal
H}_0^{KdV}\Big|_{u=\psi_x^2+\psi_{xx} } \label{mkdvlag}$$ in a straight-forward manner. But now enforcing the constraint in the full Lagrangian through the Miura transformation $${\cal L}_{0}^{KdV \; full} = {\cal L}_1^{mKdV} + \lambda ( u -
\psi_x^2 - \psi_{xx} ) \label{kdv0}$$ we shall arrive at a new Lagrangian for KdV because the Euler equation that comes from the first variation of this action will be satisfied by virtue of (\[miura2\]). Unlike (\[pavlov\]) which is a higher Lagrangian, (\[kdv0\]) is a lower Lagrangian in the sense of the action of the recursion operator on the equations of motion in the resulting Euler equation.
And the saga of KdV continues! We consider the third Hamiltonian operator for KdV $$J_3 = R^2 J_1 \label{j3kdv}$$ which is nonlocal but the relationship between differential substitutions and Hamiltonian structures of KdV [@max7] enables us to construct another new local Lagrangian for KdV. For this purpose we recall that the differential substitution $$r = \alpha q + \frac{\varepsilon}{q} + \frac{q_x}{2 q}$$ which transforms mKdV into twice modified KdV $$q_t = \left( q_{xx} - \frac{3 q_x^2}{2 q} + \frac{6
\varepsilon^2}{q} - 2 \alpha^2 q^3 \right)_x$$ is a Miura transformation for (\[j3kdv\]). This can best be seen by the expression $$J_3 = \frac{1}{2}( q^2 D + D q^2 ) - q_x D^{-1} q_x$$ for the third non-local Hamiltonian operator for KdV in terms of twice modified variable $q$. We recall that $J_3$ is fifth order in $u$. We have the Miura relation $$\begin{aligned}
r_t + 6 r^2 r_x - r_{xxx} & = & \left(
\alpha - \frac{\varepsilon}{q^2} - \frac{q_x}{2 q^2} + \frac{1}{q}
D \right) \nonumber \\ && \left[ q_t - \left( q_{xx} - \frac{3
q_x^2}{2 q} + \frac{6 \varepsilon^2}{q} - 2 \alpha^2 q^3 \right)_x
\right] \label{miura3}\\ &=& 0 \nonumber\end{aligned}$$ between modified and twice modified KdV’s. Introducing the potential for the twice modified variable $q=\chi_x$ we have $$u = \Phi(\chi_x,\chi_{xx},\chi_{xxx}) \equiv
\frac{\chi_{xxx}}{\chi_x} - \frac{\chi_{xx}^{\;2}}{\chi_x^2} + 2
\alpha \chi_{xx} + \alpha^2 \chi_{x}^2 + 2 \alpha \varepsilon +
\frac{\varepsilon^{2}}{\chi_x^2} \label{uq}$$ in terms of the original field $u$. The first Lagrangian for twice modified KdV is simply $${\cal L}_{1}^{m_2KdV} = \chi_{x} \, \chi_{t} + {\cal H}_{-1}^{KdV}
\Big|_{ u = \Phi(\chi_{x}, \chi_{xx},\chi_{xxx} ) }
\label{mmkdvlag}$$ and therefore the second lower Lagrangian for KdV is given by $${\cal L}_{-1}^{KdV \; full} = {\cal L}_1^{m_2KdV} + \lambda [ u -
\Phi(\chi_{x}, \chi_{xx}, \chi_{xxx} ) ] \label{kdvlagm2}$$ which provides another illustration of (\[reallag2\]). This process can be continued.
We note that an alternative to the Clebsch potential for KdV is the Schwartzian which was pointed out by Schiff [@sch]. We shall postpone consideration of Schwartzian potentials to future work.
Polytropic gas dynamics {#sec-gas}
=======================
The simplest examples for applying our construction of multi-Lagrangians consist of quasi-linear second order hyperbolic equations that Dubrovin and Novikov [@dn] have called equations of hydrodynamic type. The distinguished example in this set consists of the Eulerian equations of polytropic gas dynamics in $1+1$ dimensions $$\begin{aligned}
\rho _{t} + u \, \rho _{x} + \rho \, u_{x} &=&0 \label{GD1} \\
u_{t}+ u \, u_{x}+ \rho^{\gamma-2} \rho _{x} &=&0 \nonumber\end{aligned}$$ and in particular for $\gamma=-1$ we have the case of Chaplygin gas, or Born-Infeld equation that was recently shown to have a string theory antecedent [@jackiw2]. This system can be cast into quadri-Hamiltonian form [@gn1]. For the Chaplygin-Born-Infeld case the complete Hamiltonian structure can be found in [@annov] and its symmetries were given in [@hor]. In the following we shall use the labelling $u^1 =
\rho$ and $u^2=u$.
First we have three local Hamiltonian structures of first order [@n1] $$J_1 = \left( \begin{array}{cc} 0 & D \\ D & 0 \end{array} \right)
= \sigma^1 D,
\label{j1}$$ $$J_2 = \left( \begin{array}{cc} \rho \, D + D \, \rho &
(\gamma-2)\, D \, u + u \, D \\
D \, u + (\gamma-2)\, u \, D &
\rho^{\gamma-2} D + D \, \rho^{\gamma-2} \end{array} \right),
\label{j2}$$ $$J_3 = \left( \begin{array}{cc} u \, \rho \, D + D \, u \, \rho &
\begin{array}{c}
D \left[ \frac{1}{2} (\gamma-2) u^2 + \frac{1}{\gamma-1}
\rho^{\gamma-1} \right] \\ + \left[ \frac{1}{2} u^2 +
\frac{1}{\gamma-1} \rho^{\gamma-1}
\right] D \end{array} \\
\begin{array}{c}
D \left[ \frac{1}{2} u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1}
\right] \\ + \left[
\frac{1}{2} (\gamma-2) u^2 + \frac{1}{\gamma-1} \rho^{\gamma-1}
\right] D \end{array} &
u \, \rho^{\gamma-2} \, D + D \, u \, \rho^{\gamma-2} \end{array} \right)
\label{j3}$$ which form a Poisson pencil ${\cal J} = J_1 + c_1 J_2 + c_2 J_3$ with $c_1, c_2$ constants, [*i.e.*]{} these Hamiltonian operators are compatible. In eq.(\[j1\]) $\sigma^1$ is the Pauli matrix and this is the canonical Darboux form of first order Hamiltonian operators. The equations of polytropic gas dynamics admit two infinite hierarchies of conserved Hamiltonians which are in involution with respect to Poisson brackets defined by all three of these Hamiltonian operators. In the first set, which is called Eulerian [@gn1], the Hamiltonian densities are given by $$\begin{aligned}
{\cal H}_{-1}^E &=& \rho \label{casimir1} \\
{\cal H}_{0}^E &=& u\, \rho \label{momentum} \\
{\cal H}_{1}^E &=& \frac{1}{2} u^{2} \rho
+\frac{1}{\gamma (\gamma -1)} \rho^{\gamma} \label{hamiltonian} \\
{\cal H}_{2}^E &=& \frac{1}{6} u^{3} \rho
+ \frac{1}{\gamma (\gamma-1)} u\, \rho^{\gamma} \label{h5}\\
{\cal H}_{3}^E &=& \frac{1}{24} u^{4} \rho
+\frac{1}{2 \gamma (\gamma -1)} u^{2} \rho^{\gamma }
+\frac{1}{2 \gamma (\gamma -1)^{2} (2\gamma -1)}
\rho^{2\gamma -1} \label{h6}\\
& ... & \nonumber\end{aligned}$$ where (\[momentum\]) is the momentum, (\[hamiltonian\]) is the familiar Hamiltonian function, the Casimir is in (\[casimir1\]) and the rest consist of higher Hamiltonians. Therefore, the Euler series is the distinguished one in the terminology of section \[sec-main\]. The second series $$\begin{aligned}
{\cal H}_{-1}^L &=& u \label{casimir2} \\
{\cal H}_{0}^L &=& \frac{1}{2} (\gamma-2) u^{2}
+\frac{1}{\gamma -1} \rho^{\gamma-1} \label{h1l} \\
{\cal H}_{1}^L &=& \frac{1}{6} (\gamma-2) u^{3}
+ \frac{1}{\gamma-1} u\, \rho ^{\gamma-1} \label{h2l}\\
{\cal H}_{2}^L &=& \frac{1}{24} (\gamma-2) u^{4} +\frac{1}{4
(\gamma -1)} u^2 \rho^{\gamma -1}
+\frac{1}{2 (\gamma -1)^2 (2 \gamma - 3) } \rho^{2 (\gamma -1)} \label{h3l}\\
& ... & \nonumber\end{aligned}$$ is the Lagrangian series which starts with the Casimir (\[casimir2\]). Note that for $\gamma=2$ this series is no longer polynomial as logarithms will enter and the same remark holds for integer and half-integer values of $\gamma$ in both series.
Finally, we note that the recursion operator $R_{2}^{\;1} = J_2
J_1^{-1}$ can be used to write infinitely many Hamiltonian operators by letting it to act $n$ times on $ J_1 $. However, in general none of these operators will be local. In particular we note that $$R_{3}^{\;1} = J_3 \, (J_1)^{-1} \ne ( R_{2}^{\;1})^2, \qquad J_3
\ne J_2 J_1^{-1} J_2$$ except in the case of shallow water waves where $\gamma=2$ which admits extension to integrable dispersive equations.
Next, there is a third order Hamiltonian operator [@on] which was obtained from Sheftel’s remarkable recursion operator [@sheftel] $$J_4 = D U_x^{-1} \, D U_x^{-1} \, \sigma^1 D
\label{sheftel}$$ where $$U = \left( \begin{array}{cc} u & \rho \\
\frac{1}{\gamma-2} \rho^{\gamma-2} & u \end{array} \right)
\label{sheftelu}$$ which is only compatible with $J_0$. Higher conserved Hamiltonians start with the density [@sheftel], [@verosky] $$\hat {\cal H}^{SV(E)}_{-1} = \frac{\rho_x}{ u_x^{\;2} -
\rho^{\gamma - 3} \rho_x^{\;2}} \label{verosky}$$ which is part of the Eulerian series. There is also a Lagrangian series starting with $$\hat {\cal H}^{SV(L)}_{-1} = - \frac{u_x}{ u_x^{\;2} -
\rho^{\gamma - 3} \rho_x^{\;2}} \label{verosky2}$$ and both form new infinite hierarchies of conservation laws.
We will be interested in the Lagrangian formulation of the equations of polytropic gas dynamics (\[GD1\]) that correspond to all these Hamiltonian structures. Introducing the Clebsch potentials [@n3] $$u=\varphi_{x}, \qquad \rho =\psi_{x} \label{potentials}$$ we have the first Lagrangian representation for this system $${\cal L}_1^{\gamma} = {\cal H}_{-1}^L \psi_{t} + {\cal H}_{-1}^E
\varphi_{t} - 2 {\cal H}_{1}^E (\varphi _{x},\psi_{x})
\label{lag1}$$ but using the recursion operators $J_2 \, J_1^{-1}$ and $J_3 \,
J_1^{-1}$ we find two further Lagrangians $$\begin{aligned}
{\cal L}_{2}^{\gamma} & = & {\cal H}_{0}^L \psi _{t} + {\cal
H}_{0}^E \varphi_{t} - 2 {\cal H}_{2}^E (\varphi _{x},\psi_{x})
\label{lag2} \\
{\cal L}_{3}^{\gamma} & = & {\cal H}_{1}^L \psi _{t} + {\cal
H}_{1}^E \varphi_{t} - 2 {\cal H}_{3}^E (\varphi _{x},\psi_{x})
\label{lag3}\end{aligned}$$ which are local functionals of the Clebsch potentials. The Lagrangian obtained through the action of the recursion operator $J_4 \, J_1^{-1}$ is the most interesting one. Because $J_4$ is a third order operator, the fourth Lagrangian $$\begin{aligned}
{\cal L}_{4}^{\gamma} & = & {\cal H}^{SV(E)}_{-1} u_{t} + {\cal
H}^{SV(L)}_{-1} \rho_{t} - 2 {\cal H}_{-1}^E (\varphi
_{x},\psi_{x})
\label{lag4} \\
{\cal L}_{4}^{\gamma}& = & \frac{\rho_{x} u_{t} - u_{x} \rho
_{t}} {u_{x}^{2}-\rho ^{\gamma -3}\rho_{x}^{2}}- 2 \, \rho
\label{p}\end{aligned}$$ is [*local in the velocity fields*]{}. This is a general property of bi-Hamiltonian structure with a pair of first and third order Hamiltonian operators. Here we find a remarkable situation in that the number of Lagrangians that we can construct by repeated application of Sheftel’s recursion operator $J_4 J_1^{-1}$ is [*infinite*]{} in number. All of these Lagrangians will be [*local*]{} in the original field variables $\rho$ and $u$.
Now we come to lower Lagrangians that will arise from Miura transformations. The Miura transformations that bring the Hamiltonian operators (\[j2\]) and (\[j3\]) to the Darboux form of (\[j1\]) are point transformations for equations of hydrodynamic type. Dubrovin and Novikov had pointed out that first order Hamiltonian operators for equations of hydrodynamic type are given by $$J^{ik} = g^{ik} \, D - g^{im} \, \Gamma^k_{mn} u^n_x \label{dnop}$$ where $g_{ik}$ are the components of a Riemannian metric which is flat by virtue of the Jacobi identities. The Miura transformation provides manifestly flat coordinates for this metric. For example from (\[j2\]) we find the flat metric $$d s_2^2 = \frac{2}{4 \rho^{\gamma-1} - (\gamma-1)^2 u^2} \left[
\rho^{\gamma-2} d \rho^2 - (\gamma-1) u \, d \rho \, d u + \rho \,
d u^2 \right] \label{metrics}$$ and it can be verified that the Miura transformation $$\rho = r \, p \qquad u = \frac{1}{\gamma-1} \left( r^{\gamma-1}
+ p^{\gamma-1} \right)$$ brings it into the manifestly flat form $ 2 d r \, d p$. In these variables we find the first modified equations of gas dynamics $$\begin{aligned}
r_t + \frac{\gamma}{\gamma-1} (r^{\gamma-1} + p^{\gamma-1}) r_x +
\gamma r \, p^{\gamma-2} p_x =0 \label{modgas1} \\
p_t + \gamma p \, r^{\gamma-2} r_x+ \frac{\gamma}{\gamma-1}
(r^{\gamma-1} + p^{\gamma-1}) p_x =0 \nonumber\end{aligned}$$ and linear combinations of these equations with variable coefficients give eqs.(\[GD1\]) of gas dynamics. Introducing the potentials $$r = \chi_x , \qquad p = \upsilon_x$$ we have the Lagrangian $${\cal L}_{0}^{\gamma full} = \chi_{x} \upsilon_{t} +\upsilon_{x}
\chi_{t} - 2 {\cal H}_0^E + \lambda \left( u - \frac{
\chi_x^{\gamma-1} + \upsilon_x^{\gamma-1}}{\gamma-1} \right) +
\sigma \left( \rho - \chi_x \upsilon_x \right) \label{lag0gd}$$ where ${\cal H}_0^E$ is the momentum (\[momentum\]) expressed in terms of the potentials $\chi$ and $\upsilon$. Transforming to the first modified variables $r, p$ we get $\tilde{J}_1, \tilde{J}_2=
J_1$ and $\tilde{J}_3$ defining the tri-Hamiltonian structure of eqs.(\[modgas1\]). Now there is a new lower Lagrangian that we can construct from the recursion operator $\tilde{J}_1
\tilde{J}_2^{-1}$. We find $$\tilde{J}_1 = \left( \begin{array}{cc} (1-\gamma) \left[ r
p^{\gamma-2} \Delta D + D r p^{\gamma-2} \Delta \right] &
\begin{array}{c} \left[ (\gamma-2) r^{\gamma-1} + p^{\gamma-1}
\right] \Delta D \\ + D \left[
r^{\gamma-1} + (\gamma-2) p^{\gamma-1} \right] \Delta \end{array} \\
\begin{array}{c}
\left[ r^{\gamma-1} + (\gamma-2) p^{\gamma-1} \right] \Delta D\\
+ D \left[ (\gamma-2) r^{\gamma-1} + p^{\gamma-1} \right] \Delta
\end{array}
& (1 - \gamma ) \left[ p r^{\gamma-2} \Delta D + D p r^{\gamma-2}
\Delta\right]
\end{array} \right), \label{jr}$$ $$\Delta \equiv \frac{1}{(\gamma-1) (r^{\gamma-1} -
p^{\gamma-1})^{2} }$$ where the labelling of the variables is in the order $r$ and $p$. The new Lagrangian is given by $${\cal L}_{-1}^{\gamma full} = \frac{\chi_x \upsilon_{t} -
\upsilon_{x} \chi_t}{ \chi_x^{\gamma-1} - \upsilon_x^{\gamma -1}}
- {\cal H}_{-1}^E + \lambda \left( u - \frac{ \chi_x^{\gamma-1} +
\upsilon_x^{\gamma-1}}{\gamma-1} \right) + \sigma \left( \rho -
\chi_x \upsilon_x \right) \label{lowlaggas}$$ where the momenta do not belong to the polynomial series of conserved Hamiltonians. However, we can identify the lower momenta from this Lagrangian $$\begin{aligned}
{\cal H}_{-2}^{\gamma \, \pm}
= \xi_{\pm}^{\frac{3-\gamma}{\gamma-1}} \; (
\xi_+ \xi_- )^{-1/2} , \nonumber \\[2mm]
\xi^2+ (\gamma-1) u \, \xi + \rho^{\gamma-1} = 0 \nonumber\end{aligned}$$ where $\pm$ refers to Eulerian and Lagrangian series as well as the roots of the quadratic equation.
We now turn to the third Hamiltonian structure (\[j3\]) defined by the flat metric $$\begin{aligned}
d s_3^2 & = & - \frac{8 (\gamma-1)^2 }{[(\gamma-1)^2 u^{2} -4 \rho
^{\gamma -1}]^{2}}
\Big\{ u \rho^{\gamma-2} d \rho ^{2} \nonumber \\
&& - \frac{1}{2 (\gamma -1) } \left[ (\gamma-1)^2 u^{2}+ 4 \rho
^{\gamma -1} \right] d \rho \, d u + u \rho \, du^{2}
\Big\}\\
&=& 2 d q \, d w \nonumber\end{aligned}$$ and the coordinate transformation that brings it to the manifestly flat form is given by $$\begin{aligned}
q & = & \left[ (\gamma-1)^2 u^2 - 4 \rho^{\gamma
-1} \right]^{\frac{\gamma -3}{2(1-\gamma)}} \label{c1} \\
w&=& \int^z \frac{1}{\sqrt{1+\xi^2} } \, \xi^{\frac{\gamma
-3}{1-\gamma}} \;
d \xi \label{inte} \\
z&=& \sinh \left\{ \frac{1}{2} \ln \frac{ (\gamma -1) u + 2
\rho^{(\gamma-1)/2} }{ (\gamma -1) u - 2 \rho^{(\gamma-1)/2}}
\right\} \nonumber\end{aligned}$$ where, in general, the last integral cannot be done in closed form. For some specific values of $\gamma$ the integral (\[inte\]) is elementary as in the notable case of Chaplygin-Born-Infeld. But this paper is devoted to the general case of polytropic gas dynamics and we shall not consider inverting (\[c1\]), (\[inte\]) to obtain $u, \rho$ as functions of $q$ and $w$. We shall only remark that after this inversion we can obtain two more new Lagrangians.
The Lagrangians (\[lag1\]), (\[lag2\]) and (\[lag3\]) for polytropic gas dynamics are examples illustrating the general expression (\[genexp\]) for higher Lagrangians. For equations of hydrodynamic type there is no dispersion and hence ${\cal G}$ vanishes identically. We have given only two (\[lag0gd\]), (\[lowlaggas\]) of the four lower Lagrangians because the integral (\[inte\]) must be carried out before we arrive at the second modified equations of gas dynamics which will lead to two further new Lagrangians. Certainly the Lagrangian (\[p\]) which is derived from bi-Hamiltonian structure with a first and third order operators according to (\[genexp\]) is the most remarkable one because this is the first time it has been possible to write a Lagrangian for polytropic gas dynamics that is local in the original field variables, namely the density and velocity. Furthermore it is only the first element in an infinite series of such Lagrangians.
Kaup-Boussinesq system
======================
Gas dynamics with $\gamma = 2$ governs the behavior of long waves in shallow water. From the point of view of complete integrability it is a remarkable case, because in this case we find several completely integrable dispersive generalizations of eqs.(\[GD1\]). Most prominent among them is the well-known Kaup-Boussinesq system [@kaup1] $$u_{t} = \left(\frac{u^{2}}{2}+\rho \right)_x \qquad \rho_{t} =
\left(u\rho +\varepsilon ^{2}u_{xx} \right)_x \label{kbous}$$ which admits tri-Hamiltonian structure. The first Hamiltonian structure is given by the Hamiltonian operator (\[j1\]) and $$J_{2}^{KBq} = \left( \begin{array}{cc} D & \frac{1}{2} \, D \, u
\\[2mm] \frac{1}{2} \, u D & \frac{1}{2}
( \rho \, D + D \, \rho ) + \varepsilon^{2} D^{3}
\end{array} \right)
\label{j2kbq}$$ where $D^{-1}$ denotes the principal value integral, is the second Hamiltonian operator for the Kaup-Boussinesq system. In the limit $\varepsilon \rightarrow 0$ this Hamiltonian operator reduces to (\[j2\]) with $\gamma=2$. The recursion operator is given by $$R^{1 \; K Bq}_{2} =\left(
\begin{array}{cc}
\frac{1}{2}u+\frac{1}{2}u_{x} D^{-1} & 1 \\ \varepsilon ^{2}
D^{2}+\rho +\frac{1}{2}\rho _{x} D^{-1} & \frac{1}{2}u
\end{array} \right)$$ and there is a third local Hamiltonian operator obtained by the action of the recursion operator $J_{2}^{KBq}= (R^{1 \; K
Bq}_{2})^2 J_0$ as in the $\gamma=2$ case of gas dynamics.
The conserved Hamiltonians in the Eulerian and Lagrangian series are $$\begin{aligned}
{\cal H}_{-1}^{KBq} & = & \rho \label{hkb1} \\
{\cal H}_0^{KBq} & = & u \, \rho \label{hkb2} \\
{\cal H}_{1}^{KBq} &=& \frac{1}{2} \left( \rho u^{2}+\rho
^{2} + \varepsilon ^{2} u \, u_{xx} \right) \label{hkb3} \\ {\cal
H}_{2}^{KBq} &=&\frac{1}{2}\left[ \rho u^{3}+3\rho
^{2}u-\varepsilon ^{2}(4u_{x}\rho _{x}+3uu_{x}^{2} ) \right]
\label{hkb5} \\ H_{3}^{KBq}&=&\frac{1}{4}u^{4}\rho
+\frac{3}{2}u^{2}\rho ^{2}+\frac{1}{2}\rho ^{3}+\varepsilon
^{4}u_{xx}^{2} \label{hkb6} \\ && -\varepsilon
^{2}(\frac{5}{2}\rho u_{x}^{2}+4uu_{x}\rho _{x}+\rho
_{x}^{2}+\frac{3}{2}u^{2}u_{x}^{2}) \nonumber
\\ & ... & \nonumber\end{aligned}$$ and the degeneracy in the $\gamma = 2$ case of gas dynamics is repeated in its dispersive generalization. In particular, the Lagrangian and Eulerian series coincide apart from a relabelling $$\begin{aligned}
{\cal H}_{-2}^{KBq(E)} = & u & = {\cal H}_{-1}^{KBq(L)} \nonumber
\\ {\cal H}_{-1}^{KBq(E)} = & \rho & = {\cal H}_{0}^{KBq(L)} \label{degen}
\\ & ... & \nonumber \\ {\cal H}_{-2+n}^{KBq(E)} & =& {\cal H}_{-1+n}^{KBq(L)}
\nonumber\end{aligned}$$ that is dictated by the recursion operator.
With the aid of the Clebsch potentials $$u = \varphi_x, \qquad \rho = \psi_x \label{potKBq1}$$ we obtain $${\cal L}_{1}^{KBq} = {\cal H}_{-1}^{KBq} \varphi_{t} + {\cal
H}_{-2}^{KBq} \psi_{t} - 2 {\cal H}_{1}^{KBq} (\varphi _{x},\psi
_{x},\varphi _{xx},\psi _{xx},...) \label{lagKBq0}$$ for the first Lagrangian. Using the technique we have presented in section \[sec-main\] we shall now construct higher Lagrangians. These three local Hamiltonian structures enable us to construct two new Lagrangians $${\cal L}_{2}^{KBq} =( {\cal H}_{0}^{KBq} +\varepsilon ^{2}\varphi
_{xxx})\varphi _{t}+ {\cal H}_{-1}^{KBq} \psi _{t} - 2 {\cal
H}_{2}^{KBq}(\varphi _{x},\psi _{x},\varphi _{xx},\psi _{xx},...)
\label{lagKBq1}$$ and $$\begin{aligned}
{\cal L}_{3}^{KBq} & = & \left[ {\cal H}_{1}^{KBq} +\varepsilon
^{2} \left( 2 \psi_{xxx}+\varphi _{xx}^{2} + \varphi _{x}\varphi
_{xxx} \right) \right] \varphi_{t} \nonumber
\\ &&+ \left( {\cal H}_{0}^{KBq} +\varepsilon^{2}\varphi_{xxx}
\right)\psi_{t} -2 {\cal H}_{3}^{KBq}(\varphi_{x},\psi _{x},...)
\label{lagKBq2}\end{aligned}$$ for the Kaup-Boussinesq system. The determination of ${\cal
G}_{\beta ; [i]}$ is according to eq.(\[genexp\]) with $\beta=
2, 3$ and $[2] = [1] -1$ because of the relabelling difference (\[degen\]) between the Lagrangian and Eulerian series. Note that the momentum map which is the coefficient of $\phi_t$ in (\[lagKBq1\]) is exactly the same as the momentum in front of $\psi_t$ in (\[lagKBq2\]). The reason for this goes back to the degeneration of the Eulerian and Lagrangian series into one and the fact that it is the momentum map that is the important element in the general construction (\[genexp\]). In the dispersionless limit the Lagrangians (\[lagKBq0\]), (\[lagKBq1\]), (\[lagKBq2\]) reduce to the gas dynamics Lagrangians (\[lag1\]), (\[lag2\]) and (\[lag3\]) with $\gamma=2$.
Kaup-Broer System
=================
There is another completely integrable dispersive version of the $\gamma=2$ case of gas dynamics which is the Kaup-Broer system [@kaup1], [@broer]. The triangular invertible differential substitution $$\rho =\eta +\varepsilon u_{x} \label{kbqtokbr}$$ transforms the Kaup-Boussinesq system (\[kbous\]) into the Kaup-Broer system $$\begin{aligned}
u_{t} &= & u\, u_{x} +\eta_x +\varepsilon u_{xx} \nonumber
\\ \eta _{t} &=& \left( \eta u \right)_x -\varepsilon \eta _{xx}
\label{kbroer}\end{aligned}$$ which also has three local Hamiltonian structures [@kuper]. For the Kaup-Broer system the conserved Hamiltonians in the Eulerian series are given by $$\begin{aligned}
{\cal H}_{0}^{KBr} &=&u\eta \\ {\cal H}_{1}^{KBr} &=&
\frac{1}{2}[u^{2}\eta +\eta ^{2}-2\varepsilon \eta u_{x}]\\ {\cal
H}_{2}^{KBr}&=&\frac{1}{2} [u^{3}\eta +3u\eta ^{2}+6\varepsilon
\eta uu_{x}-4\varepsilon ^{2}u_{x}\eta _{x}], \\ {\cal
H}_{3}^{KBr} &=&\frac{1}{4}u^{4}\eta +\frac{3}{2}u^{2}\eta
^{2}+\frac{1}{2} \eta ^{3}+\varepsilon (\frac{3}{2}\eta
^{2}u_{x}-u^{3}\eta _{x})
\\ && +\varepsilon ^{2}(2u^{2}\eta _{xx}-\eta u_{x}^{2}-\eta
_{x}^{2})-2\varepsilon ^{3}\eta _{x}u_{xx} \nonumber\end{aligned}$$ which can be obtained from (\[hkb2\])-(\[hkb5\]) through the substitution (\[kbqtokbr\]). The first Hamiltonian operator for the Kaup-Broer system is given by (\[j1\]) and the second Hamiltonian operator $$J_1^{KBr} = \left( \begin{array}{cc} D &
\frac{1}{2} \, D \, u + \varepsilon D^{2} \\ \frac{1}{2} \, u\, D
-\varepsilon D^{2} & \frac{1}{2} ( \eta \, D + D \, \eta )
\end{array} \right)
\label{j2kbr}$$ can be obtained from (\[j2kbq\]) of the Kaup-Boussinesq system using the substitution (\[kbqtokbr\]).
For Kaup-Broer system we introduce the potentials $$\eta = w_{x}, \qquad \psi = w + \varepsilon \varphi_{x}
\label{potKBr}$$ and arrive at the first Lagrangian $${\cal L}_{1}^{KBr} = {\cal H}_{-1}^{KBq} \varphi_{t} + {\cal
H}_{-2}^{KBr} w_{t} - 2 {\cal H}_{1}^{KBr}(w_{x},\varphi
_{x},w_{xx},\varphi _{xx},...) \label{lagKBr1}$$ but now we can derive two further Lagrangians using the recursion operator obtained from the Hamiltonian operators (\[j2kbr\]) and (\[j1\]). Following our procedure of section \[sec-main\] we find the second Lagrangian $${\cal L}_{2}^{KBr} = ( {\cal H}_{0}^{KBr} -2\varepsilon
w_{xx})\varphi_{t} + {\cal H}_{-1}^{KBr} w_{t} -2 {\cal
H}_{2}^{KBr}(w_{x},\varphi _{x},w_{xx},\varphi _{xx},...)
\label{lagKBr2}$$ which is the same as the Lagrangian of Kaup-Boussinesq system (\[lagKBq2\]) subject to the differential substitution (\[kbqtokbr\]). Similarly we find $$\begin{aligned}
{\cal L}_{3}^{KBr} & = & ({\cal H}_{1}^{KBr} +2\varepsilon
^{2}w_{xxx}) \varphi _{t} \label{bk} \\ && + ( {\cal
H}_{0}^{KBr}+\varepsilon \varphi_{x}\varphi_{xx} +\varepsilon ^{2}
\varphi_{xxx})w_{t} -2 {\cal H}_{3}^{KBr}(\varphi _{x},w_{x},...)
\nonumber\end{aligned}$$ as the third Lagrangian for the Kaup-Broer equations (\[kbroer\]). As in the case of Kaup-Boussinesq, these Lagrangians reduce to $\gamma=2$ gas dynamics Lagrangians in the dispersionless limit. In the Kaup-Broer Lagrangians we find another example of the general formula (\[genexp\]) for Lagrangians.
Nonlinear Shrödinger equation
=============================
We shall consider the nonlinear Shrödinger equation in the $2$-component real version $$\begin{aligned}
\upsilon _{t} & = & \left[\frac{\upsilon ^{2}}{2} +\eta
+\varepsilon ^{2}(\frac{\eta _{xx}}{\eta }
-\frac{\eta_{x}^{2}}{2\eta ^{2}}) \right]_x \nonumber \\ \eta_{t}
& = & (\eta \upsilon )_{x}, \label{r2nls}\end{aligned}$$ which is a reaction-diffusion system. Again this reduces to the $\gamma=2$ case of gas dynamics in the dispersionless limit. This version of NLS can be obtained by another triangular differential substitution $$u = \upsilon +\varepsilon \eta _{x}/\eta \label{kbrtonls}$$ from the Kaup-Broer system.
NLS has the same first local Hamiltonian structure (\[j1\]) as in the case of Kaup-Boussinesq or Kaup-Broer systems. Once again the second Hamiltonian operator for NLS can be found by the transformation (\[kbrtonls\]) from the second Hamiltonian operator (\[j2kbr\]) of the Kaup-Broer system. Thus for the $2$-component real version of NLS the second Hamiltonian operator is given by $$J^{NLS}_2 = \left( \begin{array}{cc}
D + \varepsilon^{2} \left\{ \begin{array}{c}
\eta^{-1} \, D^3 + D^3 \, \eta^{-1} \\
-\frac{1}{2} \left[ (\eta^{-1})_{xx} \, D + D \, (
\eta^{-1})_{xx} \right]
\end{array} \right\} & \frac{1}{2} \,
D \, \upsilon \\ \frac{1}{2} \, \upsilon D
&\frac{1}{2} ( \eta \, D + D \, \eta )
\end{array} \right)
\label{j2nls}$$ and the conserved Hamiltonians are $$\begin{aligned}
{\cal H}_{-2}^{NLS} &=& \upsilon \\
{\cal H}_{-1}^{NLS} &=& \eta \\
{\cal H}_{0}^{NLS} &=& \upsilon \eta \\
{\cal H}_{1}^{NLS} &=& \frac{1}{2} \left( \eta \upsilon^2 + \eta^2 -
\varepsilon^{2} \frac{\eta_x^2}{\eta} \right)\\
{\cal H}_{2}^{NLS} &=& \frac{1}{2} \left[ \eta \upsilon^2 + 3
\upsilon \eta^2 + \varepsilon^{2} \left( \upsilon_x \eta_x - 3
\frac{\upsilon \eta_x^2}{\eta}
\right) \right] \\
{\cal H}_{3}^{NLS} &=&\frac{3}{4} \eta^2 \upsilon^2 + \frac{1}{4}
\eta^3 + \frac{1}{8} \upsilon^4 \eta +
\varepsilon^{4} \left( \frac{\eta_{xx}^{\;\;2}}{2 \eta} - \frac{
5 \eta_x^{\;4} }{ 24 \eta^3 } \right) \\
& & + \varepsilon^{2} \left( \upsilon^2 \eta_{xx}
- \frac{5}{4} \eta_{x}^{\;2}
- \frac{3}{4} \frac{\eta_{x}^{\;2}}{\eta} \upsilon^2 - \frac{1}{2}
\upsilon_{x}^{\;2} \eta \right) \nonumber
\\ &...& \nonumber\end{aligned}$$ which forms an infinite sequence combining both Eulerian and Lagrangian series according to (\[degen\]).
In order to construct the Lagrangians for NLS we introduce the potentials $$\begin{aligned}
\upsilon & = & z_{x} \nonumber \\ z & = & \varphi -\varepsilon \ln
w_{x} \label{potup}\end{aligned}$$ and the first Lagrangian $${\cal L}^{NLS}_{1}= {\cal H}^{NLS}_{-1} z_{t}+ {\cal H}^{NLS}_{-2}
w_{t} - 2 {\cal H}_{1}^{NLS}(w_{x},z_{x},...)$$ is the classical result. Once again we shall use the techniques of section \[sec-main\] to construct higher Lagrangians with the recursion operator obtained from (\[j2nls\]) and (\[j1\]). We obtain two higher Lagrangians for NLS $${\cal L}^{NLS}_{2}= {\cal H}^{NLS}_{0} z_{t}+ \left[ {\cal
H}^{NLS}_{-1} +\varepsilon ^{2} \left(\frac{w_{xxx}}{w_{x}}
-\frac{w_{xx}^{2}}{w_{x}^{2}} \right)\right] w_{t} -2 {\cal
H}_{2}^{NLS}(w_{x},z_{x},...)$$ and $$\begin{aligned}
{\cal L}^{NLS}_{3}&=& \left\{{\cal H}^{NLS}_{0}+\varepsilon^{2}
\left[ z_{xxx} + \left(\frac{z_{x}w_{xx}}{w_{x}} \right)_{x}
\right] \right\} w_{t} \\
&&+ \left( {\cal H}^{NLS}_{1} + 2 \varepsilon ^{2} w_{xxx}
\right) z_{t} -2 {\cal H}_{3}^{NLS}(z_{x},w_{x},...). \nonumber\end{aligned}$$ that are local functionals of the potentials. Here again, in the dispersionless limit we find the $\gamma=2$ gas dynamics Lagrangians. The remarkable strength of the general expression (\[genexp\]) for new Lagrangians is manifest.
Boussinesq Equation
===================
In order to discuss the bi-Hamiltonian structure and the Lagrangians for the Boussinesq equation in a unified framework we first turn to its dispersionless limit. For polytropic gas dynamics we had $$\rho _{t} = (\rho u)_{x}, \qquad u_{t} =
\left(\frac{u^{2}}{2}+\frac{\rho ^{\gamma -1}}{\gamma
-1}\right)_{x}$$ with its first nontrivial commuting flow $$\rho_{y} = u_{x}, \qquad u_{y} = \left(\frac{\rho ^{\gamma
-2}}{\gamma -2}\right)_{x}$$ both of which reduce to a second order quasi-linear wave equation [@gn1]. If we express Boussinesq equation in the form $$\rho_{yy} - \left( \frac{1}{2} \rho^2 - \varepsilon^2 \rho_{xx}
\right)_{xx} = 0 \label{boussinesq}$$ or $$\rho_{y}=u_{x}, \qquad u_{y}= \left(\frac{\rho ^{2}}{2} -
\varepsilon^2 \rho_{xx}\right)_x \label{b1}$$ as a first order evolutionary system and compare its dispersionless limit to polytropic gas dynamics, we find that it corresponds to the commuting flow for $\gamma =4$. The completely integrable dispersive equation $$\begin{aligned}
\rho _{t} & = & \left[\rho u-2\varepsilon ^{2}u_{xx} \right]_{x},
\label{commb} \\
u_{t}& = & \left[\frac{u^{2}}{2}+\frac{1}{3}\rho ^{3}-\frac{3}{2}
\varepsilon ^{2}(2\rho \rho _{xx}+\rho _{x}^{2})+2\varepsilon
^{4}\rho_{xxxx}\right]_{x} \nonumber\end{aligned}$$ is the commuting flow to the Boussinesq equation.
This system admits bi-Hamiltonian structure [@olver] with the Hamiltonian operators (\[j1\]) and $$J^{B}_{2} = \left( \begin{array}{cc} \rho D + D \rho - 8
\varepsilon^{2} D^{3} & 3 u \, D + 2 u_{x}
\\[4mm] 3 D u - 2 u_{x} & \begin{array}{c} 8 ( \rho
^{2} D + D \rho^2 ) + 8 \varepsilon ^{4} D^{5} \\ - \varepsilon
^{2} [ 5 ( \rho \, D^{3} + D^3 \rho ) - 3 ( \rho_{xx} D + D
\rho_{xx} ) ]
\end{array}
\end{array} \right)$$ which are compatible. The conserved Hamiltonian densities for the Boussinesq system are given by $$\begin{aligned}
{\cal H}_{-1}^{E} &=& \rho, \\
{\cal H}_{0}^{E} &=& \rho u, \\
{\cal H}_{1}^{E} &=&\frac{1}{4}\left[2\rho u^{2}+\frac{1}{3}\rho
^{4}+\varepsilon^{2}(6\rho \rho _{x}^{2}+4u_{x}^{2})+4\varepsilon ^{4}\rho _{xx}^{2}\right],
\nonumber \\
{\cal H}_{2}^{E} &=&\frac{1}{28}\left[\frac{14}{3}\rho
u^{3}+\frac{7}{3}\rho ^{4}u+14\varepsilon ^{2}(2uu_{x}^{2}+4\rho
^{2}\rho _{x}u_{x}+3u\rho \rho
_{x}^{2})\right. \label{bcomham} \\
&& \left. +28\varepsilon ^{4}(u\rho _{xx}^{2}+\rho
_{x}^{2}u_{xx}+4\rho \rho _{xx}u_{xx})+64\varepsilon ^{6}\rho
_{xxx}u_{xxx}\right]\end{aligned}$$ in the Eulerian sequence and we have also $$\begin{aligned}
{\cal H}_{-1}^{L} &=& u, \\
{\cal H}_{0}^{L} &=& u^{2}+\frac{1}{3}\rho ^{3}+\varepsilon
^{2}\rho_{x}^{2} , \\
{\cal H}_{1}^{L} &=& \frac{1}{3}u^{3}+\frac{1}{3}\rho
^{3}u-\varepsilon ^{2}u(4\rho \rho _{xx}+3\rho
_{x}^{2})+\frac{16}{5}\varepsilon ^{4}u_{xx}\rho _{xx}, \label{boussham}\\
{\cal H}_{2}^{L} &=& \frac{2}{3}u^{4}+\frac{4}{3}\rho ^{3}u^{2}+\frac{4}{45}%
\rho ^{6}+\varepsilon ^{2}(\frac{28}{3}\rho ^{3}\rho
_{x}^{2}+4u^{2}\rho
_{x}^{2}+32\rho u\rho _{x}u_{x}+8\rho ^{2}u_{x}^{2}) \nonumber \\
&&+\varepsilon ^{4}(\frac{136}{5}\rho ^{2}\rho
_{xx}^{2}-\frac{248}{5}\rho
_{x}^{4}+\frac{128}{5}uu_{xx}\rho _{xx}+\frac{16}{5}u_{x}^{2}\rho _{xx}+%
\frac{96}{5}\rho u_{xx}^{2}) \\
&&+\varepsilon ^{6}(32\rho \rho _{xxx}^{2}-\frac{592}{15}\rho _{xx}^{3}+%
\frac{64}{5}u_{xxx}^{2})+\frac{64}{5}\varepsilon ^{8}
\rho_{xxxx}^{2} \nonumber\end{aligned}$$ in the Lagrangian sequence. The Hamiltonian function of Boussinesq system with the first order Hamiltonian operator in Darboux form (\[j1\]) is $\frac{1}{2}H_{0}^{L}$. We note that the system (\[b1\]) for the Boussinesq equation differs from all dispersive integrable examples we encountered earlier in that its familiar Hamiltonian function (\[boussham\]) is in the Lagrangian sequence. This is because Boussinesq equation is the family of commuting flows to the regular gas dynamics hierarchy. The first commuting higher flow for the Boussinesq system (\[commb\]) has the Hamiltonian function (\[bcomham\]) in the Eulerian series.
By introducing potentials $$u = \varphi_{x}, \qquad \rho = \psi_{x}$$ we can obtain two local Lagrangian densities for the Boussinesq system. First we have the classical Lagrangian $$\begin{aligned}
{\cal L}_{1}^{B(L)} & = & {\cal H}_{-1}^{L \; \gamma=4} \psi_{y}+
{\cal H}_{-1}^{E \; \gamma=4}
\varphi_{y}-{\cal H}_{0}^{L \; \gamma=4} \label{bol1} \\
{\cal L}_{1}^{B(E)} &=& {\cal H}_{-1}^{L \; \gamma=4} \psi _{t}+
{\cal H}_{-1}^{L \; \gamma=4} \varphi_{t}-2 {\cal H}_{1}^{E \;
\gamma=4} \label{cbol1}\end{aligned}$$ for Boussinesq system and its first nontrivial commuting flow (\[commb\]). The second Lagrangians are given by $${\cal L}_{2}^{B(L)}= ( {\cal H}_{0}^{E \; \gamma=4} -4\varepsilon
^{2}\varphi _{xxx})\varphi _{y}+ [ {\cal H}_{0}^{L \; \gamma=4} -
5 \varepsilon ^{2} ( \psi _{x}\psi_{xx})_x +4\varepsilon
^{4}\psi_{xxxxx} ] \psi _{y} - {\cal H}_{1}^{L \; \gamma=4}$$ $${\cal L}_{2}^{B(E)} = ( {\cal H}_{0}^{E \; \gamma=4} -4\varepsilon
^{2}\varphi _{xxx})\varphi _{t}+ [ {\cal H}_{0}^{L \; \gamma=4} -
5 \varepsilon ^{2} ( \psi _{x}\psi_{xx})_x +4\varepsilon
^{4}\psi_{xxxxx} ] \psi _{t} - 2 {\cal H}_{2}^{E \; \gamma=4}$$ according to the general construction of Lagrangians in (\[genexp\]). Here we see also that the Lagrangian for the commuting flow is obtained by flipping the Hamiltonian functions between the Lagrangian and Eulerian series while keeping the momenta fixed. In section \[sec-gas\] we had constructed Lagrangians for gas dynamics using the Hamiltonians from the Eulerian series in the potential part of the Lagrangian. The general formula (\[genexp\]) can readily be used to construct Lagrangians for the commuting flow (\[b1\]) by this simple flip in the potential.
Conclusion
==========
This is the first time it has been possible to write a Lagrangian for polytropic gas dynamics that is local in the original field variables, namely the density and velocity. It is a result of the general expression (\[genexp\]) that serves to identify immediately multi-Lagrangians for completely integrable systems. What is even more remarkable is that this is only the first element in an infinite series of such local Lagrangians for polytropic gas dynamics.
It is worth emphasizing again that the scheme we have presented in section \[sec-main\] is a universal one for the construction of multi-Lagrangians appropriate to evolutionary systems. The expressions (\[genexp\]) and (\[reallag2\]) for Lagrangians of completely integrable systems has general validity. We note that (\[genexp\]) with $\alpha=1$ is true even in the case of non-integrable equations, provided the equations are presented in the form of conservation laws and the system admits one further conserved quantity, namely the Hamiltonian. We have discussed in detail the higher Lagrangians for the completely integrable non-linear evolution equations of polytropic gas dynamics, Kaup-Boussinesq, Kaup-Broer, NLS and Boussinesq equations all of which bear out the universal applicability of (\[genexp\]) in the construction of higher Lagrangians. We have also presented the lower Lagrangians (\[reallag2\]) fully for KdV and partially for gas dynamics owing to the difficulty of writing the second modified variables in closed form.
The invariance group of these multi-Lagrangians and their Noether currents should prove to be of interest in discovering new hidden symmetries of fluid mechanics. We did not discuss this important issue here. Recently Jackiw and co-authors [@jackiw] have used hidden symmetries in the classical Lagrangian for fluid mechanics to construct very interesting field theory models of fluid mechanics. Multi-Lagrangians may prove to be of interest in this connection also.
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[^1]: Note that there is an error in potential term of the second Lagrangian in [@nhepth]. The results that follow are stated correctly.
| ArXiv |
---
abstract: 'In coventional imaging experiments, objects are localized in a position space and such optically responsive objects can be imaged with a convex lens and can be seen by a human eye. In this paper, we introduce an experiment on a three-dimensional imaging of a pattern which is localized in a three-dimesional phase space. The phase space pattern can not be imaged with a lens in a conventional way and it can not be seen by a human eye. In this experiment, a phase space pattern is produced from object transparancies and imprinted onto the phase space of an atomic gaseous medium, of doppler broadened absorption profile at room temperature, by utilizing velocity selective hole burning in the absorption profile. The pattern is localized in an unique three dimensional phase space which is a subspace of the six dimensional phase space. Imaging of the localized phase space pattern is performed at different momentum locations. In addition, imaging of the imprinted pattern of an object of nonuniform transmittance is presented.'
author:
- Mandip Singh and Samridhi Gambhir
title: 'Three-dimensional imaging of a pattern localized in a phase space'
---
In most imaging experiments, a structure of an object is defined in a position space. The structural pattern can be stationary or for a dynamic object can be non stationary *w.r.t.* time. An image of such an optically responsive object can be produced with a convex lens therefore, such an object can be seen with a camera or by a human eye. In this paper, we go beyond the conventional notion of imaging. A structural pattern of objects in our experiment is defined in a phase space therefore, such a pattern can not be imaged with a lens or a camera and a human eye can not visualize such a pattern. In this paper, we introduce a three-dimensional (3D) imaging of a pattern localized in a phase space. The pattern is localized in an unique 3D subspace, of the six-dimensional (6D) phase space, involving two position and one momentum coordinates. However, the pattern is delocalized in a 3D position subspace and in a 3D momentum subspace of the 6D phase space, separately.
In experiment presented in this paper, the pattern of interest is produced by object transparencies and imprinted onto the phase space of an atomic gaseous medium at room temperature. Experiment is performed by utilizing velocity selective hole-burning [@lamb; @bennet; @haroche; @hughes2; @scholten1; @boudot; @schm] in doppler broadened absorption profile of an atomic gaseous medium. Tomographic images of the pattern localized in a 3D phase space are then captured with an imaging laser beam. The imaging laser beam is not interacting with actual objects used to produce the localized phase space pattern. Imaging of objects localized in a position space has been realized with quantum $\&$ classical methods with undetected photons [@zeilinger_1; @wong]. Quantum imaging with undetected photons, and unlike the ghost imaging [@bar; @imphase; @qimaging; @ghim; @boyd; @shih; @lugiato], does not rely on coincidence detection of photons. In this paper, a pattern of an object of nonuniform transmittance is also imprinted onto the phase space of an atomic medium and the pattern is then imaged at a constant location of momentum.
A localized pattern in a 3D subspace of the 6D phase space is shown in Fig. \[fig1\] (a), where a two dimensional position space is spanned by orthogonal position unit vectors $\hat{x}$ and $\hat{y}$ and a third dimension corresponds to the $z$-component of momentum, $p_{z}$.
![\[fig1\] *(a) A localized pattern in a 3D phase space and its three tomograms. (b) Experimental schematic diagram, a linearly polarized imaging laser beam is overlapped in an atomic gaseous medium with counter propagating object laser beams. A 2D transverse intensity profile of the imaging laser beam at different detunings is captured with an EMCCD camera. (c) A 2D transverse intensity profile of the overlapped object laser beams prior to their entrance into the atomic medium. All three alphabets are overlapped with each other. (d) Transmittance, for imaging laser beam, of the atomic medium in presence of object laser beams without masks. Three peaks labeled as $1$, $2$ and $3$ correspond to a velocity selective hole-burning, in doppler broadened absorption profile, produced by object laser beams of frequencies $\nu_{1}$, $\nu_{o}$ and $\nu_{2}$, respectively.*](fig1.png)
In experiment, $p_{z}$ is the $z$-component of momentum of atoms. The pattern is stationary *w.r.t.* time. A 2D planar section of a localized 3D phase space pattern at a constant $p_{z}$ represents a tomogram of the localized pattern. In Fig. \[fig1\] (a), three different tomograms at three different momenta are shown. Tomograms with an image of the English script alphabets $\bf{C}$, $\bf{A}$ and $\bf{T}$ are localized at $p_{z}$ equals to $p_{1}$, $p_{2}$ and $p_{3}$, respectively. Furthermore, in a 3D position space, spanned by orthogonal position unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$, each tomogram is completely delocalized on $z$-axis that implies, all images are overlapped with each other and distributed at all points on $z$-axis. However, in a 3D momentum space, spanned by orthogonal unit vectors $\hat{p}_{x}$, $\hat{p}_{y}$ and $\hat{p}_{z}$ of momentum components along $\hat{x}$, $\hat{y}$ and $\hat{z}$ directions, each tomogram is delocalized in all planes parallel to $p_{x}$-$p_{y}$ plane. A subspace where the pattern is completely localized is an unique 3D subspace of the 6D phase space, as shown in Fig. \[fig1\] (a). In remining 3D subspaces of the 6D phase space the pattern is delocalized. In this paper, stationary localized 3D phase space pattern of interest is produced from objects located in the position space. The pattern is then imprinted onto the phase space of an atomic gas obeying Maxwell velocity distribution, in form of difference of number density of atoms in ground state and excited state. Tomographic images of the 3D phase space pattern are then imaged with an imaging laser beam, where by varing the frequency of the laser beam the location, $p_{z}$, of the tomogram can be shifted.
In experiment, a stationary pattern in phase space of atoms is produced at room temperature (25$^{o}$C) by velocity selective hole-burning in the doppler broadened absorption profile of an atomic gaseous medium. Consider a linearly polarized object laser beam, of frequency $\nu_{p}$ and transverse intensity profile $I_{p}(x,y,\nu_{p})$, propagating in an atomic gaseous medium in a direction opposite to $z$-axis. The intensity profile $I_{p}(x,y,\nu_{p})$ represents a 2D image of an object in position space. This image information is transferred to a velocity class of the atomic gaseous medium at temperature $T$ by velocity selective atomic excitation. Consider an atomic gaseous medium where an isolated stationary atom has a ground quantum state $|g\rangle$ of energy $E_{g}$ and an excited quantum state $|e\rangle$ of energy $E_{e}$ with linewidth $\Gamma$. The object laser beam is on resonance with a velocity class of atoms of $z$-component of their velocity equals to $v_{r}=2\pi(\nu_{o}-\nu_{p})/k$, where $\nu_{o}=(E_{e}-E_{g})/h$, $k=2\pi/\lambda$ is the magnitude of the propagation vector of the object laser beam having wavelength $\lambda$. Atoms of other velocity classes are out of resonance due to the doppler shift. Transverse doppler shift is negligible beacuse of non relativistic velocity regime at room temperature. In absence of an object laser beam, all the atoms are in the ground state. Consider $n$ is the number of atoms per unit volume of the gaseous medium. According to Maxwell velocity distribution, a fraction of atoms with velocity in an interval $dv_{z}$ around $v_{z}$ at temperature $T$ is $f(v_{z}) dv_{z}=(m/2 \pi k_{B} T)^{1/2}e^{-m v^{2}_{z}/2 k_{B} T} dv_{z}$. Where, $k_{B}$ is the Boltzmann constant and $m$ is mass of an atom. Consider $L$ is length of the atomic medium along beam propagation direction. In presence of an object laser beam the ground state atoms of resonant velocity class are populated to the excited state. A steady state difference of atomic number densities in the ground state ($n_{1}$) and in the excited state ($n_{2}$) at $v_{z}$ is $n_{1}(x,y,v_{z})-n_{2}(x,y,v_{z})=n f(v_{z})/(1+I_{p}({x,y,\nu_{p}})\Gamma^{2}/(4I_{s}((2 \pi \nu_{p}-2 \pi \nu_{o}+kv_{z})^{2}+\Gamma^{2}/4)))$, where $I_{s}$ is the saturation intensity of the atomic transition. If attenuation and diffraction of the object laser beam are negligible then the transverse intensity profile of object laser beam is imprinted in the form of an atomic population difference in the resonating velocity class of atoms. This pattern is delocalized in the longitudinal direction *i.e.* in the direction of propagation of the object laser beam. However, the pattern is localized in the transverse plane of coordinates $x$, $y$ at a $z$-component of momentum of atoms, $p_{z} = m v_{r}$. If three different overlapping object laser beams of same linear polarization, different intensity profiles and frequencies are passed through the atomic medium then each beam imprints a different pattern in a different velocity class. Where each one located at a different $p_{z}$ corresponding to the resonant velocity class of atoms adressed by the resonating object laser beam. As a result a localized patten of all objects is imprinted onto a 3D subspace of the 6D phase space of atoms. The nearest frequency separation of object laser beams has to be much larger than the linewidth of the transition to reduce the overlapping of resonating velocity classes.
To image the localized phase space pattern, a counter propagating imaging laser beam of frequency $\nu$ is overlapped with the object laser beams passing through the atomic gaseous medium. The polarization of the imaging laser beam is perpendicular to the polarization of object laser beams. The total absorption coefficient $\alpha$ of the imaging laser beam at frequency detuning, $\delta\nu=\nu-\nu_{o}$, is a convolution of population difference and absortion crossection of an atom such that $\alpha(x,y,\delta\nu) =\int^{\infty}_{-\infty} [n_{1}(x,y,v_{z})-n_{2}(x,y,v_{z})] \sigma_{o}(\Gamma^{2}/4) dv_{z}/((2 \pi \delta\nu- kv_{z})^{2}+\Gamma^{2}/4)$, where $\sigma_{o}$ is the peak absorption crossection of the atomic transition. The absorption of the imaging laser beam decreases if it interacts with a velocity class of atoms excited by object laser beams *i.e.* $n_{2}(x,y,v_{z})$ is nonzero. This produces velocity selective hole-burning in the doppler broadened absorption profile of the atomic medium. For the incident transverse intensity profile of the imaging laser beam $I_{r}(x,y,\delta\nu)$, the transmitted imaging laser beam intensity profile after passing through the gaseous medium is $I_{r}(x,y,\delta\nu) \exp(-\mathrm{OD}(x,y,\delta\nu))$. Where $\mathrm{OD}(x,y,\delta\nu)=\alpha(x,y,\delta\nu) L$ is the optical density of the atomic medium. The optical density profile, at a detuning $\delta\nu$, corresponds a tomographic section of the phase space pattern at $p_{z}= 2 \pi m \delta\nu /k$. The optical denisty of the medium decreases if object laser beams are present. An image of a tomographic section can be constructed by measuring a change in the optical density profile caused by object laser beams. A 3D image of the phase space pattern can be constructed with tomograms obtained at different detunings of imaging laser beam.
In experiment, objects are three 2D transparency masks where each mask consists of an image of an alphabet of the English script, $\bf{C}$ (on mask-$1$), $\bf{A}$ (on mask-$2$) and $\bf{T}$ (on mask-$3$), as shown in Fig. \[fig1\] (b). All alphabets are transparent and remaining part of each mask is completely opaque to light. Object laser beams are initially passed through single mode (SM) polarization maintaining optical fibers to produce beams of gaussian transverse intensity profile. Where optical fibers are utilized as transverse mode filters. The mode filtered and collimated object laser beams of frequencies $\nu_{1}$, $\nu_{o}$ and $\nu_{2}$ are then passed through the mask-$1$, mask-$2$ and mask-$3$, respectively. After the masks, the transverse intensity profile of object laser beams correspond to alphabets $\bf{C}$ (at $\nu_{1}$), $\bf{A}$ (at $\nu_{o}$) and $\bf{T}$ (at $\nu_{2}$). All three object laser beams are overlapped on polarization beam splitters (PBS-$3$, PBS-$2$). The overlapped object laser beams are linearly $x$-polarized by a polarizer with its pass axis aligned along $x$-axis ($x$-polariser). Two half wave-plates are placed, before and after the PBS-$2$, to rotate the linear polarization of the object laser beams to equialize the intensity. The image of transverse intensity profile of the overlapped object laser beams, prior to their entrance into the atomic medium, is shown in Fig. \[fig1\] (c) where, images of alphabets are overlapped with each other. A different alphabet is imprinted on a light field of different frequency and momentum. Therefore, an intensity profile of each object laser beam also corresponds to a tomograph in the 3D phase space.
The overlapped object laser beams are passed through an atomic gaseous medium, which is a $10$ cm long rubidium ($^{87}$Rb) vapour cell shielded from external magnetic field. The linewidth of resonant transition of the atomic medium is broadened due to doppler shift caused by motion of atoms. An object laser beam of frequency $\nu_{o}$ is on resonance to the atomic transition of stationary $^{87}$Rb atoms where a ground quantum state is $5^{2}S_{1/2}$ with total angular momentum quantum number $F=2$ ($|g\rangle$) and an excited quantum state is $5^{2}P_{3/2}$ with $F=3$ ($|e\rangle$). For stationary atoms the wavelength of this transition is $\lambda\simeq780$ nm. Object laser is frequency locked to the transition and frequency is shifted by accousto-optic modulators. Object laser frequency $\nu_{2}$ is red detuned by $-40$ MHz and frequency $\nu_{1}$ is blue detuned by $+40$ MHz from the resonant transition for stationary atoms as shown in Fig. \[fig1\] (b). The nearest frequency separation of object laser beams is much larger than the line width, $5.75$ MHz and much lower than Doppler broadening of the resonant transition. The frequency spread of all laser beams is less than $1$ MHz. Frequency detuning of beams is measured with a resolution $0.1$ MHz. An object laser beam of frequency $\nu_{o}$ is on resonance with atomic velocity class of $v_{z}=0$. Therefore, an image of an alphabet $\bf{A}$ is imprinted in the zeroth velocity class in the form of an atomic population difference. An object laser beam of frequency $\nu_{1}$ is on resonance with a velocity class $v_{z}=-31.2$ m/sec therefore, an image of an alphabet $\bf{C}$ is imprinted in this velocity class of atoms. An object laser beam of frequency $\nu_{2}$ is on resonance with a veolcity class $v_{z}=+31.2$ m/sec therefore, an image of an alphabet $\bf{T}$ is imprinted in this velocity class of atoms. Atoms of each velocity class are uniformly distributed in the position space volume of the atomic gaseous medium. Therefore, the imprinted pattern is completely delocalized along the length of atomic gaseous medium in the beam propagation direction. All the imprinted images form a localized pattern in an unique 3D sub-space of the 6D phase space of the atomic gaseous medium.
To image of the imprinted phase space pattern, a linearly polarized imaging laser beam is passed through the atomic medium in the opposite direction relative to the propagation direction of object laser beams. Object and imaging laser beams are produced by two independent lasers. Imaging laser is also frequency locked to the same resonant transition of stationary atoms and its frequency is shifted by accouto-optic modulators. Prior to their entrance into the atomic medium, the transverse intensity profile of imaging laser beam of frequency $\nu_{r}$ and detuning $\delta\nu=\nu_{r}-\nu_{o}$ is $I_{r}(x,y,\delta\nu)$. Imaging laser beam is $y$-polarized which is perpendicular to the linear polarization of object laser beams and its peak intensity is much lower than the saturation intensity of the atomic transition. After passing through the atomic medium, imaging laser beam is reflected by PBS-$1$ and its transverse intensity distribution at different detunings is captured with an electron-multiplying-charge-coupled-device (EMCCD) camera without gain multiplication. Transmittance of the atomic vapour cell for the imaging laser beam, at different detuning $\delta\nu$, in presence of object laser beams without masks is shown in Fig. \[fig1\] (d). Three peaks labeled as $1$, $2$ and $3$ correspond to velocity selective hole-burning, in doppler broadened absorption profile, caused by object laser beams of frequencies $\nu_{1}$, $\nu_{o}$ and $\nu_{2}$, respectively. Object and imaging laser beams are counter propagating therefore, a peak in the transmittance due to a hole-burning by a higher frequency object laser beam appears at lower frequency of imaging laser beam. To measure the imaging laser beam detuning precisely, a part of object laser light is extracted and red detuned by $190$ MHz from the resonant transition.
![\[fig2\] *Three tomographic images, of a localized pattern in the 3D phase space, captured at different detunings of the imaging laser beam. Each image is a plot of a change in the optical density, $\Delta \mathrm{OD}(y,z,\delta\nu)$.*](fig2.png)
The extracted object laser light is overlapped with a part of the imaging laser light of same polarisation on a non polarization beam splitter (BS). A beating signal of two lasers is detected with a fast response photo detector and measured with a radio frequency spectrum analyzer as shown in Fig. \[fig1\] (b). Detuning is measured from frequency of the beating signal that corresponds to a frequency difference of two lasers .
The intensity profile of imaging laser beam after traversing throught the atomic medium in presence of object laser beams is $I_{on}(x,y,\delta\nu)=I_{r}(x,y,\delta\nu) \exp(-\mathrm{OD}(x,y,\delta\nu))$. The optical density $\mathrm{OD}(x,y,\delta\nu)$ is constructed at detuning $\delta\nu$. The optical density is higher in absence of object laser beams. A change in the optical density after switching-on the object laser beams is $\Delta \mathrm{OD}(x,y,\delta\nu)=-\log(I_{on}(x,y,\delta\nu)/I_{off}(x,y,\delta\nu))$. Where, $I_{off}(x,y,\delta\nu)$ is the intensity profile of the imaging laser beam after traversing through the atomic medium in absence of object laser beams. Frequency of the imaging laser beam is red detuned by $\delta\nu=-40$ MHz from the resonant transition. Its transverse intensity profile $I_{off}(x,y,\delta\nu)$ is captured with an EMCCD camera in absence of three object laser beams.. After a time delay, another image of the imaging laser beam intensity $I_{on}(x,y,\delta\nu)$ is captured in presence of three object laser beams. A $y$-polarizer is placed in front of EMCCD camera to block any back reflection of object laser beams from optical components. A change in optical density profile, $\Delta \mathrm{OD}(x,y,\delta\nu)=-\log(I_{on}(x,y,\delta\nu)/I_{off}(x,y,\delta\nu))$, of the atomic medium is evaluated. Similar measurements are performed for detuning $\delta\nu=0$ MHz and $\delta\nu=+40$ MHz. For each detuning of the imaging laser beam, a different tomographic section of the 3D phase space localized pattern is captured. A series of three tomographic images are shown in Fig. \[fig2\] for three different detunings. Imaging laser beam is reflected by PBS-$1$ therefore, images are constructed after making a reflection transformation in a plane parallel to $y$-$z$ plane and $\Delta \mathrm{OD}(x,z,\delta\nu)$ is transformed to $\Delta \mathrm{OD}(y,z,\delta\nu)$. Three tomographic images resemble to the English script alphabets of object transparancies *i.e.* $\bf{C}$ (at $\delta\nu=-40$ MHz), $\bf{A}$ (at $\delta\nu=0$ MHz) and $\bf{T}$ (at $\delta\nu=+40$ MHz). By combining all the tomographic images, a word $\bf{CAT}$ is formed as shown in Fig. \[fig2\]. Imaging laser beam detuning and $z$-component of resonating velocity class corresponding to each tomographic image are shown on top of each tomograph.
![\[fig3\] *(a) A photograph of overlapping neutral density filters. (b) An image, $\Delta \mathrm{OD}(y,z,\delta\nu=0)$, of an area enclosed by a square as shown in (a).*](fig3.png)
In another experiment, an object of nonuniform transmittance is constructed by overlapping two neutral density filters of neutral densities ($\mathrm{ND}$) $0.3$ and $0.6$ as shown in a photograph, Fig. \[fig3\] (a). Four different regions $R_{1}$ ($\mathrm{ND}=0$), $R_{2}$ ($\mathrm{ND}=0.3$), $R_{3}$ ($\mathrm{ND}=0.6$) and $R_{4}$ ($\mathrm{ND}=0.9$) are formed. This object is positioned in place of a mask-$2$ in path of an object laser beam of frequency $\nu_{o}$. The image of a part of the object enclosed by a square as shown in Fig. \[fig3\] (a) is captured with imaging laser beam. Experiment is performed with a single object laser beam. The intensity profile $I_{p}(x,y,\delta\nu=0)$ of object laser beam after passing through the object consists of four different regions of different intensity levels. Therefore, it produces four regions of different depths of hole-burning in atomic gaseous medium. Imaging laser beam is on resonance with velocity class $v_{z}=0$ and an image of $\Delta\mathrm{OD}(y,z,\delta\nu=0)$ of the atomic gaseous medium is constructed as shown in Fig. \[fig3\] (b), which is an image of the overlapping neutral density filters.
In conclusion, experiments presented in this paper provide a unique way to produce and image a 3D pattern localized only in an unique 3D subspace of the 6D phase space.
[**[Contribution of Authors:]{}** ]{}Mandip Singh (MS) created the idea, MS designed and performed the experiment, MS and PhD student Samridhi Gambhir (SG) made the masks, SG plotted data shown in Fig. 1 (d). MS wrote the paper.
[99]{}
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| ArXiv |
---
address:
- 'Department of Physics and Astronomy MSC07 4220, 1 University of New Mexico,Albuquerque NM 87131-0001'
- 'Physics Division, Los Alamos National Laboratory MS H803, P-23, Los Alamos, NM, 87545, USA'
author:
- 'N. McFadden'
- 'S. R. Elliott'
- 'M. Gold'
- 'D.E. Fields'
- 'K. Rielage'
- 'R. Massarczyk'
- 'R. Gibbons'
bibliography:
- 'mybibfile.bib'
title: 'Large-Scale, Precision Xenon Doping of Liquid Argon'
---
neutrinoless double beta decay, xenon doping, liquid argon, Birk’s constant, Geant4
| ArXiv |
---
abstract: 'We introduce a 3-manifold invariant $\hat{Z}_b^G(q)$ valued in integer-coefficient power series and an invariant of knot complements $F_K^G({\bf x},q)$ valued in multi-variable series that depend on the choice of gauge group $G$. This generalizes the earlier works of Gukov-Pei-Putrov-Vafa [@GPPV] and Gukov-Manolescu [@GM] which correspond to $G=SU(2)$ case. As in the $SU(2)$ case, there is a surgery formula relating $F_K^G({\bf x},q)$ to $\hat{Z}_b^G(q)$ of a Dehn surgery of the knot. We provide explicit calculations of $\hat{Z}_b^G(q)$ for negative definite plumbings and $F_K^G({\bf x},q)$ for torus knots. Furthermore, we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations which should satisfy a recurrence given by the quantum A-polynomial for symmetric representations.'
address: 'Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125'
author:
- Sunghyuk Park
bibliography:
- 'higher\_rank.bib'
date: 'September 19, 2019'
title: 'Higher rank $\hat{Z}$ and $F_K$'
---
Introduction
============
Categorification of the Chern-Simons theory is one of the most exciting open questions in quantum topology. While homology theories categorifying quantum link invariants are fairly well-understood by now, whether there is a homology theory categorifying the Witten-Reshetikhin-Tureav (WRT) invariants is still widely open. One approach to this problem comes from physics. According to recent works of Gukov-Putrov-Vafa [@GPV] and Gukov-Pei-Putrov-Vafa [@GPPV], we can decompose WRT invariants into categorifiable “homological blocks” often denoted by $\hat{Z}_b(q)$. These are integer coefficient $q$-series and are supposed to be the (graded) Euler characteristic of a conjectural homological invariant $\mathcal{H}^{i,j}_{b,{\rm BPS}}$ so that $$\hat{Z}_b(q) = \sum_{i,j}(-1)^i q^j \text{rank }\mathcal{H}^{i,j}_{b,{\rm BPS}}.$$ More recently Gukov-Manolescu [@GM] studied an analog of $\hat{Z}_b(q)$ for knot complements, denoted by $F_K(x,q)$, and demonstrated that $\hat{Z}_b(q)$ behaves well under cutting and gluing along a torus boundary.
More than a decade ago, Dunfield-Gukov-Rasmussen [@DGR] made a fascinating conjecture, also partly motivated from physics, that $\mathfrak{sl}(N)$ link homologies should be equipped with a family of differentials $d_N$ for $N\in \mathbb{Z}$. This conjecture revealed many hidden structures of the link homologies and could be used to compute Khovanov homology very efficiently. Now, a very natural (and ambitious) question is this : Is there an analogous story for 3-manifolds?
The purpose of this paper is to define categorifiable objects $\hat{Z}_b^G(q)$ and $F_K^G({\bf x},q)$ for arbitrary gauge group $G$. We believe that our work will serve as a stepping stone towards the analogous story of large $N$ on 3-manifold side.
Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------
In Section 2 we list a few notational conventions we will use throughout this paper.
In Section 3 we define higher rank $\hat{Z}_b(q)$ for (weakly) negative definite plumbed 3-manifolds. We provide many examples, hoping it will serve as a good reference.
In Section 4 we introduce higher rank $F_K({\bf x},q)$ and give an explicit formula for torus knots. Higher rank surgery formula (without proof) is also presented.
In Section 5 we specialize our higher rank $F_K$ to symmetric representations. We numerically check for some examples that it satisfies the quantum volume conjecture.
Appendix A is on a geometric meaning of the label $b$ of $\hat{Z}_b$, and Appendix B is a derivation of definition (\[integralZhat\]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank Sergei Gukov for his invaluable guidance and Nikita Sopenko for his kind help with Mathematica, as well as Francesca Ferrari, Sarah Harrison and Ciprian Manolescu for helpful conversations.
The author was supported by Kwanjeong Educational Foundation.
Notations and conventions
=========================
We follow the convention used in [@GM] for knots, 3-manifolds, and colored Jones polynomials. Throughout this article, $G$ is a semisimple Lie group, $Q$ is the root lattice, $P$ is the weight lattice, and $W$ is the Weyl group. $\rho$ denotes the Weyl vector (half-sum of positive roots), and $\alpha$ and $\omega$ will be reserved for roots and fundamental weights. We use $\rm B$ for the linking matrix of a plumbed 3-manifold. For a multi-index monomial, we use the following notation $$x^\beta := \prod_{1\leq i\leq r}x_i^{(\beta,\omega_i)}$$ where $r = \text{rank }G$ and $\beta\in P$. When it comes to $q$-series, we mostly don’t bother to fix the overall $q$-power, and just use the notation $\cong$ for equivalence up to sign and overall $q$-power.
Higher rank $\hat{Z}_b$
=======================
Definition for plumbings
------------------------
We present here a formula for $\hat{Z}$ for (weakly) negative definite plumbed manifolds, with arbitrary gauge group $G$. We use the same good old Gauss sum reciprocity to deduce this definition. (See appendix \[Gauss\] for derivation.) :
\[Zhatsgm\] For a plumbed 3-manifold $Y$ with weakly negative definite linking matrix $\mathrm{B}$, define[^1] $$\boxed{
\hat{Z}_b^G(Y;q) := \pm \frac{q^{\frac{3\sigma-\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\int_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq r}\frac{dx_{vi}}{2\pi i x_{vi}}\left(\sum_{w\in W}(-1)^{l(w)} x_v^{w(\rho)}\right)^{2-\deg v}\, \Theta_b^{-\mathrm{B}}(x^{-1},q) \label{integralZhat}
}$$ where $$\Theta_b^{-\mathrm{B}}(x^{-1},q) := \sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}x_v^{-\ell_v}. \label{Theta}$$ In particular, in case $G = SU(N)$, this takes the following simple form : $$\label{Zhatb}
\hat{Z}_b^{SU(N)}(Y;q) := \pm \frac{q^{\frac{3\sigma -\mathrm{Tr}\,\mathrm{B}}{2}\frac{N^3-N}{12}}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\oint_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq N-1}\frac{dx_{vi}}{2\pi i x_{vi}}\, F_{3d}(x)\Theta_{2d}^{b}(x,q)$$ with $$\begin{aligned}
F_{3d}(x) &:= \prod_{v\in V}\left(\sum_{w\in W}(-1)^{l(w)} \prod_{1\leq i\leq N-1}x_{vi}^{(\omega_i,w(\rho))}\right)^{2-\deg v}\nonumber\\
&= \prod_{v\in V}\left( \prod_{1\leq i < j\leq N}(y_{vi}^{1/2}y_{vj}^{-1/2}-y_{vi}^{-1/2}y_{vj}^{1/2}) \right)^{2-\deg v}\\
\Theta_{2d}^{b}(x,q)&:=\sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}\prod_{1\leq i\leq N-1}x_{vi}^{-(\omega_i,\ell_{v})}, \end{aligned}$$ where $x_{i} = \frac{y_{i}}{y_{i+1}}$.
Here “v.p.” denotes the principal value integral. That is, taking average over $W$ number of deformed contours, each corresponding to a Weyl chamber. For instance, the deformed contour corresponding to a permutation $\sigma\in W \cong S_{N}$ is $$|y_{\sigma(1)}| < |y_{\sigma(2)}| < \cdots < |y_{\sigma(N)}|. \label{SU(N)contour}$$ These $\hat{Z}_b^G$ are topological invariants.
$\hat{Z}_b^G$ defined above is invariant under Neumann moves (a.k.a. 3d Kirby moves).
The proof is analogous to that of the $SU(2)$ version. See Proposition 4.6 in [@GM].
Examples and explicit calculations
----------------------------------
[**Comparison with Hee-Joong Chung’s paper.**]{} We find agreement with Chung’s result [@C] for many examples. (In the following, we present $\hat{Z}$’s up to $\cong$ equivalence.)
- $Y = M(-1;\frac{1}{2},\frac{1}{3},\frac{1}{7})$. In this case $|H_1(Y)| = 1$ and there’s only one homological block.
$G$ $\hat{Z}_0(Y)$
--------- ----------------------------------------------------------------------------------------------
$SU(2)$ $1 -q -q^5 +q^{10} -q^{11} +q^{18} +q^{30} -q^{41} +q^{43} -q^{56} -q^{76} +q^{93} -\cdots$
$SU(3)$ $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$
$SU(4)$ $1 -3q +q^2 +3q^3 -3q^5 -q^6 -q^7 -5q^8 +15q^9 +5q^{10} -11q^{11} -q^{12} +\cdots$
- $Y = M(-1;\frac{1}{2},\frac{1}{5},\frac{2}{7})$. Again, $|H_1(Y)| = 1$ and there’s only one homological block.
$G$ $\hat{Z}_0(Y)$
--------- ------------------------------------------------------------------------------------------------------
$SU(2)$ $1 -q^{3} -q^{5} +q^{12} -q^{23} +q^{36} +q^{42} -q^{59} +q^{81} -q^{104} -q^{114} +q^{141} -\cdots$
$SU(3)$ $1 -2q^3 -2q^5 +2q^6 +2q^9 +q^{12} -2q^{14} +2q^{15} -2q^{18} -3q^{20} +6q^{21} -4q^{23} -\cdots$
$SU(4)$ $1 -3q^3 -3q^5 +5q^6 -q^7 +2q^8 +3q^9 -q^{10} -q^{12} +2q^{13} -6q^{14} +2q^{15} -\cdots$
- $Y = M(-1;\frac{1}{3},\frac{1}{5},\frac{3}{7})$. In this case $|H_1(Y)| = 4$.
$G$ $\hat{Z}_b(Y)$
--------- ----------------------------------------------------------------------------------------------------------------------------
$SU(2)$ $1 +q^4 +q^{16} -q^{68} +q^{144} -q^{260} -q^{320} -q^{356} +q^{484} +q^{528} +q^{612} -q^{832} +\cdots$
$-\frac{1}{2}q^{15/4}(1 +q^6 +q^{10} +q^{12} -q^{44} -q^{48} -q^{58} -q^{88} +q^{122} +q^{164} +q^{182} +q^{190} -\cdots)$
$q^{13/2}(1 -q^{32} -q^{56} -q^{72} +q^{136} +q^{160} +q^{208} -q^{344} +q^{496} -q^{696} -q^{792} -q^{848} +\cdots)$
$SU(3)$ $1 +3q^4 +2q^{12} +3q^{16} +2q^{28} +2q^{48} +2q^{52} +q^{64} +4q^{68} +4q^{80} +4q^{92} - \cdots$
$-\frac{1}{6}q^{-7/4}(2 +2q +2q^3 -4q^5 +2q^6 -2q^7 +4q^9 +2q^{10} +4q^{12} +2q^{13} -2q^{14} +\cdots)$
$-\frac{1}{3}q^{-7}(1 +2q^6 -2q^8 +2q^{12} -2q^{14} +2q^{18} -2q^{20} +q^{24} -2q^{26} +2q^{28} +4q^{30} +\cdots)$
$\frac{1}{3}q^{-21/4}(1 +q -q^2 +q^4 -q^5 +q^7 +q^{10} +q^{13} -2q^{14} -q^{15} +2q^{16} +q^{19} +\cdots) \times 2$
$SU(4)$ $1 +q^{12} +8q^{16} +3q^{20} +16q^{24} +11q^{28} +15q^{32} +4q^{36} +26q^{40} +5q^{44} +\cdots$
$-\frac{1}{12}q^{-15/4}(1 +2q -2q^2 +2q^4 -2q^5 +q^6 +4q^7 -2q^8 +3q^{10} +6q^{13} -\cdots)$
$\frac{1}{6}q^{-1}(2 -2q^2 -2q^4 -3q^6 -2q^8 -5q^{10} -14q^{12} -5q^{14} -4q^{16} -12q^{18} +\cdots)$
$\frac{1}{12}q^{-1/4}(2 +2q -2q^2 +2q^3 +q^4 -2q^5 +6q^6 +2q^7 +8q^8 +6q^9 -\cdots) \times 2$
$-q^{3/2}(1 +3q^4 +2q^8 +6q^{12} +6q^{16} +4q^{20} +9q^{24} +9q^{28} +11q^{32} +9q^{36} +\cdots) \times 2$
$q^{-2}(1 +2q^4 +3q^8 +6q^{16} +6q^{20} +11q^{24} +17q^{32} +10q^{36} +9q^{40} +14q^{48} +\cdots)$
$-\frac{1}{12}q^{-15/4}(2 +q -2q^2 +q^3 +2q^4 -q^5 +4q^6 +4q^7 +2q^8 -2q^9 +\cdots)$
$\frac{1}{6}q^{-11/2}(1 +3q^6 -4q^8 +7q^{12} -2q^{14} -q^{16} +6q^{18} +3q^{20} +7q^{22} +4q^{24} +\cdots)$
Observe that [@C] agrees with our example computations except in the $SU(4)$ case of the last example. Our homological blocks are more refined in a sense that Chung’s $\hat{Z}_{1}$ is the sum of our 2nd and 7th blocks and Chung’s $\hat{Z}_{3}$ is the sum of our 1st and 6th blocks. This example illustrates that in general we can’t simply decompose $Z_a$ into $\hat{Z}_b$’s by just collecting terms whose $q$-powers differ by an integer.[^2]
[**Other examples and higher rank false theta functions.** ]{}
- $Y=S_0^3(K_n)$. The 0-surgery on twist knots are probably the simplest examples. For instance,
$G$ $\hat{Z}_0(S_0^3(\mathbf{5}_2))$
--------- ----------------------------------------------------------------------------------------------------------------
$SU(2)$ $\frac{1}{2!}(1 -q +q^3 -q^6 +q^{10} -q^{15} +q^{21} -q^{28} +q^{36} -q^{45} +q^{55} -q^{66} +q^{78} -\cdots)$
$SU(3)$ $\frac{1}{3!}(1 -2q +2q^3 +q^4 -4q^6 +2q^9 +2q^{10} +q^{12} -2q^{13} -4q^{15} +2q^{18} +2q^{19} +\cdots)$
$SU(4)$ $\frac{1}{4!}(1 -3q +q^2 +4q^3 -2q^4 +q^5 -5q^6 -2q^7 +3q^8 +2q^9 +9q^{10} -2q^{11} -\cdots)$
$SU(5)$ $\frac{1}{5!}(1 -4q +3q^2 +6q^3 -7q^4 -2q^5 +2q^7 -2q^8 +6q^9 +15q^{10} -12q^{11} -23q^{12} +\cdots)$
Indeed, for every positive twist knot $K_p$ the following is easy to deduce from our definition (\[integralZhat\]).
\[twist knot prop\] $$\label{twist knot Zhat}
\hat{Z}_0^G(S_0^3(K_p)) \cong \frac{1}{|W|}\sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\rho)||^2} =: \frac{1}{|W|}\chi_{p,\rho}$$ where $$N_\ell := \sum_{w\in W}(-1)^{l(w)}K(w(\ell))$$ and $K(\beta)$ denotes the Kostant partition function.[^3]
Note that $\chi_{p,\rho}$ is exactly the higher rank false theta function (a character of the log-VOA $W^0(p)_Q$) given in equation (1.2) of [@BM]! Similarly for double twist knots $K_{m,n}$ with $m,n>0$,[^4] $$\label{double twist knot Zhat}
\hat{Z}_0^G(S_0^3(K_{m,n})) \cong \frac{1}{|W|}\chi_{m,\rho}\chi_{n,\rho}.$$
The 0-surgery on $K_p$ has a simple plumbing description as shown in Figure \[double twist knot figure\].
(-2,0) node\[above\][$m$]{} circle(0.5ex)– (0,0) node\[above\][$0$]{} circle(0.5ex) (2,0) node\[above\][$0$]{} circle(0.5ex)– (4,0) node\[above\][$n$]{} circle(0.5ex); (0,0) edge\[bend left\] node\[midway,above\][$+$]{} (2,0); (0,0) edge\[bend right\] node\[midway,below\][$-$]{} (2,0);
(7,0) node\[above\][$-1$]{} circle(0.5ex) (9,0) node\[above\][$0$]{} circle(0.5ex)– (11,0) node\[above\][$p$]{} circle(0.5ex); (7,0) edge\[bend left\] node\[midway,above\][$+$]{} (9,0); (7,0) edge\[bend right\] node\[midway,below\][$-$]{} (9,0);
The linking matrix and its inverse are $$\mathrm{B} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & p\end{pmatrix}
\quad,\quad
\mathrm{B}^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -p & 1 \\ 0 & 1 & 0\end{pmatrix}$$ There is a single trivalent vertex with $0$ framing. This contributes the following factor in $F_{3d}(x)$ : $$\qty(\sum_{w\in W}(-1)^{l(w)}x_0^{w(\rho)})^{-1} = \frac{1}{|W|}\sum_{\ell_0\in P_+ \cap (Q+\rho)}N_{\ell_0} \sum_{w\in W}(-1)^{l(w)}x_0^{w(\ell_0)}.$$ For $\ell = (0,\ell_0,\ell_p)^t$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2}||\sqrt{p}\ell_0 - \frac{1}{\sqrt{p}}\ell_p||^2 -\frac{1}{2p}||\ell_p||^2}.$$ Applying (\[integralZhat\]), it is straightforward to get (\[twist knot Zhat\]).
Using a plumbing description of the 0-surgery on $K_{m,n}$ (Figure \[double twist knot figure\]), it is easy to derive (\[double twist knot Zhat\]) as well.
- $Y=\Sigma(p,q,r)$. For convenience let’s define the following notation for higher rank false theta functions : $$\chi_{p,\beta}^{G} := \sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\beta)||^2}$$ Note that for $SU(2)$, this notation is related to the usual notation of false theta functions as follows : $$\chi_{p,n\rho}^{SU(2)} = \Psi_{p,p-n},\text{ for }n = 1, \cdots, p-1.$$ For every Brieskorn sphere $Y = \Sigma(p,q,r)$ with $0 < p < q < r$ pairwise relatively prime, we have
\[Brieskorn prop\] $$\hat{Z}_0^G(\Sigma(p,q,r)) \cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{pqr,qr\rho + prw_1(\rho) + pqw_2(\rho)}$$
That is, it is a sum of $|W|^2$ number of higher rank false theta functions.[^5]
The proof is analogous to that of Proposition 4.8 in [@GM].
Note that we didn’t have to treat $\Sigma(2,3,5)$ separately. In this sense, using $\chi_{p,\beta}$ as false theta functions is more natural than using $\Psi_{p,n}$.
- $Y = M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})$. Let $b_1,b_2,b_3>0$ and assume that $Y$ has negative orbifold number; i.e. $$e = a_0 + \sum_{i=1}^{3}\frac{a_i}{b_i} < 0.$$ Assume further that the central meridian is trivial in homology; i.e. $$e\, \mathrm{lcm}(b_1,b_2,b_3) = -1.$$ Then their $\hat{Z}_b$’s can be expressed as signed sum of higher rank false theta functions :
\[some Seifert prop\] $$\label{some Seifert Zhat}
\hat{Z}_b^G(M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})) \cong \sum_{(w_1,w_2)\in W^2}\mathbf{1}_b(w_1,w_2) (-1)^{l(w_1w_2)}\chi_{\frac{b_1b_2b_3}{|H_1|},\frac{b_2b_3}{|H_1|}\rho + \frac{b_1b_3}{|H_1|}w_1(\rho) + \frac{b_1b_2}{|H_1|}w_2(\rho)}$$ where $$\mathbf{1}_b(w_1,w_2) := \begin{cases} 1 &\text{ if } \ell(\rho,\rho,w_1(\rho),w_2(\rho)) \in \mathrm{B}Q^V+b\\ 0 &\text{ otherwise}\end{cases}$$
Observe that Proposition \[some Seifert prop\] is a slight generalization of Proposition \[Brieskorn prop\].
$Y$ can be described as a star-shaped plumbing with 3 legs. The only vertices whose degree is not 2 are the central vertex and the terminal vertices. Denote by $\ell(\ell_0,\ell_1,\ell_2,\ell_3)$ an element $\ell\in \mathrm{B}Q^V + b$ such that $$\ell_v = \begin{cases} \ell_0 & v \text{ is the central vertex}\\ \ell_1, \ell_2, \ell_3 & v \text{ is the corresponding terminal vertex} \\ 0 & \text{Otherwise}\end{cases}$$ Then for any $\ell$ with $\ell_1,\ell_2,\ell_3 \in W(\rho)$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2|H_1|}||\sqrt{b_1b_2b_3}\ell_0 - \frac{1}{\sqrt{b_1b_2b_3}}(b_2b_3\ell_1 + b_3b_1\ell_2 + b_1b_2\ell_3)||^2 + C}$$ for some constant $C$ independent of $\ell$. Applying the definition (\[integralZhat\]), it is straightforward to obtain (\[some Seifert Zhat\]). Note that the assumption $e\,\text{lcm}(b_1,b_2,b_3)=-1$ was introduced so that $$\ell(\rho,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b \Leftrightarrow \ell(\rho + Q,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b$$
Naturally, there are some open questions regarding modularity of these $q$-series :
What are the quantum modularity properties of the higher rank $\hat{Z}$?
More specifically we can ask
What are the “mock side” of the higher rank false theta functions? (i.e. what is the higher rank $\hat{Z}$ of the orientation reversed 3-manifold?)
Higher rank $F_K$
=================
Review of $SU(2)$ case
----------------------
In [@GM], a new knot invariant $F_K(x,q)$ was defined for weakly negative definite knots and conjectured for all knots. Roughly, $F_K$ is a series $$F_K(x,q) = \frac{1}{2!}\sum_{m=1}^{\infty} (x^{m/2}-x^{-m/2})f_m(q)$$ where $m$ is odd and each $f_m(q)$ is a Laurant series with $\mathbb{Z}$-coefficient. One of the most important property of $F_K(x,q)$ is that $$F_K(x,q) \xrightarrow{q\rightarrow 1} \frac{x^{1/2}-x^{-1/2}}{\Delta_K(x)}$$ where $\Delta_K(x) = \nabla_K(x^{1/2}-x^{-1/2})$ is the Alexander polynomial of $K$, and $\frac{1}{\Delta_K(x)}$ should be understood as an average over expansions near $x=0$ and $x=\infty$.
Higher rank $F_K$
-----------------
Let’s study higher rank $F_K(\mathbf{x},q)$. The higher rank analog of $F_K$ should be of the following form : $$F_{K}^G(\mathbf{x},q) = \frac{1}{|W|}\sum_{\beta\in P_+\cap (Q+\rho)}f_\beta^G(q)\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}.$$ This can be computed by ’t Hooft resumming the (higher rank) colored Jones polynomials and should be annihilated by the (higher rank) quantum A-polynomial.
[**Right-handed trefoil with $G = SU(3)$.**]{} The first few $f_\beta^{SU(3)}$ are (up to overall sign and $q$-power) $$\begin{aligned}
f_{(1,1)} &= -q,\quad
f_{(4,1)} = -q^2,\quad
f_{(5,2)} = -2q^3,\quad
f_{(7,1)} = -q^4,\quad
f_{(5,5)} = q^5,\quad \nonumber\\
f_{(7,4)} &= 2q^6,\quad
f_{(10,1)} = -q^7,\quad
f_{(8,5)} = q^8,\quad
f_{(11,2)} = -2q^9,\quad
f_{(7,7)} = -q^9,\quad \nonumber\\
f_{(13,1)} &= -q^{11},\quad
f_{(11,5)} = q^{12},\quad
\cdots\end{aligned}$$ where we have written $\beta$ in the fundamental weights basis. (Because $f_{(m,n)} = f_{(n,m)}$, we have only written those terms with $m \geq n$.) The $q$-power of this $f_\beta$ is, up to overall constant, $$\frac{(\beta,\beta)}{12}.\label{q-power}$$ In the $q\rightarrow 1$ limit we have $$F_{\mathbf{3}_1^r}^{SU(3)}(x_1,x_2,1) = \frac{x_1^{1/2}-x_1^{-1/2}}{x_1+x_1^{-1}-1}\frac{x_2^{1/2}-x_2^{-1/2}}{x_2+x_2^{-1}-1}\frac{x_1^{1/2}x_2^{1/2}-x_1^{-1/2}x_2^{-1/2}}{x_1x_2+x_1^{-1}x_2^{-1}-1}.\label{classicallimit}$$ Alternatively, we can start with $SU(3)$ colored Jones polynomials of the trefoil [@GMV] and ‘t Hooft resum : $$J_{\mathbf{3}_1^r,n_1,n_2}^{SU(3)}(q=e^\hbar) \xrightarrow{\text{`t Hooft resum}}\xrightarrow{\hbar \rightarrow 0} \frac{1}{x_1+x_1^{-1}-1}\frac{1}{x_2+x_2^{-1}-1}\frac{1}{x_1x_2+x_1^{-1}x_2^{-1}-1}.$$ Note that (\[q-power\]) and (\[classicallimit\]) completely characterize $F_{\mathbf{3}_1^r}(x_1,x_2,q)$.
In general the following version of Melvin-Morton conjecture was proven by D. Bar-Natan and S. Garoufalidis in [@BG] :
For arbitrary gauge group $G$ of rank $r$, the semi-classical limit of $F_K(x_1,\cdots,x_r,q)$ should be $$\boxed{
F_K^G(x_1, \cdots, x_r,1) = \prod_{\alpha\in \Delta^+}\frac{x^{\alpha/2}-x^{-\alpha/2}}{\Delta_K(x^\alpha)}
}$$
For torus knots, we found the following explicit expression :
For $K = T_{s,t}$, $f_\beta(q)$ is a monomial of degree $\frac{(\beta,\beta)}{2st}$, up to an overall $q$-power. More precisely, $$\label{torus knot F_K}
F_{T_{s,t}}^G \cong \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\mathbf{1}(\beta,w_1,w_2)N_{\frac{1}{st}(\beta + tw_1(\rho)+sw_2(\rho))} q^{\frac{(\beta,\beta)}{2st}}$$ where $$\mathbf{1}(\beta,w_1,w_2):=\begin{cases} 1 &\text{ if }\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho)) \in P_+\cap (Q+\rho)\\ 0&\text{ otherwise}\end{cases}$$
This can be derived either directly from (\[integralZhat\]) by using plumbing description or by reverse-engineering using the higher rank surgery formula that we discuss below.
Here we present a direct derivation. Recall from [@GM] that the complement of $T_{s,t}$ has a plumbing description as in Figure \[torus knot figure\], where $0<t'<t$, $0<s'<s$ are chosen such that $st'\equiv -1 (\text{mod }t)$ and $ts' \equiv -1 (\text{mod }s)$.
(-2,0) node\[above\][$-\frac{t}{t'}$]{} circle(0.5ex)– (0,0) node\[above\][$-1$]{} circle(0.5ex) – (2,0) node\[above\][$-\frac{s}{s'}$]{} circle(0.5ex); (0,-2) node\[below\][$-st$]{} circle(0.5ex)–(0,0);
The linking matrix is $$\mathrm{B} = \begin{pmatrix} -st & 1 & 0 & 0 \\ 1 & -1 & 1 & 1 \\ 0 & 1 & (-\frac{t}{t'}) & 0 \\ 0 & 1 & 0 & (-\frac{s}{s'})\end{pmatrix}$$ where $(-\frac{t}{t'})$ and $(-\frac{s}{s'})$ should be understood as block matrices corresponding to the continued fractions. To compute the integral (\[integralZhat\]) with $x_{-st}$ left unintegrated, we just have to replace the theta function $\Theta^{-\mathrm{B}}(x^{-1},q)$ with $$\Theta^{-\mathrm{B}'}(x^{-1},q) \cong \sum_{\alpha\in Q^{V'}}q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha)-(\alpha,\delta)}\prod_{v\in V'}x_v^{-(\mathrm{B}'\alpha + \delta)}\cdot x_{-st}^{-\alpha_{-1}-\rho}$$ where $V' = V \setminus \{v_{-st}\}$ and $\mathrm{B}'$ is the corresponding sub-linking matrix. Set $\beta = -\alpha_{-1}-\rho$. We need to multiply $\Theta^{-\mathrm{B}'}(x^{-1},q)$ with $$\prod_{v\in V'}\prod_{1\leq i\leq r}\qty(\sum_{w\in W}(-1)^{l(w)}x_v^{w(\rho)})^{2-\deg v}$$ and take the constant term with respect to variables $x_v$, $v\in V'$. As $2-\deg v$ is non-zero for only 3 vertices (the central vertex $v_{-1}$ and the 2 terminal vertices) it is pretty easy to compute. The only contributions come from those $\alpha$’s such that $\mathrm{B}'\alpha + \delta$ takes values $w_1(\rho), w_2(\rho)$ on the terminal vertices for some $w_1,w_2\in W$, a value in $Q+\rho$ in the central vertex, and $0$ on all the other vertices. Using simple linear algebra, it is easy to check that for those $\alpha$’s, $$q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha) - (\alpha,\delta)} = q^{\frac{(\beta,\beta)}{2st}+C}$$ for some constant $C$ independent of $\alpha$, and that $\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho))$ is the value of $\mathrm{B}'\alpha + \delta$ on the central vertex. This proves (\[torus knot F\_K\]).
Just as in [@GM], we can use surgery formula for these higher rank $F_K$ to compute higher rank $\hat{Z}_b(S_{p/r}^3(K))$. For instance, surgery on $\mathbf{3}_1^r$ gives us the following $\hat{Z}$’s ($SU(3)$ analog of the table 1 in [@GM]) :
$r$ $S_{-1/r}^3(\mathbf{3}_1^r)$ $\hat{Z}_0^{SU(3)}(S_{-1/r}^3(\mathbf{3}_1^r))$
----- ------------------------------ ----------------------------------------------------------------------------------------------------
$1$ $\Sigma(2,3,7)$ $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$
$2$ $\Sigma(2,3,13)$ $1 -2q +2q^3 -q^4 +2q^{10} -2q^{11} -2q^{14} +2q^{16} +2q^{19} -2q^{20} +4q^{21} -4q^{23} -\cdots$
$3$ $\Sigma(2,3,19)$ $1 -2q +2q^3 -q^4 +2q^{16} -2q^{17} -2q^{20} +2q^{22} +2q^{25} -2q^{26} +4q^{33} -4q^{35} -\cdots$
$4$ $\Sigma(2,3,25)$ $1 -2q +2q^3 -q^4 +2q^{22} -2q^{23} -2q^{26} +2q^{28} +2q^{31} -2q^{32} +4q^{45} -4q^{47} -\cdots$
$5$ $\Sigma(2,3,31)$ $1 -2q +2q^3 -q^4 +2q^{28} -2q^{29} -2q^{32} +2q^{34} +2q^{37} -2q^{38} +4q^{57} -4q^{59} -\cdots$
$r$ $\Sigma(2,3,6r+1)$ $\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{36r+6, 3(6r+1)w_1(\rho)+ 2(6r+1)w_2(\rho)+ 6\rho}$
In fact it is easy to check that for $K=T_{s,t}$, $$\begin{aligned}
\mathcal{L}_{-1/r}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K(\mathbf{x},q) \right] &\cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{st(rst+1),t(rst+1)w_1(\rho) + s(rst+1)w_2(\rho) +st\rho}\\
&\cong \hat{Z}_0^G(\Sigma(s,t,rst+1))\end{aligned}$$ We conjecture the following surgery formula (analogous to Conjecture 1.7 of [@GM]) :
Let $K\subset S^3$ be a knot. Then $$\boxed{
\hat{Z}_b^G(S_{p/r}^3(K)) \cong \mathcal{L}_{p/r}^{(b)}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K^G(\mathbf{x},q) \right]
}$$ whenever the RHS makes sense.
Moreover we conjecture that our 0-surgery formula in [@CGPS] holds for higher rank as well :
Let $K\subset S^3$ be a knot. Then $$\boxed{
\hat{Z}_0^G(S_{0}^3(K)) \cong \frac{1}{|W|}f_\rho^G(K)
}$$
Symmetric representations and large $N$
=======================================
Specialization to symmetric representations
-------------------------------------------
In this section we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations. We restrict our attention to $G=SU(N)$. Although the unreduced version seems to be better-behaving, to get $F_K$ with symmetric colorings (symmetric powers of the defining representation), we need to use the reduced version : $$F_K^{\rm red}(\mathbf{x},q) := \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}f_\beta(q)\frac{\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}}{\sum_{w\in W}(-1)^{l(w)}x^{w(\rho)}}$$ In particular, the symmetrically colored $F_K$ corresponds to the following specialization : $$F_K^{\rm sym}(x,q) := F_K^{\rm red}((x,q,\cdots,q),q)$$ That is, we set $x_2 = \cdots = x_r = q$. A version of quantum volume conjecture [@FGS] states that this should be annihilated by the symmetrically colored quantum A-polynomial[^6] : $$\label{QuantVolConj}
\boxed{
\hat{A}^{\rm sym}(\hat{x},\hat{y},q)F_K^{\rm sym}(x,q) = 0
}$$
[**Right-handed trefoil.**]{} For the right-handed trefoil, $F_{\mathbf{3}_1^r}^{\rm sym}(x,q)$ for $SU(N)$ with the first few values of $N$ look like the following :
- $SU(2)$ $$\begin{aligned}
F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[ (-q + q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots) \right.\\
&\quad + (x + x^{-1})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
&\quad + (x^2 + x^{-2})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
&\quad + (x^3 + x^{-3})(q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
&\quad + (x^4 + x^{-4})(-q^6 - q^8 + q^{13} + q^{16} - \cdots)\\
&\quad\left. + \cdots \right]
\end{aligned}$$
- $SU(3)$ $$\begin{aligned}
F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(-2q -2q^2 +2q^4 +4q^5 +4q^6 +4q^7 +2q^8 -2q^{10} -4q^{11} -6q^{12} +\cdots) \right.\\
&\quad + (q^{1/2}x+q^{-1/2}x^{-1})q^{1/2}(-1 -2q -q^2 +q^3 +3q^4 +4q^5 +4q^6 +3q^7 +q^8 -q^9 +\cdots)\\
&\quad + (qx^2 + q^{-1}x^{-2})(-q -q^2 +2q^4 +3q^5 +4q^6 +3q^7 + 2q^8 -2q^{10} +\cdots)\\
&\quad + (q^{3/2}x^3 + q^{-3/2}x^{-3})q^{1/2}(q^3 +2q^4 +3q^5 +3q^6 +2q^7 +q^8 +\cdots)\\
&\quad + (q^2x^4 + q^{-2}x^{-4})(q^3 +q^4 +2q^5 +2q^6 +2q^7 +q^8 +\cdots)\\
&\quad\left. +\cdots \right]
\end{aligned}$$
- $SU(4)$ $$\begin{aligned}
F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(q^{-2} +q^{-1} -2 -4q -8q^2 -7q^3 -7q^4 +\cdots) \right.\\
&\quad + (qx + q^{-1}x^{-1})(q^{-2} -1 -5q -6q^2 -8q^3 -5q^4 -2q^5 +\cdots)\\
&\quad + (q^2x^2 + q^{-2}x^{-2})(-2 -3q -6q^2 -5q^3 -5q^4 +4q^6 +\cdots)\\
&\quad + (q^3x^3 + q^{-3}x^{-3})(-q^{-1} -1 -3q -3q^2 -4q^3 -2q^4 +5q^6 +9q^7 +\cdots)\\
&\quad + (q^4x^4 + q^{-4}x^{-4})(-1 -q -2q^2 -q^3 -q^4 +2q^5 +4q^6 +8q^7 +11q^8 +\cdots)\\
&\quad\left. + \cdots \right]
\end{aligned}$$
Note that the overall factor is $\frac{1}{2}$ instead of $\frac{1}{N!}$. This is due to reduction of the Weyl symmetry to $\mathbb{Z}_2$.
It is easy to verify (\[QuantVolConj\]) numerically in this case, using the super-A-polynomial for the right-handed trefoil $$\hat{A}^{\rm super}(\hat{x},\hat{y},a,q) = a_0 + a_1\hat{y} + a_2\hat{y}^2$$ where $$\begin{aligned}
a_0 &= -\frac{(-1+\hat{x})(-1+aq\hat{x}^2)}{a\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\
a_1 &= \frac{(-1+a\hat{x}^2)(-a^2\hat{x}^2 + aq^3\hat{x}^2 +aq\hat{x}(1+\hat{x}+a(-1+\hat{x})\hat{x})-q^2(1+a^2\hat{x}^4))}{a^2q\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\
a_2 &= 1\end{aligned}$$ with $a$ specialized to $q^N$.
Future direction : large $N$
----------------------------
We end with some intriguing open questions.
Is there a HOMFLYPT version of $F_K$?
Conjecturally, there is a HOMFLYPT version $F_K^{\rm super}(x,q,a)$ such that we can recover $F_K^{SU(N),{\rm sym}}(x,q)$ as a certain specialization (e.g. $a\rightarrow q^N$), and $$\boxed{\hat{A}^{\rm super}(\hat{x},\hat{y},a,q)F_K^{\rm super}(x,q,a) = 0.}$$ We expect to have an expansion of the form $$F_{K}^{\rm super}(x,q,a) \cong \frac{1}{2}\sum_{n\in \mathbb{Z}}(q^{-1} a^{1/2} x)^n f_n(q,a)$$ with $f_{-n}(q,a)=f_{n}(q,a)$, and that it has the following Weyl symmetry : $$F_{K}^{\rm super}(x^{-1},q,a) = F_{K}^{\rm super}(a^{-1}q^2x,q,a).$$
We can also ask another, much conjectural question regarding homology theories :
What’s $\mathcal{H}_{b,{\rm BPS}}^G$ categorifying $\hat{Z}_b^G(q)$? Is there a family of differentials analogous to that of [@DGR]?
We will pursue these questions in our future work.
$\mathrm{Spin}^{T^\vee}$-structures {#spint}
===================================
In this section we define a generalization of $\mathrm{Spin}^c$-structures which we will call $\mathrm{Spin}^{T^\vee}$-structures. These are what the labels $b$ of $\hat{Z}_b^G$ are geometrically.
Let $G$ be a Lie group and let $T < G$ be a maximal torus, and let $T^\vee \cong \mathfrak{h}^*/P$ be its dual.
First, the group $\mathrm{Spin}^{T^\vee}(n)$ is defined to be $$\mathrm{Spin}^{T^\vee}(n) := \mathrm{Spin}(n) \times_{\mathbb{Z}_2}T^\vee.$$ A $\mathrm{Spin}^{T^\vee}$-structure $\mathfrak{s}$ on an $n$-manifold $M^n$ is a principal $\mathrm{Spin}^{T^\vee}(n)$-bundle with an isomorphism between the natural $SO(n)$-bundle associated to $\mathfrak{s}$ and the frame bundle of $M^n$.
Note that the Weyl group naturally acts on $\mathrm{Spin}^{T^\vee}(n)$. It is important that we regard $T^\vee \cong \mathfrak{h}^*/P$ as a variety, not just a topological torus. There’s a natural first Chern class map $$\begin{aligned}
\mathrm{Spin}^{T^\vee}(M) &\rightarrow H^2(M;P) \\
\mathfrak{s}&\mapsto c_1(\mathfrak{s}).\end{aligned}$$ The following propositions are analogous to those of $\mathrm{Spin}^c$-structures.
$\mathrm{Spin}^{T^\vee}(M)$ is affinely isomorphic to $H^2(M;Q)$
On a 4-manifold $W$ with boundary $Y$, $\mathrm{Spin}^{T^\vee}(W)$ is canonically isomorphic to the set $\mathcal{C}_W\subseteq H^2(W;P)$ of characteristic covectors of $H_2(W,Y;Q)$; i.e. $\nu\in \mathcal{C}_W\subseteq H^2(W;P)$ iff $$(\nu,x) = \frac{1}{2}\langle x,x\rangle\,\mathrm{mod}\,2$$ for any $x\in H_2(W,Y;Q)$.
Good old Gauss sum reciprocity {#Gauss}
==============================
In this section we derive a formula which generalizes appendix A of [@GPPV] to any graph manifold with weakly negative definite linking matrix, with arbitrary gauge group $G$. Let $Y$ be a graph manifold obtained by a plumbing on a connected decorated graph $\Gamma$. (Each vertex $v$ of $\Gamma$ is decorated by an integer $a_v$ ‘framing’ and a nonnegative integer $g_v$ ‘genus’. We have set $g_v=0$ in Definition \[integralZhat\].) Then the WRT invariant $Z_{G_k}(Y)$ can be computed by [^7] $$Z_{G_k}(Y) \cong \sum_{\rm colorings}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \label{eq2.1}$$ where the vertex and the edge factors are[^8] $$\mathcal{V} = t_{\lambda\lambda}^{a_v} s_{\rho\lambda}^{2-2g_v-\deg v}$$ $$\mathcal{E} = s_{\mu\lambda}.$$ Here the $s, t$ matrices are as usual [^9] $$s_{\lambda \mu} = \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda),\mu)},$$ $$t_{\lambda \mu} = \delta_{\lambda \mu} q^{\frac{1}{2}(\lambda,\lambda)}q^{-\frac{1}{2}(\rho,\rho)}$$ with $q=e^{\frac{2\pi i}{m k}}$ and set of (shifted) colors being $$\lambda,\mu\in C = \{\lambda\in P_+ + \rho \,\vert\, (\lambda,\theta^\vee) < k\}.$$ Then these $s,t$ matrices are invariant (up to sign) under the action of the affine Weyl group $$W^a = W \ltimes kQ^\vee.$$ In fact, $C$ is simply the fundamental domain $P/W^a$. Using this fact, we can manipulate the form of $Z_{G_k}(Y)$ to write it in a Gauss sum reciprocity-friendly way : $$\begin{aligned}
(\ref{eq2.1}) &= \frac{1}{|W^V|} \sum_{\text{coloring}\in W(C)^{V}}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \nonumber\\
&= \frac{1}{|W^V|} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \nonumber\\
&\quad\times \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}q^{\frac{a_v}{2}(\lambda_v,\lambda_v)} \prod_{(u_1,u_2)\in E}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda_{u_1}),\lambda_{u_2})} \nonumber\\
&= \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} \nonumber\\
&\quad \times\sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}q^{\frac{1}{2}(\lambda, \mathrm{B}_s\lambda)} \label{midstep}\end{aligned}$$ Here, the self-adjoint matrix $\mathrm{B}_s$ is the symmetrization $\frac{1}{2}(\mathrm{B}_s^0 + {\mathrm{B}_s^0}^{\dagger})$ of a bilinear form $\mathrm{B}_s^0$ on $P^{V}$ characterized by the following properties :
- For each $v$, $(\mu_v,\mathrm{B}_s^0\lambda_v) = a_v(\mu_v,\lambda_v)$
- For each $u\neq v\in V$ with $(u,v)\not\in E$, $(\mu_u, \mathrm{B}_s^0 \lambda_v) = 0$
- For each $u\neq v\in V$ with $(u,v)\in E$, $(\mu_u,\mathrm{B}_s^0\lambda_v) = (\mu_u,w_{uv}(\lambda_v))$ for some $w_{uv}\in W$
- For each cycle $c=[v_1, \cdots, v_m]\in H_1(\Gamma)$, $\prod_{i=1}^{m}w_{v_i v_{i+1}} = s_c\in W$.
So far we have expressed the WRT invariant in the following form : $$Z_{G_k}(Y) = \sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}Z_{G_k}(Y,s).$$ Each $Z_{G_k}(Y,s)$ can be considered as the contribution from almost-Abelian flat connections twisted by $s$ [@CGPS]. In the following, let’s restrict our attention to the Abelian sector ($s=\text{id}$) for simplicity. In this case $\mathrm{B}_s = \mathrm{B}$ is the usual linking matrix.
To extend the range of summation $W(C)$ to $P/kQ^\vee$, i.e. to make sense of the summation even for colors upon which the action of $W^a$ is not free, we need to regularize the linear term $\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}$. Let $\omega_1, \cdots, \omega_r\in P$ be a $\mathbb{Z}$-linear basis of $P$ (e.g. fundamental weights). We can then write $\lambda_v = \sum_{i = 1}^{r}n_{vi}\omega_i$ for some $n_{v1}, \cdots, n_{vr}\in \mathbb{Z}$. Thanks to Weyl denominator formula, we have $$\begin{aligned}
\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}
&= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(q^{\frac{(\lambda_v,\alpha)}{2}} - q^{-\frac{(\lambda_v,\alpha)}{2}}) \right)^{2-2g_v-\deg v} \nonumber\\
&= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(\prod_{1\leq i\leq r}x_{vi}^{\frac{(\omega_i,\alpha)}{2}} - \prod_{1\leq i\leq r}x_{vi}^{-\frac{(\omega_i,\alpha)}{2}}) \right)^{2-2g_v-\deg v}\bigg|_{x_{vi} = q^{n_{vi}}}\label{regularize}\end{aligned}$$ In case $\deg v > 2$, this expression can be singular only when $$\left| \prod_{1\leq i\leq r}y_i^{\frac{(\omega_i,\alpha)}{2}}\right| = 1$$ for some $\alpha\in \Delta_+$. In terms of new variables $z_i := \log |x_i|$, this is simply $$\sum_{1\leq i\leq r} (\omega_i,\alpha) z_i = 0.$$ These are precisely the walls (hyperplanes) for Weyl reflections. Deforming the origin $z_1 = \cdots = z_r = 0$ to a complement of these walls is the same as a choice of a Weyl chamber. Moreover, for each choice of such a Weyl chamber, we can expand (\[regularize\]) as a geometric series. Therefore, we can regularize the linear term by taking an average of $|W|$ number of $q$-series, each determined by a choice of a Weyl chamber. To sum up, we can re-express the linear term in the following form : $$\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}
\overset{\text{regularize}}{=\joinrel=\joinrel=} \frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}} n_\ell \,q^{(\lambda,\ell)}$$ with $\delta_v = (2-2g_v-\deg v)\rho \mod Q$ and $n_\ell\in \mathbb{Z}$.
With this regularization in hand, we extend the range of summation and apply the Gauss sum reciprocity.[^10] Let $n\in \mathbb{Z}^+$ be such that $nP\subseteq Q^\vee\subseteq P$. Then we have[^11] $$\begin{aligned}
Z_{G_k}(Y,\text{id}) &\Rightarrow \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \frac{1}{|(Q^\vee/nP)^V|} \nonumber\\
&\quad\times \sum_{\lambda \in P^{V}/nkP^{V}}q^{\frac{1}{2}(\lambda,\mathrm{B}\lambda)} \cdot \left(\frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}}n_\ell q^{(\lambda,\ell)}\right) \nonumber\\
&= \frac{(-1)^{|\Delta_+||V|}}{|W|^{|V|+b_1(\Gamma)+1}}\left(\frac{i^{|\Delta_+|}}{|P/Q^\vee|^{1/2}} \right)^{-|V|-b_1(\Gamma)+1} q^{-\frac{\sum a_i}{2}(\rho,\rho)}k^{\frac{r}{2}(b_1(\Gamma) - 1)} \nonumber\\
&\quad\times \frac{e^{\frac{\pi i}{4}\sigma(\mathrm{B})}}{|\det \mathrm{B}|^{1/2}}\sum_{a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}}e^{-\pi i k(a,\mathrm{B}^{-1}a)}\sum_{b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}}e^{-2\pi i(a,\mathrm{B}^{-1}b)} \sum_{\ell\in \mathrm{B}Q^{V}+b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\label{rawresult}\end{aligned}$$ Note that $a$ and $b$ takes values in different sets : $a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}$ while $b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}$. The $a$ labels should be understood as ‘Abelian flat connections’ and the $b$ labels are ‘$\mathrm{Spin}^{T^\vee}$-structures’ of Section \[spint\].
Generalizing the definition of $\hat{Z}_b$ in appendix A of [@GPPV], we define $\hat{Z}_b^G$ for a weakly negative definite graph manifold $Y$ with gauge group $G$ as follows : $$\hat{Z}_b^G(Y;q) :\cong |W|^{-|V|-b_1(\Gamma)}q^{-\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}\sum_{\ell\in \mathrm{B}Q^{V} + b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\in |W|^{-|V|-b_1(\Gamma)} q^{\Delta_b}\mathbb{Z}[[q]] \label{Zhatdef}$$ where $$b\in (Q^{V}+\delta)/\mathrm{B}Q^{V},$$ $$\Delta_b = -\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho) + \min_{\ell\in \mathrm{B}Q^{V}+b}-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)\in \mathbb{Q}.$$
Here $\Delta_b$ should be thought of as a relative conformal weight among different $b$’s for the same 3-manifold. This is because we ignored the framing factor in (\[eq2.1\]), which shifts the overall $q$-power. It is easy to check that this definition is equivalent to Definition \[Zhatsgm\]. Finally, it is also clear from our derivation (\[rawresult\]) that the higher rank analog of the unfolded $S_{ab}$ matrix (appearing in Conjecture 2.6 of [@GPPV]) for the Abelian sector is simply $$S_{ab} = e^{-2\pi i (a,\mathrm{B}^{-1}b)}.$$ Of course for graph manifolds with $b_1(\Gamma)>0$ there are different sectors corresponding to each $s\in W^{b_1(\Gamma)}$, and we have to combine all of them to compare with the WRT invariant. See [@CGPS].
[^1]: By considering homology orientations, we should be able to fix the sign as well.
[^2]: Still, it is possible to derive our formula (\[integralZhat\]) from Mariño’s Chern-Simons matrix model or vice versa in case of Seifert manifolds. This is because Gaussian measure is the same as Laplace transform accompanied by $S_{ab}$ matrix.
[^3]: E.g. for $SU(2)$, $N_\ell^{A_1} = \text{sgn}((\ell,\alpha_1))$, and for $SU(3)$, $N^{A_2}_\ell = \mathrm{sgn}(\prod_{\alpha\in \Delta_+}(\ell,\alpha)) \min\{|(\ell,\alpha_1)|,|(\ell,\alpha_2)|\}$.
[^4]: In our notation, $K_{m,n}$ denotes the double twist knot with $m$ and $n$ full twists.
[^5]: That $\hat{Z}$’s for Brieskorn spheres should be expressed as sums of higher rank false theta functions was envisaged earlier in [@CCFGH].
[^6]: Of course in principle there should be quantum A-polynomials for all colors, with rank number of $x$ and $y$ variables, which annihilate $F_K^G({\bf x},q)$. However for simplicity we only consider symmetric representations here.
[^7]: Up to a framing factor, which is an integer power of $\zeta^3 = e^{2\pi i \frac{(k-h^\vee)\dim G/k}{24}} = e^{\frac{\dim G}{12} \pi i} q^{-\frac{m h^\vee\dim G}{24}}$.
[^8]: Here, $2-2g_v-\deg v$ should probably be understood as the Euler characteristic of the Riemann surface $\Sigma_{g_v,\deg v}$ of genus $g_v$ with $\deg v$ punctures.
[^9]: We follow the convention and notations used in [@BK]. In particular, $P$ is the weight lattice, $Q$ is the root lattice, and $Q^\vee$ is the coroot lattice.
[^10]: In fact, this process of extending the range of colors and regularizing it is quite subtle, and in fact this may not be a strict equality for some values of $k$. We will deal with this subtlety in more detail elsewhere. Anyway the end result will be a topological invariant.
[^11]: See [@DT] for the version of Gauss sum reciprocity formula we use.
| ArXiv |
---
abstract: 'The correlation-driven Mott transition is commonly characterized by a drop in resistivity across the insulator-metal phase boundary; yet, the complex permittivity provides a deeper insight into the microscopic nature. We investigate the frequency- and temperature-dependent dielectric response of the Mott insulator $\kappa$-(BEDT-TTF)$_{2}$Cu$_2$(CN)$_3$ when tuning from a quantum spin liquid into the Fermi-liquid state by applying external pressure and chemical substitution of the donor molecules. At low temperatures the coexistence region at the first-order transition leads to a strong enhancement of the quasi-static dielectric constant $\epsilon_1$ when the effective correlations are tuned through the critical value. Several dynamical regimes are identified around the Mott point and vividly mapped through pronounced permittivity crossovers. All experimental trends are captured by dynamical mean-field theory of the single-band Hubbard model supplemented by percolation theory.'
author:
- 'A. Pustogow'
- 'R. Rösslhuber'
- 'Y. Tan'
- 'E. Uykur'
- 'M. Wenzel'
- 'A. Böhme'
- 'A. Löhle'
- 'R. Hübner'
- 'Y. Saito'
- 'A. Kawamoto'
- 'J. A. Schlueter'
- 'V. Dobrosavljević'
- 'M. Dressel'
date:
-
-
-
title: 'Low-Temperature Dielectric Anomalies at the Mott Insulator-Metal Transition'
---
[^1]
[^2]
[^3]
The insulator-metal transition (IMT) remains the main unresolved basic science problem of condensed-matter physics. Especially intriguing are those IMTs not associated with static symmetry changes, where conventional paradigms for phase transitions provide little guidance. Early examples of such behavior are found in certain disorder-driven IMTs [@Shklovskii1984]. In recent years, IMTs with no symmetry breaking were also identified around the Mott transition [@Imada1998], which bears close connection to exotic states of strongly-correlated electron matter such as superconductivity in the cuprates. From a theoretical point of view, the single-band Hubbard model is at present well understood [@Georges1996; @Vollhardt2012], and is found to be in excellent agreement with experiments [@Limelette2003; @*Limelette2003a; @Hansmann2013; @Kagawa2005; @*Kagawa2004; @*Furukawa2015]. While commonly concealed by antiferromagnetism, recent development in the field of organic quantum spin liquids (QSL) enabled us to study the low-temperature Mott IMT in absence of magnetic order [@Kurosaki2005; @*Shimizu2003; @Shimizu2016; @Itou2017; @Li2019; @Furukawa2018; @Pustogow2018], revealing finite-frequency precursors of the metal already on the insulating side [@Pustogow2018].
The Mott insulator and the correlated metal converge at the critical endpoint $T_{\rm crit}$ (Fig. \[fig:phase-diagram\]). The former is bounded by a quantum-critical region along the quantum Widom line (QWL) [@Terletska2011; @*Vucicevic2013; @Furukawa2015; @Pustogow2018; @Dobrosavljevic1997]. On the metallic side, resistivity maxima at the “Brinkman-Rice” temperature $T_{\rm BR}$ signal the thermal destruction of resilient quasiparticles [@Radonjic2012; @*Deng2013] and the crossover to semiconducting transport. Below $T_{\rm crit}$, the IMT is of first order and comprises an insulator-metal coexistence regime [@Georges1996; @Terletska2011; @Vucicevic2013]. It is currently debated whether electrodynamics is dominated by closing of the Mott gap or by spatial inhomogeneity, fueled by recent low-temperature transport studies [@Furukawa2018].
![(a) Tuning the bandwidth $W$, for instance by chemical or physical pressure, transforms a Mott insulator to a correlated metal. Dynamical mean-field theory predicts a first-order transition with phase coexistence up to the critical endpoint [@Georges1996], and a quantum-critical regime associated with the quantum Widom line (QWL) above $T_{\rm crit}$ [@Vucicevic2013]. The metallic state is confined by the Brinkman-Rice temperature $T_{\rm BR}$, the coherent Fermi-liquid regime by $T_{\rm FL}$. When interactions $U$ are comparable to $W$, and $ T \gg T_{\rm crit}$, semiconducting behavior prevails; neither a gap nor a quasiparticle peak are stabilized. (b-d) Resistivity signatures of the crossovers.[]{data-label="fig:phase-diagram"}](phase-diagram38.pdf){width="0.8\columnwidth"}
Phase coexistence around the first-order line ($T<T_{\rm crit}$) emerges from bistability of the insulating and metallic phases between closing of the Mott gap at $U_{c1}$ and demise of the metal at $U_{c2}$ [@Georges1996]. One generally expects hysteretic behaviors when tuning across a first-order transition. Seminal transport and susceptibility experiments indeed found a pronounced hysteresis in Mott insulators with a magnetically-ordered ground state [@Lefebvre2000; @Limelette2003; @*Limelette2003a]. Unfortunately, analogue measurements with continuous pressure tuning are not feasible on QSL compounds, such as , due to the low temperatures ($T<20$ K) and high pressures ($p>1$ kbar) required to cross the first-order IMT.
A more direct insight into the coexistence region was provided by spatially resolved optical spectroscopy [@Sasaki2004]. The most compelling results came from near-field optical experiments on vanadium oxides by Basov and collaborators [@Qazilbash2007; @*Huffman2018; @McLeod2016] where a spatial separation of metallic and insulating regions upon heating could be visualized, in accord with x-ray studies [@Lupi2010; @Hansmann2013]; the range with hysteresis in $\rho(T)$ coincides with the observed phase coexistence. Although recent developments in cryogenic near-field instrumentation are rather promising [@McLeod2016; @Post2018; @*Pustogow2018s], they fall short of covering the regime $T<T_{\rm crit} \approx 15$ K required here and do not allow for pressure tuning. For this reason, we suggest dielectric spectroscopy as novel bulk-sensitive method in order to reveal the coexistence regime, distinguish the individual phases and obtain a deeper understanding of the dynamics around the IMT. The complex conductivity $\sigma_1+{\rm i}\sigma_2$ not only reveals the closing of the Mott gap but yields insight into the growth of metallic regions and the formation of quasiparticles as correlation effects decrease. In this Letter we tackle the fundamental question whether the electrodynamic response around the Mott IMT is overwhelmed by the gradual decrease of the Mott-Hubbard gap within a homogeneous insulating phase, or whether the effects of phase coexistence dominate. Furthermore, is it possible to distinguish on the metallic side between the coherent (quasiparticle) low-$T$ regime and incoherent transport at high-$T$? To answer these questions, we present temperature- and frequency-dependent dielectric measurements on a genuine Mott compound that is bandwidth-tuned across its first-order IMT. In addition to hydrostatic pressure we developed a novel approach of chemically substituting the organic donor molecules. The experimental findings are complemented by dynamical mean-field theory (DMFT) calculations, incorporating spatial inhomogeneities in a hybrid approach. We conclude that electronic phase segregation plays a crucial role, leading to percolative phenomena due to the separation of insulating and metallic regions, also allowing clear and precise mapping of different dynamical regimes around the IMT.
![The dielectric conductivity and permittivity of were measured as a function of temperature and frequency for various applied (a-e) pressures and (f-j) chemical substitutions \[introduction of Se-containing BEDT-STF molecules illustrated in (h) and (k)\] that drive the system across the Mott transition. (a,f) Starting from the insulator, $\sigma_1(T)$ grows with increasing $p$ or $x$; a metallic phase is stabilized below $T_{\rm BR}$, in accord with dc transport [@Kurosaki2005; @Furukawa2015; @Furukawa2018]. (b,c,g,h) In the Mott-insulating state $\epsilon_1(T)$ exhibits relaxor-ferroelectric behavior similar to the parent compound [@Abdel-Jawad2010; @Pinteric2014]. Extrinsic high-temperature contributions are subtracted. (d,i) The strong enhancement of $\epsilon_1(T)$ at the transition is a hallmark of a percolative IMT, as sketched in (l-o). (e,j) When screening becomes dominant in the metal, $\epsilon_1$ turns negative; $\sigma_1$ exhibits Fermi-liquid behavior below $T_{\rm FL}$. []{data-label="fig:sigma-eps_T"}](Fig_2_44.pdf){width="1\columnwidth"}
We have chosen single crystals for our investigations because this paradigmatic QSL candidate is well characterized by electric, optical and magnetic measurements [@Kurosaki2005; @*Shimizu2003; @Kezsmarki2006; @Kanoda2011; @*Zhou2017; @Furukawa2018; @Pustogow2018; @Culo2019]. Although the dimerized charge-transfer salt possesses a half-filled conduction band, strong electronic interaction $U \approx 250$ meV stabilizes a Mott-insulating state below the QWL ($T_{\rm QWL}\approx 185$ K at ambient pressure [@Pustogow2018; @Furukawa2015]). The effective correlation strength $U/W$ can be reduced by increasing the bandwidth $W$; for pressure $p > 1.4$ kbar the metallic state is reached at low temperatures [^4], with $T_{\rm crit} \approx 15$-20 K. In addition, we exploited a novel route of partially replacing the S atoms of the donor molecules by Se, where more extended orbitals lead to larger bandwidth \[see sketches in Fig. \[fig:sigma-eps\_T\](h,k)\]. The substitutional series ($0\leq x \leq 1$, abbreviated ) spans the interval ranging from a Mott insulator to a Fermi-liquid metal [^5]. Details on the sample characterization and experimental methods are given in Refs. . Here we focus on the out-of-plane dielectric response measured from $f=7$ kHz to 5 MHz down to $T=5$ K. Both physical pressure and STF-substitution allow us to monitor the permittivity while shifting the system across the first-order IMT.
Fig. \[fig:sigma-eps\_T\] displays the temperature-dependent conductivity and permittivity data of when $p$ rises (a-e) and $x$ increases in (f-j). The insulating state ($p<1.4$ kbar, $x<0.1$), characterized by ${\rm d}\sigma_1/{\rm d}T>0$, generally features small positive $\epsilon_1\approx 10$. The relaxor-like response previously observed in the parent compound below 50 K has been subject of debate [@Abdel-Jawad2010; @Pinteric2014]. The metallic state ($p>3$ kbar, $x>0.2$) is defined by ${\rm d}\sigma_1/{\rm d}T<0$ and, concomitantly, $\epsilon_1<0$ that becomes very large at low $T$ as itinerant electrons increasingly screen [^6]. This onset of metallic transport identifies $T_{\rm BR}$ [@Radonjic2012]; while thermal fluctuations prevail at higher $T$, the quasiparticle bandwidth is the dominant energy scale for $T<T_{\rm BR}$. Below $T_{\rm FL}$ the resistivity $\rho(T)\propto T^2$ indicates the Fermi-liquid state.
Right at the first-order IMT, however, the dielectric behavior appears rather surprising. When approaching the low-temperature phase boundary, $\epsilon_1$ rapidly increases by several orders of magnitude. This colossal permittivity enhancement is more pronounced in the quasi-static limit, $\epsilon_1\approx 10^5$ at $f=7.5$ kHz, and the peak value approximately follows a $f^{-1.5}$ dependence. The overall range in $T$ and $p$/$x$ of the divergency is robust and does not depend on the probing frequency; detailed analysis on the dynamic properties is given in [@Rosslhuber2019; @Saito2019].
In Fig. \[fig:sigma-eps\_p\_x\_DMFT\](a,b) the pressure evolution of $\sigma_1$ and $\epsilon_1$ is plotted for fixed $T$. At $T=10$ K, $\sigma_1(p)$ rises by six orders of magnitude in the narrow range of 1 kbar and $\Delta x = 0.1$. This behavior flattens to a gradual transition above 20 K, associated with the quantum-critical crossover at the QWL. The inflection point shifts to higher $p$, in accord with the positive slope of the phase boundary [@Pustogow2018] associated with the rising onset of metallicity at $T_{\rm BR}$. The series exhibits similar behavior \[Fig. \[fig:sigma-eps\_p\_x\_DMFT\](c,d)\]: around the critical concentration of $x\approx 0.12$ a drastic increase in $\sigma_1$ is observed at low $T$ that smears out as $T$ rises. The maximum in $\epsilon_1(x)$ is reached for $x=0.16$ but broadens rapidly upon heating.
![(a) The Mott IMT of $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ appears as a rapid increase of $\sigma_1(p)$ that smoothens at higher $T$; above $T_{\rm crit}$ a gradual crossover remains. (b) $\epsilon_1(p)$ exhibits a sharp peak below $T_{\rm crit}$. The results at $f=380$ kHz are plotted on a logarithmic scale. (c,d) Similar behavior is observed for chemical BEDT-STF substitution. (e,f) Fixed-temperature line cuts of our hybrid DMFT simulations (see text) as a function of correlation strength $U/W$ and $T/W$ [@Vucicevic2013; @Pustogow2018] resemble the experimental situation in minute detail, including the shift of the IMT with $T$. The lack of saturation of $\sigma_1(T\rightarrow 0)$ seen in DMFT modeling reflects the neglect of elastic (impurity) scattering in the metal (outside the coexistence region).[]{data-label="fig:sigma-eps_p_x_DMFT"}](Fig_3_41.pdf){width="1\columnwidth"}
{width="90.00000%"}
The Mott IMT is based on the idea that a reduction of electronic correlations, i.e. rise of $W/U$, gradually closes the Mott-Hubbard gap: a coherent charge response develops, causing a finite metallic conductivity. Pressure-dependent optical studies on several organic Mott insulators actually observe this behavior [@Faltermeier2007; @*Merino2008; @*Dumm2009; @Li2019]. It was pointed out [@Aebischer2001] that even in the case of certain second-order phase transitions, a continuously vanishing charge gap might induce an enhancement in $\epsilon_1$, perhaps leading to a divergence at low $T$. Probing the optical response at THz frequencies is actually a convenient method to monitor the gap contribution to the permittivity. From $p$ and $T$ sweeps across the Mott IMT of very different materials an increase by a factor of 10 is consistently reported [@Qazilbash2007; @Li2019]; in the case of we find it even smaller [@Saito2019]. Hence, the dielectric catastrophe of $\epsilon_1 \approx 10^5$ observed in our pressure and substitution-dependent dielectric experiments evidences an additional effect.
Treating the fully-frustrated model at half filling, even at $T=0$ DMFT finds metallic and insulating solutions coexisting over an appreciable range of $U/W$ [@Georges1996]. This may result in a spatial segregation of these distinct electronic phases. Such a picture resembles composite materials, such as microemulsions, composites or percolating metal films [@vanDijk1986; @Clarkson1988a; @*Clarkson1988b; @Pecharroman2000; @Nan2010; @Hovel2010]. Classical percolation is not a thermodynamic phase transition, but a statistical problem that has been studied theoretically for decades by analytical and numerical methods in all details; one of the key predictions is the divergency of $\epsilon_1$ when approaching the transition from either side, with characteristic scaling [@Dubrov1976; @Efros1976; @Bergman1977; @*Bergman1978]. Over a large parameter range the dielectric properties are well described by the Maxwell-Garnett or Bruggeman effective medium approaches [@Choy2015].
To further substantiate this physical picture in quantitative details, we carried out theoretical modeling of the systems under study. We calculated $\epsilon_1+{\rm i}\epsilon_2$ using DMFT for a single-band Hubbard model, and obtained the respective responses for both the insulating and the metallic phase around the Mott point [@Vucicevic2013]. The DMFT phase diagram (Fig. \[fig:phase-diagram\]) also features an intermediate coexistence region below $ k_{B} T_{\rm crit} \approx 0.02 W$. In accord with the analysis of our experimental results, we assumed a smoothly varying metallic fraction $x$ within the phase coexistence region. To obtain the total dielectric response, we solved an appropriate electrical-network model representing such spatial inhomogeneity utilizing the standard effective-medium approximation (for details, see Ref. ).
Our hybrid DMFT simulation yields excellent agreement with experiment – both pressure-tuning and chemical substitution – as illustrated in Fig. \[fig:sigma-eps\_p\_x\_DMFT\](e,f) and in the false-color plots in Fig. \[fig:contour-plots\]. We find that the colossal peak in $\epsilon_1$ is confined to the spatially inhomogeneous coexistence regime, exactly as observed in experiments. As correlation effects diminish, the dynamical conductivity (upper panels) increases from the Mott insulator to the Fermi liquid. The step in $\sigma_1$ and drop in $\epsilon_1$ appear abruptly in the model but more smoothly in experiment, most likely due to inhomogeneities which broaden the coexistence regime by providing nucleation seeds for the incipient phase. Note, the percolative transition region narrows for $T\rightarrow T_{\rm crit}$ and vanishes above that; the metallic fraction of the simulation is indicated in Fig. \[fig:contour-plots\](f).
The inset of panel (f) clearly demonstrates that the colossal permittivity enhancement appears exclusively for a percolating mixture of metallic and insulating regions, and not for the pure phases. These results render the gap closing irrelevant for the electrodynamics at the low-temperature Mott IMT. While the transition is of first-order at half filling, doping [@Hebert2015] or disorder [@Gati2018; @Urai2019] effectively move $T_{\rm crit}\rightarrow 0$, eventually turning it into a true quantum-critical point. We further point out that the discussed mechanism of a percolative enhancement of $\epsilon_1$ may also apply to related organic compounds subject to first-order transitions. Similar dielectric anomalies in and were previously assigned to ferroelectricity [@Gati2018a] and multiferroicity [@Lunkenheimer2012], respectively. The former exhibits a peak in $\epsilon_1$ right at its charge-order IMT, where phase coexistence is evident [@Gati2018a; @Hassan2019]. The latter is located extremely close to the Mott IMT [@Limelette2003a], so the coexistence region is likely entered already at ambient pressure [^7].
While previous transport and optical studies [@Furukawa2015; @Pustogow2018] already provided hints favoring the DMFT scenario, they do not map out the predicted dynamical regimes, especially regarding a well-defined coexistence region at $T < T_{\rm crit}$. Our new dielectric data, however, reveal all phases in vivid detail and in remarkable agreement with the respective crossover lines obtained from dc transport. Indeed, we recognize the gapped Mott insulator by essentially constant $\epsilon_1$ (light red) bounded precisely by the QWL [@Vucicevic2013], while also below $T_{\rm BR}$ [@Radonjic2012; @*Deng2013] the response clearly follows the dielectric behavior expected for a metal ($\epsilon_1 < 0$, blue). Most remarkably, these two crossover lines converge towards $T_{\rm crit}$, which marks the onset of the coexistence region, just as anticipated from Fig. \[fig:phase-diagram\]. The emergence of phase segregation is evidenced by the huge peak of $\epsilon_1$ in excellent agreement with our current DMFT-based modeling. The sharply defined boundaries of this dielectric anomaly imply that the corresponding inhomogeneities are [*the consequence and not the cause*]{} of phase separation, the latter resulting from strong correlation effects inherent to Mottness. Our findings leave little room for doubt that the DMFT scenario offers a rather accurate picture of the Mott IMT, in contrast to other theoretical viewpoints [@Senthil2008] which focus on the spin degrees of freedom in the QSL. This also confirms recent experimental and theoretical results [@Lee2016; @Pustogow2018spinons] suggesting that such gapless spin excitations, while dominant deep within the low-temperature Mott-insulating phase, are quickly damped away by the onset of charge fluctuations close to the IMT.
We also demonstrated that the novel chemical method of partially substituting the organic donor molecules of the fully-frustrated Mott insulator by BEDT-STF yields similar bandwidth-tuning like physical pressure. By comparing the boundaries of the Mott state and the correlated metal (QWL, $T_{\rm BR}$, $T_{\rm FL}$) we find that 1 kbar is equivalent to $\Delta x \approx 0.06$. The pronounced divergency in $\epsilon_1$ evidences a spatial coexistence of metallic and insulating electronic phases around the first-order IMT that can be circumstantially described by percolation theory. Our results yield that the Mott gap has a minor effect on the dielectric properties while the effects of phase coexistence dominate.
We appreciate discussions with S. Brown, B. Gompf and I. Voloshenko. We acknowledge support by the DFG via DR228/52-1. A.P. acknowledges support by the Alexander von Humboldt Foundation through the Feodor Lynen Fellowship. Work in Florida was supported by the NSF Grant No. 1822258, and the National High Magnetic Field Laboratory through the NSF Cooperative Agreement No. 1157490 and the State of Florida. E.U. receives support of the European Social Fund and the Ministry of Science Research and the Arts of Baden-Württemberg. J.A.S. acknowledges support from the Independent Research/Development program while serving at the National Science Foundation. A.P., R.R. and Y.T. contributed equally to this work.
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[^1]: authors contributed equally
[^2]: authors contributed equally
[^3]: authors contributed equally
[^4]: The superconducting state at $T\approx 4$ K [@Kurosaki2005; @Furukawa2018] is below the temperature accessible to us here.
[^5]: BEDT-TTF stands for bis-ethylene-dithio-tetrathiafulvalene. Substituting two of the inner sulfur atoms by selenium leads to bis-ethylenedithio-diselenium-dithiafulvalene, abbreviated BEDT-STF [@Saito2019].
[^6]: Comparison of the results in Fig. \[fig:sigma-eps\_T\](j) with optical data measured on the same substitution yields fair agreement of the metallic values $\epsilon_1 < 0$ [@Saito2019]. Technical details of the dielectric experiments can be found in Ref. [@Rosslhuber2019].
[^7]: While $\epsilon_1$ initially increases upon cooling in , it peaks at the antiferromagnetic transition and reduces at lower $T$. This could be a consequence of the metallic fraction first increasing as the insulator-metal phase boundary approaches the ambient-pressure position, but then reducing below $T_{\rm N}$ because of the negative slope of the boundary between antiferromagnet and metal which moves the IMT further away from $p=0$.
| ArXiv |
---
abstract: 'We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.'
author:
- |
Dmitri Burago [^1]\
Bruce Kleiner[^2]
bibliography:
- 'refs.bib'
title: Separated nets in Euclidean space and Jacobians of biLipschitz maps
---
Introduction
============
A subset $X$ of a metric space $Z$ is a [*separated net*]{} if there are constants $a,\,b>0$ such that $d(x,x')>a$ for every pair $x,\,x'\in X$, and $d(z,X)<b$ for every $z\in Z$. Every metric space contains separated nets: they may be constructed by finding maximal subsets with the property that all pairs of points are separated by some distance $a>0$. It follows easily from the definitions that two spaces are quasi-isometric if and only if they contain biLipschitz equivalent separated nets. One may ask if the choice of these nets matters, or, in other words, whether any two separated nets in a given space are biLipschitz equivalent. To the best of our knowledge, this problem was first posed by Gromov [@asyinv p.23]. The answer is known to be yes for separated nets in non-amenable spaces (under mild assumptions about local geometry), see [@Gromov; @McMullen; @Whyte]; more constructive proofs in the case of trees or hyperbolic groups can be found in [@Papas; @Bogop].
In this paper, we prove the following theorem:
\[T1\] There exists a separated net in the Euclidean plane which is not biLipschitz equivalent to the integer lattice.
The proof of Theorem \[T1\] is based on the following result:
\[T2\] Let $I:=[0,1]$. Given $c>0$, there is a continuous function $\rho:I^2{\rightarrow}[1,1+c]$, such that there is no biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with
$$Jac(f):=Det(Df)=\rho \quad a.e.$$
1\. Although we formulate and prove these theorems in the 2-dimensional case, the same proofs work with minor modifications in higher dimensional Euclidean spaces as well. We only consider the 2-dimensional case here to avoid cumbersome notation.
2\. Theorem \[T2\] also works for Lipschitz homeomorphisms; we do not use the lower Lipschitz bound on $f$.
3\. After the first version of this paper had been written, Curt McMullen informed us that he also had a proof of Theorems \[T1\] and \[T2\]. See [@McMullen] for a discussion of the the linear analog of Theorem \[T2\], and the Hölder analogs of the mapping problems in Theorems \[T1\] and \[T2\].
The problem of prescribing Jacobians of homeomorpisms has been studied by several authors. In [@DacMos] Dacorogna and Moser proved that every ${\alpha}$-Holder continuous function is locally the Jacobian of a $C^{1,{\alpha}}$ homeomorphism, and they then raised the question of whether any continuous function is (locally) the Jacobian of a $C^1$ diffeomorphism. [@RivYe; @Ye] consider the prescribed Jacobian problem in other regularity classes, including the cases when the Jacobian is in $L^{\infty}$ or in a Sobolev space.
Overview of the proofs
[*Theorem \[T2\] implies Theorem \[T1\].*]{} Let $\rho:I^2{\rightarrow}{\mathbb R}$ be measurable with $0<\inf\rho\leq \sup\rho <\infty$. We will indicate why $\rho$ would be the Jacobian of a biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ if all separated nets in ${\mathbb E}^2$ were biLipschitz equivalent. Take a disjoint collection of squares $S_i\subset{\mathbb E}^2$ with side lengths $l_i$ tending to infinity, and “transplant” $\rho$ to each $S_i$ using appropriate similarities ${\alpha}_i:I^2{\rightarrow}S_i$, i.e. set $\rho_i\defeq \rho\circ{\alpha}_i^{-1}$. Then construct a separated net $L\subset {\mathbb E}^2$ so that the “local average density” of $L$ in each square $S_i$ approximates $\rho_i^{-1}$. If $g:L{\rightarrow}{\mathbb Z}^2$ is a biLipschitz homeomorphism, consider “pullbacks” of $g{\mbox{\Large \(|\)\normalsize}}_{S_i}$ to $I^2$, i.e. pre and post-compose $g{\mbox{\Large \(|\)\normalsize}}_{S_i}$ with suitable similarities so as to get a sequence of uniformly biLipschitz maps $g_i:I^2\supset Z_i{\rightarrow}{\mathbb E}^2$. Then extract a convergent subsequence of the $g_i$’s via the Arzela-Ascoli theorem, and obtain a limit map $f:I^2{\rightarrow}{\mathbb E}^2$ with Jacobian $\rho$.
[*Theorem \[T2\].*]{} The observation underlying our construction is that if the Jacobian of $f:I^2{\rightarrow}{\mathbb E}^2$ oscillates in a rectangular neighborhood $U$ of a segment $\ol{xy}\subset I^2$, then $f$ will be forced to stretch for one of two reasons: either it maps $\ol{xy}$ to a curve which is far from a geodesic between its endpoints, or it maps $\ol{xy}$ close to the segment $\ol{f(x)f(y)}$ but it sends $U$ to a neighborhood of $\ol{f(x)f(y)}$ with wiggly boundary in order to have the correct Jacobian. By arranging that $Jac(f)$ oscillates in neighborhoods of a hierarchy of smaller and smaller segments we can force $f$ to stretch more and more at smaller and smaller scales, eventually contradicting the Lipschitz condition on $f$.
We now give a more detailed sketch of the proof.
We first observe that it is enough to construct, for every $L>1,\,\bar c>0$, a continuous function $\rho_{L,\bar c}:I^2{\rightarrow}[1,1+\bar c]$ such that $\rho_{L,\bar c}$ is not the Jacobian of an L-biLipschitz map $I^2{\rightarrow}{\mathbb E}^2$. Given such a family of functions, we can build a new continuous function $\rho:I^2{\rightarrow}[1,1+c]$ which is not the Jacobian of any biLipschitz map $I^2{\rightarrow}{\mathbb E}^2$ as follows. Take a sequence of disjoint squares $S_k\subset I^2$ which converge to some $p\in I^2$, and let $\rho:I^2{\rightarrow}[1,1+c]$ be any continuous function such that $\rho{\mbox{\Large \(|\)\normalsize}}_{S_k}=\rho_{k,\min(c,\frac{1}{k})}\circ{\alpha}_k$ where ${\alpha}_k:S_k{\rightarrow}I^2$ is a similarity.
Also, note that to construct $\rho_{L,\bar c}$, we really only need to construct a measurable function with the same property: if $\rho^k_{L,\bar c}$ is a sequence of smoothings of a measurable function $\rho_{L,\bar c}$ which converge to $\rho_{L,\bar c}$ in $L^1$, then any sequence of $L$-biLipschitz maps $\phi_k:I^2{\rightarrow}{\mathbb E}^2$ with $Jac(\phi_k)=\rho^k_{L,\bar c}$ will subconverge to a biLipschitz map $\phi:I^2{\rightarrow}{\mathbb E}^2$ with $Jac(\phi)=\rho_{L,\bar c}$.
We now fix $L>1,\,c>0$, and explain how to construct $\rho_{L,c}$. Let $R$ be the rectangle $[0,1]\times [0,\frac{1}{N}]\subset {\mathbb E}^2$, where $N\gg 1$ is chosen suitably depending on $L$ and $c$, and let $S_i=[\frac{i-1}{N},\frac{i}{N}]\times[0,\frac{1}{N}]$ be the $i^{th}$ square in $R$. Define a “checkerboard” function $\rho_1:I^2{\rightarrow}[1,1+c]$ by letting $\rho_1$ be $1+c$ on the squares $S_i$ with $i$ even and $1$ elsewhere. Now subdivide $R$ into $M^2N$ squares using $M$ evenly spaced horizontal lines and $MN$ evenly spaced vertical lines. We call a pair of points [*marked*]{} if they are the endpoints of a horizontal edge in the resulting grid.
The key step in the proof (Lemma \[MLE\]) is to show that any biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with Jacobian $\rho_1$ must stretch apart a marked pair quantitatively more than it stretches apart the pair $(0,0),\,(1,0)$; more precisely, there is a $k>0$ (depending on $L,\, c$) so that $\frac{d(f(p),f(q))}{d(p,q)}>(1+k)d(f(0,0),f(1,0))$ for some marked pair $p,\,q$. If this weren’t true, then we would have an $L$-biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ which stretches apart all marked pairs by a factor of at most $(1+\eps)d(f(0,0),f(0,1))$, where $\eps\ll 1$. This would mean that $f$ maps horizontal lines in $R$ to “almost taut curves”. Using triangle inequalities one checks that this forces $f$ to map most marked pairs $p,\,q$ so that vector $f(q)-f(p)$ is $\approx d(p,q)(f(1,0)-f(0,0))$; this in turn implies that for some $1\leq i\leq N$, [*all*]{} marked pairs $p,\,q$ in the adjacent squares $S_i,\,S_{i+1}$ are mapped by $f$ so that $f(q)-f(p)\approx d(p,q)
(f(1,0)-f(0,0))$. Estimates then show that $f(S_i)$ and $f(S_{i+1})$ have nearly the same area, which contradicts the assumption that $Jac(f)=\rho_1$, because $\rho_1$ is $1$ on one of the squares and $1+c$ on the other.
Our next step is to modify $\rho_1$ in a neighborhood of the grid in $R$: we use thin rectangles (whose thickness will depend on $L,\, c$) containing the horizontal edges of our grid, and define $\rho_2:I^2{\rightarrow}[1,1+c]$ by letting $\rho_2$ be a “checkerboard” function in each of these rectangles and $\rho_1$ elsewhere. Arguing as in the previous paragraph, we will conclude that some suitably chosen pair of points in one of these new rectangles will be stretched apart by a factor $>d(f(0,0),f(1,0))(1+k)^2$ under the map $f$. Repeating this construction at smaller and smaller scales, we eventually obtain a function which can’t be the Jacobian of an $L$-biLipschitz map.
The paper is organized as follows. In Section 2 we prove that Theorem \[T2\] implies Theorem \[T1\]. Section 3 is devoted to the proof of Theorem \[T2\].
Reduction of Theorem \[T1\] to Theorem \[T2\]
=============================================
Recall that every biLipschitz map is differentiable a.e., and the area of the image of a set is equal to the integral of the Jacobian over this set. We formulate our reduction as the following Lemma:
Let $\rho:I^2{\rightarrow}[1,1+c]$ be a measurable function which is not the Jacobian of any biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with $$Jac(f):=det(Df)=\rho \quad a.e.$$ Then there is a separated net in ${\mathbb E}^2$ which is not biLipschitz homeomorphic to ${\mathbb Z}^2$.
In what follows, the phrase “subdivide the square $S$ into subsquares will mean that $S$ is to be subdivided into squares using evenly spaced lines parallel to the sides of $S$. Let ${\cal S}=\{S_k\}_{k=1}^\infty$ be a disjoint collection of square regions in ${\mathbb E}^2$ so that each $S_k$ has integer vertices, sides parallel to the coordinate axes, and the side length $l_k$ of $S_k$ tends to $\infty$ with $k$. Choose a sequence $m_k\in(1,\infty)$ with $\lim_{k{\rightarrow}\infty}m_k=\infty$ and $\lim_{k{\rightarrow}\infty}\frac{m_k}{l_k}=0$. Let $\phi_k:I^2{\rightarrow}S_k$ be the unique affine homeomorphism with scalar linear part, and define $\rho_k:S_k{\rightarrow}[1,1+c]$ by $\rho_k=(\frac{1}{\rho})\circ \phi_k^{-1}$. Subdivide $S_k$ into $m_k^2$ subsquares of side length $\frac{l_k}{m_k}$. Call this collection ${\cal T}_k=\{T_{ki}\}_{i=1}^{m_k^2}$. For each $i$ in $\{1,\ldots,m_k^2\}$, subdivide $T_{ki}$ into $n_{ki}^2$ subsquares $U_{kij}$ where $n_{ki}$ is the integer part of $\sqrt{\int_{T_{ki}}\rho_kd{\cal L}}$. Now construct a separated net $X\subset{\mathbb E}^2$ by placing one point at the center of each integer square not contained in $\cup S_k$, and one point at the center of each square $U_{kij}$.
We now prove the lemma by contradiction. Suppose $g:X{\rightarrow}{\mathbb Z}^2$ is an $L$-biLipschitz homeomorphism. Let $X_k=\phi_k^{-1}(X)\subset I^2$, and define $f_k:X_k{\rightarrow}{\mathbb E}^2$ by $$f_k(x)=\frac{1}{l_k}(g\circ\phi_k(x)-g\circ\phi_k(\star_k))$$ where $\star_k$ is some basepoint in $X_k$. Then $f_k$ is an $L$-biLipschitz map from $X_k$ to a subset of ${\mathbb E}^2$, and the $f_k$’s are uniformly bounded. By the proof of the Arzela-Ascoli theorem we may find a subsequence of the $f_k$’s which “converges uniformly” to some biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$. By the construction of $X$, the counting measure on $X_k$ (normalized by the factor $\frac{1}{l_k^2}$) converges weakly to $\frac{1}{\rho}$ times Lebesgue measure, while the (normalized) counting measure on $f_k(X_k)$ converges weakly to Lebesgue measure. It follows that $f_*((\frac{1}{\rho}){\cal L})={\cal L}{\mbox{\Large \(|\)\normalsize}}_{f(I^2)}$, i.e. $Jac(f)=\rho$.
501em $\square$=0
Construction of a continuous function which is not a Jacobian of a biLipschitz map
==================================================================================
The purpose of this section is to prove Theorem \[T2\]. As explained in the introduction, Theorem \[T2\] follows from
\[ML\] For any given $L$ and $c>1$, there exists a continuous function $\rho:S=I^2{\rightarrow}[1,1+c]$, such that there is no $L$-biLipschitz homeomorphism $f:I^2{\rightarrow}{\mathbb E}^2$ with
$$Jac(f)=\rho \quad a.e.$$
From now on, we fix two constants $L$ and $c$ and proceed to construct of a continuous function $\rho: I^2 \rightarrow [1,1+c]$ which is not a Jacobian of an $L$-biLipschitz map. By default, all functions which we will describe, take values between $1$ and $1+c$.
We say that two points $x,\,y\in I^2$ are $A$-stretched (under a map $f:I^2 \rightarrow {\mathbb E}^2$) if $d(f(x),f(y)) \geq Ad(x,y)$.
For $N\in{\mathbb N}$, $R_N$ be the rectangle $[0,1]\times [0,\frac{1}{N}]$ and define a “checkerboard” function $\rho_N:R_N \rightarrow [1,1+c]$ by $\rho_N(x,y)=1$ if $[Nx]$ is even and $1+c$ otherwise. It will be convenient to introduce the squares $S_i=[\frac{i-1}{N},\frac{i}{N}] \times [0, \frac{1}{N}]$, $i=1, \dots,N$; $\rho_N$ is constant on the interior of each $S_i$.
The cornerstone of our construction is the following lemma:
\[MLE\] There are $k>0,\,M,\,\mu$, and $N_0$ such that if $N\geq N_0$, $\eps\leq \frac{\mu}{N^2}$ then the following holds: if the pair of points $(0,0)$ and $(1,0)$ is $A$-stretched under an $L$-biLipschitz map $f:R_N \rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho_N$ on a set of area no bigger than $\epsilon$ , then at least one pair of points of the form $((\frac{p}{NM}, \frac{s}{NM}), (\frac{q}{NM}, \frac{s}{NM}))$ is $(1+k)A$-stretched (where $p$ and $q$ are integers between $0$ and $NM$ and $s$ is an integer between $0$ and $M$).
We will need constants $k,\,l,\,m,\,\eps\in(0,\infty)$ and $M,\,N\in {\mathbb N}$, which will be chosen at the end of the argument. We will assume that $N>10$ and $c,\,l<1$. Let $f:R_N{\rightarrow}{\mathbb E}^2$ be an $L$-biLipschitz map such that $Jac(f)=\rho_N$ off a set of measure $\eps$. Without loss of generality we assume that $f(x)=(0,0)$ and $f(y)=(z,0)$, $z \geq A$.
We will use the notation $x_{pq}^i:=(\frac{p+M(i-1)}{NM}, \frac{q}{NM})$, where $i$ is an integer between 1 and $N$, and $p$ and $q$ are integers between $0$ and $M$. We call these points [*marked*]{}. Note that the marked points in $S_i$ are precisely the vertices of the subdivision of $S_i$ into $M^2$ subsquares. The index $i$ gives the number of the square $S_i$, and $p$ and $q$ are “coordinates” of $x_{pq}^i$ within the square $S_i$.
We will prove Lemma \[MLE\] by contradiction: we assume that all pairs of the form $x_{pq}^i, x_{sq}^j$ are no more than $(1+k)A$-stretched.
If $x^i_{pq}\in S_i$ is a marked point, we say that $x^{i+1}_{pq}\in S_{i+1}$ is the [*marked point corresponding to $x^i_{pq}$*]{}; corresponding points is obtained by adding the vector $(\frac{1}{N},0)$, where $\frac{1}{N}$ is the side length of the square $S_i$. We are going to consider vectors between the images of marked points in $S_i$ and the images of corresponding marked points in the neighbor square $S_{i+1}$. We denote these vectors by $W_{pq}^i:=f(x_{pq}^{i+1})-f(x_{pq}^i)$. We will see that most of the $W_{pq}^i$’s have to be extremely close to the vector $W:=(A/N,0)$, and, in particular, we will find a square $S_i$ where $W_{pq}^i$ is extremely close to $W$ for [*all*]{} $0\leq p,q\leq M$. This will mean that the areas of $f(S_i)$ and $f(S_{i+1})$ are very close, since $f(S_{i+1})$ is very close to a translate of $f(S_i)$. On the other hand, except for a set of measure $\eps$, the Jacobian of $f$ is $1$ in one of the square $S_i,\,S_{i+1}$ and $1+c$ in the other. This allows us to estimate the difference of the areas of their images from below and get a contradiction.
If $l\in (0,1)$, we say that a vector $W_{pq}^i=f(x_{pq}^{i+1})-f(x_{pq}^i)$, (or the marked point $x_{pq}^i$), is [*regular*]{} if the length of its projection to the x-axis is greater than $\frac{(1-l)A}{N}$. We say that a square $S_i$ is [*regular*]{} if all marked points $x_{pq}^i$ in this square are regular.
There exist $k_1=k_1(l)>0$, $N_1=N_1(l)$, such that if $k \leq k_1,\,N\geq N_1$, there is a regular square.
Reasoning by contradiction, we assume that all squares are irregular. By the pigeon-hole principle, there is a value of $ s $ (between 0 and $M$) such that there are at least $\frac{N}{2M+2}$ irregular vectors $W_{p_js}^{ i_j}$, $j=1,2,\dots J \geq \frac{N}{2M+2}$, where $i_j$ is an increasing sequence with a fixed parity. This means that we look for $l$-irregular vectors between marked points in the same row $s$ and only in squares $S_i$’s which have all indices $i$’s even or all odd. The latter assumption guarantees that the segments $[x_{p_js}^{ i_j},x_{p_js}^{ i_j+1}]$ do not overlap. We look at the polygon with marked vertices $$(0,0),x_{0s}^0= (0, s/MN),x_{p_1s}^{ i_1},
x_{p_1s}^{ i_1+1},x_{p_2s}^{ i_2},x_{p_2s}^{ i_2+1} \dots ,x_{p_Js}^{
i_J},
x_{p_Js}^{ i_J+1}, x_{Ms}^N=(1, s/MN), (1,0)$$ The image of this polygon under $f$ connects $(0,0)$ and $(z,0)$ and, therefore, the length of its projection onto the x-axis is at least $z \geq A$. On the other hand, estimating this projection separately for the images of $l$-irregular segments $[x_{p_js}^{ i_j},x_{p_js}^{ i_j+1}]$, the “horizontal” segments $[x_{p_js}^{ i_j+1},x_{p_{j+1}s}^{ i_{j+1}}]$ and the two “vertical” segments $[(0,0), x_{0s}^0]$ and $[x_{Ms}^N,
(1,0)]$ , one gets that the lengths of this projection is no bigger than
$$\label{projlength}
( \frac{N}{2M+2}) (\frac{(1-l)A}{N}) +
(\frac{(1+k)A}{N})(N-\frac{N}{2M+2})+2\frac{L}{N}.$$
The first term in (\[projlength\]) bounds the total length of projections of images of irregular segments by the definition of irregular segments and total number of them. The second summand is maximum stretch factor $(1+k)A$ between marked points times the total length of remaining horizontal segments. The third summand estimates the lengths of images of segments $[(0,0), x_{0s}^0]$ and $[x_{Ms}^N,
(1,0)]$ just by multiplying their lengths by our fixed bound $L$ on the Lipschitz constant.
Recalling that this projection is at least $z$, which in its turn is no less than $A$, we get $$( \frac{N}{2M+2}) (\frac{(1-l)A}{N}) +
(\frac{(1+k)A}{N})(N-\frac{N}{2M+2})+2\frac{L}{N}\geq A.$$ One easily checks that this is impossible when $k$ is sufficiently small and $N$ is sufficiently large. This contradiction proves Claim 1.
501em $\square$=0
Let $W=(\frac{A}{N}, 0)$.
Given any $m>0$, there is an $l_0=l_0(m)>0$ such that if $l \leq l_0$ and $k \leq l$, then $|W-W_{pq}^i| \leq \frac{m}{N}$ for every regular vector $W_{pq}^i$.
Consider a regular vector $W_{pq}^i=(X,Y)$. Since $W_{pq}^i$ is regular, $X \geq \frac{(1-l)A}{N}$. On the other hand, $X^2+Y^2 \leq \frac{(1+k)^2A^2}{N^2}$ and $X \leq
\frac{(1+k)A}{N}$. Thus the difference of the $x$-coordinates of $W_{pq}^i$ and $W$ is bounded by $\frac{(l+k)A}{N} <\frac{2lA}{N}$. Substituting the smallest possible value $\frac{(1-l)A}{N}$ for $X$ into $X^2+Y^2 \leq \frac{(1+k)^2A^2}{N^2}$, we get $Y^2 \leq \frac{2(l+k)A^2}{N^2} \leq
\frac{4lA^2}{N^2}$ . This implies that
$$\label{w'sclose}
N|W-W_{pq}^i| \leq 2A\sqrt{l^2+l}\leq 2L\sqrt{l^2+l}.$$
The right-hand side of (\[w’sclose\]) tends to zero with $l$, so Claim 2 follows.
501em $\square$=0
There are $m_0>0$, $M_0$ such that if $m<m_0$ and $M>M_0$, then the following holds: if for some $1\leq i\leq N$ and every $p,\,q$ we have $|W-W_{pq}^i|\leq\frac{m}{N}$, then $$\label{areasclose}
|Area(f(S_{i+1}))-Area(f(S_i))|<\frac{c}{2N^2}.$$
We assume that $i$ is even and therefore $\rho$ takes the value $1$ on $S_i$ and $1+c$ on $S_{i+1}$; the other case is analogous. We let $Q:=f(S_i)$ and $R=f(S_{i+1})$.
$Q$ is bounded by a curve (which is the image of the boundary of $S_i$). Consider the result $\tilde R :=Q+W$ of translating $Q$ by the vector $W=(A/N, 0)$. The area of $\tilde R$ is equal to the area of $Q$.
The images of the marked points on the boundary of $S_i$ form an $\frac{L}{NM}$-net on the boundary of $Q$, and the images of marked points on the boundary of $S_{i+1}$ form an $\frac{L}{NM}$-net on $R$. By assumption the difference between $W$ and each vector $W_{pq}^i$ joining the image of a marked point on the boundary of $S_i$ and the image of the corresponding point on the boundary of $S_{i+1}$ is less than $\frac{m}{N}$. We conclude that the boundary of $\tilde R$ lies within the $\frac{m}{N}+\frac{2L}{MN}$-neighborhood of the boundary of $R$. Since $f$ is $L$-Lipschitz, the length of the boundary of $R$ is $\leq 4L/N$. Using a standard estimate for the area of a neighborhood of a curve, we obtain:
$$|Area(R)-Area(Q)|=|Area(R)-Area(\tilde R)|
\leq \frac{2L}{N}(\frac{m}{N}+\frac{2L}{MN})+\pi(\frac{m}{N}+\frac{2L}{MN})^2.$$ Therefore (\[areasclose\]) holds if $m$ is sufficiently small and $M$ is sufficiently large.
501em $\square$=0
Now assume $m<m_0$, $M>M_0$, $l\leq l_0(m)$, $k\leq\min(l,k_1(l))$, $N\geq N_1(l)$, and $\eps\leq\frac{c}{8N^2L^2}$. Combining claims 1, 2, and 3, we find a square $S_i$ so that (\[areasclose\]) holds. On the other hand, since $Jac(f)$ coincides with $\rho$ off a set of measure $\eps$, $Area(f(S_i))\leq 1/N^2+\eps L^2$ and $Area(f(S_{i+1})\geq (1+c)(1/N^2-\epsilon)$. Using the assumption that $\eps\leq\frac{c}{8N^2L^2}$ we get $$Area(R)-Area(Q)\geq \frac{c}{2N^2},$$ contradicting (\[areasclose\]). This contradiction proves Lemma \[MLE\].
501em $\square$=0
We will use an inductive construction based on Lemma \[MLE\]. Rather than dealing with an explicit construction of pairs of points as in Lemma \[MLE\], it is more convenient to us to use the following lemma, which is an obvious corollary of Lemma \[MLE\]. (To deduce this lemma from Lemma \[MLE\], just note that all properties of interest persist if we scale our coordinate system.)
\[IL\] There exists a constant $k>0$ such that, given any segment $\ol{xy}\subset I^2$ and any neighborhood $\ol{xy}\subset U\subset I^2$, there is a measurable function $\rho:U \rightarrow [1,1+c]$, $\eps>0$ and a finite collection of non-intersecting segments $\ol{l_kr_k}\subset U$ with the following property: if the pair $x,\,y$ is $A$-stretched by an $L$-biLipschitz map $f:U
\rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho$ on a set of area $<\epsilon$ , then for some $k$ the pair $l_k,\,r_k$ is $(1+k)A$-stretched by $f$. The function $\rho$ may be chosen to have finite image.
We will prove Lemma \[ML\] by induction, using the following statement. (It is actually even slightly stronger than Lemma \[ML\] since it not only guarantees non-existence of $L$-biLipschitz maps with a certain Jacobian, but also gives a finite collection of points, such that at least one distance between them is distorted more than by factor $L$.)
\[FF\] For each integer $i$ there is a measurable function $\rho_i: I^2 \rightarrow [1, 1+c]$ , a finite collection ${\cal S}_i$ of non-intersecting segments $\ol{l_kr_k}\subset I^2$, and $\epsilon_i>0$ with the following property: For every $L$-biLipschitz map $f:I^2 \rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho_i$ on a set of area $<\epsilon_i$ , at least one segment from ${\cal S}_i$ will have its endpoints $\frac{(1+k)^i}{L}$-stretched by $f$.
The case $i=0$ is obvious. Assume inductively that there are $\rho_{i-1},\,\eps_{i-1}$, and a disjoint collection of segments ${\cal S}_{i-1}=\{\ol{l_kr_k}\}$ which satisfy the conditions of the lemma. Let $\{U_k\}$ be a disjoint collection of open sets with $U_k\supset\ol{l_kr_k}$ and with total area $<\frac{\eps_{i-1}}{2}$. For each $k$ apply Lemma \[IL\] to $U_k$ to get a function $\hat\rho_k:U_k{\rightarrow}[1,1+c],\,\hat\eps_k>0$, and a disjoint collection $\hat{\cal S}_k$ of segments. Now let $\rho_i:I^2{\rightarrow}[1,1+c]$ be the function which equals $\hat\rho_k$ on each $U_k$ and equals $\rho_{i-1}$ on the complement of $\cup U_k$; let ${\cal S}_i=\cup \hat{\cal S}_k$, and $\eps_i=\min\hat\eps_k$. The required properties follow immediately.
501em $\square$=0
Lemma \[ML\] and (Theorem \[T2\]) follows from Lemma \[FF\].
[^1]: Supported by a Sloan Foundation Fellowship and NSF grant DMS-95-05175.
[^2]: Supported by a Sloan Foundation Fellowship and NSF grants DMS-95-05175 and DMS-96-26911.
| ArXiv |
---
abstract: 'I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.'
address: '*Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano (Italy); e-mail: [[email protected]]{}; ORCID: [0000-0003-3310-3455]{}*'
author:
- 'G. Gaeta'
title: |
On the geometry of twisted prolongations,\
and dynamical systems
---
Introduction
============
Sophus Lie created what is nowadays known as the theory of Lie groups and algebras first and foremost to study (nonlinear) differential equations. The theory has then been extended in several directions, in particular generalizing the set of admitted vector fields. On the other hand, it remained clear that once we have defined how the basic (independent and dependent) variables are acted upon by our transformations, the action on derivatives is given – by a natural action, known in geometrical terms as the *prolongation* operation.
More recently, it has been realized that one can also deform the action on derivatives, i.e. deform the prolongation operation (in this case one usually speaks of “twisted prolongation” and “twisted symmetry”), and still obtain useful concepts and results – where useful is meant in the sense of “useful to get solutions of the equations under study”, beside the abstract geometrical interest.
It happens that in these cases the deformation is assigned at the level of first derivatives, while deformations on higher derivative sees no different action. This means that – as is often the case in symmetry theory of differential equations – in the case of first order equations, even more so for Dynamical Systems, one has “too much freedom” (a standard euphemism to mean there is no algorithmic way to proceed). Despite this fact, the theory can also be used in the context of dynamical systems (a special attention in this direction was paid in the development of $\s$-symmetries, see below).
In this paper I will review the theory of twisted symmetries, paying special attention to geometrical aspects – in particular to the connection between the usual Lie reduction and Lie-Frobenius one – and to results which can be applied in the realm of ODEs and Dynamical Systems. The Bibliography will provide the interested reader with indications on how to go beyond these short notes.
Standard symmetries of differential equations
=============================================
I will consider differential equations[^1] with independent variables $x^i$ ($i=1,...,n$) and dependent variables $u^a$ ($a = 1,..., m$); partial derivatives will be denoted by $u^a_J$, where $J$ is a multi-index $J = \{ j_1
, ... , j_n \}$ of order $|J| = j_1 + ... + j_n$ and u\^a\_J = (here and somewhere in the following we moved the vector index of the $x$ for typographical convenience). We denote by $u_{(k)}$ the set of all partial derivatives of order $k$, and by $u_{[n]}$ the set of all partial derivatives of order $k \le n$.
Geometrical description of differential equations and solutions
---------------------------------------------------------------
The $x$ are local coordinates in a manifold $B$, while $u$ are local coordinates in a manifold $U$; we consider the phase manifold $M = B \times U$, which has a natural structure of bundle $(M,\pi_0,B)$ over $B$ with fiber $U$.
As well known we can associate to $(M,\pi_0,B)$ its Jet bundles $J^k M$ of any order; these have a structure of fiber bundle $(J^k M, \pi_k , B)$ over $B$ (with projection $\pi_k$) but also of fiber bundle over those of lower order (with projection $\chi_{kq}$, for $q < k$), i.e. $(J^k M , \chi_{kq} ,
J^q M)$ with $\pi_k = \pi_q \circ \chi_{kq}$. Natural local coordinates in $J^k M$ are provided by $\(x,u, u_{(1)} , ... ,
u_{(k)} \)$.
We also recall that the Jet bundle is equipped with a *contact structure* $\Omega$ [@ArnGM; @God; @Olv2; @Sha; @Stern]; this can be encoded in the *contact forms*[^2] \^a\_J := u\^a\_J - u\^a\_[J,i]{} x\^i .
Functions $u = f(x)$ are naturally identified with sections in $M$ (elements of $\Sigma (M)$); the function $u=f(x)$ corresponds to the section \_f = { (x,u) M : u = f(x) } . Note that in this way we have established a correspondence between an *analytical* object (the function) and a *geometrical* one (the section).
A section $\ga_f \in \Sigma (M)$ identifies naturally a section $\ga_f^{(k)}$ in $\Sigma (J^k M)$, with of course \_f\^[(k)]{} = { (x,u\_[\[k\]]{}) J\^k M : u\^a\_J = f\^a\_J (x) J : |J| k } .
Given a differential equation $\Delta$ of order $k$, written in local coordinates as F\^i $ x,u,...,u_{(k)} $ = 0 , we consider its *solution manifold* $S_\Delta
\ss J^k M$, S\_ = { (x,u\_[(1)]{} , ... , u\_[(k)]{}) : F $ x,u_{[k]}$ = 0 } . This is just the set of points in $J^k M$ where the relation described by $\Delta$ between independent, dependent variables and derivatives is satisfied; but now we have again transformed an *analytic* object (the differential equation) into a *geometric* one.
The same can be done for the concept of solutions to $\De$: a function $u = f (x)$ identifies, as mentioned above, a section $\ga_f$ in $(M,\pi_0,B)$, and this in turn identifies a section $\ga_f^{(k)}$ in $(J^k M,\pi_k , B)$, which is just the set of points $(x,u_{[k]})$ with $u^a = f^a(x)$ and $u^a_J = f^a_J (x)$. Now $u = f(x)$ is a solution to $\De$ if and only if $\ga_f^{(k)}
\ss S_\De$.
Vector fields and prolongations
-------------------------------
Let us now consider a vector field $X$ in $M$ and its *prolongation* to $J^k M$. In local coordinates, we write X = \_i \^i (x,u) + \_a \^a (x,u) = \^i \_i + \^a \_a ; the prolongation is then described –in the same local coordinates – by X\^[(k)]{} = X + \_a \_[|J|=1]{}\^k \^a\_J := X + \^a\_J \_a\^J , where we introduced the shorthand notations \_i := , \_a := ; \_a\^J := . We will also write $\psi^a_0 = \vphi^a$.
The coefficient $\psi^a_J$ of the Jet components, i.e. of the components making the prolongation of $X$, are computed by the *prolongation formula*, which is more conveniently expressed in recursive form: \^a\_[J,i]{} = D\_i \^a\_J - u\^a\_[J,k]{} D\_i \^k . Here and in the following $\wt{J} = \{ J,i \}$ is the multi-index with components $\wt{j}_m
= j_m + \delta_{m,i}$.
It is maybe worth recalling that this is easily obtained in analytic terms (we assume the reader to be familiar with this derivation, which is however provided in [@Gtwist1; @Olv1; @Olv2]), but the prolongation of a vector field can also be defined geometrically.
The following Lemmas are well known; see e.g. [@Gtwist1; @Olv1; @Olv2] for proofs and details.
[**Lemma 1.**]{} [*The prolonged vector field $X^{(n)}$ is the unique vector field in $J^n M$ which: $(i)$ is projectable to each $J^k M$ for $0 \le k \le n$; $(ii)$ coincides with $X$ when restricted to $M$; $(iii)$ preserves the contact structure on $J^n
M$.*]{}
[**Lemma 2.**]{} [*The prolongation of the commutator of two vector fields is the commutator of their prolongations; in other words, = Z $$X^{(n)} , Y^{(n)}$$ = Z\^[(n)]{} .* ]{}
We will now consider *differential invariants* (DIs) for a vector field $X$; these are invariants for the action of the prolongation of $X$ in $J^k M$. If a differential invariant $\zeta$ depends only on variables belonging to $J^k M$ (that is, no dependence on derivatives of order higher than $k$, and effective dependence on at least one derivative of order $k$), we say it is a DI of order $k$. DIs of order zero are ordinary invariants for the $X$ action in $M$.
[**Lemma 3.**]{} [*Let $\eta : M \to \R$ be a differential invariant of order zero and $\zeta : J^k M \to \R$ a differential invariant of order $k$ for $X^{(n)}$ ($n > k$). Then $\chi = (D_i
\zeta / D_i \eta): J^{k+1} M \to \R$ is a differential invariant of order $k+1$ for $X^{(n)}$.*]{}
[**Remark 1.**]{} Lemma 2 is also formulated saying that prolongation preserves Lie algebra structures.
[**Remark 2.**]{} The property stated in Lemma 3 is also known as “invariant by differentiation property” (IBDP); if we start with a set of invariants of order 0 and 1, we can generate differential invariants of all orders. In the case of ODEs, if we start with a complete set of DIs of order zero and one, we can generate in this way a complete set of DIs of any order, basically because if $U$ is $q$-dimensional, we have $k \cdot q$ DIs of order $k$ (these includes those of lower orders), as follows at once from $J^k M$ being of dimension $d_k = (k +1) q + 1$. The situation is different for PDEs, as the dimension of $J^k M$ grows combinatorially; see e.g. the discussion in [@Olv1].
Lie-point symmetries
--------------------
A (Lie-point[^3]) symmetry, or more precisely a Lie-point symmetry generator, is a vector field $X$ on $M$ such that its *prolongation* $X^{(k)}$ to $J^k M$ is tangent to $S_\De$. For a given $\Delta$ in the form , this condition is expressed in local coordinates as $$X^{(k)} \(
F^i \)$$\_[S\_]{} = 0 . In these equations, also called *determining equations*, the $F$ are given and one looks for $\xi$, $\vphi$ (i.e. components of the vector field $X$) satisfying them. As the components $\psi^a_J$ of $X^{(k)}$ along $u_{J}$ are given in terms of $\xi,\vphi$ and their derivatives, all dependencies of $u^a_J$ (with $|J| \not= 0$) are fully explicit, and hence decouple into a system of simpler equations, one for each monomial in the $u^a_J$; each of these is a *linear* PDE for the $\xi$ and $\vphi$, and they can be solved algorithmically – usually with the help of a symbolic manipulation program, as the dimension of the system can be quite large. This fails in the case of *first order* equations.
[**Remark 3.**]{} The symmetry relation requires the vector field to be tangent to the manifold representing the equation; this means that only *integral curves* of vector fields are relevant, and not the speed on these [@CGWs1; @CGWs3; @Gtwist2; @PuS].
[**Remark 4.**]{} A generic equation will have *no* symmetries; symmetry is a non-generic property – albeit it may become generic in a given class of equations: e.g., equations for isolated physical systems are invariant under time and space translations, and space rotations; as well known, conservation of Energy, Momentum and Angular Momentum is related to these invariances via Noether theorem [@Arn; @YKS; @Olv1].
[**Remark 5.**]{} There can be vector fields $X$ such that is satisfied without the restriction to $S_\De$, i.e. such that $X^{(k)} \( F \) = 0$; in this case one speaks of *strong* (Lie-point) symmetries. The relation between strong and standard symmetries was clarified by Carinena, Del Olmo and Winternitz [@CDW] (CDW theorem); roughly speaking – and up to some cohomological considerations – if a differential equation $\De$ admits $X$ as a symmetry, there is always a differential equation $\wt{\De}$ which admits $X$ as a *strong* symmetry and such that $\De$ and $\wt{\De}$ have the same set of solutions.
[**Remark 6.**]{} In a nineteenth-century language, the advantage of knowing symmetries of a differential equation is that its analysis, and the search for its solutions, are much easier if one uses the “right” coordinates, i.e. *symmetry-adapted* coordinates – pretty much as analyzing rotationally invariant problems is easier using spherical coordinates.
Symmetry of ODEs {#sec:sode}
----------------
The use of symmetry for ODEs is quite simple; let us focus for simplicity (and for ease of comparison in the case of $\la$-symmetries to be considered below) on $\De$ a *scalar* ODE of order $N>1$, F $ x,,u,...,u_{(N)} $ = 0 . Suppose we were able to determine a Lie-point symmetry $X = \xi
\pa_x + \vphi \pa_u$ for it, and say it is a strong symmetry (if not we can use the CDW theorem and consider the equivalent equation $\wt{\De}$, see above); outside singular points of $X$, we can pass to coordinates $(y,v)$ such that $X = \pa_v$ (flow box theorem [@ArnODE; @ArnGM]); but if in these coordinates $X$ is written in this way, its prolongation will be $X^{(N)} = \pa_v$, as follows immediately from .
The equation $\De$ will be written in the new coordinates in some different way, i.e. $\De$ now reads as G $ y,v,...,v_{(N)}
$ = 0 ; however the fact that it is invariant under $X^{(N)}$ *and* the peculiar form of $X^{(N)}$ in these coordinates imply that $G$ does not depend on $v$, i.e. we actually have G $ y,v_{(1)},...,v_{(N)} $ = 0 . It now suffices to make a new change of variables, introducing $ w := v_y$, to have a differential equation of lower order, H (y, w\_[\[N-1\]]{}) G $
y,w,...,w_{(N-1)} $ = 0 . The procedure can be iterated if this has some further symmetry (see also the Remarks below). In this way we obtained a (symmetry) *reduction* of our ODE.
If we are able to solve , say to determine a solution $w = g (y)$, we can reconstruct a solution $v = v(y)$ to simply by an integral – in this context one speaks of a *quadrature* – i.e. by v(y) = w(y) d y . Inverting the original change of coordinates we obtain a function $u = u(x)$, which is a solution to the original equation.
Note that our general notation is redundant for ODEs; in this case all derivatives are w.r.t. a single variable $x$, and we can accordingly just write $u^a_{(k)} = d^k u^a / d x^k$, and similarly $\psi^a_{(k)}$ for the coefficient of $d/du^a_{(k)}$ in $X^{(m)}$. The prolongation formula takes then the simpler form \^a\_[(k+1)]{} = D\_x \^a\_[(k)]{} - u\^a\_[(k+1)]{} D\_x .
[**Remark 7.**]{} Changing variables from $(x,u)$ to $(y,v)$ also means changing the contact forms from $\om^a_J = \d u^a_J - u^a_{J,i} \d
x^i$ to $\eta^a_J = \d v^a_J - v^a_{J,i} \d y^j$.
[**Remark 8.**]{} If the equation has several symmetries, i.e. not only the one we are using for reduction but some other ones as well, it is not guaranteed that these will still be present after the reduction. In general, one can fully use only a (maximal) *solvable* subalgebra of the symmetry algebra of the equation, and this provided the generators are used for reduction in the “right” order; see e.g. .
[**Remark 9.**]{} On the other hand, the reduced equation could have symmetries which were not present in the original equation. These symmetries appearing upon reduction can be “predicted”, and the features behind their appearance go under the name of “solvable structures”; see e.g. [@BP; @BH; @CFM1; @CFM2; @HaA; @ShP] for details.
[**Remark 10.**]{} The possibility of effectively operating symmetry reductions as sketchily described above depends on the “invariants by differentiation property”; we refer again to standard texts for details.
Symmetry of PDEs
----------------
The use of symmetries in the analysis of PDEs is rather different; actually even the *aim* of using symmetry is different. In fact, for ODEs one can hope of determining the most general solution, or at least (as we have seen above) to determine a reduced equation whose solutions are in correspondence – via a quadrature – to solutions to the original equation.
For nonlinear PDEs looking for the general solution is in general (i.e. except for integrable equations) a hopeless task, and one should instead aim at determining at least some solutions. Again the parallel with the familiar case of rotational symmetries makes things quite clear: one looks first for *symmetric solutions*, and such solutions can be determined by solving a (usually) simpler equation, i.e. one depending on fewer variables. E.g., rotationally invariant solutions depend just on the radial coordinate $r$, and hence are determined by an ODE. In the case of a nonlinear equation this will in general be a nonlinear ODE, and its solution can still be rather hard, but we definitely face a simpler problems than the original one – and correspondingly if we completely solve it, we have only a partial solution to the original one.
Thus, while in the ODE case we were looking for new coordinates in which the symmetry vector field $X$ was along one of the *dependent* coordinates, in the PDE case we want new coordinates in which the symmetry vector field $X$ (or vector fields $X_i$) is (are) along one (several) of the *independent* coordinates $y^i$. We will then look for solutions $v = f (y)$ which are invariant under the $X_i$ ($i=1,...,r$), i.e. which do not depend on the $(y^1,...,y^r)$; correspondingly we will have to solve a PDE for $v$ being a function of the $(y^{r+1},...,y^n)$ variables, i.e. in less independent variables than the original one.
[**Remark 11.**]{} From the geometric point of view, an $X$-invariant solution $u = f(x)$ is a section $\ga_f \in \Sigma
(M,\pi_0,B)$ such that $\ga_f^{(n)} \in S_\De$ (which ensures it is a solution) and also such that $X (\ga_f ) = 0$, which of course also implies $X^{(n)} ( \ga_f^{(n)} ) = 0$. If we consider a different vector field $\wt{X}$ such that $\wt{X} = \mu X$ on $\ker (X)$ (here $\mu$ is a smooth function on $M$), such solutions will also be $\wt{X}$-invariant.
Simple twisted symmetries
=========================
In recent years, starting from the seminal work of Muriel and Romero in 2001 [@MuRom1; @MuRom2] (see also ), several kinds of *twisted symmetries* have been considered in the literature [@Gtwist1; @Gtwist2].
The name originated from the fact here one considers a Lie-point vector field $X$ in $M$, but the prolongation operation is deformed in a way which depends on an auxiliary object. In different realizations this can be a scalar function ($\lambda$-symmetries [@MuRom1; @MuRom2]), a matrix-valued one form satisfying the horizontal Maurer-Cartan equations – i.e. a set of matrices satisfying a compatibility condition ($\mu$-symmetries [@CGMor]), or a matrix acting in an auxiliary space ($\sigma$-symmetries [@CGWs1]).[^4]
It should also be stressed that twisted symmetries are more easily used for *higher order* differential equations (ordinary or partial), while the case of first order equations is in some sense degenerate from this point of view, and presents several additional problems.
$\la$-symmetries {#sec:lambda}
----------------
The first type of twisted symmetries to be introduced was $\la$-symmetries (the name $C^\infty$ symmetries also appears in the literature). These (originally) considered scalar ODEs of any order, and the name refers to the auxiliary $C^\infty$ function $\la (t,x,\xd)$ defining the twisted prolongation, which in this case is called $\la$-prolongation. In fact, this is recursively defined as $$\begin{aligned}
\psi^a_{(k+1)} &=& D_x \psi^a_{(k)} \ - \
u^a_{(k+1)} \ D_x \, \xi \ + \ \la \, \( \psi^a_{(k)} \ - \
u^a_{(k)} \, \xi \) \nonumber \\ &=& (D_x \, + \, \la )
\psi^a_{(k)} \ - \ u^a_{(k+1)} \ (D_x \, + \, \la ) \, \xi \ .
\label{eq:laprol} \end{aligned}$$ We will denote the $\la$-prolongation of order $k$ of the vector field $X$ in $M$ as $X^{(k)}_{\la}$.
The vector field $X$ in $M$ is said to be a *$\la$-symmetry* of the equation $\De$ (of order $k$) if X\^[(k)]{}\_ : S\_ S\_ . Note that in general a vector field is a $\la$-symmetry of a given equation *only* for a specific choice of the function $\la$.
[**Lemma 4.**]{} [In general, the commutator of the $\la$-prolongations of two vector fields $X,Y$ in $M$ is *not* the $\la$-prolongation of their commutator, i.e. if $Z
= [X,Y]$ then (in general, for $\la \not= 0$) $$X^{(n)}_\la , Y^{(n)}_\la$$ = Z\^[(n)]{}\_ . ]{}
[**Proof.**]{} Consider e.g. $X = x \pa_u$, $Y = u \pa_u$; in this case $Z = [X,Y] = x \pa_u$, and $\delta := [ X^{(1)}_\la ,
Y^{(1)}_\la ] - Z^{(1)}_\la = x \la + ( u - x u_x )
\la_{u_x}$.
[**Lemma 5.**]{} [*The IBDP holds for $\la$-prolonged vector fields.*]{}
[**Proof.**]{} See e.g. [@MuRom1; @MuRom2], or [@Gtwist1].
[**Remark 12.**]{} Lemma 5 makes $\la$-symmetries “as useful as standardly prolonged ones”, as we will see below in Section \[sec:use\].
[**Remark 13.**]{} It was pointed out by Pucci and Saccomandi [@PuS] that $\la$-prolonged vector fields can be characterized as the *only* vector fields in $J^k M$ with the property that their integral lines are the same as the integral lines of some vector field which is the standard prolongation of some vector field in $M$. This remark was fully understood only some time after their paper, and was the basis for many of the following developments, discussed below.
$\mu$-symmetries {#sec:mu}
----------------
The $\la$-prolongation is specifically designed to deal with ODEs (or systems thereof); a generalization of it aiming at tackling PDEs (or systems thereof) is the $\mu$-prolongation. This can of course also be applied to ODEs and Dynamical Systems.
### PDEs
Now the relevant object is not a single matrix, but an array of matrices $\La_i$, one for each independent variable. These are better encoded as a ($GL(n,\R)$-valued) *horizontal one-form* = \_i (x,u,u\_x) x\^i . The matrices $\La_i$ should satisfy a compatibility condition, i.e. D\_i \_j - D\_j \_i + $$\La_i ,
\La_j$$ = 0 ; this is immediately recognized as the *horizontal Maurer-Cartan equation*,or equivalently as a *zero-curvature condition* for the connection on $\T U$ identified by \_i = D\_i + \_i .
If $\mu$ satisfies , we can define $\mu$-prolongations in terms of a modified prolongation formula, called of course *$\mu$-prolongation formula* (and which represents now an actual twisting of the familiar prolongation operation): $$\begin{aligned}
\psi^a_{J,i} &=& D_i \psi^a_{J} \ -
\ u^a_{J,k} \ D_i \, \xi^k \ + \ (\La_i)^a_{\ b} \, \( \psi^b_{J}
\ - \ u^b_{J,k} \, \xi^k \) \nonumber \\ &=& (D_i \, I \, + \, \La_i )^a_{\
b} \, \psi^b_{J} \ - \ u^b_{J,k} \ (D_i \, I \, + \, \La_i )^a_{\
b} \, \xi^k \ . \label{eq:muprol} \end{aligned}$$
We will denote the $\mu$ prolongation (of order $k$) of the vector field $X$ in $M$ as $X^{(k)}_\mu$. The vector field $X$ in $M$ is said to be a *$\mu$-symmetry* of the equation $\De$ (of order $k$) if X\^[(k)]{}\_ : S\_ S\_ . Note that in general a vector field is a $\mu$-symmetry of a given equation *only* for a specific choice of the one-form $\mu$.
[**Remark 14.**]{} In $\la$-prolongations the prolongation operation is modified, but it acts separately on the different vectorial components in $\T U$ (and in $\T U_J$). In $\mu$-prolongations, instead, the different vector components of $\T U$ (and of $\T
U_J$) are “mixed” by the prolongation operation.
[**Remark 15.**]{} It is known that $\mu$-symmetries (and hence $\la$-symmetries, which are a special case of the latter) are related to *nonlocal* symmetries; we will not discuss this relation here [@CF; @MuRom5; @MuRom12].
### ODEs
In the case of ODEs one just replaces the scalar function $\la :
J^1 M \to \R$ with a *matrix* function $\La : J^1 M \to
\mathtt{Mat} (n)$ (more generally, $\La : J^1 M \to \T U$) and define a “$\La$-prolongation” (which is just a special case of $\mu$-prolongation, for $\mu = \La \d x$) $$\begin{aligned}
\psi^a_{(k+1)} &=& D_x
\psi^a_{(k)} \ - \ u^a_{(k+1)} \ D_x \, \xi \ + \ \La^a_{\ b} \,
\( \psi^b_{(k)} \ - \ u^b_{(k)} \, \xi \) \nonumber \\ &=& (D_x \, I \, + \,
\La )^a_{\ b} \, \psi^b_{(k)} \ - \ u^b_{(k+1)} \ (D_x \, I \, +
\, \La )^a_{\ b} \, \xi \ . \label{eq:Laprol} \end{aligned}$$
In this ODE case we just have $\mu = \La \, \d x $ (only one component), and is identically satisfied.
[**Remark 16.**]{} The IBDP property is in general not holding for $\mu$-prolonged vector fields, not even in the ODEs framework; the exception is the case where the $\La_i$ are diagonal matrices.
### Recursion formula
The $\mu$-prolongation $X^{(k)}_\mu$, which we will now write in components as $X^{(k)}_\mu = \xi^i \pa_i + (\psi^a_J)_{(\mu)}
\pa_a^J $, of a vector field $X$ in $M$ is defined through ; however in some cases and applications it is relevant to characterize these in terms of the difference F\^a\_J := $ \psi^a_J $\_ - $ \psi^a_J $\_0 . It can be shown [@CGMor; @GMor] that the $F^a_J$ satisfy the recursion relation F\^a\_[J,i]{} = \^a\_b $$D_i
\, \( \Ga^J \)^b_c$$ (D\_i Q\^c ) + $ \La_i $\^a\_b $$\(\Ga^J \)^b_c \ (D_J Q^c ) \ + \ D_j Q^b$$ , where we have written Q\^a = \^a - u\^a\_i \^i , and the $\Ga^J$ are certain matrices whose detailed expression can be computed [@CGMor; @GMor] but is not essential.
[**Remark 17.**]{} With the notation , the set $I_X$ of $X$-invariant functions is characterized by $Q^a |_{I_X} = 0$. It follows from that $X^{(k)}_\mu$ coincides with $X^{(k)}_0$ on $I_X$.
Collective twisted symmetries: $\s$-symmetries {#sec:sigma}
==============================================
Let us consider the vector structure in $\T U$ and more generally in $\T^k U$. We have seen that in $\la$-prolongations different components (in terms of this structure) of a vector field “do not mix” under the prolongation operation, while in $\mu$-prolongations they do indeed “mix”.
As mentioned above, Pucci and Saccomandi [@PuS] observed that (in the scalar case) $\la$-prolongations are the *only* vector fields in $J^k M$ which have the same characteristics as *some* standardly prolonged vector field.
One can extend this approach to *distributions* generated by sets – in involution *à la Frobenius* – of standardly prolonged vector fields, and wonder if there is some deformation of the prolongation operations such that a set of vector fields obtained by this generate the same distribution in $J^k M$ as some other set of vector fields prolonged in the standard way.
This problem was tackled by Cicogna *et al.* in a series of papers [@CGWs1; @CGWs2; @CGWs3; @CGWs4] and the answer is that the most general class of systems with this property is provided by so called *$\s$-prolonged* sets of vector fields[^5]. Note that here the deformation of the prolongation operation involves *sets* (more precisely, an involutive system) of vector fields, and not a single one. We also stress that we are working in the frame of ODEs, hence with only one independent variable $x$.[^6]
Given vector fields $X_\a$ ($\a = 1 , ... , r $) in $M$, written in local coordinates as X\_ = \_ \_x + \^a\_ \_a , and satisfying the Frobenius involution relations $$X_\a , X_\b$$ = f\_\^ X\_with $f_{\a \b}^\ga : M \to \R$ smooth functions on $M$, the $\s$-prolonged vector fields $Y_\a$ on $J^k
M$ are written as Y\_ = \_ \_x + $
\psi^a_k $\_ \_a\^k where $(\psi^a_0)_\a = \vphi^a_\a$ and $ \psi^a_{k+1} $\_ = $ D_x (\psi^a_k
)_\a \ - \ u^a_{k+1} \ D_x \, \xi_\a $ + \_\^[ ]{} $
(\psi^a_k)_\b \ - \ u^a_{k+1} \, \xi_\b $ .
[**Lemma 6.**]{} Let $X_\a$ satisfy , and *assume* their $\s$-prolongations $Y_\a$ are in involution. Then the set $\{ Y_\a \}$ has the IBDP property[^7].
[**Remark 18.**]{} For fields $X_\a$ satisfying and $Y_\a$ their $\s$-prolongations, in general, $$Y_\a , Y_\b$$ = f\_\^ Y\_ . However, the $Y_\a$ can happen to be still in involution, or to be embedded in set of vector fields in involution of non-maximal rank. This is why in Lemma 6 the involution property of the $Y_\a$ has to be assumed.
[**Remark 19.**]{} While the $\mu$-prolongations mix different vector components of the same vector field, here corresponding components of different vector fields are mixed. For $r=1$ we are back to the case of $\la$-prolongations.
[**Remark 20.**]{} If $\s = \mathtt{diag} (\la_1 , ... , \la_n )$ is a diagonal matrix (but not a multiple of the identity) we have different vector fields undergoing $\la$-prolongations with different functions $\la_i$. In the case of $\s = \la I$ we are back to the case of $\la$-prolongations (in general, applied to a set of vector fields in multidimensional space).
The use of twisted symmetries {#sec:use}
=============================
We have so far discussed the definition of different types of twisted prolongations and hence of twisted symmetries. We would now like to discuss how these are applied in the study of differential equations. In doing this one should distinguish between ODEs and PDEs, recalling that – as also stressed above – the very aim of symmetry theory is different in these two contexts.
We will always assume that the vector field $X$ is a twisted symmetry (of different types) of the equations under study.
The use of $\la$-symmetries
---------------------------
If $X$ is a symmetry for an equation $\Delta$ of order $n$, this means that $\Delta$ can be written in terms of the differential invariants for $X^{(n)}_\mu$. On the other hand, as we have seen above (Lemma 5), $\la$-prolonged vector fields enjoy the IBDP. This implies that passing to $\la$-symmetry-adapted coordinates, one can indeed rewrite the equation in terms of differential invariants of order zero and one and their total derivatives, implementing the reduction procedure sketched in Section \[sec:sode\].
In other words, the usual symmetry reduction algorithm can be applied also in the case of $\la$-symmetries, which are as useful as standard ones in the study of ODEs.[^8]
The use of $\mu$-symmetries
---------------------------
As mentioned above, $\mu$-symmetries were intended for application in the study of PDEs. Here the key fact is (see also Remark 17); in fact, in studying PDEs by the symmetry approach one is aiming at determining invariant solutions, and shows that when restricting to $X$-invariant solutions it makes no difference to consider standard prolongations or $\mu$-prolongations. This also entails that we can use the same methods and techniques familiar from the case $X$ is a standard symmetry also in the case $X$ is a $\mu$-symmetry (and in general not a proper symmetry).
[**Remark 21.**]{} This also shows that $\mu$-symmetries of a given equation are strong candidates for being also (standard) conditional symmetries [@LeWin; @PuSweak], or partial symmetries [@CGpart], for the same equation.
[**Remark 22.**]{} The situation is different in the case of ODEs (this case was studied by Cicogna (he speaks in this case of $\rho$-symmetries, the $\rho$ standing for “reducing”, see below) [@Cds1; @Cds2]). In this case one can proceed pretty much as in the standard reduction procedure up to a (relevant) feature: that is, the reconstruction equation, which in the standard case amounts to a quadrature, is now a proper differential equation, and its solution may very well be very hard, or turn out to be impossible. See [@Cds1; @Cds2] for details.
The use of $\s$-symmetries
--------------------------
It follows immediately by Lemma 6 that $\s$-symmetries can also be used to reduce (systems of) ODEs in the same way and with the same procedure as in the standard case. Once again, the key fact is that this standard reduction procedure is actually based on the IBDP.
It should be stressed, however, that in this case there is a further condition to be satisfied, i.e. that the $Y_\a$ are (or can be completed in a nontrivial way – that is, without spanning the whole tangent space – to a system) in involution.
[**Remark 23.**]{} In this context, it should also be mentioned that determining $\s$-symmetries is in general a nontrivial task (recall that the determination of standard symmetries is often computationally demanding, but always algorithmic); but when one considers as differential equation a perturbation of a system for which symmetries are known, $\s$-symmetries can be sought for as deformation of the symmetries for the unperturbed system; see [@CGWs3] (and Section \[sec:pert\] below) for details.
Twisted symmetries and gauge transformations {#sec:gauge}
============================================
It appears that twisted symmetries are related to gauge transformations, and indeed the operators $\nabla_i = D_i + \La_i$ appearing in $\mu$-prolongations look very much like a covariant derivative. We will now make this relation more precise. For this, it is convenient to consider just vertical vector fields, including evolutionary representatives of general vector fields in $M$.
[**Lemma 7.**]{} Let $X = Q^a \pa_a $ and $\wt{X} = \wt{Q}^a \pa_a$ be vertical vector fields on $(M,\pi_0,B)$, $A : M \to
\mathtt{Mat}(\R,q)$ (with $q = \mathtt{dim} (U)$) a nowhere zero smooth matrix function, and $\wt{Q}^a = A^a_{\ b} Q^b$. Then A $ X^{(n)}_\mu $ = \^[(n)]{}\_0 ; = (D A) A\^[-1]{} .
[**Remark 24.**]{} The relation between $X^{(n)}_\mu$ and $\wt{X}^{(n)}_0$ in should be meant as follows: if $X^{(n)}_\mu = \psi^a_J \pa_a^J$, with $\psi^a_0 = Q^a$ and the $\psi^a_J$ for $1 \le |J| \le n$ obtained by the $\mu$-prolongation formula, and $\wt{X}^{(n)} = \wt{\psi}^a_J$ with $\wt{\psi}^a_0 = \wt{Q}^a$ and the $\wt{\psi}^a_J$ for $1 \le
|J| \le n$ obtained by the standard prolongation formula, then the relation \^a\_J = A\^a\_[ b]{} \^b\_J holds for any $a$ and $J$, $0 \le |J| \le n$.
[**Remark 25.**]{} The relations stated by Lemma 7 can be encoded in a commutative diagram:
X & &\
& &\
X\^[(n)]{}\_& & \^[(n)]{}\_0
where $A$ and $\mu$ are related by $\mu = (D A) A^{-1}$.
[**Remark 26.**]{} Lemma 7 is *not* stating that any $\mu$-prolonged vector field is obtained as the gauge transformed of a standardly prolonged one; this relation only holds for vertical vector fields. If $X = Q^a \pa_a$ is the evolutionary representative of a generic vector field $X_g = \xi^i \pa_i +
\vphi^a \pa_a$, hence $Q^a = \vphi^a - u^a_i \xi^i$, its components $Q^a$ satisfy the relations $(\pa Q^a / \pa u^b_i ) = -
\delta^a_b \xi^i$. These are obviously not satisfied in general by the components $\wt{Q}^a = A^a_{\ b} Q^b$ of $\wt{X}$ (now $(\pa
\wt{Q}^a / \pa u^b_i ) = - A^a_b \xi^i$), hence we cannot interpret the gauge-transformed vector field as the evolutionary representative of a vector field in $M$ [@Gtwist2].
[**Remark 27.**]{} The previous Remark also means that the connection between $\mu$-prolongations and gauge transformations is only transparent when we consider the action of vector fields on $\Sigma (M)$, the set of sections on $M$. It also explains why we can have twisted symmetries for equations having no standard symmetries.
[**Remark 28.**]{} More details on the interrelations between (different kinds of) twisted symmetries and gauge transformations are provided e.g. in [@Ggauge1; @Ggauge2; @Ggauge3].
Twisted symmetries and Frobenius theory
=======================================
The existence of a relation between twisted symmetries and gauge transformations was more and less evident since the introduction of $\la$-symmetries by Muriel and Romero, and so the generalization of $\la$-symmetries to $\mu$-symmetries was, in this sense, not surprising.
It was much less obvious that symmetries and twisted symmetries could be generalized in a different direction, focusing on *sets* (actually, involutive systems) of vector fields rather than on single ones[^9]. As already mentioned, the key step in this direction was provided by Pucci and Saccomandi [@PuS], who stressed the symmetry relation involves the integral lines of (prolongations of) symmetry vector fields, not the way in which the flow defined by the vector field travels on these.
The geometrical idea behind $\s$-symmetries is that (standard) symmetry vector fields for the equation $\De$ of order $n$ are the vector fields on $M$ whose (standard) prolongation to $J^n M$ belongs to the *distribution* tangent to $S_\De$ in $\T J^n
M$. Focusing on the distribution – rather on the single vector fields, i.e. the “usual” generators of the distribution – has an obvious consequence: we can change the generators of the distribution.
In particular, if we are dealing with ODEs, we would like to be able to change the generators of the distribution (to have more freedom), but at the same time be sure that the key tool for ODE reduction, i.e. the IBDP, is still at work.
The idea of $\s$ symmetries is exactly this: the $\s$-prolongation is the most general way of twisting prolongation by mixing *different* vector fields in such a way that the IBDP still holds, and hence so that twisted symmetries can be of use for the reduction of ODEs.
[**Lemma 8.**]{} Let $\mathcal{X} = \{ X_1 , ... , X_r \}$ be a set of vector fields on $M$; and let the vector fields $\mathcal{Y} =
\{ Y_1 , ... , Y_r \}$ on $J^n M$ be their $\s$-prolongation. Consider also $A : M \to \mathtt{Mat} (\R,q)$ (where $q =
\mathtt{dim} (U)$) a nowhere singular matrix function on $M$, such that $\s = A^{-1} (D_x A)$; and the set $\mathcal{W} = \{ W_1 ,
... , W_r \}$ of vector fields on $M$ given by $W_\a = A_\a^{\ \b}
X_\b$, with $\mathcal{Z} = \{ Z_1 , ... , Z_r \}$ on $J^n M$ their standard prolongation. Then, $Z_\a = A_\a^{\ \b} Y_\b$.
[**Remark 29.**]{} The relations stated by Lemma 8 can be encoded in a commutative diagram:
{X\_i } & & { W\_i }\
& &\
{ Y\_i } & & { Z\_i }
where $A$ and $\s$ are related by $\s = A^{-1} (D A)$.
[**Remark 30.**]{} Traditionally, in the symmetry analysis of differential equations one focuses on the Lie algebra structure of symmetry vector fields. Passing to consider Frobenius reduction means one is instead focusing on the *Lie module* structure.[^10]
[**Remark 31.**]{} The determination of standard symmetries is algorithmic for equations of order $n \ge 2$, but is especially difficult for first order equations – even more so for first order ODEs, i.e. Dynamical Systems – albeit in general we always have infinitely many symmetries in this case. In the case of Dynamical Systems, an interesting possibility was noted by Cicogna [@CGWs3]: if we consider the perturbations of a symmetric Dynamical Systems, $\s$-symmetries can be looked for by building $\s$ as a perturbation to the identity matrix. See Section \[sec:pert\] below.
[**Remark 32.**]{} The possibility of using sequentially different symmetries for Frobenius reduction rests – like in the case of standard reduction – on a suitable involution structure; that is, we should have a solvable Lie-module.
[**Remark 33.**]{} More details on $\s$-prolongations and symmetries, including their geometrical meaning, is provided in the papers [@CGWs1; @CGWs2; @CGWs3; @CGWs4]; see also [@Gtwist2].
Twisted symmetries and variational problems {#sec:variational}
===========================================
The theory of twisted symmetries was developed mainly referring to ODEs and evolution PDEs. But we know that in Physics a special role is played by problems admitting a *variational formulation*. In this case, the relation between symmetries and conservation laws is embodied by the classical Noether theorem [@Arn; @YKS; @Olv1]. It is thus natural to wonder if there is a version of Noether theorem applying to *twisted* symmetries.
Only partial results exist in this direction; these are concerned with $\la$-symmetries [@MRO; @RMO] and with $\mu$-symmetries [@CGNoether], while it seems no result is available dealing with $\s$-symmetries.
Variational problems and $\la$-symmetries
-----------------------------------------
The relation between $\la$-symmetries and Euler-Lagrange equations has been considered in a by now classical work of Muriel, Romero and Olver [@MRO]; lately new results in this direction have been obtained by Ruiz, Muriel and Olver [@RMO].
### Single $\la$-symmetry of a variational problem
We consider variational problems defined by a Lagrangian density $L$, hence by S \[u\] = L (x,u\_[\[n\]]{}) d x ; here $x \in \R$, $u \in \R$. To this problem are associated the Euler-Lagrange equations E\[L;u\] = \_[k=0]{}\^n $ - \, D_x $\^k $ \frac{\pa L}{\pa u_k} $ = 0 .
A vector field $X = \xi (x,u) \pa_x + \vphi (x,u) \pa_u$ is a standard variational symmetry [@Olv1] if there is a function $F : J^n M \to \R$ such that X\^[(n)]{} ( L) + L (D\_x ) = D\_x F . This definition is generalized by saying that $X$ is a *variational $\la$-symmetry* if there is a function $F : J^n M \to \R$ such that X\^[(n)]{}\_[()]{} ( L) + L \[(D\_x + ) ) = (D\_x +) F . If $X$ is a variational $\la$ symmetry for $L$, it is such also for $\wt{L}
= L + (D_x f)$, for any $f : J^n M \to \R$, i.e. for any equivalent Lagrangian [@MRO].
Variational $\la$-symmetries lead to reduction of order for the variational problem in the same way as standard variational symmetries. More precisely, Muriel, Romero and Olver prove the following result (Theorem 1 in [@MRO]).
[**Lemma 9.**]{} Let $S[u]$ as in be an n$^{th}$-order variational problem with Euler-Lagrange equation $E[L;u] = 0$ of order $2n$. Let $X$ be a variational $\la$-symmetry, where $\la : J^1 M \to \R$ is smooth. Then there exists a variational problem Ŝ \[w\] = L (,w\_[\[n-1\]]{}) d of order $n - 1$, with Euler-Lagrange equation $E[\^L;w] = 0$ of order $2n-2$, such that a $(2n-1)$-parameter family of solutions of $E[L;u] = 0$ can be found by solving a first-order equation from the solutions of the Euler-Lagrange reduced equation $E[\^L;w] = 0$.
As for the Noether theorem, this essentially follows (for standard symmetries) from X\^[(n)]{} (L) = Q E\[L\] + D\_x F where $F$ is some function $F: J^n M \to \R$, and $Q =
\vphi - u_x \xi$ is the characteristic of the (evolutionary representative of) $X$.
In the case of $\la$-prolongations, one can prove [@MRO] that there is some $F$ such that X\^[(n)]{}\_(L) = Q E\[L\] + (D\_x + ) F . Then the following result (which is Theorem 2 in [@MRO]) follows.
[**Lemma 10.**]{} Let $X$ be a variational $\la$-symmetry of the variational problem , and $Q$ the characteristic of $X$. Then there exists $P[u] : J^n M \to \R$ such that Q E\[L\] = (D\_x + ) P .
[**Remark 34.**]{} While standard variational symmetries of a variational problem are symmetries of the corresponding Euler-Lagrange equations, $\la$-symmetries of the variational problem are in general *conditional* symmetries of the Euler-Lagrange equations [@MRO] (see also Remark 17 in this respect).
[**Remark 35.**]{} If $X$ is a variational $\la$-symmetry of , and $P[u]$ is the functional given in Lemma 10, then $X$ is a $\la$-symmetry of the equation $P[u] = 0$. The reduction of this equation through $X$ is (up to multipliers) the reduced equation of the Euler–Lagrange equation corresponding to $X$, according to Lemma 9 [@MRO].
### Multiple $\la$-symmetry of a variational problem
In more recent work, Ruiz, Muriel and Olver [@RMO] studied variational problems systems which admit several $\la$-symmetries $X_i$, where for each of them a *different* function $\la_i$ defines $\la$-prolongation.
They considered in particular the case of two such $\la$-symmetries $\{ X, Y \}$, subject to the “solvability condition” $$X^{(n)}_{\la_1} , Y^{(n)}_{\la_2}$$ = h X\^[(n)]{}\_[\_1]{} .
Then the $\la$-symmetries can be used (in the proper order!) to perform *two* symmetry reductions of the variational problem.[^11] In particular, one can prove [@RMO] that:
[**Lemma 11.**]{} Let be an $n$-th order variational problem with Euler-Lagrange equation $E[L;u] = 0$ of order $2n$. Let $(X_1, \la_1), (X_2, \la_2)$ be variational $\la$-symmetries that form a solvable pair, as in . Then there exists a variational problem $\^S = \int \^L [x,z_{n-2}] d x$ of order $n-2$ such that a $(2n-2)$-parameter family of solutions of $E[L;u] = 0$ can be reconstructed from the solutions of the associated $(2n-4)$-th order Euler-Lagrange equation $E[\^L;z]
= 0$ by solving two successive first order ordinary differential equations.
[**Remark 36.**]{} Albeit $\la$-prolongations with different functions $\la_i$ for different vector fields fit within $\s$-prolongations – see in particular Remark 20 – it should be stressed that variational problems have never been studied in that framework. Thus Lemma 11 calls for a full study of Frobenius reduction in the variational context.
Variational problems and $\mu$-symmetries
-----------------------------------------
A different approach to twisted symmetries (in particular, $\mu$-symmetries) in variational problems was considered by Cicogna [*et al.*]{} [@CGNoether]. In this work they show that $\mu$-symmetries are associated to so called $\mu$-conservation laws[^12] and in the end, for variational problems with a single independent variable (dynamical variational problems) and $\La = \la I$, to conditionally conserved quantities [@LC; @PuRos; @PuRos2; @SarCan; @SarCan2].
These are quantities such that only *some* of their level sets correspond to invariant manifolds – while for proper conserved quantities *all* the level sets are dynamically invariant; note that the result is strictly related to the one mentioned in Remark 34.
There is a different way of looking at variational problems, in particular dynamical variational problems, with $\mu$-symmetries; this descends from the relation between $\mu$-prolongations and gauge transformations.
If we think of such a problem as arising – through a gauge transformation – from one in which the $\mu$-prolonged vector field was prolonged in the standard way, i.e. perform the needed inverse gauge transformation (see section \[sec:gauge\]), we should think that the original variational problem was a different one, and Euler-Lagrange equations were also different. In particular, the role of $D_i$ in the Euler-Lagrange operator would be taken by $\nabla_i = D_i + \La_i$. Thus, e.g., in the case of Mechanics the $\mu$-Euler-Lagrange equations read - = $ \La^T $\_i\^[ j]{} . One can then show that (for further details, proofs and examples, see [@CGNoether]):
[**Lemma 12.**]{} If $L$ is a first-order Lagrangian admitting the vector field $X = \vphi^a \pa_a$ as a $\mu$-symmetry, then ${\bf
P}$ of components $P^i = \vphi^a \pi^i_a$ (where $\pi^i_a = \pa L
/ \pa u^a_i$) defines a *standard* conservation law, $D_i P^i
= 0$, for the flow of the associated $\mu$-Euler-Lagrange equations.
Twisted symmetries and perturbations of Dynamical Systems {#sec:pert}
=========================================================
As mentioned above (see Remarks 23 and 31), $\s$-symmetries turn out to be specially suited for the investigation of perturbations of symmetric Dynamical Systems. It should be stressed that in this case also the determination of $\s$-symmetries (which is, as for all types of twisted symmetries, a non-algorithmic task) turns out to be facilitated.
The type of result one can obtain in this direction is illustrated by the following result (Theorem 4 in [@CGWs3]):
[**Lemma 13.**]{} Let the dynamical system = f\^i (x) admit the vector fields $X_\a = \vphi^i_\a \pa_i
$ as standard symmetries, and let these span a Lie algebra[^13], $$X_\a , X_\b$$ = c\_\^ X\_ . Consider moreover the vector fields $Y_\a$ in $J M$ obtained as $\s$-prolongations of the $X_\a$ with \_\^[ ]{} = c\_\^ F\^ + X\_(F\^) . Then: $(i)$ The $Y_\a$ are in involution and satisfy the same commutation relations as the $X_\a$; $(ii)$ any dynamical system of the form = f\^i (x) + \_[=1]{}\^r F\^(x) \^i\_(x) admits the set of $X_\a$ as $\s$-symmetries – with $\s$ given by – and hence can be reduced via these.
[**Remark 37.**]{} The form of systems which can be dealt with in this way may seem too specific, but it includes at least one relevant class, i.e. that of systems in Poincaré-Dulac *normal form* [@ArnGM; @CGbook]. In fact, let $f(x) = A x$ with $A$ a semi-simple matrix; then we consider $\vphi_\a (x) =
B_\a x$ with matrices such that $[A,B_\a] = 0$ (these obviously form a Lie algebra $\mathcal{G}$, and $B_0 = A$ is always in the set). Then the polynomial vector fields which admit $X_0 = (A x)^i
\pa_i $ as symmetry are just those written in the form , with $F^\a (x)$ generators for the ring of $X_0$-invariant functions. See [@CGWs3] for details and examples, as well as [@GGSW; @GWNF; @GWNF2] for related matters.
[**Remark 38.**]{} Orbital reduction of dynamical systems [@HaWa; @Wal99; @Wal99b] can be dealt with in a similar manner; we will not discuss this here.
Conclusions
===========
The theory of twisted symmetries of differential equations has been created in 2001; it passed from a smart observation by its creators [@MuRom1; @MuRom2] to a coherent set of results, and from an analytic formulation to a geometrical one. In particular, in the course of this travel several relations with *gauge transformations* and with the Frobenius theory of vector fields have been uncovered, and it has been realized how twisted symmetries become rather natural if one looks not at the standard theory of (Lie) reduction, but at *Lie-Frobenius reduction* for differential equations.
It has also been realized that twisted symmetries – in the form of “perturbed prolongations” – can be used to study perturbations of symmetric equations and in particular symmetric Dynamical Systems; this part of the theory definitely awaits further developments.
Similarly, the study of twisted symmetries (and their use) for variational problems is in its initial phase, and is worth receiving further attention.
Albeit we have not touched this topic at all, I would like to mention also that in the recent wave of interest for symmetries of *stochastic* differential equations (see [@GGPR] and references therein) there is not yet any work studying the role (if any) of twisted symmetries in that context.
I hope these pages can help attracting mathematicians to this nice and promising field; it would be even nicer if our young friend Juergen Scheurle himself could contribute to the topic.
Acknowledgements {#acknowledgements .unnumbered}
================
Work partially supported by SMRI and GNFM-INdAM.
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[^1]: For the moment, ODEs or PDEs will not make a difference, and differential equations, are always possibly vector ones, i.e. systems; similarly, functions are always possibly vector ones – albeit in some cases I will use vector indices explicitly to avoid possible confusion.
[^2]: Here and in the following we adhere to Einstein summation convention over repeated indices (and multi-indices).
[^3]: More general classes of symmetry can be (and indeed are, in the literature) considered, but we will only consider these.
[^4]: It should be said that actual “twisting” only occurs in the latter cases, not for $\lambda$-symmetries, but I find it convenient to use this collective name [@Gtwist1; @Gtwist2].
[^5]: This refers to the $(r \times r)$ matrix $\s$, where $r$ is the cardinality of the set of vector fields, which appears in the $\s$-prolongation formula , see below.
[^6]: Actually $\s$-prolongations and symmetries can also be defined in the framework of PDEs (they go then under the name of $\chi$-symmetries), but here we are mainly concerned with Dynamical Systems.
[^7]: We specify that in this case the IBDP property should be meant as follows: if $\eta$ and $\zeta_{(k)}$ are independent common differential invariants for all of the $Y_\a$, then so are the $\zeta_{(k+1)} := (D_x \zeta_{(k)})/(D_x \eta)$.
[^8]: We mention in passing that $\la$-symmetries have also been used for the reduction of *discrete* equations [@LNR12; @LR10]; this lies outside our scope here.
[^9]: One speaks therefore of “collective” twisted symmetries. Actually here we will only deal with the case of ODEs and Dynamical Systems ($\s$-symmetries), rather than general PDEs ($\chi$-symmetries). For the latter, the interested reader is referred to [@CGWs3; @Gtwist2].
[^10]: Here we mean a module over the algebra $C^\infty (M,R)$ of (smooth) real functions on $M$. Note that while some equations (in particular all equations which are linear or can be linearized by a change of variables) have an infinite dimensional Lie algebra of symmetries, their set of symmetries is always finitely generated as a Lie module.
[^11]: It appears that the result can be extended to $k$ $\la_i$-symmetries with a suitable solvability condition, i.e. generating a solvable Lie module.
[^12]: A conservation law is a relation of the type $D_i \cdot {\bf P}^i = 0$ for some vector ${\bf P}$; a $\mu$-conservation law reads $\mathtt{Tr} \( \nabla_i \cdot {\bf
P^i} \) = 0$, with $\nabla_i = D_i +\Lambda_i$.
[^13]: The real constants $c_{\a \b}^\ga$ being the structure constants.
| ArXiv |
---
abstract: |
[\
]{} Heavy neutrinos with additional interactions have recently been proposed as an explanation to the MiniBooNE excess. These scenarios often rely on marginally boosted particles to explain the excess angular spectrum, thus predicting large rates at higher-energy neutrino-electron scattering experiments. We place new constraints on this class of models based on neutrino-electron scattering sideband measurements performed at MINER$\nu$A and CHARM-II. A simultaneous explanation of the angular and energy distributions of the MiniBooNE excess in terms of heavy neutrinos with light mediators is severely constrained by our analysis. In general, high-energy neutrino-electron scattering experiments provide strong constraints on explanations of the MiniBooNE observation involving light mediators.
author:
- 'Carlos A. Argüelles'
- Matheus Hostert
- 'Yu-Dai Tsai'
bibliography:
- 'main.bib'
title: |
Testing New Physics Explanations of MiniBooNE Anomaly\
at Neutrino Scattering Experiments
---
**Introduction –** Non-zero neutrino masses have been established in the last twenty years by measurements of neutrino flavor conversion in natural and human-made sources, including long- and short-baseline experiments. The overwhelming majority of data supports the three-neutrino framework. Within this framework, we have measured the mixing angles that parametrize the relationship between mass and flavor eigenstates to few-percent-level precision [@Esteban:2018azc]. The remaining unknowns are the absolute scale of neutrino masses and their origin, the CP-violating phase, and the mass ordering of the neutrinos. Nevertheless, anomalies in short-baseline accelerator and reactor experiments [@Athanassopoulos:1996jb; @Aguilar:2001ty; @AguilarArevalo:2007it; @Aguilar-Arevalo:2018gpe] challenge this framework and are yet to receive satisfactory explanations. Minimal extensions of the three-neutrino framework to explain the anomalies introduce the so-called sterile neutrino states, which do not participate in Standard Model (SM) interactions in order to agree with measurements of the Z-boson invisible decay width [@ALEPH:2010aa]. Unfortunately, these minimal scenarios are disfavoured as they fail to explain all data [@Collin:2016aqd; @Capozzi:2016vac; @Dentler:2018sju; @Diaz:2019fwt]. This has led the community to explore non-minimal scenarios. Along this direction, it is interesting to study well-motivated neutrino-mass models that can also explain the short-baseline anomalies and are testable in the laboratory. In this work, we will investigate a class of neutrino-mass-related models that have been proposed as an explanation of the anomalous observation of $\nu_e$-like events in MiniBooNE [@Aguilar-Arevalo:2018gpe].
MiniBooNE is a mineral oil Cherenkov detector located in the Booster Neutrino Beam (BNB), at Fermilab [@AguilarArevalo:2008yp; @AguilarArevalo:2008qa]. Using data collected between 2002 and 2017, the experiment has observed an excess of $\nu_e$-like events that is currently in tension with the standard three-neutrino prediction and is beyond statistical doubt at the $4.7 \sigma$ level [@Aguilar-Arevalo:2018gpe]. While it is possible that the excess is fully or partially due to systematic uncertainties or SM backgrounds (see, *e.g.*, [@AguilarArevalo:2008rc; @Aguilar-Arevalo:2012fmn; @Hill:2010zy]), many Beyond the Standard Model (BSM) explanations have been put forth. These new physics (NP) scenarios typically require the existence of new particles, which can: participate in short-baseline oscillations [@Murayama:2000hm; @Strumia:2002fw; @Barenboim:2002ah; @GonzalezGarcia:2003jq; @Barger:2003xm; @Sorel:2003hf; @Barenboim:2004wu; @Zurek:2004vd; @Kaplan:2004dq; @Pas:2005rb; @deGouvea:2006qd; @Schwetz:2007cd; @Farzan:2008zv; @Hollenberg:2009ws; @Nelson:2010hz; @Akhmedov:2010vy; @Diaz:2010ft; @Bai:2015ztj; @Giunti:2015mwa; @Papoulias:2016edm; @Moss:2017pur; @Carena:2017qhd], change the neutrino propagation in matter [@Liao:2016reh; @Liao:2018mbg; @Asaadi:2017bhx; @Doring:2018cob], or be produced in the beam or in the detector and its surroundings [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. These models either increase the conversion of muon- to electron-neutrinos or produce electron-neutrino-like signatures in the detector, where in the latter category one typically exploits the fact that the LSND and MiniBooNE are Cherenkov detectors that cannot distinguish between electrons and photons. Although many MiniBooNE explanations lack a connection to other open problems in particle physics, recent models [@Bertuzzo:2018ftf; @Bertuzzo:2018itn; @Ballett:2018ynz; @Ballett:2019cqp; @Ballett:2019pyw] are motivated by neutrino-mass generation via hidden interactions in the heavy-neutrino sector. In particular, a common prediction of these models is the upscattering of a light neutrino into a heavy neutrino, usually with masses in the tens to hundreds of MeV, which subsequently decays into a pair of electrons. To reproduce the MiniBooNE excess angular distribution either the heavy neutrino must have moderate boost factors and the pair of electrons produced need to be collimated [@Bertuzzo:2018itn], or the heavy neutrino two-body decays must be forbidden [@Ballett:2018ynz].
In this article, we introduce new techniques to probe models that rely on the ambiguity between photons and electrons to explain the MiniBooNE observation, using the dark neutrino model from [@Bertuzzo:2018itn; @Bertuzzo:2018ftf] as a benchmark scenario. Our analysis extends to all models with new marginally boosted particles produced in coherent-like neutrino interactions, as they predict large number of events at higher energies [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. Thus, our analysis uses high-energy neutrino-electron scattering measurements [@Auerbach:2001wg; @Deniz:2009mu; @Bellini:2011rx; @Park:2013dax; @Valencia:2019mkf; @Park:2015eqa; @Valencia-Rodriguez:2016vkf; @DeWinter:1989zg; @Geiregat:1992zv; @Vilain:1994qy]. This process is currently used to normalize the neutrino fluxes, due to its well-understood cross section, and has been a fertile ground for light NP searches [@Pospelov:2017kep; @Lindner:2018kjo; @Magill:2018tbb]. Here, however, we expand the capability of these measurements to probe BSM-produced photon-like signatures, by developing a new analysis using previously neglected sideband data. Our technique is complementary to recent searches for coherent single-photon topologies [@Abe:2019cer]. Since the upscattering process has a threshold of tens to hundreds of MeV, we focus on two high-energy neutrino experiments: [@Park:2013dax; @Valencia:2019mkf; @Park:2015eqa; @Valencia-Rodriguez:2016vkf], a scintillator detector in the Neutrinos at the Main Injector (NuMI) beamline at Fermilab, and CHARM-II [@DeWinter:1989zg; @Geiregat:1992zv; @Vilain:1994qy], a segmented calorimeter detector at CERN along the Super Proton Synchrotron (SPS) beamline. These experiments are complementary in the range of neutrino energies they cover and have different background composition. In all cases a relevant sideband measurement exists, allowing us to take advantage of the excellent particle reconstruction capabilities of and the precise measurements at CHARM-II to constrain NP.
![[*Upscattering cross section compared to the quasi-elastic.*]{} The quasi-elastic cross section on Carbon ($6p^+$) is shown as a function of the neutrino energy (solid black line). The coherent (solid blue) and incoherent (dashed blue) scattering NP cross sections are also shown for the benchmark point of [@Bertuzzo:2018itn]. In the background, we show the BNB flux of $\nu_\mu$ at MiniBooNE (light gray), and the NuMI beam neutrino flux at MINER$\nu$A for the LE (light golden) and ME (light blue) runs in neutrino mode.\[fig:cross\_section\]](cross_sections.pdf){width="49.00000%"}
**Model –** We consider a minimal realisation of dark neutrino models [@Bertuzzo:2018ftf; @Bertuzzo:2018itn; @Ballett:2018ynz; @Ballett:2019cqp; @Ballett:2019pyw] that can explain MiniBooNE. This comprises of one Dirac heavy neutrino[^1], $\nu_4$, with its associated flavor state, $\nu_D$. The dark neutrino $\nu_D$ is charged under a new local U$(1)^\prime$ gauge group, which is part of the particle content and gauge structure needed for mass generation. The dark sector is connected to the SM in two ways: kinetic mixing between the new gauge boson and hypercharge, and neutrino mass mixing. We start by specifying the kinetic part of the NP Lagrangian $$\mathscr{L}_{\rm kin} \supset
\;\; \frac{1}{4} \hat{Z}^{\prime}_{\mu \nu} \hat{Z}^{\prime \mu \nu} + \frac{\sin{\chi}}{2} \hat{Z}^{\prime}_{\mu \nu} \hat{B}^{\mu \nu} + \frac{m_{\hat{Z}^\prime}^2}{2} \hat{Z}^{\prime \mu} \hat{Z}^\prime_{\mu},$$ where $\hat{Z}^{\prime \mu}$ stands for the new gauge boson field, $\hat{Z}^{\prime \mu\nu}$ its field strength tensor, and $\hat{B}^{\mu \nu}$ the hypercharge field strength tensor. After usual field redefinitions [@Chun:2010ve], we arrive at the physical states of the theory. Working at leading order in $\chi$ and assuming $m_{Z^\prime}^2/m_{Z}^2$ to be small, we can specify the relevant interaction Lagrangian as $$\mathscr{L}_{\rm int} \supset \;\;g_D \overline{\nu}_D \gamma_\mu \nu_D Z^{\prime \mu}
+ e \varepsilon Z'^{\mu}J^{\rm EM}_{\mu},$$ where $J^{\rm EM}_{\mu}$ is the SM electromagnetic current, $g_D$ is the U$(1)^\prime$ gauge coupling assumed to be $\mathcal{O}(1)$, and $\varepsilon \equiv c_{\rm w} \chi$, with $c_{\rm w}$ being the cosine of the weak angle. Additional terms would be present at higher orders in $\chi$ and mass mixing with the SM $Z$ is also possible, though severely constrained. After electroweak symmetry breaking, $\nu_D$ is a superposition of neutrino mass states. The flavor and mass eigenstates are related via $$\nu_\alpha = \sum^{4}_{i=1} U_{\alpha i}\nu_{i}, \quad (\alpha=e,\mu,\tau,D),$$ where $U$ is a $4\times4$ unitary matrix. It is expected that $|U_{\alpha 4}|$ is small for $\alpha = e, \mu, \tau$, but $|U_{D4}|$ can be of $\mathcal{O}(1)$ [@Parke:2015goa; @Collin:2016aqd].
![*New physics prediction at ME and CHARM-II.* Neutrino-electron scattering data in $dE/dx$ at (top) and in $E\theta^2$ at CHARM-II (bottom). Error bars are too small to be seen. For both experiments, we show the $\nu-e$ signal and total background prediction quoted (after tuning at MINER$\nu$A), as well as the NP prediction (divided by 10 at CHARM-II). The cuts in our analysis our shown as vertical lines. \[fig:NP\_events\]](both_cartoon.pdf){width="50.00000%"}
**MiniBooNE signature and region of interest–** The heavy neutrino is produced from an active flavour state upscattering on a nuclear target $A$, $\nu_\alpha A \to \nu_4 A$. The upscattering cross section is proportional to $\alpha_D \alpha_\textsc{qed}\varepsilon^2 |U_{\alpha 4}|^2$, dominated by $|U_{\mu 4}|$ since all current accelerator neutrino beams are composed mainly of muon neutrinos. This production can happen off the whole nucleus in a coherent way or off individual nucleons. For $m_{Z^\prime} \lesssim 100$ MeV, the production will be mainly coherent, but for heavier masses, such as the ones considered in [@Ballett:2018ynz], incoherent upscattering dominates. In Fig. \[fig:cross\_section\], we show the NP cross section at the benchmark point of [@Bertuzzo:2018itn] and compare it with the quasi-elastic cross section. By superimposing the cross section on the neutrino fluxes of and MiniBooNE, we make it explicit that the larger energies at and CHARM-II are ideal to produce $\nu_4$. Once produced, $\nu_4$ predominantly decays into a neutrino and a dielectron pair, $\nu_4 \to \nu_\alpha e^+ e^-$, either via an on-shell [@Bertuzzo:2018itn] or off-shell [@Ballett:2018ynz] $Z^\prime$ depending on the choice of $m_4$ and $m_{Z^\prime}$. In this work, we restrict our discussion to the $m_4 > m_{Z^\prime}$ case, where the upscattering is mainly coherent and is followed by a chain of prompt two body decays $\nu_4 \to \nu_\alpha (Z^\prime \to e^+ e^-)$. The on-shell $Z^\prime$ is required to decay into an overlapping $e^+e^-$ pair, setting a lower bound on its mass of a few MeV. Experimentally, however, $m_{Z^\prime} > 10$ MeV for $e \epsilon \sim 10^{-4}$ to avoid beam dump constraints [@Bauer:2018onh]. Increasing $m_{Z^\prime}$ increases the ratio of incoherent to coherent events and makes the electron pair less overlapping. Even though we focus on overlapping $e^+e^-$ pairs, we note that a significant fraction of events would appear as well-separated showers or as a pair of showers with large energy asymmetry, similarly to neutral current (NC) $\pi^0$ events. The asymmetric events also contribute to the MiniBooNE excess and offer a different target for searches in $\nu-e$ scattering data.
A fit to the neutrino energy spectrum at MiniBooNE was performed in [@Bertuzzo:2018itn] and is reproduced in [Fig. \[fig:final\_plot\]]{}. We have performed our own fit to the MiniBooNE energy spectrum using the data release from [@Aguilar-Arevalo:2018gpe], and our results agree with [@Bertuzzo:2018itn], when we simulate the signal at MiniBooNE and the analysis cuts in the same way. This fit leads to preferred values of $m_4$ close to 100 MeV and $|U_{\mu 4 }| \sim 10^{-4}$. Unfortunately, this energy-only fit neglects the distribution of the excess events as a function of their angle $\theta$ with respect to the beam. This is important, as the total observed excess contains only $\approx 50\%$ of the events in the most forward bin ($0.8 < \cos{\theta} < 1.0$), with a statistical uncorrelated uncertainty of 5% on this quantity.
As was recently pointed out in [@Jordan:2018qiy], few NP scenarios can reproduce the angular distribution of the MiniBooNE excess. Among these are models where new unstable particles are produced in inelastic collisions in the detector, such as the present case. Here, large $\theta$ can be achieved by tweaking the mass of the heavy neutrino; the signal becomes less forward as $\nu_4$ becomes heavier. To show this, we use our dedicated Monte Carlo (MC) simulation to asses the values of $m_4$ preferred by MiniBooNE data [^2]. For $m_{Z^\prime} = 30$ MeV and $m_4 = 100$, $200$, and $400$ MeV, we find that 98%, 87%, and 70% of the NP events would lie in the most forward bin, respectively. We find the predicted angular distribution to be more forward than [@Bertuzzo:2018itn] due to an improved MiniBooNE simulation; see Supplementary Material for details. This simulation discrepancy is understood and only strengthens our conclusions. Thus the relevant region for the MiniBooNE angular distribution is $m_4 \gtrsim 400$ MeV for $m_{Z^\prime} = 30$ MeV.
[**Our analysis –**]{} Neutrino-electron scattering measurements predicate their cuts in the following core ideas: no hadronic activity near the interaction vertex, small opening angle from the beam, $E_e \theta^2 \lesssim 2 m_e$, and the requirement that the measured energy deposition, $dE/dx$, be consistent with that of a single electron. For the NP events, when the coherent process dominates and the mass of the $Z^\prime$ is small, the first two conditions are often satisfied. However, the requirement of a single-electron-like energy deposition removes a significant fraction of the new-physics induced events. This presents a challenge, as the NP events are mostly overlapping electron pairs and will potentially be removed by the $dE/dx$ cut. In order to circumvent this problem, we perform our analysis not at the final-cut level, but at an intermediate one. This is done differently for CHARM-II and MINER$\nu$A: the CHARM-II experiment provides data as a function of $E_e \theta^2$ without the $dE/dx$ cut, and provides data as a function of the measured $dE/dx$ after analysis cuts on $E_e \theta^2$.
We have developed our own MC simulation for candidate electron pair events in MiniBooNE, MINER$\nu$A and CHARM-II; see the Supplementary Material for more details on detector resolutions, precise signal definition, and resulting distributions. We only consider the coherent part of the cross section to avoid hadronic-activity cuts, which is conservative. We also select only events with small energy asymmetries and small opening electron angles. When required, we assume the mean $dE/dx$ in plastic scintillator to follow the same shape as the NC $\pi^0$ prediction. Our prediction for new physics events for the BP point is show in Fig. \[fig:NP\_events\] on top of the ME and CHARM-II data and MC prediction.
The CHARM-II analysis is mostly based on Fig. 1 of [@Vilain:1994qy]. This sample is shown as a function of $E\theta^2$ and does not have any cuts on $dE/dx$. It contains all events with shower energies between $3$ and $24$ GeV, and our final cut on $E\theta^2$ is fixed at $28$ MeV. For , the event selection is identical for the low-energy (LE) and medium-energy (ME) analyses [@Park:2015eqa; @Valencia:2019mkf]. The minimum shower energy required is $0.8$ GeV in order to remove the $\pi^0$ background and have reliable angular and energy reconstruction. Events are kept only when they meet the following angular separation criterion: $E_e \theta^2 < 3.2\times 10^{-3}~{\rm ~GeV ~rad^2}$. A final cut is applied, ensuring $dE/dx < 4.5~{\rm MeV} / 1.7~{\rm cm}$. The analyses use the data outside the previous $dE/dx$ cut to constrain backgrounds. This sideband is defined by all events with $E_e\theta^2 > 5 \times 10^{-3} {\rm ~GeV ~rad^2}$ and $dE/dx < 20~{\rm MeV}/1.7~{\rm cm}$. Using this sideband measurement, the collaboration tunes their backgrounds by ($0.76$, $0.64$, $1.0$) for ($\nu_e$CCQE, $\nu_\mu$NC, $\nu_\mu$CCQE) processes in the LE mode. Our LE analysis uses the data shown in Fig. 3 of [@Park:2015eqa] where all the cuts are applied except for the final $dE/dx$ cut. In our final event selection, we require that the sum of the energy deposited be more than $4.5$ MeV$/ 1.7$ cm, compatible with an $e^+e^-$ pair and yielding an efficiency of $90\%$.
The recent ME data contains an excess in the region of large $dE/dx$ [@Valencia:2019mkf], where the NP events would lie. However, this excess is attributed to NC $\pi^0$ events, and grows with the shower energy undershooting the rate require to explain the MiniBooNE anomaly. With normalization factors as large as 1.7, the collaboration tunes primarily the NC $\pi^0$ prediction in an energy dependent way. After tuning, the total NC $\pi^0$ sample corresponds to $20\%$ of the total number of events before the $dE/dx$ cut.
To place our limits, we perform a rate-only analysis by means of a Pearson’s $\chi^2$ as test statistic; detailed definition is given in the Supplementary Material. We incorporate uncertainties in background size and flux normalization as nuisance parameters with Gaussian constraint terms. For the neutrino-electron scattering and BSM signal, we allow the normalization to scale proportionally to the same flux uncertainty parameter. The background term also scales with the flux-uncertainty parameter but has an additional nuisance parameter to account for its unknown size. We obtain our constraint as a function of heavy neutrino mass $m_4$, and mixing $|U_{\mu 4}|$ assuming a $\chi^2$ with two degrees of freedom [@Tanabashi:2018oca].
In our nominal LE (ME) analysis, we allow for 10% uncertainty on the flux [@Aliaga:2016oaz], and 30% (40%) uncertainty on the background motivated by the amount of tuning performed on the original backgrounds. Note that the nominal background predictions in the LE (ME) analysis overpredicts (underpredicts) the data before tuning, and that tuning parameters are measured at the 3% (5%) level [@Park:2013dax; @Valencia:2019mkf]. We also perform a background-ignorant analysis in which we assume 100% uncertainty for the background normalization, which changes our conclusions by only less than a factor of two. This emphasizes the robustness of our bound, since the NP typically overshoots the low number of events in the sideband. For the benchmark point of [@Bertuzzo:2018itn], we predict a total signal of 232 (4240) events for LE (ME).
For CHARM-II, the NP signal lies mostly in a region with small $E\theta^2$. Thus, we constrain backgrounds using the data from $28 < E\theta^2 < 60$ MeV rad$^2$. This sideband measurement constrains the normalization of the backgrounds in the signal region at the level of $3\%$. The extrapolation of the shape of the background to the signal region introduces the largest uncertainty in our analysis. For this reason, we raise the uncertainty of the background normalization from $3\%$ to a conservative $10 \%$ when setting the limits. Flux uncertainties are assumed to be $4.7\%$ and $5.2\%$ for neutrino and antineutrino mode [@Allaby:1987bb], respectively, and are applicable to the new-physics signal, $\nu-e$ scattering prediction, and backgrounds. Uncertainties in the $\nu-e$ scattering cross sections are expected to be sub-dominant and are neglected in the analysis [@deGouvea:2006hfo]. For CHARM-II, the NP also yields too many events in the signal region, namely $\approx 2.2\times10^{5}$ events for the benchmark point of [@Bertuzzo:2018itn] in antineutrino mode. If we lower $|U_{\mu4}| = 10^{-4}$ and $m_4 = 100$ MeV, CHARM-II would still have $\approx 3 \times 10^3$ new physics events.
![[*New constraints on dark neutrinos as a MiniBooNE explanation.*]{} The fit to the MiniBooNE energy distribution from [@Bertuzzo:2018itn] is shown as closed yellow (orange) region for one (three) sigma C.L., together with the benchmark point (${\bf\odot}$). Our constraints are shown at 90% C.L. for LE in blue (solid – 30% background normalization uncertainty, dashed – conservative 100% case), for ME in cyan (solid – 40% background normalization uncertainty, dashed – conservative 100% case), and for CHARM-II in red (solid – 3% background normalization from the sideband constraint, dashed – conservative 10% case). Vertical lines show the percentage of excess events at MiniBooNE that lie in the most forward angular bin. Exclusion from heavy neutrino searches is shown as a hatched background. Other relevant assumed parameters are shown above the plot; changing them does not alter our conclusion.\[fig:final\_plot\]](bounds.pdf){width="3.38in"}
[**Results and conclusions –**]{} The resulting limits on dark neutrinos using neutrino-electron scattering experiments are shown in the $|U_{\mu 4}|$ vs $m_4$ plane at 90% confidence level (CL) in [Fig. \[fig:final\_plot\]]{}. The MiniBooNE fit from [@Bertuzzo:2018itn] is shown, together with vertical lines indicating the percentage of events at MiniBooNE that populate the most forward angular bin. We have chosen the same values of $\varepsilon$, $\alpha_D$, and $m_{Z^\prime}$ as used in [@Bertuzzo:2018itn], and shown their benchmark point ($m_4 = 420$ MeV and $|U_{\mu 4}|^2 = 9 \times 10^{-7}$) as a dotted circle. For these parameters, we can conclude that a good angular distribution at MiniBooNE is in large tension with neutrino-electron scattering data. We note that the MiniBooNE event rate scales identically to our signal rate in all the couplings, and the dependence on $m_{Z^\prime}$ is subleading due to the typical momentum transfer to the nucleus, provided $m_{Z^\prime} \lesssim 100$ MeV . This implies that changing the values of these parameters does not modify the overall conclusions of our work. In addition, for this realization of the model, larger $m_{Z^\prime}$ implies larger values of $m_4$, increasing their impact on neutrino-electron scattering data. Our and CHARM-II results are mutually reinforcing given that they impose similar constraints for $m_4 \lesssim 200 $ MeV. For larger masses, the kinematics of the signal becomes less forward and the production thresholds start being important. This explains the upturns visible in our bounds, where we observe it first in and later in CHARM-II as we increase $m_4$, since CHARM-II has higher beam energy.
We emphasize that our analysis is general, and can be adapted to other models. In fact, any MiniBooNE explanation with heavy new particles faces severe constraints from high-energy neutrino-electron scattering data if the signal is free from hadronic activity. This is realised, for instance, in scenarios with heavy neutrinos with dipole interactions [@Gninenko:2009ks; @Gninenko:2010pr; @Dib:2011jh; @McKeen:2010rx; @Masip:2012ke; @Masip:2011qb; @Gninenko:2012rw; @Magill:2018jla]. Our bounds can also be adapted to other scenarios with dark neutrinos and heavy mediators [@Ballett:2018ynz; @Ballett:2019pyw]. For those, however, we do not expect our bounds to constrain the region of parameter space where the MiniBooNE explanation is viable, since most of the signal at MiniBooNE contains hadronic activity which would be visible at and CHARM-II.
In the near future, our new analysis strategy could be used in the up-coming ME results on antineutrino-electron scattering. The NP cross section, being the same for neutrino and antineutrinos, is thus more prominent on top of backgrounds. This class of analyses will also greatly benefit from improved calculations and measurements of coherent $\pi^0$ production and single-photon emitting processes. This is particularly important given the excess seen in the ME analysis. A new result can also be obtained by neutrino-electron scattering measurements at NO$\nu$A, which will sample a different kinematic regime as its off-axis beam peaks at lower energies and expects fewer NC $\pi^0$ events per ton. Beyond neutrino-electron scattering, the BSM signatures we consider could be lurking in current measurements of $\pi^0$ production, *e.g.*, at MINOS [@Adamson:2016hyz] and MINER$\nu$A [@Wolcott:2016hws] [^3], and in analyses like the single photon search performed by T2K [@Abe:2019cer]. Thus, if dark neutrinos are indeed present in current data, our technique will be crucial to confirm it.
To summarize, a variety of measurements are underway to further lay siege to this explanation of the MiniBooNE observation and, simultaneously, start probing testable neutrino mass generation models, as well as other similar NP signatures.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Janet Conrad, Kareem Farrag, Alberto Gago, Gordan Krnjaic, Trung Le, Pedro Machado, Kevin Mcfarland, and Jorge Morfin for useful discussions, and Jean DeMerit for carefully proofreading our work. The authors would like to thank Fermilab for the hospitality at the initial stages of this project. Also, the authors would like to thank Fermilab Theory Group and the CERN Theory Neutrino Platform for organizing the conference “Physics Opportunities in the Near DUNE Detector Hall,” which was essential to the completion of this work. CAA would especially like to thank Fermilab Center for Neutrino Physics summer visitor program for funding his visit. MH’s work was supported by Conselho Nacional de Ciência e Tecnologia (CNPq). CAA is supported by U.S. National Science Foundation (NSF) grant No. PHY-1801996. This document was prepared by YDT using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359.
**Supplementary Material**
Our analysis discussed in the main text is now described in more detail and all assumptions used in our simulations are summarized. We start by discussing our statistical method, and then discuss our Monte Carlo (MC) simulation, stating our signal definitions more precisely. Later, we show a few kinematical distributions from our dedicated MC simulation, including the angular distributions at MiniBooNE used to obtain the vertical lines in [Fig. \[fig:final\_plot\]]{}. In order to furher aid reproducibility of our result, we also make our Monte Carlo events for some parameter choices available on <span style="font-variant:small-caps;">g</span>it<span style="font-variant:small-caps;">h</span>ub [^4].
Statistical Analysis
====================
Our statistical analysis uses the Pearson-$\chi^2$ as a test statistic, where the expected number of events is scaled by nuisance parameters to incorporate systematic uncertainties. Our test statistic reads
$$\begin{aligned}
\chi^2(\vec\theta, \alpha, \beta) = \frac{ (N_{\rm data} - N_\mathrm{pred}(\vec\theta, \alpha, \beta) )^2 }{ N_\mathrm{pred}(\vec\theta, \alpha, \beta)} + \left(\frac{\alpha}{\sigma_\alpha}\right)^2 +
\left(\frac{\beta}{\sigma_\beta}\right)^2,\end{aligned}$$
with the number of predicted events given by $$\begin{aligned}
N_\mathrm{pred}(\vec\theta, \alpha, \beta) = (1+\alpha + \beta) \mu_{\rm MC}^{\rm BKG} + (1+\alpha) \mu_{\rm MC}^{\nu-e} + (1 + \alpha) \mu_{\rm BSM}(\vec\theta),\end{aligned}$$
where $\vec\theta$ are the model parameters, while $\alpha$ and $\beta$ are nuisance parameters that incorporate uncertainties from the overall rate and the background rate, respectively. Here, $N_{\rm data}$ stands for the total rate observed in the experiment, $\mu_{\rm MC}^{\rm BKG}$ the quoted background rates, $\mu_{\rm MC}^{\nu-e}$ the quoted $\nu-e$ events, and $\mu_{\rm BSM}(\vec\theta)$ the predicted number of BSM events calculated using our MC. We discuss the choice of these systematic uncertainties, namely the choice of $\sigma_\alpha$ and $\sigma_\beta$, when describing the simulation of each experiment below. To obtain our results we use the test statistic profiled over the nuisance parameters, namely $\chi^2(\vec\theta) = \min_{(\alpha, \beta)} \left(\chi^2(\vec\theta, \alpha, \beta) \right)$, and use the test-statistic thresholds given in [@pdg].
Simulation Details
==================
Experiment Detector Resolution Overlapping Analysis Cuts
------------- ----------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- -----------------------------------------------------------------------------------------
MiniBooNE
$\sigma_E/E = 12\%/\sqrt{E_e/{\rm GeV}}$ $\sigma_\theta = 4{^\circ}$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 13^\circ$ N/A
MINER$\nu$A
$\sigma_E/E = 5.9\%/\sqrt{E_e/{\rm GeV}} + 3.4\%$ $\sigma_\theta = 1{^\circ}$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 8^\circ$ $E_{\rm vis} > 0.8$ GeV $E_{\rm vis} \theta^2 < 3.2$ MeV $Q^2_{\rm rec} < 0.02$ GeV$^2$
CHARM-II
$\sigma_E/E = 9\%/\sqrt{E/{\rm GeV}} + 11\%$ $\sigma_\theta/{\rm mrad} = \frac{27 (E/{\rm GeV})^2 +14}{\sqrt{E/{\rm GeV}}} + 1$ $E_{+} > 30$ MeV $E_{-} > 30$ MeV $\Delta \theta_{\pm} < 4^\circ$ $3$ GeV $<E_{\rm vis} < 24$ GeV $E_{\rm vis} \theta^2 < 28$ MeV
We generate events distributed according to the upscattering cross section for the process $\nu_\mu A \to \nu_4 A$, where $A$ is a nuclear target. We only discuss upscattering on nuclei, as the number of incoherent scattering on protons is much smaller for the relevant $Z^\prime$ masses; see [Fig. \[fig:cross\_section\]]{}. We then implement the chain of two-body decays: $\nu_4 \to \nu_\mu Z^\prime$ followed by $Z^\prime \to e^+ e^-$. To go from our MC true quantities to the predicted experimental observables, we perform three procedures. First, we smear the energy and angles of the $e^+$ and $e^-$ originating from the decay of the $Z^\prime$ according to detector-dependent Gaussian resolutions. Next, we select all events with that satisfy the $e^+e^-$ overlapping condition given in [Table \[tab:parameters\]]{}. Namely, if the condition is satisfied they are assumed to be reconstructed as a single electromagnetic (EM) shower. This guarantees that the events behave like a photon shower inside the detector [^5]. Finally, for and CHARM-II, these samples are subject to analysis-dependent kinematic cuts to determine if they contribute to the $\nu-e$ scattering sample. Detector resolutions, requirements for the dielectron pair to be overlapping, and analysis-dependent cuts are summarized in [Table \[tab:parameters\]]{}. We now list the experimental parameters used in our simulations for each individual detector.
#### CHARM-II
The CHARM-II experiment is simulated using the CERN West Area Neutrino Facility (WANF) wide band beam [@Vilain:1998uw]. The total number of protons-on-target (POT) is $2.5 \times 10^{19}$ for the $\nu$ and $\overline{\nu}$ runs combined. We assume glass to be the main detector material, (SiO$_2$), such that we can treat neutrino scattering off an average target with $\langle Z\rangle=11$ and $\langle A \rangle = 20.7$ [@DeWinter:1989zg; @Vilain:1993sf]. The fiducial volume in our analysis is confined to a transverse area of $320$cm$^2$, which corresponds to a fiducial mass of $547$t, and the detection efficiency is taken to be $76\%$; efficiency for $\pi^0$ sample is quoted at $82\%$ [@Vilain:1992wx]. We reproduce the total number of $\nu-e$ scattering events with $3$ GeV $< E_{\rm vis} <24$ GeV, namely $2677+2752$, to within a few percent level when setting the number of POTs in $\nu$ mode to be $1.69$ of that in the $\overline{\nu}$ mode [@Geiregat:1991md]. We assume a flux uncertainty of $\sigma_\alpha = 4.7\%$ for neutrino, and $\sigma_\alpha = 5.2\%$ for antineutrino beam [@Vilain:1992wx]. The background uncertainty is constrained to be $\sigma_\beta = 3\%$ using the data with $E_{\rm vis} \theta^2 > 28$ MeV, where the number of new physics events is negligible.
#### MINER$\nu$A
For our MINER$\nu$A simulation, we use the low-energy (LE) and medium-energy (ME) NuMI neutrino fluxes [@AliagaSoplin:2016shs]. The total number of POT is $3.43\times 10^{20}$ for LE data, and $11.6\times10^{20}$ for ME data. The detector is assumed to be made of CH, with a fiducial mass of $6.10$ tons and detection efficiencies of $73\%$ [@Parke:2015goa; @Valencia:2019mkf]. We assume a flux uncertainty of $\sigma_\alpha = 10\%$ for both the LE and ME modes [@Aliaga:2016oaz]. Due to the tuning performed in the sideband of interest, the uncertainties on the background rate are much larger. For the LE, we take $\sigma_\beta = 30\%$, while for the ME data $\sigma_\beta = 50\%$. Although tuning is significant for the coherent $\pi^0$ production sample, the overall rate of backgrounds in the sideband with large $dE/dx$ does not vary by more than $20\%$ ($40\%$) in the LE (ME) tuning.
#### MiniBooNE
To simulate MiniBooNE, we use the Booster Neutrino Beam (BNB) fluxes from [@AguilarArevalo:2008yp]. Here, we only discuss the neutrino run, although the predictions for the antineutrino run are very similar. We assume a total of $12.84 \times 10^{20}$ POT in neutrino mode. The fiducial mass of the detector is taken as $450$t of CH$_2$. In order to apply detector efficiencies, we compute the reconstructed neutrino energy under the assumption of CCQE scattering $$\begin{aligned}
E_\nu^{CCQE} = \frac{E_{\rm vis} m_p}{m_p - E_{\rm vis} (1 - \cos{\theta}) },\end{aligned}$$ where $E_{\rm vis} = E_{e^+} + E_{e^-}$ is the total visible energy after smearing. Under this assumption, we can apply the efficiencies provided by the MiniBooNE collaboration [@Aguilar-Arevalo:2012fmn]. Using our MC we can reproduce well the distributions obtained using the MiniBooNE Monte Carlo data release provided for oscillation analyses.
Kinematic Distributions
=======================
As an important check of our calculation and of the explanation of the MiniBooNE excess within the model of interest, we plot the MiniBooNE neutrino data from 2018 [@Aguilar-Arevalo:2018gpe] against our MC prediction in Suppl. Fig. \[fig:MB\_distributions\]. We do this for three different new physics parameter choices. We set $m_{Z^\prime} = 30$ MeV, $\alpha \epsilon^2 = 2\times10^{-10}$ and $\alpha_D = 1/4$ for all points, but vary $|U_{\mu 4}|^2$ and $m_4$ so that the final number of excess events predicted by the model at MiniBooNE equals 334.
{width="49.00000%"} {width="49.00000%"}
To verify that the new physics signal is important in neutrino-electron studies, we also plot kinematical distributions for the benchmark point (BP) discussed in the main text for different detectors. This corresponds to $m_{Z^\prime} = 30$ MeV, $\alpha \epsilon^2 = 2\times10^{-10}$, $\alpha_D = 1/4$, $|U_{\mu 4}|^2 = 9\times10^{-7}$ and $m_4 = 420$ MeV. The interesting variables are the energy asymmetry of the dielectron pair $$|E_{\rm asym}| = \frac{|E_+ - E_-|}{E_+ + E_-},$$ as well as the separation angle $\Delta \theta_{e^+e^-}$ between the two electrons. These variables are plotted in Suppl. Fig. \[fig:other\_distributions\] at MC truth level, before any smearing or selection takes place. We also plot the total reconstructed energy $E_{\rm vis} = E_{e^+} + E_{e^-}$ and the quantity $E_{\rm vis} \theta^2$, where $\theta$ stands for the angle formed by the reconstructed EM shower and the neutrino beam. The visible energy, $E_{\rm vis}$, and angle, $\theta$, are computed after smearing, but before the selection into overlapping pairs takes place.
{width="49.00000%"} {width="49.00000%"}\
{width="49.00000%"} {width="49.00000%"}
[^1]: Models with the decay of Majorana particles will lead to greater tension with the angular distribution at MiniBooNE due to their isotropic nature [@Formaggio:1998zn; @Balantekin:2018ukw].
[^2]: Since the released MiniBooNE data do not provide the correlation between angle and energy, and their associated systematics, an energy-angle fit is not possible.
[^3]: This $\nu_e$CCQE measurement by observes a significant excess of single photon-like showers attributed to diffractive $\pi^0$ events. These are abundant in similar realizations of this NP model [@Ballett:2018ynz].
[^4]: [[github.com/mhostert/DarkNews]{}](https://github.com/mhostert/DarkNews).
[^5]: For MiniBooNE, we also include events that are highly asymmetric in energy, *i.e.*, $E_{\pm} > 30$ MeV and $E_{\mp} < 30$ MeV, where the most energetic shower defines the angle with respect to the beam.
| ArXiv |
---
abstract: 'Person Re-identification (ReID) is to identify the same person across different cameras. It is a challenging task due to the large variations in person pose, occlusion, background clutter, How to extract powerful features is a fundamental problem in ReID and is still an open problem today. In this paper, we design a Multi-Scale Context-Aware Network (MSCAN) to learn powerful features over full body and body parts, which can well capture the local context knowledge by stacking multi-scale convolutions in each layer. Moreover, instead of using predefined rigid parts, we propose to learn and localize deformable pedestrian parts using Spatial Transformer Networks (STN) with novel spatial constraints. The learned body parts can release some difficulties, pose variations and background clutters, in part-based representation. Finally, we integrate the representation learning processes of full body and body parts into a unified framework for person ReID through multi-class person identification tasks. Extensive evaluations on current challenging large-scale person ReID datasets, including the image-based Market1501, CUHK03 and sequence-based MARS datasets, show that the proposed method achieves the state-of-the-art results.'
author:
- |
Dangwei Li$^{1,2}$, Xiaotang Chen$^{1,2}$, Zhang Zhang$^{1,2}$, Kaiqi Huang$^{1,2,3}$\
$^{1}$CRIPAC$\;\&\;$NLPR, CASIA $^{2}$University of Chinese Academy of Sciences\
$^{3}$CAS Center for Excellence in Brain Science and Intelligence Technology\
[@nlpr.ia.ac.cn]{}
bibliography:
- 'egbib.bib'
title: |
Learning Deep Context-aware Features over Body and Latent Parts\
for Person Re-identification
---
Introduction
============
Person re-identification aims to search for the same person across different cameras with a given probe image. It has attracted much attention in recent years due to its importance in many practical applications, such as video surveillance and content-based image retrieval. Despite of years of efforts, it still has many challenges, such as large variations in person pose, illumination, and background clutter. In addition, similar appearance of clothes among different people and imperfect pedestrian detection results further increase its difficulty in real applications.
![The schematic of typical feature learning framework with deep learning. As shown in black dashed boxes, current approaches focus on the full body or rigid body parts for feature learning. Different from them, we use the spatial transformer networks to learn and localize pedestrian parts and use multi-scale context-aware convolutional networks to extract full-body and body-parts representations for ReID. Best viewed in color. []{data-label="fig:framework_simple"}](figs/PipelineNetwork.pdf){width="0.35\paperwidth"}
Most existing methods for ReID focus on developing a powerful representation to handle the variations of viewpoint, body pose, background clutter, [@xu2014person; @GrayECCV08; @FarenzenaCVPR10; @YangyangECCV14; @Yang2017aaai; @KviatkovskyPAMI13color; @ZhaoruiCVPR13unsupervised; @zhaoruiCVPR14learning; @LiaoshengcaiCVPR15; @MatsukawaCVPR16; @LidangweiArxiv16], or learning an effective distance metric [@KostingerCVPR12; @ProsserBMVC10person; @ZhengweishiPAMI13reid; @LizhenCVPR13; @LiaoshengcaiCVPR15; @ZhangLiCVPR16; @ChendapengCVPR16similarity]. Some of existing methods learn both of them jointly [@LiWeiCVPR14; @YiICPR14DML; @AhmedCVPR15improved; @Shihanlin2016Embedding]. Recently, deep feature learning based methods [@DingPR15deep; @Chengde2016person; @Varior2016Siamese; @VariorECCV16Gated], which learn a global pedestrian feature and use Euclidean metric to measure two samples, have obtained the state-of-the-art results. With the increasing sample size of ReID dataset, the learning of features from multi-class person identification tasks [@XiaotongCVPR16Domain; @ZhengliangECCV16; @XiaotongARXIV16end; @ZhengliangArxiv16; @SchumannArxiv16deep], denoted as ID-discriminative Embedding (IDE) [@ZhengliangArxiv16], has shown great potentials on current large-scale person ReID datasets, such as MARS [@ZhengliangECCV16] and PRW [@ZhengliangArxiv16], where the IDE features are taken from the last hidden layer of Deep Convolutional Neural Networks (DCNN). In this paper, we aim to learn the IDE feature for person ReID using DCNN.
Existing DCNN models for person ReID typically learn a global full-body representation for input person image (Full body in Figure \[fig:framework\_simple\]), or learn a part-based representation for predefined rigid parts (Rigid body parts in Figure \[fig:framework\_simple\]) or learn a feature embedding for both of them. Although these DCNN models have obtained impressive results on existing ReID datasets, there are still two problems. **First**, for feature learning, current popular DCNN models typically stack single-scale convolution and max pooling layers to generate deep networks. With the increase of the number of layers, these DCNN models could easily miss some small scale visual cues, such as sunglasses and shoes. However, these fine-grained attributes are very useful to distinguish the pedestrian pairs with small inter-class variations. Thus these DCNN models are not the best choice for pedestrian feature learning. **Second**, due to the pose variations and imperfect pedestrian detectors, the pedestrian image samples may be misaligned. Sometimes they may have some backgrounds or lack some parts, legs. In these cases, for part-based representation, the predefined rigid grids may fail to capture correct correspondence between two pedestrian images. Thus the rigid predefined grids are far from robust for effective part-based feature learning.
In this paper, we propose to learn the features of full body and body parts jointly. **To solve the first problem**, we propose a Multi-Scale Context-Aware Network (MSCAN). As shown in Figure \[fig:framework\_simple\], for each convolutional layer of the MSCAN, we adopt multiple convolution kernels with different receptive fields to obtain multiple feature maps. Feature maps from different convolution kernels are concatenated as current layer’s output. To decrease the correlations among different convolution kernels, the dilated convolution [@YuKoltun2016] is used rather than general convolution kernels. Through this way, multi-scale context knowledge is obtained at the same layer. Thus the local visual cues for fine-grained discrimination is enhanced. In addition, through embedding contextual features layer-by-layer (convolution operation across layers), MSCAN can obtain more context-aware representation for input image. **To solve the second problem**, instead of using rigid body parts, we propose to localize latent pedestrian parts through Spatial Transform Networks (STN) [@JaderbergNIPS15spatial], which is originally proposed to learn image transformation. To adapt it to the pedestrian part localization task, we propose three new constraints on the learned transformation parameters. With these constraints, more flexible parts can be localized at the informative regions, so as to reduce the distraction of background contents.
Generally, the features of full body and body parts are complementary to each other. The full-body features pay more attention to the global information while the body-part features care more about the local regions. To better utilize these two types of representations, in this paper, features of full body and body parts are concatenated to form the final pedestrian representation. In test stage, the Euclidean metric is adopted to compute the distance between two L2 normalized person representations for person ReID.
The contributions of this paper are summarized as follows: (a) We propose a multi-scale context-aware network to enhance the visual context information for better feature representation of fine-grained visual cues. (b) Instead of using rigid parts, we propose to learn and localize pedestrian parts using spatial transformer networks with novel prior spatial constraints. Experimental results show that fusing the global full-body and local body-part representations greatly improves the performance of person ReID.
Related Work {#relatedwork}
============
Typical person ReID methods focus on two key points: developing a powerful feature for image representation and learning an effective metric to make the same person be close and different persons far away. Recently, deep learning approaches have achieved the state-of-the-art results for person ReID [@XiaotongCVPR16Domain; @ZhengliangECCV16; @VariorECCV16Gated; @zheng2016personreview; @zhang2015bit]. Here we mainly review the related deep learning methods.
Deep learning approaches for person ReID tend to learn person representation and similarity (distance) metric jointly. Given a pair of person images, previous deep learning approaches learn each person’s features followed by a deep matching function from the convolutional features [@LiWeiCVPR14; @AhmedCVPR15improved; @Chen2017aaai; @Chen2017cvprid] or the Fully Connected (FC) features [@YiICPR14DML; @wang2016dari; @Shihanlin2016Embedding]. In addition to the deep metric learning, some work directly learns image representation through pair-wise contrastive loss or triplet ranking loss, and use Euclidean metric for comparison [@DingPR15deep; @Chengde2016person; @Varior2016Siamese; @VariorECCV16Gated].
With the increasing sample size of ReID dataset, the IDE feature which is learned through multi-class person identification tasks, has shown great potentials on current large-scale person ReID datasets. Xiao [@XiaotongCVPR16Domain] propose the domain guided dropout to learn features over multiple datasets simultaneously with identity classification loss. Zheng [@ZhengliangECCV16] learn the IDE feature for the video-based person re-identification. Xiao [@XiaotongARXIV16end] and Zheng [@ZhengliangArxiv16] learn the IDE feature to jointly solve the pedestrian detection and person ReID tasks. Schumann [@SchumannArxiv16deep] learn the IDE feature for domain adaptive person ReID. The similar phenomenon has also been validated on face recognition [@SunyiCVPR14deep1]. As we know, previous DCNN models usually adopt the layer-by-layer single-scale convolution kernels to learn the context information. Some DCNN models [@YiICPR14DML; @Chengde2016person; @Shihanlin2016Embedding] adopt rigid body parts to learn local pedestrian features. Different from them, we improve the classical models in two ways. Firstly, we propose to enhance the context knowledge through multi-scale convolutions at the same layer. The relationship among different context knowledge are learned by embedding feature maps layer-by-layer (convolution or FC operation). Secondly, instead of using rigid parts, we utilize the spatial transformer networks with proposed prior constraints to learn and localize latent human parts.
{width="0.70\paperwidth"}
Proposed Method {#proposedmethod}
===============
The focus of this approach is to learn powerful feature representations to describe pedestrians. The overall framework of the proposed method is shown in Figure \[fig:framework\_all\]. In this section, we introduce our model from four aspects: a multi-scale context-aware network for efficient feature learning (Section \[MSCAN\]), the latent parts learning and localization for better local part-based feature representation (Section \[LatentPartLoc\]), the fusion of global full-body and local body-part features for person ReID (Section \[FeatureFusion\]), and our final objective function in Section \[ObjectiveFunction\].
Multi-scale Context-aware Network {#MSCAN}
---------------------------------
Visual context is an important component to assist visual-related tasks, such as object recognition [@LintsungyiECCV14microsoft] and object detection [@ZhengPAMI12context; @ZengArxiv16crafting]. Typical convolutional neural networks model context information through hierarchical convolution and pooling [@KrizhevskyNIPS12; @HekaimingCVPR16Residual]. For person ReID task, the most important visual cues are visual attribute knowledge, such as clothes color and types. However, they have large variations in scale, shape and position, such as the hat/glasses at small local scale and the cloth color at the larger scale. Directly using bottom-to-up single-scale convolution and pooling may not be effective to handle these complex variations. Especially, with the increase number of layers, the small visual regions, such as hat, will be easily missed in top layers. To better learn these diverse visual cues, we propose the Multi-scale Context-Aware Network.
layer dilation kernel pad \#filters output
------- ---------- ------------ ------- ----------- -------------------------
input - - - - 3$\times$160$\times$64
conv0 1 5$\times$5 2 32 32$\times$160$\times$64
pool0 - 2$\times$2 - - 32$\times$80$\times$32
conv1 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$80$\times$32
pool1 - 2$\times$2 - - 96$\times$40$\times$16
conv2 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$40$\times$16
pool2 - 2$\times$2 - - 96$\times$20$\times$8
conv3 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$20$\times$8
pool3 - 2$\times$2 - - 96$\times$10$\times$4
conv4 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$10$\times$4
pool4 - 2$\times$2 - - 96$\times$5$\times$2
: Model architecture of MSCAN.[]{data-label="tab:mscan"}
The architecture of the proposed MSCAN is shown in Tabel \[tab:mscan\]. It has an initial convolution layer with kernel size $5\times5$ to capture the low-level visual features. Then we use four multi-scale convolution layers to obtain the complex image context information. In each multi-scale convolution layer, we use a convolution kernel with size $3\times3$. To obtain multi-scale receptive fields, we adopt dilated convolution [@YuKoltun2016] for the convolution filters. We use three different dilation ratios, i.e. 1,2 and 3, to capture different scale context information. The feature maps from different dilation ratios are concatenated along the channel axis to form the final output of the current convolution layer. Thus, the visual context information are enhanced explicitly. To integrate different context information together, the feature maps of current convolution layer are embedded through layer-by-layer convolution or FC operation. As a result, the visual cues at different scales are fused in a latent way. Besides, we adopt Batch Normalization [@Ioffe15batch] and ReLU neural activation units after each convolution layer.
In this paper, we use the dilated convolutions with dilation ratios 1, 2 and 3 instead of the classic convolution filters with kernel size $3\times3$, $5\times5$ and $7\times7$. The main reason is that the classic convolution filters with kernel size $3\times3$, $5\times5$ and $7\times7$ overlap with each other at the same output position and produce redundant information. To make it clearer, we show the dilated convolution kernel (size $3\times3$) with dilation ratio ranging from $1$ to $3$ in Figure \[fig:DilationCov\]. For the same output position which is shown in red circle, the convolution kernel with larger dilation ratio has larger receptive field, while only the center position is overlapped with other convolution kernels. This can reduce the redundant information among filters with different receptive fields.
In summary, as shown in Figure \[fig:framework\_all\], we use MSCAN to learn the multi-scale context representation for full body and body parts. In addition, it is also used for feature learning in spatial transformer networks mentioned below.
![Example of dilated convolution for the same input feature map. The convolutional kernel is $3\times3$ and the dilation ratio from left to right is 1, 2, and 3. The blue boxes are effective positions for convolution at the red circle. Best viewed in color. []{data-label="fig:DilationCov"}](figs/DilationCov.pdf){width="0.35\paperwidth"}
Latent Part Localization {#LatentPartLoc}
------------------------
Pedestrian parts are important in person ReID. Some existing work [@GrayECCV08; @LiaoshengcaiCVPR15; @YiICPR14DML; @Chengde2016person] has explored rigid body parts to develop robust features. However, due to the unsatisfying pedestrian detection algorithms and large pose variations, the method of using rigid body parts for local feature learning is not the optimal solution. As shown in Figure \[fig:framework\_simple\], when using rigid body parts, the top part consists of large amount of background. This motivates us to learn and localize the pedestrian parts automatically.
We integrate STN [@JaderbergNIPS15spatial] as the part localization net in our proposed model. The original STN is proposed to explicitly learn the image transformation parameters, such as translation and scale. It has two main advantages: (1) it is fully differentiable and can be easily integrated into existing deep learning frameworks, (2) it can learn to translate, scale, crop or warp an interesting region without explicit region annotations. These facts make it very suitable for pedestrian parts localization.
STN includes two components, the spatial localization network to learn the transformation parameters, and the grid generator to sample the input image using an image interpolation kernel. More details about STN can be seen in [@JaderbergNIPS15spatial]. In our implementation of STN, the bilinear interpolation kernel is adopted to sample the input image. And four transformation parameters $\theta=[s_x, t_x, s_y, t_y]$ are used, where $s_x$ and $s_y$ are the horizontal and vertical scale transformation parameters, and $t_x$ and $t_y$ are the horizontal and vertical translation parameters. The image height and width are normalized to be in $[-1, 1]$. Only scale and translation parameters are learned because these two types of transformations serve enough to crop the pedestrian parts effectively. The transformation is applied as an inverse warping to generate the output body part regions: $$\begin{gathered}
\begin{pmatrix} x^{in}_{i} \\ y^{in}_{i} \end{pmatrix} =
\begin{bmatrix} s_x & 0 & t_x \\ 0 & s_y & t_y \end{bmatrix}
\begin{pmatrix} x^{out}_{i} \\ y^{out}_{i} \\ 1 \end{pmatrix}\end{gathered}$$
where $x^{in}$ and $y^{in}$ are the input image coordinates, $x^{out}$ and $y^{out}$ are the output part image coordinates, and $i$ indexes the pixels in the output body part image.
In this paper, we expect STN to learn three parts corresponding to the head-shoulder, upper body and lower body. Each part is learned by an independent STN from the original pedestrian image. For the spatial localization network, firstly we use MSCAN to extract the global image feature maps. Then we learn the high-level abstract representation by a 128-dimension FC layer (FC\_[loc]{} in Figure \[fig:framework\_all\]). At last, we learn the transformation parameters $\theta$ with a 4-dimension FC layer based on the FC\_loc. The MSCAN and FC\_[loc]{} are shared among three spatial localization networks. The grid generator can crop the learned pedestrian parts based on the learned transformation parameters. In this paper, the resolution of the cropped part image is $96\times64$.
For the part localization network, it is hard to learn three groups of parameters for part localization. There are three problems. First, the predicted parts from STN can easily fall into the same region, , the center region of a person, and result in redundance. Second, the scale parameters can easily become negative and the pedestrian part will be mirrored vertically or horizontally or both. This is not consistent with general human cognition. Because few person will stand upside down in surveillance scenes. At last, the cropped parts may fall out of the person image, thus the network would be hard to converge. To solve the above problems, we propose three prior constraints on the transformation parameters in the part localization network.
The first constraint is for the positions of predicted parts. We expect the predicted parts to be near the prior center points, so that the learned parts would be complementary to each other. This is termed as the center constraint, which is formalized as follows: $$%L_{cen} = \frac{1}{2}\{(t_x-C_x)^2 + (t_y-C_y)^2\}
L_{cen} = \frac{1}{2}\max\{0, (t_x-C_x)^2 + (t_y-C_y)^2 - \alpha\}$$ where $C_x$ and $C_y$ are prior center points for each part. $\alpha$ is the threshold to control the translation between estimated and prior center points. In our experiments, we set the prior center point ($C_x, C_y$) to $(0, 0.6)$, $(0, 0)$, and $(0, -0.6)$ for each part. The threshold $\alpha$ is set to $0.5$.
The second one is the value range constraint on the predicted scale parameter. We hope the scale to be positive, so that the predicted parts have a reasonable extent. The value range constraint on the scale parameter is formalized as follows: $$L_{pos} = \max\{0, \beta - s_x \} + \max\{0, \beta - s_y \}$$ where $\beta$ is threshold parameter and is set to 0.1 in this paper.
The last one is to make the localization network focus on the inner region of an image. It is formalized as follows: $$\renewcommand\arraystretch{1.5}
\begin{array}{r}
L_{in} = \frac{1}{2}\max\{0, ||s_x \pm t_x||^2 - \gamma \} \\
+ \frac{1}{2}\max\{0, ||s_y \pm t_y||^2 - \gamma \}
\end{array}$$ where $\gamma$ is the boundary parameter. $\gamma$ is set to 1.0 in our paper, which means the cropped parts should be inside the pedestrian image.
Finally the loss for the transformation parameters in the part localization network is described as follows: $$\label{equ:los_loc}
L_{loc} = L_{cen} + \xi_1 L_{pos} + \xi_2 L_{in}$$ where $\xi_1$ and $\xi_2$ are hyperparameters. The hyperparameters $\xi_1$ and $\xi_2$ are both set to 1.0 in our experiments.
Feature Extraction and Fusion {#FeatureFusion}
-----------------------------
The features of full body and body parts are learned by separate networks and then are fused in a unified framework for multi-class person identification tasks. For the body-based representation, we use MSCAN to extract the global feature maps and then learn a 128-dimension feature embedding (denoted as FC\_body in Figure \[fig:framework\_all\]). For the part-based representation, first, for each body part, we use the MSCAN to extract its feature maps and learn a 64-dimension feature embedding (denoted as FC\_part1, FC\_part2, FC\_part3). Then, we learn a 128-dimension feature embedding (denoted as FC\_part) based on features of each body part. The Dropout [@srivastava2014dropout] is adopted after each FC layer to prevent overfitting. At last, the features of global full body and local body parts are concatenated to be a 256-dimension feature as the final person representation.
Objective Function {#ObjectiveFunction}
------------------
In this paper, we adopt the softmax loss as the objective function for multi-class person identification tasks. $$\label{equ:los_cls}
L_{cls} = -\sum_{i=1}^{N}log\frac{\exp(W_{y_i}^{T}x_i+b_{y_i})}{\sum\nolimits_{j=1}^{C}\exp(W_j^Tx_i+b_j)}$$ where $i$ is the index of person images, $x_i$ is the feature of $i$-th sample, $y_i$ is the identity of $i$-th sample, $N$ is the number of person images, $C$ is the number of person identities, $W_j$ is the classifier for $j$-th identity.
For the overall network training, we use the classification and localization loss jointly. The final objective function is as follows. $$\label{equ:los_all}
L = L_{cls} + \lambda L_{loc}$$ where the $\lambda$ is the hyperparameter, which is set to 0.1 in our experiments.
Experiments
===========
In this paragraph, the datasets and evaluation protocols are introduced in Section \[exp:dataset\]. Implementation details are described in Section \[exp:detail\]. Comparisons with state-of-the-art methods are discussed in Section \[exp:stateoftheart\]. The effectiveness of proposed model is analyzed in Section \[exp:effmscan\] and Section \[exp:efflpl\]. Cross-dataset evaluation is described in Section \[exp:discussion\].
Datasets and Protocols {#exp:dataset}
----------------------
**Datasets.** In this paper, we evaluate our proposed method on current largest person ReID datasets, including Market1501 [@ZhengliangICCV15], CUHK03 [@LiWeiCVPR14] and MARS [@ZhengliangECCV16]. We do not directly train our model on small datasets, such as VIPeR [@Gray07VIPeR]. It would be easily overfitting and insufficient to learn such a large capacity network on small datasets from scratch. However, we give some results through fine-tuneing the model from Market1501 to VIPeR and make cross-dataset ReID on VIPeR for generalization validation. Related experimental results are discussed in Section \[exp:discussion\].
Market1501: It contains 1,501 identities which are captured by six manually set cameras. There are 32,368 pedestrian images in total. Each person has 3.6 images on average at each viewpoint. It provides two types of images, including cropped and automatically detected pedestrians by the Deformable Part based Model (DPM) [@FelzenszwalbPAMI10object]. Following [@ZhengliangICCV15], 751 identities are used for training and the rest 750 identities are used for testing.
CUHK03: It contains 1,360 identities which are captured by six surveillance cameras in campus. Each identity is captured by two disjoint cameras. Totally it consists of 13,164 person images and each identity has about 4.8 images at each viewpoint. This dataset provides two types of annotations, including manually annotated bounding boxes, and bounding boxes detected using DPM. We validate our proposed model on both types of data. Following [@LiWeiCVPR14], we use 1,260 person identities for training and the rest 100 identities for testing. Experiments are conducted 20 times and the mean result is reported.
MARS: It is the largest sequence-based person ReID dataset. It contains 1,261 identities with each identity captured by at least two cameras. It consists of 20,478 tracklets and 1,191,003 bounding boxes. Following [@ZhengliangECCV16], we use 625 identities for training and the rest 631 identities for testing.
**Protocols.** Following original evaluation protocols in each dataset, we adopt three evaluation protocols for fair comparison with existing methods. The first one is Cumulated Matching Characteristics (CMC) which is adopted on the CUHK03 and MARS datasets. The second one is Rank-1 identification rate on the Market1501 dataset. The third one is mean Average Precision (mAP) on the Market1501 and MARS datasets. mAP considers both precision and recall rate, which could be complementary to CMC.
Implementation Details {#exp:detail}
----------------------
**Model:** We try to learn the pedestrian representation through multi-class person identification tasks using full body and body parts. To evaluate the effectiveness of full body and body parts independently, we extract two sub-models from the whole network of Figure \[fig:framework\_all\]. The first one only uses the full body to learn the person representation with identity classification loss. The second one only uses the parts to learn the person representation with identity classification and body parts localization loss. For person re-identification, we use the L2 normalized person representation and Euclidean metric to measure the distance between two pedestrian samples.
**Optimization:** Our model is implemented based on Caffe [@JiaMM14caffe]. We use all the available training identities for training and randomly select one sample for each identity for validation. As the dataset can be quite large, in practice we use a stochastic approximation of the objective function. Training data is randomly divided into mini-batches with a batch size of 64. The model performs forward propagation on each mini-batch and computes the loss. Backpropagation is then used to compute the gradients on each mini-batch and the weights are updated with stochastic gradient descent. We start with a base learning rate of $\eta = 0.01 $ and gradually decrease it after each $1\times10^4$ iterations. It should be noted that the learning rate of part localization network is 1% of that in feature learning network. We use a momentum of $\mu = 0.9$ and weight decay $\lambda = 5\times10^{-3}$. For overall network training, we initialize the network using pretrained body-based and part-based model and then follow the same training strategy as described above. We use the model at $5\times10^4$ iterations for testing.
**Data Preprocessing:** For each image, we resize it to $160\times64$, subtract the mean value on each channel (B, G and R), and then normalize it with scale $1.0/256$ for network training. To prevent overfitting, we randomly reflect each image horizontally in the training stage.
Comparison with State-of-the-art Methods {#exp:stateoftheart}
----------------------------------------
**Market1501:** For the Market1501 dataset, several state-of-the-art methods are compared, including Bag of Words (BOW) [@ZhengliangICCV15], Weighted Approximate Rank Component Analysis (WARCA) [@Jose2016scalable], Discriminative Null Space (DNS) [@ZhangLiCVPR16], Spatially Constrained Similarity function on Polynomial feature map (SCSP) [@ChendapengCVPR16similarity], and deep learning based approaches, such as PersonNet [@Wulin2016Personnet], Comparative Attention Network (CAN) [@Liu2016end], Siamese Long Short-Term Memory (S-LSTM) [@Varior2016Siamese], Gated Siamese Convolutional Neural Network (Gate-SCNN) [@VariorECCV16Gated]. The experimental results are shown in Table \[tab:marketresults\].
Compared with existing full body-based convolutional neural networks, such as CAN and Gate-SCNN, the proposed network structure can better capture pedestrian features with multi-class person identification tasks. Our full-body representation improves Rank-1 identification rate by 9.57% on the state-of-the-art results produced by the Gate-CNN in single query. Compared with the full body, our body-part representation increase 0.80%. The main reason is that the pedestrians detected by DPM consists much more background information and the part-based representation can better reduce the influences of background clutter.
The full-body and body-part representations are complementary to each other. The full-body representation cares more about the global information, such as the background and body shape. The body-part representation pays more attention to parts, such as head, upper body and lower body. As shown in Table \[tab:marketresults\], the fusion model of full body and body parts improves Rank-1 identification rate by more than 4.00% compared with the body and parts-based models separately in single query. The mAP improves about 17.98% compared with the best result produced by Gate-CNN.
Query
--------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics R1 mAP R1 mAP
BOW [@ZhengliangICCV15] 34.38 14.1 42.64 19.47
BOW + HS [@ZhengliangICCV15] - - 47.25 21.88
WARCA [@Jose2016scalable] 45.16 - - -
PersonNet [@Wulin2016Personnet] 37.21 26.35 - -
S-LSTM [@Varior2016Siamese] - - 61.6 35.3
SCSP [@Chengde2016person] 51.9 26.35 - -
CAN [@liu2017end] 48.24 24.43 - -
DNS [@ZhangLiCVPR16] 55.43 29.87 71.56 46.03
Gate-SCNN [@VariorECCV16Gated] 65.88 39.55 76.04 48.45
Our-Part 76.25 53.33 84.12 62.90
Our-Body 75.45 52.41 83.43 62.03
Our-Fusion **80.31** **57.53** **86.79** **66.70**
: Experimental results on the Market1501 dataset. - means that no reported results are available.[]{data-label="tab:marketresults"}
**CUHK03:** For the CUHK03 dataset, we compare our method with many existing approaches, including Filter Pair Neural Networks (FPNN) [@LiWeiCVPR14], Improved Deep Learning Architecture (IDLA) [@AhmedCVPR15improved], Cross-view Quadratic Discriminant Analysis (XQDA) [@LiaoshengcaiCVPR15], PSD constrained asymmetric metric learning (denoted as MLAPG) [@LiaoshengcaiICCV15], Sample-Specific SVM (SS) [@ZhangCVPR16sample], Single image and Cross image representation (SI-CI) [@WangfaqiangCVPR16JSC], Embedding Deep Metric (EDM) [@Shihanlin2016Embedding], Domain Guided Dropout (DGD) [@XiaotongCVPR16Domain], DNS, S-LSTM and Gate-SCNN. On this dataset, we conduct experiments on both the detected and the labeled datasets. As presented in previous work [@LiWeiCVPR14], we use the CMC curve in the single shot case to evaluate the performance. The overall results are shown in the Table \[tab:cuhk03detectedresults\] and Table \[tab:cuhk03labeledresults\]. The full CMC curves are shown in supplementary materials.
Compared with metric learning methods, such as the state-of-the-art approach DNS, the proposed fusion model improves the Rank-1 identification rate by 11.66% and 13.29% on the labeled and detected datasets respectively. Compared with the similar multi-class person identification network DGD, the Rank-1 identification rate improves by 1.63% using our fusion model on the labeled dataset. It should be noted that we only use the labeled sets for training, while the DGD is trained on both the labeled and detected datasets. This demonstrates the effectiveness of the proposed model.
Dataset
-------------------------------- ----------- ----------- ----------- -----------
Rank 1 5 10 20
FPNN [@LiWeiCVPR14] 19.89 50.00 64.00 78.50
IDLA [@AhmedCVPR15improved] 44.96 76.01 83.47 93.15
XQDA [@LiaoshengcaiCVPR15] 46.25 78.90 88.55 94.25
MLAPG [@LiaoshengcaiICCV15] 51.15 83.55 92.05 96.90
SS-SVM [@ZhangCVPR16sample] 51.20 80.80 89.60 95.50
SI-CI [@WangfaqiangCVPR16JSC] 52.17 84.30 92.30 95.00
DNS [@ZhangLiCVPR16] 54.70 84.75 94.80 95.20
S-LSTM [@Varior2016Siamese] 57.30 80.10 88.30 -
Gate-SCNN [@VariorECCV16Gated] 61.80 80.90 88.30 -
EDM [@Shihanlin2016Embedding] 52.09 82.87 91.78 97.17
Our-Part 62.74 88.53 93.97 97.21
Our-Body 64.95 89.82 94.58 97.56
Our-Fusion **67.99** **91.04** **95.36** **97.83**
: Experimental results on the CUHK03 detected dataset.[]{data-label="tab:cuhk03detectedresults"}
Dataset
------------------------------------ ----------- ----------- ----------- -----------
Rank 1 5 10 20
FPNN [@LiWeiCVPR14] 20.65 51.50 66.50 80.00
IDLA [@AhmedCVPR15improved] 54.74 86.50 93.88 98.10
XQDA [@LiaoshengcaiCVPR15] 52.20 82.23 92.14 96.25
MLAPG [@LiaoshengcaiICCV15] 57.96 87.09 94.74 98.00
Ensemble [@PaisitkriangkraiCVPR15] 62.10 89.10 94.80 98.10
SS-SVM [@ZhangCVPR16sample] 57.00 85.70 94.30 97.80
DNS [@ZhangLiCVPR16] 62.55 90.05 94.80 98.10
EDM [@Shihanlin2016Embedding] 61.32 88.90 96.44 **99.94**
DGD [@XiaotongCVPR16Domain] 72.58 91.59 95.21 97.72
Our-Part 69.41 92.68 96.68 99.02
Our-Body 71.88 93.66 97.46 99.18
Our-Fusion **74.21** **94.33** **97.54** 99.25
: Experimental results on the CUHK03 labeled dataset.[]{data-label="tab:cuhk03labeledresults"}
**MARS:** This dataset is the largest sequence-based person ReID dataset. On this dataset, we compare the proposed method with several classical methods, including Keep It as Simple and straightforward Metric (KISSME) [@KostingerCVPR12], XQDA [@LiaoshengcaiCVPR15], and CaffeNet [@KrizhevskyNIPS12]. Similar to previous work [@ZhengliangECCV16], both single query and multiple query are evaluated on MARS. The overall experimental results on the MARS are shown in Table \[tab:marsresults\_single\] and Table \[tab:marsresults\_mutiple\]. The full CMC curves are shown in supplementary materials.
Compared with CaffeNet, a similar multi-class person identification network, our body-based model improves the Rank-1 identification rate by 2.93% and mAP by 4.22% using XQDA in single query. It should be noticed that our network does not use any pre-training with additional data. Usually deep learning network can obtain better results when pretrained with on ImageNet classification task. Our fusion model improves Rank-1 identification rate and mAP by 6.47% and by 8.45% in single query. This illustrates the effectiveness of our model.
Query
---------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics 1 5 20 mAP
CNN+Eulidean [@ZhengliangECCV16] 58.70 77.10 86.80 40.40
CNN+KISSME [@ZhengliangECCV16] 65.00 81.10 88.90 45.60
CNN+XQDA [@ZhengliangECCV16] 65.30 82.00 89.00 47.60
Our-Fusion+Eulidean 68.38 84.19 91.52 51.13
Our-Fusion+KISSME 69.24 85.15 92.17 53.00
Our-Part+XQDA 66.62 82.07 90.76 49.74
Our-Body+XQDA 68.23 83.99 92.17 51.82
Our-Fusion+XQDA **71.77** **86.57** **93.08** **56.05**
: Experimental results on the MARS with single query.
\[tab:marsresults\_single\]
Query
----------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics 1 5 20 mAP
CNN+KISSME+MQ [@ZhengliangECCV16] 68.30 82.60 89.40 49.30
Our-Fusion+Euclidean+MQ 78.28 91.97 96.87 61.62
Our-Fusion+KISSME+MQ 80.51 93.18 97.22 63.50
Our-Fusion+XQDA+MQ **83.03** **93.69** **97.63** **66.43**
: Experimental results on the MARS with multiple query.
\[tab:marsresults\_mutiple\]
Effectiveness of MSCAN {#exp:effmscan}
----------------------
To determine the effectiveness of MSCAN, we explore four variants of MSCANs to learn IDE feature based on the whole body image, which is denoted as MSCAN-$k$, $k = \{1,2,3,4\}$. $k$ is the number of dilation ratios. For example, MSCAN-$3$ means for each convolution layer in Conv1-Conv4, there are three convolution kernels with dilation ratio 1, 2, and 3 respectively. With the increase of $k$, the MSCAN captures larger context information at the same convolution layer.
The experimental results based on these four types of MSCANs on the Market1501 dataset are shown in Table \[tab:mscanmarket\]. As we can see, with the increase of the number of dilation ratios, the Rank-1 identification rate and mAP improve stably in single query case. For multiple query case, which means fusing all images belonging to the same query person at the same camera through average pooling in feature space, the Rank-1 identification rate and mAP also improves step by step. However, the Rank-1 identification rate and mAP increase not much when $K$ increase from 3 to 4. We think there is a suitable number of dilation ratios for feature learning. Considering the model complexity and accuracy improvements in Rank-1 identification rate, we adopt the MSCAN-3 as our final MSCAN model in this paper.
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
MSCAN-1 65.38 41.85 75.21 51.14
MSCAN-2 72.21 49.19 82.22 59.03
MSCAN-3 75.45 52.41 83.43 62.03
MSCAN-4 **76.25** **53.14** **84.09** **62.95**
: Experimental results of four types of MSCAN using body-based representation for ReID on the Market1501 dataset.
\[tab:mscanmarket\]
Effectiveness of Latent Part Localization {#exp:efflpl}
-----------------------------------------
**Learned parts rigid parts** To compare with popular rigid pedestrian parts, we divide the pedestrian into three overlapped regions as predefined rigid parts. We use the rigid body parts instead of the learned latent body parts for part-based feature learning. Experimental results with rigid and learned body parts are shown in Table \[tab:partmarket\]. Compared with rigid body parts, the learned body parts improve Rank-1 identification rate and mAP by 3.27% and 3.73% in single query, and by 1.70% and by 2.67% in multiple query. This validate the effectiveness of learned person parts.
For better understanding the learned pedestrian parts, we visualize the localized latent parts in Figure \[fig:visualizationloc\] using our fusion model. For these detected person with large background (the first row in Figure \[fig:visualizationloc\]), the proposed model can learn foreground information with complementary latent pedestrian parts. As we can see, the learned parts consist of three main components, including upper body, middle body (combination of upper body and lower body), and lower body. Similar results can be achieved when original detection pedestrians contain less background or occlusion (the second row in Figure \[fig:visualizationloc\]). It is easy to see that, the automatically learned pedestrian parts are not strictly head-shoulder, upper body and lower-body. But it indeed consists of these three parts with large overlap. Compared with rigid parts, the proposed model can automatically localize the appropriate latent parts for feature learning.
![Six samples of original image, rigid predefined parts and learned latent pedestrian parts. Samples in each column are the same person with different backgrounds. Best viewed in color. []{data-label="fig:visualizationloc"}](figs/part_loc.pdf){width="0.35\paperwidth"}
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
Rigid parts 72.98 49.60 82.42 60.23
Latent parts **76.25** **53.33** **84.12** **62.90**
: Experimental results of rigid parts and learned parts for ReID on the Market1501 dataset.
\[tab:partmarket\]
**Effectiveness of localization loss** To evaluate the effectiveness of the proposed constraints on the latent part localization network, we conduct additional experiments by adding or deleting proposed $L_{loc}$ in the training stage of body parts network for ReID. Experimental results are shown in Table \[tab:stnconstrictmarket\]. As we can see, with the additional $L_{loc}$, the Rank-1 accuracy increases by 9.03%. We owe the improvements to the effectiveness of the proposed constraints on the part localization network.
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
$L_{cls}$ 67.22 45.27 77.55 55.40
$L_{cls}+L_{loc}$ **76.25** **53.33** **84.12** **62.90**
: The influences of $L_{loc}$ on part-based network on the Market1501 dataset.
\[tab:stnconstrictmarket\]
Cross-dataset Evaluation {#exp:discussion}
------------------------
Similar with typical image classification task with CNN, our approach requires large scale of data, not only more identities, but also more instances for each identity. So we do not train the proposed model on each single small person ReID dataset, such as VIPeR. Instead, we conduct cross-dataset evaluation from the pretrained model on the Market1501, CUHK03 and MARS datasets to the VIPeR dataset. The experimental results are shown in Table \[tab:crossviper\]. Compared with other methods, such as Domain Transfer Rank Support Vector Machines [@MaICCV13domain] and DML [@YiICPR14DML], the models trained on large-scale datasets have better generalization ability and have better Rank-1 identification rate.
To take further analysis of the proposed method, we also fine-tune the model from large dataset Market1501 to small dataset VIPeR. Experimental results are shown in Table \[tab:markettransfertoviper\]. Our fusion-based model obtains better Rank-1 identification rate than existing deep models, IDLA [@AhmedCVPR15improved] (34.8%), Gate-SCNN [@VariorECCV16Gated] (37.8%), SI-CI [@WangfaqiangCVPR16JSC] (35.8%), and achieves comparable results with DGD [@XiaotongCVPR16Domain] (38.6%).
Methods Training Set 1 10 20 30
-------------------------- ----------------- ----------- ----------- ----------- -----------
DTRSVM [@MaICCV13domain] i-LIDS 8.26 31.39 44.83 53.88
DTRSVM [@MaICCV13domain] PRID 10.90 28.20 37.69 44.87
DML [@YiICPR14DML] CUHK Campus 16.17 45.82 57.56 64.24
Ours-Fusion CUHK03 detected 17.30 44.58 55.51 61.77
Ours-Fusion CHUK03 labeled 19.44 **49.99** **60.78** **66.74**
Ours-Fusion MRAS 18.46 43.65 52.96 59.34
Ours-Fusion Market1501 **22.21** 47.24 57.13 62.26
: Cross-dataset person ReID on the VIPeR dataset
\[tab:crossviper\]
Method Rank-1 Rank-5 Rank-10 Rank-20
------------ ----------- ----------- ----------- -----------
Our-Part 32.70 57.49 67.62 78.90
Our-Body 33.12 60.23 72.05 82.59
Our-Fusion **38.08** **64.14** **73.52** **82.91**
: Experimental results on VIPeR through fine-tuneing the model from Market1501 to VIPeR.
\[tab:markettransfertoviper\]
Conclusion {#conclusions}
==========
In this work, we have studied the problem of person ReID in three levels: 1) a multi-scale context-aware network to capture the context knowledge for pedestrian feature learning, 2) three novel constraints on STN for effective latent parts localization and body-part feature representation, 3) the fusion of full-body and body-part identity discriminative features for powerful pedestrian representation. We have validated the effectiveness of the proposed method on current large-scale person ReID datasets. Experimental results have demonstrated that the proposed method achieves the state-of-the-art results.
**Acknowledgement** This work is funded by the National Key Research and Development Program of China (2016YFB1001005), the National Natural Science Foundation of China (Grant No. 61673375, Grant No. 61403383 and Grant No. 61473290), and the Projects of Chinese Academy of Science (Grant No. QYZDB-SSW-JSC006, Grant No. 173211KYSB20160008).
| ArXiv |
---
abstract: 'We show that a tetragonal lattice of weakly interacting cavities with uniaxial electromagnetic response is the photonic counterpart of topological crystalline insulators, a new topological phase of atomic band insulators. Namely, the frequency band structure stemming from the interaction of resonant modes of the individual cavities exhibits an omnidirectional band gap within which gapless surface states emerge for finite slabs of the lattice. Due to the equivalence of a topological crystalline insulator with its photonic-crystal analog, the frequency band structure of the latter can be characterized by a $Z_{2}$ topological invariant. Such a topological photonic crystal can be realized in the microwave regime as a three-dimensional lattice of dielectric particles embedded within a continuous network of thin metallic wires.'
author:
- Vassilios Yannopapas
title: 'Gapless surface states in a lattice of coupled cavities: a photonic analog of topological crystalline insulators'
---
Introduction
============
The frequency band structure of artificial periodic dielectrics formally known as photonic crystals is the electromagnetic (EM) counterpart of the electronic band structure in ordinary atomic solids. Recently, a new analogy between electron and photon states in periodic structures has been proposed by Raghu and Haldane, [@haldane] namely the one-way chiral edge states in two-dimensional (2D) photonic-crystal slabs which are similar to the corresponding edge states in the quantum Hall effect. [@one_way] The photonic chiral edge states are a result of time-reversal (TR) symmetry breaking which comes about with the inclusion of gyroelectric/ gyromagnetic material components; these states are robust to disorder and structural imperfections as long as the corresponding topological invariant (Chern number in this case) remains constant.
In certain atomic solids, TR symmetry breaking is not prerequisite for the appearance of topological electron states as it is the case in the quantum Hall effect. Namely, when spin-orbit interactions are included in a TR symmetric graphene sheet, a bulk excitation gap and spin-filtered edge states emerge [@mele_2005] without the presence of an external magnetic field, a phenomenon which is known in literature as quantum spin Hall effect. Its generalization to three-dimensional (3D) atomic solids lead to a new class of solids, namely, topological insulators. [@ti_papers] The latter possess a spin-orbit-induced energy gap and gapless surface states exhibiting insulating behavior in bulk and metallic behavior at their surfaces. Apart from topological insulators where the spin-orbit band structure with TR symmetry defines the topological class of the corresponding electron states, other topological phases have been proposed such as topological superconductors (band structure with particle-hole symmetry), [@ts_papers] magnetic insulators (band structure with magnetic translation symmetry), [@mi_papers] and, very recently, topological crystalline insulators. [@fu_prl] In the latter case the band structure respects TR symmetry as well as a certain point-group symmetry leading to bulk energy gap and gapless surface states.
In this work, we propose a photonic analog of a topological crystalline insulator. Our model photonic system is a 3D crystal of weakly interacting resonators respecting TR symmetry and the point-symmetry group associated with a given crystal surface. As a result, the system possesses an omnidirectional band gap within which gapless surface states of the EM field are supported. It is shown that the corresponding photonic band structure is equivalent to the energy band structure of an atomic topological crystalline insulator and, as such, the corresponding states are topological states of the EM field classified by a $Z_{2}$ topological invariant.
The frequency band structure of photonic crystals whose (periodically repeated) constituent scattering elements interact weakly with each other can be calculated by a means which is similar to the tight-binding method employed for atomic insulators and semiconductors. Photonic bands amenable to a tight-binding-like description are e.g., the bands stemming from the whispering-gallery modes of a lattice of high-index scatterers [@lido] the defect bands of a sublattice of point defects, within a photonic crystal with an absolute band gap, [@bayindir] the plasmonic bands of a lattice of metallic spheres [@quinten] or of a lattice of dielectric cavities within a metallic host. [@stefanou_ssc] In the latter case, the frequency band structure stems from the weak interaction of the surface plasmons of each individual cavity [@stefanou_ssc] wherein light propagates within the crystal volume by a hopping mechanism. Such type of lattice constitutes the photonic analog of a topological crystalline insulator presented in this work whose frequency band structure will be revealed based on a photonic tight-binding treatment within the framework of the coupled-dipole method. [@cde] The latter is an exact means of solving Maxwell’s equations in the presence of nonmagnetic scatterers.
Tight-binding description of dielectric cavities in a plasmonic host
====================================================================
We consider a lattice of dielectric cavities within a lossless metallic host. The $i$-th cavity is represented by a dipole of moment ${\bf P}_{i}=(P_{i;x},P_{i;y},P_{i;z})$ which stems from an incident electric field ${\bf E}^{inc}$ and the field which is scattered by all the other cavities of the lattice. This way the dipole moments of all the cavities are coupled to each other and to the external field leading to the coupled-dipole equation $${\bf P}_{i}= \boldsymbol\alpha_{i}(\omega) [{\bf E}^{inc} +
\sum_{i' \neq i} {\bf G}_{i i'}(\omega) {\bf P}_{i'}].
\label{eq:cde}$$ ${\bf G}_{i i'}(\omega)$ is the electric part of the free-space Green’s tensor and ${\bf \boldsymbol\alpha}_{i}(\omega)$ is the $3
\times 3$ polarizability tensor of the $i$-th cavity. Eq. (\[eq:cde\]) is a $3N \times 3N$ linear system of equations where $N$ is the number of cavities of the system. We assume that the cavities exhibit a uniaxial EM response, i.e., the corresponding polarizability tensor is diagonal with $\alpha_{x}=\alpha_{y}=\alpha_{\parallel}$ and $\alpha_{z}=\alpha_{\perp}$. For strong anisotropy, the cavity resonances within the $xy$-plane and along the $z$-axis can be spectrally distinct; thus, around the region of e.g., the cavity resonance $\omega_{\parallel}$ within the $xy$-plane, $\alpha_{\perp} \ll \alpha_{\parallel}$ (see appendix). In this case, one can separate the EM response within the $xy$-plane from that along the $z$-axis and Eq. (\[eq:cde\]) becomes a $2N
\times 2N$ system of equations, $${\bf P}_{i}= \alpha_{\parallel}(\omega) [\sum_{i' \neq i} {\bf
G}_{i i'}(\omega) {\bf P}_{i'}]. \label{eq:cde_no_field}$$ where we have set ${\bf E}^{inc}={\bf 0}$ since we are seeking the eigenmodes of the system of cavities. Also, now, ${\bf
P}_{i}=(P_{i;x},P_{i;y})$.
For a particle/cavity of electric permittivity $\epsilon_{\parallel}$ embedded within a material host of permittivity $\epsilon_{h}$, the polarizability $\alpha_{\parallel}$ is given by the Clausius-Mossotti formula $$\alpha_{\parallel}=\frac{3 V}{4 \pi}
\frac{\epsilon_{\parallel}-\epsilon_{h}}{\epsilon_{\parallel}+
2\epsilon_{h}} \label{eq:cm}$$ where $V$ is the volume of the particle/ cavity. For a lossless plasmonic (metallic) host in which case the electric permittivity can be taken as Drude-type, i.e., $\epsilon_{h}=1-\omega_{p}^{2} /
\omega^{2}$ (where $\omega_{p}$ is the bulk plasma frequency), the polarizability $\alpha_{\parallel}$ exhibits a pole at $\omega_{\parallel}=\omega_{p} \sqrt{2/ (\epsilon_{\parallel}
+2)}$ (surface plasmon resonance). By making a Laurent expansion of $\alpha_{\parallel}$ around $\omega_{\parallel}$ and keeping the leading term, we may write $$\alpha_{\parallel}= \frac{F} {\omega - \omega_{\parallel}} \equiv
\frac{1} {\Omega} \label{eq:a_laurent}$$ where $F=(\omega_{\parallel}/2) (\epsilon_{\parallel} -
\epsilon_{h})/ (\epsilon_{\parallel}+2)$. For sufficiently high value of the permittivity of the dielectric cavity, i.e., $\epsilon_{\parallel}
> 10$, the electric field of the surface plasmon is much localized at the surface of the cavity. As a result, in a periodic lattice of cavities, the interaction of neighboring surface plasmons is very weak leading to much narrow frequency bands. By treating such a lattice in a tight binding-like framework, we may assume that the Green’s tensor ${\bf G}_{i i'}(\omega)$ does not vary much with frequency and therefore, ${\bf G}_{i i'}(\omega) \simeq {\bf G}_{i
i'}(\omega_{\parallel})$. In this case, Eq. (\[eq:cde\_no\_field\]) becomes an eigenvalue problem $$\sum_{i' \neq i} {\bf G}_{i i'}(\omega_{\parallel}) {\bf P}_{i'}=
\Omega {\bf P}_{i} \label{eq:cde_eigen}$$ where $$\begin{aligned}
{\bf G}_{i i'}(\omega_{\parallel})=q_{\parallel}^{3} \Bigl[
C(q_{\parallel} | r_{ii'}|) {\bf
I}_{2} + J(q_{\parallel} | r_{ii'}|) \left(%
\begin{array}{cc}
\frac{x_{ii'}^2}{r_{ii'}^{2}} & \frac{x_{ii'}y_{ii'}}{r_{ii'}^{2}} \\
\frac{x_{ii'}y_{ii'}}{r_{ii'}^{2}} & \frac{y_{ii'}^2}{r_{ii'}^{2}} \\
\end{array}%
\right) \Bigr]. \nonumber \\ \label{eq:g_tensor}\end{aligned}$$ with ${\bf r}_{ii'}={\bf r}_{i}-{\bf r}_{i'}$, $q_{\parallel}=\sqrt{\epsilon_{h}}\omega_{\parallel}/c$ and ${\bf
I}_{2}$ is the $2 \times 2$ unit matrix. The form of functions $C(q_{\parallel} | r_{ii'}|)$, $J(q_{\parallel} | r_{ii'}|)$ generally depends on the type of medium hosting the cavities (isotropic, gyrotropic, bi-anisotropic, etc). [@eroglu; @dmitriev] The Green’s tensor of Eq. (\[eq:g\_tensor\]) describes the electric interactions between two point dipoles ${\bf P}_{i}$ and ${\bf P}_{i'}$ each of which corresponds to a single cavity. The first term of ${\bf
G}_{i i'}$ describes an interaction which does not depend on the orientation of the two dipoles whilst the second one is orientation dependent.
![(Color online) (a) Tetragonal crystal with two cavities within the unit cell. (b) The bulk Brillouin zone and (c) the surface Brillouin zone corresponding to the (001) surface.[]{data-label="fig1"}](Fig1.eps){width="8cm"}
For an infinitely periodic system, i.e., a crystal of cavities, we assume the Bloch ansatz for the polarization field, i.e., $${\bf P}_{i}={\bf P}_{n \beta}=\exp (i {\bf k} \cdot {\bf R}_{n})
{\bf P}_{0 \beta} \label{eq:bloch}$$ The cavity index $i$ becomes composite, $i \equiv n \beta$, where $n$ enumerates the unit cell and $\beta$ the positions of inequivalent cavities in the unit cell. Also, ${\bf R}_{n}$ denotes the lattice vectors and ${\bf k}=(k_{x},k_{y},k_{z})$ is the Bloch wavevector. By substituting Eq. (\[eq:bloch\]) into Eq. (\[eq:cde\_eigen\]) we finally obtain $$\sum_{\beta'} \tilde{{\bf G}}_{\beta
\beta'}(\omega_{\parallel},{\bf k}) {\bf P}_{0 \beta'}= \Omega
{\bf P}_{0 \beta} \label{eq:cde_eigen_periodic}$$ where $$\tilde{{\bf G}}_{\beta \beta'}(\omega_{\parallel}, {\bf k}) =
\sum_{n'} \exp [i {\bf k} \cdot ({\bf R}_{n}-{\bf R}_{n'})] {\bf
G}_{n \beta; n' \beta'}(\omega_{\parallel}).
\label{eq:green_fourier}$$ Solution of Eq. (\[eq:cde\_eigen\_periodic\]) provides the frequency band structure of a periodic system of cavities.
![Frequency band structure for tetragonal lattice of resonant cavities within a plasmonic host (see Fig. \[fig1\]) corresponding to the Green’s tensor of Eq. (\[eq:G\_elem\]) with $s^{A}_{1}=-s^{B}_{2}=1.2, s^{A}_{2}=-s^{B}_{2}=0.5,
s'_{1}=2.5,s'_{2}=0.5,s_{z}=2$.[]{data-label="fig2"}](Fig2.eps){width="8cm"}
Topological frequency bands
===========================
Since Eq. (\[eq:cde\_eigen\_periodic\]) is equivalent to a Hamiltonian eigenvalue problem, we adopt the crystal structure of Ref. . Namely, a tetragonal lattice with a unit cell consisting of two same cavities at inequivalent positions $A$ and $B$ \[see Fig. \[fig1\](a)\] along the $c$-axis. In this case, the index $\beta$ in Eq. (\[eq:cde\_eigen\_periodic\]) assumes the values $\beta=A,B$ for each sublattice (layer) of the crystal. The above lattice is characterized by the $C_{4}$ point-symmetry group. In order to preserve the $C_{4}$ symmetry [@fu_prl] in the Green’s tensor matrix of Eq. (\[eq:cde\_eigen\_periodic\]) we assume that the interaction between two cavities within the same layer (either $A$ or $B$) depends on the relative orientation of the point dipole in each cavity whilst the interaction between cavities belonging to adjacent layers is orientation independent. Also, we take into account interactions up to second neighbors in both inter- and intra-layer interactions. Taking the above into account, the lattice Green’s tensor assumes the form $$\begin{aligned}
\tilde{{\bf G}}({\bf k})
=\left(%
\begin{array}{cc}
\tilde{{\bf G}}^{AA}({\bf k}) & \tilde{{\bf G}}^{AB}({\bf k}) \\
\tilde{{\bf G}}^{AB \dagger}({\bf k}) & \tilde{{\bf G}}^{BB}({\bf k}) \\
\end{array}%
\right) \nonumber \\\end{aligned}$$ where $$\begin{aligned}
\tilde{{\bf G}}^{\beta \beta}({\bf k})= 2 s^{\beta}_{1}
\left(
\begin{array}{cc}
\cos (k_{x} \alpha) & 0 \\
0 & \cos (k_{y} \alpha) \\
\end{array}
\right)+
\nonumber && \\
2 s^{\beta}_{2}\left(%
\begin{array}{cc}
\cos (k_{x} \alpha)\cos (k_{y} \alpha) & -\sin (k_{x} \alpha)\sin (k_{y} \alpha) \\
-\sin (k_{x} \alpha)\sin (k_{y} \alpha) & \cos (k_{x} \alpha)\cos (k_{y} \alpha) \\
\end{array}%
\right), %\ \ \beta=A,B
\nonumber && \\
\tilde{{\bf G}}^{A B}({\bf k})=[s'_{1}+2 s'_{2} (\cos(k_{x}
\alpha) + \cos (k_{y} \alpha)) +s'_{z} \exp( i k_{z} \alpha)] {\bf
I}_{2}. \nonumber && \\ \label{eq:G_elem}\end{aligned}$$
![Frequency band structure for a finite slab $ABAB \cdots
ABB$ of the crystal of Fig. \[fig1\] made from 80 bilayers. []{data-label="fig3"}](Fig3.eps){width="8cm"}
The lattice Green’s tensor of Eq. (\[eq:G\_elem\]) is completely equivalent to the lattice Hamiltonian of Ref. . $s^{\beta}_{1},s^{\beta}_{2},s'_{1},s'_{2},s_{z}$ in Eq. (\[eq:G\_elem\]) generally depend on $q_{\parallel}$, the lattice constant $a$ and the interlayer distance $c$ but hereafter will be used as independent parameters. Namely we choose $s^{A}_{1}=-s^{B}_{2}=1.2, s^{A}_{2}=-s^{B}_{2}=0.5,
s'_{1}=2.5,s'_{2}=0.5,s_{z}=2$. In Fig. \[fig2\] we show the (normalized) frequency band structure corresponding to Eq. (\[eq:G\_elem\]) along the symmetry lines of the Brillouin zone shown in Fig. \[fig1\](b). It is evident that an omnidirectional frequency band gap exists around $\Omega=0$ which is prerequisite for the emergence of surface states. In order to inquire the occurrence of surface states we find the eigenvalues of the Green’s tensor of Eq. (\[eq:G\_elem\]) in a form appropriate for a slab geometry. The emergence of surface states depends critically on the surface termination of the finite slab, i.e., for different slab terminations different surface-state dispersions occur (if occur at all). Namely, we assume a finite slab parallel to the (001) surface (characterized by the $C_{4}$ symmetry group) consisting of 80 alternating $AB$ layers except the last bilayer which is $BB$, i.e., the layer sequence is $ABAB
\cdots ABB$. The corresponding frequency band structure along the symmetry lines of the surface Brillouin zone of the (001) surface \[see Fig. \[fig1\](c)\] is shown in Fig. \[fig3\]. It is evident that there exist gapless surface states within the band gap exhibiting a quadratic degeneracy at the $\overline{M}$-point. In this case, the corresponding doublet of surface states can be described by an effective theory [@chong_prb] similarly to the doublet states at a point of linear degeneracy (Dirac point). [@sepkhanov]
We note that the equivalence of the Green’s tensor ${\bf G}$ with the atomic Hamiltonian of Ref. as well as the form of the time-reversal $T$ and (geometric) $C_{4}$-rotation $U$ operators for the EM problem [@haldane] which are the same as for spinless electrons, allows to describe the photonic band structure with the $Z_{2}$ topological invariant $\nu_{0}$ $$(-1)^{\nu_{0}}=(-1)^{\nu_{\Gamma M}} (-1)^{\nu_{A Z}}
\label{eq:z2_def}$$ where for real eigenvectors of $\tilde{{\bf G}}({\bf k})$ we have [@fu_prl] $$(-1)^{{\bf k}_{1} {\bf k}_{2}} = {\rm Pf}[w({\bf k}_{2})]/ {\rm
Pf}[w({\bf k}_{1})] \label{eq:pf_frac}$$ and $$w_{mn}({\bf k}_{i}) = \langle u_{m} ({\bf k}_{i}) | U | u_{n}({\bf
k}_{i}) \rangle. \label{eq:w_def}$$ ${\rm Pf}$ stands for the Pfaffian of a skew-symmetric matrix, i.e., ${\rm Pf}[w]^{2}=\det(w)$. Due to the double degeneracy of the band structure at the four special momenta points $\Gamma, M,
A, Z$ the frequency bands come in doublets. Since frequency eigenvectors with different eigenfrequencies are orthogonal, all the inter-pair elements of the $w$-matrix are zero and the latter is written as: $$w({\bf k}_{i})=\left(%
\begin{array}{cccc}
w^{1}({\bf k}_{i}) & 0 & 0 & 0 \\
0 & w^{2}({\bf k}_{i}) & 0 & 0 \\
0 & 0 & \ddots & 0 \\
0 & 0 & 0 & w^{N}({\bf k}_{i}) \\
\end{array}%
\right) \label{eq:w_reduced_form}$$ where $w^{j}({\bf k}_{i})$ are anti-symmetric $SU(2)$ matrices, [@wang_njp] i.e., $w^{j}({\bf k}_{i})=A_1$ or $A_2$, where $$A_1= \left(%
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}%
\right), A_2= \left(%
\begin{array}{cc}
0 & -1 \\
1 & 0 \\
\end{array}%
\right). \label{eq:alpha_matr_def}$$ In this case, ${\rm Pf}[w({\bf k}_{i})]=w^{1}_{12} w^{2}_{12}
\cdots w^{N}_{12}=\pm 1$. Therefore, $(-1)^{{\bf k}_{1} {\bf
k}_{2}} = \pm 1$ and $\nu_{0}=1$ which ensures the presence of gapless surface states.
We note that the above analysis relies on the assumption of real frequency bands. The presence of losses in the constituent materials renders the frequency bands complex, i.e., the Bloch wavevector possesses both a real and an imaginary part. However, even in this case, one can still speak of real frequency bands if the imaginary part of the Bloch wavevector is at least [*hundred*]{} times smaller than the corresponding real part. This a common criterion used in calculations of the complex frequency band structure by on-shell electromagnetic solvers such as the layer-multiple scattering method [@comphy] or the transfer-matrix method. [@tmm]
![(Color online) A possible realization of a photonic structure with gapless surface states: dielectric particles of square cross section, joined together with cylindrical coupling elements and embedded within a 3D network of metallic wires (artificial plasma).[]{data-label="fig4"}](Fig4.eps){width="6cm"}
Blueprint for a photonic topological insulator
==============================================
A possible realization of the photonic analogue of topological insulator in the laboratory is depicted in Fig. \[fig4\]. Since our model system requires dielectric cavities within a homogeneous plasma, a lattice of nano-cavities formed within a homogeneous Drude-type metal, e.g., a noble metal (Au, Ag, Cu), would be the obvious answer. [@stefanou_ssc] However, the plasmon bands are extremely lossy due to the intrinsic absorption of noble metals in the visible regime. A solution to this would be the use of an [*artificial*]{} plasmonic medium operating in the microwave regime where metals are perfect conductors and losses are minimal. Artificial plasma can be created by a 3D network of thin metallic wires of a few tens of $\mu$m in diameter and spaced by a few mm. [@art_plasma] A lattice of dielectric particles within an artificial plasma can be modelled with the presented tight-binding Green’s tensor. Since the interaction among first and second neighbors within the same bilayer ($A$ or $B$) should depend on the dipole orientation (in order to preserve the $C_{4}$ symmetry), the dielectric particles in each layer are connected with cylindrical waveguiding elements (different in each layer $A$ or $B$ - see Fig. \[fig4\]). In contrast, between two successive bilayers there are no such elements since interactions between dipoles belonging to different layers should be independent of the dipole orientations (s orbital-like). Another advantage of realizing the photonic analog in the microwave regime is the absence of nonlinearities in the EM response of the constituent materials since photon-photon interactions may destroy the quadratic degeneracy of the surface bands in analogy with fermionic systems. [@sun]
Finally, we must stress that a photonic topological crystalline insulator can be also realized with purely dielectric materials if the host medium surrounding the cavities is not a plasmonic medium but a photonic crystal with an absolute band gap: the cavities would be point defects within the otherwise periodic photonic crystal and the tight-binding description would be still appropriate. [@bayindir; @k_fang] In this case, Maxwell’s equations lack of any kind of characteristic length and the proposed analog would be realized in any length scale.
Conclusions
===========
In conclusion, a 3D lattice of weakly interacting cavities respecting TR symmetry and a certain point-group symmetry constitutes a photonic analog of a topological crystalline insulator by demonstrating a spectrum of gapless surface states. A possible experimental realization would be a 3D lattice of dielectric particles within a continuous network of thin metallic wires with a plasma frequency in the GHz regime.
This work has been supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 228455-NANOGOLD (Self-organized nanomaterials for tailored optical and electrical properties).
The $z$-component of the polarizability need not be zero but can assume finite values as far as the frequency band stemming from the surface-plasmon resonance corresponding to the z-component is spectrally distinct (no overlap) with the bands stemming from the $xy$-components of the polarizability. In this case, one can treat separately the two frequency bands (doublet) stemming from the xy-resonance from the band coming from the z-resonance (singlet). The above requirements can be quantified as follows, $$C_{\parallel} \ll \frac{\omega_{\parallel} -
\omega_{\perp}}{\omega_{p}} \label{eq:app_1}$$ where $C_{\parallel}$ is the width of the $xy$-frequency bands (in dimensionless frequency units). $C_{\parallel}$ is obtained from the first term of the right-hand side of Eq. (\[eq:g\_tensor\]) of the paper. To a first approximation, it is given by [@cde] $$C_{\parallel} \sim \frac{\exp(-q_{\parallel} a_{\parallel})}
{q_{\parallel} a_{\parallel}} \label{eq:app_2}$$ where $q_{\parallel}= \sqrt{|\epsilon_{h}|} \omega_{\parallel} /
c$. Therefore, the condition (\[eq:app\_1\]) is written as $$\frac{\exp(-q_{\parallel} a_{\parallel})} {q_{\parallel}
a_{\parallel}} \ll \frac{\omega_{\parallel} -
\omega_{\perp}}{\omega_{p}} \label{eq:app_3}$$ where $a_{\parallel}$ is the lattice constant in the $xy$-plane. Given that $\omega_{\parallel} = \omega_{p}
\sqrt{2/(\epsilon_{\parallel}+2)}$, Eq. (\[eq:app\_3\]) becomes $$\frac{\exp({-\sqrt{|\epsilon_{h}|} \omega_{\parallel}
a_{\parallel} /c})}{\sqrt{|\epsilon_{h}|} \omega_{\parallel}
a_{\parallel} /c} \ll \sqrt{\frac{2
(\epsilon_{\perp}-\epsilon_{\parallel})}
{(\epsilon_{\parallel}+2)(\epsilon_{\perp}+2)}} \label{eq:app_5}$$ From the above equation it is evident that for a given value of the dielectric anisotropy $\epsilon_{\perp} -
\epsilon_{\parallel}$, one can always find a suitably large lattice constant $\alpha_{\parallel}$ such that Eq. (\[eq:app\_5\]) is fulfilled. The latter allows the easy engineering of the photonic analog of a crystalline topological insulator since there is practically no restriction on the choice of the (uniaxial) material the cavities are made from. It can also be easily understood that if Eq. (\[eq:g\_tensor\]) holds, the same equation is true for the width $C_{\perp}$ of the singlet frequency band (resulting from the $z$-resonance).
[*Numerical example*]{}. Suppose that the cavities are made from a nematic liquid crystal which is a uniaxial material. Typical values of the permittivity tensor $\epsilon$ are e.g., $\epsilon_{\parallel}=1.5$, $\epsilon_{\perp}=1.8$. In this case, $\omega_{\parallel} \approx 0.75 \omega_{p}$ and $\epsilon_{h}=1-\omega_{p}^{2} / \omega_{\parallel}^{2} \approx
-0.777$. By choosing a large lattice constant, i.e., $a_{\parallel} = 4 c / \omega_{p}$ , Eq. (\[eq:app\_5\]) is clearly fulfilled $$\frac{\exp{(-\sqrt{|\epsilon_{h}|} \omega_{\parallel}
a_{\parallel} /c)}}{\sqrt{|\epsilon_{h}|} \omega_{\parallel}
a_{\parallel} /c} \approx 0.026866 \ll 0.2213 \approx
\sqrt{\frac{2 (\epsilon_{\perp}-\epsilon_{\parallel})}
{(\epsilon_{\parallel}+2)(\epsilon_{\perp}+2)}}. \label{eq:app_6}$$
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| ArXiv |
---
abstract: 'A prefix grammar is a context-free grammar whose nonterminals generate prefix-free languages. A prefix grammar $G$ is an ordinal grammar if the language $L(G)$ is well-ordered with respect to the lexicographic ordering. It is known that from a finite system of parametric fixed point equations over ordinals one can construct an ordinal grammar $G$ such that the lexicographic order of $G$ is isomorphic with the least solution of the system, if this solution is well-ordered. In this paper we show that given an ordinal grammar, one can compute (the Cantor normal form of) the order type of the lexicographic order of its language, yielding that least solutions of fixed point equation systems defining algebraic ordinals are effectively computable (and thus, their isomorphism problem is also decidable).'
address: 'University of Szeged, Hungary'
author:
- Kitti Gelle
- Szabolcs Iván
bibliography:
- 'biblio.bib'
title: The ordinal generated by an ordinal grammar is computable
---
Algebraic ordinals; Ordinal grammars; Parametric fixed-point equations over ordinals; Isomorphism of algebraic well-orderings
Introduction
============
Least solutions of finite systems of fixed points equations occur frequently in computer science. Some very well-known instances of this are the regular and context-free languages, rational and algebraic power series, well-founded semantics of generalized logic programs, semantics of functional programs, just to name a few. A perhaps less-known instance is the notion of the algebraic linear orders of [@Bloom:2010:MTC:1655414.1655555]. A linear ordering is algebraic if it is (isomorphic to) the first component of the least solution of a finite system of fixed point equations of the sort $$F_i(x_0,\ldots,x_{n_i-1})=t_i,\quad i=1,\ldots,n,$$ where $n_1=0$ and each $t_i$ is an expression composed of the function variables $F_j$, $j=1,\ldots,n$, the variables $x_0,\ldots,x_{n_i-1}$ which range over linear orders, the constant $1$ and the sum operation $+$. As an example, consider the following system from [@DBLP:journals/fuin/BloomE10]: $$\begin{aligned}
F_0 &= G(1)\\
G(x)&=x+G(F(x))\\
F(x)&=x+F(x)\end{aligned}$$ In this system, the function $F$ maps a linear order $x$ to $x+x+\ldots = x\times\omega$, the function $G$ maps a linear order $x$ to $x+G(x\times \omega)=x+x\times\omega+G(x\times\omega^2)+\ldots
=x\times\omega^\omega$, thus the first component of the least solution of the system is $F_0=G(1)=\omega^\omega$.
If the system in question is parameterless, that is, $n_i=0$ for each $i$, then the ordering which it defines is called a regular ordering. An ordinal is called algebraic (regular, respectively) if it is algebraic (regular, resp.) as a linear order. It is known [@ITA_1980__14_2_131_0; @BLOOM2001533; @10.1007/978-3-540-73859-6_1; @DBLP:journals/fuin/BloomE10; @10.1007/978-3-642-29344-3_25] that an ordinal is regular if and only if it is smaller than $\omega^\omega$ and is algebraic if and only if it is smaller than $\omega^{\omega^\omega}$.
To prove the latter statement, the authors of [@DBLP:journals/fuin/BloomE10] applied a path first used by Courcelle [@ITA_1978__12_4_319_0]: every countable linear order is isomorphic to the frontier of some (possibly) infinite (say, binary) tree. Frontiers of infinite binary trees in turn correspond to prefix-free languages over the binary alphabet, equipped with the lexicographic ordering. Moreover, algebraic (regular, resp.) ordinals are exactly the lexicographic orderings of context-free (regular, resp.) prefix-free languages [@COURCELLE198395] (prefix-free being optional here as each language can be effectively transformed to a prefix-free order-isomorphic one for both the regular and the algebraic case). Thus, studying lexicographic orderings of prefix-free regular or context-free languages can give insight to regular or algebraic linear orders. The works [@BLOOM2001533; @6a0baa0d4e6744d38956d22057b410ce; @BLOOM200555; @10.1007/978-3-540-73859-6_1; @COURCELLE198395; @ITA_1980__14_2_131_0; @LOHREY201371; @ITA_1986__20_4_371_0] deal with regular linear orders this way, in particular [@LOHREY201371] shows that the isomorphism problem for regular linear orders is decidable in polynomial time The study of the context-free case was initiated in [@10.1007/978-3-540-73859-6_1], and further developed in [@DBLP:journals/fuin/BloomE10; @doi:10.1142/S0129054111008155; @ESIK2011107; @10.1007/978-3-642-22321-1_19; @10.1007/978-3-642-29344-3_25; @CARAYOL2013285; @KUSKE201446].
Highlighting the results from these works that are tightly connected to the current paper: the case of regular linear orders is well-understood, even their isomorphism problem (that is, whether two regular linear orders, given by two finite sets of fixed-point equations, are isomorphic) is decidable. For algebraic linear orders, there are negative results: it is already undecidable whether an algebraic linear ordering is dense, thus (as there are exactly four dense countable linear orders up to isomorphism) the isomorphism problem of algebraic linear orders is undecidable. On the other hand, deciding whether an algebraic linear order is scattered, or a well-order, is decidable. The frontier of decidability of the isomorphism problem of algebraic linear orderings is an interesting question: for the general case it is undecidable, while for the case of regular ordinals it is known to be decidable by [@LOHREY201371] and [@Khoussainov:2005:ALO:1094622.1094625]. In [@DBLP:journals/fuin/BloomE10], it was shown that a system of equations defining an algebraic ordering can be effectively transformed (in polynomial time) to a so-called prefix grammar $G$ (a context-free grammar whose nonterminals each generate a prefix-free language), such that the lexicographic order of the language generated by $G$ is isomorphic to the algebraic ordering in question. If the ordering is a well-ordering (i.e. the system defines an algebraic ordinal), then the grammar we get is called an ordinal grammar, that is, a prefix grammar generating a well-ordered language with respect to the lexicographic ordering.
In this paper we show that given an ordinal grammar, the order type of the lexicographic ordering of the language it generates is computable (that is, we can effectively construct its Cantor normal form). Hence, applying the above transformation we get that the Cantor normal form of any algebraic ordinal is computable from its fixed-point system presentation, thus in particular, the isomorphism problem of algebraic ordinals is decidable.
Notation
========
When $n\geq 0$ is an integer, $[n]$ denotes the set $\{1,\ldots,n\}$. (Thus, $[0]$ is another notation for the empty set $\emptyset$.)
Linear orders, ordinals {#linear-orders-ordinals .unnumbered}
-----------------------
In this paper we consider countable linear orderings. A good reference on the topic is [@rosenstein]. A linear ordering $(I,<)$ is a set $I$ equipped with a strict linear order: an irreflexive, transitive and trichotome relation $<$. When the order $<$ is clear from the context, we omit it. Set-theoretic properties of $I$ are lifted to $(I,<)$, thus we can say that a linear order is finite, countable etc. When $(I_1,<_1)$ and $(I_2,<_2)$ are linear orders, their (ordered) sum is $(I_1,<_1)+(I_2,<_2)=(I_1\uplus I_2,<)$ with $x<y$ if and only if either $x\in I_1$ and $y\in I_2$, or $x,y\in I_1$ and $x<_1y$, or $x,y\in I_2$ and $x<_2y$. A linear ordering $(I',<')$ is a subordering of $(I,<)$ if $I'\subseteq I$ and $<'$ is the restriction of $<$ onto $I'$. In order to ease notation, we usually use $<$ in these cases in place of $<'$ and so we will simply write $(I_1,<)+(I_2,<)=(I,<)$ or even $I_1+I_2=I$ in the case of sums.
A linear ordering $I$ is called a *well-ordering* if there are no infinite descending chains $\ldots<x_2<x_1<x_0$ in $I$. Clearly, well-orderings are closed under (finite) sums and suborderings, and they are also closed under $\omega$-sums: if $I_1,I_2,\ldots$ are pairwise disjoint linear orderings, then their sum $I=I_1+I_2+\ldots$ is the ordering with underlying set $\bigcup_i I_i$ and order $x<y$ if and only if $x\in I_i$ and $y\in I_j$ for some $i<j$, or $x,y\in I_i$ for some $i$ and $x<_iy$, which is well-ordered if so is each $I_i$.
Two linear orders $(I,<_i)$ and $(J,<_j)$ are called *isomorphic* if there is a bijection $h:I\to J$ with $x<_iy$ implying $h(x)<_jh(y)$. An *order type* is an isomorphism class of linear orderings. The order type of the linear order $I$ is denoted by $o(I)$. Clearly, if two orderings are isomorphic and one of them is a well-ordering, then so is the other one. The *ordinals* are the order types of well-orderings (for a concise introduction see e.g. the lecture notes of J. A. Stark [@jalex]). The order types of the finite ordered sets are identified with the nonnegative integers. The order type of the natural numbers themselves (whose set is $\mathbb{N}_0=\{0,1,\ldots\}$, equipped by their usual ordering) is denoted by $\omega$, while the order types of the integers and rational numbers are respectively denoted by $\zeta$ and $\eta$. Since if $o(I)=o(I')$ and $o(J)=o(J')$, then $o(I+J)=o(I'+J')$, the sum operation can be lifted to order types, even for $\omega$-sums. For example, $\omega+\omega$ is the order type of $\{0,1\}\times\mathbb{N}$, equipped with the lexicographic ordering $(b_1,n_1)<(b_2,n_2)$ if and only if either $b_1<b_2$ or ($b_1=b_2$ and $n_1<n_2$). Note that $1+\omega=\omega$ but $\omega+1\neq\omega$.
The ordinals themselves are also equipped with a relation $<$ so that each set of ordinals is well-ordered by $<$, namely $o_1<o_2$ if $o_1\neq o_2$ and there are linear orderings $I$ and $J$ such that $o(I)=o_1$, $o(J)=o_2$ and $I$ is a subordering of $J$. With respect to this relation, every set $\Omega$ of ordinals have a least upper bound (a *supremum*) $\bigvee\Omega$ (which is also an ordinal), moreover, for each ordinal $\alpha$, the ordinals smaller than $\alpha$ form a set.
Each ordinal $\alpha$ is either a *successor ordinal* in which case $\alpha=\beta+1$ for some smaller ordinal $\beta$, or a *limit ordinal* in which case $\alpha=\mathop\bigvee\limits_{\beta<\alpha}\beta$, the supremum of all the ordinals smaller than $\alpha$. These two cases are disjoint. For an example, $0=\bigvee\emptyset$ is a limit ordinal, and it is the smallest ordinal; $1$, $2$ and $42$ are successor ordinals, $\omega$ is a limit ordinal, $\omega+1$ is again a successor ordinal, $\omega+\omega$ is a limit ordinal and so on.
Since every set of ordinals is well-ordered, and to each ordinal $\alpha$ the ordinals smaller than $\alpha$ form a set, the principle of *(well-founded) induction* is valid for ordinals: if $P$ is a property of ordinals, and
- whenever $P$ holds for $\alpha$, then $P$ holds for $\alpha+1$ and
- whenever $\alpha$ is a limit ordinal and $P$ holds for each ordinal $\beta<\alpha$, then $P$ holds for $\beta$,
then $P$ holds for all the ordinals. (In practice we usually separate the case of $\alpha=0$ from the rest of the limit ordinals.)
Over ordinals, the operations of (binary) product and exponentiation are defined via induction as follows: $$\begin{aligned}
\alpha\times 0 &=0 & \alpha\times(\beta+1)&=\alpha\times\beta+\alpha& \alpha\times\beta^*&=\mathop\bigvee\limits_{\beta'<\beta^*}\left(\alpha\times\beta'\right)\\
\alpha^0 &=1 & \alpha^{\beta+1}&=\alpha^\beta\times\alpha&\alpha^{\beta^*}&=\mathop\bigvee\limits_{\beta'<\beta^*}\alpha^{\beta'}\end{aligned}$$ where the equations of the last column hold for limit ordinals $\beta^*$.
Every ordinal $\alpha$ can be uniquely written as a finite sum $$\alpha=\omega^{\alpha_1}\times n_1+\omega^{\alpha_2}\times n_2+\ldots+\omega^{\alpha_k}\times n_k$$ where $k\geq 0$ and for each $1\leq i\leq k$, $n_i>0$ are integers, and $\alpha_1>\alpha_2>\ldots>\alpha_k$ are ordinals. The ordinal $\alpha_1$ in this form is called the *degree* of $\alpha$, denoted by $\deg(\alpha)$, and the sum itself is called the *Cantor normal form* of $\alpha$. The operations $+$ and $\times$ are associative, and the above operations satisfy the identities $$\begin{aligned}
\alpha\times(\beta+\gamma)&=\alpha\times\beta+\alpha\times\gamma&\alpha^\beta\times\alpha^\gamma&=\alpha^{\beta+\gamma}&(\alpha^\beta)^\gamma&=\alpha^{\beta\times\gamma}\\
\deg(\alpha+\beta)&=\max\{\deg(\alpha),\deg(\beta)\}&\deg(\alpha\times\beta)&=\deg(\alpha)+\deg(\beta)&\deg(\alpha^\beta)&=\deg(\alpha)\times\beta,\end{aligned}$$ the last one being valid only when $\alpha\geq\omega$. From $\deg(\alpha+\beta)=\max\{\deg(\alpha),\deg(\beta)\}$ we get that if $o_1\leq o_2\leq \ldots$ are ordinals with $\deg(o_i)<\alpha$ for some ordinal $\alpha$, then $\deg(o_1+o_2+\ldots)\leq \alpha$ and equality holds if and only if $\bigvee\deg(o_i)=\alpha$ is a limit ordinal, in which case $o_1+o_2+\ldots=\omega^{\alpha}$.
The following theorem from [@doi:10.1112/plms/s3-4.1.177] gives lower and upper bounds for the order type of the union of two well-ordered sets:
\[thm-union\] Let $(I,<)$ be a countable well-ordered set and $I=A\cup B$. Let us write the order types of $A$ and $B$ as $$\begin{aligned}
o(A) &= \omega^{\alpha_1}\times a_1+\ldots \omega^{\alpha_n}\times a_n\\
o(B) &= \omega^{\alpha_1}\times b_1+\ldots \omega^{\alpha_n}\times b_n
\end{aligned}$$ for an integer $n\geq 0$, ordinals $\alpha_1>\alpha_2>\ldots> \alpha_n$ and integer coefficients $a_1,\ldots,a_n,b_1,\ldots,b_n\geq 0$ such that $\max\{a_i,b_i\}\geq 1$ for each $1\leq i\leq n$.
Then $$\begin{aligned}
o(I) &= \omega^{\alpha_1}\times c_1+\omega^{\alpha_2}\times c_2+\ldots +\omega^{\alpha_n}\times c_n
\end{aligned}$$ for some integer coefficients $0\leq c_1,\ldots,c_n$ with $c_i\leq a_i+b_i$ for each $1\leq i\leq n$, and $c_1\geq\max\{a_1,b_1\}$.
Observe that the Theorem can be applied as follows: if $o(A)<\omega^\alpha\times N$ and $o(B)<\omega^\beta\times M$, then $o(A\cup B)< \omega^{\max\{\alpha,\beta\}}\times(N+M-1)$: writing out the Cantor normal forms explicitly for $o(A)$ and $o(B)$ we would get the coefficients for $\omega^{\max\{\alpha,\beta\}}$ can be at most $N-1$ and $M-1$, respectively, making its coefficient in $o(I)$ to be at most $M+N-2$, thus (as the main term cannot be larger than $\omega^{\max\{\alpha,\beta\}}$ in either one of $o(A)$ and $o(B)$) we get $o(I)<\omega^{\max\{\alpha,\beta\}}\times(M+N-1)$. In particular, $\deg(o(A\cup B))=\max\{\deg(o(A)),\deg(o(B))\}$.
Order types of context-free languages {#order-types-of-context-free-languages .unnumbered}
-------------------------------------
For a nonempty finite set (an *alphabet*) $\Sigma$ of terminal symbols, also called *letters* equipped with a total ordering $<$, let $\Sigma^*$ denote the set of all finite words $a_1a_2\ldots a_n$, with $\varepsilon$ standing for the case $n=0$, the *empty word*, and let $\Sigma^\omega$ denote the set of all *$\omega$-word*s $a_1a_2\ldots$. The set of all finite and $\omega$-words is $\Sigma^{\leq\omega}=\Sigma^\omega\cup\Sigma^*$. When $u=a_1\ldots a_n$ is a finite word and $v=b_1b_2\ldots$ is either a finite or an $\omega$-word, then their product is the word $u\cdot v=a_1\ldots a_nb_1b_2\ldots$, also written $uv$. Also, when $u=a_1\ldots a_n$ is a finite word, then its *$\omega$-power* is the word $u^\omega=a_1\ldots a_na_1\ldots a_na_1\ldots$ which is $\varepsilon$ if $u=\varepsilon$ and is an $\omega$-word whenever $u$ is nonempty.
Two (strict) partial orderings, the *strict ordering* $<_s$ and the *prefix ordering* $<_p$ are defined over $\Sigma^{\leq\omega}$ as follows:
- $u<_sv$ if and only if $u=u_1au_2$ and $v=u_1bv_2$ for some words $u_1\in\Sigma^*$, $u_2,v_2\in\Sigma^{\leq\omega}$ and terminal symbols $a<b$
- $u<_pv$ if and only if $v=uw$ for some nonempty word $w\in\Sigma^{\leq\omega}$ (in particular, this implies $u\in\Sigma^*$).
The union of these partial orderings, the *lexicographical ordering* $<_\ell~=~<_s\cup<_p$, simply written as $<$ when it is clear from the context, is a total ordering on $\Sigma^{\leq\omega}$, which is a complete lattice with respect to $<_\ell$.
A *language* is an arbitrary set $L\subseteq\Sigma^*$ of *finite* words. The *supremum* of $L$, viewed as a subset of $\bigl(\Sigma^{\leq \omega},<_\ell\bigr)$ is denoted by $\bigvee L$ and is either a finite word $u\in L$, or an $\omega$-word. The *order type* $o(L)$ of $L$ is the order type of the linear ordering $(L,\leq_\ell)$. As an example, the order types of the languages $a^*$, $a^*\cup\{b\}$ and $b^*a^*$ are $\omega$, $\omega + 1$ and $\omega^2$, respectively. We say that $L$ is *well-ordered* if so is $(L,<_\ell)$. For example, the previous three languages are well-ordered but $a^*b$ is not (as it contains an infinite descending chain $\ldots<aab<ab<b$).
When $K$ and $L$ are languages, then their product is $K\cdot L=\{uv:u\in K,v\in L\}$ and if $u\in\Sigma^*$, then the *left quotient of $L$ with respect to $u$* is $u^{-1}L=\{v\in\Sigma^*:uv\in L\}$, and of course, $K\cup L=\{u:u\in K\hbox{ or }u\in L\}$ is their *union*. We write $K<_\ell L$ if $u<_\ell v$ for each $u\in K$ and $v\in L$. Thus, if $K<_\ell L$, then viewing them as the linear orderings $(K,<_\ell)$ and $(L,<_\ell)$ we get their *sum* $K+L~=~(K\cup L,<_\ell)$. We put an emphasis here on the fact that taking the sum of two languages $K$ and $L$ is a *partial* operation, defined only if $K<_\ell L$.
When $L$ is a language and $u$ is a (possibly infinite) word, then let $L^{<u}$ and $L^{\geq u}$ respectively denote the languages $\{v\in L:v<u\}$ and $\{v\in L:v\geq u\}$. Then clearly, $L=L^{<u}+L^{\geq u}$ for any $L$ and $u$. Note that $L^{<\varepsilon}=\emptyset$ and $L^{\geq\varepsilon}=L$, and also $L^{<a\cdot u}=L^{<a}+ a\bigl((a^{-1}L)^{<u}\bigr)$, $L^{\geq a\cdot u}=a\bigl((a^{-1}L)^{\geq u}\bigr) + L^{\geq b}$ for the least letter $b$ with $a<b$, if such a letter exists and $L^{\geq a\cdot u}=a\bigl((a^{-1}L)^{\geq u}\bigr)$ if $a$ is the last letter of the alphabet. Moreover, $(K\cup L)^{>u}=K^{>u}\cup L^{>u}$ and $(K\cup L)^{\geq u}=K^{\geq u}\cup L^{\geq u}$.
A *context-free grammar* is a tuple $G=(N,\Sigma,P,S)$ with $N$ and $\Sigma$ being the disjoint alphabets of nonterminal and terminal symbols respectively, $S\in N$ is the *start symbol* and $P$ is a finite set of *productions* of the form $A\to \balpha$ with $A\in N$ being a nonterminal and $\balpha$ being a *sentential form*, i.e. $\balpha=X_1\ldots X_n$ for some $n\geq 0$ and $X_1,\ldots,X_n\in N\cup\Sigma$. If $\balpha=uX\bbeta$ for some $u\in\Sigma^*$, $X\in N$ and $\bbeta\in(N\cup\Sigma)^*$, and $X\to\boldsymbol{\gamma}$ is a production, then $\balpha$ can be rewritten to $u\boldsymbol{\gamma}\bbeta$, which is denoted by $\balpha\Rightarrow u\boldsymbol{\gamma}\bbeta$. The reflexive-transitive closure of the relation $\Rightarrow$ is denoted by $\Rightarrow^*$. For any set $\Delta$ of sentential forms, the *language generated by $\Delta$* is $L(\Delta)~=~\{u\in\Sigma^*:\balpha\Rightarrow^*u\hbox{ for some }\balpha\in\Delta\}$. For brevity, when $\Delta=\{\balpha_1,\ldots,\balpha_n\}$ is finite, we simply write $L(\balpha_1,\ldots,\balpha_n)$. Moreover, $o(\Delta)$ denotes $o(L(\Delta))$.
The *language $L(G)$ generated by $G$* is $L(S)$. Languages generated by context-free grammars are called context-free languages. Two context-free grammars $G$ and $G'$ over the the same terminal alphabet are *equivalent* if $L(G)=L(G')$ and *order-equivalent* if $o(L(G))=o(L(G'))$. Any context-free grammar generating a nonempty language of nonempty words can be effectively transformed into a *Greibach normal form* in which the following all hold:
- each production has the form $X\to aX_1\ldots X_n$ for some $a\in\Sigma$,
- each nonterminal $X$ is *productive*, i.e., $L(X)\neq\emptyset$, and *accessible*, i.e., $S\Rightarrow^*uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$.
Also, considering the grammar $G'=(N\cup\{S'\},\Sigma,P\cup\{S'\to aS\},S')$ for a fresh symbol $S'$, we get $L(G')=a\cdot L(G)$, and the order type of each $X\in N$ is the same in both cases, and of course $o(S)=o(S')$. Thus, to each grammar $G$ one can effectively construct another one $G'$ in Greibach normal form, with $o(L(G))=o(L(G'))$.
Suppose $\balpha=aX_1\ldots X_n$ is a sentential form of a context-free grammar $G=(N,\Sigma,P,S)$ in Greibach normal form and $b$ is a terminal symbol. Then we define $\balpha^{<b}$, $\balpha^{\geq b}$ and $b^{-1}\balpha$ as the following finite sets of sentential forms: $$\begin{aligned}
\balpha^{<b}&=\begin{cases}
\{\balpha\}&\hbox{if }a<b\\
\emptyset&\hbox{otherwise}
\end{cases}
&
\balpha^{\geq b}&=\begin{cases}
\emptyset&\hbox{if }a<b\\
\{\balpha\}&\hbox{otherwise}
\end{cases}\end{aligned}$$
$$\begin{aligned}
b^{-1}\balpha&=\begin{cases}
\{\varepsilon\}&\hbox{if }a=b\hbox{ and }n=0\\
\{X_1\ldots X_n\}&\hbox{if }a=b,~n>0\hbox{ and }X_1\in\Sigma\\
\{\boldsymbol{\delta} X_2\ldots X_n:X_1\to \boldsymbol{\delta}\in P\}&\hbox{if }a=b,~n>0\hbox{ and }X_1\in N\\
\emptyset&\hbox{otherwise}
\end{cases}\end{aligned}$$
Then clearly, $L(\balpha^{<b})=L(\balpha)^{<b}$, $L(\balpha^{\geq b})=L(\balpha)^{\geq b}$ and $L(b^{-1}\balpha)=b^{-1}L(\balpha)$. Extending these definitions with $\varepsilon^{<b}=\{\varepsilon\}$, $\varepsilon^{\geq b}=b^{-1}\varepsilon=\emptyset$ and the recursion $$\begin{aligned}
\balpha^{<b\cdot u}&=\balpha^{<b}\cup \{b\cdot(\boldsymbol{\gamma}^{<u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}&
\alpha^{\geq b\cdot u}&=\{b\cdot(\boldsymbol{\gamma}^{\geq u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}\cup\balpha^{\geq c}\end{aligned}$$ where $c\in\Sigma$ is the first letter with $b<c$ if such a $c$ exists, otherwise $$\begin{aligned}
\balpha^{\geq b\cdot u}&=\{b\cdot(\boldsymbol{\gamma}^{\geq u}):\boldsymbol{\gamma}\in b^{-1}\balpha\}\end{aligned}$$ and $(a\cdot u)^{-1}\balpha=\bigcup\bigl(u^{-1}\boldsymbol{\gamma}:\boldsymbol{\gamma}\in a^{-1}\balpha\bigr)$ we have $L(\balpha^{<u})=L(\balpha)^{<u}$, $L(\balpha^{\geq u})=L(\balpha)^{\geq u}$ and $L(u^{-1}\balpha)=u^{-1}L(\balpha)$ for any sentential form $\balpha$ not beginning with a nonterminal and word $u$, moreover, each member of any of these sets is still a sentential form not beginning with a nonterminal. Clearly, $\balpha^{<u}$, $\balpha^{\geq u}$ and $u^{-1}\balpha$ are all computable for any $u$ and $\balpha$.
A context-free grammar $G=(N,\Sigma,P,S)$ is called an *ordinal grammar* if $o(X)$ is an ordinal and $L(X)$ is a *prefix-free language* (that is, there are no words $u,v\in L(X)$ with $u<_pv$) for each nonterminal $X\in N$. It is known [@DBLP:journals/fuin/BloomE10] that to each well-ordered context-free language $L$ there exists an ordinal grammar $G$ generating $L$. It is also known that for *regular* grammars (in which each production has the form $A\to uB$ or $A\to v$) generating a well-ordered language $L$, order equivalence is decidable [@DBLP:journals/ita/Thomas86], while for general context-free grammars, it is undecidable whether $o(L(G))=o(L(G'))$ for two grammars $G$ and $G'$: it is already undecidable whether $o(L(G))=\eta$ holds (or that whether $o(L(G))$ is dense) [@ESIK2011107]. In contrast, it is decidable whether $L(G)$ is well-ordered [@10.1007/978-3-642-22321-1_19].
It is unknown whether the order-equivalence problem is decidable for two grammars generating well-ordered languages.
In this paper we show that it is decidable whether $o(L(G))=o(L(G'))$ for two *ordinal* grammars $G$ and $G'$. Thus, if there is an algorithm that constructs an ordinal grammar $G'$ for an input context-free grammar $G$ generating a well-ordered language (it is known that such an ordinal grammar $G'$ exists but the proof is nonconstructive), then the order-equivalence problem is decidable for well-ordered context-free languages. As any finite system $E$ of fixed point equations over variables taking ordinals as values can effectively by transformed into an ordinal grammar $G$ such that $o(L(G))$ coincides with the least fixed point of the first component of $E$ [@DBLP:journals/fuin/BloomE10], we also get as a byproduct that the Cantor normal form of an algebraic ordinal, given by a finite system of fixed point equations, is effectively computable. Thus, the isomorphism problem of algebraic ordinals is decidable.
Ordinal grammars
================
In this section we recall some known properties of ordinal grammars and then we prove that the order type of the lexicographic ordering of a language, given by an ordinal grammar, is computable.
It is known from [@DBLP:journals/fuin/BloomE10; @10.1007/978-3-642-29344-3_25] that the following are equivalent for an ordinal $\alpha$:
1. $\alpha<\omega^{\omega^\omega}$.
2. $\alpha=o(L(G))$ for a context-free grammar $G$.
3. $\alpha=o(L)$ for a deterministic context-free language $L$.
4. $\alpha=o(L(G))$ for an ordinal grammar $G$.
If $G=(N,\Sigma,P,S)$ is a context-free grammar, we define the relation $\preceq$ on $N\cup\Sigma$ as follows: $Y\preceq X$ if and only if $X\Rightarrow^*\balpha Y\bbeta$ for some $\balpha,\bbeta\in(N\cup\Sigma)^*$. Clearly, $\preceq$ is reflexive and transitive (a preorder): $X\approx Y$ denotes that $X\preceq Y$ and $Y\preceq X$ holds. An equivalence class of $\approx$ is called a *component* of $G$. If $Y\preceq X$ and they do not belong to the same component, we write $Y\prec X$. As an extension, when $\balpha=X_1\ldots X_n$ is a sentential form with $X_i\prec X$ for each $i\in[n]$, we write $\balpha\prec X$. Productions of the form $X\to\balpha$ with $\balpha\prec X$ are called *escaping* productions, the others (when $X_i\approx X$ for some $i\in[n]$) are called *component* productions.
A nonterminal $X$ is called *recursive* if $X\Rightarrow^+\balpha X\bbeta$ for some $\balpha,\bbeta\in(N\cup\Sigma)^*$.
The following are known for ordinal grammars having only usable nonterminals:
\[lem-product\] If $G$ is an ordinal grammar, then for any word $X_1\ldots X_n\in(\Sigma\cup N)^*$, $o(X_1\ldots X_n)=o(X_n)\times o(X_{n-1})\times\ldots\times o(X_1)$.
We will frequently use the above Lemma in the following form: if $X\to X_1\ldots X_n$ is a production of the ordinal grammar $G$ (and thus $L(X_1\ldots X_n)\subseteq L(X)$), then $o(X_n)\times o(X_{n-1})\times\ldots\times o(X_1)\leq o(X)$.
\[lem-u0\] To each recursive nonterminal $X$ there exists a nonempty word $u_X$ such that if $X\Rightarrow^+ uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$, then $u\in u_X^+$.
Moreover, whenever $X\Rightarrow^*w$ for some word $w$, then $w<_s u_X^\omega$.
\[lem-monotone\] If $Y\preceq X$ for the symbols $X,Y\in N\cup\Sigma$, then $o(Y)\leq o(X)$. So if $X\approx Y$, then $o(X)=o(Y)$.
For the rest of the section, let $G=(N,\Sigma,P,S)$ be an ordinal grammar. Since it is decidable whether $L(G)$ is finite, and in that case its order type $o(G)=|L(G)|$ is computable, we assume from now on that $L(G)$ is infinite.
Without loss of generality we can assume that $G$ is in *normal form*:
- $G$ has only usable nonterminals: for each $X$, there are words $u,v,w\in\Sigma^*$ with $S\Rightarrow^*uXv$ and $X\Rightarrow^*w$.
- $L(X)$ is infinite for each nonterminal $X$;
- Each production in $P$ has the form $A\to a\balpha$ for some $A\in N$, $a\in\Sigma$ and $\balpha\in(N\cup\Sigma)^*$;
- All nonterminals different from $S$ are recursive.
To see that such a normal form is computable, consider the following sequence of transformations, starting from an ordinal grammar $G$:
1. Unusable nonterminals are eliminated applying the usual algorithm [@Hopcroft+Ullman/79/Introduction].
2. If $L(A)$ is finite for some nonterminal $A$, then $A$ gets replaced by all the members of $L(A)$ on each right-hand side and gets erased from the set of nonterminals. The result of this transformation is still an ordinal grammar.
3. In particular, if $A\Rightarrow^*\varepsilon$, then by prefix-freeness of $L(A)$ we get that $L(A)=\{\varepsilon\}$, so after this step no $\varepsilon$-transitions remain.
4. Chain rules of the form $A\to B$ with $A,B\in N$ also get eliminated by the usual algorithm which still outputs an ordinal grammar as the generated languages do not change.
5. By Lemma \[lem-u0\], there are no left-recursive nonterminals, that is, no $A\in N$ with $A\Rightarrow^+A\balpha$ for some $\balpha\in(N\cup\Sigma)^*$. Hence, the relation $B<A$ if $A\Rightarrow^+B\balpha$ for some $\balpha\in(N\cup\Sigma)^*$ is a partial ordering. Thus, if we replace each rule of the form $A\to B\balpha$ by $A~\to~\bbeta_1\balpha~|~\bbeta_2\balpha~|~\ldots~|~\bbeta_k\balpha$ where $\bbeta_1,\ldots,\bbeta_k$ are all the alternatives of $B$, the process eventually terminates.
6. Finally, if $X\neq S$ is a nonrecursive nonterminal with $X~\to~\balpha_1~|~\ldots~|~\balpha_n$ being all the alternatives of $X$, let us erase $X$ from $N$ and replace $X$ by one of the $\balpha_i$’s in all possible ways in the right-hand sides of the productions. Clearly, this transformation does not change $L(Y)$ for any $X\neq Y$ and reduces the number of nonterminals in $G$. Applying this transformation for each nonrecursive nonterminal different from $S$ in some arbitrary order now results in an ordinal grammar in normal form.
Clearly, for each $X$ it is decidable whether it is recursive, and if so, then an $u\in\Sigma^+$ can be computed for which $X\Rightarrow^+uX\balpha$ for some $\balpha\in(N\cup\Sigma)^*$. Thus, $u_X$ can be chosen as the (still computable) primitive root [@shyr1991free] of $u$.
We can show also the following:
\[lem-normal-form\] If $G=(N,\Sigma,P,S)$ is an ordinal grammar in normal form, then for each rule $X\to X_1\ldots X_n$ in $P$ one of the following holds:
1. either the production is an escaping one (clearly, for a nonrecursive nonterminal this is the only option),
2. or $X_i\approx X$ for a unique index $i\in[n]$, and $X_j\in\Sigma$ for each $j<i$.
Assume that there is a production $X\to X_1\ldots X_n$ for which none of the conditions hold. This can happen in the following two cases:
1. If there are at least two indices $i<j$ with $X_i\approx X_j\approx X$, then by Lemma \[lem-product\] we get $\alpha\times o(X)\times \beta\times o(X)\times \gamma\leq o(X)$ for some nonzero ordinals $\alpha,\beta$ and $\gamma$, which is nonsense since if $G$ is in normal form, $L(X)$ is infinite, thus $o(X)>1$.
2. Similarly, assume there is a unique index $i\in[n]$ with $X_i\approx X$ (thus, $X_j\prec X$ for each $j\neq i$) and $X_j$ is a nonterminal for some $j<i$. Then again by Lemma \[lem-product\] we get $\alpha\times o(X_i)\times\beta\times o(X_j)\times\gamma\leq o(X)=o(X_i)$ for some nonzero ordinals $\alpha,\beta$ and $\gamma$. Since with $X_j$ being a nonterminal we have $o(X_j)>1$, this is again a contradiction.
Operations on languages
-----------------------
In this subsection we aim to show that whenever $\balpha\in(N\cup\Sigma)^*$ for some ordinal grammar $G=(N,\Sigma,P,S)$ in normal form, both the supremum $\bigvee L(\balpha)$ and whether $\bigvee L(\balpha)$ is a member of $L(\balpha)$ or not, are computable and also a technical decidability lemma which will be used in the proof of Theorem \[thm-computable-recursive-themingeszishere\].
Let $X$ be a recursive nonterminal. By Lemma \[lem-u0\], for each $X\Rightarrow^+w$ we have $w<_s u_X^\omega$, so $u_X^\omega$ is an upper bound of $L(X)$. It is also clear that if $X\Rightarrow^+u_X^tXv$, then $X\Rightarrow^+u_X^{t\cdot k}Xv^k$ for every $k\geq 0$. Hence for any integer $N>0$ there is a word $w\in L(X)$ (say, $w=u_X^{N\cdot t}w'v^N$ where $w'\in L(X)$ is an arbitrary fixed word) such that $u_X^N<_\ell w$, and as $\mathop\bigvee\limits_{N\geq 0}u_X^N=u_X^\omega$, we immediately get:
\[lem-recursive-supremum\] Suppose $X$ is a recursive nonterminal. Then $\mathop\bigvee L(X)=u_X^\omega$. (Thus in particular, there is no largest element in $L(X)$, since $L(X)$ consists of finite words only.)
It is obvious that for any $a\in\Sigma$ we have $\bigvee L(a)=a$ and $a\in L(a)$. For the case of nonrecursive nonterminals (that can be at most $S$) we need to handle the operations union and product. For union, we of course have $\bigvee (K\cup L)~=~\bigvee K\vee\bigvee L$ and this element $u$ belongs to $K\cup L$ if and only if $u=\bigvee K$ and $u\in K$, or $u=\bigvee L$ and $u\in L$ holds.
For product, we state a useful property first:
\[prop-strict\] If $L$ is prefix-free and $\bigvee L$ exists, then either $L<_s\bigvee L$, or $\bigvee L\in L$ holds.
Assume neither of the two cases hold for the supremum of $L$. Then, since $\bigvee L\notin L$, we have $L<_\ell \bigvee L$. Thus, since $L\nless_s\bigvee L$, there is a word $u\in L$ with $u\nless_s\bigvee L$ and $u<_\ell \bigvee L$, hence $u<_p\bigvee L$. But since $L$ is prefix-free, there is no word $v\in L$ with $u<_pv$, thus – as there is no largest element in $L$ by $\bigvee L\notin L$ – there is a word $v\in L$ with $u<_s v$. But as $u<_p\bigvee L$, this yields $\bigvee L<_s v$, a contradiction since $v<_\ell\bigvee L$ has to hold.
This proposition entails the following:
\[cor-strict\] If $K$ and $L$ are nonempty prefix-free languages and both $\bigvee L$ and $\bigvee K$ exist, then $$\begin{aligned}
\bigvee (KL) &= \begin{cases}
\bigvee K&\hbox{ if }K<_s\bigvee K;\\
\bigvee K\cdot \bigvee L&\hbox{ otherwise},
\end{cases}\end{aligned}$$ and $\bigvee(KL)\in KL$ if and only if $K\in \bigvee K$ and $L\in\bigvee L$.
If $K<_s\bigvee K$, then $K\Sigma^*<_s\bigvee K$, so $\bigvee K$ is an upper bound of $KL$ in that case. To see it’s the smallest one, assume $u<_\ell\bigvee K$. Since $\bigvee K$ is the supremum of $K$ with respect to the total ordering $<_\ell$, this means $u<_\ell v$ for some $v\in K$. But for this $v$ and an arbitrary $w\in L$ we still have $u<_\ell vw$, hence $u$ cannot be an upper bound of $KL$. Thus, $\bigvee K=\bigvee(KL)$.
If $u=\bigvee K\in K$, then for any word $v\in K$ and $w\in L$ we have either $v<_s u$, in which case $vw<_s ux$ for any word $x\in\Sigma^{\leq\omega}$, or $v=u$, in which case $vw\leq_\ell u\bigvee L$ since $w\leq_\ell \bigvee L$. Thus, $\bigvee K\cdot\bigvee L$ is an upper bound of $\bigvee(KL)$. Again, if $v<_\ell u\bigvee L$ for some $v$, then either $v<_\ell u$ in which case $v<_\ell uw\in KL$ for any $w\in L$, thus $v$ cannot be the supremum of $KL$, or $u<_pv$ in which case $v=uw$ for some $w$ with $w<_\ell\bigvee L$. This in turn implies the existence of some $w'\in L$ with $w<_\ell w'$, thus $v=uw<_\ell uw'\in KL$, hence $v$ cannot be an upper bound of $KL$, showing the claim.
The statement on membership is clear.
\[cor-sup-computable\] For any ordinal grammar $G=(N,\Sigma,P,S)$ in normal form and $\balpha\in(N\cup\Sigma)^*$, the supremum $\bigvee L(\balpha)$ is computable and one of the following cases holds:
- $\bigvee L(\balpha)=u$ for some finite $u\in\Sigma^*$, and $u\in L(\balpha)$;
- $\bigvee L(\balpha)=uv^\omega$ for some finite $u\in\Sigma^*$ and $v\in\Sigma^+$, and (of course) $uv^\omega\notin L(\balpha)$.
We already established $\bigvee L(X)=u_X^\omega$ when $X$ is a recursive nonterminal and that $\bigvee L(a)=a\in L(a)$ for terminals $a\in \Sigma$.
Also, for any $\balpha=X_1X_2\ldots X_n\in (N\cup\Sigma)^+$ we can compute $\bigvee L(\balpha)$ with the recursion $$\begin{aligned}
\bigvee L(X_1\ldots X_n) &=\begin{cases}
\varepsilon&\hbox{if }n=0\\
\bigvee(X_1)&\hbox{if }n>0\hbox{ and }\bigvee X_1=uv^\omega\hbox{ for some }u\in\Sigma^*,v\in\Sigma^+\\
u\cdot\bigvee L(X_2\ldots X_n)&\hbox{if }n>0\hbox{ and }\bigvee X_1=u\in\Sigma^*
\end{cases}
\end{aligned}$$ using Corollary \[cor-strict\].
Then, if $X=S$ is a nonrecursive nonterminal and $X~\to~\balpha_1~|~\balpha_2~|\ldots|~\balpha_n$ are all the alternatives for $X$, then we have $\bigvee L(X)=\bigvee\limits_{i=1}^n L(\balpha_i)$, which yields an inductive proof for the only possible nonrecursive nonterminal $S$.
Concluding the subsection, we show the following technical lemma:
\[lem-transducer\] It is decidable for any context-free language $L\subseteq\Sigma^*$ and words $u,v$, whether there exists an integer $N\geq 0$ such that $uv^N\Sigma^*\cap L~=~\emptyset$. (If so, then $uv^M\Sigma^*\cap L=\emptyset$ for each $M\geq N$.)
Let us define the following generalized sequential mappings $f,g:\Sigma^*\to a^*$: let $$\begin{aligned}
f(x)&=\begin{cases}
g(y)&\hbox{if }x=uy\\
\varepsilon&\hbox{otherwise,}
\end{cases}
&
g(x)&=\begin{cases}
a\cdot g(y)&\hbox{if }x=vy\\
\varepsilon&\hbox{otherwise.}
\end{cases}
\end{aligned}$$ We have that if $uv^N\Sigma^*\cap L$ is nonempty, then $f(L)$ contains some word of length at least $N$, and also, if $a^N\in f(L)$, then $uv^N\Sigma^*\cap L$ is nonempty. Thus, there is such an integer $N$ satisfying the condition of the lemma if and only if $f(L)$ is finite, which is decidable, since the class of context-free languages is effectively closed under generalized sequential mappings [@Ginsburg:1966:MTC:1102023].
The order type of recursive nonterminals
----------------------------------------
In this subsection we show that $o(X)$ is computable, whenever $X$ is a recursive nonterminal of an ordinal grammar $G=(N,\Sigma,P,S)$.
Clearly, for each $a\in\Sigma$ we have $o(L(a))=1$. We will apply induction on the *height* of $X$, defined as the length of the longest chain $X_1\prec X_2\prec\ldots\prec X_n=X$ with each $X_i$ in $N\cup\Sigma$. (Thus, the height of the terminals is $0$, nonterminals have positive height.)
Since $X$ is a recursive nonterminal, by Lemma \[lem-u0\] there is a (shortest, computable) nonempty word $u_X$ such that
1. $w<_su_X^\omega$ for each $w\in L(X)$;
2. whenever $X\Rightarrow^+uX\balpha$ for some $u\in\Sigma^*$ and $\balpha\in(N\cup\Sigma)^*$, then $u\in u_X^+$.
This also implies that whenever $X$ and $Y$ are nonterminals belonging to the same component, then there is a unique word $u_{(X,Y)}<_pu_X$ such that $u_X^\omega = u_{(X,Y)}u_Y^\omega$. Moreover we have:
\[prop-betakisebb\] If $Y\to\bbeta$ is an escaping production for $X\approx Y$, then $u_{(X,Y)}\cdot L(\bbeta)<_su_X^\omega$.
In this case, $X\Rightarrow^+u_{(X,Y)}Y\balpha$ for some sentential form $\balpha$. Since $L(\bbeta)\subseteq L(Y)<_s u_Y^\omega$, we get $u_{(X,Y)}\cdot L(\bbeta)<_s u_{(X,Y)}u_Y^\omega=u_X^\omega$.
Now by Lemma \[lem-normal-form\] we can deduce that any (leftmost) derivation from $X$ has the form $$\begin{aligned}
\label{eq-leftmost}
X&\Rightarrow~u_1X_1\balpha_1~\Rightarrow~u_1u_2X_2\balpha_2\balpha_1~\Rightarrow~\ldots\\&\Rightarrow~u_1u_2\ldots u_nX_n\balpha_n\ldots\balpha_2\balpha_1~\Rightarrow~u_1u_2\ldots u_n\bbeta\balpha_n\ldots\balpha_2\balpha_1
~\Rightarrow^*~w\nonumber\end{aligned}$$ for some integer $n\geq 0$, nonempty words $u_1,\ldots,u_n\in\Sigma^+$ with $u_1\ldots u_n<_pu_X^\omega$, sentential forms $\balpha_1,\ldots,\balpha_n, \bbeta \in(N\cup\Sigma)^*$ with $\bbeta\prec X$, $X_i\approx X$ and $\balpha_i\prec X$ for each $i\in[n]$.
By induction, $o(\bbeta)$ is computable (applying Lemma \[lem-product\]) for each possible $\bbeta\prec X_i$ with $X_i\approx X$ and production $X_i\to \bbeta$. Moreover, $o(\balpha)$ is also computable for each $\balpha\prec X_i$ with a production $X_i\to u_iX_{i+1}\balpha$, $X_{i+1}\approx X_i$ as there are only finitely many such productions.
Let $v_1<_s v_2<_s\ldots<_sv_\ell$ be the complete enumeration of those words $v_i$ with $v_i<_s u_X$ having the form $v_i=ua$ with $u<_pu_X$.
Observe that $L=L(X)$ is the disjoint union of languages of the form $u_X^Nv_i\Sigma^*~\cap~L(X)$, with $N\geq 0$ and $1\leq i\leq \ell$. Moreover, whenever $u\in u_X^Nv_i\Sigma^*$ and $v\in u_X^Mv_j\Sigma^*$, then $N<M$ or ($N=M$ and $i<j$) implies $u<_sv$. Thus, these languages form an $\omega$-sequence with respect to the lexicographic ordering and we can write $L$ as $$L~=~L_1+L_2+L_3+\ldots$$ We will construct an increasing sequence of ordinals $$o_1~\leq~o_2~\leq~o_3~\leq~\ldots$$ such that the following hold:
- for each $i\geq 1$, there is a $j\geq 1$ with $o(L_i)\leq o_j$ and
- for each $j\geq 1$, there is an $i\geq 1$ with $o_j\leq o(L_i)$.
This implies $o(L)~=~o_1~+~o_2~+~o_3~+~\ldots$. Indeed: by the first condition we have $$o(L) ~=~o(L_1)~+~o(L_2)~+~\ldots\\
~\leq~o_{f(1)}~+~o_{f(2)}~+~\ldots$$ for some indices $f(1)$, $f(2)$ and so on. Let us define for each $j$ the index $g(j)$ as follows: $g(1)=f(1)$ and for each $j>1$, let $g(j)=\max\{g(j-1)+1,f(j)\}$. Then we have $o(L)\leq o_{g(1)}+o_{g(2)}+\ldots$ and $g(1)<g(2)<\ldots$. Thus, $o(L)\leq o_1+o_2+\ldots$ holds (as the former order type is a sub-order type of the latter), the other direction being symmetric.
Let us now consider one such language $L_t$. Then, $L_t$ is a finite union of languages of the form $$\begin{aligned}
\label{eqn-theyhavethisform}
u_1u_2\ldots u_nL'L(\balpha_n)\ldots L(\balpha_2)L(\balpha_1)\end{aligned}$$ where $u_1\ldots u_n<_p (u_X)^N$ for some $N$ depending only on $t$, moreover, applying Proposition \[prop-betakisebb\] we get that each such $L'$ has the form $\bigl((u_1\ldots u_n)^{-1}u_X^Nv_j\bigr)\Sigma^*~\cap~L(\bbeta)~=~u_{X'}^{M}v\Sigma^*~\cap~L(\bbeta)$, and for each $i\geq 0$ there is a production of the form $X_i\to u_iX_{i+1}\balpha_i$ (recall that due to the normal form each $u_i$ is nonempty) for some nonterminals $X_i\approx X$, $X_1=X$ and $X_{n+1}\to\bbeta$ with $\bbeta\prec X$. Clearly, for any fixed $N$ and $v_i$, there are only finitely many such choices.
We do not have to explicitly compute the order type of each such $L_t$, due to the following lemma:
\[lem-uniodeg\] Assume $o_1\leq o_2\leq \ldots$ is a sequence of ordinals and $K$, $L$ are languages with $\deg(o(L))$, $\deg(o(K))<\deg\bigl(\bigvee o_i\bigr)$. Then $o(K\cup L)<o_j$ for some index $j$.
Without loss of generality, let $o(L)\leq o(K)$. By Theorem \[thm-union\] we have that $o(K\cup L)<\omega^{\deg(o(K))}\times T$ for some integer $T$. It suffices to show that for each integer $T>0$, there exists an $o_i$ with $o_i> \omega^{\deg(o(K))}\times T$. Assume to the contrary that each $o_i$ is at most $\omega^{\deg(o(K))}\times T'$ for some integer $T'$. But then, $\bigvee o_i\leq \omega^{\deg(o(K))}\times T'$ and thus $\deg(\bigvee o_i)\leq \deg(o(K))$, a contradiction.
Equipped by our lemmas we are ready to prove the (technically most involved) main result of the subsection:
\[thm-computable-recursive-themingeszishere\] Assume $G$ is an ordinal grammar in normal form and $X$ is a recursive nonterminal. Let $o_\balpha$ be the maximal order type of some $L(\balpha)$ for which a component production of the form $X'\to uX''\balpha$ exists in $G$ for some $X\approx X'$, and $o_\bbeta$ be the maximal order type of some $L(\bbeta)$ with $X'\to \bbeta$ being an escaping production of $G$ with $X'\approx X$.
Then the order type of $L(X)$ is:
1. ${(o_\balpha)}^\omega$ if $o_\bbeta<{( o_\balpha)}^\omega$;
2. $o_\bbeta$ if $o_\bbeta=\omega^{\deg(o_\bbeta)}$ and for each escaping production $X'\to\bbeta$ with $\deg(o(L(\bbeta)))=\deg(o_\bbeta)$, the language $u_{X'}^N\Sigma^*\cap L(\bbeta)$ is nonempty for infinitely many integers $N\geq 0$;
3. $o_\bbeta\times\omega$, otherwise.
So let $o_\balpha$ be the ordinal $\max\{o(L(\balpha)):~X'\to uX''\balpha\hbox{ is a production for some }X'\approx X''\approx X\}$. Since there are only finitely many such $\balpha$, and $\balpha\prec X$ holds for each of them, $o_\balpha$ is well-defined and computable by induction.
Also, let $o_\bbeta$ be $\max\{o(L(\bbeta)):~X'\to \bbeta\hbox{ is a production for some }X'\approx X,\bbeta\prec X\}$. This ordinal $o_\bbeta$ is well-defined and computable as well.
We also use the shorthands $\gamma=\deg(o_\balpha)$ and $\delta=\deg(o_\bbeta)$. These ordinals are also computable (as an ordinal “being computable” means in our context that the Cantor normal form of the ordinal is computable).
Now we apply a case analysis, based on $\delta$ and $\gamma$. We note next to these (sub, subsub)cases to which case of the theorem they correspond.
Case 1: $\delta<\gamma\times\omega$ {#case-1-deltagammatimesomega .unnumbered}
-----------------------------------
This case corresponds to Case $1$ of the theorem. We claim that in this case $o(X)={(o_\balpha)}^\omega$. To see this, it suffices to show for each integer $N\geq 0$ that ${(o_\balpha)}^N<o(X)$ and that there is an $L_i$ with $o(L_i)<{(o_\balpha)}^N$.
For ${(o_\balpha)}^N<o(X)$, let $X'\to uX''\balpha$ be a component production with $o(L(\balpha))=o_\balpha$ and let $u_0,v_0,u_1,v_1\in\Sigma^*$ be so that $X''\Rightarrow^* u_1X'v_1$ and $X\Rightarrow^* u_0X'v_0$. Finally, let $w\in L(X')$. Then we have $$X\Rightarrow^*u_0(uu_1)^Nw(v_1\balpha)^Nv_0.$$ Since by Lemma \[lem-product\] the order type of the language generated by this sentential form is at least ${(o_\balpha)}^N$, and this language is a subset of $L(X)$, this direction is proved.
For the other direction, note that $\deg({(o_\balpha)}^\omega)=\gamma\times\omega$. Thus, since each $L_i$ is a finite union of languages of the form \[eqn-theyhavethisform\], in which $L'\subseteq L(\bbeta)$ for some $\bbeta$, by Lemma \[lem-uniodeg\] it suffices to show that $$\deg(o(u_1\ldots u_nL(\bbeta)L(\balpha_n)L(\balpha_{n-1})\ldots L(\balpha_1)))<\gamma\times\omega.$$ But, as each $o(\balpha_i)$ is at most $o_\balpha$ and $o(\bbeta)\leq o_\bbeta$, we get that this sentential form has the order type at most ${(o_\balpha)}^n\times o_\bbeta$.
We have that $\deg({(o_\balpha)}^n\times o_\bbeta)=\gamma\times n+\delta$ which is smaller than $\gamma\times\omega$ if so is $\delta$ and the claim is proved.
Case 2: $\gamma\times\omega\leq \delta$ {#case-2-gammatimesomegaleq-delta .unnumbered}
---------------------------------------
Observe that this case applies if and only if $\deg(\gamma)<\deg(\delta)$ and that this cannot happen within Case $1$ of the theorem. We split the analysis of this case to several subcases. For each escaping production $X'\to\bbeta$ with $\deg(o(\bbeta))=\delta$, we decide whether there exists an $N\geq 0$ such that $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$. By Lemma \[lem-transducer\], this is decidable.
### Subcase 2.1: $\gamma\times\omega\leq\delta$ and there exists a $\bbeta$ such that $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$ for some $N$ {#subcase-2.1-gammatimesomegaleqdelta-and-there-exists-a-bbeta-such-that-u_xnsigmacap-lbbetaemptyset-for-some-n .unnumbered}
This subcase rules out Case $2$ of the theorem by the condition $u_{X'}^N\Sigma^*\cap L(\bbeta)~=~\emptyset$, so this subcase falls under Case $3$ of the theorem, and we claim $o(X)=o_\bbeta\times\omega$ in this subcase.
In this subcase, $L(\bbeta)$ is a finite union of languages of the form $K_{N,v}~=~u_{X'}^Nv\Sigma^*~\cap~L(\bbeta)$ for some word $v=v'a<_su_{X'}$ with $v'<_p u_{X'}$ (see Figure \[fig-tree\]). Thus, there is one $K_{N,v}$ among these languages with $\deg(o(K_{N,v}))=\delta$ (since the degree of this finite union is $\delta$). Such a language is a subset of a factor $L'$ of a language of the form (\[eqn-theyhavethisform\]), moreover, such an $L'$ occurs as a factor in infinitely many languages $L_i$: if $X'\Rightarrow^+ u_{X'}^tX'\boldsymbol{\alpha}$, and $K_{N,v}$ is a subset of one of the languages $L'$ belonging to $L_i$, then it also belongs to the same factor $L'$ of $L_{i+t}$. Hence, we have the lower bound $\omega^\delta\times\omega=\omega^{\delta+1}={o_\bbeta}\times\omega\leq o(X)$.
To see that this is an upper bound as well, it suffices to show that each language of the form (\[eqn-theyhavethisform\]) has an order type less than $o_\bbeta\times\omega$, that is, has a degree at most $\delta$. Again, similarly to Case 1 we get that the order type of such a language is upperbounded by ${(o_\balpha)}^n\times o_\bbeta$ whose degree is $\gamma\times n+\delta$ which is $\delta$ since the degree of $\gamma$ is smaller than the degree of $\delta$. (In this case it can happen that $o_\balpha<\omega$ but for finite powers, $\deg(\alpha^n)=\deg(\alpha)\times n$ still holds.)
Thus, in this subcase the order type of $L(X)$ is $o_\bbeta\times\omega$.
(0,10) node\[anchor=south\][$X$]{} – (3,7) node\[anchor=north\] – (-3,7) node\[anchor=south\] – cycle; (-3,7) – (0,7) node \[black,midway,yshift=-0.6cm\] [$u^M_Xu_{X,X'}$]{}; (0,7) – (3,7) node \[black,midway,yshift=-0.6cm\] [$\balpha_m\ldots\balpha_1$]{}; (0,7) node\[anchor=south\][$X'$]{} – (1.5,4) node\[anchor=north\] – (-1.5,4) node\[anchor=south\] – cycle; (-0.5,6) – node\[anchor=north\][$\bbeta$]{} (0.5,6); (-1.5,4) – (1,4) node \[black,midway,yshift=-0.6cm\] [$(u_{X'}^N)v$]{};
### Subcase 2.2: $\gamma\times\omega\leq\delta$ and for all $\bbeta$ and $N$, $u_{X'}^N\Sigma^*\cap L(\bbeta)~\neq~\emptyset$ {#subcase-2.2-gammatimesomegaleqdelta-and-for-all-bbeta-and-n-u_xnsigmacap-lbbetaneqemptyset .unnumbered}
In this subcase, the order type of each such $\bbeta$ can be written as an infinite sum of nonempty ordinals $o_\bbeta=o_{\bbeta_1}+o_{\bbeta_2}+\ldots$, $L(\bbeta)$ being the ordered disjoint union of the nonempty languages $K_{N,v}$. Now again, we have two subsubcases: either $o_\bbeta=\omega^\delta$ (this subsubcase corresponds to Case $2$ of the theorem) or $o_\bbeta>\omega^\delta$ (which in turn falls under Case $3$ of the theorem as well).
[**If $o_\bbeta=\omega^\delta$**]{}, then the degree of each such $o_{\bbeta_i}$ is strictly smaller than $\delta$. In this case, each language of the form (\[eqn-theyhavethisform\]) has an order type at most ${(o_\balpha)}^n\times o$ for some $o$ with $\deg(o)=\delta'<\delta$, the degree of which ordinal is $\gamma\times n+\delta'$. Since $\deg(\gamma\times n)<\deg(\delta)$, we have $\gamma\times n+\delta'<\gamma\times n+\delta=\delta$, thus each such language $L_i$ has a degree still strictly smaller than $\delta$. Thus, $o_\bbeta=\omega^\delta$ is an upper bound for $o(X)$ in this case. Since $o(\bbeta)$ occurs as a subordering in $o(X)$, we also have $o_\bbeta\leq o(X)$, thus $o(X)=o_\bbeta$ in this subsubcase.
[**If $o_\bbeta>\omega^\delta$**]{}, then there exists an $o_{\bbeta_i}$ with degree $\delta$. Proceeding with the argument exactly as in Subcase 2.1, we get that $o(L)=o_\bbeta\times\omega$ in this subsubcase.
Thus in particular, as each condition is decidable if the order types $o(\bbeta)$ and $o(\balpha)$ are computable, which are, applying the induction hypothesis, we get decidability:
\[thm-recursive\] Assume $G$ is an ordinal grammar in normal form and $X$ is a recursive nonterminal.
Then $o(X)$ is computable.
The order type of nonrecursive nonterminals
-------------------------------------------
Recall that if $G$ is an ordinal grammar in normal form, then its only nonrecursive nonterminal can be its starting symbol $S$. Thus, if $\balpha_1,\ldots,\balpha_n$ are all the alternatives of $S$, then $L(G)=\mathop\bigcup\limits_{i=1}^nL(\balpha_i)$ and all the $\balpha_i$s consist of terminal symbols and recursive nonterminals, whose order type is already known to be computable.
Hence we only have to show that the following problem is computable:
- [**Input:**]{} An ordinal grammar $G=(N,\Sigma,P,S)$ (in normal form), and a finite set $\{\balpha_1,\ldots,\balpha_n\}$ of sentential forms such that for each symbol $X$ occurring in the set, $o(X)$ is known.
- [**Output:**]{} The order type of $L=\mathop\bigcup\limits_{i=1}^nL(\balpha_i)$.
We claim that the following algorithm $A$ solves this problem:
``` {mathescape="true" style="myScalastyle"}
function $A(\{\balpha_1,\ldots,\balpha_n\})$
if( $n$ == $0$ ) return $0$
$\Right$ := $\{\balpha_1,\ldots,\balpha_n\}$
$\Left$ := $\emptyset$
$u$ := $\varepsilon$
while( true ) {
$w$ := $\max\{\bigvee L(\balpha) :\balpha\in\mathrm{Right} \}$
$\Right_1$ := $\{\balpha\in\Right:~\bigvee L(\balpha)<w\}$
$\Right_2$ := $\{\balpha\in\Right:~\bigvee L(\balpha)=w\}$
$o$ := $\max\{o(L(\balpha)):\balpha\in\mathrm{Right}_2\}$
if( $o=\omega^\gamma$ for some $\gamma$ )
Let $w'$ be a finite prefix of $w$ such that for each $\balpha\in\Right_1$, $L(\balpha)<w'$ already holds.
return $A(\mathrm{Left})+A\Bigl(\Right_1\cup\bigl\{(\balpha^{<{w'}}):\balpha\in\Right_2\bigr\}\Bigr)~+~\omega^\gamma$
Let $a$ be the largest letter of $\Sigma$ such that there exists some $a\balpha\in\mathrm{Right}$
$\mathrm{Left}$ := $\mathrm{Left}\cup\{u\cdot\balpha:\balpha\in\mathrm{Right},~\mathrm{First}(\balpha)\neq a\}$
$\mathrm{Right}$ := $a^{-1}\mathrm{Right}$
$u$ := $u\cdot a$
$\mathrm{Right}$ := $\{\boldsymbol{\delta}\balpha':\exists X\to\boldsymbol{\delta}\in P,X\balpha'\in\mathrm{Right}\}~\cup~\{\balpha:\alpha\in\mathrm{Right},~\mathrm{First}(\balpha)\notin N\}$.
}
```
In the above algorithm, for a sentential form $\balpha=X\cdot\balpha'$, $\mathrm{First}(\balpha)=X$ and $\mathrm{First}(\varepsilon)=\varepsilon$.
We use induction on $o(L)$ to show that the algorithm always terminates, and it does so with the right answer. Since $G$ is in normal form, we can restrict the proof to those cases when each $\balpha_i$ is either $\varepsilon$ or starts with a terminal symbol.
If this order type is $0$, then (since each nonterminal is productive as $G$ is in normal form) $n=0$ has to hold, in which case the algorithm indeed returns $0$. Now assume $o(L)>0$, thus $n>0$.
For the sake of convenience, let $L(\mathrm{Left})$ stand for the language $\bigcup_{\bbeta\in\mathrm{Left}}L(\bbeta)$ and similarly for $L(\Right)$. We claim that the following invariants are preserved in the loop of the algorithm: $$\begin{aligned}
L(\Left)&< u&
&\hbox{and}&
L&=L(\Left)~\cup~u\cdot L(\Right).\end{aligned}$$ Also, $\Right\neq\emptyset$ and after each execution of Line 7, $u\cdot w=\bigvee L$ .
Upon entering the loop, $\Left=\emptyset$ and from $u=\varepsilon$ we have $u\cdot L(\Right)=L(\Right)=L$. Within the loop, if $L=L(\Left)\cup u\cdot L(\Right)$ and $L(\Left)< u$ before executing Line 7, then $\bigvee L~=~\bigvee \bigl(u\cdot L(\Right)\bigr)~=~u\cdot \bigvee L(\Right)~=~u\cdot\max\{\bigvee L(\balpha):\balpha\in\Right\}$, thus indeed, $u\cdot w=\bigvee L$.
Now assuming $L(\Left)< u$ holds when we start an iteration of the loop, we have to see that $L(\Left)\cup u\cdot\bigl(\bigcup L(\balpha):\balpha\in\Right,\First(\balpha)\neq a\bigr)<u\cdot a$ for the letter $a$ chosen during Line 14. The part $L(\Left)<u<u\cdot a$ is clear. The latter part is equivalent to $L(\balpha)<a$ holds for each $\balpha\in\Right$ with $\First(\balpha)\neq a$, which holds since if such an $\balpha$ begins with a terminal symbol $b$ then by the choice of $a$ we have $b<a$, and if $\balpha=\varepsilon$, then also $\varepsilon<a$, showing preservation of the property $L(\Left)<u$. It is also clear that the operation in Line 16 can’t make $\Right$ empty by the choice of $a$ (also, since $\Right$ is nonempty and by assumption, each $\balpha\in\Right$ begins with a terminal symbol, such a letter $a$ always exists: if $\Right=\{\varepsilon\}$, then $o=1=\omega^0$ and the algorithm terminates at Line 13).
Assuming $L=L(\Left)~\cup~u\cdot L(\Right)$ when starting an iteration, after executing Line 17 we have to show that $L=L(\Left)\cup\{u\cdot L(\balpha):\balpha\in\Right,\First(\balpha)\neq a\}~\cup~u\cdot a\cdot L(a^{-1}\Right)$ for the original values of $\Left$ and $\Right$ to see preservance of this property. But this clearly holds for arbitrary set of sentential forms $\Left$ and $\Right$, thus this property is again a loop invariant.
After executing Line 16, it may happen that $\Right$ contains some sentential form(s) starting with a nonterminal; executing Line 18 does not change $L(\Right)$ but restores the property of $\Right$ that each $\balpha\in\Right$ begins with a terminal symbol (or $\balpha=\varepsilon$).
Now by the first two properties we have $o(L)~=~o(L(\Left))+o(u\cdot L(\Right))~=~o(L(\Left))+o(L(\Right))$.
We show that this is exactly the ordinal we return in Line 13, should the condition of Line 11 hold. Consider the sets $\Right_1$ and $\Right_2$ of sentential forms. By the definition of $w$, $\Right_2$ is nonempty and $\Right=\Right_1\uplus\Right_2$. By the choice of $w'$, we have that $L(\Right_1)<w'$ and of course $L(\Right_2)=L({\Right_2}^{<w'})+L({\Right_2}^{\geq w'})$, thus $$L~=L(\Left)+\Bigl(L(\Right_1)\cup L({\Right_2}^{<w'})\Bigr)+L({\Right_2}^{\geq w'}).$$ Observe that such a $w'$ is computable as $w$ is a computable word (possibly having the form $xy^\omega$ for some finite words $x,y$ by Corollary \[cor-sup-computable\]), so its prefixes can be enumerated and for each prefix $w_0$, the emptiness of the context-free language $L(\balpha^{\geq w_0})$ can be decided for each $\balpha\in\Right_1$; as for these strings $\balpha$ we have $\bigvee L(\balpha)<w$, there is a finite prefix $w_0$ of $w$ with $L(\balpha)$ being already smaller than $w_0$. Thus, even the shortest such prefix $w'$ of $w$ can be computed.
Since $L({\Right_2}^{\geq w'})$ is nonempty (as $w'<w=\bigvee L(\Right_2)$) we get that $o(L(\Left))$ and $o\Bigl(L(\Right_1)\cup L({\Right_2}^{<w'})\Bigr)$ are both strictly smaller than $o(L)$, thus applying the induction hypothesis we get that the algorithm terminates with a correct answer in Line 13 if $o(L({\Right_2}^{\geq w'}))=\omega^\gamma$. Since each nonempty suffix of $\omega^\gamma$ is itself, and $\omega^\gamma$ is the order type of at least one $L(\balpha)$ with $\balpha\in\Right_2$ by the choice of $o$, we have $o(\balpha^{\geq w'})=\omega^\gamma$, thus $\omega^\gamma\leq o(L({\Right_2}^{\geq w'}))$. For the lower bound, note that $L({\Right_2}^{\geq w'})$ is a finite union of languages $L_i$ such that for each $i$, $\bigvee L_i=w$ and $o(L_i)\leq \omega^\gamma$. If $\gamma=0$, then all these languages are singletons containing the word $w$ and the claim holds. Otherwise, none of the languages $L_i$ have a largest element and so for any word $v\in L({\Right_2}^{\geq w'})$ we have $o({L_i}^{< v})<\omega^\gamma$ (by that $v<\bigvee L({\Right_2}^{\geq w'})=\bigvee L_i=w$ and so ${L_i}^{\geq v}$ is nonempty) and so $L({\Right_2}^{<v})$ is a finite union of languages, each having an order type strictly less than $\omega^\gamma$, thus the union itself also has an order type less than $\omega^\gamma$. So each prefix of $o(L(\Right_2))$ is less than $\omega^\gamma$ which makes $o(L(\Right_2))\leq \omega^\gamma$ and the claim holds.
Thus, if the algorithm makes a recursive call in Line 13, then it returns with a correct answer.
We still have to show that the algorithm eventually terminates. To see this, observe that $u\cdot w=\bigvee L$ holds after each iteration of the loop and $u$ gets longer by one letter in each iteration. Hence, if the algorithm does not terminate, then the supremum of the values of the variable $u$ is $\bigvee L$. Since $o(L)\neq 0$, say $o(L)=\omega^{\gamma_1}\times n_1+\ldots +\omega^{\gamma_k}\times n_k$ for some integer $k>0$, integer coefficients $n_i>0$ and ordinals $\gamma_1>\gamma_2>\ldots>\gamma_k$, so there exists some word $x\in L$ with $o(L^{\geq x})=\omega^{\gamma_k}$. Clearly, after some finite number (say, $|x|$) of iterations we have $x<u$, this makes $o(L^{\geq u})=\omega^{\gamma_k}$, and by $u\cdot L(\Right)$ being a nonempty suffix of $L^{\geq u}$, we get that $o(L(\Right))=\omega^{\gamma_k}$: as $L(\Right_2)\subseteq L(\Right)$ is a finite union of languages, we have $o(L(\balpha))\leq \omega^{\gamma_k}$ for each $\balpha\in\Right_2$ and equality holds for at least one of them. Hence, the loop terminates in at most $|x|$ steps, finishing the proof of termination as well.
Theorem \[thm-recursive\], in conjunction with the correctness of the above algorithm yields the main result of the paper:
\[thm-main\] Given an ordinal grammar $G$, one can compute the order type $o(G)$ in Cantor normal form.
Applying the construction of [@DBLP:journals/fuin/BloomE10], we get the following corollary:
\[cor-main\] The Cantor normal form of an algebraic ordinal, given by a finite system of fixed point equations, is effectively computable. Thus, the isomorphism problem of algebraic ordinals is decidable.
Conclusion and acknowledgement
==============================
We have shown that the isomorphism problem of algebraic ordinals is decidable, studying the order types of well-ordered context-free languages, given by an ordinal grammar. It is an interesting question whether the proof can be lifted to scattered linear orders: in many cases, scattered linear orders behave almost as well as well-orders. Also, it would be interesting to analyze the runtime of our algorithm: we only know that by well-founded induction the computation eventually terminates.
The authors wish to thank Prof. Zoltán Fülöp for the discussion on generalized sequential mappings on context-free languages.
| ArXiv |
---
abstract: 'The magnetic nature of Cs$_{2}$AgF$_{4}$, an isoelectronic and isostructural analogue of La$_2$CuO$_4$, is analyzed using density functional calculations. The ground state is found to be ferromagnetic and nearly half metallic. We find strong hybridization of Ag-$d$ and F-$p$ states. Substantial moments reside on the F atoms, which is unusual for the halides and reflects the chemistry of the Ag(II) ions in this compound. This provides the mechanism for ferromagnetism, which we find to be itinerant in character, a result of a Stoner instability enhanced by Hund’s coupling on the F.'
author:
- 'Deepa Kasinathan,$^1$ A. B. Kyker,$^1$ and D. J. Singh$^2$'
title: 'Origin of ferromagnetism in Cs$_2$AgF$_4$: importance of Ag - F covalency'
---
Cs$_2$AgF$_4$ is a member of a family of Ag(II) fluorides that form in perovskite and layered perovskite structures. The distinguishing feature is the presence of Ag(II), which is a powerful oxidizing agent. [@hoppe1; @hoffmann] This compound was first synthesized in 1974 by Odenthal and co-workers. [@hoppe] It occurs in the tetragonal K$_{2}$NiF$_{4}$ layered perovskite structure. This is the same structure as the parent of the high temperature superconducting cuprates, La$_2$CuO$_4$. Cs$_2$AgF$_4$ shows no tilts or rotations of the octahedra, which are common in oxide layered perovskites. Synthesis of isostructural Na$_2$AgF$_4$ and K$_2$AgF$_4$ was also reported and these compounds also have the K$_{2}$NiF$_{4}$ structure. All three compounds are reported as being blue or purple in appearance and ferromagnetic. While transport measurements have not been reported for these compounds, it is known that the related distorted perovskite compound KAgF$_3$ is metallic at high temperatures, and then has a metal insulator transition coincident with an antiferromagnetic ordering temperature. [@g-kagf3]
In the doped high-T$_c$ cuprates, superconductivity develops from a paramagnetic metallic phase, with Fermi surfaces coming from hybridized Cu $d$ - O $p$ bands. These are formally antibonding bands of $d_{x^2-y^2}$ - $p_\sigma$ character. [@pickett] While the theory of high temperature cuprate superconductivity remains to be established, it is widely held that the phenomenon is associated with the physics of the undoped compounds, which are antiferromagnetic Mott insulators. Specifically, it is thought that there is a relationship between superconductivity and the antiferromagnetic fluctuations associated with the correlated $d$ electrons of cuprates. Cs$_2$AgF$_4$ has interesting similarities to the high-T$_c$ cuprates. As mentioned, it is isostructural, featuring AgF$_2$ sheets in place of CuO$_2$ sheets, it has a transition element with a $d^9$ configuration, and it is magnetic. Moreover, related compounds have been shown both in band structure calculations and X-ray photoelectron spectroscopy experiments to display significant Ag - F covalency, reminiscent of the Cu - O hybridization in the cuprates. [@hoffmann; @jaron; @g2] These similarities and other considerations have led to speculations about possible high temperature superconductivity in Ag(II) fluorides. [@hoffmann; @g3] One puzzling difference between the cuprates and the layered Ag(II) fluorides is that the undoped cuprates are antiferromagnetic, while the argentates are ferromagnetic. One possible explanation would be an orbital ordering that favors ferromagnetism within a superexchange framework, as was recently suggested. However, neutron measurements did not detect the symmetry lowering that would occur in this case. [@mclain]
Here we use electronic structure calculations to elucidate the electronic structure of Cs$_2$AgF$_4$ and the origin of its magnetic properties. A previous density functional calculation for this material found it to be a covalent metal,[@hoffmann] with a substantial density of states (DOS) at the Fermi level (E$_{F}$) in the absence of magnetism.
We did electronic structure calculations within the local spin density approximation (LSDA) and the generalized gradient approximation (GGA), [@pw; @pbe] using the general potential linearized augmented planewave method, with local orbitals, [@lapw; @lo] as implemented in the WIEN2K program. [@wien] The augmented planewave plus local orbital extension was used for the Ag $d$ and semicore levels. [@apw] The valence states were treated in a scalar relativistic approximation, while the core states were treated relativistically. Well converged basis set sizes and Brillouin zone samplings were employed. Except as noted otherwise, the LAPW sphere radii were 2.0 $a_0$ and 1.85 $a_0$ for the metal and fluorine atoms, respectively. The basis set cut-off was chosen to be $RK_{max}$=7.0, where $R$ is the radius of the F sphere. We tested the convergence by comparison of LSDA results with an independent code, employing the LAPW augmentation with local orbitals and with higher basis set cut-offs as well as different sphere radii.
The structural data were obtained from the report[@hoppe] of Odenthal and co-workers: $a$ = 4.58Å, $c$ = 14.19Å, including the two internal parameters corresponding to the Cs and apical F heights above the AgF$_2$ square planar sheets. Minimization of the forces in the LDA approximation yielded a value of z$_{\rm Cs}$=0.361 and z$_{\rm F}$=0.147, in close agreement with the experimental values of z$_{\rm Cs}$=0.36 and z$_{\rm F}$=0.15.
Within the LSDA we find a Cs$_2$AgF$_4$ to be a metal on the borderline of ferromagnetism. Fixed spin moment calculations showed a non-spin-polarized ground state, but with a 1 $\mu_B$ per formula unit fully polarized solution only 35 meV higher in energy. We also did LSDA calculations applying fields only inside the Ag LAPW spheres, which were chosen to be 2.1 $a_0$ in radius for this purpose. With 5 mRy fields of this type in a ferromagnetic pattern, moments of 0.35 $\mu_B$ were induced in the Ag spheres, and moments also appeared in the F spheres, for a total spin magnetization of 0.62 $\mu_B$. Application of the same field in an in-plane $c$(2x2) antiferromagnetic pattern yielded induced Ag moments in the spheres of only 0.17 $\mu_B$, with a small moment also appearing on the apical F, but no moments on the in-plane F, as is required by symmetry. This shows the system to be much closer to ferromagnetism than antiferromagnetism at the LSDA level, and suggests an important role for the in-plane F in the magnetism.
Within the GGA, we obtain a ferromagnetic ground state, with spin magnetization, $M=0.9 \mu_B$ and energy 6 meV below the non-spin polarized solution. However, we do not find any metastable antiferromagnetic solution, implying itinerant magnetism, in particular, the absence of stable local moments. The calculated electronic density of states (DOS) for the ferromagnetic ground state is shown in Fig. \[dos\]. The band structure is shown in Fig. \[bands\], and the Fermi surface in Fig. \[fermi\]. The band structure is expected to be two dimensional, due to the bonding topology, which has 180$^\circ$ Ag-F-Ag links in the AgF$_2$ sheets, but no direct Ag-F-Ag connections in the $c$-axis direction. This in fact is the case. [@disp-note] As may be seen, Cs$_2$AgF$_4$ is close to a half metal, with the Fermi energy being near a band edge in the majority channel, but not in the minority channel. The minority spin Fermi surface consists of small hole cylinders running along the zone corner (from the $d_{x^2-y^2}$ band) and electron cylinders around the zone center i(from the $d_{z^2}$ band). The majority spin Fermi surface consists of a single large square cylindrical electron surface that almost fills the Brillouin zone, leaving a small region of holes around the zone boundary.
Cs$_{2}$AgF$_{4}$ has two type of F sites forming distorted Ag centered octahedra; one is in the AgF$_2$ sheets (referred as F1 in this paper), and the other is the apical F along the $c$ - axis (referred as F2 in this paper). The apical Ag - F2 distance is slightly smaller than the in-plane Ag - F1 distance. A key point is that the F1 atoms bridge the Ag atoms in the sheets, with 180$^\circ$ bonds, while the apical, F2 atoms connect to only one Ag atom and therefore are not bridging.
Examining the DOS and projections in more detail, one may note that the valence bands have substantially mixed Ag $d$ - F $p$ character, especially near the bottom and top of the manifold where $e_g$ - $p_\sigma$ bonding and antibonding combinations occur. This hybridization involves both F1 and F2 atoms, and is consistent with previous results for Ag(II) fluorides. [@hoffmann; @jaron; @g2] The result is a very stable metallic electronic structure, with substantial F character at the Fermi energy, $E_F$, and a valence band width of $\sim$ 5.5 eV. This strong hybridization can be understood in chemical terms considering the very strongly oxidizing character of Ag(II), which in this compound partially oxidizes F$^{-}$. Thus covalency in this compound is a consequence of the unusual valence state of Ag. Turning to the band structure, there are two bands crossing $E_F$ in the minority spin channel. These are the $d_{x^2-y^2}$, which hybridizes with the in-plane F, and the $d_{z^2}$ hybridized with the apical F. The $d_{x^2-y^2}$ - F1 combination has greater dispersion because of the in-plane topology, mentioned above. However, the $d_{z^2}$ - F2 combination is higher lying, with the result that the two band maxima nearly coincide. The higher lying position of the $d_{z^2}$ - F2 is readily explained by the fact that these bands are antibonding $e_g$ - $p_\sigma$ and the Ag - F2 bond is shorter. In the minority spin channel, the $d_{z^2}$ band extends from from -0.5 to 0.7 eV (relative to $E_F$), while the $d_{x^2-y^2}$ band extends from -2.1 to 0.5 eV. In the absence of the lighter $d_{x^2-y^2}$ band, one would have a half-filled $d_{z^2}$ band. Because the $d_{x^2-y^2}$ is in fact present, the $d_{z^2}$ is slightly electron doped away from half filling. This is in contrast to the cuprates, where only a $d_{x^2-y^2}$ band is active, and this band is hole doped away from half-filling in the highest $T_c$ compounds.
The small size of F$^-$ relative to O$^{2-}$ emphasizes the effect of the perovskite bonding topology in the band structure. This is because direct F - F hopping is reduced by its small size, relative to O in oxides, and the strongly oxidizing nature of F and Ag(II) push the Cs conduction bands to high energy, reducing the assisted hopping via Cs for the valence bands. Thus, the hopping is dominated by nearest neighbor Ag-F channels, so for example, the $d_{xz} - p_\pi$ and $d_{yz} - p_\pi$ derived bands take strong one dimensional character and are seen to be almost perfectly flat along some directions as seen in Fig. \[bands\].
The mixed character of the bands is reflected in the distribution of the magnetic moments in the ferromagnetic ground state. Of the total spin moment of 0.9 $\mu_{B}$, only 0.5 $\mu_{B}$ lies within the Ag LAPW sphere, radius 2.0 $a_0$. The remaining $\sim$ 40% of the magnetization is F derived, approximately equally divided between the F1 and F2 sites. This is important for understanding the itinerant ferromagnetic ground state that we find. First of all, the large moments on the in-plane F1 atoms seen both in the GGA ferromagnetic ground state and in the LSDA calculations with ferromagnetic fields in the Ag (but not the F) spheres, mean that there is a contribution to the energy from the F polarization. F$^-$ is a relatively small anion, at the end of the first row of the periodic table. Thus, when magnetic, it can have a strong Hund’s coupling. This provides a generalized double exchange mechanism for favoring ferromagnetism, similar to the mechanism in SrRuO$_3$. [@singh-ru; @mazin-ru] In the ferromagnetic case, the F1 atoms take moments due to the hybridized character of the bands around $E_F$ and contribute to the Stoner instability through their Hund’s coupling. With antiferromagnetic ordering, no induced moments can be present on the F1 atoms by symmetry, and therefore the Hund’s coupling on these sites cannot stabilize the magnetism. Thus, the fact that the moments become unstable in an in-plane antiferromagnetic configuration supports this picture. Different from the ruthenates, the hybridized states in Cs$_2$AgF$_4$ involve $e_g$ instead of $t_{2g}$ states, and the F$^{-}$ ion is much smaller than O$^{2-}$.
We studied the stability of this ferromagnetic, two-band electronic structure, using LDA+U calculations and treating the Coulomb $U$ as a parameter. We found, as expected, that a local moment, insulating state could be obtained. However this only happened when using a very high value $U$=7 eV. This is an unreasonably large value for a 4$d$ ion in a screening environment. The reason for the weak effect of more realistic values of $U$ is that the bands are strongly hybridized, and are really mixed F $p$ - Ag $d$ bands, and not narrow bands built from the Ag $d$ orbitals. Thus we conclude that on-site Coulomb correlations do not have a large effect on the electronic structure or magnetism of this compound. This is in contrast to the undoped cuprates, where the LSDA and GGA approximations incorrectly predict non-magnetic metallic ground states, and the Hubbard $U$ is crucial for obtaining moment formation and an insulating ground state.
To summarize, density functional calculations of the electronic structure of Cs$_2$AgF$_4$ show strong covalency between Ag $d$ and F $p$ states. Within the GGA, the ground state is ferromagnetic, and is stabilized by Hund’s coupling on the in-plane F atoms which occurs due to F participation in the magnetism resulting from the $e_g$ - $p_\sigma$ hybridization. The electronic structure is nearly half-metallic, and not insulating. It would be of interest to experimentally test the prediction of a metallic electronic structure.
The resulting picture of the electronic structure and magnetism is very different from the undoped cuprates. (1) Cs$_2$AgF$_4$ has two active bands: $d_{x^2-y^2}$ and $d_{z^2}$; neither is exactly half-filled; (2) moment formation in Cs$_2$AgF$_4$ is due to a Stoner type mechanism as opposed to on-site Coulomb repulsions that are crucial in the cuprates; (3) the magnetism has strong itinerant character due to F participation, as opposed to the local moment superexchange mediated character of cuprate antiferromagnetism; and (4) we find ferromagnetism with the ideal tetragonal structure; orbital ordering to obtain ferromagnetism within a superexchange mediated framework is not needed. We note that the predicted F contributions to the magnetism are large enough to be detected using neutron scattering.
Finally, we note that the mechanism for ferromagnetism in Cs$_2$AgF$_4$ is quite robust, and would expected to occur in other Ag(II) fluorides with similar bond lengths and topologies. Since it does not rely on small structural effects, it provides a ready explanation for the observed ferromagnetism in the other $A_2$AgF$_4$ compounds. Furthermore, the above picture of itinerant magnetism may be more generally applicable to other Ag(II) fluoride compounds. For example, KAgF$_3$ shows a metal insulator transition coincident with an antiferromagnetic ordering. [@g-kagf3] This is much more natural in an itinerant system than in a strongly correlated local moment system, which would tend to be insulating on both sides of the ordering temperature at odd integer band fillings. The structure of that compound shows compressed octahedra and Ag-F-Ag chains along the $c$-axis direction with short bond lengths. Assuming that the above mechanism applies also to this compound, one may expect ferromagnetic chains along $c$. Considering that the ground state is known to be antiferromagnetic, one may anticipate a C-type ordering of antiferromagnetic $a-b$ planes, stacked ferromagnetically in that case. In any case, in perovskite derived Ag(II) fluorides, the mechanism that we propose would generally favor ferromagnetism or complex antiferromagnetic states, with ferromagnetic interactions along some bonding directions, as opposed to simple G-type ordering.
We are grateful for helpful discussions with W.E. Pickett and J. Turner. Research at ORNL was sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy, under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC. Work at UC Davis was supported by DOE contract DE-FG03-01ER45876.
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This can be seen for the valence bands in Fig. \[bands\] from the $\Gamma -M$ and $X - P$ lines, which are along $k_z$ in the body centered tetragonal zone. The two-dimensional character is also evident in the Fermi surfaces. The higher lying (mainly Cs) derived states above the gap are more three dimensional in character.
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| ArXiv |
[**Soft photon emission as a sign of sharp transition in quark-gluon plasma**]{}[^1]
I.V.Andreev\
\
[**Abstract**]{}
Photon emission arising in the course of transition between the states of quark-gluon and hadron plasma has been considered. Single-photon distributions and two-photon correlations in central rapidity region have been calculated for heavy ion collisions at high energies. It has been found that opposite side two-photon correlations can serve as a sign of sharp transitions between the states of strongly interaction matter.
Introduction
============
Forty years ago E.S.Fradkin and his students had calculated the photon polarization operator in relativistic plasma at finite temperatures [@F]. These results will be used here for estimation of a new specific mechanism of photon production which may appear effective for identification of transitions between the states of quark and hadron matter in heavy ion collisions.
The phenomenon under consideration is the photon production in the course of evolution of strongly interacting matter. Let us consider photons existing in the medium at initial moment $t_0$ having momentum $\bf k$ and energy $\omega_{in}$. Let the properties of the medium (its dielectric penetrability) change within time interval $\delta\tau$ so that the final photon energy is $\omega_f$. As a result of the energy change the production of extra photons with momenta $\pm\bf k$ takes place these photons having specific two-photon correlations. Analogous processes were considered for mesons [@AW; @AC; @A; @ACG] and applied to pion production in high-energy heavy ion collisions [@A1].
The conditions for a strong effect are the following: first, the ratio of the energies ${\omega_{in}/{\omega_f}}$ must not be too close to unity and second, the transition should be fast enough.
Basic formulation
=================
Time evolution of the transverse photon creation and annihilation operators is given by canonical Bogoliubov transformation [@B] which represents solution of the Hamilton equations and contains two modes with momenta $\pm\vk$ : a(,t)=u()a(,0)+v()a\^(-,0),\
a\^(-,t)=v()a(,0)+u()a\^(-,0), \[eq:1\] (polarizations are omitted for a moment). Here Bogoliubov coefficients $u,v$ satisfy equation |u()|\^[2]{}-|v()|\^[2]{}=1 \[eq:2\] preserving canonical commutation relations and the limit $t\rightarrow\infty$ must be taken. Physically the process under consideration is analogous to parametric excitation of quantum oscillators. It was considered in more details earlier [@A]. The Bogoliubov coefficients $u(\vk)$, $v(\vk)$ are taken to be real valued and $k=|\vk|$ dependent. So we use a parametrization u()=r(k), v()=r(k) \[eq:3\] thus introducing evolution parameter $r(k)$.
To get feeling of the main features of the evolution effect (and for further references and comparison) let us formulate a simple model – fast simultaneous break-up of large homogeneous system at rest [@A1]. In this case the resulting single-particle momentum distribution can be written in a simple form =a\^()a()|\_[t]{} =\[eq:4\] (for single polarization) where $V$ is the volume of the system and $n(k)$ is the level occupation number at $t=0$. The first term in [*rhs*]{} of Eq.(4) describes amplification of existed particles and the second term describes the contribution arising due to rearrangement of the ground state of the system in the course of the transition.
The transition effect is better seen in particle correlations.Two-particle inclusive cross-section is given here by =a\^\_[1]{}a\^\_[2]{}a\_[1]{}a\_[2]{}=a\^\_[1]{}a\_[1]{}a\^\_[2]{}a\_[2]{}+a\^\_[1]{}a\_[2]{}a\^\_[2]{}a\_[1]{}+a\^\_[1]{}a\^\_[2]{}a\_[1]{}a\_[2]{}\[eq:5\] The fist term in [*rhs*]{} of Eq.(5) is the product of single-particle distributions, the second term gives the usual Hanbury Brown-Twiss effect (HBT) and the third term is essential if time evolution takes place giving opposite side photon correlations (see below). The correlators in Eq.(5) in the case under consideration have the form: a\^()a()=G(-), \[eq:6\] a()b()=2r(k) G(+) \[eq:7\] where $G(\vko\pm\vkt)$ represents normalized Fourier transform of the source volume at break-up stage $(G(0)=1)$. It is sharply peaked function of $\vko\pm\vkt$ (at zero momentum) having characteristic scale of the order of inverse size of the source, this scale being much less than characteristic scales of photon momentum distribution $n(k)$ and evolution parameter $r(k)$. So the last two functions may be evaluated at any of momenta $\vko,\vkt\approx\pm\vk$ (we suggest that the process is $\vk\to -\vk$ symmetric).
Relative correlation function which is measured in experiment is now given by C(,)=1+G\^[2]{}(-)+R\^[2]{}(k)G\^[2]{}(+) \[eq:8\] with R(k)= \[eq:9\] As it can be seen from Eqs.(8-9), HBT effect is given simply by the form- factor $G(\vko-\vkt)$ in this model whereas the transition effect depends strongly on evolution parameter $r(k)$. In turn $r(k)$ depends on time duration $\delta\tau$ of the transition. For very small characteristic times $\delta\tau$ the expression for $r(k)$ is universal [@AW], r(k)=(), 1 \[eq:10\] where $\omega_{in}$ and $\omega_f$ are particle energies before and after the transition. For larger $\delta\tau$ the evolution parameter lessens. In general we expect that it falls exponentially at large $\omega\delta\tau $ if the time dependence of the energy in the course of transition has no singularities at real times. So for large $\omega\delta\tau$ we shall use an exponentially falling expression motivated by solvable model expression [@A]. Below, after necessary modification, we apply the above consideration to photon production in heavy ion collisions.
Photons in plasma
=================
Spectrum of photons in plasma is given by dispersion equation \^[2]{}\_[k]{}=k\^[2]{}+(\_[k]{},k,T,,m) \[eq:11\] Here $\Pi$ is the polarization operator for transverse photons dependent on temperature $T$, chemical potential $\mu$ and the mass $m$ of charged particles. Below we use an approximate form extracted from original expression [@F]: =\^[2]{}\_[a]{}\[eq:12\] with \^[2]{}\_[a]{}=\_[m/T]{}\^dx (x\^[2]{}-)\^[1/2]{}n\_[F]{}(x,/T) \[eq:13\] where $\alpha=1/137$, $v^2$ is the averaged velocity squared of the charged particles in the plasma, factor $g$ takes into account the number of the particle kinds and their electric charges ($g=5/3$ for $u,d$ quarks) and $n_{F}$ is the occupation number of the charged particles (Fermi distribution). Polarization operator for scalar charged particles is approximately a half of that for fermions with substitution of Bose distribution for Fermi distribution. Evidently the polarization operator plays the role of (momentum dependent) photon termal mass squared $m^{2}_{\gamma}$.
We calculated the polarization operator and photon spectrum for three possible kinds of plasma: quark-gluon plasma (QGP) with $u,d$ light quarks, constituent quark ($m=350 MeV$)-pion plasma and hadronic (pions and nucleons) plasma. Chemical potential (baryonic one) was taken to be equal to $100 MeV$ per quark corresponding to typical value for SPS energies. The temperature was taken to be equal to $140 MeV$ (see below).
The evolution parameter $r(k)$ for photons is determined through photon energy $\omega(k)$. For small momenta $k$ the parameter $r(k)$ is well approximated by simple expression (to be used for $k_{T}<40 MeV$) r(k)== , k1 \[eq:14\] where $m^{2}_{\gamma i}$ are photon termal masses squared at both sides of the transition and $\langle m^{2}_{\gamma}\rangle$ is their average mass squared ,cf Eq.(10). At $k=0$ the values of $\delta m^2_\gamma$ are equal to 289, 178 and 106 (in $MeV^2$ units) for QGP-hadron, QGP-valon and valon-hadron transitions correspondingly. Corresponding values of zero momentum evolution parameter $r(0)$ are 0.330, 0.154 and 0.178.
Higher momentum behaviour of $r(k)$ (to be used for $k_{T}>40 MeV$) is taken in the form r(k)=(-k) \[eq:15\] where $\delta\tau$ is time duration of the transition. The Eq.(15) is a simple version of the expression given by the solvable model [@A] which is sewed together with Eq.(14) giving a monotonically decreasing function of the momentum. Below Eqs.(14-15) will be used for estimation of the transition effect in heavy ion collisions. Only QGP-hadron transition will be calculated. In view of fact that evolution parameter $r(k)$ appeared to be small number at all momenta $k$, all expressios will be taken in the lowest order in $r(k)$.
Transition effect in heavy ion collisions
=========================================
Let us now apply the above consideratins to photon production in heavy ion collisions. Let us suggest that the quark-gluon plasma is formed at the initial stage of the ion collision. Let the plasma undergoes expansion and cooling. The expansion is taken to be longitudinal and boost invariant [@BJ]. Recent lattice calculations [@LAT] show rather low critical temperature of the deconfinement and chiral phase transition, $T_c\approx 150 MeV$ as well as sharp drop of the pressure when the temperature approachs $T_c$ thus provocing instability in the presence of overcooling. So we do not expect long-living mixed phase and consider fast transition from quark to hadron matter with characteristic transition proper time duration $\delta\tau$ of the order of $1 fm/c$.
To calculate the transition effect one must shift to rest frame of each moving element of the system and integrate over proper times and space-time rapidities of the elements. Then single-photon distribution in central rapidity region $y=0$ reads:
$$\begin{aligned}
\frac{dN}{d^{2}k_{T}dy}\Bigl|_{y=0}\Bigr.=I_{QGP}+I^{(1)}_{tr}\nonumber\\
=\int\tau d\tau\int d\eta\int d^{2}x_{T}(p_{0}\frac{dR_{\gamma}}{d^{3}p})
+\int d\eta\int d^{2}x_{T}\frac{2p\tau_{c}}{(2\pi)^{3}}r^{2}(p)
\label{eq:16}\end{aligned}$$
with $p=k_{T}\cosh\eta$.
The first term in rhs of Eq.(16) describes photon production from hot quark-gluon plasma. Here $R_{\gamma}$ is the QGP production rate per unit four-volume in the rest frame of the matter [@KLS]:
$$p_{0}\frac{dR_{\gamma}}{d^{3}p}=\frac{5\alpha\alpha_{s}}{18\pi^{2}}T^{2}
\exp(-p/T)\ln(1+\frac{\kappa p}{T})
\label{eq:17}$$
with $\alpha=1/137$, $\alpha_{s}=0.4$, $\kappa=0,58$. It can be used also for hadron gas as its uncertainity is larger than the difference between the first-order QGP and hadron gas production rates [@NKL]. Contribution from hadronic resonances are not considered here. The second term in [*rhs*]{} of Eq.(16) describes photon production due to transition from QGP to hadrons in the vicinity of proper time $\tau_{c}$, cf Eq.(4). The time duration of the transition is taken to be small in this term in comparison with total time duration of photon production process.
The photon production rate in Eq.(16) can be expressed through photon occupation number $n(k)$: $$p_{0}\frac{dR_{\gamma}}{d^{3}p}=\frac{2k_{T}}{(2\pi)^3}\frac{dn(k)}{d\tau}
\label{eq:18}$$ (with two polarizations included). Taking Eq.(18) into account one can see that if the velocities of the volume elements, as well as proper time interval in the first term in [*rhs*]{} of Eq.(16) are small then Eq.(16) is reduced to Eq.(4) as it should be. The photon occupation number $n(k)$ in Eq.(18) appears to be small numerically, $$n(k)\ll 1$$ That means in particular that transition radiation is dominated not by the photon amplification but by the ground state rearrangement (cf Eqs.(4-8)).
As the last step one must specify temperature evolution. We suggest that the temperature depends on proper time of the volume element with power-like dependence: $$(T/T_{0})=(\tau/\tau_{0})^{-1/b}
\label{eq:19}$$ where $\tau_{0}$ and $T_{0}$ are initial proper time and initial temperature. For final estimation we use $b=3$ typical for hydrodynamical picture and choose low transition temperature $T_{c}=140 MeV$. After transition the photons live some time in hadronic medium and we suggest termal momentum distribution of the hadrons (modified by the expansion of the system). We neglect termal photon production below $T_{c}$ and do not introduce a special freese-out temperature.
Below we will be interested in rather low photon transverse momenta $k_{T}$ (up to $500 MeV$) where transition effect is more pronounced. In this momentum region the QGP production term $I_{QGP}$ depends mainly on final temperature $T_c$. For two main variants of initial conditions used in literature [@NFS]: $\tau_{0}T_{0}=1, T_{0}/T_{c}=3/2{}, \tau_{0}=1 fm/c $ and $\tau_{0}T_{0}=1/3, T_{0}/T_{c}=5/2{}, \tau_{0}=0.2 fm/c$ a variable factor in $I_{QGP}$ changes inessentially (from 5.06 to 4.37 for $b=3$) and we will use for this factor an average value 4.7 in our estimations. The transition proper time $\tau_c$ also changes inessentially for these two variants of initial conditions being $3.00 fm/c$ and $3.02 fm/c$ correspondingly.
The transition contribution $I^{(1)}_{tr}$ in Eq.(16) appears essential only at very small momenta $k_T$. So dealing with single-photon distributions one can use Eq.(14) for evolution parameter $r(k)$. The resulting relative strength of transition radiation $$R_{1}(k_{T})=I^{(1)}_{tr}/I_{QGP}
\label{eq:20}$$ appears sizable only in the momentum region $k_{T}\leq (15-20) Mev$ independently of the time duration of the transition. $R_{1}$ reachs 4.44 at $k_{T}=0$ and falls down to 0.06 at $k_{T}=40 MeV$. Because of high background effects in this momentum region the single- photon transition effect should be difficult to observe experimentally. Mach better the transition effect is seen in photon correlations (cf Eqs.(5-9)) where it is first order effect with respect to $r(k)$. Let us note that HBT effect for photons has now more complicated form than that in Eq.(8) because of finite time duration [@APW] of the process of photon emission from QGP and it will not be exposed here. We consider only the transition effect (the third term in Eqs.(5,8) which gives opposite side correlations) estimating its contribution to two-photon correlation function in central rapidity region. Suggesting fast transition we can evaluate the contribution in the vicinity of fixed proper time $\tau_c$. So one only has to shift to the rest frame of each element of the expanding volume and perform $\eta$-integration. Then the extention of the correlator in Eq.(7) to the case of expanding volume takes the form: $$2k\langle a({\bf k}_{1})a({\bf k}_{2})\rangle=G({\bf k}_{1T}+{\bf k}_{2T})
I^{(2)}_{tr}
\label{eq:21}$$ with $$I^{(2)}_{tr}=\int d^{2}x_{T}\int d\eta \frac{2\tau_{c}k_{T}\cosh\eta}{(2\pi)^3}
r(k_{T}\cosh\eta)
\label{eq:22}$$ where we neglected $n(k)$ in comparison with unity (see above). Therefore the normalized two-photon correlation function is given by (cf Eqs.(8-9)) $$C({\bf k}_{1T},{\bf k}_{2T})|_{y_{1}=y_{2}=0} =1+C_{HBT}
+R^{2}(k_{T})G^{2}({\bf k}_{1T}+{\bf k}_{2T})
\label{eq:23}$$ with $$R(k_{T})=\frac{I^{(2)}_{tr}}{I_{QGP}+I^{(1)}_{tr}}
\label{eq:24}$$
We calculated the ratio $R(k_{T})$ for different transition times $\delta\tau
=0 fm/c,\\ 0.5 fm/c, 1 fm/c$ up to $k_{T}=500 MeV$. In the region $k_{T}<100 MeV$ the ratio $R$ is sizable for all these $\delta\tau$ being equal 4.94 at $k_{T}=0$, reaching maximal value $R\sim 6$ at $k_{T}\sim
20 MeV$ and falling down at $k_{T}=100 MeV$ to $R=1.78$ for $\delta\tau=0$, $R=0.95$ for $\delta\tau=0.5 fm/c$, $R=0.55$ for $\delta\tau=1 fm/c$ . At larger transverse momenta the behaviour of the ratio $R$ depends strongly on transition time $\delta\tau$: in $k_{T}$ interval $(200-500) MeV$ the ratio $R(k_{T})$ rises from 1.3 to 3.9 for $\delta\tau=0$, it is approximately constant ($R$=0.3) for $\delta\tau=0.5 fm/c$ and it decreases from 0.15 to 0.03 for $\delta\tau=1 fm/c$. On the whole it seems that the measurement of $R(k_{T})$ gives a possibility to identify the transition effect, especially if $\delta\tau\leq 0.5 fm/c $
Conclusion
==========
Estimation of photon emission accompaning transition between quark-gluon and hadron states of matter in heavy ion collisions shows that opposite side photon correlations can serve as a sign of the transition if transition time is small enough.
[22]{} E.S.Fradkin, Proceedings of P.N.Lebedev Inst. [**29**]{}, Nauka, Moscow, 1965. I.V.Andreev and R.M.Weiner, [*Phys. Lett.*]{} [**B373**]{} (1996) 159. M.Asakawa and T.Csorgo, [*Heavy Ion Phys.*]{} [**4**]{} (1966) 233. I.V.Andreev, [*Mod. Phys. Lett.*]{} [**A14**]{} (1999) 459. M.Asakawa, T.Csorgo and M.Gyulassy, [*Phys. Rev. Lett.*]{} [**83**]{} (1999) 4019. I.V.Andreev, hep-ph/99-06439; [*Phys. Atomic. Nucl.*]{} [**63**]{} (2000) N.N.Bogoliubov, Proc. Acad. Sci.USSR, Ser. Phys. [**11**]{} (1947) 77. J.D.Bjorken, [*Phys. Rev.*]{} [**D27**]{} (1983) 140. T.Blum et al, [*Phys. Rev.*]{} [**D51**]{} (1995) 5153. J.Kapusta, R.Lichard and D.Seibert, [*Phys. Rev.*]{} [**D44**]{} (1991) 2774. H.Nadeau, J.Kapusta and P.Lichard, [*Phys. Rev.*]{} [**C45**]{} (1992) 3034. J.J.Neumann, G.Fai and D.Seibert, Proceedings of the 4-th Rio de Janeiro Int. Workshop on Relativistic Aspects of Nuclear Physics, eds T.Codama et al, World Scientific, Singapore, 1995. I.V.Andreev, M.Plumer and R.M.Weiner, [*Int. Journ. Mod. Phys.*]{} [**A8**]{} (1993) 4577.
[^1]: Talk presented at E.S.Fradkin memorial conference, Moscow, July 2000.\
This work was supported by Russian Fund for Fundamental Research,\
grant 00-02-16101a
| ArXiv |
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abstract: |
We consider [*monotone*]{} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on $n$ points can be embedded into $l_2^n$, while, (in a sense to be made precise later), for almost every $n$-point metric space, every monotone map must be into a space of dimension $\Omega(n)$ (Lemma \[momad2\]).\
It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on [*sphericity*]{} of graphs, which suggest one example of such a metric space - that defined by a complete bipartite graph. We prove that an $\delta n$-regular graph of order $n$, with bounded diameter has sphericity $\Omega(n/(\lambda_2+1))$, where $\lambda_2$ is the second largest eigenvalue of the adjacency matrix of the graph, and $0 < \delta \leq {\frac 1 2}$ is constant (Theorem \[our-bound\]). We also show that while random graphs have linear sphericity, there are [*quasi-random*]{} graphs of logarithmic sphericity (Lemma \[alex\]).\
For the above bound to be linear, $\lambda_2$ must be constant. We show that if the second eigenvalue of an $n/2$-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only $o(n^2)$ entries (Theorem \[main1\]). Furthermore, for any $0 < \delta < {\frac 1 2}$, and $\lambda_2$, there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$ (Corollary \[no-graphs\]).
author:
- 'Yonatan Bilu and Nati Linial [^1]'
bibliography:
- 'bib.bib'
title: 'Monotone Maps, Sphericity and Bounded Second Eigenvalue'
---
0.2cm
\[section\] \[section\] \[section\] \[section\] \[section\] \[THEOREM\][Problem]{}
=msbm10
[*Keywords:*]{} Embedding, Finite Metric Space, Graphs, Sphericity, Eigenvalues, Bipartite Graphs, Second Eigenvalue.
Introduction
============
Euclidean embeddings of finite metric spaces have been extensively studied, with the aim of finding an embedding that doesn’t distort the metric too much. We refer the reader to the survey papers of Indyk ([@Indyk]) and Linial ([@Nati]), as well as chapter 15 of Matou[š]{}ek’s Discrete Geometry book [@Matousek]. Here we focus on a different type of embeddings. Namely, those that preserve the order relation of the distances. We call such embeddings [*monotone*]{}. There are quite a few applications that make this concept natural and interesting, since there are numerous algorithmic problems whose solution depends only on the order among the distances. Specifically, questions that concern nearest neighbors. The notion of monotone embeddings suggests the following general strategy toward the resolution of such problems. Namely, embed the metric space at hand monotonically into a “nice” space, for which good algorithms are known to solve the problem. Solve the problem in the “nice” space - the same solution applies as well for the original space. “Nice” often means a low dimensional normed space. Thus, we focus on the minimal dimension which permits a monotone embedding.\
In section \[dope\] we observe that any metric on $n$ points can be monotonically embedded into an $n$-dimensional Euclidean space, and that the bound on the dimension is asymptotically tight. The embedding clearly depends only on the order of the distances (Lemma \[dope-lemma\]). We show that for almost every ordering of the ${n \choose 2}$ distances among $n$ points, the host space of a monotone embedding must be $\Omega(n)$-dimensional. Similar bounds are given for embeddings into $l_\infty$, and some bounds are also deduced for other norms.\
Next we consider embeddings that are even less constrained. Given a metric space $(X,\delta)$ and some threshold $t$, we seek a mapping $f$ that only respects this threshold. Namely, $||f(x)-f(y)||<1$ iff $\delta(x,y)<t$. The input to this problem can thus be thought of as a graph (adjacency indicating distances below the threshold $t$). The minimal dimension $d$, such that a graph $G$ can be mapped this way into $l_2^d$ is known as the [*sphericity*]{} of G, and denoted $Sph(G)$. Reiterman, R[ö]{}dl and [Š]{}i[ň]{}ajov[á]{} ([@RRS89a]) show that the sphericity of $K_{n,n}$ is $n$. This is, then, an explicit example of a metric space which requires linear dimension to be monotonically embedded into $l_2$. Other than that, the best lower bounds previously known to us are logarithmic. In section \[prox-graph-sec\] we prove a novel lower bound, namely that for $0 < \delta \leq {\frac 1 2}$, $Sph(G) = \Omega(\frac n {\lambda_2 + 1})$, for any $n$-vertex $\delta n$-regular graph, with bounded diameter. Here $\lambda_2$ is the second largest eigenvalue of the graph. We also show examples of quasi-random graphs of logarithmic sphericity. This is somewhat surprising since quasi-random graphs tend to behave like random graphs, yet the latter have linear sphericity.\
In our search for further examples of graphs of linear sphericity, we investigate in section \[lambda2-cons\] families of graphs whose second eigenvalue is bounded by a constant (for which the aforementioned lower bound is linear). We show that such graphs are close to being complete bipartite, in the sense that one needs to modify only $o(n^2)$ entries in the adjacency matrix to get the latter from the former. As a corollary, we get that for $0 < \delta < {\frac 1 2}$, and $\lambda_2$ there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$.
{#dope}
Definitions
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Let $X = ([n],\delta)$ be a metric space on $n$ points, such that all pairwise distances are distinct. Let $||\;||$ be a norm on $\R^d$. We say that $\phi:X \rightarrow (\R^d,||\;||)$ is a [**]{} if for every $w,x,y,z \in X$, $\delta(x,y) < \delta(w,z) \Leftrightarrow ||\phi (x) - \phi (y)|| <
||\phi (w) - \phi (z)||$.
We denote by $d(X,||\;||)$ the minimal $t$ such that there exists a from $X$ to $(\R^t,||\;||)$. We denote by $d(n,||\;||) = \max_X d(X,||\;||)$, the smallest dimension to which every $n$ point metric can be mapped monotonically.
The first thing to notice is that we are actually concerned only with the [*order*]{} among the distances between the points in the metric space, and not with the actual distances. Let $(X,\delta)$ be a finite metric space, and let $\rho$ be a linear order on $X \choose 2$. We say that $\rho$ and $(X,\delta)$ are [*consistent*]{} if for every $w,x,y,z \in X$, $\delta(x,y) < \delta(w,z) \Leftrightarrow (x,y) <_\rho (w,z)$.
We start with an easy, but useful observation.
\[dope-lemma\] Let $X$ be a finite set. For every strict order relation $\rho$ on $X \choose 2$, there exists a distance function $\delta$ on $X$, that is consistent with $\rho$.
Let $\{\epsilon_{ij}\}_{(i,j)\in {X \choose 2}}$ be small, non-negative numbers, ordered as per $\rho$. Define $\delta(i,j) = 1 + \epsilon_{ij}$. It is obvious that $\delta$ induces the desired order on the distances of $X$, and, that if the $\epsilon$’s are small, the triangle inequality holds.
When we later (Section \[l2\]) use this observation, we refer to it as a [*standard $\epsilon$-construction*]{}, where $\epsilon = \max
\epsilon_{ij}$. It is not hard to see that this metric is Euclidean, that is, the resulting metric can be isometrically embedded into $l_2$, see Lemma \[momad2\] below.
We say that an order relation $\rho$ on $[n] \choose 2$ is [*realizable*]{} in $(\R^d,||\;||)$ if there exists a metric space $(X,\delta)$ on $n$ points which is consistent with $\rho$, and a $\phi:X \rightarrow \R^d$. We say that $\phi$ is a realization of $\rho$. (In other words, $d(n,||\;||)$ is the minimal $d$ such that any linear order on $[n] \choose 2$ is realizable in $(\R^d,||\;||)$.)
We denote by $J = J_n$ the $n \times n$ all ones matrix, and by $PSD_n$ the cone of real symmetric $n \times n$ positive semidefinite matrices. We omit the subscript $n$ when it is clear from the context.
Finally, for a graph $G$, and $U, V$ subsets of its vertices, we denote by $e(U,V) = |\{(u,v) \in E(G): u \in U, v\in V\}|$, and $e(U) = |\{(u,u') \in E(G): u,u' \in U\}|$.
into $l_\infty$.
----------------
$\frac n 2 - 1 \leq d(n,l_\infty) \leq n$
It is well known that any metric $X$ on $n$ points can be embedded into $l_\infty^n$ isometrically, hence $d(n,l_\infty) \leq n$.
For the lower bound, we define a metric space $(X,\delta)$ with $2n+2$ points that cannot be realized in $l_\infty^n$. By lemma \[dope-lemma\], it suffices to define an ordering on the distances. In fact, we define only a partial order, any linear extension of which will do. The $2n+2$ points come in $n+1$ pairs, $\{x_i,y_i\}_{i=1,\ldots,n+1}$. If $z \notin \{x_i,y_i\}$, we let $\delta(x_i,y_i) > \delta(x_i,z), \delta(y_i,z)$. Assume for contradiction that a $\phi$ into $l_\infty^n$ does exist. For each pair $(x,y)$ define $j(x,y)$ to be some index $i$ for which $|\phi(x)_i - \phi(y)_i|$ is maximized, that is, an index $i$ for which $|\phi(x)_i - \phi(y)_i|=\|\phi(x)-\phi(y)\|_{\infty}$.
By the pigeonhole principle there exist two pairs, say $(x_1,y_1)$ and $(x_2,y_2)$, for which $j(x_1,y_1) = j(x_2,y_2)=j$. It is easy to verify that our assumptions on the four real numbers $\phi(x_1)_j$, $\phi(x_2)_j$, $\phi(y_1)_j$, $\phi(y_2)_j$, are contradictory. Thus $d(n,l_\infty) \geq \frac n 2 - 1$.
into $l_2$. {#l2}
-----------
\[momad2\] $\frac n 2 \leq d(n,l_2) \leq n$. Furthermore, for every $\delta > 0$, and every large enough $n$, almost no linear orders $\rho$ on ${[n] \choose 2}$ can be realized in dimension less than $\frac n {2+\delta}$.
The upper bound is apparently folklore. As we could not find a reference for it, we give a proof here.
Let $\rho$ be a linear order on ${[n] \choose 2}$. Let $\epsilon$ be a real symmetric matrix with the following properties:
- $\epsilon_{ii} = 0$ for all $i$.
- $\frac{1}{n} > \epsilon_{ij} > 0$, for all $i \neq j$.
- The numbers $\epsilon_{i,j}$ are consistent with the order $\rho$.
Since the sum of each row is strictly less than one, all eigenvalues of $\epsilon$ are in the open interval $(-1,1)$. It follows that the matrix $I - \epsilon$ is positive definite. Therefore, there exists a matrix $V$ such that $V V^t = I - \epsilon$. Denote the $i$’th row of $V$ by $v_i$. Clearly, the $v_i$’s are unit vectors, and $<v_i,v_j> = - \epsilon_{i,j}$ for $i \neq j$. Therefore, $||v_i - v_j||_2^2 = <v_i,v_i> + <v_j,v_j> - 2<v_i,v_j> = 2 + 2 \epsilon_{i,j}$. It follows that the map $\phi(i) = v_i$ is a realization of $\rho$, and the upper bound is proved. In fact, one can add another point without increasing the dimension, by mapping it to $0$, and perturbing the diagonal.\
For the lower bound, it is essentially known that if $X$ is the metric induced by $K_{n,n}$, then $d(X,l_2) \geq n$. We discuss this in more detail in the next section.
For the second part of the lemma we need a bound on the number of [*sign-patterns*]{} of a sequence of real polynomials. Let $p_1,...,p_m$ be real polynomials in $l$ variables of (total) degree $d$, and let $x \in \R^l$ be a point where none of them vanish. The sign-pattern at $x$ is $(\sgn(p_1(x)),...,\sgn(p_m(x)))$. Denote the total number of different sign-patterns that can be obtained from $p_1,...,p_m$ by $s(p_1,...,p_m)$. A variation of the Milnor-Thom theorem [@Milnor] due to Alon, Frankl and Rödl [@AFR85] shows:
\[Milnor\][@AFR85] Let $p_1,...,p_m$ be real polynomials as above. Then for any integer $k$ between 1 and $m$: $$\begin{aligned}
s(p_1,...,p_m) \leq 2kd \cdot (4kd - 1) ^ {l + \frac m k - 1}\end{aligned}$$
Set $n = c \cdot d$, for some constant $c$, and $l = n \cdot d$. Consider a point $x \in R^l$, and think of it as an $n \times d$ matrix. Denote the $i$th row of this matrix by $x_i$. As before, $x$ [*realizes*]{} an order $\rho$ on ${[n] \choose 2}$ if the distances $||x_i - x_j||$ are consistent with $\rho$.
For two different pairs, $(i_1,j_1)$ and $(i_2,j_2)$, define the polynomial $$\begin{aligned}
p_{(i_1,j_1),(i_2,j_2)}(x) = ||x_{i_1} - x_{j_1}||^2 -
||x_{i_2} - x_{j_2}||^2.\end{aligned}$$ The list contains $m = {{n \choose 2} \choose 2}$ polynomials of degree 2. Note that there is a $1:1$ correspondence between orders on ${[n]} \choose {2}$ and sign-patterns of $p_1,...,p_m$, thus no more than $s=s(p_1,...,p_m)$ orders may be realized in $l_2^d$.
Take $k=\mu n^2$, for some large constant $\mu$. Then $\log s$ is approximately $2 c d^2 \log d$. By contrast, that total number of orders is ${n \choose 2}!$, so its log is about $c^2d^2 \log d$. If $c$ is bigger than 2, almost all order relations can not be realized.
In fact, the same proof shows that for any positive integer $t$, almost all orders on $n \choose 2$ require linear dimension to be realized, and in particular that $d(n,l_{2t}) = \Omega(n)$ (where the constant of proportionality depends only t): Simply repeat the argument above with polynomials of degree $2t$ rather than quadratic polynomials.
Other Norms
-----------
We conclude this section with two easy observations about into other normed spaces. The first gives an upper bounds on the dimension required for embedding into $l_p$:
$d(n,l_p) \leq {n \choose 2}$.
By Lemma \[momad2\], any metric space on $n$ points can be mapped monotonically into $l_2$. It is known (see [@DeLa] and also chapter 15 of [@Matousek]) that any $l_2$ metric on $n$ points can be isometrically embedded into ${n \choose 2}$-dimensional $l_p$. The composition of these mappings is a monotone mapping of the metric space into ${n \choose 2}$-dimensional $l_p$.
The second observation gives a lower bound for arbitrary norms. We first note the following:
Let $||\;||$ be an arbitrary $n$-dimensional norm and let $x_1,...,x_{5^n}$ be points in $\R^n$, such that $||x_i-x_j||>1$ for all $i \neq j$. Then there exits a pair $(x_i,x_j)$ that $||x_i-x_j|| \geq 2$
Denote by $v$ the volume of $B$, the unit ball in $(\R^n,||\;||)$. The translates $x_i + {\frac 1 2}B$ are obviously non intersecting, so the volume of their union is $(\frac{5}{2})^n v$. Assume for contradiction that all pairwise distances are less than $2$, then all these balls are contained in a single ball of radius less than $\frac 5 2$. But this is impossible, since the volume of this ball is less than $(\frac{5}{2})^n v$. Note that the $l_\infty$ norm shows that indeed an exponential number of points is required for the lemma to follow. We do not know, however, the smallest base of the exponent for which the claim holds. The determination of this number seems to be of some interest.
There exists an $n$-point metric spaces $(X,\delta)$ such that for any norm $||\;||$, $d(n,||\;||) = \Omega(\log n)$.
We construct a distance function on $5^n+1$ points which can not be realized in any $n$-dimensional norm. By lemma \[dope-lemma\] it suffices to define a partial order on the distances. Denote the points in the metric space $0,\ldots,5^n$. Let the distance between $0$ and any other point be smaller than any distance between any two points $i \neq j > 0$. Consider a monotone map $\phi$ of the metric space into $n$-dimensional normed space. Assume, w.l.o.g., that $\min_{i,j=1,\ldots,5^n}||\phi(i)-\phi(j)|| = 1$. By the previous lemma there exists a pair of points, $i,j \neq 0$, such that $||\phi(i)-\phi(j)||>2$. But for $\phi$ to be monotone it must satisfy $||\phi(0)-\phi(i)||<1$ and $||\phi(0)-\phi(j)||<1$, contradicting the triangle inequality.
Sphericity {#prox-graph-sec}
==========
So far we have concentrated on embeddings of a metric space into a normed space, that preserve the order relations between distances. However, in the examples that gave us the lower bounds for $l_\infty$ and for arbitrary norms, we actually only needed to distinguish between “long” and “short” distances. This motivates the introduction of a broader class of maps, that need only respect the distinction between short and long distances. More formally, let $X=([n],\delta)$ be a metric space. Its [*proximity graph*]{} with respect to some threshold $\tau$, is a graph on $n$ vertices, with an edge between $i$ and $j$ iff $\delta(i,j) \leq \tau$. An embedding of a proximity graph, is a mapping $\phi$ of its vertices into normed space, such that $||\phi (i) - \phi (j) || < 1$ iff $(i,j)$ is an edge in the proximity graph (We assume that no distance is exactly 1). The minimal dimension in which a graph can be so embedded (in Euclidean space) was first studied by Maehara in [@Maehara] under the name [*sphericity*]{}, and denoted $Sph(G)$. Following this terminology, we call such an embedding [*spherical*]{}.\
The sphericity of graphs was further studied by Maehara and Frankl in [@FraMa], and then by Reiterman, R[ö]{}dl and [Š]{}i[ň]{}ajov[á]{} in [@RRS89a], [@RRS89b], [@RRS92]. Breu and Kirkpatrick have shown in [@BK93] that it is NP-hard to recognize graphs of sphericity 2 (also known as [*unit disk graphs*]{}) and graphs of sphericity 3. We refer the reader to [@RRS89b] for a survey of most known results regarding this parameter, and mention only a few of them here.\
\[list\] Let $G$ be graph on $n$ vertices with minimal degree $\delta$. Let $\lambda_n$ the least eigenvalue of its adjacency matrix.
1. $Sph(K_{m,n}) \leq m + \frac n 2 - 1$ [@Maehara].
2. $Sph(G) = O(\lambda_n^2 \log n)$ [@FraMa].\[it-lam\]
3. $Sph(G) = O((n-\delta) \log (n-\delta))$ [@RRS89b].
4. $Sph(K_{n,n}) \geq n$ [@RRS89a].
5. All but a $\frac 1 n$ fraction of graphs on $n > 37$ vertices have sphericity at least $\frac n {15} - 1$ [@RRS89b].
6. $Sph(G) \geq \frac {\log \alpha(G)} {\log (2r(G)+1)}$, where $\alpha(G)$ is the independence number of $G$, and $r(G)$ is its radius [@RRS89a].
The first thing to notice is that any lower bower on the sphericity of some graph on $n$ vertices is also a lower bound on $d(n,l_2)$. In particular, the fact that $Sph(K_{n,n}) \geq n$ proves the lower bound in Lemma \[momad2\]. (Similarly, any upper bound on the former also applies to the latter.)\
In this section we are interested in graphs of large sphericity. The above results tell us that they exist in abundance, yet that graphs of very small or very large degree have small sphericity (the maximal degree is an upper bound on $|\lambda_n|$, hence by (\[it-lam\]) the sphericity is small if all degrees are small). Other than the complete bipartite graph, the above results do not point out an explicit graph with super-logarithmic sphericity.
Upper Bound on Margin
---------------------
Following Frankl and Maehara [@FraMa], consider an embedding of a proximity graph where there is a large margin between short and long distances. In such a situation, the Johnson-Lindenstrauss Lemma ([@JoLi84]) would yield a spherical embedding into lower dimension: It allows reducing the dimension at the cost of some distortion. If the distortion is small with respect to the margin, the short and long distances remain separated. Alas, we show that for most regular graphs this margin is not large enough for the method to be useful:
Let $G$ be a $\delta n$-regular graph, with second eigenvalue $\lambda_2 > \frac 2 n$. Let $\phi$ be an embedding of $G$ as a proximity graph. Denote $a=\max_{u \sim v}||\phi(u) - \phi(v)||_2^2$, and $b=\min_{u \not \sim v}||\phi(u) - \phi(v)||_2^2$. Then $b-a = O(\frac {\lambda_2 + \delta} {\delta n})$.
Denote $m=\min\{1-a, b-1\}$, and for a vertex $i$, denote $v_i = \phi(i)$. The largest value $m$ can attain, over all embeddings $\phi$, is given by the following quadratic semidefinite program: $$\begin{aligned}
& \max m & \\
& s.t. \forall (i,j) \in E(G) & ||v_i - v_j||^2 \leq 1-m\\
& \forall (i,j) \notin E(G) & ||v_i - v_j||^2 \geq 1+m\end{aligned}$$ Its dual turns out to be: $$\begin{aligned}
& \min {\frac 1 2}tr A &\\
& s.t. & A \in PSD \\
& \forall (i,j) \in E(G) & A_{ij} \leq 0 \\
& \forall (i,j) \notin E(G), i\neq j & A_{ij} \geq 0 \\
& \forall i & \sum_{j=1,...,n} A_{ij} = 0 \\
& & \sum_{i \neq j} |A_{ij}| = 1\end{aligned}$$ Equivalently, we can drop the last constraint, and change the objective function to $ \min \frac {tr A} {\sum_{i \neq j} |A_{ij}|} $. Next we construct an explicit feasible solution for the dual program, and conclude from it a bound on m.
Let $M$ be the adjacency matrix of $G$. Define $A = I + \alpha J - \beta M$. To satisfy the constraints we need: $$\begin{aligned}
& & A \in PSD \\
& & \beta \geq \alpha \geq 0\\
& & 1 + \alpha n - \beta \delta n = 0\\\end{aligned}$$ The last condition implies $\alpha = \beta \delta - \frac 1 n$, so it follows that $\beta \geq \alpha$, and the constraint on $\beta$ is $\beta \geq \frac {1} {\delta n}$.
Now, since we assume that the graph is $\delta n$-regular, its Perron eigenvector is $\vec{1}$, corresponding to eigenvalue $\delta n$. Therefore, we can consider the eigenvectors of $M$ to be eigenvectors of $J$ and $I$ as well, and hence also eigenvectors of $A$. If $\lambda \neq \delta n$ is an eigenvalue of $M$, then $1-\beta \lambda$ is an eigenvalue of $A$, corresponding to the same eigenvector. Denote by $\lambda_2$ the second largest eigenvalue of $M$, then in order to satisfy the condition $ A \in PSD$ it is enough to set $\beta = \frac {1} {\lambda_2}$, in which case all the constraints are fulfilled.
We conclude that: $$\begin{aligned}
m & \leq & \frac {tr A} {\sum_{i \neq j} |A_{ij}|} =
\frac {n(1+\alpha)}
{\delta n^2 (\beta - \alpha) + ((1-\delta) n^2 - n) \alpha} \\
& = &
\frac {n+ \frac {\delta n} {\lambda_2} - 1}
{\delta n (\frac {n+\delta n} {\lambda_2} - 1) + ((1-\delta) n - 1)
(\frac {\delta n} {\lambda_2} - 1)}
<
4 \frac {1 + \frac {\delta} {\lambda_2}} {\frac {\delta n} {\lambda_2}}
= 4 \frac {\lambda_2 + \delta} {\delta n}.\end{aligned}$$ In particular, $b-a = O(\frac {\lambda_2 + \delta} {\delta n})$.
In order to derive a non trivial result from Johnson-Lindenstrauss lemma, we need that $\frac 1 {m^2} \log n = o(n)$, and in particular that $\lambda_2=\Omega(\delta \sqrt{n \log n})$. The above shows that this can happen only if $\lambda_2 = \omega(\delta \sqrt{n \log n})$. On the other hand, Frankl and Maehara show that their method does give a non trivial bound when $\lambda_n = o(\sqrt{\frac n {\log n}})$. Consequently, we get that a $\delta n$-regular graph (think of $\delta$ as constant) can’t have both $\lambda_2=o(\sqrt{n \log n})$ and $\lambda_n = o(\sqrt{\frac n {\log n}})$. This is a bit more subtle than what one gets from the second moment argument, namely, that the graph can’t have both $\lambda_2=o(\sqrt{n})$ and $\lambda_n = o(\sqrt{n})$.
Lower Bound on Sphericity
-------------------------
\[our-bound\] Let $G$ be a $d$-regular graph with diameter $D$ and $\lambda_2$, the second largest eigenvalue of $G$’s adjacency matrix, is at least $d - {\frac 1 2}n$. Then $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))})$.
In the interesting range where $d \leq \frac n 2$, and $\lambda_2 \geq 1$ the bound is $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2 \lambda_2})$. It will be useful to consider the following operation on matrices. Let $A$ be an $n \times n$ symmetric matrix, and denote by ${\vec{a}}$ the vector whose $i$-th coordinate is $A_{ii}$. Define $R(A)$ to be the $n \times n$ matrix with all rows equal to ${\vec{a}}$, and $C(A) = R(A)^t$. Define: $$\begin{aligned}
\breve{A} = 2A - C(A) - R(A) + J\end{aligned}$$ First note that the rank of $\breve{A}$ and that of $A$ can differ by at most 3. Now, consider the case where $A$ is the Gram matrix of some vectors $v_1,...,v_n \in R^d$. Then all diagonal entries of $\breve{A}$ equal one, and the $(i,j)$ entry is 2$<v_i,v_j> - <v_i,v_i> - <v_j,v_j> + 1 = 1 - ||v_i-v_j||^2$.
We will need the following lemma (see [@HoJo], p.175):
\[d-lemma\] Let $X$ be a real symmetric matrix, then $rank(X) \geq \frac {(tr X)^2} {\sum_{i,j}X_{i,j}^2}$
Applying this to $\breve{A}$, we conclude that: $$\begin{aligned}
\label{d-bound}
rank(\breve{A}) \geq \frac {n^2} {n +
\sum_{i \neq j} (1 - ||v_i - v_j||^2)^2 }\end{aligned}$$
Let $v_1,...,v_n \in \R^d$ be an embedding of $G$. By the discussion above it is enough to show that $$\begin{aligned}
\label{denom}
\sum_{i \neq j} (1-||v_i - v_j||^2)^2 = O(D^2 n^2 \frac {\lambda_2}
{d - \lambda_2}).\end{aligned}$$ By the triangle inequality $||v_i - v_j|| \leq D$ for any two vertices. So the LHS of (\[denom\]) is bigger by at most a factor of $D^2$ than: $$\begin{aligned}
&& \sum_{(i,j) \notin E} (||v_i - v_j||^2 - 1) +
\sum_{(i,j) \in E} (1 - ||v_i - v_j||^2) =\end{aligned}$$ $$\begin{aligned}
\label{hs-norm}
&& \sum_{(i,j) \notin E} ||v_i - v_j||^2 -
\sum_{(i,j) \in E} ||v_i - v_j||^2 - {n \choose 2} + nd\end{aligned}$$ We can bound this sum from above, by solving the following SDP: $$\begin{aligned}
& \max & \sum_{(i,j) \notin E} (V_{ii} + V_{jj} - 2V_{ij}) +
\sum_{(i,j) \in E} (- V_{ii} - V_{jj} + 2V_{ij}) - {n \choose 2} + nd\\
& s.t. & V \in PSD \\
& \forall (i,j) \in E & V_{ii} + V_{jj} - 2V_{ij} \leq 1 \\
& \forall (i,j) \notin E & V_{ii} + V_{jj} - 2V_{ij} \geq 1\end{aligned}$$ The dual problem is: $$\begin{aligned}
& \min & {\frac 1 2}tr A \\
& s.t. & A \in PSD \\
& \forall (i,j) \in E & A_{ij} \leq -1 \\
& \forall (i,j) \notin E, i\neq j & A_{ij} \geq 1 \\
& \forall i \in [n] & \sum_{j=1,...,n} A_{ij} = 0\end{aligned}$$ Let $M$ by the adjacency matrix of the graph, and set $A = (\alpha d - n)I + J - \alpha M$, where $\alpha \geq 2$ will be determined shortly. This takes care of the all constraints except for $A \in PSD$. Note that since $M$ is regular, its eigenvectors are also eigenvectors of $A$. Moreover, if $M u = \lambda u$ for a non constant $u$, then $A u = \alpha d - n - \alpha \lambda$ (and $A \vec{1} = 0$). So take $\alpha = \frac {n} {d - \lambda_2}$, and by our assumption on $\lambda_2$, $\alpha \geq 2$.
Now $A$ gives an upper bound on (\[hs-norm\]): $$\begin{aligned}
{\frac 1 2}tr A =
{\frac 1 2}n(\alpha d - n + 1) = {\frac 1 2}n^2 \frac d {d-\lambda_2} - {\frac 1 2}n^2
+ {\frac 1 2}n= {\frac 1 2}n^2 \frac {\lambda_2} {d-\lambda_2} + {\frac 1 2}n.\end{aligned}$$ This, by (\[d-bound\]), shows that the dimension of the embedding is $\Omega\left(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))} \right)$.
A Quasi-random Graph of logarithmic Sphericity
----------------------------------------------
It is an intriguing problem to construct new examples of graphs of linear sphericity. Since random graphs have this property, it is natural to search among quasi-random graphs. There are several equivalent definitions for such graphs (see [@AlSp]). The one we adopt here is:
A family of graphs is called [*quasi-random*]{} if the graphs in the family are $(1+o(1))\frac n 2$-regular, and all their eigenvalues except the largest one are (in absolute value) $o(n)$ .
Counter-intuitively, perhaps, quasi-random graphs may have very small sphericity.
\[alex\] Let $\G$ be the family of graphs with vertex set $\{0,1\}^k$, and edges connecting vertices that are at Hamming distance at most $\frac k 2$. Then $\G$ is a family of quasi-random graphs of logarithmic sphericity.
The fact that the sphericity is logarithmic is obvious - simply map each vertex to the vector in $\{0,1\}^n$ associated with it. To show that all eigenvalues except the largest one are $o(2^k)$ we need the following facts about Krawtchouk polynomials (see [@vL]). Denote by $K_s^{(k)}(i) = \sum_{j=0}^s(-1)^j{i \choose j}{{k-i} \choose {s-j}}$ the Krawtchouk polynomial of order $s$ over $\Z_2^k$. For simplicity we assume that $k$ is odd.
1. For any $x \in \Z_2^k$ with $|x|=i$, $\sum_{z \in \Z_2^k\\|z|=s}(-1)^{<x,z>} = K_s^{(k)}(i)$.
2. $\sum_{s=0}^l K_s^{(k)}(i) = K_l^{(k-1)}(i-1)$.
3. For any $s$ and $k$, $\max_{i=0,\dots,n} |K_s^{(k)}(i)| = K_s^{(k)}(0) = {k \choose s}$.
Observe that $G$ is a Cayely graph for the group $\Z_2^k$ with generator set $\{g \in \Z_2^k : |g| \leq \frac k 2\}$. Since $\Z_2^k$ is abelian, the eigenvectors of the graphs are independent of the generators, and are simply the characters of the group written as the vector of their values. Namely, corresponding each $y \in \Z_2^k$ we have an eigenvector $v^y$, such that $v^y_x = (-1)^{<x,y>}$. For every $y$, $v^y_0 = 1$, so to figure out the eigenvalue corresponding to $v^y$, we simply need to sum the value of $v^y$ on the neighbors of $0$. Note that for $y=0$ we get the all $1$s vector, which corresponds to the largest eigenvalue. So we are interested in $y$’s such that $|y| > 0$. By the first two facts above we have: $$\lambda_y = \sum_{g \in \Z_2^k, |g| \leq \frac k 2}(-1)^{<y,g>} = \sum_{s=0}^{\frac {k-1} 2} K_s^{(k)}(|y|) = K_{\frac {k-1} 2}^{(k-1)}(|y|-1).$$ By the third fact, this is at most ${{k-1} \choose {\frac {k-1} 2}}
\approx \frac {2^{k-1}} {\sqrt{k-1}} =
o(2^{k-1})$.
Graphs with bounded $\lambda_2$ {#lambda2-cons}
===============================
Theorem \[our-bound\] suggests families of graphs that have linear sphericity. Namely, for $0 < \delta \leq {\frac 1 2}$, and $\lambda_2 > 0$, the theorem says that $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$ have linear sphericity. In this section we characterize such graphs. We prove that for $\delta = {\frac 1 2}$ such graphs are nearly complete bipartite, and that for other values, only finitely many graphs exist.\
It is worth noting that graphs with bounded second eigenvalue have been previously studied. The apex of these works is probably that of Cameron, Goethals, Seidel and Shult, who characterize in [@CGSS] graphs with second eigenvalue at most 2.
$n/2$-regular graphs
--------------------
In this section we consider the family $\G$ of $n/2$-regular graphs, and second largest eigenvalue $\lambda_2$ bounded by a constant. We prove that, asymptotically, they are nearly complete bipartite.
Let $G$ and $H$ be two graphs on $n$ vertices. We say that $G$ and $H$ are [*close*]{}, if there is a labeling of their vertices such that $|E(G) \bigtriangleup E(H)| = o(n^2)$.
\[main1\] Every $G \in \G$ is close to $K_{n/2,n/2}$, where $n$ is the number of vertices in $G$.
By passing to the complement graph, if $\lambda_n = O(1)$, then $G$ is close to the disjoint union of two cliques, $K_{n/2} \dot{\cup} K_{n/2}$.
We need several lemmas. The first is the well-known expander mixing lemma [@FriPi]. The second is a special case of Simonovitz’s stability theorem ([@Sim]), for which we give a simple proof here. The third is a commonly used corollary of Szemeredi’s Regularity Lemma. We shall also make use of the Regularity Lemma itself (see e.g. [@Diestal]).
\[no-clique\] Let $G$ be an $\frac n 2$-regular graph on $n$ vertices with second largest eigenvalue $\lambda_2$. Then every subset of vertices with $k$ vertices has at most $\frac 1 4 k^2 + {\frac 1 2}\lambda_2 k$ internal edges.
\[close-bi\] Let $R$ be a triangle-free graph on $n$ vertices, with $n^2/4 - o(n^2)$ edges. Then $R$ is close to $K_{n/2,n/2}$. Furthermore, all but $o(n)$ of the vertices have degree $\frac n 2 \pm o(n)$.
Denote by $d_i$ the degree of the $i$th vertex in $R$, and by $m$ the number of edges. Then: $$\sum_{(i,j) \in E(R)}(d_i+d_j) = \sum_{i \in V(R)} d_i^2 \geq \frac 1 n
(\sum_{i \in V(R)} d_i)^2 = \frac {4m^2} n.$$ Thus, there is some edge $(i,j) \in E(R)$ such that $d_i + d_j \geq \frac {4m} n = n - o(n)$. Let $\Gamma_i$ and $\Gamma_j$ be the neighbor sets of $i$ and $j$. Since $i$ and $j$ are adjacent, and $R$ has no triangles, the sets $\Gamma_i$ and $\Gamma_j$ are disjoint and independent. If we delete the $o(n)$ of vertices in $V\backslash (\Gamma_i \cup \Gamma_j)$ we obtain a bipartite graph. We have deleted only $o(n^2)$ edges, so the remaining graph still has $n^2/4 - o(n^2)$ edges. But this means that $|\Gamma_i|, |\Gamma_j| = \frac n 2 - o(n)$, and that the degree of each vertex in these sets is $\frac n 2 \pm o(n)$
Recall that the Regularity Lemma states that for every $\epsilon > 0$ and $m \in \N$ there’s an $M$, such that the vertex set of every large enough graph can be partitioned into $k$ subsets, for some $m \leq k \leq M$ with the following properties: All subsets except one, the “exceptional” subset, are of the same size. The exceptional subset contains less than an $\epsilon$-fraction of the vertices. All but an $\epsilon$-fraction of the pairs of subsets are $\epsilon$-regular.\
The regularity graph with respect to such a partition and a threshold $d$, has the $k$ subsets as vertices. Two subsets, $U_1$ and $U_2$ are adjacent, if they are $\epsilon$-regular, and $e(U_1,U_2) > dn^2$.
\[[@Diestal], Lemma 7.3.2\] Let $G$ be a graph on $n$ vertices, $d \in (0,1]$, $\epsilon = d^{-4}$. Let $R$ be an $\epsilon$-regularity graph of $G$, with (non exceptional) sets of size at least $\frac s {\epsilon}$, and threshold $d$. If $R$ contains a triangle, then $G$ contains a complete tripartite subgraph, with each side of size $s$.
If $G \in \G$, and $R$ is as in the lemma, with $s = 10 \lambda_2$, then $R$ is triangle free. In this case, if $R$ has $\frac {k^2} 4 - o(k^2)$ edges, then $R$ is close to complete bipartite.
If $R$ contains a triangle, then $G$ contains a complete tripartite subgraph, with $s$ vertices on each side. Let $U$ be the set vertices in this subgraph. Then $e(U) = 3s^2 = 300 \lambda_2^2$, but by lemma \[no-clique\] $e(U) \leq 50 \lambda_2^2$ - a contradiction. The second part now follows from Lemma \[close-bi\].
(Theorem) We would like to apply the Regularity Lemma to graphs in $\G$, and have $\epsilon = o(1)$, and $k = \omega(1)$ as well as $k = o(n)$. Indeed, this can be done. Since $M$ depends only on $m$ and $\epsilon$, choose $d=o(1)$, and $m = \omega(1)$, such that the $M$ given by the lemma satisfies $\frac n {(M+1)} \geq \frac s {\epsilon}$. As $M$ depends only on $m$ and $\epsilon$, $\frac M {\epsilon}$ can be made small enough, even with the requirements on $d$ and $m$.
Let $R$ be the regularity graph for the partition given by the Regularity Lemma, with threshold $d$ as above. Denote by $k$ the number of sets in the partition, and their size by $l$ (so $k\cdot l = n(1-\eta)$, for some $\eta \leq \epsilon$). We shall show that $R$ is close to complete bipartite, and that $G$ is close the graph obtained by replacing each vertex in $R$ with $l$ vertices, and replacing each edge in $R$ by a $K_{l,l}$.
Call an edge in $G$ $(i)$ “irregular” if it belongs to an irregular pair; $(ii)$ “internal” if it connects two vertices within the same part; $(iii)$ “redundant” if it belongs to a pair of edge density smaller than $d$, or touches a vertex in the exceptional set. Otherwise $(iv)$, call it “good”.\
Recall that $\epsilon = o(1)$, so only $o(k^2)$ pairs of sets are not $\epsilon$-regular. Thus, $G$ can have only $o(l^2k^2) = o(n^2)$ irregular edges. Also, $d = o(1)$, so the number of redundant edges is $k^2 \cdot o(l^2) + o(l) \frac n 2 = o(n^2)$. Finally, the number of internal edges is at most ${\frac 1 2}l^2 k$, hence there are $\frac {n^2} 4 - o(n^2)$ good edges.\
The number of edges between two sets is at most $l^2$, so $R$ must have at least $$\frac {n^2 - o(n^2)} {4l^2} = \frac {k^2} 4 - o(k^2)$$ edges. The corollary implies that it is close to complete bipartite. By lemma \[close-bi\], the valency of all but $o(k)$ of the vertices in $R$ is indeed $\frac k 2 \pm o(k)$. This means that every edge in $R$ corresponds to $l^2 - o(l^2)$ good edges in $G$ (as the number of edges in $R$ is also no more than $\frac {k^2} 4 + o(k^2)$).\
To see that $G$ is close to complete bipartite, let’s count how many edges need to be modified. First, delete $o(n^2)$ edges that are not “good”. Next add all possible $o(n^2)$ new edges between pairs of sets that have “good” edges between them. As $R$ is close to complete bipartite, we need to delete or add all edges between $o(k^2)$ pairs. Each such step modifies $l^2$ edges, altogether $o(l^2k^2) = o(n^2)$ modifications. Finally, divide the $o(n)$ vertices of the exceptional set evenly between the two sides of the bipartite graph, and add all the required edges, and the tally remains $o(n^2)$.
In essence, the proof shows that a graph with no dense induced subgraphs is close to complete bipartite. This claim is similar in flavor to Bruce Reed’s [*Mangoes and Blueberries*]{} theorem [@Reed99]. Namely, that if every induced subgraph $G'$ of $G$ has an independent set of size ${\frac 1 2}|G'| - O(1)$, then $G$ is close to being bipartite. The conclusion in Reed’s theorem is stronger in that only a [*linear*]{} number of edges need to be deleted to get a bipartite graph.
In fact, the proof gives something a bit stronger. Let $t_r(n)$ be the number of edges in an $n$-vertex complete $r$-partite graph, with parts of equal size. Using the general Stability Theorem ([@Sim]) instead of Lemma \[close-bi\], the same proof shows that if a graph has $t_n - o(n^2)$ edges and no dense induced subgraphs, then it is close to being complete $r$-partite.
$\delta n$-regular graphs
-------------------------
In Theorem \[main1\] we required that the degree is $n/2$. We can deduce from the theorem that this requirement can be relaxed:
Let $\G$ be a family of $d$-regular graphs, with $d \leq \frac n 2$, ($n$ being the number of vertices in the graph) and bounded second eigenvalue, then every $G \in \G$ is close to a complete bipartite graph.
Let $M \in \M_n$ be the adjacency matrix of such a $d$-regular graph, and denote $\bar{M} = J - M$, where $J$ is the all ones matrix. Consider the graph $H$ corresponding to the following matrix: $$\begin{aligned}
N =
\left(
\begin{array}{cc}
M & \bar{M} \\
\bar{M}^t & M
\end{array}
\right)\end{aligned}$$ Clearly $H$ is an $n$-regular graph on $2n$ vertices. Denote by $(x, y)$ the concatenation of two $n$-dimensional vectors, $x$, $y$, into a $2n$ dimensional vector. Let $v$ be an eigenvector of $M$ corresponding to eigenvalue $\lambda$. It is easy to see that $v$ is also an eigenvalue of $\bar{M}$: If $v = \vec{1}$ (and thus $\lambda = d$) it corresponds to eigenvalue $n - \lambda$, otherwise to $(-\lambda)$.\
Thus, $(v, v)$ and $(v, -v)$ are both eigenvectors of $N$. If $v = \vec{1}$ they correspond to eigenvalues $n$, $2d-n$, respectively, otherwise to $0$, $2\lambda$. Since the $v$’s are linearly independent, so are the $2n$ vectors of the form $(v, v)$ and $(v, -v)$: Consider a linear combination of these vectors that gives $0$. Both the sum and the difference of the coefficients of each pair have to be $0$, and thus both are 0. So we know the entire spectrum of $N$, and see, since $d \leq \frac n 2$, that theorem \[main1\] holds for it.\
Let $H'$ be a complete bipartite graph that is close to $H$. Since $H$ differs from $H'$ by $o(n^2)$ edges, the same holds for subgraphs over the same set of vertices. In particular, $G$ is close to the subgraph of $H'$ spanned by the first $n$ vertices. Obviously, every such subgraph is itself complete bipartite.
\[no-graphs\] For every $0 < \delta < {\frac 1 2}$ and $c$, there are only finitely many $\delta n$-regular graphs with $\lambda_2 < c$.
Consider such a graph with $n$ large. By the previous corollary it is close to complete bipartite. Since it is also regular, it must be close to $K_{\frac n 2, \frac n 2}$, which contradicts the constraint $\delta < {\frac 1 2}$.
Graphs with both $\lambda_2$ and $\lambda_{n-1}$ bounded by a constant
----------------------------------------------------------------------
Theorem \[main1\] can loosely be stated as follows: A regular graph with spectrum similar to that of a bipartite graph ($\lambda_1$ being close to $n/2$ and $\lambda_2$ being close to $0$) is close to being complete bipartite. We conclude this section by noting that if we strengthen the assumption on how close the spectrum of a graph is to that of a bipartite graph, we get a stronger result as to how close it is to a complete bipartite graph.
\[main2\] Let $\G$ be a family of $\frac n 2$-regular graphs on $n$ vertices, with both $\lambda_2$ and $\lambda_{n-1}$ bounded by a constant. Then every $G \in \G$ is close to a $K_{\frac n 2,\frac n 2 }$, in the sense that such a graph can be obtained from $G$ by modifying a linear number of edges for $O(\sqrt{n})$ vertices of $G$, and $O(\sqrt{n})$ edges for the rest.
First note that it follows that $\lambda_n(G) = -\frac n 2 + O(1)$. Take $G \in \G$, and let $A$ be its adjacency matrix. Clearly $tr(A^2) = \frac {n^2} 2$. If $\lambda_{n-1}(G) = - O(1)$, then $$\frac {n^2} 2 = tr(A^2) =
\lambda_1^2 + \lambda_n^2 + \sum_{i=2,...,n-1}\lambda_i^2$$ Since $\lambda_1 = \frac n 2$ $$\lambda_n^2 =
\frac {n^2} 2 - \left(\frac n 2\right)^2 - \sum_{i=2,...,n-1}\lambda_i^2$$ As $\lambda_2,\ldots,\lambda_{n-1} = O(1)$ we have $$\lambda_n^2 = \frac {n^2} 4 + O(n)$$ And since $\lambda_n$ is negative, and is smaller than $\lambda_1$ in absolute value: $$\lambda_n = -\frac n 2 + O(1).$$
Let $x$ be an eigenvector corresponding to $\lambda_n$. Suppose, w.l.o.g that $||x||_{\infty} = 1$ and that $x_v = 1$. Denote $A = \{u : x_u \leq -(1-\frac 1 {\sqrt{n}})\}$, and $B = \{w : x_w \geq (1-\frac 1 {\sqrt{n}})\}$. The eigenvalue condition on $v$ entails: $$\begin{aligned}
\frac n 2 - O(1) = -\sum_{u:(u,v) \in E} x_u.\end{aligned}$$ Thus, there is a vertex $u$ such that $x_u \leq -(1-O(\frac 1 n))$. It is not hard to verify that $v$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $A$, and that $u$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $B$.
Now denote $A' = \{u : x_u \leq -{\frac 1 2}\}$, and $B' = \{w : x_w \geq {\frac 1 2}\}$. Again, it is not hard to check that each vertex in $A$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $B'$, and vice versa. Thus, delete the $O(\sqrt{n})$ vertices that are neither in $A$ nor in $B$. For each remaining vertex in $A$ (similarly in $B$), its degree is at most $\frac n 2$, and at least $\frac n 2 - O(\sqrt{n})$. It has $\frac n 2 - O(\sqrt{n})$ neighbors in $B$, so the number of its neighbors in $A$, and the number of its non-neighbors in $B$ is $O(\sqrt{n})$. By deleting and adding $O(\sqrt{n})$ edges to each vertex, we get a complete bipartite graph.
Alternatively, we could have defined $\G$ as a family of $\frac n 2$-regular graphs with $\lambda_2$ bounded, and $\lambda_n(G) = -\frac n 2 + O(1)$. It’s interesting to note that in this case it follows that $\lambda_{n-1}$ is bounded. For $G \in G$, if $G$ is bipartite, then it is complete bipartite, and $\lambda_{n-1}(G) = 0$. Otherwise, $\chi(G) > 2$, and by a theorem of Hoffman ([@Hoff]) $\lambda_n(G) + \lambda_{n-1}(G) + \lambda_1(G) \geq 0$. By our assumption, $\lambda_n(G) + \lambda_1(G) = O(1)$, and since $\lambda_{n-1}(G) < 0$ (otherwise the eigenvalues won’t sum up to 0), it follows that $\lambda_{n-1}(G) = -O(1)$.
Conclusion and Open Problems
============================
The only explicit examples known so far for graphs that have linear sphericity are $K_{n,n}$ and small modifications of it. We conjecture that more complicated graphs, such as the Paley graph, also have linear sphericity. Note that the lower bound presented here only shows a bound of $\Omega(\sqrt{n})$. It is also interesting to know if the bound can be improved, either as a pure spectral bound, or with some further assumptions on the structure of the graph.\
What is the largest sphericity, $d=d(n)$, of an $n$-vertex graph? We know that $\frac n 2 \leq d \leq n-1$. Can this gap be closed? For a seemingly related question, the smallest dimension required to realize a sign matrix (see [@AFR85]) the answer is known to be $\frac n 2 \pm o(n)$. We have also seen a similar gap for $d(n,l_2)$ and $d(n,l_\infty)$. Can this be closed? Can some kind of interpolation arguments generalize the bounds we know for these two numbers to bounds on $d(n,l_p)$ for $p>2$?\
Our interest in sphericity arose from a search for a lower bound on $d(n,l_2)$. But why limit the discussion to Euclidean space? What can be said of spherical embeddings into $l_1$ or $l_{\infty}$? The former may be particularly interesting, as it will give a non-trivial lower bound on $d(n,l_1)$.\
We have seen that $\frac n 2$-regular graphs with bounded second eigenvalue are $o(n^2)$-close to complete bipartite. However, the only example we know of such graphs are constructed by taking a complete bipartite graph, and changing a constant number of edges for each vertex. These graphs are $O(n)$-close to being complete bipartite. Are there examples of such families which are further from complete bipartite graphs, or can a stonger notion of closeness be proved?
Acknowledgments
===============
We would like to thank Alex Samorodnitsky for showing us how to prove Lemma \[alex\] and Noga Alon for sending us Lemma \[d-lemma\].
[^1]: Institute of Computer Science, Hebrew University Jerusalem 91904 Israel . This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.
| ArXiv |
---
abstract: 'The thermal and magnetic properties of spin-$1$ magnetic chain compounds with large single-ion and in-plane anisotropies are investigated via the integrable $su(3)$ model in terms of the quantum transfer matrix method and the recently developed high temperature expansion method for exactly solved models. It is shown that large single-ion anisotropy may result in a singlet gapped phase in the spin-$1$ chain which is significantly different from the standard Haldane phase. A large in-plane anisotropy may destroy the gapped phase. On the other hand, in the vicinity of the critical point a weak in-plane anisotropy leads to a different phase transition than the Pokrovsky-Talapov transition. The magnetic susceptibility, specific heat and magnetization evaluated from the free energy are in excellent agreement with the experimental data for the compounds Ni(C$_2$H$_8$N$_2$)$_2$Ni(CN)$_4$ and Ni(C$_{10}$H$_8$N$_2$)$_2$Ni(CN)$_4$$\cdot$H$_2$O.'
author:
- 'M.T. Batchelor, Xi-Wen Guan and Norman Oelkers'
title: 'Thermal and magnetic properties of spin-$1$ magnetic chain compounds with large single-ion and in-plane anisotropies'
---
Introduction
============
Haldane’s [@Hald] conjecture that spin-$S$ chains exhibit an energy gap in the lowest magnon excitation for $2S$ even with no significant gap for $2S$ odd inspired a great deal of experimental and theoretical investigation. Rich and novel quantum magnetic effects, including valence-bond-solid Haldane phases and dimerized phases [@AFF1; @LADD1], fractional magnetization plateaux [@FPL] and spin-Peierls transitions [@SPT] have since been found in low-dimensional spin systems. In this light, the spin-$1$ Heisenberg magnets have been extensively studied in Haldane gapped materials [@SP1C1; @SP1C2]. The valence-bond-solid ground state and the dimerized state form the Haldane phase with an energy gap [@AFF1]. The Haldane gap in integer spin chains may close in the presence of additional biquadratic terms or in-plane anisotropies. In particular a large single-ion anisotropy may result in a singlet ground state [@AFF3; @Tsvelik] which is significantly different from the standard Haldane phase.
The difference between the two gapped phases appears to arise from the ground state and excitations. In the Haldane nondegenerate ground state, a single valence bond connects each neighbouring pair to form a singlet. An expected excitation comes from breaking down the valence bond solid state where a nonmagnetic state $S_i=0$ at site $i$ is substituted for a state $S_i=1$. In this way a total spin $S=1$ excitation causes an energy gap referred to as the Haldane gap. Whereas the large-anisotropy-induced gapped phase in the spin-$1$ chain is caused by trivalent orbital splitting. For a large single-ion anisotropy, the singlet can occupy all states such that the ground state lies in the nondegenerate gapped phase. The lowest excitation arises as the lower component of the doublet is involved in the ground state. This excitation results in the energy gap.
A number of spin-1 magnetic chain compounds have been identified as planar Heisenberg magnetic chains with large anisotropy. These include Ni(C$_2$H$_8$N$_2$)$_2$Ni(CN)$_4$ (abbreviated NENC), Ni(C$_{11}$H$_{10}$N$_2$O)$_2$Ni(CN)$_4$ (abbreviated NDPK) [@NENC; @sus] and Ni(C$_{10}$H$_8$N$_2$)$_2$Ni(CN)$_4$$\cdot$H$_2$O (abbreviated NBYC) [@NBYC]. This kind of system exhibits a nondegenerate ground state which can be separated from the lowest excitation. This gapped phase also occurs in some nickel salts with a large zero-field splitting, such as NiSnCl$_6\cdot 6$H$_2$O [@PRB3488], \[Ni(C$_5$H$_5$NO)$_6$\](ClO$_4$)$_2$ [@PRB3523] and Ni(NO$_3$)$_2\cdot 6$H$_2$O [@PRB4009]. The theoretical study of these compounds relies on a molecular field approximation for the Van Vleck equation [@Carlin]. To first-order Van Vleck approximation, the exchange interaction is neglected. To obtain a good fit to the experimental data an effective crystalline field has to be incorporated. This approximation causes uncertainties and discrepancies in fitting the experimental data. Here we take a new approach via the theory of integrable models.
It recently has been demonstrated [@HTE1] that integrable models can be used to study real ladder compounds via the thermodynamic Bethe Ansatz (TBA) [@TBA] and the exact high temperature expansion (HTE) method [@HTE2; @ZT]. In this paper we present an integrable spin-$1$ chain with additional terms to account for planar single-ion anisotropy and in-plane anisotropy. The ground state properties and the thermodynamics of the chains are studied via the TBA and HTE. We show that a large planar single-ion anisotropy results in a nondegenerate singlet ground state which is significantly different from the Haldane phases found in Haldane gapped materials [@SP1C1; @SP1C2]. We examine the thermal and magnetic properties of the compounds NENC [@NENC; @sus] and NBYC [@NBYC]. Excellent agreement between our theoretical results and the experimental data for the magnetic susceptibility, specific heat and magnetization confirms that the strong single-ion anisotropy, which is induced by an orbital splitting, can dominate the low temperature behaviour of this class of compounds. Our exact results for the integrable spin-$1$ model may provide widespread application in the study of thermal and magnetic properties of other real compounds, such as NDPK [@NENC; @sus] and certain nickel salts [@PRB3488; @PRB3523; @PRB4009; @Carlin].
The integrable spin-$1$ model
=============================
In contrast to the standard Heisenberg spin-$1$ materials, experimental measurements on the new spin-$1$ compound LiVGe$_2$O$_6$ [@SP1] and the compounds NENC and NBYC [@NENC; @NBYC] exhibit unexpected behaviour, possibly due to the presence of biquadratic interaction and a strong single-ion anisotropy, making it very amenable to our approach. The axial distortion of the crystalline field in the compounds NENC and NBYC results from the triplet $^3A_{2g}$ splitting. Specifically, the triplet orbit splits into a low-lying doublet ($d_{xy}, d_{yz}$) and a singlet orbital ($d_{xz}$) at an energy $\Delta_{CF}$ above the doublet. Inspired by the high temperature magnetic properties of this kind of material, we consider an integrable spin-$1$ chain with Hamiltonian $$\begin{aligned}
{\cal H}&=&J\,{\cal H}_0+D\sum_{j=1}^N(S_j^z)^2+E\sum_{j=1}^N((S_j^x)^2-(S_j^y)^2) \nonumber\\
& &
-\mu_Bg H\sum_{j=1}^N S^z_j, \label{Ham1}\\
{\cal H}_0&=& \sum_{j=1}^{N}\left\{\vec{S}_j\cdot \vec{S}_{j+1}+(\vec{S}_j\cdot
\vec{S}_{j+1})^2\right\}. \nonumber\end{aligned}$$ ${\cal H}_0$ is the standard $su(3)$ integrable spin chain, which is well understood [@U; @BA; @Fujii; @sun]. Here $\vec{S}_i$ denotes the spin-$1$ operator at site $i$, $N$ is the number of sites and periodic boundary conditions apply. The constants $J$, $D$ and $E$ denote exchange spin-spin coupling, single-ion anisotropy and in-plane anisotropy, respectively. The Bohr magneton is denoted by $\mu_B$ and $g$ is the Land$\acute{e}$ factor. We consider only antiferromagnetic coupling, i.e. $J>0$ and $D>0$.
The ground state at zero temperature {#sec:TBA}
------------------------------------
For the sake of simplicity in analyzing the ground state properties at zero temperature, we first take $E=0$, i.e., no in-plane anisotropy. In this case Hamiltonian (\[Ham1\]), which can be derived from the $su(3)$ row-to-row quantum transfer matrix with appropriate chemical potentials in the fundamental basis, is integrable by the Bethe Ansatz. The energy is given by $${\cal E}=-J\sum_{j=1}^{M_1}\frac{1}{(v_j^{(1)})^2+\frac{1}{4}}-DN_0-\mu_BgH(N_+-N_-),$$ where the parameters $v_j^{(1)}$ satisfy the Bethe equations [@U; @BA] $$\begin{aligned}
\prod_{i=1}^{M_{k-1}}\frac{v_j^{(k)}\!-\!v_i^{(k-1)}\!+\!\frac{\mathrm{i}}{2}}
{v_j^{(k)}\!-\!v_i^{(k-1)}\!-\!\frac{\mathrm{i}}{2}}
%
%
&=&
%
%
\prod^{M_k}_{\stackrel{\scriptstyle l=1}{l\neq j}}
\frac{v_j^{(k)}-v_l^{(k)}+\mathrm{i}}{v_j^{(k)}-v_l^{(k)}-\mathrm{i}}\nonumber\\
%
%
&% \!\!\!
\times & % \!\!\! &
%
%
\!\! \prod^{M_{k+1}}_{l=1}\!\!\frac{v_j^{(k)\!\!}-\!v_l^{(k+1)}\!-\!\frac{\mathrm{i}}{2}}
{v_j^{(k)}\!\!-\!v_l^{(k+1)}\!+\!\frac{\mathrm{i}}{2}}.\label{BE}\end{aligned}$$ In the above, $k=1,2$ and $ j=1,...,M_k$ and the conventions $v_j^{(0)}=v_j^{(3)}=0,\, M_3=0$ apply. $N_{+},N_{0},N_{-}$ denote the number of sites with spin $S^z=1,0,-1$ in the Bethe eigenstates. In the thermodynamic limit, the Bethe ansatz equations (\[BE\]) admit complex string solutions [@TBA] from which the TBA equations can be derived [@TBA2; @ying]. Following the standard TBA analysis, we find that the ground state in the zero temperature limit is gapped if the single-ion anisotropy $D \!>\! 4J$. The singlet ground state is separated from the lowest spin excitation by an energy gap $\Delta =D-4J$. This energy gap is decreased by the external magnetic field $H$. At the critical point $H_{c1}=(D-4J)/\mu_Bg$, the singlet ground state breaks down. Due to the magnon excitation, the magnetization almost linearly increases with the magnetic field. Once the magnetic field is increased beyond the second critical point $ H_{c2}=( D+4J)/\mu_Bg$ the ground state is fully polarized, i.e., in the $M\!=\!M_s$ plateau region. The magnetization derived from the TBA is shown in figure \[fig:SP1SZ\]. We remark that a gapped phase exists only for anisotropy values satisfying the ‘strong anisotropy’ condition $D> 4J$. As shown in Ref. [@TBA2; @ying] for the spin ladders, the magnetization in the vicinity of the critical fields $H_{c1}$ and $H_{c2}$ depends on the square root of the field, indicating a Pokrovsky-Talapov transition. In this regime, the anisotropy effects overwhelm the contribution from the biquadratic interaction and open a gapped phase in the ground state.
When $E\neq 0$, the in-plane anisotropy $x^2-y^2$ breaks the $z^2$ symmetry and weakens the energy gap. In the presence of the in-plane anisotropy term $E$, the energies split into three levels with respect to the new basis $\phi_0 \!\! = \, \mid \!\!\! 0 \, \rangle$ and $\phi_{\pm} \!\! = a_{\pm} \!\!\! \mid\!\!\! -1 \, \rangle + \!\!\! \mid\!\!\!1\, \rangle$, with $a_{\pm}=[\mu_BgH\pm \sqrt{(\mu_BgH)^2+E^2}\,]/E$. In this basis the eigenvalues of the underlying permutation operator are the same as the eigenvalues using the fundamental basis. The model thus remains integrable. We find that if $E<D$, there is still a gapped phase with gap $\Delta =D-4J-\sqrt{(\mu_BgH)^2+E^2}$ for the region $H<H_{c1}$. Here the critical field $H_{c1}=\sqrt{(D-4J)^2-E^2}/\mu_Bg$. In this gapped phase the ground state is the non-degenerate singlet. Subsequently, when $H>H_{c1}$ the state $\phi_-$ gets involved in the ground state. At the critical point $H_{c1}$, the phase transition is not of the Pokrovsky-Talapov type due to the mixture of a doublet state in the $\phi_-$ state. The magnetization increases as the magnetic field increases. Past the second critical point $H_{c2}=\sqrt{(D+4J)^2-E^2}/\mu_Bg$, the singlet state is no longer involved in the ground state. The state $\phi _-$ fully occupies the ground state. As the magnetic field is increased beyond $H_{c2}$, the (normalized) magnetization $M={H}/{\sqrt{H^2+(E/\mu_Bg)^2}}$ gradually approaches $M_s=1$. These novel phase transition may be observed from the low temperature magnetization curve, which can be evaluated from the TBA equations at $T=0$, as per the example in figure \[fig:SP1SZ\]. It shows that the gap sensitively depends on the single-ion anisotropy and the in-plane anisotropy. These phase transitions disappear at high temperatures. In addition, the inflection point at $H=\sqrt{D^2-E^2}/\mu_Bg$ and $M=\frac{1}{2}\sqrt{1-({E}/{D})^2}$ indicates that the probabilities of the components $\phi_0$ and $\phi_-$ are equal. Moreover, if the exchange interaction decreases, the magnetization in the vicinity of the critical point $H_{c1}$ increases steeply. For $J=0$, i.e. the case of independent spins, the critical points $H_{c1}$ and $H_{c2}$ merge into one point, at which a discontinuity in the magnetization occurs. For $D<4J+E$, there is no gapped phase.
Magnetic properties at high temperature
---------------------------------------
In order to study thermodynamic properties, we adopt the Quantum-Transfer-Matrix (QTM) approach [@QTM]. Explicitly, following [@ZT] the eigenvalue of the QTM for the model (\[Ham1\]) (up to a constant) is given by $$\begin{aligned}
T^{(1)}_1(v,\left\{v^{(a)}_i\right\})&=&e^{\beta \mu_1}\phi _-(v-\mathrm{i})
\phi _+(v)\frac{Q_1(v+\frac{\mathrm{i}}{2})}{Q_1(v-\frac{\mathrm{i}}{2})}\nonumber\\
&+&
e^{\beta \mu_2}\phi _-(v)\phi _+(v)
\frac{Q_1(v-\frac{3\mathrm{i}}{2})Q_2(v)}{Q_1(v-\frac{\mathrm{i}}{2})Q_2(v-\mathrm{i})}\nonumber\\
&+&e^{\beta \mu_3}\phi _-(v)\phi _+(v+\mathrm{i})
\frac{Q_2(v-2\mathrm{i})}{Q_2(v-\mathrm{i})}.
\label{EQTM1}\end{aligned}$$
In the above equation the chemical potential terms are $$\mu_1=\mu_BgH,\,\,\mu_2=D,\,\, \mu_3=-\mu_BgH,
\label{cp1}$$ where for the moment we take $E=0$. We have adopted the notation from [@ZT] with $\phi _{\pm}(v)=(v\pm
\mathrm{i}u_N)^{\frac{N}{2}}$, $Q_a(v)=\prod_{i=1}^{M_a}(v-v_i^{(a)})$ for $a=1,2$, and $Q_0(v)=1$. Here $u_N=-J\beta/N$ where $N$ is the Trotter number. Following the HTE scheme [@ZT], we derive the high temperature expansion for the free energy of model (\[Ham1\]) in powers of ${J}/{T}$. Because the expansion parameter ${J}/{T}$ is small for weak intrachain coupling $J$, we may expect the free energy to accurately describe the thermodynamic quantities at sufficiently high temperatures, even for a small number of terms. To third order, the result is $$\begin{aligned}
%\hspace{-1cm}
-\frac{1}{T}f(T,H)&=&\ln{C_0} + C^1_{1,0}\, \frac{J}{T}
+ C^1_{2,0} \left(\!\frac{J}{T}\!\right)^{\!\!2} \nonumber
\\
&+&\, C^1_{3,0} \left(\!\frac{J}{T}\!\right)^{\!\!3}+...
\label{fe}\end{aligned}$$ The coefficients $C_{b,0}^{1},\, b=1,2,3$, are given by [@ZT] $$\begin{aligned}
C^1_{1,0} &=& 2A_+,\nonumber\\
C^1_{2,0} &=& 3A_+(1-2A_+)+3A_-,\\
C^1_{3,0} &=& {\textstyle{\frac{10}{3}}\displaystyle} A_+
(1-{\textstyle{\frac{27}{5}}\displaystyle} A_+ + 8 A_+^2)+8A_-(1-3A_+), \nonumber\label{Fcoef}\end{aligned}$$ with $$\begin{aligned}
C_0&=& B_{0,D},\nonumber\\
A_+ &=& B_{D,0}/B_{0,D}^2,\nonumber\\
A_-&=&\exp(D/T)/B_{0,D}^3, \nonumber\\
B_{x,y}&=& 2\exp(x/T)\cosh(\mu_BgH/T)+\exp(y/T).\label{SP1c1}\end{aligned}$$
For later use, we also give the HTE free energy with in-plane rhombic anisotropy $E$. If the external magnetic field is parallel to the $z$-axis, the chemical potentials in Eq. (\[EQTM1\]) become $$\mu_1=h,\,\,\mu_2=D,\,\, \mu_3=-h,$$ where $h=\sqrt{E^2+(\mu_Bg_{\parallel}H)^2}$. In this case the function $B_{x,y}$ in Eq. (\[SP1c1\]) changes to $$B_{x,y}= 2\exp(x/T)\cosh(h/T)+\exp(y/T).\label{SP1c2}$$
On the other hand, if we apply a perpendicular magnetic field to the Hamiltonian (\[Ham1\]), the chemical potential terms in Eq. (\[EQTM1\]) are replaced by $$\begin{aligned}
\mu_1&=&\frac{1}{2}(D-E+h'),\nonumber\\
\mu_2&=&E,\\
\mu_3&=&\frac{1}{2}(D-E-h'), \nonumber
\label{case3}\end{aligned}$$ where $h'= \sqrt{(D+E)^2+4g_{\perp}^2\mu_B^2H_{a}^2}$ with $a=x$ or $y$ [@foot1]. Subsequently, we have $$\begin{aligned}
C_0&=& B_{(D-E)/2,D},\nonumber\\
A_+&=&B_{(D+E)/2,D-E}/B_{(D-E)/2,D}^2, \label{SP1c3}\\
A_-&=&\exp(D/T)/B_{(D-E)/2,D}^3, \nonumber\end{aligned}$$ with now $B_{x,y}=2\exp(x/T)\cosh(h'/T)+\exp(y/T)$.
Eq. (\[fe\]) for the free energy $f(T,H)$ is our key result. Physical properties such as the susceptibility, magnetization and the specific heat follow in the usual way by differentiation. We also use $f(T,H)$ to calculate the phase diagram for both $T \simeq 0$ K and finite temperatures (see figure \[PD-NENC\]). We find that considering up to 3rd order in $J/T$ is sufficient as higher orders are negligibly small. This is in stark contrast to other series expansions which need many orders to accurately describe physical properties. This is mainly because here the coefficents are not just constants, but functions of the external model parameters, e.g., the magnetic field and the coupling strength.
Spin-$1$ compounds
==================
The compound NENC
-----------------
It is known that antiferromagnetic spin-$1$ chains [@SP1C1; @SP1C2] with weak planar anisotropy can exhibit a non-magnetic gapped phase. The large $D$ gapped phase has been observed in the compounds NENC, NDPK and NBYC [@NENC; @NBYC]. In these compounds the in-plane anisotropy $x^2-y^2$ breaks the $z^2$ symmetry and weakens the planar anisotropy. From experimental analysis, it was inferred that the in-plane anisotropy $E$ in NENC [@NENC] is negligible in comparison with the large $D$ single-ion anisotropy, where the Nickel(II) $z^2$ orbit along the c-axis forms a strong crystalline field. As a result the low temperature physics is dominated by this strong crystalline field. The antiferromagnetic exchange interaction further lowers the energy but its contribution to the ground state as well as the low-lying excitations is minimal. As a consequence, the Hamiltonian (\[Ham1\]) can be expected to describe this compound quite well. Experimentally, the specific heat was measured up to a temperature around $10$ K in the absence of magnetic field [@NENC]. A typical round peak for short range ordering at $T\approx 2.4$ K is observed, see Figure \[FIGheat\]. An exponential decay is detected for temperatures below approx $2.4$ K. Our calculated HTE specific heat for the Hamiltonian (\[Ham1\]) with best visual fit constants $J=0.17$ K and $D=6.4$ K in the case (\[cp1\]) (the solid line in Figure \[FIGheat\]) is in excellent agreement with the experimental curve in the temperature region $T>0.8$ K. In particular, the analytic result for the specific heat gives a better fit with experimental data than the result from perturbation theory [@NENC]. For low temperatures (below $0.8$ K), paramagnetic impurities and a small rhombic distortion are the main reasons for the discrepancy. The inset of Figure \[FIGheat\] shows that the inclusion of a small rhombic anisotropy $E=0.7$ K gives a better fit for low temperatures than with $E=0$. However, at high temperature this rhombic anisotropy is negligible.
As far as we know, the susceptibility was measured only for powdered samples of this compound. Moreover, the experimental susceptibility of NENC was studied only in the temperature range 50 mK - 18 K under a static magnetic field $H=0.1$ mT. From the data shown in Ref. [@NENC] we cannot accurately estimate the contributions for the Curie-Weiss term and the paramagnetic impurity. In Figure \[FIGSZ\] we present our theoretical curves for the susceptibility with parallel and perpendicular field evaluated from the free energy associated with different chemical potentials. A susceptibility estimation for powdered samples using $\chi _{{\rm Powder}} \approx \frac{1}{3}\chi_{\parallel}+\frac{2}{3}\chi_{\perp}$ [@Carlin] does not fit the experimental data very well at low temperatures due to the Curie-Weiss contribution and paramagnetic impurities. A visual fit with the experimental susceptibility suggests that the contribution from the Curie-Weiss term is not negligible. We find that our theoretical susceptibility $\chi _{{\rm Powder}}$ for powder with a Curie-Weiss contribution $c/(T-\theta)$ gives a satisfactory agreement with the experimental curves, where $c \approx 0.045 $ cm$^{3}$ K/mol and $\theta \approx -0.9$ K. This fit suggests the values $J=0.17$ K and $D=6.4$ K, with $g_{\perp}= 2.18$ and $g_{\parallel}=2.24 $. From the TBA analysis we find an energy gap $\Delta \approx 5.72 $ K with a parallel external magnetic field at zero temperature for these coupling constants. The typical antiferromagnetic behaviour of the susceptibility with a parallel magnetic field to the axis of quantization follows from our results. This is in accordance with the behaviour of the specific heat given in Figure \[FIGheat\]. The inset of Figure \[FIGSZ\] shows the magnetization of a powdered sample at $T=4.27$ K. It is obvious that the singlet is supressed by the temperature. Fitting suggests the empirical relation $M_{\rm Powder} \approx \frac{1}{3}M_{\parallel}+\frac{2}{3}M_{\perp}$ for the powdered magnetization with the same constants as before. Here $M_{\parallel}$ and $M_{\perp}$ denote the magnetization with the field parallel and perpendicular to the axis of quantization.
The compound NBYC
-----------------
We now turn to the properties of the compound NBYC, which has also been experimentally investigated [@NBYC]. In particular, in-plane anisotropy $E$ and a large anisotropy $D$ are present, suggesting that the model Hamiltonian (\[Ham1\]) may again be a good microscopic model for this type of compound. Theoretical studies based on strong-coupling expansion methods [@Spathis] suggest that the anisotropy of this compound might lie in the vicinity of the boundary between the Haldane and field-induced gapped phases [@NBYC]. However, due to the validity of the strong-coupling expansion method, the fits for specific heat, susceptibility and magnetization become increasingly inconsistent with each other as the rhombic anisotropy increases. Figure \[FIGNBYC\] presents the susceptibility for this compound. The theoretical susceptibility curve for the powdered sample is evaluated from the free energy (\[fe\]) with parallel and perpendicular fields (see Eq. (\[case3\])) via the empirical formula $\chi_{{\rm Powder}} \approx \frac{1}{3}\chi_{\parallel}+\frac{2}{3}\chi_{\perp}$. A good fit for the susceptibility suggests the values $D=2.62$ K, $E=1.49$ K, $J=0.35 $ K, with $g_{\parallel}=g_{\perp}=2.05$. A small discrepancy at low temperature can be attributed to a Curie-Weiss contribution term. From the TBA analysis we conclude that the ground state is gapless.
The inset of Figure \[FIGNBYC\] shows the magnetization for powdered samples at $5$ K, $10$ K and $20$ K. Again our theoretical curves are evaluated using the empirical relation $M_{\rm Powder} \approx \frac{1}{3}M_{\parallel}+\frac{2}{3}M_{\perp}$ for the powdered magnetization. An overall agreement in magnetization for different temperatures gives a consistent parameter setting for the susceptibility. The singlet state is now supressed by the in-plane rhombic anisotropy and the temperature.
The specific heat was measured up to a temperature of $6$ K in absence of magnetic field [@NBYC]. The theoretical specific heat evaluated from the model Hamiltonian (\[Ham1\]) (the solid line in Figure \[FIGNBYCheat\]), with the same parameters used before, is in good agreement with the experimental curve in the temperature region $0.5$ K to $6$ K. For temperatures below $0.5$ K the high temperature expansion does not converge and thus cannot provide valid predictions.
Conclusion
==========
We have investigated the thermal and magnetic properties of spin-$1$ compounds with large single-ion anisotropy, such as NENC and NYBC, via the thermodynamic Bethe Ansatz and the high temperature expansion for the integrable model (1). Excellent agreement was found with the experimental magnetic properties of these compounds [@foot2]. The large single-ion anisotropy results in a nondegenerate singlet ground state which is different from the valence bond solid Haldane phase. The in-plane anisotropy weakens the energy gap.
Finally, we give the full phase diagram of the compound NENC in Fig. \[PD-NENC\]. We see that the gapped phase is quickly exhausted as the temperature increases. The magnetic ordered Luttinger liquid phase lies between the curves defined by $H_{c1}$ and $H_{c2}$. The ferromagnetic polarized phase is above the $H_{c2}$ curve. The intersection of the critical curves and the $H$-axis indicates the estimated values $H_{c1} \approx 3.8$ T and $H_{c2}\approx 4.7$ T, which coincide with the TBA results at $T=0$ K discussed in section \[sec:TBA\]. However, for the case where the in-plane anisotropy $E\neq 0$, the critical behaviour is different from the phase diagram of Fig. \[PD-NENC\]. In this case the fully-polarized phase appears for $H>>H_{c2}$ because the in-plane anisotropy mixes the doublet components $\mid \!\! S^z=\pm 1 \rangle$. We anticipate that the exact results for the susceptibility and the magnetization of the powdered samples as well as for the compounds with parallel and perpendicular magnetic fields may find widespread use in the study of their magnetic properties and for identifying the quantum effects resulting from single-ion anisotropy. Our analytic approach via the Hamiltonian (1) may thus describe the thermal and magnetic properties of other compounds, such as NDPK [@NENC; @sus], NiSnCl$_6\cdot 6$H$_2$O [@PRB3488], \[Ni(C$_5$H$_5$NO)$_6$\](ClO$_4$)$_2$ [@PRB3523], and Ni(NO$_3$)$_2\cdot 6$H$_2$O [@PRB4009].
[*Acknowledgements.*]{} This work has been supported by the Australian Research Council. N. Oelkers also thanks DAAD for financial support. We thank Z. Tsuboi, A. Foerster and H.-Q. Zhou for helpful discussions. We also thank M. Orendác for providing us with experimental results and helpful discussions.
[99]{}
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We find similar agreement for the compound NDPK.
![Magnetization versus magnetic field $H$ in units of saturation magnetization for Hamiltonian (\[Ham1\]) with $J=0.2$ K, $D=6$ K, $E=3.5$ and $0$ K, $g=2.0$ with parallel magnetic field. The solid and dashed lines denote the magnetization derived from the TBA with $E=3.5$ K and $E=0$ K, respectively. The magnetization curve for $E=3.5$ K indicates different quantum phase transitions in the vicinity of $H_{c1}$ and $H_{c2}$ than the $E=0$ K square root field-dependent critical behaviour in the absence of the in-plane anisotropy. []{data-label="fig:SP1SZ"}](figure1.eps){width="0.95\linewidth"}
![ Comparison between theory and experiment [@NENC] for the magnetic specific heat versus temperature of the compound NENC. The conversion constant is $C_{{\rm HTE}} \approx 8 C_{{\rm EXP}}$ (J/mol-K). The solid line denotes the specific heat evaluated directly from the free energy (\[fe\]) with the paramet ers $J=0.17$ K, $D=6.4$ K, $g=2.24$ and $\mu_B=0.672$ K/T. The inset shows the low temperature specific heat. Clearly the inclusion of in-plane rhombic anisotropy $E=0.7$ K (dashed line) gives a better fit than without rhombic anisotropy (solid line).[]{data-label="FIGheat"}](figure2.eps){width="0.95\linewidth"}
![ Comparison between theory and experiment [@sus] for the susceptibility versus temperature of the compound NENC at $H=0.1$ mT. The fitting curve (solid line) is obtained via the empirical relation $\chi _{{\rm Powder}} \approx \frac{1}{3}\chi_{\parallel}+\frac{2}{3}\chi_{\perp}$ together with a Curie-Weiss (CW) contribution. The inset shows the comparison between theory and experiment [@NENC] for the magnetization versus magnetic field of NENC at the temperature $T=4.27K$. A good fit for both the susceptibility and magnetization suggests the coupling constants $J=0.17$ K, $D=6.4$ K, $g_{\perp}= 2.18$ and $g_{\parallel}=2.24$. The conversion constants are $\chi_{{\rm HTE}}\approx 0.8123 \chi_{{\rm EXP}}$(cgs/mol) and $M_{{\rm HTE}}\approx 8.5 M_{{\rm EXP}}$ ($10^3$ cgs/mol). []{data-label="FIGSZ"}](figure3.eps){width="0.95\linewidth"}
![ Comparison between theory and experiment [@NBYC] for the susceptibility versus temperature of the compound NBYC. The conversion constants are the same as for NENC. The solid line is the susceptibility for the powdered samples with coupling constants $D=2.62$ K, $E=1.49$ K and $J=0.35 $ K, with $g_{\parallel}=g_{\perp}=2.05$. The small discrepancy at low temperature might be attributed to a Curie-Weiss contribution. The inset shows the magnetizations for powdered samples at $5$ K, $10$ K and $20$ K. In each case the theoretical results verify the existence of weak exchange coupling and in-plane rhombic anisotropy, with a strong single-ion anisotropy. []{data-label="FIGNBYC"}](figure4.eps){width="0.95\linewidth"}
![ Comparison between theory and experiment [@NBYC] for the magnetic specific heat versus temperature of the compound NBYC. The conversion constant is $C_{{\rm HTE}} \approx 10 C_{{\rm EXP}}$ (J/mol-K). The solid line denotes the specific heat at $H=0.1$ mT evaluated directly from the free energy (\[fe\]) with the same parameters as in Figure \[FIGNBYC\]. []{data-label="FIGNBYCheat"}](figure5.eps){width="0.95\linewidth"}
![ Phase diagram for the compound NENC with parameters $J=0.17$ K, $D=6.4$ K, $g=2.24$ and $\mu_B=0.672$ K/T. []{data-label="PD-NENC"}](figure6.eps){width="0.95\linewidth"}
| ArXiv |
---
abstract: 'In this paper we continue our studies of the phase space geometry and dynamics associated with index $k$ saddles ($k > 1$) of the potential energy surface. Using Poincaré-Birkhoff normal form theory, we give an explicit formula for a “dividing surface” in phase space, i.e. a co-dimension one surface (within the energy shell) through which all trajectories that “cross” the region of the index $k$ saddle must pass. With a generic non-resonance assumption, the normal form provides $k$ (approximate) integrals that describe the saddle dynamics in a neighborhood of the index $k$ saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having 4 minima; two symmetry related pairs of minima are connected by low energy index one saddles, with the pairs themselves connected via higher energy index one saddles and an index two saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing (“hilltop crossing”) isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional “bath modes” on the dynamics, which is a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.'
author:
- Peter Collins
- 'Gregory S. Ezra'
- Stephen Wiggins
title: 'Index $k$ saddles and dividing surfaces in phase space, with applications to isomerization dynamics'
---
Introduction {#sec:intro}
============
Transition state theory has long been, and continues to be, a cornerstone of the theory of chemical reaction rates [@Wigner38; @Keck67; @Pechukas81; @Truhlar83; @Anderson95; @Truhlar96]. A large body of recent research has shown that index one saddles [@saddle_footnote1] of the potential energy surface [@Mezey87; @Wales03] give rise to a variety of geometrical structures in [*phase space*]{}, enabling the realization of Wigner’s vision of a transition state theory constructed in *phase space* [@Wiggins90; @wwju; @ujpyw; @WaalkensBurbanksWiggins04; @WaalkensWiggins04; @WaalkensBurbanksWigginsb04; @WaalkensBurbanksWiggins05; @WaalkensBurbanksWiggins05c; @SchubertWaalkensWiggins06; @WaalkensSchubertWiggins08; @MacKay90; @Komatsuzaki00; @Komatsuzaki02; @Wiesenfeld03; @Wiesenfeld04; @Wiesenfeld04a; @Komatsuzaki05; @Jaffe05; @Wiesenfeld05; @Gabern05; @Gabern06; @Shojiguchi08].
Following these studies, attention has naturally focussed on phase space structures associated with saddles of index greater than one, and their possible dynamical significance [@Ezra09; @Haller10; @Haller10a]. In previous work we have described the phase space structures and their influence on transport in phase space associated with [*index two saddles*]{} of the potential energy surface for $n$ degree-of-freedom (DoF) deterministic, time-independent Hamiltonian systems [@Ezra09]. (The case of higher index saddles has also been investigated by Haller et al. [@Haller10; @Haller10a]; see also refs .)
The phase space manifestation of an index one saddle of the potential energy surface in an $n$ DoF system is an equilibrium point of the associated Hamilton’s equations of saddle-center-$\ldots$-center stability type. This means that the matrix associated with the linearization of Hamilton’s equations about the equilibrium point has one pair of real eigenvalues of equal magnitude, but opposite in sign ($\pm \lambda$) and $n-1$ pairs of purely imaginary eigenvalues, $\pm i \omega_j$, $j=2, \ldots , n$.
The phase space manifestation of an index $k$ saddle is an equilibrium point of saddle stability type: $$\label{eq:saddle_center}
\underbrace{\text{saddle}\times\ldots\times\text{saddle}}_{k \;\; \text{times}}
\times \underbrace{\text{center} \times \ldots \times\text{center}}_{n-k \;\; \text{times}}.$$ The matrix associated with the linearization of Hamilton’s equations about the equilibrium point then has $k$ pairs of real eigenvalues of equal magnitude, but opposite in sign ($\pm \lambda_i$, $i=1,\ldots, k$) and $n-k$ pairs of purely imaginary eigenvalues, $\pm i \omega_j$, $j=k+1, \ldots , n$ [@saddle_footnote2]. Informally, an index $k=2$ saddle on a potential surface corresponds to a maximum or “hilltop” in the potential [@Heidrich86; @Mezey87; @Wales03].
Although it has been argued on the basis of the Murrell-Laidler theorem [@Murrell68; @Wales03] that critical points of index $2$ and higher are of no direct chemical significance [@Mezey87; @Minyaev91], many instances of index 2 (and higher) saddles of chemical significance have been identified [@Heidrich86]. For example, Heidrich and Quapp discuss the case of face protonated aromatic compounds, where high energy saddle points of index two prevent proton transfer across the aromatic ring, so that proton shifts must occur at the ring periphery [@Heidrich86]. Index two saddles are found on potential surfaces located between pairs of minima and index one saddles, as in the case of internal rotation/inversion in the H$_2$BNH$_2$ molecule [@Minyaev97] or in urea [@Bryantsev05], or connected to index one saddles connecting four symmetry related minima, as for isomerization pathways in B$_2$CH$_4$ [@Fau95].
Saddles with index $>1$ might well play a significant role in determining system properties and dynamics for low enough potential barriers [@Carpenter04; @Bachrach07] or at high enough energies [@Meroueh02]. The role of higher index saddles in determining the behavior of supercooled liquids and glasses [@Cavagna01; @Cavagna01a; @Doye02; @Wales03a; @Angelini03; @Shell04; @Grigera06; @Coslovich07; @Angelini08] is a topic of continued interest, as is the general relation between configuration space topology (distribution of saddles) and phase transitions [@Kastner08].
Several examples of non-MEP (minimum energy path) reactions [@Mann02; @Sun02; @Debbert02; @Ammal03; @Carpenter04; @Lopez07; @Lourderaj08] and “roaming” mechanisms [@Townsend04; @Bowman06; @Shepler07; @Shepler08; @Suits08; @Heazlewood08] have been identified in recent years; the dynamics of these reactions is not mediated by a single conventional transition state associated with an index one saddle. Higher index saddles can also become mechanistically important for structural transformations of atomic clusters [@Ball96] when the range of the pairwise potential is reduced [@Berry96]. We note for example the work of Shida on the importance of high index saddles in the isomerization dynamics of Ar$_7$ clusters [@Shida05].
The role of index two saddles in the (classical) ionization dynamics of the Helium atom in an external electric field [@Eckhardt01; @Sacha01; @Eckhardt06] has recently been studied from a phase space perspective by Haller et al. [@Haller10; @Haller10a].
In previous work we have discussed phase space structures and their influence on phase space transport in some detail for the case of an index two saddle of the potential energy surface corresponding to an equilibrium point of saddle–saddle–center-$\ldots$–center stability type [@Ezra09]. In this paper we extend our analysis in several respects.
One motivation for the work presented here is the possibility of developing a dynamically based characterization of *concerted* and *sequential* reaction pathways [@Carpenter04; @Bachrach07] in phase space. Consider isomerization dynamics on the model potential shown in Fig. \[fig:seq\_con\]. (This potential is discussed in more detail in Sec. \[sec:model\_potential\].) The potential has 4 minima; two symmetry related pairs of minima are connected by low energy index one saddles, while the pairs themselves are connected via higher energy index one saddles. An index two saddle (hilltop, denoted $\ddagger\ddagger$) is located at the origin. In Figure \[fig:seq\_con\], we indicate schematically two possible pathways between the lower left hand well (designated $(--)$; the symbolic code is discussed further in Sec. \[sec:crossing\]) and the upper right hand well, $(++)$ (see also Fig. 7 of ref. ). At energies above that corresponding to the index two saddle, there are, qualitatively speaking, two possible isomerization routes: a *sequential* path, shown in Fig. \[fig:seq\_con\]a, in which a trajectory passes through an intermediate well (either $(+-)$ or $(-+)$) in the course of the isomerization, and a *concerted* route, shown in Fig. \[fig:seq\_con\]b, in which the trajectory effectively passes directly from the reactant well to product well without entering a well corresponding to an ‘intermediate’ species.
It is natural to ask whether the above qualitative distinction between sequential and concerted reaction pathways [@Carpenter04; @Bachrach07], made on the basis of considerations of the nature of isomerizing trajectories in configuration space, can be given a rigorous dynamical formulation in phase space. In the present work we provide such a formulation, based on the properties of the normal form in the vicinity of the index two saddle point. In particular, we show that it is possible to define phase space dividing surfaces for higher index saddles which generalize the now-familiar dividing surfaces defined for index one saddles [@Wiggins90; @wwju; @ujpyw; @WaalkensBurbanksWiggins04; @WaalkensWiggins04; @WaalkensBurbanksWigginsb04; @WaalkensBurbanksWiggins05; @WaalkensBurbanksWiggins05c; @SchubertWaalkensWiggins06; @WaalkensSchubertWiggins08; @MacKay90; @Komatsuzaki00; @Komatsuzaki02; @Wiesenfeld03; @Wiesenfeld04; @Wiesenfeld04a; @Komatsuzaki05; @Jaffe05; @Wiesenfeld05; @Gabern05; @Gabern06; @Shojiguchi08] to the case of higher indices, and that phase points can be chosen on such dividing surfaces with prescribed dynamical character (e.g., concerted crossing trajectories).
The structure of this paper is as follows. In Sec. \[sec:ndof\] we review and extend our previous analysis [@Ezra09] of the phase space structure in the vicinity of a (non-resonant) index $k$ saddle in terms of the normal form. Particular emphasis is given to discussion of the dynamical significance of the values of the associated action integrals, and to the symbolic representation of the qualitatively distinct classes of trajectory behavior in the vicinity of the saddle-saddle equilibrium. For generic non-resonance conditions on the eigenvalues of the matrix associated with the linearization of Hamilton’s equations about the equilibrium point, the normal form Hamiltonian is integrable. Integrability provides all of the advantages that separability provides for quadratic Hamiltonians: the saddle dynamics can be described separately and the integrals associated with the saddle DoFs can be used to characterize completely the geometry of trajectories passing through a neighborhood of the equilibrium point. As for the case of index one saddles, normally hyperbolic invariant manifolds (NHIMs) [@Wiggins90; @Wiggins94] associated with index two saddles are an important phase space structure and we give a brief discussion of their existence and the role they play in phase space transport in the vicinity of index $k$ saddles, following our work in [@Ezra09].
In Sec. \[sec:crossing\] we introduce the concept of concerted crossing trajectories. The concerted crossing trajectories are defined in terms of the symbolic code introduced previously [@Ezra09], and realize the inuitive notion of direct, concerted passage between wells via the hilltop region. Those trajectories that are not concerted crossing are potentially, but not necessarily, sequentially isomerizing trajectories.
Sec. \[sec:DS\] defines the dividing surface (DS) for index $k$ saddles. This is a key aspect of the present work; we show that it is possible to define a codimension one surface in phase space through which all concerted crossing trajectories must pass, and which is everywhere transverse to the flow. We also introduce a parametrization of the dividing surface. This parametrization together with the use of the normal form enables us to sample the DS and select phase points on trajectories having specified dynamical character.
In Section \[sec:model\_potential\] we present a numerical study of the phase space structure in the vicinity of an index two saddle in the context of a problem of chemical dynamics, namely, isomerization in a multiwell potential. For isomerization on this model potential energy surface, which has multiple (four) symmetry equivalent minima, analysis of the phase space structure in the vicinity of the index two saddle enables a rigorous distinction to be made between concerted crossing (“hilltop crossing”) isomerizing trajectories and those trajectories that are not concerted (potentially sequentially isomerizing) [@Carpenter04; @Bachrach07]. Our normal form based procedure for sampling the DS enables us to determine phase points lying on concerted crossing trajectories. Numerical propagation forwards and backwards in time shows that such trajectories do indeed have the properties intuitively associated with the concerted isomerizing pathway. Sec. \[sec:summary\] concludes.
The Poincaré-Birkhoff normal form in a phase space neighborhood of an index $k$ saddle {#sec:ndof}
======================================================================================
We begin by considering the normal form in the neighborhood of an equilibrium point of saddle- $\ldots$-saddle-center-$\ldots$-center stability type, where there are $k$ saddle degrees-of-freedom (DoF) and $n-k$ center degrees-of-freedom (eq. ). We assume the usual non-resonance condition on the eigenvalues of the matrix associated with the linearization of the Hamiltonian vector field about the equilibrium point (see, e.g. [@WaalkensSchubertWiggins08]). In particular, we assume that the purely imaginary eigenvalues satisfy the non-resonance condition $k_{k+1} \omega_{k+1} + \ldots + k_n \omega_n \ne 0$ for any $(n-k)$-vector of integers $(k_{k+1}, \ldots, k_n)$ with [*not all*]{} the $k_i=0$ (that is, $(k_{k+1}, \ldots , k_n) \in {\mathbb Z}^{n-k}-\{0\}$) and the real eigenvalues satisfy the (independent) non-resonance condition $k_1 \lambda_1 + \ldots + k_k \lambda_k\ne 0$ for any $k$-vector of integers $(k_1, \ldots, k_k)$ with [*not all*]{} the $k_i=0$, $i=1, \ldots, k$. In this case the normal form transformation transforms the Hamiltonian to an even order polynomial in the variables
\[ndof\_ints\] $$\begin{aligned}
I_i & = q_i p_i, \, i=1, \ldots, k, \\
I_j & = \frac{1}{2} \left( q_j^2 + p_j^2 \right), \, j=k+1, \ldots, n.\end{aligned}$$
In other words, we can express the normal form Hamiltonian as: $$H(I_1, I_2, I_3, \ldots, I_n),
\label{nf_ham}$$ with associated Hamilton’s equations:
\[nf\_hameq\] $$\begin{aligned}
\dot{q}_i& = \frac{\partial H}{\partial p_i}=
\frac{\partial H}{\partial I_i} \frac{\partial I_i}{\partial p_i} = \Lambda_i q_i, \\
\dot{p}_i& = -\frac{\partial H}{\partial q_i}
= -\frac{\partial H}{\partial I_i}\frac{\partial I_i}{\partial q_i}= -\Lambda_i p_i, \qquad i=1, \ldots, k,\\
\dot{q}_j & = \frac{\partial H}{\partial p_j}= \frac{\partial H}{\partial I_j}\frac{\partial I_j}{\partial p_j} =
\Omega_j p_j, \\
\dot{p}_j & = -\frac{\partial H}{\partial q_j}= -\frac{\partial H}{\partial I_j}\frac{\partial I_k}{\partial q_j}=
- \Omega_j q_j,
\qquad j=k+1, \ldots, n,\end{aligned}$$
where we have defined the frequencies
$$\begin{aligned}
\Lambda_i(\mathcal{I}) & \equiv \frac{\partial H}{\partial I_i}(\mathcal{I}), \;\; i=1,\ldots, k,\\
\Omega_j(\mathcal{I}) & \equiv \frac{\partial H}{\partial I_j}(\mathcal{I}), \;\; j=k+1,\ldots, n,\end{aligned}$$
where $\mathcal{I} = (I_1, I_2, I_3, \ldots, I_n)$ and it can be verified by a direct calculation that the $I_j$, $j=1, \ldots, n$, are integrals of the motion for .
The integrability of the normal form equations provides us with a very straightforward way of characterizing the dynamics in a neighborhood of the index $k$ saddle. The coordinates $(q_i, p_i)$, $i=1, \ldots, k$, describe saddle-type or ‘reaction dynamics’ , and the dynamics in the $k$ saddle planes can be further characterized by the [*saddle integrals*]{}, $I_i$, $i=1, \ldots, k$. The coordinates $(q_i, p_i)$, $i=k+1, \ldots, n$, describe bounded motions (center-type dynamics) or ‘bath modes’, which can be further characterized by integrals $I_i$, $i=k+1, \ldots, n$.
We now introduce a canonical transformation of the saddle variables [@Ezra09]. Passage of trajectories over the saddle is more naturally described in terms of these new coordinates, and computation of codimension one dividing surfaces for index $k$ saddles is also facilitated. The transformation is given by:
\[trans1\] $$\begin{aligned}
q_i & = \frac{1}{\sqrt{2}} \left( \bar{q}_i + \bar{p}_i \right), \\
p_i & = \frac{1}{\sqrt{2}} \left( \bar{p}_i - \bar{q}_i \right), \quad i=1, \ldots, k,\end{aligned}$$
with inverse
\[inv\_trans1\] $$\begin{aligned}
\bar{q}_i & = \frac{1}{\sqrt{2}} \left( q_i - p_i \right), \\
\bar{p}_i & = \frac{1}{\sqrt{2}} \left( p_i + q_i \right), \quad i=1, \ldots k.\end{aligned}$$
The transformation of variables given by eq. , where $i=1, \ldots, k$, and the remaining variables are transformed by the identity transformation, is canonical. The variables $\bar{q}_i$, $i=1,\ldots,k$, are naturally identified with physical configuration space coordinates in the vicinity of the saddle. The Hamiltonian is given by eq. , with action variables
\[ndof\_ints\_2\] $$\begin{aligned}
I_i & = q_i p_i = \frac{1}{2} \left( \bar{p}_i^2 - \bar{q}_i^2 \right), \, i =1, \ldots, k,\\
I_j & = \frac{1}{2} \left( q_j^2 + p_j^2 \right), \, j=k+1, \ldots, n,\end{aligned}$$
and Hamilton’s equations then take the following form:
\[nf\_hameq\_2\] $$\begin{aligned}
\dot{\bar{q}}_i & = \frac{\partial H}{\partial \bar{p}_i}
= \frac{\partial H}{\partial I_i}\frac{\partial I_i}{\partial \bar{p}_i}
= \Lambda_i \bar{p}_i, \\
\dot{\bar{p}}_i & = -\frac{\partial H}{\partial \bar{q}_i}= -\frac{\partial H}{\partial I_i}\frac{\partial I_i}{\partial \bar{q}_i}
= - \Lambda_i \bar{q}_i, \qquad i=1, \ldots, k,\\
\dot{q}_j & = \frac{\partial H}{\partial p_j}= \frac{\partial H}{\partial I_j}\frac{\partial I_j}{\partial p_j}
= \Omega_j p_j, \\
\dot{p}_j & = -\frac{\partial H}{\partial q_j}= -\frac{\partial H}{\partial I_j}\frac{\partial I_j}{\partial q_j}
= - \Omega_j q_j,
\qquad j=k+1, \ldots, n,\end{aligned}$$
A Normally Hyperbolic Invariant Manifold {#sec:NHIM}
----------------------------------------
As noted previously [@Ezra09], for $n>k$ the $2n-2k-1$ dimensional surface: $${\mathcal{M}}\equiv \left\{ \bar{q}_1=\bar{p}_1= \cdots =\bar{q}_k = \bar{p}_k=0, \, H(0, \ldots, 0, I_{k+1}, \ldots, I_n ) =E>0 \right\},
\label{eq:NHIM}$$ is a normally hyperbolic invariant manifold (NHIM) in the energy surface $H(I_1, \ldots, I_n)=E>0$. Moreover, this NHIM has $2n-k-1$ dimensional stable and unstable manifolds (within the fixed energy surface). Note that these stable and unstable manifolds are codimension one in the energy surface only for $k=1$, i.e., index one saddles. Nevertheless, in the construction of dividing surfaces for index $k$ saddles we will see that the NHIM (but not its stable and unstable manifolds) plays a similar role for all $k$.
The case $n=k$, i.e. a $n$ degree-of-freedom system with an index $n$ saddle requires special consideration. In this case the NHIM corresponds to an equilibrium point and exists [*only*]{} on the $E=0$ energy surface. Thus, to get a non-trivial NHIM existing for $E>0$ we must have $n>k$.
Accuracy of the Normal Form {#sec:accuracy}
---------------------------
Here we briefly address some of the issues associated with the accuracy of the normal form. The transformation to normal form is implemented by an algorithm that operates in an iterative fashion by simplifying terms in the Taylor expansion about the equilibrium point order by order, i.e., the order $M$ terms are normalized, then the order $M+1$ terms are normalized, etc [@WaalkensSchubertWiggins08]. The algorithm is such that normalization at order $M$ does not affect any of the lower order terms (which have already been normalized). The point here is that although the algorithm can be carried out to arbitrarily high order, in practice we must stop the normalization (i.e., truncate the Hamiltonian) at some order $M$, after which we make a restriction to some neighborhood of the saddle in which the resulting computations achieve some desired accuracy. It is therefore necessary to determine the accuracy of the normal form as a power series expansion truncated at order $M$ in a neighborhood of the equilibrium point by comparing the dynamics associated with the normal form to the dynamics of the original system. There are several independent tests that can be carried out to verify accuracy of the normal form, such as the following:
- Examine the extent to which integrals associated with the normal form are conserved along trajectories of the full Hamiltonian (the integrals will be constant on trajectories of the normal form Hamiltonian).
- Check invariance of specific invariant manifolds (e.g., the NHIM, its stable and unstable manifolds, the energy surface) under dynamics determined by the full Hamiltonian.
Both of these tests require us to use the transformation between the original coordinates and the normal form coordinates. Software for computing the normal form as well as the transformation (and inverse transformation) between the original coordinates and the normal form coordinates can be downloaded from <http://lacms.maths.bris.ac.uk/publications/software/index.html>. Specific examples where the accuracy of the normal form and its relation to $M$, the fixed neighborhood of the saddle, and the constancy of integrals of the truncated normal form on trajectories of the full Hamiltonian can be found in refs . A general discussion of accuracy of the normal form can be found in ref. .
Crossing and concerted crossing trajectories {#sec:crossing}
============================================
In this section we introduce the notion of [crossing]{} trajectories. *Crossing* trajectories pass through a neighborhood of the index $k$ saddle in such a way that *all* the saddle coordinates $\bar{q}_i$ change sign, and constitute a dynamically well-defined subset of trajectories. Our treatment of crossing trajectories provides an excellent illustration of the power of the normal form (NF) and the utility of the integrals of the motion in analyzing dynamics in the vicinity of the saddle.
The definition of crossing trajectories given here is purely local, in that it relies on the values of the integrals computed using the NF in the vicnity of the saddle. In our discussion of the isomerization dynamics for a model multi-well system given below in Sec. \[sec:model\_potential\], we establish a connection between the local crossing property and the more global mechanistic notion of *concerted* (as opposed to sequential) isomerization trajectories; in this context, the crossing trajectories defined here are usefully described as *concerted crossing* (CC) trajectories.
In the normal form coordinate system the ‘crossing’ of a saddle is analyzed by considering only the saddle degrees-of-freedom, $(\bar{q}_i, \bar{p}_i)$, $i=1, \ldots, k$, since the remaining coordinates remain bounded. Crossing occurs when all coordinates $\bar{q}_i$, $i=1, \ldots, k$, change sign as the trajectory passes through a neighborhood of the index $k$ saddle. Whether or not a given coordinate $\bar{q}_i$ changes sign depends on the value of the integral $I_i$. If $\bar{q}_i$ changes sign in the vicinity of the saddle then $I_i >0$. If $I_i =0$ then $\bar{q}_i$ is zero or evolves to zero either in forward or negative time (depending on the initial condition). We consider the boundary of the region of crossing trajectories, characterised by $I_i =0$ in more detail below.
More precisely, crossing and other trajectories are characterized as follows:
1. If $I_i > 0$ for *all* $i=1, \ldots , k$, then the trajectory is a crossing trajectory.
2. If $I_i = 0$ and $\bar{q}_i = 0$ for *all* $i=1, \ldots , k$, then the trajectory is on the NHIM and is not a crossing trajectory.
3. If $I_i = 0$ for any $i=1, \ldots , k$ and $\bar{q}_i \neq 0$ for any point on the trajectory then $\bar{q}_i \neq 0$ for all points on the trajectory (for finite time); the coordinate $\bar{q}_i$ therefore does not change sign, and the trajectory is not a crossing trajectory.
The signs of the integrals provide a rather coarse descriptor for crossing trajectories. In ref. a symbolic description of saddle crossing trajectories for index $k=2$ saddles was introduced based on the sign change of the $\bar{q}_i$. This symbolic description distinguishes all qualitatively distinct classes of crossing trajectories. Here we recall the discussion of the symbolic description of crossing trajectories given in ref. .
Consider the index $k=2$ case. The symbolic description of the behavior of a trajectory as it passes through a neighborhood of the index 2 saddle with respect to the coordinates ${\bar{q}}_k, \, k=1, 2$, is expressed by the following four symbols, $(f_1 f_2; i_1 i_2)$, where $i_1 = \pm$, $i_2 = \pm$, $f_1 = \pm$, $f_2 = \pm$. Here $i_k$, $k=1,2$, refers to the “initial” sign of ${\bar{q}}_k$ as it enters the neighborhood of the index 2 saddle and $f_k$, $k=1,2$, refer to the “final” sign of ${\bar{q}}_k$, as it leaves the neighborhood of the index 2 saddle. For example, trajectories of type $(--;+-)$ pass over the barrier from ${\bar{q}}_1 >0$ to ${\bar{q}}_1 <0$, but remain on the side of the barrier with ${\bar{q}}_2 <0$.
Based on the number of distinct sequences of length four of $+$ and $-$, there are $2^4=16$ qualitatively distinct classes of trajectory, However, there are only four types of trajectories for which there is a change of sign of [*both*]{} coordinates ${\bar{q}}_1$ and ${\bar{q}}_2$ as they pass through a neighborhood of the index 2 saddle, and these have symbolic descriptions $(++;--)$, $(-+;+-)$, $(+-;-+)$ and $(--;++)$. We will see that this symbolic description of crossing trajectories can be directly related to the geometry of the dividing surface.
As noted previously [@Ezra09], codimension one surfaces separate the different types of trajectory. For index 2 saddles these are the codimension one invariant manifolds. given by ${\bar{q}}_1={\bar{p}}_1$, ${\bar{q}}_2 ={\bar{p}}_2$ (i.e., $I_1 =0$, $I_2 =0$). For example, the codimension one surface ${\bar{q}}_1={\bar{p}}_1$ forms the boundary between trajectories of type $(++; +-)$ and $(-+; +-)$, and so on.
An obvious generalization of this symbolic construction can be carried out for the case of index $k$ saddles, $k >2$, where there are $2^{2k}$ possible classes of trajectory.
The dividing surface associated with index $k$ saddles {#sec:DS}
======================================================
An essential component of the analysis of reaction dynamics from the phase space perspective is the construction of a dividing surface (DS) through which all reactant trajectories must pass, and which (locally) has the essential no-recrossing property [@WaalkensSchubertWiggins08]. In this section we discuss construction of a codimension one (within the energy surface) DS for an index $k$ saddle using the equations of motion associated with the normal form Hamiltonian of Sec. \[sec:ndof\]. This construction again demonstrates the utility of an analysis in terms of the ‘physical’ saddle coordinates $\bar{q}_i$. In addition to constructing the DS, using the normal form we are able to (locally) classify reactive trajectories, thereby obtaining a rigorous phase space characterization of the subset of barrier crossing trajectories associated with concerted isomerization dynamics.
Definition of dividing surface in the general case {#sec:index_k}
--------------------------------------------------
We define a measure of the distance from the origin (i.e., the saddle) in the configuration space of the saddle degrees-of-freedom as: $${\mathcal{D}}\equiv \frac{1}{2} \sum_{i=1}^{k} \bar{q}_i^2.
\label{eq:distance}$$ The dividing surface we define contains the set of phase space points corresponding to the minimum distance from the origin attained by crossing trajectories. At any turning point in the variable ${\mathcal{D}}$ we have $$\dot{{\mathcal{D}}} = \sum_{i=1}^{k} \bar{q}_i \dot{\bar{q}}_i = \sum_{i=1}^{k} \Lambda_i \bar{q}_i \bar{p}_i =0.
\label{eq:min_1}$$ This equation defines a critical point for ${\mathcal{D}}$, but it is in fact a minimum since $$\ddot{{\mathcal{D}}} = \sum_{i=1}^{k} \Lambda_i \left( \dot{\bar{q}}_i \bar{p}_1 + \bar{q}_i \dot{\bar{p}}_i \right)
= \sum_{i=1}^{k} \Lambda_i^2 \left( \bar{q}_i^2 + \bar{p}_i^2 \right)\geq0,
\label{eq:min_2}$$ and where we have used $\frac{d}{dt} \Lambda_i =0$ for dynamics under the normal form Hamiltonian. Note that $\ddot{{\mathcal{D}}}=0$ precisely when $\bar{q}_i = \bar{p}_i=0, \, i=1, \ldots , k$, i.e., it is zero on the NHIM.
The dividing surface at constant $E$ is defined by the intersection of the following two $2n-1$ dimensional surfaces:
\[eq:DS\_1\] $$\begin{aligned}
S_1(\bar{q}_1,\bar{p}_1, \ldots ,\bar{q}_k, \bar{p}_k, q_{k+1}, p_{k+1}, \ldots, q_n, p_n) & \equiv H(I_1, \ldots, I_n) - E =0, \\
S_2(\bar{q}_1,\bar{p}_1, \ldots ,\bar{q}_k, \bar{p}_k, q_{k+1}, p_{k+1}, \ldots, q_n, p_n) &
\equiv \sum_{i=1}^{k} \Lambda_i \bar{q}_i \bar{p}_i =0.\end{aligned}$$
Points on the DS lie on crossing trajectories and must therefore satisfy the additional conditions:
$$I_i > 0, \quad i=1, \ldots , k . \quad
\iffalse
\begin{cases}
\text{if} \, I_i =0 \; \text{then} \; \bar{q}_i = 0 \\
I_i \neq 0 \, \text{for at least one} \,\; i=1, \ldots , k.
\end{cases}
\fi
\label{eq:constraint}$$
The surface defined by without the constraint in includes trajectories which are not crossing trajectories and we will refer to it as the *extended dividing surface*. The boundary between the dividing surface and the rest of the extended dividing surface consists of phase points satisying the conditions $I_1 =0$, and/or $I_2 =0$ and is codimension two, the intersection of with the codimension one invariant manifolds $I_1=0$, $I_2=0$.
It should be clear that the surface defined by is codimension one, and is restricted to the energy surface $H(I_1, \ldots, I_n) = E$ by construction. However, we need to prove that it has the properties of a DS, that is,
1. [**The vector field is transverse to the DS.**]{}
The DS is codimension two in the full phase space (and codimension one restricted to the energy surface) and therefore has two vectors transverse to it at each point.
The Hamiltonian vector field ${\boldsymbol{v}}_H$ associated with the normal form Hamiltonian $H$ is: $${\boldsymbol{v}}_H = \left(\Lambda_1 \bar{p}_1, \Lambda_1 \bar{q}_1, \ldots \Lambda_k \bar{p}_k, \Lambda_k \bar{q}_k,
\Omega_{k+1} p_{k+1} , - \Omega_{k+1} q_{k+1} , \ldots, \Omega_n p_n, - \Omega_n q_n \right).
\label{eq:vf}$$ We will show that the Hamiltonian vector field ${\boldsymbol{v}}_H$ is transverse to the DS on the energy surface $S_1=0$.
The rate of change of $S_1$ along ${\boldsymbol{v}}_H$ is necessarily zero: $$dS_1({\boldsymbol{v}}_H) = \{H, H\} =0,$$ where $\{ \cdot, \cdot\}$ denotes the usual Poisson bracket. The rate of change of $S_2$ along ${\boldsymbol{v}}_H$ is
$$\begin{aligned}
dS_2({\boldsymbol{v}}_H) & = \{S_2, H\} = \frac{d}{d t} S_2 \\
& = \sum_{i=1}^{k} \Lambda_i^2 \left( \bar{q}_i^2 + \bar{p}_i^2 \right).\end{aligned}$$
from . Clearly, this expression is greater than or equal to zero. It is zero precisely when $\bar{q}_j = \bar{p}_j=0, \, j=1, \ldots , k$, i.e. it is zero on the NHIM.
2. [**All crossing trajectories pass through the dividing surface.**]{}
It is evident that all trajectories of that enter and leave a neighborhood of the origin necessarily achieve such a minimum distance from the origin. In particular, this is true for all trajectories satisfying $I_i > 0, \, i=1, \ldots, k$. Therefore all crossing trajectories pass through the dividing surface.
Index 1 Saddles {#sec:index1}
---------------
It is instructive to see how our construction of a dividing surface for index $k$ saddles reduces to the familiar case of index $1$ saddles [@wwju; @ujpyw]. In this case conditions become
\[eq:DS\_1\_ind1\] $$\begin{aligned}
H(I_1, \ldots, I_n) & = E \\
\bar{q}_1\bar{p}_1& =0,
\end{aligned}$$
where points on the dividing surface are also subject to the additional constraint: $$I_1 = \frac{1}{2} \left( \bar{p}_1^2 - \bar{q}_1^2\right)>0,
\label{eq:constraint_ind1}$$ Now, in order to have $\bar{q}_1\bar{p}_1 =0$ [*and*]{} $I_1 = \frac{1}{2} \left( \bar{p}_1^2 - \bar{q}_1^2\right)>0$ we must have $$\bar{q}_1=0,$$ which is the DS for index 1 saddles expressed in terms of the normal form coordinates ${\bar{q}}$.
Explicit parametrization of the DS for quadratic Hamiltonians {#sec:param}
-------------------------------------------------------------
For the case of a quadratic Hamiltonian it is possible to give an explicit parametrization of the dividing surface. For simplicity we consider the case of an index 2 saddle for a system with 2 DoF. For the two degree-of-freedom index 2 saddle the dividing surface is 2 dimensional in the 3 dimensional energy surface, and visualization of the dividing surface is therefore possible. Such visualization provides insight into the crossing dynamics for an index 2 saddle, and we therefore discuss this topic in some detail. Moreover, we will see that the dividing surface in the quadratic Hamiltonian case plays an important role in our algorithm for sampling the dividing surface associated with the full normal form for general (non-quadratic) Hamiltonians.
The discussion given below is in the spirit of ref. , where the geometry associated with the quadratic Hamiltonian was used to to examine several different approaches to visualizing the phase space structures that govern reaction dynamics for index 1 saddles. A quadratic approximation to the Hamiltonian is expected to accurately capture the relevant geometrical features for energies ‘close’ to the energy of the saddle.
We examine an $n$ DoF quadratic Hamiltonian with an index 2 saddle as motivation for the parameterisation we will develop but the value of the index $k$ in our formulas is kept general for use later. In the quadratic case, $\Lambda_i \equiv \lambda_i =\mbox{constant}$, $\Omega_i \equiv \omega_i =\mbox{constant}$, and the Hamiltonian reduces to:
$$\begin{aligned}
H(I_1, I_2, I_3, \ldots, I_n) &= \sum_{i=1}^{k} \lambda_i I_i + \sum_{j=k+1}^{n} \omega_j I_j \\
& = \sum_{i=1}^{k} \frac{1}{2} \lambda_i \left( \bar{p}_i^2 - \bar{q}_i^2 \right)
+ \sum_{j=k+1}^{n} \frac{1}{2} \omega_j \left( q_j^2 + p_j^2 \right)\end{aligned}$$
which defines the energy surface: $$H(I_1, I_2, I_3, \ldots, I_n) = E,
\label{energy_phys_1_fnf}$$ where we will consider the case $E>0$. We define the energy $E_s$ associated with the saddle DoF to be $$E_{s} = \sum_{i=1}^{k} \frac{1}{2}\lambda_i \left( \bar{p}_i^2 - \bar{q}_i^2 \right)
= E - \sum_{j=k+1}^{n} \frac{1}{2} \omega_j \left( q_j^2 + p_j^2 \right).
\label{energy_phys_2_fnf}$$ ($\lambda_i$ and $\omega_i$ constants) and rewrite this equation as follows:
$$\begin{aligned}
E_{s} + \sum_{i=1}^{k} \frac{1}{2}\lambda_i \bar{q}_i^2 & =
\sum_{i=1}^{k} \frac{1}{2}\lambda_i \bar{p}_i^2 \\
& =
E + \sum_{i=1}^{k} \frac{1}{2}\lambda_i \bar{q}_i^2
- \sum_{j=k+1}^{n} \frac{1}{2} \omega_j \left( q_j^2 + p_j^2 \right)\end{aligned}$$
Two ellipsoids are defined in the phase space $(\bar{q}_i, \bar{p}_i)$, $i=1, \ldots , k$ as follows. For some parameter $R \geq 0$ the condition $$R = \sum_{i=1}^{k} \frac{1}{2}\lambda_i \bar{q}_i^2,
\label{R_eq_fnf}$$ defines a $(k-1)$-dimensional ellipsoidal surface in the ${\bar{q}}$ configuration space and $$E_{s}+R = \sum_{i=1}^{k} \frac{1}{2}\lambda_i \bar{p}_i^2,
\label{E+R_eq_fnf}$$ defines a $(k-1)$-dimensional ellipsoidal surface in the ${\bar{p}}$ momentum space.
For the quadratic Hamiltonian, the condition for a trajectory to achieve a minimum distance from the origin in the ${\bar{q}}$ configuration space is (cf. ) $$\dot{{\mathcal{D}}} = \sum_{i=1}^{k} \bar{q}_i \dot{\bar{q}}_i =
\sum_{i=1}^{k} \lambda_i \bar{q}_i \bar{p}_i =0
\label{eq:r_dot_fnf}$$
The case of an index 2 saddle with no center degrees-of-freedom corresponds to a two degree-of-freedom quadratic Hamiltonian system having an equilibrium point at the origin, where the matrix associated with the Hamiltonian vector field has two pairs of purely real eigenvalues, $\pm \lambda _1$, $\pm \lambda_2$, with $\lambda_1, \lambda_2 >0$. The two dimensional dividing surface in the three dimensional energy surface given by $H=E_s=E$ is then given parametrically by:
\[DS\_param\_2D\_fnf\] $$\begin{aligned}
(\bar{q}_1, \bar{q}_2) &= \sqrt{2R} \left( \frac{\sin \theta}{\sqrt{\lambda_1}},
\frac{\cos \theta}{\sqrt{\lambda_2}} \right), \\
(\bar{p}_1, \bar{p}_2) &= \pm \sqrt{2(E+R)} \left( \frac{\cos \theta}{\sqrt{\lambda_1}},
\frac{-\sin \theta}{\sqrt{\lambda_2}} \right)\end{aligned}$$
where $0 \leq R \leq \infty$, $0 \leq \theta \leq 2 \pi$. It is straightforward to check that points on surface satisfy conditions , , and , and that the Hamiltonian vector field is transverse to this surface. However, as we have not yet imposed condition , the extended dividing surface contains phase points in addition to those on crossing trajectories.
Projections of the full surface into various 3 dimensional subspaces of phase space are shown in Figure \[fig:index2plot3d1\], without the constraints on $R$ and $\theta$ implied by the conditions . Consequently, not all phase points on this surface lie on crossing trajectories, but the true DS is embedded in it. The parameter values chosen are $\lambda_1 =1$, $\lambda_2 = \sqrt{3}$, $E=1.0$, and we show both signs of the square root in , with $0 \leq R \leq 1$.
Figures \[fig:index2plot3d1\]a and \[fig:index2plot3d1\]b show $\bar{p}_1$ and $\bar{p}_2$, respectively, as functions of $\bar{q}_1$ and $\bar{q}_2$. Note that $\bar{p}_1$, $\bar{p}_2$ are apparently continuous as $\bar{p}_1$, $\bar{p}_2$ pass through zero. This is however deceptive. If we examine equation we see that for any configuration $(\bar{q}_1, \bar{q}_2)$, the two parts of the surface given by the positive and negative square roots in the second equation must be distinct, since from equation even when $R$ is zero the values of $\bar{p}_1$, $\bar{p}_2$ cannot both be zero. Hence, although in Figure \[fig:index2plot3d1\]a the values of $\bar{p}_1$ are continuous on the axis $\bar{q}_2 = 0$, $\bar{p}_1 = 0$ the values of $\bar{p}_2$ are not continuous as can be seen from Figure \[fig:index2plot3d1\]b, so that the two surface components are disjoint.
Also note that, following the discussion in Section \[sec:crossing\], there are codimension one surfaces separating the different types of trajectory. These are the four codimension one invariant manifolds defined by particular values of the integrals and given by $\bar{q}_1=\bar{p}_1$, $\bar{q}_1=-\bar{p}_1$, $\bar{q}_2 =\bar{p}_2$, $\bar{q}_2 =-\bar{p}_2$ (i.e., $I_1 =0$, $I_2 =0$). For example, the codimension one surface $\bar{q}_1=\bar{p}_1$ forms the boundary between trajectories of type $(++; +-)$ and $(-+; +-)$, and so on.
Figure \[fig:index2plot3d2\] shows only the parts of the dividing surface satisfying the constraints given by equation . In other words, we only plot points on for which $I_1 > 0$, $I_2 > 0$.
In order to obtain explicit constraints we substitute the expressions for the phase points in terms of $R$ and $\theta$ into the integrals to obtain
\[eq:i\_1\] $$\begin{aligned}
I_1 & = \frac{\bar{p}_1^2 - \bar{q}_1^2}{2} , \\
& = \frac{E}{\lambda_1} \cos^2 \theta + \frac{R}{\lambda_1} \left(\cos^2 \theta - \sin^2 \theta \right), \\
& = \frac{(E+2R)}{\lambda_1} \cos^2 \theta - \frac{2R}{\lambda_1}\end{aligned}$$
and
\[eq:i\_2\] $$\begin{aligned}
I_2 & = \frac{\bar{p}_2^2 - \bar{q}_2^2}{2}, \\
& = \frac{E}{\lambda_2} \sin^2 \theta + \frac{R}{\lambda_2} \left(\sin^2 \theta - \cos^2 \theta \right), \\
& = \frac{(E+2R)}{\lambda_2} \sin^2 \theta - \frac{R}{\lambda_2}.\end{aligned}$$
Imposing the conditions $I_1 > 0$ and $I_2 > 0$ we then obtain the constraint $$\label{eq:constraint_1}
\frac{R}{E+2R} < \sin^2\theta < \frac{E+R}{E+2R}\, .$$ Any point $(R, \theta)$ on the dividing surface satisfying condition lies on a crossing trajectory.
Figures \[fig:index2plot3d2\]a–\[fig:index2plot3d2\]d show four almost disjoint, apparently continuous components of the surface which meet only at points $\bar{p}_1 = 0$, $\bar{p}_2 = 0$ where $R = 0$ ($\bar{q_1} = \bar{q_2} = 0$) and $\bar{p_1}$, $\bar{p_2}$ satisfy equation .
It is apparent from equation that, for points on the DS with $0 < \theta < \frac{\pi}{2}$ (quadrant $\bar{q}_1 > 0$, $\bar{q}_2 > 0$), for the positive square root we have $\bar{p}_1 > 0$, $\bar{p}_2 < 0$ and CC trajectories of the type $(+-;-+)$, while for the negative square root we have $\bar{p}_1 < 0$, $\bar{p}_2 > 0$ and CC trajectories of the type $(-+;+-)$.
Similarly for $ \pi < \theta < \frac{3 \pi}{2} $ with points in the quadrant $\bar{q}_1 < 0$, $\bar{q}_2 < 0$, for the positive square root we have $\bar{p}_1 < 0$, $\bar{p}_2 > 0$ and trajectories of type $(-+;+-)$, while for the negative square root we have $\bar{p}_1 > 0$, $\bar{p}_2 < 0$ and points corresponding to CC trajectories of type $(+-;-+)$. There are therefore four disjoint pieces of the dividing surface, two of type $(+-;-+)$ and two of type $(-+;+-)$.
For the angle ranges $ \frac{\pi}{2} < \theta < \pi$ and $ \frac{3 \pi}{2} < \theta < 2 \pi$ there are similarly four disjoint pieces of the dividing surface: two of the type $(++;--)$ and two of type $(--;++)$.
Figures \[fig:index2plot3d11\]a and \[fig:index2plot3d11\]b show $\bar{p}_1(\bar{q}_1, \bar{q}_2)$ and $\bar{p}_2(\bar{q}_1, \bar{q}_2)$, respectively, for the portion of the DS corresponding to CC trajectories of type $(++;--)$.
Sampling the dividing surface {#sec:samp}
-----------------------------
Sampling the DS for an index 2 saddle in the NF coordinate set and then using the sample points as initial conditions enables us to obtain directly concerted crossing trajectories of particular dynamical (symbolic) type. (See Sec. \[sec:model\_potential\].)
Although we have obtained a parametrization of the DS for the index 2 saddle for the quadratic Hamiltonian case, we would like to be able to sample the DS for general Hamiltonians. Sampling the DS in the quadratic case is in principle straightforward: we need to sample points parametrized by $(R, \theta)$ in eq. and use the NF coordinate transformation to convert these points to the original set of physical coordinates. The resulting phase points can be used as initial conditions for trajectory integration.
For the case of a general Hamiltonian, there are however several complications:
1. The frequencies $\Lambda_1$, $\Lambda_2$ in are no longer constant, but are in general functions of the phase space coordinates.
2. The energy $E$ in is in general a nonlinear function of the action variables.
3. The truncated NF coordinate transformations and the associated Hamiltonian are approximate, and become less accurate the farther we move from the saddle point.
4. In addition to sampling the saddle coordinates, for $n \geq 3$ DoF it is necessary to sample the center planes, essentially partitioning the total energy between the center and saddle degrees of freedom but subject to the constraint $H = E$.
5. To use the sampled points as initial conditions for trajectories in microcanonical (constant $E$) simulations we require them to lie on the energy surface of the full Hamiltonian.
6. Ultimately, one would like to sample the DS uniformly with respect to some prescribed probability density.
For the full nonlinear normal form one of the difficulties is that we need to sample the actions $I_i$, $ i=1, \ldots , k$, and $I_j$, $j=k+1, \ldots, n$, subject to the nonlinear energy constraint .
First, consider the problem of sampling phase points in the center planes. The phase space of the center DoF can be sampled uniformly, either in a rectangular grid in physical coordinates $(\bar{q}_j, \bar{p}_j)$ or in action-angle variables $(I_j\, \phi_j)$, but we need to restrict the range of these variables so that the constraints implied by the condition on the total energy are obeyed.
There are two possibilities.
First, with zero energy in the saddle planes ($I_i=0$, $i = 1, 2$), we could calculate the allowed range for each phase space coordinate pair $(\bar{q}_j,
\bar{p}_j)$ or each action $I_j$ from . These values are however non-linearly interdependent and the calculation generally involves an iterative numerical procedure such as Newton’s method.
Alternatively, we can sample the center mode phase space over a sufficiently large but fixed region and then simply accept points for which $H(I_1, \ldots, I_n) < E$ with $I_i=0$ for $ i=1, \ldots, k$.
For each set of center DoF variables so obtained we need to sample points in the saddle planes such that $H(I_1, \ldots, I_n) = E$. For $k=1$ (index 1) there are no free saddle plane parameters. For index $k=2$ we sample values for parameters $\theta$ and $R$. It is then necessary to solve and numerically for the values of $(\bar{q}_i, \bar{p}_i)$.
For the general (non-quadratic) index 2 saddle we see that a point with given $(R, \theta)$ in lies (by construction) on the surface $S_1$, but does not in general lie on $S_2$ (although it might approximately do so). Such a phase point satisfies , but now does not include the nonlinear (higher order) terms in actions $I_1 \ldots, I_n$ in an expansion of $H$. We define the energy correction $\Delta E$ via the relation
$$\begin{aligned}
H(I_1, I_2, I_3, \ldots, I_n) &= \sum_{i=1}^{k} \Lambda_i I_i +
\sum_{j=k+1}^{n} \Omega_j I_j + \Delta E \\
& = \sum_{i=1}^{k} \frac{1}{2} \Lambda_i \left( \bar{p}_i^2 - \bar{q}_i^2 \right)
+ \sum_{j=k+1}^{n} \frac{1}{2} \Omega_j \left( q_j^2 + p_j^2 \right) + \Delta E \,,\end{aligned}$$
where $\Delta E $ includes the higher order terms from but will also include NF errors which are in general functions of $\bar{q}_i, \bar{p}_i$.
We now postulate a generalised parameterisation of appropriate for an index 2 saddle with $n-2$ bath modes:
\[DS\_param\_kD\_fnf\] $$\begin{aligned}
(\bar{q}_1, \bar{q}_2) &= \sqrt{2R} \left( \frac{\sin \theta}{\sqrt{\Lambda_1}},
\frac{\cos \theta}{\sqrt{\Lambda_2}} \right), \\
(\bar{p}_1, \bar{p}_2) &= \pm \sqrt{2(E_{s}+R)} \left( \frac{\cos \theta}{\sqrt{\Lambda_1}},
\frac{-\sin \theta}{\sqrt{\Lambda_2}} \right)\end{aligned}$$
where $R \geq 0$, $0 \leq \theta \leq 2 \pi$. Note that the $\Lambda_i$ are in general no longer constants so that the relations are implicit equations for ${\bar{\boldsymbol{q}}}\equiv (\bar{q}_1, \ldots, \bar{q}_k)$ and ${\bar{\boldsymbol{p}}}\equiv (\bar{p}_1, \ldots, \bar{p}_k)$. It is a simple matter to check that phase points parametrized by satisfy generalised forms of , , and (with the $\lambda$ replaced by $\Lambda$) if we note that $\sum_{i=1}^{k} \Lambda_i I_i$ formally defines the saddle energy $E_{s}$, with $$E_{s} = \sum_{i=1}^{k} \Lambda_i I_i = H(I_1, I_2, I_3, \ldots, I_n) -
\sum_{j=k+1}^{n} \Omega_j I_j - \Delta E \,.$$ Such phase points satisfy both equations and and so lie on $S_1$ and $S_2$.
For given parameters $R, \theta$ the phase space coordinates $(\bar{q}_i, \bar{p}_i)$ must be calculated in combination with the $\Lambda_i, E_{s}$, which are non-linear functions of the $(\bar{q}_i, \bar{p}_i)$. While this is in general a difficult task, for the calculations reported here, the required values can be found by an iterative procedure based on the quadratic approximation to the Hamiltonian.
Extension of eq. to the index $k$ saddle case with $n-k$ bath modes is possible but cumbersome. Although we do not go into details here, a similar technique can be applied.
Once we have a parametrization of the DS, we can use it to choose points on the surface and integrate associated trajectories forward and backward in time. Quantitative calculation of, for example, fluxes requires sampling the DS according to a prescribed density or properly weighting the samples.
Index-2 saddles: model potentials {#sec:model_potential}
=================================
In this section we consider isomerization dynamics in a 2 DoF model potential exhibiting an index 2 saddle. Using the definition of the dividing surface in the vicinity of the index 2 saddle discussed above, we are able to sample phase points on crossing trajectories via the normal form and the associated symbolic code. It is found that the trajectories defined in this way have the dynamical attributes one would intuitively associate with trajectories following a concerted isomerization mechanism; in this section we therefore refer to such trajectories as *concerted crossing* (CC) trajectories.
2 DoF 4 well model potential
----------------------------
We consider a non-separable 2 DoF 4-well potential of the form $$\label{eq:4_well_pot_1}
v({\bar{Q}}_1, {\bar{Q}}_2) = -\alpha {\bar{Q}}_1^2 + {\bar{Q}}_1^4 - {\bar{Q}}_2^2 + {\bar{Q}}_2^4 + \beta {\bar{Q}}_1^2 {\bar{Q}}_2^2.$$ In our numerical computations we use parameter values $\alpha =2$, $\beta = 0.4$. Contours of the potential for these parameters are shown in Fig. \[plot\_1a\]. Note that in this figure the horizontal axis is ${\bar{Q}}_2$ and the vertical axis is ${\bar{Q}}_1$.
The associated Hamiltonian is: $$H = \frac{{\bar{P}}_1^2}{2} + \frac{{\bar{P}}_2^2}{2} + v({\bar{Q}}_1, {\bar{Q}}_2),
\label{ham_1a}$$ with equations of motion:
\[hameq\_1a\] $$\begin{aligned}
\dot{{\bar{Q}}}_1 & = \frac{\partial H}{\partial {\bar{P}}_1} = {\bar{P}}_1, \\
\dot{{\bar{Q}}}_2 & = \frac{\partial H}{\partial {\bar{P}}_2} = {\bar{P}}_2, \\
\dot{{\bar{P}}}_1 & = -\frac{\partial H}{\partial {\bar{Q}}_1} = 2 \alpha {\bar{Q}}_1 - 4 {\bar{Q}}_1^3 - 2 \beta {\bar{Q}}_1 {\bar{Q}}_2^2, \\
\dot{{\bar{P}}}_2 & = -\frac{\partial H}{\partial {\bar{Q}}_2} = 2 {\bar{Q}}_2 - 4 {\bar{Q}}_2^3 - 2 \beta {\bar{Q}}_1^2 {\bar{Q}}_2.
\end{aligned}$$
For the range of parameter values of interest, the potential $v$ has the following set of critical points:
1. Minima (4) $$({\bar{Q}}_1, {\bar{Q}}_2) = \left(\pm \sqrt{\frac{2 \alpha - \beta}{4 - \beta^2}},
\pm \sqrt{\frac{2 - \alpha\beta}{4-\beta^2}} \right),
\;\;
v = -\frac{\alpha^2 +1 - \alpha\beta}{4-\beta^2}$$
2. Index-1 saddle (2) $$({\bar{Q}}_1, {\bar{Q}}_2) = \left(\pm \frac{\sqrt{\alpha}}{\sqrt{2}}, 0 \right),
\;\; v = -\frac{\alpha^2}{4}$$
3. Index-1 saddle (2) $$({\bar{Q}}_1, {\bar{Q}}_2) = \left(0, \pm -\frac{1}{\sqrt{2}} \right),
\;\; v = -\frac{1}{4}$$
4. Index-2 saddle (1) $$({\bar{Q}}_1, {\bar{Q}}_2) = \left(0, 0 \right),
\;\; v=0.$$
The basic problem of interest associated with a potential of the form concerns the nature of the isomerization dynamics (i.e., well-to-well transformations). In particular, we wish to distinguish in a dynamically rigorous and useful way between ‘concerted’ and ‘sequential’ isomerization processes.
Taking $\alpha > \beta/2$ for definiteness, we have the following rough classification of dynamics as a function of energy:
- Confined regime: $$-\frac{\alpha^2 +1 - \alpha\beta}{4-\beta^2} \leq E \leq -\frac{\alpha^2}{4}$$ Trajectories are trapped in the vicinity of one of 4 possible minima, and no isomerization is possible.
- Restricted isomerization: $$-\frac{\alpha^2}{4} < E \leq \frac{1}{4}$$ Trajectories can pass between pairs of wells connected by the lowest energy index 1 saddles. Passage between wells connected by higher energy saddles is not possible.
In this case the standard phase space picture can be applied to analyze isomerizations, with reactant regions identified in the usual way. There are 2 symmetry related NHIMs, and for each isomerization reaction either a 2-state (RRKM) or 3-state (Gray-Rice [@Gray87; @Rice96]) model can be used to describe the isomerization kinetics for each pair of wells.
- Unrestricted isomerization: $$-\frac{1}{4} <E < 0.$$ Trajectories have sufficient energy to pass from any well to any well, but do not have enough energy to reach the index-2 saddle.
In this regime, the only possible way for the system to pass from the lower left well to the upper right well, say, is via *sequential* isomerization routes that proceed through either of the intervening wells (top left or lower right). Concerted passage via hilltop crossing is not energetically feasible.
In this case there are 4 relevant NHIMs. A standard RRKM model could presumably be be used to analyze the kinetics (flux over saddles), or a 5-state generalized Gray-Rice model can be applied [@Gray87; @Rice96]. In the latter case, there are 4 reactant regions (wells) with boundaries consising of broken separatrices, and a 5th region lying outside the reactant region. We do not pursue such an analysis here.
- “Roaming/concerted crossing” regime: $E \geq 0$.
Trajectories can wander freely over a single connected region of configuration space that encompasses all four wells. In this regime, a direct *concerted* isomerization route exists that connects, for example, the lower left and upper right wells. In principle this route coexists with the sequential routes discussed above; the dynamical problem addressed here then concerns the possibility of making a rigorous distinction between trajectories associated with the concerted and sequential isomerization pathways. This classification will be made in phase space using the normal form computed in the vicinity of the index two saddle.
Sampling trajectories on the DS
-------------------------------
We wish to sample trajectories having prescribed dynamical character (CC and non-CC) for the 2 DoF system with Hamiltonian using the DS specified by with crossing trajectories subject to the constraint . The CC trajectories constitute a dynamically well-defined subset of trajectories that are associated with concerted well-to-well transitions. Isomerizing trajectories that enter the vicinity of an index $k$ saddle and which are not crossing (non-CC) trajectories are potentially associated with sequential well-to-well transformations.
The DS is sampled as outlined in Section \[sec:samp\]. A regular grid in cartesian coordinates $(\xi = R \cos[\theta], \eta = R \sin[\theta])$ provides an associated set of $(R, \theta)$ values; the quadratic frequencies $\lambda_1, \lambda_2$ are substituted for the exact, nonlinear frequencies $\Lambda_1, \Lambda_2$ in as a first approximation to give a sample point in the space of NF coordinates. New values for $\Lambda_1, \Lambda_2$ are then computed from , and simple iteration gives a point satisfying .
In general, the phase point thus obtained is not on the energy surface, so we must solve numerically for the value of $E_{s}$ for which the condition on the total energy is satisfied. This is done by appropriate scaling of the momenta $\bar{p}$. Each calculation of the energy error involves a separate iteration to obtain the nonlinear frequencies $\Lambda_i$; for the calculations reported here a single iteration proves sufficient to achieve convergence.
A further difficulty arises as a consequence of the inherent inaccuracy of the truncated NF transformations. If we solve for phase points with fixed NF energy as defined by and then use the NF coordinate transformation to map the point into the original physical coordinates, the resulting point obtained no longer lies exactly on the energy surface defined by the original physical Hamiltonian; the associated error increases with distance from the saddle. We must therefore determine the value of $E_{s}$ yielding a NF point which, after transformation, lies on the energy surface defined by the original Hamiltonian (expressed in the original physical coordinates). This procedure yields points which we use as initial conditions for integrating trajectories of the full Hamiltonian in the original physical coordinates.
In Figures \[fig:cc\_traj\_0.01\], \[fig:cc\_traj\_0.1\],\[fig:cc\_traj\_0.5\] we show trajectories in $(\bar{Q}_1, \bar{Q}_2)$ space obtained by sampling points on the DS as described above. The spacing of the sampling grid in $(\xi, \eta)$ is set at $0.01$ with $R \leq 0.1$. Our sampling procedure leads to a sparsity of points on the DS near the origin in configuration space, $({\bar{Q}}_1, {\bar{Q}}_2)=(0,0)$. Note also that $\theta$ is undefined at $R=0$.
The total energies for the 2 DoF system are set at $E = 0.01, 0.1, 0.5$, respectively. As we do not impose condition , we sample the extended dividing surface and so obtain points in addition to those on concerted crossing trajectories.
At each sample point a trajectory is integrated forwards and backwards in time, classified by symbolic code (as above) and plotted. Each of the Figures \[fig:cc\_traj\_0.01\], \[fig:cc\_traj\_0.1\] and \[fig:cc\_traj\_0.5\] has 4 panels, where each panel shows trajectories associated with points on the DS having the following symbolic classifications: concerted crossing (classes $(++;--)$ and $(+-;-+)$), and non-CC trajectories (classes $(+-;--)$ and $(++;-+)$).
The results shown in Figs \[fig:cc\_traj\_0.01\]–\[fig:cc\_traj\_0.5\] are noteworthy in several respects. First, it is clear that, for all three values of the energy considered, the symbolic classification of trajectories as obtained using the NF in the vicinity of the index-2 saddle corresponds precisely to the observed dynamical behavior of the numerically integrated trajectories. That is, the NF enables us to sample a subset of trajectories of a given dynamical type, e.g., concerted crossing. Second, we note that, although the ensemble of CC trajectories for a given energy appears to consist of two disjoint pencils or bundles, the set of initial consitions is in fact connected (there is a CC trajectory passing through the origin). The form of the configuration space projections of the various trajectory classes is by no means inuitively obvious: at all three energies studied, CC trajectories tend to be concentrated away from the hilltop itself. The non-CC trajectories passing through the vicinity of the index 2 saddle tend to ‘bounce’ off the saddles as they pass between the wells, while CC trajectories appear to ‘graze’ the hilltop.
As mentioned previously, the boundary between CC and non-CC trajectories on the extended dividing surface is composed of those points on the DS satisfying $I_1=0$ and/or $I_2=0$. Projected into configuration space, the boundary consists of a number of lines emanating from the origin. In Fig. \[fig:boundary\_0.01\] we show the projection into configuration space of a segment of a boundary between CC and non-CC trajectories defined by the condition $I_2=0$ with $0 \leq R \leq 0.1$ and $E =0.01$. Each phase point on the boundary is propagated forward and backward in time. It can be seen that trajectories associated with boundary points start in the lower left hand well, pass through the vicinity of the saddle and end up (at short times) tending towards the vertical axis, i.e., neither left (non-CC) nor right (CC). Again, the phase points on the boundary are obtained using the NF, yet show precisely the expected dynamical behavior when propagated numerically.
Lastly, it is natural to consider the fraction of CC trajectories in a given ensemble at a prescribed energy. To provide an unambigous determination of this quantity, we must carefully define the relevant ensemble, such as we have done above in terms of the sampling procedure on the DS, and also specify the relevant weighting factor (measure) for trajectories. As the coordinates $(R, \theta)$ used to parametrize the DS are not canonical coordinates, a weighting factor enters that is a non-trivial function of $(R, \theta)$. Rather than calculate the fraction of CC trajectories by sampling the DS using this weight function, in the next subsection we consider a different sampling procedure where trajectories are initiated on the plane $\bar{Q}_1 = 0$ with $\bar{P}_1 >0$. In this case the associated density at constant energy is just the natural measure (area) in the $(\bar{Q}_2, \bar{P}_2)$ plane.
Trajectory studies of isomerization dynamics
--------------------------------------------
We now study the isomerization dynamics in the concerted crossing regime using an approach complementary to that of the previous subsection. These computations provide additional insight into the way in which the presence of the index-2 saddle affects trajectories in the neighborhood of the saddle, and further confirm the accuracy of the NF in the vicinity of the index 2 saddle.
Initial conditions are chosen as follows. In the configuration space $({\bar{Q}}_1, {\bar{Q}}_2)$, we define a regular grid of points along the line ${\bar{Q}}_1=0$ in an interval symmetric about $\bar{Q}_2 =0$. At each point $({\bar{Q}}_1, {\bar{Q}}_2)$ along this line, a number of momentum pairs $({\bar{P}}_1, {\bar{P}}_2)$ are chosen such that the phase space points $({\bar{Q}}_1, {\bar{Q}}_2, {\bar{P}}_1, {\bar{P}}_2)$ lie on the fixed energy surface, $H=E$. We consider the same 3 energies as in the previous subsection, $H({\bar{Q}}_1, {\bar{Q}}_2, {\bar{P}}_1, {\bar{P}}_2)=0.01$, $0.1$ and $0.5$, and we set ${{\rm d}}{{\bar{Q}}}_1/{{\rm d}}t \geq 0$, so that trajectories start out moving from the lower to the upper half of the potential (the potential is symmetric by construction).
A symbolic code can be assigned to each initial condition in either of two ways. Trajectories can be integrated forward and backward in time until the first well is reached. In practice this means that we integrate trajectories until a turning point is reached in either of the coordinates $\bar{Q}_{1,2}$. In forward time the trajectory can enter one of 2 possible wells, and in backward time the trajectory can enter 2 possible wells, so that there are 4 qualitatively different type of trajectories. Using our symbolic classification [@Ezra09], the trajectory $(f_1, f_2; i_1, i_2)$, where $i_k=\pm$, $f_k = \pm$, $k=1, 2$, denotes the trajectory that passes from well $(i_1, i_2)$ in the immediate past to the well $(f_1, f_2)$ in the immediate future. Each initial condition on the horizontal line can therefore be labelled according to its symbolic description $(f_1, f_2; i_1, i_2)$, i.e., the first well visited in the past and future, as determined by the exact trajectory dynamics. We can also use the NF to classify phase points in the $\bar{Q}_1=0$ plane, and this symbolic classification can be compared with the trajectory results.
Initial conditions at constant energy with ${\bar{Q}}_1 =0$, ${\bar{P}}_1 \geq 0$ are uniquely specified by the values of the phase space variables $({\bar{Q}}_2, {\bar{P}}_2)$. We therefore assign the symbolic trajectory code (4 possibilities) as a function of coordinates $({\bar{Q}}_2, {\bar{P}}_2)$. Initial conditions are sampled uniformly in the $(\bar{Q}_2, \bar{P}_2)$ plane.
Our results are presented in Fig. \[fig:q2p2\_f\] in which we plot the fraction $F$ of crossing trajectories as a function of coordinate ${\bar{Q}}_2$. Results are shown for three energies $E=0.01$, $0.1$ and $0.5$ across the interval $-0.5 \leq {\bar{Q}}_2 \leq 0.5$. This range of the coordinate $\bar{Q}_2$ corresponds approximately to the constraint $0 \le R \leq 0.1$ employed when sampling the DS as described in the previous subsection. On the same graph we show the corresponding trajectory fractions determined by taking each initial condition, converting it to NF coordinates, and using the NF to predict the symbolic trajectory code. In Figure \[fig:q2p2\_f\] the sample spacing for trajectory initialisation was $\Delta \bar{Q}_2 = 0.01$, while for the faster normal form sampling we used a 5 times finer spacing, $\Delta \bar{Q}_2 = 0.002$. Use of the NF results in a smoother curve. If the same sampling spacing is used as for the trajectory calculations, the NF results are essentially identical to those obtained by trajectory integration. Discrepencies between the trajectory and NF results are most pronounced near the boundaries of the interval, and this is not unexpected as the accuracy of the NF presumably deteriorates as one moves away from the saddle. What is perhaps striking about our results is the size of the region of the $\bar{Q}_2$ axis over which the NF predictions do accurately match the trajectory results.
Saddle crossing in the presence of bath modes
---------------------------------------------
Finally, we briefly explore the nature of the saddle crossing dynamics in the presence of additional degrees of freedom (bath modes). We consider a 4 DoF model in which 2 bath modes are added to the 4-well 2 DoF system studied above, and bilinear coupling terms are introduced between saddle and bath modes.
### System-bath Hamiltonian
We consider a 4 DoF system with an index 2 saddle point. The relevant system-bath Hamiltonian $H_{\text{sb}}$ is obtained by adding two ‘bath’ modes, coordinates $(x, y)$, to the 2 DoF system with Hamiltonian given by eq. : $$\label{sb_ham}
H_{\mbox{sb}} =
{\frac{{{\bar{P}}_1}^2}{2} + \frac{{{\bar{P}}_2}^2}{2} +
V({\bar{Q}}_1, {\bar{Q}}_2)}
+
{\frac{1}{2} \left[ p_{x}^2 +
\left(\omega_x x -\frac{c_{1} {\bar{Q}}_1}{\omega_x} \right)^2
\right]}
+
{ \frac{1}{2} \left[ p_{y}^2 + \left(
\omega_y y - \frac{c_{2} {\bar{Q}}_2}{\omega_x} \right)^2
\right]},$$ where mode $x$ is coupled to system coordinate $\bar{Q}_1$ and mode $y$ is coupled to system coordinate $\bar{Q}_2$.
Hamilton’s equations are given by:
\[hameq\_3a\] $$\begin{aligned}
\dot{{\bar{Q}}}_1 & = \frac{\partial H}{\partial {\bar{P}}_1} = {\bar{P}}_1, \\
\dot{{\bar{Q}}}_2 & = \frac{\partial H}{\partial {\bar{P}}_2} = {\bar{P}}_2, \\
\dot{x} & = \frac{\partial H}{\partial p_x} = p_x,\\
\dot{y} &= \frac{\partial H}{\partial p_y} = p_y, \\
\dot{{\bar{P}}}_1 & = -\frac{\partial H}{\partial {\bar{Q}}_1} = 2 \alpha {\bar{Q}}_1 - 4 {\bar{Q}}_1^3 - 2 \beta {\bar{Q}}_1 {\bar{Q}}_2^2
+ \frac{c_1}{\omega_x} \left(\omega_x x
-\frac{c_{1} {\bar{Q}}_1}{\omega_x} \right), \\
\dot{{\bar{P}}}_2 & = -\frac{\partial H}{\partial {\bar{Q}}_2} = 2 {\bar{Q}}_2 - 4 {\bar{Q}}_2^3 - 2 \beta {\bar{Q}}_1^2 {\bar{Q}}_2
+ \frac{c_2}{\omega_y} \left(
\omega_y y - \frac{c_{2} {\bar{Q}}_2}{\omega_y} \right), \\
\dot{p}_x & = -\frac{\partial H}{\partial x} =-\omega_1\left(\omega_x x -\frac{c_{1} {\bar{Q}}_1}{\omega_x} \right),\\
\dot{p}_y & = -\frac{\partial H}{\partial y}=-\omega_y \left(
\omega_y y - \frac{c_{2} {\bar{Q}}_2}{\omega_y} \right).
\end{aligned}$$
For our numerical calculations we use bath parameters $ \omega_x = 1.0$, $\omega_y = \sqrt{2}$ and set $c_1 = c_2$.
### Sampling trajectories on the extended DS for 4 DoF
We now consider the effect of additional degrees of freedom on the dynamics, in particular, our ability to sample concerted crossing trajectories on the (extended) DS for the 4 DoF system-bath Hamiltonian, eq. .
The energy shell for the 4 DoF system-bath model is 7 dimensional, while the DS is 6 dimensional. Fully sampling the DS according to the procedures described above is a numerically intensive task. For purposes of illustration, we demonstrate that computation of the NF for the 4 DoF model allows us to sample CC trajectories on the DS, and that bundles of such trajectories, when projected into the $(\bar{Q}_1, \bar{Q}_2)$ subspace, appear simply as ‘fattened’ versions of the 2DoF saddle trajectories described above with an appropriate value of the saddle energy. This inherent apparent simplicity of the system-bath dynamics is a consequence of the integrability of the NF in the vicinity of the saddle.
In order to sample the DS, we first set the saddle energy $E_s = 0.1$ (no excitation in center modes) and sample phase space coordinates in the saddle planes; for each saddle plane point we sample phase points in the center DoF setting $x = \pm 0.5$, $y = \pm 0.5$, $p_x = \pm 0.5$, and $p_y = \pm 0.5$, and scale the center DoF coordinates and momenta to obtain a fixed value of the total energy $E_{\text{sb}} = 0.5$.
Figure \[fig:ccb\_traj\_0.1\_0.1\] shows $(\bar{Q}_1, \bar{Q}_2)$ projections of trajectories obtained using the sampling procedure just described with saddle-bath coupling parameter $c_1 = c_2 = 0.1$.
Each initial condition in the saddle planes is therefore associated with a ‘bundle’ of trajectories for the 4 DoF full system.
Figure \[fig:ccb\_traj\_0.1\_0.5\] shows analogous results for larger system-bath coupling parameters $ c_1 = c_2 = 0.5$. For the larger coupling we obtain a thicker bundle associated with the fiducial initial condition in the saddle planes.
Summary and conclusions {#sec:summary}
=======================
In this paper we have extended our earlier analysis of the phase space structure in the vicinity of an equilibrium point associated with an index $k$ saddles [@Ezra09]. We have shown that Poincaré-Birkhoff normal form theory provides a constructive procedure for obtaining an integrable approximation to the full Hamiltonian in the vicinity of the equilibrium, provided a generic non-resonance condition is satisfied independently for both the real eigenvalues and the complex eigenvalues of the matrix associated with the linearization of Hamilton’s equations about the index $k$ saddle. As a consequence there are $k$ independent integrals associated with the saddle degrees-of-freedom. These integrals provide a precise tool for classifying trajectories that pass through a neighborhood of the saddle. In particular, they provide a symbolic classification of the trajectories into $2^{2k}$ distinct types of trajectory that pass through a neighborhood of the index $k$ saddle.
The normal form also provides an algorithm for constructing a dividing surface, i.e., a co-dimension one surface (restricted to the energy surface) through which all trajectories in a neighborhood of the index $k$ saddle must pass (with the exception of a set of zero measure). We provide a parametrization of this dividing surface which, when using the integrals associated with the saddle degrees-of-freedom, can be sampled in such a way that we can choose initial conditions corresponding to any particular type of trajectory described by the symbolic classification.
We illustrated our analytical and computational techniques by analyzing a problem that brings to light a fundamental mechanistic role played by index two saddles in chemical dynamics. Namely, we consider isomerization on a potential energy surface with multiple symmetry equivalent minima. In the two degree-of-freedom example we computed the normal form and the dividing surface and showed that the different classes of reactive trajectories in the vicinity of the index two saddle (for three different energies) could be computed by our sampling routine. Our procedure enables a rigorous definition of concerted crossing trajectories to be given in terms of local phase space structure. We then considered a simplified system-bath model (one harmonic oscillator mode for each degree-of-freedom), and showed that our approach could be applied to this four degree-of-freedom system.
PC and SW acknowledge the support of the Office of Naval Research Grant No. N00014-01-1-0769. PC, GSE, and SW would like to acknowledge the stimulating environment of the NSF sponsored Institute for Mathematics and its Applications (IMA) at the University of Minnesota, where this work was begun.
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| ArXiv |
Atomic parity nonconservation (PNC) has now been measured in bismuth [@Bi], lead [@Pb], thallium [@Tl], and cesium [@Cs]. Analysis of the data provides an important test of the Standard Electroweak model and imposes constraints on new physics beyond the model, see Ref. [@RPP]. The analysis is based on the atomic many-body calculations for Tl, Pb, and Bi [@Dzuba1] and for Cs [@Dzuba2; @Blundell]. Both the experimental and the theoretical accuracy is best for Cs. Therefore, this atom provides the most important information on the Standard model in the low energy sector. The analysis performed in Ref. [@Cs] has indicated a deviation of the measured weak charge value from that predicted by the Standard model by 2.5 standard deviations $\sigma$.
In the many-body calculations [@Dzuba1; @Dzuba2; @Blundell] the Coulomb interaction between electrons was taken into account, while the magnetic interaction was neglected. The contribution of the magnetic (Breit) electron-electron interaction was calculated in the recent papers [@Der; @Dzuba3]. It proved to be much larger than a naive estimate, and it shifted the theoretical prediction for PNC in Cs. As a result, the deviation from the Standard model has been reduced. The calculations [@Der; @Dzuba3] have already been used to get new restrictions on possible modifications of the Standard model, see, e. g., Ref. [@Ros]. The reason for the enhancement of the Breit correction has been explained in Ref. [@Sushkov]. In the case of the Coulomb residual interaction the effect of the many-body polarization is maximum for the outer electronic subshell and quickly drops down inside the atom [@Dzuba1; @Dzuba2; @Blundell]. The Breit interaction is more singular at small distances than the Coulomb one. Hence, the polarization is maximum for the lowest subshell ($1s^2$) and quickly drops down towards the outer shells. The estimate of the relative effect of the magnetic polarization gives $Z\alpha^2$ instead of naive $\alpha^2$, where $Z$ is the nuclear charge and $\alpha$ is the fine structure constant. To find the Breit correction there is no need to repeat the involved many-body calculations performed in Refs. [@Dzuba1; @Dzuba2; @Blundell]. Indeed, the Breit correction comes from small distances, $r\sim a_B/Z$ ($a_B$ is the Bohr radius), while all the Coulomb polarization and correlation corrections come from large distances, $r\sim a_B$. Therefore, it is sufficient to calculate the relative Breit correction to some PNC mixing matrix element (say $6s_{1/2}-6p_{1/2}$ mixing in Cs) in the simplest Hartree-Fock or RPA approximation. The relative Breit correction to the PNC effect with account of all many-body Coulomb polarization and correlation corrections is exactly the same.
The Breit correction to PNC is just a part of the effect. This part is related to the virtual excitations of the $1s^2$ subshell. Another contribution comes from the vacuum fluctuations, i.e. from the radiative corrections. Attempts to estimate this effect were made in Ref. [@Lynn] and gave very small values of the corrections. It has been pointed out recently [@Sushkov] that the strong electric field of the nucleus enhances the radiative corrections, and they may be comparable with the Breit correction. Very recently this suggestion has been confirmed by the numerical calculation of the vacuum polarization correction in Cs [@W].
In the present paper we consider PNC in heavy atoms and calculate radiative corrections enhanced by the strong electric field of the nucleus. We calculate analytically the leading term in the correction and we also estimate other terms. It turns out that in Cs, Tl, Pb, and BI the radiative correction compensates the Breit correction calculated in Refs. [@Der; @Dzuba3]. Thus, we return back to the result of the experimental data analysis made in Ref. [@Cs]: deviation from the Standard model is 2.2 - 2.3 $\sigma$.
In the Standard model it is accepted to normalize the Weinberg angle at the W-boson mass $M_W$. Atomic experiments correspond to a very low momentum transfer compared to $M_W$. The renormalization from $M_W$ to zero momentum transfer was performed in Refs.[@Mar1; @Mar2]. This renormalization is reduced to the logarithmically enhanced single loop corrections $\propto \alpha/\pi$ that lead to what is usually called the radiative correction to the nuclear weak charge. Account of this very important correction gives the nuclear weak charge $Q_{W}$ measured in on-mass-shell electron scattering at zero momentum transfer, where $p^2=m^2$ and $q=0$. However, atomic PNC corresponds to a different situation. The electron in the strong nuclear electric field is off the mass-shell, $p^2 \sim 1/r_0^2 \gg m^2$ and, besides, the typical momentum transfer is of the order of the inverse nuclear radius, $q\sim 1/r_0$. In a precise calculation these effects must be taken into account. It is convenient to use $Q_{W}$ calculated in Refs. [@Mar1; @Mar2] as a reference point. Then the renormalization procedure is the same as that in quantum electrodynamics and the correction vanishes on mass shell at zero momentum transfer. In this approach it is clear that the correction we are talking about is somewhat similar to the radiative correction to the hyperfine constant in a heavy atom [@KK].
The wave function of the external electron is of the form $$\label{Dirac}
u({\bf r})=
\left(
\begin{array}{c}
F(r)\Omega\\
iG(r)\tilde{\Omega}
\end{array}
\right),$$ where $\Omega$ and $\tilde{\Omega}=-({{\mbox{\boldmath $\sigma$ \unboldmath}}}\cdot{\bf n})\Omega$ are spherical spinors[@BLP]. At small distances $r \ll Z\alpha\lambda_C$, where $\lambda_C$ is the electron Compton wave-length, the electron mass is small compared to the nuclear Coulomb potential, and the radial wave functions obey the equations $$\begin{aligned}
\label{fg}
&&{{d(rF)}\over{dr}}+{{\kappa}\over{r}}(rF)-{{Z\alpha}\over{r}}(rG)=0,\\
&&{{d(rG)}\over{dr}}-{{\kappa}\over{r}}(rG)+{{Z\alpha}\over{r}}(rF)=0.\nonumber\end{aligned}$$ For PNC effect we need to consider only $s_{1/2}$ ($\kappa=-1$) and $p_{1/2}$ ($\kappa=+1$) electron states. Solution of Eqs. (\[fg\]) reads $$\label{fg1}
F= Ar^{\gamma-1},\ \ \
G=A{{Z\alpha}\over{\kappa-\gamma}}r^{\gamma-1},$$ where $\gamma=\sqrt{1-Z^2\alpha^2}$ and $A$ is some constant dependent on the wave function behavior at large distances ($r\sim a_B$) [@Kh]. In the leading approximation the PNC interaction
=8.cm
related to the weak charge is due to Z-bozon exchange, see Fig.1a. Calculation of the corresponding weak interaction matrix element gives [@Kh] $$\label{pnc}
<p_{1/2}|H_{W}|s_{1/2}>_0=M_0\propto (F_sG_p-G_sF_p)|_{r=r_0}.$$ At $r_0 \to 0$ this matrix element is divergent, $M_0\propto r_0^{2\gamma-2}$. As a result, the relativistic enhancement factor is $R \approx$3 for Cs and $R \approx 9$ for Tl, Pb, and Bi [@Kh]. In the present paper we show that this divergence results in the double logarithmic enhancement of the radiative corrections.
The first correction is shown in Fig.1b. It corresponds to a modification of the electron wave function because of the vacuum polarization. In the leading $Z\alpha$ approximation the vacuum polarization results in the Uehling potential [@Ueh]. At $r \ll \lambda_C$, this potential is of the form $V(r)\approx 2Z\alpha^2[\ln(r/\lambda_C)+C+5/6]/(3\pi r)$, where $C\approx 0.577$ is the Euler constant. Account of higher in $Z\alpha$ corrections in the vacuum polarization leads to a modification of the constant: $C \to C + 0.092Z^2\alpha^2+...$, see Ref. [@Mil]. However, this correction is small and can be neglected even for $Z\alpha \sim 1$. The potential $V(r)$ modifies the Coulomb interaction in Eqs. (\[fg\]) $-Z\alpha/r \to -Z\alpha/r + V(r)$. It is convenient to search for solution of the modified Eqs. (\[fg\]) in the following form ${\cal F}=F(1+F^{(1)})$, ${\cal G}=G(1+F^{(1)})$, where $F$ and $G$ are given by (\[fg1\]). The functions $F_{s,p}^{(1)}$ and $G_{s,p}^{(1)}$ satisfy the following equations $$\begin{aligned}
\label{fg2}
&& \frac{1}{\gamma+\kappa}\,\frac{d F^{(1)}}{d x}- F^{(1)}+
G^{(1)}=-\frac{2\alpha}{3\pi}\, x \nonumber \\
&&\frac{1}{\gamma-\kappa}\frac{d G^{(1)}}{d x}-G^{(1)}+ F^{(1)}
=-\frac{2\alpha}{3\pi}\, x \quad ,\end{aligned}$$ where $x=\ln(\lambda/r)$, and $\lambda=\lambda_C\exp(-C-5/6)$. The solution of these equations reads $$\begin{aligned}
\label{final}
&& F^{(1)}=\frac{\alpha}{3\pi}\left[\frac{(Z\alpha)^2}{\gamma}x^2+
\frac{\kappa(\gamma+\kappa)}{\gamma^2}x+\frac{\kappa}{2\gamma^2}+a
\right]\nonumber\\
&& G^{(1)}=\frac{\alpha}{3\pi}\left[\frac{(Z\alpha)^2}{\gamma}x^2-
\frac{\kappa(\gamma-\kappa)}{\gamma^2}x-\frac{\kappa}{2\gamma^2}+a\right],\end{aligned}$$ where $a$ is some constant. Using ${\cal F}$ and ${\cal G}$ instead of $F$ and $G$ in Eq. (\[pnc\]) we obtain the PNC matrix element in the form $$\label{pnc1}
<p_{1/2}|H_{W}|s_{1/2}>=M_0\left(1+\delta\right),$$ where $\delta$ due to the diagram Fig.1b is $$\label{ffgg}
\delta_b={{1+\gamma}\over{2}}\left(F_s^{(1)}+G_p^{(1)}\right)+
{{1-\gamma}\over{2}}\left(F_p^{(1)}+G_s^{(1)}\right).$$ To find the correction $\delta_b$ with the logarithmic accuracy there is no need to calculate $a$ in Eqs. (\[final\]), it is enough to substitute the logarithmic terms from (\[final\]) into (\[ffgg\]). An analysis that also includes a consideration of distances $r\sim \lambda_C$ gives $$\label{db}
\delta_b=\alpha\left({1\over{4}}Z\alpha+ {{2(Z\alpha)^2}\over{3\pi\gamma}}
\left[\ln^2(b\lambda_C/r_0)+B\right]\right),$$ where $b=\exp(1/(2\gamma)-C-5/6)$, and $B\sim 1$ is some smooth function of $Z\alpha$ independent of $r_0$. A numerical calculation of $\delta_b$ for Cs was performed recently in Ref. [@W]. The result is in a good agreement with Eq. (\[db\]). Comparison of Eq. (\[db\]) with the result of Ref. [@W] allows also to determine $B$: $B\approx 1$. We would like to emphasize that Eq. (\[db\]) does not assume that $Z\alpha \ll 1$, it is valid for any $Z\alpha <1$. Note that the $(Z\alpha)^2$ term in (\[db\]) is larger than the $Z\alpha$ one at $Z> 10$.
We already pointed out that the weak charge calculated in Refs.[@Mar1; @Mar2] corresponds to zero momentum transfer. On the other hand, it is clear from Eq. (\[pnc\]) that the weak interaction matrix element is determined by the momentum transfer $q\sim 1/r_0$. The renormalization of the weak charge from $q=0$ to $q\sim 1/r_0$ is described by diagrams c and d in Fig.1. A simple calculation gives the following correction $\delta Q_W/Q_W=\delta_{cd}$ related to this renormalization $$\label{cd}
\delta_{cd}={{4\alpha Z}\over{3\pi Q_W}}
(1-4\sin^2\theta_W)\ln(\lambda_C/r_0)\approx -0.1\%.$$ Where $\theta_W\approx $ is the Weinberg angle, $\sin^2\theta_W\approx
0.2230$, see Ref. [@RPP]. Note that this correction is practically independent of $Z$ because $Z/Q_W \approx -Z/N\approx -0.7$, where $N$ is the number of neutrons. One can also obtain the correction (\[cd\]) using Eqs. (2a,b), and (3b) from Ref.[@Mar1].
Next we consider the contribution of the electron self-energy operator $\Sigma$. This operator being substituted to the Dirac equation, $m \to m+\Sigma$, leads to the Lamb shift of the energy level and to the modification of the electron wave function, see, e.g., Ref. [@BLP]. As shown in Fig.1e, this modification influences the matrix element of the weak interaction. The diagram Fig.1e is not invariant with respect to the gauge transformations of the electromagnetic field. However, the sum of the diagrams Fig.1e and Fig.1f (the vertex correction) is gauge invariant. It is convenient to represent the self energy operator as a series in powers of the Coulomb field of the nucleus, $\Sigma=\Sigma_0+\Sigma_1+\Sigma_2+...$, see Fig.2.
=8.cm
We need $\Sigma({\bf r},{\bf r'}|\epsilon)$ at $r \sim r' \ll \lambda_C$, $\epsilon\approx m$. In this limit a calculation in the Feynman gauge with logarithmic accuracy gives $$\label{ss}
\Sigma_0={\hat p}{{\alpha}\over{4\pi}}\ln(p^2/m^2), \ \ \
\Sigma_1={{Z\alpha^2}\over{4\pi r}}\gamma_0\ln(p^2/m^2),$$ where ${\hat p}=p^{\mu}\gamma_{\mu}$, $\gamma_{\mu}$ is the Dirac matrix, and $p^{\mu}$ is the momentum operator. All the higher terms are not logarithmically enhanced. Further calculation in Feynman gauge is rather involved because the diagram Fig.1f is also logarithmically enhanced and there is a delicate cancellation between a part of the $\Sigma$-contribution and the logarithmic part of Fig.1f. To avoid all these complications it is convenient to use the Landau gauge. In this gauge neither $\Sigma_n$ ($n=0,1,2...$) nor the contribution of Fig.1f contain the large logarithm $\ln(\lambda_C/r_0)$. Therefore, after the renormalization, taking zero momentum transfer as a reference point, the contribution of Fig.1f is of the form $\delta_f \sim Z\alpha^2(1+fZ\alpha/\pi+...)$, where we expect $f\sim 1$. Since $\Sigma $ does not contain logarithms, at $r \ll \lambda_C$ we have $\Sigma u =({{Z\alpha^2}/{\pi r}}){\cal D} u$, where ${\cal D} \sim1 $ is some matrix dependent on $Z\alpha$, and $u$ is the electron wave function. Substitution of $\Sigma u$ into the Dirac equation results in the $x$-independent terms of the order of $\sim \alpha/\pi$ in the right hand sides of Eqs. (\[fg2\]). As a result, the relative correction to the matrix element of the weak interaction due to the diagram Fig.1e contains logarithmically enhanced $Z^2\alpha^3$ terms. There is also a $Z\alpha^2$ contribution coming from the distances $r\sim \lambda_C$. All in all the total contribution of diagrams Fig.1e and Fig.1f is of the form $$\label{ef}
\delta_{ef} =Z\alpha^2\left[a_1+ a_2{{Z\alpha}\over{\pi}}
\ln(\lambda_C/r_0)\right].$$ Our preliminary estimate gives $a_1 \approx 0.15$, and we expect $a_2 \sim 1$. Therefore, the value of $\delta_{ef}$ for Cs is $\delta_{ef}\sim 0.1\%$. The calculation of the coefficient $a_2$ in Eq. (\[ef\]) is a very interesting and challenging problem. At this stage we can claim only that the contribution (\[ef\]) is much smaller than that of the Uehling potential (\[db\]) because it does not contain the big logarithm squared.
There is one more radiative correction that has never been considered before. This contribution is due to the virtual excitation of the nuclear giant dipole resonance $A^*$ shown in Fig.3.
=3.cm
Our estimate gives $$\label{giant}
\delta_{A^*}\approx -0.1Z^{2/3}\alpha^2.$$ So it is completely negligible.
Considering all the corrections, one has to remember about the contribution of the electron-electron weak interaction. Although this is not a radiative correction, we still denote it $\delta_{e-e}$. According to Ref. [@SF] this correction is $$\label{ee}
\delta_{e-e} \approx {{0.56}\over{ZR(Z)}}(1-4\sin^2\theta_W),$$ where $R(Z)$ is the relativistic enhancement factor defined in Ref. [@Kh]. Values of $R$ are presented after Eq. (\[pnc\]). For Cs the $\delta_{e-e}$-correction is +0.04%, so it is also negligible.
In summary, we have calculated radiative corrections to the parity nonconservation effects in atoms due to the strong electric field of the nucleus. The structure of the correction is somewhat similar to that for the radiative correction to the hyperfine interaction [@KK]. However there are two essential differences. 1) The corrections to the PNC effects are enhanced by the large logarithm dependent on the nuclear radius. 2) The contribution of the Uehling potential to the PNC radiative correction is dominating because it contains the second power of the logarithm, see Eq. (\[db\]). The vertex and the self-energy corrections do not vanish, but they are not enhanced by the second power of the logarithm. The vertex, self-energy, nuclear excitations, and the electron-electron corrections, see Eqs. (\[cd\]), (\[ef\]), (\[giant\]), and (\[ee\]), contribute at the level $\pm 0.1\%$.
The correction (\[db\]) has been derived for the electron-nucleus weak interaction independent of the nuclear spin, i. e. for the weak charge $Q_W$. However, the same formula is valid for the PNC effect dependent on the nuclear spin, i. e. for the anapole moment. This is obvious because the anapole moment interaction is also local, see Ref. [@Kh].
The radiative correction (\[db\]) is $+0.4\%$ for Cs and $+0.9\%$ for Tl, Pb and Bi. The value of the Breit correction in Cs is $-0.6\%$, see Refs. [@Der; @Dzuba3]. The Breit correction scales practically linearly with Z, see Ref.[@Sushkov]. Therefore, for Tl, Pb, and Bi the value of the Breit correction is $-0.9\%$. Thus, the radiative correction compensates the great part of the Breit correction in Cs, and practically exactly compensates it in Tl, Pb, and Bi. With account of both corrections, the difference between the Standard model prediction and the experimental result in Cs, see Ref. [@Cs], stays at the level of $2.2\sigma$.
Concluding we would like to say that both the radiative and the Breit corrections presently are known with high accuracy and this accuracy can be even further improved. The present limitation of the accuracy in the atomic calculations comes from many-body effects with pure Coulomb electron-electron interaction considered in Refs. [@Dzuba2; @Blundell]. An improvement of the accuracy of these calculations is a feasible but a very challenging problem.
A.I.M. would like to thank M. G. Kozlov for helpful discussions, and the School of Physics at the University of New South Wales for warm hospitality and financial support (Gordon Godfrey fund) during a visit. O.P.S. gratefully acknowledges discussions and correspondence with W. R. Johnson and A. S. Yelkhovsky.
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| ArXiv |
---
abstract: |
Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy’s gradient curves, convergence of Newton’s flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka-Łojasiewicz inequality. All examples are planar.\
These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved $C^k$ convex compact sets in the plane, we provide a level set interpolation of a $C^k$ smooth convex function where $k\geq2$ is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore, given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.
author:
- 'Jérôme Bolte[^1] and Edouard Pauwels[^2]'
date: Draft of
title: Curiosities and counterexamples in smooth convex optimization
---
Introduction
============
Questions and method
--------------------
One of the goals of convex optimization is to provide a solution to a problem with stable and fast algorithms. The quality of a method is generally assessed by the convergence of sequences, rate estimates, complexity bounds, finite length of relevant quantities and other quantitative or qualitative ways.
Positive results in this direction are numerous and have been the object of intense research since decades. To name but a few: gradient methods e.g., [@newmirovsky1983problem; @Nesterov; @Boyd], proximal methods e.g., [@PLC], alternating methods e.g., [@Beck; @wright2015coordinate], path following methods e.g., [@Aus99; @NN], Tikhonov regularization e.g. [@Golub], semi-algebraic optimization e.g., [@jon; @BNPS], decomposition methods e.g., [@PLC; @Beck], augmented Lagrangian methods e.g., [@Bertsek] and many others.
Despite this vast literature, some simple questions remain unanswered or just partly tackled, even for [*smooth convex coercive*]{} functions. Does the alternating minimization method, aka Gauss-Seidel method, converge? Does the steepest descent method with exact line search converge? Do mirror descent or Bregman methods converge? Does Newton’s flow converge? Do central paths converge? Is the gradient flow directionally stable? Do smooth convex functions have the Kurdyka-Łojasiewicz property?
In this article we provide a negative answer to all these questions.
Our work draws inspiration from early works of de Finetti [@definetti1949stratificazioni], Fenchel [@fenchel51], on convex interpolation, but also from Torralba’s PhD thesis [@torralba96] and the more recent [@bolte2010characterization], where some counterexamples on the Tikhonov path and Łojasiewicz inequalities are provided. The convex interpolation problem, see [@definetti1949stratificazioni], is as follows: given a monotone sequence of convex sets[^3] may we find a convex function interpolating each of these sets, i.e., having these sets as sublevel sets? Answers to these questions for [*continuous*]{} convex functions were provided by de Finetti, and improved by Fenchel [@fenchel51], Kannai [@kannai77], and then used in [@torralba96; @bolte2010characterization] for building counterexamples.
We improve this work by providing, for $k\geq 2$ arbitrary, a general $C^k$ interpolation theorem for positively curved convex sets, imposing at the same time the positive definiteness of its Hessian out of the solution set. An abridged version could be as follows.
Let $\left( T_i \right)_{i\in {\mathbb{Z}}}$ be a sequence of compact convex subsets of ${\mathbb{R}}^2$, with positively curved $C^k$ boundary, such that $T_i\subset{\mathrm{int}\,}T_{i+1}$ for all $i$ in ${\mathbb{Z}}$. Then there exists a $C^k$ convex function $f$ having the $T_i$ as sublevel sets with positive definite Hessian outside of the set: $$\operatorname*{argmin}f=\bigcap_{i\in{\mathbb{Z}}} T_i.$$
We provide several additional tools (derivatives computations) and variants (status of the solution set, Legendre functions, Lipschitz continuity). Whether our result is generalizable to general smooth convex sequences, i.e., with vanishing curvature, seems to be a very delicate question whose answer might well be negative.
Our central theoretical result is complemented by a discrete approximate interpolation result “of order one" which is particularly well adapted for building counterexamples. Given a nested collection of polygons, one can indeed build a smooth convex function having level sets interpolating its vertices and whose gradients agree with prescribed normals.
Our results are obtained by blending parametrization techniques, Minkovski summation, Bernstein approximations and convex analysis.
As sketched below, our results offer the possibility of building counterexamples in convex optimization by restricting oneself to the construction of countable collections of nested convex sets satisfying some desirable properties. In all cases failures of good properties are caused by some curvature oscillations.
A digest of counterexamples
---------------------------
Counterexamples provided in this article can be classified along three axes: structural counterexamples[^4], counterexamples for convex optimization algorithms and ordinary differential equations.
In the following, the term “nonconverging” sequence or trajectory means, a sequence or a trajectory with at least two distinct accumulation points. Unless otherwise stated, convex functions have full domain.
[*The following results are proved for $C^k$ convex functions on the plane with $k\geq 2$.*]{}
#### Structural counterexamples
- **Kurdyka-Łojasiewicz:** There exists a $C^k$ convex function whose Hessian is positive definite outside its solution set and which does not satisfy the Kurdyka-Łojasiewicz inequality. This is an improvement on [@bolte2010characterization].
- **Tikhonov regularization path:** There exists a $C^k$ convex function $f$ such that the regularization path $$\begin{aligned}
x(r)= \operatorname*{argmin}\left\{ f(y) + r \|y\|^2:y\in {\mathbb{R}}^2\right\}, \,\,r\in(0,1)
\end{aligned}$$ has infinite length. This strengthens a theorem by Torralba [@torralba96].
- **Central path:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, and $c$ in ${\mathbb{R}}^2$ such that the central path $$\begin{aligned}
x(r) =
\operatorname*{argmin}\left\{ \left\langle c, y \right\rangle + r h(y):y\in D\right\}
\end{aligned}$$ does not have a limit as $r \to 0$.
#### Algorithmic counterexamples:
- **Gauss-Seidel method (block coordinate descent):** There exists a $C^k$ convex function with positive definite Hessian outside its solution set and an initialization $ (u_0,v_0)$ in ${\mathbb{R}}^2$, such that the alternating minimization algorithm $$\begin{aligned}
u_{i+1} &= \operatorname*{argmin}_{u \in {\mathbb{R}}} f(u, v_i) \\
v_{i+1} &= \operatorname*{argmin}_{v \in {\mathbb{R}}}f(u_{i+1}, v)
\end{aligned}$$ produces a bounded nonconverging sequence $((u_i,v_i))_{i\in {\mathbb{N}}}$.
- **Gradient descent with exact line search:** There exists a $C^k$ convex function $f$ with positive definite Hessian outside its solution set and an initialization $x_0$ in ${\mathbb{R}}^2$, such that the gradient descent algorithm with exact line search $$\begin{aligned}
x_{i+1} &= \operatorname*{argmin}_{t \in {\mathbb{R}}} f(x_i + t \nabla f(x_i))
\end{aligned}$$ produces a bounded nonconvergent sequence.
- **Bregman or mirror descent method:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, a vector $c$ in ${\mathbb{R}}^2$ and an initialization $x_0$ in $(-1,1)^2$ such that the Bregman recursion $$\begin{aligned}
x_{i+1} = \nabla h^*(\nabla h(x_i) - c)
\end{aligned}$$ produces a nonconverging sequence. The couple $(h,\langle c,\cdot\rangle$) satisfies the smoothness properties introduced in [@bauschke2016descent].
#### Continuous time ODE counterexamples:
- **Continuous time Newton method:** There exists a $C^k$ convex function with positive definite Hessian outside its solution set, and an initialization $x_0$ in ${\mathbb{R}}^2$ such that the continuous time Newton’s system $$\begin{aligned}
\dot{x}(t) &= - \left[\nabla^2 f(x(t))\right]^{-1} \nabla f(x(t)), \,\,t\geq 0,\\
x(0) &= x_0
\end{aligned}$$ has a solution approaching $\operatorname*{argmin}f$ which does not converge.
- **Directional convergence for gradient curves:** There exists a $C^k$ convex function with $0$ as a unique minimizer and a positive definite Hessian on ${\mathbb{R}}^2\setminus\{0\}$, such that for any non stationary solution to the gradient system $$\begin{aligned}
\dot{x}(t) &= - \nabla f(x(t))
\end{aligned}$$ the direction $x(t) / \|x(t)\|$ does not converge.
- **Hessian Riemannian gradient dynamics:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, a linear function $f$ and a nonconvergent solution to the following system $$\begin{aligned}
\dot{x}(t) = - \nabla_H f(x(t)),
\end{aligned}$$ where $H = \nabla^2 h$ is the Hessian of $h$ and $\nabla_H f = H^{-1} \nabla f$ is the gradient of $f$ in the Riemannian metric induced by $H$ on $(-1,1)^2$.
#### Pathological sequences and curves
Our counterexamples lead to sequences or paths in ${\mathbb{R}}^2$ which are related to a function $f$ by a certain property (see examples above) and have a certain type of pathology. For illustration purposes, we provide sketches of the pathological behaviors we met in Figure \[fig:pathologicalbehaviors\].
Preliminaries
=============
Let us set ${\mathbb{N}^*}={\mathbb{N}}\setminus\{0\}$. For $p$ in ${\mathbb{N}}^*$, the Euclidean scalar product in ${\mathbb{R}}^p$ is denoted by $\langle\cdot,\cdot\rangle$, otherwise stated the norm is denoted by $\|\cdot\|$. Given subsets $S,T$ in ${\mathbb{R}}^p$, and $x$ in ${\mathbb{R}}^p$, we define $${\mathrm{dist}}(x,S):=\inf \left\{\|x-y\|:y\in S\right\},$$ and the Hausdorff distance between $S$ and $T$, $${\mathrm{dist}}(S,T)=\max\left(\sup_{x\in S}{\mathrm{dist}}(x,T),\sup_{x\in T}{\mathrm{dist}}(x,S)\right).$$
Throughout this note, the assertion “$g$ is $C^k$ on $D$” where $D$ is not an open set is to be understood as “$g$ is $C^k$ on an open neighborhood of $D$”. Given a map $G \colon X \mapsto A \times B$ for some space $X$, $[G]_1 \colon X \mapsto A$ denotes the first component of $G$.
Continuous convex interpolations
--------------------------------
We consider a sequence of compact convex subsets of ${\mathbb{R}}^p$, $\left(T_i \right)_{i \in {\mathbb{Z}}}$ such that $T_{i+1} \subset \mathrm{int}\ T_i$. Finding a continuous convex interpolation of $\left(T_i \right)_{i \in {\mathbb{Z}}}$ is finding a convex continuous function which makes the $T_i$ a sequence of level sets. We call this process [*continuous convex interpolation*]{}. This questioning was present in Fenchel [@fenchel51] and dates back to de Finetti [@definetti1949stratificazioni], let us also mention the work of Crouzeix [@crouzeix80] revolving around this issue.
Such constructions have been shown to be realizable by Torralba [@torralba96], Kannai [@kannai77], using ideas based on Minkowski sum. The validity of this construction can be proved easily using the result of Crouzeix [@crouzeix80] which was already present under different and weaker forms in the works of de Finetti and Fenchel.
Let $f \colon {\mathbb{R}}^p \to {\mathbb{R}}$ be a quasiconvex function. The functions $$\begin{aligned}
F_x \colon \lambda \mapsto \sup\left\{ \langle z,x\rangle: f(z)\leq \lambda\right\}
\end{aligned}$$ are concave for all $x$ in ${\mathbb{R}}^p$, if and only $f$ is convex. \[th:crouzeix\]
Our goal is to build smooth convex interpolation for sequences of smooth convex sets. To make such a construction we shall use [*nonlinear*]{} [*Minkowski*]{} [*interpolation*]{} between level sets.
We shall also rely on Bernstein approximation which we now describe.
Bernstein approximation
-----------------------
We refer to the monograph [@lorentz1954bernstein] by G.G. Lorentz.
The main properties of Bernstein polynomials to be used in this paper are the following:
- Bernstein approximation is linear in its functional argument $f$ and “monotone” which allows to construct an approximation using only positive combination of a finite number of function values.
- There are precise formulas for derivatives of Bernstein approximation. They involve repeated finite differences. So approximating piecewise affine function with high enough degree leads to an approximation for which corner values of derivatives are controlled while the remaining derivatives are vanishing (up to a given order).
- Bernstein approximation is shape preserving, which means in particular that approximating a concave function preserves concavity.
The main idea to produce a smooth interpolation which preserves level sets is depicted in Figure \[fig:ideaBernstein\] where we use Bernstein approximation to interpolate smoothly between three points and controlling the successive derivatives at the end points of the interpolation.
Let us now be specific. Given $f$ defined on the interval $[0,1]$, the [*Bernstein polynomial*]{} of order $d \in {\mathbb{N}^*}$ associated to $f$ is given by $$\begin{aligned}
B_d(x) = B_{d,f}(x) = \sum_{k=0}^d f\left( \frac{k}{d} \right) {d \choose k} x^k(1-x)^{d-k}, \mbox{ for $x\in [0,1].$}
\label{eq:bernsteinPolynomial}\end{aligned}$$
#### Derivatives and shape preservation:
For any $h $ in $ (0,1)$ and $x $ in $ [0,1-h]$, we set $\Delta_h^1 f(x) = f(x+h) - f(x)$ and recursively for all $k $ in $ {\mathbb{N}^*}$, $\Delta_h^k f(x) = \Delta\left( \Delta_h^{k-1} f(x) \right)$. We fix $d \neq0$ in ${\mathbb{N}}$ and for $h = \frac{1}{d}$ write $\Delta_h^k = \Delta^k$. Then for any $m \leq d$, we have $$\begin{aligned}
B_d(x)^{(m)} &= d(d-1)\ldots(d-m+1) \sum_{k=0}^{d - m} \Delta^m f\left( \frac{k}{d} \right) {{d-m} \choose k} x^k(1-x)^{d-k-m},
\label{eq:bernsteinDerivative}\end{aligned}$$ for any $x$ in $[0,1]$. If $f$ is increasing (resp. strictly increasing), then $\Delta^1f(x) \geq 0$ (resp. $\Delta^1f(x) > 0$) for all $x$ and $B'_d$ is positive (resp. strictly positive) and $B_d$ is increasing (resp. strictly increasing). Similarly, if $f$ is concave, then $\Delta^2 f(x) \leq 0$ for all $x$ so that $B^{(2)}_d \leq 0$ and $B_d$ is concave. From , we infer $$\label{e:born}
\left|B_d(x)^{(m)}\right|\leq d(d-1)\ldots(d-m+1)\sup_{k\in\{0,\ldots,d-m\}} \left|\Delta^m f\left( \frac{k}{d} \right)\right|$$ for $x$ in $[0,1]$.
#### Approximation of piecewise affine functions:
The following lemma will be extensively used throughout the proofs.
\[lem:convexInterpol\] Let $q_0,q_1 \in {\mathbb{R}}^p$, $\lambda_-<\lambda_0< \lambda_1 < \lambda_+$ and $0<e_1,e_0<1$. Set $\Theta = \left( q_0,q_1,,\lambda_-,\lambda_0,\lambda_1,\lambda_+, e_0,e_1 \right)$ and define $\gamma_\Theta \colon [0,1] \mapsto {\mathbb{R}}^p$ through $$\begin{aligned}
\small
\gamma_\Theta(t)=
\begin{cases}
q_0 \left( 1 + \frac{e_0}{\lambda_0 - \lambda_-} \left( t (\lambda_+- \lambda_-) \right) \right) & \text{ if } 0\leq t \leq \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-} \\
q_0(1 + e_0) \left( \frac{\lambda_1 - \lambda_- - t (\lambda_+ - \lambda_-)}{\lambda_1 - \lambda_0} \right) + q_1(1 - e_1) \left( \frac{ \lambda_- - \lambda_0 + t (\lambda_+ - \lambda_-)}{\lambda_1 - \lambda_0} \right)& \text{ if } \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-} \leq t \leq \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \\
q_1\left( 1 + \frac{(t-1)(\lambda_+-\lambda_-)e_1}{\lambda_+ - \lambda_1} \right) & \text{ if } \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \leq t \leq 1.
\end{cases}
\end{aligned}$$ The curve $\gamma_\Theta$ in ${\mathbb{R}}^{p+1}$ is the affine interpolant between the points $q_0$, $ (1 + e_0) q_0$, $(1 - e_1) q_1$ and $ q_1$. For any $m $ in $ {\mathbb{N}}$, we choose $d$ in ${\mathbb{N}}^*$ such that $$\begin{aligned}
\label{e:deg} \frac{m}{d} \leq \min\left\{ \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-}, 1 - \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \right\}.
\end{aligned}$$ We consider a Bernstein-like reparametrization of $\tilde\gamma_\Theta $ given by $$\begin{aligned}
\tilde\gamma_\Theta \colon [\lambda_-,\lambda_+] &\mapsto {\mathbb{R}}^p\\
\lambda & \mapsto \sum_{k=0}^d \tilde\gamma_\Theta \left( \frac{k}{d} \right){d \choose k} \left(\frac{\lambda - \lambda_-}{\lambda_+ - \lambda_-} \right)^k\left(1 - \frac{\lambda - \lambda_-}{\lambda_+ - \lambda_-} \right)^{d-k}.
\end{aligned}$$ Then the following holds, for any $2 \leq l \leq m$, $\tilde\gamma_\Theta $ is $C^m$ and $$\begin{aligned}
\tilde\gamma_\Theta (\lambda_-)&=q_0&
\tilde\gamma_\Theta (\lambda_+)&=q_1\\
\tilde\gamma_\Theta '(\lambda_-)&=\frac{e_0}{\lambda_0- \lambda-} q_0&
\tilde\gamma_\Theta '(\lambda_+)&=\frac{e_1}{\lambda_+ - \lambda_1} q_1\\
\tilde\gamma_\Theta ^{(l)}(\lambda_-) &=0 &\tilde\gamma_\Theta ^{(l)}(\lambda_+) &= 0.
\end{aligned}$$ Furthermore, if $\gamma_\Theta $ has monotone coordinates (resp. strictly monotone, resp. concave, resp. convex), then so has $\tilde\gamma_\Theta $.
Note that the dependence of $\tilde\gamma_\Theta $ in $(q_0,q_1)$ is linear so that the dependence of $\tilde\gamma_\Theta $ in $(q_0,q_1)$ is also linear. Hence $\tilde\gamma_\Theta $ is of the form $\lambda \mapsto a(\lambda) q_0 + b(\lambda)q_1$. We can restrict ourselves to $p = 1$ since the general case follows from the univariate case applied coordinatewise.
If $p=1$, then $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ A$, where $A \colon \lambda \mapsto \frac{\lambda - \lambda_-}{\lambda_+-\lambda_-}$. We have $\tilde\gamma_\Theta (0) = q_0$, $\tilde\gamma_\Theta (1) = q_1$ and $\Delta^1 f_{\Theta}(0) = \frac{\lambda_+ - \lambda_-}{\lambda_0 - \lambda_-}\frac{e_0}{d} q_0$, $\Delta^1 f_{\Theta}\left( 1 - \frac{1}{d} \right) = \frac{\lambda_+ - \lambda_-}{\lambda_+ - \lambda_1}\frac{e_1}{d} q_1$ and $\Delta^{(l)} f_{\Theta}(0) = \Delta^{(l)} f_{\Theta}\left( 1 - \frac{l}{d} \right) =0$. The results follow from the expressions in and and the chain rule for $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ A$.
The last property of $\tilde\gamma_\Theta $ is due to the shape preserving property of Bernstein approximation and the fact that $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ L$.
\[rem:alignedInterpolation\] [(a) [**\[Affine image\]**]{} Using the notation of Lemma \[lem:convexInterpol\], if $(\lambda_-,q_0)$, $(\lambda_0, (1 + e_0) q_0)$, $(\lambda_1, (1 - e_1) q_1)$ and $(\lambda_+, q_1)$ are aligned, then the interpolation is actually an affine function.\
(b) [**\[Degree of the interpolants\]**]{} Observe that the degree of the Bernstein interpolant is connected to the slopes of the piecewise path $\lambda$ by .]{}
Smooth convex interpolation
===========================
Being given a subset $S$ of ${\mathbb{R}}^p$, we denote by $\mathrm{int}(S)$ its interior, $\bar S$ its closure and ${\mathrm{bd}\,}S=\bar S\setminus \mathrm{int}(S)$ its boundary. Let us recall that the [*support function*]{} of $S$ is defined through $$\sigma_S(x)=\sup\left\{\langle y,x\rangle :y\in S\right\}\in {\mathbb{R}}\cup\{+\infty\}.$$
Smooth parametrization of convex rings
--------------------------------------
A [*convex ring*]{} is a set of the form $C_1\setminus C_2$ where $C_1\subset C_2$ are convex sets. Providing adequate parameterizations for such objects is key for interpolating $C_1$ and $C_2$ by some (regular) convex function.
The following assertion plays a fundamental role.
Let $T_-,T_+ \subset {\mathbb{R}}^2$ be convex, compact with $C^k$ boundary ($k \geq 2$) and positive curvature. Assume that, $T_- \subset \mathrm{int} (T_+)$ and $0 \in \mathrm{int}(T_-)$. \[ass:curvature\]
The positive curvature assumption ensures that the boundaries can be parametrized by their normal, that is, for $i=-,+$, there exists $$\begin{aligned}
c_i \colon {\mathbb{R}}/ 2 \pi {\mathbb{Z}}\mapsto {\mathrm{bd}\,}(T_i)\end{aligned}$$ such that the normal to $T_i$ at $c_i(\theta)$ is the vector $n(\theta) = (\cos(\theta),\sin(\theta))^T$ and $\dot{c}_i(\theta) = \rho_{i}(\theta) \tau(\theta)$ where $\rho_i > 0$ and $\tau(\theta) = (-\sin(\theta),\cos(\theta))$. In this setting, it holds that $c_i(\theta) = \mbox{argmax}_{y \in T_i} \left\langle n(\theta), y\right\rangle$. The map $c_i$ is the inverse of the Gauss map and is $C^{k-1}$ (see [@schneider1993convex] Section 2.5).
\[l:curv\] Let $T_-,T_+$ be as in Assumption \[ass:curvature\] with normal parametrizations as above. For $a,b\geq0$ with $a+b>0$ set $T=aT_-+bT_+$.
Then $T$ has positive curvature and its boundary is given by $${\mathrm{bd}\,}T=\{a\,c_-(\theta)+b\,c_+(\theta): \theta \in {\mathbb{R}}/ 2 \pi {\mathbb{Z}}\},$$ with the natural parametrization ${{\mathbb{R}}/2\pi{\mathbb{Z}}\,}\ni \theta \to a\,c_-(\theta)+b\,c_+(\theta).$
We may assume $ab>0$ otherwise the result is obvious. Let $x$ be in ${\mathrm{bd}\,}T$ and denote by $n(\theta)$ the normal vector at $x$ for a well chosen $\theta$, so that $x={\mathrm{argmax}\,}\left\{\langle y,n(\theta)\rangle\right\}: y \in T\}$. Observe that the definition of the Minkowski sum yields $$\begin{aligned}
\max_{y\in T}\langle y,n(\theta)\rangle & = & \max_{(v,w)\in T_-\times T_+}\langle a v+bw,n(\theta)\rangle \\
&=& a \max_{v\in T_-} \langle v,n(\theta)\rangle+b\max_{w\in T_+} \langle w,n(\theta)\rangle
\end{aligned}$$ so that $$\begin{aligned}
\langle x,n(\theta)\rangle & =& a \langle c_-(\theta),n(\theta)\rangle+b\langle c_+(\theta),n(\theta)\rangle \\
& = & \langle a c_-(\theta)+b c_+(\theta),n(\theta)\rangle \end{aligned}$$ which implies by extremality of $x$ that $x= a c_-(\theta)+b c_+(\theta)$. Conversely, for any such $x$, $n(\theta)$ defines a supporting hyperplane to $T$ and $x$ must be on the boundary of $T$. The other results follow immediately.
In the following fundamental proposition, we provide a smooth parametrization of the convex ring $T_+\setminus {\mathrm{int}\,}T_-$. The major difficulty is to control tightly the derivatives at the boundary so that the parametrizations can be glued afterward to build smooth interpolants.
Let $T_-,T_+$ be as in Assumption \[ass:curvature\] with their normal parametrization as above. Fix $k\geq 2$, $\lambda_-< \lambda_0 < \lambda_1 < \lambda_+$ and $0 < e_0,e_1<1$. Choose $d $ in $ {\mathbb{N}}^*$, such that $$\begin{aligned}
\frac{k}{d} \leq \min\left\{ \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-}, 1 - \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \right\}.
\end{aligned}$$ Consider the map $$\begin{aligned}
G \colon [\lambda_-,\lambda_+] \times {\mathbb{R}}/2\pi{\mathbb{Z}}&\mapsto {\mathbb{R}}^2 \nonumber\\
(\lambda, \theta) \mapsto \tilde\gamma_\Theta(\lambda)
\label{eq:mapG}
\end{aligned}$$ with $\Theta = (c_-(\theta), c_+(\theta), \lambda_-, \lambda_0, \lambda_1, \lambda_+, e_0,e_1)$ and $\tilde\gamma$ as given by Lemma \[lem:convexInterpol\]. Assume further that:
${(\mathcal{M})}$ For any $\theta \in {\mathbb{R}}/ 2\pi{\mathbb{Z}}$, $\lambda \mapsto \left\langle G(\lambda, \theta), n(\theta)\right\rangle$ has strictly positive derivative on $[\lambda_-,\lambda_+]$.
Then the image of $G$ is ${\mathcal{R}}:=T_+ \setminus \mathrm{int}(T_-)$, $G$ is $C^k$ and satisfies, for any $2 \leq l \leq k$ and any $m $ in $ {\mathbb{N}}^*$, $$\begin{aligned}
\frac{\partial^m G}{\partial \theta^m}(\lambda_-,\theta) &= c_-^{(m)}(\theta)&
\frac{\partial^m G}{\partial \theta^m}(\lambda_+,\theta) &= c_+^{(m)}(\theta)\\
\frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (\lambda_-,\theta) &= c_-^{(m)}(\theta) \frac{e_0}{\lambda_0 - \lambda_-}&
\frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (\lambda_+,\theta) &= c_+^{(m)}(\theta) \frac{e_1}{\lambda_+-\lambda_1}\\
\frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_-,\theta) &= 0&
\frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_+,\theta) &= 0.
\end{aligned}$$ Besides $G$ is a diffeomorphism from its domain on to its image. Set ${\mathcal{R}}\ni x \mapsto (f(x), \theta(x))$ to be the inverse of $G$. Then $f$ is $C^k$ and in addition, for all $x$ in $ {\mathbb{R}}$, $$\begin{aligned}
\nabla f(x) =\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, n(\theta(x))\nonumber\\
\nabla \theta(x) =\quad& \frac{1}{\left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle} \tau(\theta(x)) \nonumber\\
&- \frac{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), \tau(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), n(\theta(x))\right\rangle \left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle } n(\theta(x)) \nonumber\\
\nabla^2 f(x)=\quad&\frac{\left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle}
\nabla \theta(x) \nabla \theta (x)^T \nonumber\\
&- \frac{\left\langle \frac{\partial^2 G}{\partial \lambda^2}(f(x),\theta(x)), n(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \nabla f(x) \nabla f(x)^T,
\label{eq:gradient}
\end{aligned}$$ where all denominators are positive. \[prop:diffeoMorphism\]
[ Note that $G$ is actually well defined and smooth on an open set containing its domain. As we shall see it is also a diffeomorphism from an open set containing its domain onto its image.]{}
Note that by construction, we have $G(\lambda,\theta) = a(\lambda) c_-(\theta) + b(\lambda)c_+(\theta)$ for some polynomials $a$ and $b$ which are nonnegative on $[\lambda_-,\lambda_+]$. The formulas for the derivatives follow easily from this remark, the form of $a$ and $b$ and Lemma \[lem:convexInterpol\].
Set, for any $\lambda $ in $ [\lambda_-,\lambda_+]$, $T_\lambda = a(\lambda) T_- + b(\lambda) T_+$. The resulting set $T_\lambda$ is convex and has a positive curvature by Lemma \[l:curv\], and for $\lambda$ fixed $G(\lambda,\cdot)$ is the inverse of the Gauss map of $T_\lambda$, which constitutes a parametrization by normals of the boundary.
Assume that $\lambda < \lambda'$, using the monotonicity assumption ${(\mathcal{M})}$, we have for any $\theta,\theta'$, $$\begin{aligned}
\left\langle n(\theta'), G(\lambda,\theta) \right\rangle& \leq \sup_{y \in T_{\lambda}} \left\langle n(\theta'),y \right\rangle \\
& = \left\langle n(\theta'), G(\lambda,\theta') \right\rangle\\
&< \left\langle n(\theta'), G(\lambda',\theta') \right\rangle
\end{aligned}$$ so that $G(\lambda,\theta) \neq G(\lambda',\theta')$. Furthermore, we have by definition of $G(\lambda',\theta')$ $$\begin{aligned}
G(\lambda,\theta) \in \bigcap_{\theta' \in {\mathbb{R}}/2\pi{\mathbb{Z}}} \left\{ y,\; \left\langle y, n(\theta')\right\rangle \leq \left\langle n(\theta'), G(\lambda',\theta') \right\rangle \right\} = T_{\lambda'},
\end{aligned}$$ where the equality follows from the convexity of $T_{\lambda'}$. By convexity and compactness, this entails that $T_\lambda = \mathrm{conv}({\mathrm{bd}\,}(T_\lambda)) \subset {\mathrm{int}\,}T_{\lambda'}$.
Let us show that the map $G$ is bijective, first consider proving surjectivity. Let $f$ be defined on $T_+ \setminus \mathrm{int}(T_-)$ through $$\begin{aligned}
f \colon x \mapsto \inf_{}\left\{ \lambda : \lambda\geq \lambda_-, \; x \in T_\lambda = a(\lambda) T_- + b(\lambda) T_+\right\}.
\label{eq:defIntropol}
\end{aligned}$$ Since $a(\lambda_+)=0, b(\lambda_+)=1$ this function is well defined and by compactness and continuity the infimum is achieved. It must hold that $x $ belongs to $ {\mathrm{bd}\,}(T_{f(x)})$, indeed, if $f(x) = \lambda_-$, then $x $ belongs to $ {\mathrm{bd}\,}(T_-)$ and otherwise, if $x $ is in $ \mathrm{int}(T_{\lambda'})$ for $\lambda' > \lambda_-$, then $f(x) < \lambda'$. We deduce that $x$ is of the form $G(f(x),\theta)$ for a certain value of $\theta$, so that $G$ is surjective.
As for injectivity, we have already seen a first case, the monotonicity assumption ${(\mathcal{M})}$ ensures that $\lambda \neq \lambda'$ implies $G(\lambda,\theta) \neq G(\lambda',\theta')$ for any $\theta,\theta'$. Furthermore, we have the second case, for any $\lambda $ in $ [\lambda_-,\lambda_+]$ and any $\theta$, $G(\lambda,\theta) = \arg\max_{y\in T_\lambda} \left\langle y,n(\theta) \right\rangle$ so that $\theta \neq \theta'$ implies $G(\lambda,\theta) \neq G(\lambda,\theta')$. So in all cases, $(\lambda,\theta) \neq (\lambda',\theta')$ implies that $G(\lambda,\theta) \neq G(\lambda',\theta')$ and $G$ is injective.
Let us now show that the map $G$ is a local diffeomorphism by estimating its Jacobian map.
Since $0 \in \mathrm{int}(T_-)$, we have for any $\lambda,\theta$, $$\begin{aligned}
0 &< \sup_{y \in T_-} \left\langle y, n(\theta) \right\rangle \\
&= \left\langle G(\lambda_-,\theta),n(\theta) \right\rangle \\
& \leq a(\lambda) \left\langle c_-(\theta), n(\theta)\right\rangle + b(\lambda) \left\langle c_+(\theta), n(\theta) \right\rangle,
\end{aligned}$$ and both scalar products are positive so that $a(\lambda) + b(\lambda) > 0$. Hence, for any $\theta $ in $ {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned}
\label{eq:derivativeGtheta}
\frac{\partial G}{\partial \theta}(\lambda,\theta) = (a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \tau(\theta),
\end{aligned}$$ with $a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta) > 0$. Furthermore by assumption $\lambda\to \max_{y \in T_\lambda} \left\langle y, n(\theta) \right\rangle=\left\langle G(\lambda,\theta),n(\theta) \right\rangle$ has strictly positive derivative in $\lambda$, whence $$\begin{aligned}
\left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), n(\theta)\right\rangle > 0.
\end{aligned}$$ We deduce that for any fixed $\theta$, in the basis $(n(\theta), \tau(\theta))$, the Jacobian of $G$, denoted $J_G$, is triangular with positive diagonal entries. More precisely fix $\lambda, \theta$ and set $x = G(\lambda, \theta)$ such that $\lambda = f(x)$, $\theta = \theta(x)$. In the basis $(n(\theta), \tau(\theta))$, we deduce from that the Jacobian of $G$ is of the form $$\begin{aligned}
J_G(\lambda, \theta) =
\begin{pmatrix}
\alpha &0\\
\gamma&\beta
\end{pmatrix},
\end{aligned}$$ where $$\begin{aligned}
\alpha &= \left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), n(\theta)\right\rangle>0 \\
\beta &= \left\langle \frac{\partial G}{\partial \theta} (\lambda,\theta), \tau(\theta)\right\rangle = a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta) >0\\
\gamma &= \left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), \tau(\theta)\right\rangle.
\end{aligned}$$ It is thus invertible and we have a local diffeomorphism. We deduce that $$\begin{aligned}
J_G(\lambda, \theta)^{-1} =
\begin{pmatrix}
\alpha^{-1} &0\\
\frac{-\gamma}{\alpha\beta}&\beta^{-1}
\end{pmatrix}.
\end{aligned}$$ We have $J_G(\lambda,\theta)^{-1} = J_{G^{-1}}(x)$ so that the first line is $\nabla f(x)$ and second line is $\nabla \theta(x)$, which proves the claimed expressions for gradients.
We also have $d n(\theta) / d\theta = \tau(\theta)$ so that $$\begin{aligned}
J_{n \circ \theta}(x) = \tau(\theta(x)) \nabla \theta(x)^T.
\end{aligned}$$ Differentiating the gradient expression, we obtain ($\nabla$ denotes gradient with respect to $x$): $$\begin{aligned}
&\nabla^2 f(x)\\
=\quad& \frac{\partial}{\partial x} \left(\frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, n(\theta(x))\right)\\
=\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, J_{n\circ \theta}(x) \\
&+ n(\theta(x)) \nabla \left(\frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle }\right)^T \\
=\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, J_{n\circ \theta}(x) \\
&- n(\theta(x)) \nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle^2} \\
=\quad & \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \left( \tau(\theta) \nabla \theta(x)^T - \nabla f(x) \nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T\right).
\end{aligned}$$ We have $$\begin{aligned}
=\quad&\nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T \\
=\quad & \frac{\partial G}{\partial \lambda} (f(x), \theta(x))^T J_{n\circ\theta}(x) + n(\theta(x))^T J_{\frac{\partial G}{\partial \lambda}}(f(x),\theta(x)) J_{G^{-1}} (x) \\
=\quad & \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla \theta(x)^T + \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x)^T,
\end{aligned}$$ where, for the last identity, we have used the fact that $$\begin{aligned}
n(\theta(x))^T J_{\frac{\partial G}{\partial \lambda}}(f(x),\theta(x)) &= \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle n(\theta(x))^T + \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda \partial \theta}(\lambda,\theta)\right\rangle \tau(\theta(x))^T\\
&= \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle n(\theta(x))^T\\
n(\theta(x))^T J_{G^{-1}}(x) &= n(\theta(x))^T J_G(f(x),\theta(x))^{-1}\\
&= \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), n(\theta(x))\right\rangle} n(\theta(x))^T = \nabla f(x)^T.
\end{aligned}$$ We deduce that $$\begin{aligned}
&\nabla^2 f(x)\\
=\quad & \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle}\\
&\Bigg( \tau(\theta(x)) \nabla \theta(x)^T - \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla f(x) \nabla \theta(x)^T \\
&- \nabla f(x) \nabla f(x)^T \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \Bigg).
\end{aligned}$$ We have $$\begin{aligned}
& \tau(\theta(x)) \nabla \theta(x)^T - \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla f(x) \nabla \theta(x)^T \\
=\quad& \left(\tau(\theta(x)) - \frac{\left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle}{\left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), n(\theta(x)) \right\rangle} n(\theta(x)) \right) \nabla \theta(x)^T\\
=\quad& (a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \nabla \theta(x) \nabla \theta (x)^T
\end{aligned}$$ So that we actually get $$\begin{aligned}
&\nabla^2 f(x) \left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle\\
=\quad &
(a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \nabla \theta(x) \nabla \theta (x)^T - \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x) \nabla f(x)^T\\
=\quad &\left\langle\frac{\partial G}{\partial \theta}(\lambda,\theta),\tau(\theta(x))\right\rangle
\nabla \theta(x) \nabla \theta (x)^T - \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x) \nabla f(x)^T.
\end{aligned}$$ This concludes the proof.
Smooth convex interpolation of smooth positively curved convex sequences
------------------------------------------------------------------------
In this section, we consider an indexing set $I$ with either $I = {\mathbb{N}}$ or $I = {\mathbb{Z}}$, and an increasing sequence of compact convex sets $\left( T_i \right)_{i\in I}$ such that for any $i $ in $ I$, the couple $T_+ := T_{i+1}$, $T_-: = T_i$ satisfies Assumption \[ass:curvature\]. In particular, for each $i $ in $ I$, $T_i$ is compact, convex with $C^k$ boundary and positive curvature. We denote by $c_i$ the corresponding parametrization by the normal, $T_i \subset \mathrm{int}\, T_{i+1}$. With no loss of generality we assume $\displaystyle 0 \in \cap_{i \in I} T_i$.
This is our main theoretical result.
\[th:smoothinterp\] Let $I = {\mathbb{N}}$ or $I = {\mathbb{Z}}$ and $\left( T_i \right)_{i\in I}$ such that for any $i \in I$, $T_i \subset {\mathbb{R}}^2$ and the couple $T_+ := T_{i+1}$, $T_-: = T_i$ satisfies Assumption \[ass:curvature\]. Then there exists a $C^k$ convex function $$\begin{aligned}
f \colon {\mathcal{T}}:= \mathrm{int}\left(\bigcup_{i\in I} T_i \right) \mapsto {\mathbb{R}}\end{aligned}$$ such that
\(i) $T_i$ is a sublevel set of $f$ for all $i $ in $ I$.
\(ii) We have $$\begin{aligned}
\operatorname*{argmin}f= \begin{cases}
\bigcap_I T_i &\quad \text{ if } I = {\mathbb{Z}}\\
\{0\}&\quad \text{ if } I = {\mathbb{N}}.
\end{cases}
\end{aligned}$$
\(iii) $\nabla^2 f $ is positive definite on ${\mathcal{T}}\setminus \operatorname*{argmin}f$, and if $I = {\mathbb{N}}$, it is positive definite throughout ${\mathcal{T}}$.
$\:$\
[*Preconditionning.*]{} We have that $0 \in \cap_{i \in I } \mathrm{int}(T_i)$. Hence for any $i $ in $ I$ and $0 \leq \alpha <1$, $\alpha T_i \in \mathrm{int}(T_i)$. Furthermore, for $\alpha>0$, small enough, $(1 + \alpha)T_{i} \subset (1-\alpha) \mathrm{int}(T_{i+1})$. Set $\alpha_0$ such that $(1 + \alpha_0)T_0 \subset (1-\alpha_0) \mathrm{int} (T_1)$. By forward (and backward if $I = {\mathbb{Z}}$) induction, for all $i $ in $ I$, we obtain $\alpha_i>0$ such that $$\begin{aligned}
(1 + \alpha_i) T_i &\subset (1-\alpha_{i}) \mathrm{int} (T_{i+1}).
\end{aligned}$$ Setting for all $i $ in $ I$, $\epsilon_{i+1} = \min\{\alpha_i,\alpha_{i+1}\}$ ($\epsilon_0 = \alpha_0$ if $I = {\mathbb{N}}$), we have for all $i $ in $ I$ $$\begin{aligned}
(1 + \epsilon_i) T_i \subset (1 + \alpha_i) T_i &\subset (1-\alpha_{i}) \mathrm{int} (T_{i+1}) \subset(1-\epsilon_{i+1}) \mathrm{int} (T_{i+1}).
\end{aligned}$$ For all $i $ in $ I$, we introduce $$\begin{aligned}
S_{3i} &= T_i,\\
S_{3i + 1} &= (1 + \epsilon_i) T_i\\
S_{3i + 2} &= (1 - \epsilon_{i+1}) T_{i+1}.\end{aligned}$$
We have a new sequence of strictly increasing compact convex sets $\left( S_i\right)_{i \in I }$.
[*Value assignation.*]{} For each $i $ in $ I $, we set $$\begin{aligned}
K_i = \max_{\|x\| = 1} \frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{ \sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} \in \left( 0, + \infty \right).\end{aligned}$$ Note that for all $i $ in $ I$, $K_{3i} = 1$. We choose $\lambda_{1} = 2$, $\lambda_0 = 1$ and for all $i $ in $ I $, $$\label{value}
\lambda_{i+1} = \lambda_i + K_i(\lambda_i - \lambda_{i-1}).$$ By construction, we have for all $i $ in $ I $ and all $\theta \in {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$, $$\begin{aligned}
\frac{\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta))}{\lambda_{i+1} - \lambda_{i}} \leq \frac{\sigma_{S_{i}}(n(\theta)) - \sigma_{S_{i-1}}(n(\theta)) }{\lambda_{i} - \lambda_{i-1}}.\end{aligned}$$ If $I = {\mathbb{Z}}$, this entails $$\begin{aligned}
0 < \lambda_i - \lambda_{i - 1} \leq \frac{\lambda_{1}- \lambda_0}{\sigma_{S_{1}}(n(\theta)) - \sigma_{S_{0}}(n(\theta))} (\sigma_{S_{i}}(n(\theta)) - \sigma_{S_{i-1}}(n(\theta))),\end{aligned}$$ and the right-hand side is summable over negative indices $i\leq 0$, so that $\lambda_{i} \to \underline{\lambda} \in {\mathbb{R}}$ as $i \to -\infty$. In all cases $(\lambda_i)_{i\in I}$ is an increasing sequence bounded from below.
[*Local interpolation.*]{} We fix $i $ in $ I $ and consider the function $G_i$ described in Proposition \[prop:diffeoMorphism\] with $T_+ = S_{3i + 3} = T_{i+1}$, $T_- = S_{3i} = T_i$, $\lambda_+ = \lambda_{3i+3}$, $\lambda_{1} = \lambda_{3i+2}$, $\lambda_0 = \lambda_{3i+1}$, $\lambda_- = \lambda_{3i}$, $e_0 = \epsilon_i$, $e_1 = \epsilon_{i+1}$. By linearity, we have for any $(\lambda,\theta) \in [\lambda_-,\lambda_+] \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned}
\left\langle G_i(\lambda,\theta),n(\theta)\right\rangle = \tilde \gamma_{\Theta}(\lambda)\end{aligned}$$ where $ \tilde \gamma_{\Theta}$ is as in Lemma \[lem:convexInterpol\] with input data $q_0 = \left\langle c_{3i}(\theta), n(\theta)\right\rangle = \sigma_{S_{3i}}(n(\theta))$, $q_1 = \left\langle c_{3i+3}(\theta), n(\theta)\right\rangle = \sigma_{S_{3i+3}}(n(\theta))$, and $\lambda_-,\lambda_0,\lambda_1,\lambda_+,e_1,e_0$ as already described. This corresponds to the Bernstein approximation of the piecewise affine interpolation between the points $$\begin{aligned}
&\left( \lambda_{3i}, \sigma_{S_{3i}}(n(\theta))\right)\nonumber\\
&\left( \lambda_{3i+1}, \sigma_{S_{3i+1}}(n(\theta))\right),\nonumber \\
&\left( \lambda_{3i+2}, \sigma_{S_{3i+2}}(n(\theta))\right),\nonumber \\
&(\lambda_{3i+3}, \sigma_{3i+3}(n(\theta))), \label{eq:piecewiseAffineInterpolExplicit}\end{aligned}$$ By construction of $\left( K_i \right)_{i \in I }$, we have for all $\theta$, $$\begin{aligned}
0 < \frac{\sigma_{S_{3i+3}}(n(\theta)) - \sigma_{S_{3i+2}}(n(\theta)) }{\lambda_{3i+3} - \lambda_{3i + 2}} \leq \frac{\sigma_{S_{3i+2}}(n(\theta)) - \sigma_{S_{3i+1}}(n(\theta)) }{\lambda_{3i+2} - \lambda_{3i + 1}} \leq \frac{\sigma_{S_{3i+1}}(n(\theta)) - \sigma_{S_{3i}}(n(\theta)) }{\lambda_{3i+1} - \lambda_{3i}}.\end{aligned}$$ Whence the affine interpolant between points in is strictly increasing and concave, and by using the shape preserving properties of Bernstein polynomials, $\left\langle G_i(\lambda,\theta),n(\theta)\right\rangle$ has strictly positive derivative. As a consequence $G_i$ is a diffeomorphism and its derivatives are as in Proposition \[prop:diffeoMorphism\]. Furthermore $$\begin{aligned}
\lambda \mapsto \left\langle G_i(\lambda,\theta),n(\theta)\right\rangle \end{aligned}$$ is a $C^k$ concave function of $\lambda$.
[*Global interpolation.*]{} Recall that $\underline{\lambda} = \inf_{i \in I} \lambda_i>-\infty$ and set $\bar{\lambda} = \sup_{i \in I} \lambda_i\in (-\infty,+\infty]$. For any $\lambda \in (\underline{\lambda},\bar{\lambda})$, there exists a unique $i_\lambda \in I$ such that $\lambda \in [\lambda_{3i_\lambda}, \lambda_{3i_\lambda+3})$. Define $$\begin{aligned}
G \colon (\underline{\lambda}, \bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}&\mapsto {\mathbb{R}}^2 \\
(\lambda,\theta) &\mapsto G_{i_\lambda} (\lambda, \theta).\end{aligned}$$ Fix $i$ in $I$. The boundary of $T_{i+1}$ is given by $G_{i+1}(\lambda_{3i+3}, {\mathbb{R}}/ 2 \pi {\mathbb{Z}}) = G_{i}(\lambda_{3i+3}, {\mathbb{R}}/ 2 \pi {\mathbb{Z}})$ with actually $$\label{coinc}
G_{i+1}(\lambda_{3i+3}, \theta) = G_{i}(\lambda_{3i+3}, \theta ), \mbox{ for all }\theta \mbox{ in }{\mathbb{R}}/ 2 \pi {\mathbb{Z}}.$$
Since $K_{3i} = 1$, we have $$\begin{aligned}
\lambda_{3i+1} - \lambda_{3i} = \lambda_{3i} - \lambda_{3i-1}.\end{aligned}$$ The expressions of the derivatives in Proposition \[prop:diffeoMorphism\] and ensure that the derivatives of $G_{i+1}$ and $G_{i}$ agree on ${\lambda_{3i+3}} \times {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$ up to order $k$. Hence $G$ is a local diffeomorphism. Bijectivity of each $G_i$ ensure that $G$ is also bijective and thus $G$ is a diffeomorphism. Furthermore $$\begin{aligned}
\lambda \mapsto \left\langle G(\lambda,\theta),n(\theta)\right\rangle \end{aligned}$$ is $C^k$ piecewise concave and thus concave.
[*Extending $G$.*]{} If $I = {\mathbb{N}}$, we may assume without loss of generality that $S_0 = B$ the Euclidean ball and $S_1 = 5/3 S_0$, which corresponds to $\epsilon_0 = 2/3$, eventually after adding a set in the list and rescaling. Let $\phi$ denote the function described in Lemma \[lem:interpolationAroundZero\] and $G_{-1}$ be described as in Lemma \[lem:diffGauge2\]. This allows to extend $G$ for $\lambda \in [0,1]$, $G$ is then $C^k$ on $(0,\bar{\lambda}) \times{\mathbb{R}}/ 2\pi {\mathbb{Z}}$ by using Lemma \[lem:diffGauge2\] and Proposition \[prop:diffeoMorphism\]. This does not affect the differentiability, monotonicity and concavity properties of $G$.
[*Defining the interpolant $f$.*]{} We assume without loss of generality that $\underline{\lambda} = 0$. We set $f$ to be the first component of the inverse of $G$ so that it is defined on $G^{-1}\left( (0,\bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}\right)$. We extend $f$ as follows:
- $f(0) = 0$ if $I = {\mathbb{N}}$,
- $f = 0$ on $\cap_{i \in I } T_i$ if $I = {\mathbb{Z}}$.
Since $G$ is $C^k$ and non-singular on $ (0,\bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, the inverse mapping theorem ensures that $f$ is $C^k$ on $\mathrm{int}({\mathcal{T}}) \setminus \arg\min_{{\mathcal{T}}} f$.
[*Convexity of $f$.*]{} For any $\theta $ in $ {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned}
(0, \bar{\lambda}) &\mapsto {\mathbb{R}}_+\\
\lambda &\mapsto \sup_{z\in[f \leq \lambda]} n(\theta)^Tz \end{aligned}$$ is equal to $\left\langle G(\lambda,\theta),n(\theta)\right\rangle$ which is concave. It can be extended at $\lambda = 0$ by continuity. This preserves concavity hence, using Theorem \[th:crouzeix\], we have proved that $f$ is convex and $C^k$ on ${\mathcal{T}}\setminus \operatorname*{argmin}_{{\mathcal{T}}}f$.
[*Smoothness around the argmin and Hessian positivity.*]{} If $I = {\mathbb{N}}$, then the interpolant defined in Lemma \[lem:diffGauge2\] ensures that $f$ is proportional to the norm squared around $0$. Hence it is $C^k$ around $0$ with positive definite Hessian. We may compose $f$ with the function $g \colon t \mapsto \sqrt{t^2 + 1} + t$ which is increasing and has positive second derivative. This ensures that the resulting Hessian is positive definite outside $\operatorname*{argmin}f$ and thus everywhere since $$\begin{aligned}
\nabla^2 g \circ f = g' \nabla^2 f + g'' \nabla f \nabla f^T\end{aligned}$$ is positive definite thanks to the expressions for the Hessian of $f$ in Proposition \[prop:diffeoMorphism\]
If $I = {\mathbb{Z}}$, we let all the derivatives of $f$ vanish around the solution set. The smoothing Lemma \[lem:CkSmoothing\] applies and provides a function $\phi$ with positive derivative on $(0,+\infty)$, such that $\phi \circ f$ is convex, $C^k$ with prescribed sublevel sets. Furthermore, we remark that $$\begin{aligned}
\nabla^2 \phi \circ f = \phi' \nabla^2 f + \phi'' \nabla f \nabla f^T.\end{aligned}$$ We may compose $\phi \circ f$ with the function $g \colon t \mapsto \sqrt{t^2 + 1} + t$ which is increasing with positive second derivative, the expressions for the Hessian of $f$ in Proposition \[prop:diffeoMorphism\] ensure that the resulting Hessian is positive definite out of $\operatorname*{argmin}f$.
\[rem:alignedLevelSets\][In view of Remark \[rem:alignedInterpolation\], if we have $T_{i+1} = \alpha T_i$ for some $0<\alpha < 1$ and $i $ in $ {\mathbb{Z}}$, then the interpolated level sets between $T_i$ and $T_{i+1}$ are all of the form $s T_i$ for $\alpha \leq s \leq 1$.]{}
\[rem:strictConvexity\][ Recall that strict convexity of a differentiable function amounts to the injectivity of its gradient. In Theorem \[th:smoothinterp\] if there is a unique minimizer, then the invertibility of the Hessian outside argmin $f$ ensures that our interpolant is strictly convex (note that this is automatically the case if $I = {\mathbb{N}}$).]{}
Considerations on Legendre functions {#s:legendre}
------------------------------------
The following proposition provides some interpolant with additional properties as global Lipschitz continuity and finiteness properties for the dual function. At this stage these results appear as merely technical but they happen to be decisive in the construction of counterexamples involving Legendre functions. The properties of Legendre functions can be found in [@rockafellar1970convex Chapter 6]. We simply recall here that, given a convex body $C$ of ${\mathbb{R}}^p$, a convex function $h:C\to {\mathbb{R}}$ is [*Legendre*]{} if it is differentiable on ${\mathrm{int}\,}C$ and if $\nabla h$ defines a bijection from ${\mathrm{int}\,}C$ to $\nabla h({\mathrm{int}\,}C)$ with in addition $$\lim_{\begin{array}{l}
x\in {\mathrm{int}\,}C\\
x\to z\end{array}} \|\nabla h(x)\|=+\infty,$$ for all $z$ in ${\mathrm{bd}\,}C$. We also assume that $\mbox{epi}\,f:=\{(x,\lambda):f(x)\leq \lambda\}$ is closed in ${\mathbb{R}}^{p+1}$. The [*Legendre conjugate*]{} or [*dual function*]{} of $h$ is defined through $$h^*(z)=\sup\left\{\langle z,x\rangle -h(x):x \in C\right\},$$ for $z$ in ${\mathbb{R}}^p$, and its domain is $D:=\left\{z\in{\mathbb{R}}^p:h(z)<+\infty\right\}.$ The function $h^*$ is differentiable on the interior of $D$, and the inverse of $\nabla h:{\mathrm{int}\,}C \to {\mathrm{int}\,}D$ is $\nabla h^*:{\mathrm{int}\,}D \to {\mathrm{int}\,}C$.
We start with a simple technical lemma on the compactness of the domain of a Legendre function.
Let $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ be a globally Lipschitz continuous Legendre function, and set $D = \mathrm{int}(\mathrm{dom}(h^*))$ where $h^* \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ is the conjugate of $h$. For each $\lambda \geq \min_{{\mathbb{R}}^2}h$ let $\sigma_\lambda$ be the support function associated to the set $\left\{ z \in {\mathbb{R}}^2, \, h(z) \leq \lambda \right\}$. The following are equivalent
- $h^*(x)\leq 0$ for all $x \in D$.
- For all $y \in {\mathbb{R}}^2$, $\sigma_{h(y)}(\nabla h(y)) \leq h(y)$.
In both cases $h^*$ has compact domain. \[lem:legendreBounded\]
Let us establish beforehand the following formula $$\begin{aligned}
\label{e:h}
h^*(z) &= \sigma_{h(y)}(\nabla h(y))- h(y),
\end{aligned}$$ with $y = \nabla h^* (z)$ and $y\in{\mathbb{R}}^2$. Since $h^*$ is Legendre, we have for all $y$ in $D$, $$\begin{aligned}
h^*(z) &= \sup_{y \in {\mathbb{R}}^2} \left\langle z, y \right\rangle - h(y) = \left\langle z, \nabla h^*(z) \right\rangle - h(\nabla h^*(z)).
\end{aligned}$$ We have, setting $y = \nabla h^*(z)$ $$\begin{aligned}
\left\langle z, \nabla h^*(z) \right\rangle = \left\langle \nabla h(y),y \right\rangle = \sigma_{h(y)}(\nabla h(y))
\end{aligned}$$ because $\nabla h(y)$ is normal to the sublevel set of $h$ which contains $y$ in its boundary. Hence we have $h^*(z) = \sigma_{h(y)}(\nabla h(y))- h(y)$ with $y = \nabla h^* (z)$, that is holds. Since $\nabla h^* \colon D \mapsto {\mathbb{R}}^2$ is a bijection, the equivalence follows. In this case the domain of $h^*$ is closed because $h^*$ is bounded and lower semicontinuous. The domain is also bounded by the Lipschitz continuity of $h$, whence compact.
\[th:globallyLipshitz\] Let $\left( S_i \right)_{i\in {\mathbb{N}}}$ be such that for any $i $ in $ I$, $T_- = S_{i}$, $T_+ = S_{i+1}$ satisfy Assumption \[ass:curvature\] and there exists a sequence $\left( \epsilon_i \right)_{i \in {\mathbb{N}}}$ in $(0,1)$ such that for all $i\geq 1$, $(1 - \epsilon_i)^{-1} S_{3i-1} = (1 +\epsilon_i)^{-1} S_{3i+1}=S_{3i}$.\
Assume in addition that, $$\begin{aligned}
\inf_{\|x\| = 1}& \sigma_{S_i}(x) - \sigma_{S_{i-1}}(x) =1 + O\left(\frac{1}{i^3}\right)&\mbox{ (non degeneracy)}\label{nd}, \\
\sup_{\|x\|=1}& \left|\frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{\sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} - 1\right| = O\left( \frac{1}{i^3} \right)& \mbox{ (moderate growth)}\label{mg}.
\end{aligned}$$ Then there exists a convex $C^k$ function $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$, such that
- For all $i $ in $ {\mathbb{N}}$, $S_{3i}$ is a sublevel set of $h$,
- $h$ has positive definite Hessian,
- $h$ is globally Lipschitz continuous,
- $h^*$ has a compact domain $D$ and is $C^k$ and strictly convex on $\mathrm{int}(D)$.
The construction of $h$ follows the exact same principle as that of Theorem \[th:smoothinterp\]. This ensures that the first two points are valid. Note that equation implies that the sets sequence grows by at least a fixed amount in each direction as $i$ grows. Hence we have ${\mathcal{T}}= {\mathbb{R}}^2$.\
*Global Lipschitz continuity of $h$:* The values of $h$ are defined through $$\begin{aligned}
\lambda_{i+1}-\lambda_{i}&=K_i(\lambda_{i}-\lambda_{i-1}),\;\forall i\in {\mathbb{N}}^*\\
K_i &= \max_{\|x\| = 1} \frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{ \sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} \in \left( 0, + \infty \right),
\end{aligned}$$ so that $K_i = 1 + O(1/i^3)$ thanks to equation . Note that the moderate growth assumption entails $$\begin{aligned}
\sup_{i \in {\mathbb{N}}} \sigma_{S_{i+1}}(x) - \sigma_{S_{i}}(x) = O\left( \prod_{i \in {\mathbb{N}}^*} K_i\right) = O(1).
\label{eq:suportDiffBounded}
\end{aligned}$$ For $i\geq 1$, one has $$\label{for}
\lambda_{i+1}-\lambda_{i}=\prod_{1 \leq j \leq i} K_j(\lambda_1-\lambda_0).$$
On the other hand using the bounds , and the identity , there exists a constant $\kappa>0$ such that for all $i \geq 1$, all $\theta $ in $ {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$, $$\begin{aligned}
\frac{\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta)) }{\lambda_{i+1} - \lambda_{i}} = \frac{\left(\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta))\right) }{\prod_{j=1}^i K_j (\lambda_{1} - \lambda_{0})} \geq \kappa>0.
\label{eq:tempDerivLambda}\end{aligned}$$
By the interpolation properties described in Lemma \[lem:convexInterpol\] the function $\left\langle G(\lambda,\theta),n(\theta)\right\rangle$ constructed in Theorem \[th:smoothinterp\] has derivative with respect to $\lambda$ greater than $\kappa$. Recalling the expression of the gradient as given in Proposition \[prop:diffeoMorphism\] (and the concavity of $G$ with respect to $\lambda$), this shows that $$\|\nabla h(x)\|\leq \frac{1}{\kappa}$$ for all $x $ in $ {\mathbb{R}}^2$, and by the mean value theorem, $h$ is globally Lipschitz continuous on ${\mathbb{R}}^2$.
*Properties of the dual function:* $h$ is Legendre, its conjugate $h^*$ is therefore Legendre. From the definiteness of $\nabla^2 h$ and the fact that $\nabla h \colon {\mathbb{R}}^2 \mapsto \mathrm{int}(D)$ is a bijection, we deduce that $h^*$ is $C^k$ by the inverse mapping theorem. So the only property which we need to establish is that $h^*$ has a compact domain, in other words, using Lemma \[lem:legendreBounded\], it is sufficient to show that $\sup_{x \in \mathrm{int} D} h^*(x) \leq 0$.
Using the notation of the proof of Theorem \[th:smoothinterp\], we will show that it is possible to verify that, for all $\lambda,\theta$ in the domain of $G$ $$\begin{aligned}
\label{eq:toBeCheckedForBoundedness}
\frac{\left\langle n(\theta), G(\lambda,\theta)\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle} \leq \lambda.\end{aligned}$$ Equation is indeed the coordinate form of the characterization given in Lemma \[lem:legendreBounded\]. Let us observe that $$\begin{aligned}
\label{e:int}\frac{\partial}{\partial \lambda} \left( \left\langle n(\theta), G(\lambda,\theta)\right\rangle - \lambda \left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle \right) = -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle,\end{aligned}$$ and since $G$ is concave, the right hand side is positive.
Assume that we have proved that, $$\label{e:lim}
\lambda \mapsto \sup_\theta -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle$$ has finite integral as $\lambda \to \infty$. Since the function $$\label{e:funTheta}
\theta \mapsto \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle$$ is continuous on ${\mathbb{R}}/2\pi{\mathbb{Z}}$ for any $\lambda$, Lebesgue dominated convergence theorem would ensure that $$\begin{aligned}
\theta \mapsto \int_{\lambda \geq \lambda_0} -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle d \lambda\end{aligned}$$ is continuous in $\theta$, so that: $$\begin{aligned}
\sup_{\theta} \left[\lim_{\lambda \to \infty} \left\langle n(\theta), G(\lambda,\theta)\right\rangle - \lambda \left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle\right]<+\infty.\end{aligned}$$ Shifting values if necessary, we could assume that this upper bound is equal to zero to obtain equation . The latter being the condition required in Lemma \[lem:legendreBounded\], we would have reached a conclusion.
Let us therefore establish that is integrable over ${\mathbb{R}}_+$. Recall that $G$ is constructed using the Bernstein interpolation given in Lemma \[lem:convexInterpol\] between successive values of $\lambda$. As a result, for a fixed $\theta$, the function $\left\langle n(\theta), G(\lambda,\theta)\right\rangle $ is the interpolation of the piecewise affine function interpolating $$\begin{aligned}
&\left( \lambda_{3i}, \sigma_{S_{3i}}(n(\theta))\right)\nonumber\\
&\left( \lambda_{3i+1}, \sigma_{S_{3i+1}}(n(\theta))\right),\nonumber \\
&\left( \lambda_{3i+2}, \sigma_{S_{3i+2}}(n(\theta))\right),\nonumber \\
&(\lambda_{3i+3}, \sigma_{3i+3}(n(\theta))),
\label{eq:piecewiseAffineInterpolExplicit2}\end{aligned}$$ as in equation . This interpolation is concave and increasing.
Assumption ensures that $K_j = 1 + O(1/j^3)$. Then $$\begin{aligned}
\prod_{j=1}^m K_j = \bar{K} + O(1 / j^{2})\end{aligned}$$ where $\bar{K}$ is the finite, positive limit of the product (we can for example perform integral series comparison after taking the logarithm).
The recursion on the values writes for all $i\geq1$ $$\begin{aligned}
\lambda_{i+1} = \lambda_i + K_i(\lambda_i - \lambda_{i-1}),\end{aligned}$$ so that $$\begin{aligned}
\lambda_{i+1} - \lambda_i= (\lambda_1 - \lambda_0) \prod_{j=1}^i K_i = (\lambda_1 - \lambda_0) \bar{K} + O(1/i^{2}).\end{aligned}$$
This means that the gap between consecutive values tends to be constant. Thus by in Lemma \[lem:convexInterpol\], see also Remark \[rem:alignedInterpolation\], the degree of the Bernstein interpolants is bounded. Using this bound together with inequality , providing bounds for the derivatives of Bernstein’s polynomial, ensure that, for all $\lambda $ in $ [\lambda_{3i}, \lambda_{3i+3})$: $$\begin{aligned}
&\left|\left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\right| \\
=\,& O\left( \max _{j = 3i+2, 3i+1}\left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\lambda_{j+1} - \lambda_{j}} - \frac{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) }{\lambda_{j} - \lambda_{j-1}} \right| \right).\end{aligned}$$ Now for any $j = 3i+2, 3i+1$, $$\begin{aligned}
&\left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\lambda_{j+1} - \lambda_{j}} - \frac{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) }{\lambda_{j} - \lambda_{j-1}} \right| \\
=\,& \frac{1}{\lambda_{j+1} - \lambda_{j}} \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - \frac{\lambda_{j+1} - \lambda_{j}}{\lambda_{j} - \lambda_{j-1}} (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\\
=\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\
&\times \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - (1 + O(1/i^2)) (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\\
=\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\
&\times \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\end{aligned}$$ where the last identity follows from the triangle inequality because using $\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) = O(1)$ in . Hence $$\begin{aligned}
&\left|\left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\right| \\
=\,&
\left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\
&\times O\left( \max _{j = 3i+2, 3i+1}\left| \sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta)) - (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) ) \right| \right)\\
=\,&
\left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\
&\times O\left( \max _{j = 3i+2, 3i+1}\left|\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) \right| \times \left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) } - 1 \right| \right)\\
=\,& O(1/i^3),\end{aligned}$$ where the last inequality follows from and . Now as $i \to \infty$, $\lambda_{3i} \sim \lambda_{3i + 3} \sim i c$ for some constant $c>0$ and $$\begin{aligned}
\sup_{\lambda \in [\lambda_{3i}, \lambda_{3i+3}], \theta \in [0, 2 \pi]} - \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle = O(1 / i^2)\end{aligned}$$ and $$\begin{aligned}
\sup_{ \theta \in [0, 2 \pi]} - \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\end{aligned}$$ has finite integral as $\lambda \to \infty$. This implies and it concludes the proofs.
Smooth convex interpolation for sequences of polygons
=====================================================
Given a sequence of points $A_1,\ldots,A_n$, we denote by $A_1\ldots A_n$ the polygon obtained by joining successive points ending the loop with the segment $[A_n,A_1]$. In the sequel we consider mainly convex polygons, so that the vertices $A_1,\ldots,A_n$ are also the extreme points.
The purpose of this section is first to show that polygons can be approximated by smooth convex sets with prescribed normals under weak assumptions. Figure \[fig:illustrPolySmooth\] illustrates the result we would like to establish: given a target polygon with prescribed normals at its vertices, we wish to construct a smooth convex set interpolating the vertices with the desired normals and whose distance to the polygon is small.
Then given a sequence of nested polygons, we provide a smooth convex function which interpolates the polygons in the sense described just above.
Given a closed nonempty convex subset $S$ of ${\mathbb{R}}^p$ and $x$ in $S$, we recall that the [*normal cone to $S$ at $x$*]{} is $$N_S(x)=\left\{z\in {\mathbb{R}}^p:\langle z,y-x\rangle \leq 0,\forall y\in S\right\}.$$ Such vectors will often simply called normals (to $S$) at $x$.
Smooth approximations of polygons
---------------------------------
\[lem:approxSegment\] For any $r_-,r_+ > 0$, $t_- > 0$, $t_+<0$ and $\epsilon > 0$, $m \in {\mathbb{N}}$, $m \geq 3$, there exists a strictly concave polynomial function $p \colon [0,1] \mapsto [0,\epsilon]$ such that $$\begin{aligned}
p(0) &= 0 &p(1) &= 0\\
p'(0) &= t_-& p'(1) &= t_+\\
p''(0) &= -r_- &p''(1) &= -r_+.\\
p^{(q)}(0) & = 0 \quad q \in\{3, \ldots, m\}.
\end{aligned}$$
![Arrows designate the prescribed normals. We construct a strictly convex set with smooth boundary entirely contained in the auxiliary (blue) polygon. This set interpolates the normals and the distance to the original (red) polygon can be chosen arbitrarily small. The degree of smoothness of the boundary can be chosen arbitrarily high.[]{data-label="fig:illustrPolySmooth"}](illustrPolySmooth){width=".7\textwidth"}
Let us begin with preliminary remarks. Consider for any $a,b $ in $ {\mathbb{R}}$, the function $$\begin{aligned}
f \colon t \mapsto a(t-b)^2.
\end{aligned}$$ For any $t $ in $ {\mathbb{R}}$, $q $ in $ {\mathbb{N}}$, $q > 2$, and any $c > 0$, we have $$\begin{aligned}
f(t + c) - f(t) &= \Delta^1_c f(t) = a c ( 2(t-b) + c)\nonumber\\
f(t + 2 c) - 2 f(t+c) + f(t) &= \Delta^2_c f(t) = a c ( 2 (t + c - b) + c - 2(t-b) - c) = 2ac^2\nonumber\\
\Delta_c^q f(t) &= 0
\label{eq:approxSegment00}
\end{aligned}$$
[*Choosing the degree $d$ and constructing the polynomial.* ]{} For any $d $ in $ {\mathbb{N}}$, $d \geq 2m + 1$, we set $$\begin{aligned}
a_-(d) &= \frac{-dr_-}{2(d - 1)}<0&b_-(d) &= \frac{1}{2d} \left( 1 + 2 t_-\frac{d-1}{r_-} \right)> 0\nonumber\\
a_+(d) &= \frac{-dr_+}{2(d - 1)}<0&b_+(d) &= 1 + \frac{1}{2d} \left( -1 + 2t_+\frac{d-1}{r_+} \right)< 1,
\label{eq:approxSegment0}
\end{aligned}$$ and define the functions $$\begin{aligned}
f_d \colon s &\mapsto
\begin{cases}
a_-(d) (( s - b_-(d))^2 - b_-(d)^2) & \text{ if } s\leq b_-(d)\\
- a_-(d) b_-(d)^2& \text{ if } s\geq b_-(d)
\end{cases}\\
g_d \colon t &\mapsto
\begin{cases}
a_+(d) (( s - b_+(d))^2 - (1-b_+(d))^2) & \text{ if } s\geq b_+(d)\\
- a_+(d) (1-b_+(d))^2& \text{ if } s\leq b_+(d).
\end{cases}
\end{aligned}$$ Furthermore, we set $$\begin{aligned}
f \colon t &\mapsto
\begin{cases}
\frac{r_-}{2} \left( \left( \frac{t_-}{r_-} \right)^2- \left(s - \frac{t_-}{r_-} \right)^2 \right) & \text{ if } s\leq \frac{t_-}{r_-}\\
\frac{r_-}{2} \left( \frac{t_-}{r_-} \right)^2 & \text{ if } s\geq \frac{t_-}{r_-}
\end{cases}\\
g \colon t &\mapsto
\begin{cases}
\frac{r_+}{2} \left(\left( \frac{t_+}{r_+} \right)^2 - \left(s -1 - \frac{t_+}{r_+} \right)^2 \right) & \text{ if } s\geq 1 + \frac{t_+}{r_+}\\
\frac{r_+}{2} \left( \frac{t_+}{r_+} \right)^2 & \text{ if } s\leq 1 + \frac{t_+}{r_+}.
\end{cases}
\end{aligned}$$ Note that $b_-(d) \to t_- / r_-$, $b_+(d) \to 1 + t_+ / r_+$, $a_-(d) \to - r_-/2$ and $a_+(d) \to - r_+/2$ as $d \to \infty$ so that $f_d \to f$ and $g_d \to g$ uniformly on $[0,1]$. For any $d$, $f_d$ is concave increasing and $g_d$ is concave decreasing and all of them are Lipschitz continuous on $[0,1]$ with constants that do not depend on $d$. Note also that $f(0) = 0 < g(0)$ and $g(1) = 0 < f(1)$. We choose $d \geq 2m + 1$ such that $$\begin{aligned}
f_d\left( \frac{m}{d} \right) &\leq \min\left( \epsilon, g_d\left( \frac{m}{d} \right) \right)\nonumber\\
g_d\left(1 - \frac{m}{d} \right) &\leq \min\left( \epsilon, f_d\left(1 - \frac{m}{d} \right) \right).
\label{eq:approxSegment1}
\end{aligned}$$ Such a $d$ always exists because in both cases, the left hand side converges to $0$ and the right hand side converges to a strictly positive term as $d$ tends to $\infty$. For such a $d$, we set $h \colon s \mapsto \min\left\{ f_d(s), g_d(s),\epsilon \right\}$. By construction, $h$ is concave, agrees with $f_d$ on $\left[ 0,\frac{m}{d} \right] \subset [0,1/2]$ and with $g_d$ on $\left[ 1- \frac{m}{d},1 \right] \subset [1/2,1]$. Using equation with $c = 1/d$, we deduce that $$\begin{aligned}
d (d-1) \Delta^2 h(0) &= d (d-1) \Delta^2 f_d(0) = d(d-1)\frac{2 a_-(d)}{d^2} = -r_-\\
d \Delta h(0) &= d \Delta f_d(0) = a_-(d)\left( \frac{1}{d} -2b_-(d)\right) = t_-\\
\Delta^{q}h(0) &= 0 = \Delta^{q}f_d(0) \quad \forall m \geq q \geq 3\\
d (d-1) \Delta^2 h\left(1 - \frac{2}{d}\right) &= d (d-1) \Delta^2 g_d\left(1 - \frac{2}{d}\right)= d(d-1)\frac{2 a_+(d)}{d^2} = -r_+\\
d \Delta h\left( 1 - \frac{1}{d} \right) &= d \Delta g_d\left( 1 - \frac{1}{d} \right) = a_+(d) \left( \frac{-1}{d} + 2 (1 - b_+(d))\right) = t_+\\
\Delta^{q}h\left( 1 - \frac{m}{d} \right) &= \Delta^{q}g_d\left( 1 - \frac{m}{d} \right) = 0 \quad \forall m \geq q \geq 3.
\end{aligned}$$ From the concavity of $h$ and the derivative formula , we deduce that the polynomial $B_{h,d}$ satisfies the desired properties.
![Illustration of the approximation result of Lemma \[lem:approxSegment\] with $\epsilon = 0.1$, $m = 3$, $t_- =0.7$, $t_+ = -2.2$, $r_- = 2$ and $r_+ = 0.2$. The resulting polynomial is of degree 66. Numerical estimations of the first and second order derivatives at 0 and 1 match the required values up to 3 precision digits. The polynomial is strongly concave, however, this is barely visible because the strong concavity constant is extremely small. []{data-label="fig:illustrApproxSeg"}](illustrApproxSeg){width="\textwidth"}
We deduce the following result
Let $a > 0$, $r > 0$, $\epsilon > 0$, and an integer $m \geq 3$. Consider two unit vectors: $v_-$ with strictly positive entries and $v_+$ with first entry strictly positive and second entry strictly negative. Then there exists a $C^m$ curve $\gamma \colon [0,M] \mapsto {\mathbb{R}}^2$, such that
1. $\|\gamma'\| = 1$.
2. $\gamma(0) = (-a, 0) := A$ and $\gamma(1) = (0,a) : = B$.
3. $\gamma'(0) = v_-$ and $\gamma'(1) = v_+$.
4. $\|\gamma''(0)\| = \|\gamma''(-1)\| = r$.
5. $\mathrm{det}(\gamma',\gamma'') < 0$ along the curve.
6. $\gamma^{(q)}(0) = \gamma^{(q)}(1) = 0$ for any $3 \leq q \leq m$.
7. ${\mathrm{dist}}(\gamma([0,M]), [A,B]) \leq \epsilon$.
\[lem:existenceCurve\]
Consider the graph of a polynomial as given in Lemma \[lem:approxSegment\] with $t_- = v_-[2] / v_-[1]> 0$, $t_+ = v_+[2] / v_+[1]< 0$ and $r_- = \frac{r}{2a} (1 + t_-^2)^{\frac{3}{2}}$, $r_+ = \frac{r}{2a} (1 + t_+^2)^{\frac{3}{2}}$ and $\epsilon/2a$ as an approximation parameter. This graph is parametrized by $t$. It is possible to reparametrize it by arclength to obtain a $C^m$ curve $\gamma_0$ whose tangents at $0$ is $T_-$, at $1$ is $T_+$, and whose curvature at $0$ and $1$ is $- \frac{r}{2a}$. Furthermore, $\gamma_0$ has strictly negative curvature whence item 5. Consider the affine transform: $x \mapsto 2a(x - 1/2)$, $y \mapsto 2a y$. This results in a $C^m$ curve $\gamma$, parametrized by arclength which satisfies the desired assumptions.
Let $S=A_1...A_n$ be a convex polygon. For each $i$, let $V_i$ be in $ N_S(A_i)$ such that the angle between $V_i$ and each of the two neighboring faces is within $\left( \frac{\pi}{2},\pi \right)$. Then for any $\epsilon > 0$ and any $m \geq 2$, there exists a compact convex set $C \subset {\mathbb{R}}^2$ such that
- the boundary of $C$ is $C^m$ with non vanishing curvature,
- $S \subset C$,
- for any $i = 1,\ldots, n$, $A_i $ in $ {\mathrm{bd}\,}(C)$ and the normal cone to $C$ at $A_i$ is given by $V_i$,
- $\max_{y \in C} {\mathrm{dist}}(y,S) \leq \epsilon$.
\[lem:polyGon\]
We assume without loss of generality that $A_1,\ldots,A_n$ are ordered clockwise. For $i = 1, \ldots, n-1$, and each segment $[A_i,A_{i+1}]$, we may perform a rotation and a translation to obtain $A_i = -(a,0)$ and $A_{i+1} = (a,0)$. Working in this coordinate system, using the angle condition on $V_i$, we may choose $v^-_i$, $v^+_{i+1}$ satisfying the hypotheses of Lemma \[lem:existenceCurve\] respectively orthogonal to $V_i$ and $V_{i+1}$. Choosing $r = 1$, we obtain $\gamma_i \colon [0,M_i] \mapsto {\mathbb{R}}^2$ as given by Lemma \[lem:existenceCurve\]. Rotation and translations affect only the direction of the derivatives of curves, not their length. Hence, it is possible to concatenate curves $\left( \gamma_i \right)_{i=1}^{n-1}$ and to preserve the $C^m$ properties of the resulting curve. At end-points, tangents and second order derivatives coincide while higher derivatives vanish. Furthermore the curvature has constant sign and does not vanish. We obtain a closed $C^m$ curve which defines a convex set which satisfies all the requirements of the lemma.
[Given any polygon, choosing normal vectors as given by the direction of the bissector of each angles ensure that the above assumptions are satisfied. Hence all our approximation results hold given polygon without specifying the choice of outer normals.]{} \[rem:noNormal\]
Smooth convex interpolants of polygonal sequences
-------------------------------------------------
[For $n\geq3$, let $A_1\ldots A_n $ be a convex polygon $S$ and $V_i $ be in $ N_S(V_i)$ for $i=1,\ldots,n$. We say that $\left( A_i,V_i \right)_{i=1}^n $ is [*interpolable*]{} if for each $i = 1,\ldots,n$, the angle between $V_i$ and each of the two neighboring faces of the polygon is in $\left( \frac{\pi}{2},\pi \right)$. The collection $\left( A_i,V_i \right)_{i=1}^n $ is called [*a polygon-normal pair*]{}.]{} \[def:interpolable\]
Let $I = {\mathbb{Z}}$ or $I = {\mathbb{N}}$. Let $\left( PN_i \right)_{i \in I}$ be a sequence of interpolable polygon-normal pairs. Setting for $i $ in $ I$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_j $ is in $ {\mathbb{N}}$ and denoting by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, we say that the sequence $\left( PN_i \right)_{i \in I}$ is strictly increasing if for all $i $ in $ I$, $T_i \subset \mathrm{int}(T_{i+1})$.
Let $\left( PN_i \right)_{i \in I}$ be a strictly increasing sequence of interpolable polygon-normal pairs. A sequence $(\epsilon_i)_{i \in I}$ in $(0,1)$ is said to be [*admissible*]{} if $0 \in \mathrm{int}(T_i)$ for each $i $ in $ I$ and $$\gamma T_i\subset{\mathrm{int}\,}T_{i+1}$$ for all $\gamma \in [1-\epsilon_i,1+\epsilon_i].$ We have the following corollary of Theorem \[th:smoothinterp\].
Let $I = {\mathbb{Z}}$ or $I = {\mathbb{N}}$. Let $\left( PN_i \right)_{i \in I}$ be a strictly increasing sequence of interpolable polygon-normal pairs and $(\epsilon_i)_{i \in I}$ be admissible. Set $\displaystyle\mathcal{T}:=\mathrm{int}\left(\cup_{i\in I}T_i \right).$
Then for any $k $ in $ {\mathbb{N}}$, $k \geq 2$ there exists a $C^k$ convex function $f \colon \mathcal{T} \mapsto {\mathbb{R}}$, and an increasing sequence $(\lambda_i)_{i \in I}$, with $\inf_{i \in I} \lambda_i > -\infty$, such that for each $i $ in $ I$
- $T_i \subset \left\{ x,\,f(x) \leq \lambda_i \right\}$.
- ${\mathrm{dist}}(T_i, \left\{ x,\,f(x) \leq \lambda_i \right\}) \leq \epsilon_i$.
- For each $i $ in $ I$, $j$ in $\{1,\ldots,n_i\}$, we have $f(A_{i,j}) = \lambda_i$ and $\nabla f(x)$ is colinear to $V_{i,j}$.
- $\nabla^2 f$ is positive definite outside $\operatorname*{argmin}f$. When there is a unique minimizer then $\nabla^2f$ is positive definite throughout ${\mathcal{T}}$ [(]{}this is the case when $I = {\mathbb{N}}$ or when $I={\mathbb{Z}}$ and $\cap_{i \in I} T_i$ is a singleton[)]{}.
\[cor:polygonDecrease\]
We add two remarks which will be useful for directional convergence issues and the construction of Legendre functions:
- If two consecutive elements of the sequence of interpolable polygon-normal pairs are homothetic with center $0$ in the interior of both polytopes, then the restriction of the resulting convex function to this convex ring can be constructed such that all the sublevel sets within this ring are homothetic with the same center.
- If further conditions are imposed on the elements of a strictly increasing interpolable polygon-normal pair, then the resulting function can be constructed to be Legendre and globally Lipschitz continuous (that is, its Legendre conjugate has bounded support). This is a consequence of Proposition \[th:globallyLipshitz\] and will be detailled in the next section.
More on Legendre functions and a pathological function with polyhedral domain
-----------------------------------------------------------------------------
Using intensively polygonal interpolation, we build below a finite continuous Legendre function $h$ on an $\ell^\infty$ square with oscillating “mirror lines": $t\to \nabla h^*(\nabla h(x_0)+tc)$.
We start with the following preparation proposition related to the Legendre interpolation of Proposition \[th:globallyLipshitz\].
Let $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ be a strictly increasing sequence of interpolable polygon-normal pairs. Setting for $i$ in ${\mathbb{N}}^*$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_j$ is in ${\mathbb{N}}^*$ and denoting by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, we assume that $$T_i=3i P,\,\forall i \in {\mathbb{N}}^*,$$ where $P$ is a fixed polygon which contains the unit Euclidean disk.\
Then for any $l$ in ${\mathbb{N}}$, $l \geq 2$, there exists a strictly increasing sequence of sets $\left( S_i \right)_{i \in {\mathbb{N}}, \,i\geq 2}$, such that for $j \geq 1$,
- $S_{3j}$ interpolates the normals of $PN_j$ in the sense of Lemma \[lem:polyGon\] with $\mathrm{dist}(S_{3j}, T_j) \leq 1 / (4(3j+2)^l)$
- $S_{3j - 1} = \frac{3j - 1}{3_j} S_{3j}$
- $S_{3j + 1} = \frac{3j + 1}{3_j} S_{3j}$
This sequence has the following properties
- there exists $c>0$, such that for all $j $ in ${\mathbb{N}}$, $j \geq 3$ and for all unit vector $x$, $$\begin{aligned}
c\geq\sigma_{S_{j+1}}(x) - \sigma_{S_j}(x) \geq 1 - \frac{1}{(j+1)^l}.
\label{eq:interpolGauge1}
\end{aligned}$$
- for all unit vector $x$, $$\begin{aligned}
\left| \frac{\sigma_{S_{j+1}}(x) - \sigma_{S_j}(x)}{ \sigma_{S_{j}}(x) - \sigma_{S_{j-1}}(x)} -1\right| \leq \frac{1}{j^l},\:\forall j\geq 3.
\label{eq:interpolGauge2}
\end{aligned}$$
\[lem:interpolGauge\]
Set for all $j$ in ${\mathbb{N}}^*$, $\delta_j = \frac{1}{4(3j+2)^l}$ and let $S_{3j}$ be the $\delta_j$ interpolant of $T_j=3jP$ as given by Lemma \[lem:polyGon\] so that ${\mathrm{dist}}(S_{3j}, 3jP) \leq \delta_j$. Since $P$ contains the unit ball, $$\label{inc} 3j P \subset S_{3j} \subset (3j + \delta_j) P.$$ Now set $$\begin{aligned}
S_{3j - 1} &= \frac{3j - 1}{3_j} S_{3j}\\
S_{3j + 1} &= \frac{3j + 1}{3_j} S_{3j}.
\end{aligned}$$ For any $j$ in ${\mathbb{N}}^*$, it is clear that $S_{3j-1} \subset \mathrm{int}(S_{3j})$ and $S_{3j} \subset \mathrm{int}(S_{3j+1})$. Furthermore, by , we have $$\begin{aligned}
S_{3j+1} \subset \frac{3j+1}{3j} ( 3j + \delta_j) P \subset ((3j+1) + 2\delta_j) P \subset \mathrm{int}((3j+2) P) \subset \mathrm{int} (S_{3j+2})
\end{aligned}$$ so that we indeed have a strictly increasing sequence of sets. We obtain from the construction, for any $j$ in ${\mathbb{N}}^*$, and any unit vector $x$, $$\begin{aligned}
\sigma_{S_{3j}}(x) - \sigma_{S_{3j-1}}(x) &=\sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)\notag \\
&= \frac{1}{3j} \sigma_{S_{3j}}(x) \in \left[ \sigma_P(x), \left( 1 + \frac{\delta_j}{3j} \right) \sigma_P(x) \right] \subset \left[ 1, \left(1+\frac{\delta_j} {3j} \right) \sigma_P(x) \right]\notag \\
\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x) &\leq \sigma_P(x) (3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - \sigma_P(x)(3j+1)\notag\\
\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x) &\geq \sigma_P(x) (3j +2) - \sigma_P(x)(3j+1)\left( 1 + \frac{\delta_j}{3j} \right) \notag\\
& = \sigma_P(x) \left( 1 - \delta_j \frac{3j+1}{3j} \right)\notag\\
&\geq 1 - \delta_j \frac{4}{3} \geq 1 - \frac{1}{(3j+2)^l}\label{intermed}
\end{aligned}$$ This proves . We deduce that for all $j$ in ${\mathbb{N}}^*,$ $$\begin{aligned}
\frac{\sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)}{ \sigma_{S_{3j}}(x) - \sigma_{S_{3j-1}}(x)}& = 1, \mbox{ for all nonzero vector $x$,} \\
\max_{\|x\| = 1} \frac{\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)}{ \sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)} &\leq (3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - (3j+1)\\
&= 1 + \frac{3j+2}{3j+3} \delta_{j+1} \leq 1 + \frac{1}{(3j +1)^l},\\
\min_{\|x\| = 1} \frac{\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)}{ \sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)} &\underset{\eqref{intermed}}{\geq} \frac{1 - \delta_j \frac{3j+1}{3j} }{\left( 1 + \frac{\delta_j}{3j} \right) }\\
&\geq \left( 1 - \delta_j \frac{3j+1}{3j} \right) \left( 1 - \frac{\delta_j}{3j} \right) \\
& \geq 1 - \delta_j\left( \frac{3j+1}{3j} + \frac{1}{3j} \right) \geq 1 - \delta_j \frac{5}{3} \geq 1 - \frac{1}{(3j+1)^l}.
\end{aligned}$$ Furthermore, using the fact that $t \mapsto \frac{1+t}{1-t}$ is increasing on $(-\infty,1)$ and the fact that $\delta_{j+1} \leq \delta_j$, $$\begin{aligned}
\max_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\underset{\eqref{intermed}}{\leq} \frac{1 + \frac{\delta_{j+1}}{3j + 3}}{1 - (3j+1) \frac{\delta_j}{3j}} \notag\\
&\leq\frac{1 + (3j+1)\frac{\delta_{j}}{3j}}{1 - (3j+1) \frac{\delta_j}{3j}} \notag\\
&\leq\frac{1 + \delta_j\frac{4}{3}}{1 - \delta_j \frac{4}{3}}\label{borntobewild}.
\end{aligned}$$ Setting $s(t)= (1+t)/(1-t)$, we have, for all $t \leq 1/2$ $$\begin{aligned}
s'(t) &= \frac{2}{(1-t)^2}, \, s(0)=1\\
s''(t) &= \frac{4}{(1-t)^3} \leq 24,\, s'(0)=2.
\end{aligned}$$ Thus $s(t) \leq 1 + 2 t + 12 t^2$ on $(-\infty,1/2]$. Since $\frac{4}{3} \delta_j \leq \frac{4}{75} \leq \frac{1}{2}$, we deduce from the previous remark and above: $$\begin{aligned}
\max_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\leq 1 + \frac{8}{3} \delta_j + \frac{64}{3} \delta_j^2 = 1 + \delta_j\left( \frac{8}{3} + \frac{64}{3} \delta_j \right)\\
&\leq 1 + \delta_j\left(3 + \frac{64}{3 \times 25}\right) \leq 1 + 4 \delta_j = 1 + \frac{1}{(3j+2)^l}.
\end{aligned}$$ Finally using again, $$\begin{aligned}
\min_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\geq \frac{1 }{(3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - (3j+1) } \\
&= \frac{1 }{1 + \delta_{j+1}\frac{3j+2}{3j+3} } \\
&\geq 1 - \delta_{j+1}\frac{3j+2}{3j+3} \geq 1 - \delta_{j+1} \geq 1 - \frac{1}{(3j+2)^l}. \end{aligned}$$ This proves the desired result.
Combining Lemma \[lem:interpolGauge\] and Proposition \[th:globallyLipshitz\], we obtain the following result.
Let $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ be a strictly increasing sequence of interpolable polygon-normal pairs. Set for $i$ in ${\mathbb{N}}^*$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_i$ is in ${\mathbb{N}}^*$, denote by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, and assume that for all $i $ in $ {\mathbb{N}}^*$, $T_i=3i P$ where $P$ is a fixed polygon which contains the unit disk in its interior.\
Then for any $k $ in $ {\mathbb{N}}$, $k \geq 2$ and all $l \geq 3$, there exists a $C^k$ globally Lipschitz continuous Legendre function, $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$, and an increasing sequence $(\lambda_i)_{i \in {\mathbb{N}}}$, such that for each $i $ in $ {\mathbb{N}}$:
- $T_i \subset \left\{ x,\,h(x) \leq \lambda_i \right\}$,
- ${\mathrm{dist}}(T_i, \left\{ x,\,h(x) \leq \lambda_i \right\}) \leq \frac{1}{4(3i+2)^l}$,
- For any $x$ with $h(x) = \lambda_i$ and $\nabla h(x)$ is colinear to $V_i$ for each vertex $x$ of $T_i$,
- $h$ has positive definite Hessian and is globally Lipschitz continuous,
- $h^*$ has compact domain and is $C^k$ on the interior of its domain.
\[th:Legendre\]
The function $h^*$ constructed in Theorem \[th:Legendre\] has compact polygonal domain and is continuous on this domain. \[cor:legendreContinuity\]
Since $P$ is a polygon and contains the unit Euclidean disk, the gauge function of $3P$ is polyhedral with full domain ${\mathbb{R}}^2$, call it $\omega$. Denote by $P^{\circ}$ the polar of $P$. This is a polytope and since $\omega$ is the gauge of $P$, we actually have $\omega = \sigma_{P^{\circ}}$, the support function of the polar of $P$ [@schneider1993convex Theorem 1.7.6]. Hence the the convex conjugate of $\omega$ is the indicator of the polytope $P^\circ$ [@rockafellar1970convex Theorem 13.2].
It can be easily seen from the proof of Proposition \[th:globallyLipshitz\] that $\lambda_i = \alpha i + r_i$ with $r(i) = O(1)$ as $i \to \infty$. Without loss of generality, we may suppose that $\alpha = 1$ (this is a simple rescaling) so that there is a positive constant $c$ such that $|\lambda_i - i| \leq c$ for all $i$.
Let $h$ be given as in Theorem \[th:Legendre\], fix $i \geq 1$ and $x \in {\mathbb{R}}^2$ such that $\lambda_{i-1} \leq h(x) \leq \lambda_i$. We have in ${\mathbb{R}}^2$ $$\begin{aligned}
\{y:\, \omega(y) \leq i-1 \} \subset \{y:\, h(y) \leq \lambda_{i-1} \} \subset \{y:\, h(y) \leq \lambda_{i} \} \subset \{y:\, \omega(y) \leq i+1 \}
\end{aligned}$$ and hence $$\begin{aligned}
i - 1 \leq \omega(x) \leq i+1
\end{aligned}$$ and we deduce that $$\begin{aligned}
\omega(x) - 2 - c \leq i - 1 -c \leq \lambda_{i-1} \leq h(x) \leq \lambda_i \leq i+c \leq \omega(x) + c + 1.
\end{aligned}$$ Since $i$ was arbitrary, this shows that there exists a constant $C>0$ such that $|h(x) - \omega(x)|\leq C$ for all $x \in {\mathbb{R}}^2$. Recall that $z \mapsto \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y)$ is the indicator function of $P^\circ$, hence, $$\begin{aligned}
z \in P^{\circ} & \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y) = 0 \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - h(y) \leq C < +\infty \\
z \not\in P^{\circ} & \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y) = +\infty \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - h(y) = +\infty
\end{aligned}$$ which shows that the domain of $h^*$ is actually $P^{\circ}$ which is a polytope. Now, $h^*$ is convex and lower semicontinuous on $P^{\circ}$, invoking the results of [@gale68convex], it is also upper semicontinuous on $B^*$ and finally it is continuous on $B^*$.
For any $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$ there exists a Legendre function $h\colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ whose domain is a closed square, continuous on this domain and $C^k$ on its interior, such that for all $i \in {\mathbb{N}}^*$, $\nabla h^*(i, 0)$ is proportional to $(\cos(\theta), (-1)^i \sin(\theta))$. \[cor:legendreOscillating\]
For $x=(u,v)$, set $\|x\|_1=|u|+|v|$, and let $P = \left\{ x \in {\mathbb{R}}^2,\, \|x\|_1 \leq 2 \right\}$. Let us construct a strictly increasing sequence of interpolable polygon-normal pairs $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ as follows, we fix $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$ and set for all $i\in {\mathbb{N}}^*$ :
- $T_i = 3i P$, the polygon associated to the $i$-th term $PN_i$ of the sequence,
- except at the rightmost corner, consider the normals given by the canonical basis vectors and their opposite,
- at the rightmost corner, $(6i,0)$, one chooses the normal given by the vector $$(\cos(\theta), (-1)^i \sin(\theta)).$$
We now invoke Theorem \[th:Legendre\] to obtain a Lipschitz continuous Legendre function, denoted $h^*$, with full domain having all the $T_i$ as sublevel sets and satisfying the hypotheses of the corollary. Rescaling by a factor $6$ and setting $h = h^{**}$ gives the result.
Counterexamples in continuous optimization
==========================================
We are now in position to apply our interpolation results to build counterexamples to classical problems in convex optimization. We worked on situations ranging from structural questions to qualitative behavior of algorithms and ODEs. Through 9 counterexamples we tried to cover a large spectrum but there are many more possibilities that are left for future research. Some example are constructed from decreasing sequences of convex sets, they can be interpolated using Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$, indexing the sequence with negative indices and adding artificially additional sets for positive indices. Nonetheless we sometimes depart from the notations of the first sections and index these sequences by ${\mathbb{N}}$ even though they are decreasing for simplification purposes.
Kurdyka-Łojasiewicz inequality may not hold
-------------------------------------------
The following result is proved in [@bolte2010characterization], it was crucial to construct a $C^2$ convex function which does not satisfy Kurdyka-Łojasiewicz (KL) inequality.
[[@bolte2010characterization Lemma 35]]{} There exists a decreasing sequence of compact convex sets $\left( T_i \right)_{i\in {\mathbb{N}}}$ such that for any $i $ in $ {\mathbb{N}}$, $T_- = T_{i+1}$ and $T_+ = T_i$ satisfy Assumption \[ass:curvature\] and $$\begin{aligned}
\sum_{i=0}^{+\infty} {\mathrm{dist}}(T_i, T_{i+1}) = +\infty
\end{aligned}$$ \[lem:tamsLoja\]
As a corollary, we improve the counterexample in [@bolte2010characterization] and provide a $C^k$ convex counterexamples for any $k\geq 2 $ in $ {\mathbb{N}}$.
There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ which does not satisfy KL inequality. More precisely, for any $r > \inf f$ and $\varphi \colon [\inf f, r] \mapsto {\mathbb{R}}$ continuous and differentiable on $(\inf f, r)$ with $\varphi' > 0$ and $\varphi(\inf f) = 0$, we have $$\begin{aligned}
\inf \{\| \nabla (\varphi \circ f)(x) \|: \: x\in {\mathbb{R}}^2, \,\inf f < f(x) < r \} = 0.
\end{aligned}$$
Using [@schneider1993convex Theorem 1.8.13], each $T_i$ can be approximated up to arbitrary precision by a polygon. Hence we may assume that all $T_i$ are polygonal while preserving the property of Lemma \[lem:tamsLoja\] as well as Assumption \[ass:curvature\]. Furthermore, using Lemma \[lem:polyGon\] and Remark \[rem:noNormal\] each $T_i$ can in turn be approximated with arbitrary precision by a convex set with $C^k$ boundary and positive curvature. Hence we may also assume that all $T_i$ satisfy both the result of Lemma \[lem:tamsLoja\] and have $C^k$ boundary with nonvanishing curvature. Reversing the order of the sets and adding additional sets artificially, we are in the conditions of application of Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$ and the resulting $f$ follows from the same argument as in [@bolte2010characterization Theorem 36].
Block coordinate descent may not converge
-----------------------------------------
![Illustration of the alternating minimization (resp. exact line search) example: on the left, the sublevel sets in gray and the corresponding alternating minimization (resp. exact line search) sequence in dashed lines. On the right the interpolating polygons together with their normal vectors as in Lemma \[lem:polyGon\].[]{data-label="fig:illustrAltMin"}](illustrAltMin1 "fig:"){width=".45\textwidth"}![Illustration of the alternating minimization (resp. exact line search) example: on the left, the sublevel sets in gray and the corresponding alternating minimization (resp. exact line search) sequence in dashed lines. On the right the interpolating polygons together with their normal vectors as in Lemma \[lem:polyGon\].[]{data-label="fig:illustrAltMin"}](illustrAltMin2 "fig:"){width=".4\textwidth"}
The following polygonal construction is illustrated in Figure \[fig:illustrAltMin\]. For any $n\geq2$ in ${\mathbb{N}}$, we set $$\begin{aligned}
A_n &= \left(\frac{1}{4} + \frac{1}{n}, \frac{1}{4} + \frac{1}{n}\right)\\
B_n &= \left( \frac{1}{4} + \frac{1}{2(
n-1)} + \frac{1}{2n}, \frac{1}{4} + \frac{1}{2n} + \frac{1}{2(n+1)} \right)\\
C_n &= \left(\frac{1}{4} + \frac{1}{n}, - \frac{1}{4} - \frac{1}{n}\right)\\
D_n &= \left(-\frac{1}{4} - \frac{1}{n}, - \frac{1}{4} - \frac{1}{n}\right)\\
E_n &= \left(-\frac{1}{4} - \frac{1}{n}, + \frac{1}{4} + \frac{1}{n}\right).\end{aligned}$$ This defines a convex polygon. We may choose the normals at $A_n, C_n, D_n, E_n$ to be bisectors of the corresponding corners and the normal at $B_n$ to be horizontal (see Figure \[fig:illustrAltMin\]). Rotating by an angle of $-\frac{n\pi}{2}$ and repeating the process indefinitely, we obtain the sequence of polygons depicted in Figure \[fig:illustrAltMin\]. It can be checked that the polygons form a strictly decreasing sequence of sets, as for $n>1$, the polygon $A_nB_nC_nD_nE_n$ is contained in the interior of the square $A_{n-1}C_{n-1}D_{n-1}E_{n-1}$. This fulfills the requirement of Corollary \[cor:polygonDecrease\].
There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initialization $x_0=(u_0,v_0)$ such that the recursion, for $i \geq 1$ $$\begin{aligned}
u_{i+1} &\in \operatorname*{argmin}_{u} f(u,v_i) \\
v_{i+1} &\in \operatorname*{argmin}_{v} f(u_{i+1},v)
\end{aligned}$$ produces a non converging sequence $\displaystyle (x_i)_{i\in{\mathbb{N}}}=\left((u_i,v_i)\right)_{i\in{\mathbb{N}}}$. \[cor:altMin\]
We apply Corollary \[cor:polygonDecrease\] to the proposed decreasing sequence and by choosing $(u_0,v_0) = B_2$ for example. This requires to shift indices (start with $i = 2$) and use Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$. Note that the optimality condition for partial minimization and the fact that level sets have nonvanishing curvature ensure that the partial minima are unique.
In the nonsmooth convex case cyclic minimization is known to fail to provide the infimum value, see e.g., [@auslender p. 94]. Smoothness is sufficient for establishing value convergence (see e.g. [@beck2013convergence; @wright2015coordinate] and references therein), whether it is enough or not for obtaining convergence was an open question. Our counterexample closes this question and shows that cyclic minimization does not yield converging sequences even for $C^k$ convex functions. This result also closes the question for the more general nonconvex case for which we are not aware of a nontrivial counterexample for convergence of alternating minimization. Let us mention however Powell’s example [@powell1973search] which shows that cyclic minimization with three blocks does not converge for smooth functions.
It would also be interesting to understand how our result may impact dual methods and counterexamples in that field, as for instance the recent three blocks counterexample in [@chen].
Gradient descent with exact line search may not converge
--------------------------------------------------------
Gradient descent with exact line search is governed by the recursion: $$x^+\in \operatorname*{argmin}\left\{f(y):y=x-t\nabla f(x),\,t\in{\mathbb{R}}\right\},$$ where $x$ is a point in the plane.
Observe that the step coincides with partial minimization when the gradient $\nabla f(x)$ is colinear to one of the axis of the canonical basis. From the previous section, we thus deduce the following.
There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initialization $z_0$ in the plane such that the recursion, for $i \geq 1$ $$\begin{aligned}
x_{i+1} \in \operatorname*{argmin}\left\{f(y): y=x_i-t\nabla f(x_i),\, t\in{\mathbb{R}}\right\}
\end{aligned}$$ produces a well defined non converging sequence $\left(x_i \right)_{i \in {\mathbb{N}}}$. \[cor:exactLineSearch\]
Convergence failure for gradient descent with exact line search is new up to our knowledge. Let us mention that despite non convergence, the constructed sequence satisfy sublinear convergence rates in function values [@beck2013convergence].
Tikhonov regularization path may have infinite length
-----------------------------------------------------
![Illustration of the Tikhonov regularization example, on the left in gray, polygons used to build the sublevel sets of the constructed $f$ and the corresponding solutions to for some values of $r$ (solutions are joined by dotted lines). On the right the normal to be chosen to apply Lemma \[lem:polyGon\] (for $n = 1$, see main text for details). The point $P$ represents $\bar x$, it sits on the $x$-axis and is constantly contained in the normal cone at $B_n$ for any $n \geq 1$.[]{data-label="fig:regTorralba"}](regTorralba1 "fig:"){width=".38\textwidth"}![Illustration of the Tikhonov regularization example, on the left in gray, polygons used to build the sublevel sets of the constructed $f$ and the corresponding solutions to for some values of $r$ (solutions are joined by dotted lines). On the right the normal to be chosen to apply Lemma \[lem:polyGon\] (for $n = 1$, see main text for details). The point $P$ represents $\bar x$, it sits on the $x$-axis and is constantly contained in the normal cone at $B_n$ for any $n \geq 1$.[]{data-label="fig:regTorralba"}](regTorralba2 "fig:"){width=".55\textwidth"}
Following [@torralba96], we consider for any $r > 0$ $$\begin{aligned}
x(r) = \operatorname*{argmin}\left\{ f(x) + r \|x - \bar{x}\|_2^2:x\in {\mathbb{R}}^2\right\}
\label{eq:tikhonovRegularization}\end{aligned}$$ where $f$ is $C^k$ convex and where $\bar x$ is any anchor point. We would like to show that the curve $r \mapsto x(r)$ may have infinite length. Torralba provided a counterexample in his PhD Thesis for [*continuous*]{} convex functions, see [@torralba96]. This work extends his result to smooth $C^k$ convex functions in ${\mathbb{R}}^2$.
For any $n $ in $ {\mathbb{N}}^*$, we set $$\begin{aligned}
A_n &= \left(\frac{2}{n}, \frac{2}{n}\right)\\
B_n &= \left(\frac{2}{n} + \frac{1}{n^2}, -\frac{1}{n} \right)\\
C_n &= \left(\frac{2}{n}, - \frac{2}{n}\right)\\
D_n &= \left(- \frac{2}{n}, - \frac{2}{n}\right)\\
E_n &= \left(- \frac{2}{n}, \frac{2}{n}\right).\end{aligned}$$ This is depicted in Figure \[fig:regTorralba\]. For all $n \geq 1$, denote by $M_n$ the point on the $x$ axis above $B_n$ and $N_n$, the intersection of the normal cone at $B_n$ and the $x$ axis. We have $$\begin{aligned}
\frac{M_nN_n}{M_nB_n} = n \times M_nN_n= \frac{A'_nB_n}{A_nA'_n} = \frac{3/n}{1/n^2} = 3 n,\end{aligned}$$ so that for all $n \geq 1$, $M_nN_n = 3$ and $N_n = (3 + 2/n + 1 / n^2, 0)$. Choosing $P = (7,0)$, since for $n \geq 1$, $3 + 2/n + 1 / n^2 \leq 6 < 7$ , this shows that $P$ constantly belongs to the interior of the normal cone at $B_n$ for all $n \geq 1$. The sequence of level sets is constructed as in Figure \[fig:regTorralba\] by considering alternating symmetries with respect to the $x$-axis of the sequence of polygons above. It can be checked that the polygons form a strictly decreasing sequence of sets, as for $n>1$, the polygon $A_nB_nC_nD_nE_n$ is contained in the interior of the square $A_{n-1}C_{n-1}D_{n-1}E_{n-1}$. We choose the normal at $A_n, C_n, D_n, E_n$ to belong to the bisector at the corner and the normal at $B_n$ to be proportional to the vector $B_n P$. Applying Corollary \[cor:polygonDecrease\], we construct $f$ and choose $\bar{x} = P$ in to obtain the following:
There exists a $C^k$ strictly convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and $\bar{x} \in {\mathbb{R}}^2$ such that the curve $x((0,1))$ given by has infinite length. \[cor:homotopy\]
We apply Corollary \[cor:polygonDecrease\] with $I = {\mathbb{Z}}$ and revert the indices set to match the sequence that we have described. For any $n \geq 1$ there exists a value of $\lambda_n$ such that $f(B_n)= \lambda_n$ and $\nabla f(B_n)$ is colinear to the vector $B_n P$. Set $$\begin{aligned}
r = \frac{\|\nabla f(B_n)\|}{2 B_nP}
\end{aligned}$$ we have $\nabla f(B_n) + 2r(B_n - P) = 0$ which is the optimality condition in with $\bar{x} = P$. Hence we have shown that there exists a value of $r$ such that $B_n$ is the solution to . Since $n$ was arbitrary this is true for all $n$ and the curve $r \mapsto x(r)$ has to go through a sequence of points whose second coordinate is of the form $(-1)^n/n$ for all $n \geq 1$. Since this sequence is not absolutely summable, the curve has infinite length.
This result is in contrast with the definable case for which we have finite length by the monotonicity lemma, since the whole trajectory is definable and bounded.
Secants of gradient curves at infinity may not converge
-------------------------------------------------------
#### Thom’s gradient conjecture and Kurdyka-Mostowski-Parusinski’s theorem
A theorem of Łojasiewicz [@lojasiewicz1984trajectoires] asserts that bounded solutions to the gradient system $$\begin{aligned}
\dot{x}(t) = - \nabla f(x(t))\end{aligned}$$ converge when $f$ is a real analytic potential. Thom conjectured in [@thom1989problemes] that this convergence should occur in a stronger form: trajectories converging to a given $\bar{x}$ should admit a tangent at infinity, that is $$\begin{aligned}
\label{secants}
\frac{x(t) - \bar{x}}{\|x(t) - \bar{x}\|}\end{aligned}$$ should have a limit as $t \to \infty$. Lines passing through $\bar x$ and having as a slope are called [*secants of $x$ at $\bar x$*]{}. This conjecture was proved to be true in [@kurdyka2000proof]. In the convex world, it is well known that solutions to the gradient system converge for general potentials (this is a Féjer monotonicity argument due to Bruck); see also the original approaches by Manselli and Pucci [@Manselli] and Daniilidis et al. [@dan]. It is then natural to wonder whether this convergence satisfies higher order rigidity properties as in the analytic case. The answer turns out to be negative in general yielding a quite mysterious phase portrait.
#### Absence of tangential convergence for convex potentials
![$A = (-5,0)$, $B=\left( - \frac{5}{3} - \frac{25}{16}, \frac{10}{3} + \frac{5}{4} \right)$, $C=\left( \frac{-5}{2},5 \right)$, $D=\left( 0,5 \right)$, $E = \left( 5,0 \right)$, $F = \left( 0, -5 \right)$. All normals are chosen to be bisectors except $w$ which is parallel to the line $(DE)$. The vector $v$ is orthogonal to the segment $[BC]$. The point $C'$ is obtained by considering the intersection between the line $(Bw)$ (starting from $B$ with direction $w$), and the segment $[OC]$. The points $A'$, $B'$, $D'$, $E'$, $F'$ are obtained by performing a scaling of $A,B,D,E,F$ of a factor $\frac{OC'}{OC}$. The polygon $A''B''C''D''E''F''$ is $ABCDEF$ scaled by a factor $\frac{OC' + OC}{2 OC}$.[]{data-label="fig:thom"}](thom){width=".6\textwidth"}
The construction given in this paragraph is more complex than the previous ones, we start with a technical lemma which will be the basic building block for our counterexample.
Let $S$ be a convex set with $C^k$ boundary interpolating $ABCDEF$ in Figure \[fig:thom\] and let $g$ be the gauge function associated to $S$. The function $g$ is differentiable outside the origin. Consider any initialization $x_0 $ in $ [BC]$ with corresponding trajectory solution to the equation $$\begin{aligned}
\dot{x}(t) &= - \nabla g(x(t)), \, t\geq 0,\\
x(0) &= x_0.
\end{aligned}$$ Set $\bar{t} = \sup_{x(t) \in OBC} t$, we have $\bar{t} < + \infty$ and $x(\bar{t}) $ in $ [CC']$. \[lem:techThom\]
The fact that $g$ is differentiable comes from the fact that its subgradient is uniquely determined by the normal cone to $S$ which has dimension one because of the smoothness of the boundary of $S$. Since $S$ is interpolating the polygon, we have $g(B) = g(C) = 1$. Furthermore, we have for all $t$, $\frac{d}{dt} g(x(t)) = -\|\nabla g(x(t))\|^2= -1$, thence $\bar{t} \leq 1 - g(C')$. By homogeneity, for any $x \neq 0$ and $s > 0$, $\nabla g (sx) = \nabla g(x)$. For any $x $ in $ [BC]$, by convexity $$\begin{aligned}
0 \leq \left\langle C-x,\nabla g(C) - \nabla g(x) \right\rangle = \left\langle C-B,\nabla g(C) - \nabla g(x) \right\rangle \frac{\|C-x\|}{\|C-B\|},
\end{aligned}$$ and therefore $$\begin{aligned}
-\left\langle C - B, \nabla g(x) \right\rangle \geq -\left\langle C - B, \nabla g(C) \right\rangle
\label{eq:thomGradient1}
\end{aligned}$$ By homogeneity of $g$, is true for any $x$ in the triangle $OCB$ (different from $0$) and thus in the triangle $C'CB$ . Denote by $y$ the solution to the equation $$\begin{aligned}
\dot{y} &= - \nabla g(C)\\
y(0) &= B,
\end{aligned}$$ which integrates to $y(t) = B - t w$ for all $t$. Equation ensures that for any $0\leq t \leq \bar{t}$ $$\begin{aligned}
\frac{d}{dt} (\left\langle C - B, x(t) \right\rangle) &\geq \frac{d}{dt} (\left\langle C - B, y(t) \right\rangle)
\end{aligned}$$ Hence, we have for any $0\leq t \leq \bar{t}$, integrating on $[0,t]$ $$\begin{aligned}
\left\langle C - B, x(t) \right\rangle &\geq \left\langle C - B, y(t) \right\rangle + \left\langle C - B, x_0 - B \right\rangle \nonumber\\
&\geq \left\langle C - B, y(t) \right\rangle.
\label{eq:thomGradient2}
\end{aligned}$$ Furthermore, for all $x $ in $ [BC]$, we have $$\begin{aligned}
1 = \|\nabla g(x)\|^2 = \frac{1}{\|C - B\|^2}\left\langle C - B, \nabla g(x) \right\rangle^2 + \left\langle v, \nabla g(x) \right\rangle^2,
\end{aligned}$$ because $v$ is orthogonal to $C-B$. The first term is maximal for $x = C$ and thus the second term is minimal for $x = C$, we have thus for all $x $ in $ [BC]$ $$\begin{aligned}
0 < \left\langle \nabla g(C), v \right\rangle = \left\langle - \nabla g(C), -v \right\rangle \leq \left\langle \nabla g(x), v \right\rangle = \left\langle -\nabla g(x), -v \right\rangle \leq 1.
\label{eq:thomGradient3}
\end{aligned}$$ Equation holds for all $x$ in $OCB$ different from $O$ by homogeneity. We deduce that for all $0 \leq t \leq \bar{t}$, we have $$\begin{aligned}
\frac{d}{dt} (\left\langle -v, x(t) \right\rangle) &\geq \frac{d}{dt} (\left\langle -v, y(t) \right\rangle)
\end{aligned}$$ and by integration $$\begin{aligned}
\left\langle -v, x(t) \right\rangle &\geq \left\langle -v, y(t) \right\rangle + \left\langle -v, x_0 - B \right\rangle \nonumber\\
&= \left\langle -v, y(t) \right\rangle.
\label{eq:thomGradient4}
\end{aligned}$$ Hence, in the coordinate system $(C - B, -v)$, which is orthogonal, for all $t $ in $ [0,\bar{t}]$, $x(t)$ has larger coordinates than $y(t)$.
The trajectory $y(t)$, of equation $t \mapsto B - t w$ is the line going from $B$ to $C'$. From equations and , we may write for all $t$ in $ [0,\bar{t}]$, $x(t) = y(t) + \alpha (t) (C-B) + \beta(t) (-v)$ where $\alpha$ and $\beta$ are positive functions. Since $y(t)$ belongs to the line $(BC')$, this shows that $x(t)$ has to be above this line for all $t \geq 0$, $t\leq \bar{t}$ and actually, $x(\bar{t}) $ in $ BCC'$. Hence at time $\bar{t}$, we have $x(\bar{t}) $ in $ [CC']$. This holds true because $x(\bar{t})$ is on the boundary of $OCB$ and on the boundary of $BCC'$. Hence either $x(\bar{t}) $ in $ [CC']$, either $x(\bar{t}) $ in $ [BC]$. Equation ensures that if $x(\bar{t}) $ in $ [BC]$ then $x(\bar{t}) = C$ which concludes the proof.
There exists a $C^k$ strictly convex function on ${\mathbb{R}}^2$ with a unique minimizer $\bar{x}$, such that any nonconstant solution to the gradient flow equation $$\begin{aligned}
\dot{x}(t) = -\nabla f(x(t))
\end{aligned}$$ is such that $$\begin{aligned}
\frac{x(t) - \bar{x}}{\|x(t) - \bar{x}\|}
\end{aligned}$$ does not have a limit as $t \to \infty$.\
The function $f$ has a positive definite Hessian everywhere except at $0$. \[cor:Thom\]
We assume without loss of generality that $\bar{x} = O$ is the origin. Writting $x(t) = (r(t),\theta(t))$ in polar coordinate, we will construct a function $f$ such that each solution to the ODE produces nonconverging trajectories $\theta(t)$.
We start with an interpolating set $S_0 = ABCDE$ as in Lemma \[lem:techThom\] and let $S_1 = A'B'C'D'E'$ be its scaled version as described in Figure \[fig:thom\].
Let $\alpha$ be the value of the angle $\widehat{BOC}$ and $m = \left\lceil \frac{2 \pi}{\alpha} \right\rceil + 1$. We have $$\begin{aligned}
\frac{2\pi}{m} < \alpha.
\end{aligned}$$ To obtain $S_{2}$, we rotate $S_0$ by an angle $2\pi / m$, we denote $S'_0$ the resulting set. We rescale $S'_0$ by a factor $\beta $ in $ (0,1)$ so that $\beta S'_0$ lies in the interior of $S_1$. Call the resulting set $S_2$ and $S_3$ is obtained from $S_2$ exactly the same way as $S_1$ is obtained from $S_0$. We repeat the same process indefinitely to obtain a strictly decreasing sequence of $C^k$ sets. Note that for any $k $ in $ {\mathbb{N}}$, $S_{2km}$ and $S_{2km + 1}$ are homothetic to $S_0$.
We now invoke Corollary \[cor:polygonDecrease\] (with $I = {\mathbb{Z}}$ and revert the indices) to obtain a $C^k$ function $f$ with those prescribed level sets. Using Remark \[rem:alignedLevelSets\] it turns out that the level sets of $f$ between $S_0$ and $S_1$ are simple scalings of $S_0$. Hence the gradient curves of $f$ and those of the gauge function of $S$ are the same between $S_0$ and $S_1$, up to time reparametrization.
Using Lemma \[lem:techThom\] any trajectory crossing $[BC]$ in Figure \[fig:thom\], must also be crossing $[CC']$ and leave the triangle $BOC$ in finite time. The same statement holds after scaling the level sets and since for all $k $ in $ {\mathbb{N}}$, $S_{2km}$ and $S_{2km+1}$ are homothetic to $S_0$, this shows that no solution stays indefinitely in the triangle $BOC$.
Lemma \[lem:techThom\] still holds after rotations and by our construction, for any triangle $T$ obtained by rotating $BOC$ by a multiple of $2\pi/m$, no trajectory stays indefinitely within $T$. Since $2\pi / m < \alpha$, the union of these triangles $U$ contains $O$ in its interior.
Note first that any gradient curve converges to $\bar x$. Let us argue by contradiction and assume that there exists a continuous gradient curve $t \mapsto z(t)$ distinct from the stationary solution $\bar{x}$, such that $$\begin{aligned}
\frac{z(t) - \bar{x}}{\|z(t) - \bar{x}\|}
\end{aligned}$$ converges. This exactly means that the angle $\theta(t)$ of the curve has a limit in $[0,2\pi)$ as $t$ goes to infinity. There is a rotation of $BOC$ by a multiple of $2\pi/m$ whose interior intersects the half line given by the direction $\theta$, call this triangle $T$. The directional convergence entails that there exists $t_0 \geq 0$ such that $z(t)$ belongs to $T$ for all $t \geq t_0$. Hence $z$ can not be a gradient curve. To complete the proof, we may add disks of increasing size to the list of sets to obtain a full domain function and invoke Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$.
Newton’s flow may not converge
------------------------------
Given a twice differentiable convex function $f$, we define the open set $\Omega:=\{x\in{\mathbb{R}}^2:\nabla^2f \mbox{ is invertible}\}$ and we consider maximal solutions to the differential equation $$\begin{aligned}
\dot{x}(t) = - \nabla^2 f(x(t))^{-1} \nabla f(x(t)),
\label{eq:newton}\end{aligned}$$ on $\Omega$. This is the continuous counterpart of Newton’s method, it has been studied in [@aubin1984differential] and [@alvarez1998dynamical]. Let $x_0$ be in $\Omega$, there exists a unique maximal nontrivial interval $I$ containing $0$ and a unique solution $x$ to on $I$ with $x(0) = x_0$. Equation may be rewritten as $$\begin{aligned}
\frac{d}{dt} \nabla f(x(t)) = - \nabla f(x(t))\end{aligned}$$ and thus for all $t $ in $ I$, we have $$\begin{aligned}
\nabla f(x(t)) = e^{-t} \nabla f(x_0).
\label{eq:newtonIntegrated}\end{aligned}$$ If we could ensure that $I = {\mathbb{R}}$ and $f$ has oscillating gradients close to its minimum, then entails that the direction of the gradient is constant along the solution, which requires oscillations in space to compensate for gradient oscillations.
For any $k\geq2$, there exists a $C^k$ convex coercive function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initial condition $x_0 $ in $ {\mathbb{R}}^2$ such that the solution to is bounded, defined on ${\mathbb{R}}$ and has at least two distinct accumulation points.\[cor:newton\]
The counterexample is sketched in Figure \[fig:illustrNewton\], the construction is the same as for Corollary \[cor:altMin\] but instead of doing quarter rotations, we use symmetry with respect to the first axis. We can then call for Corollary \[cor:polygonDecrease\] to construct the function $f$ and equation ensures that the solution interval is unbounded.
![Illustration of the continuous time Newton’s dynamics. On the left, the “skeletons" of the sublevel sets in gray and a sketch of the corresponding curve. On the right, the normals to be chosen in order to apply Lemma \[lem:polyGon\].[]{data-label="fig:illustrNewton"}](illustrNewton "fig:"){width=".45\textwidth"}![Illustration of the continuous time Newton’s dynamics. On the left, the “skeletons" of the sublevel sets in gray and a sketch of the corresponding curve. On the right, the normals to be chosen in order to apply Lemma \[lem:polyGon\].[]{data-label="fig:illustrNewton"}](illustrAltMin2 "fig:"){width=".4\textwidth"}
Bregman descent (mirror descent) may not converge
-------------------------------------------------
The mirror descent algorithm was introduced in [@newmirovsky1983problem] as an efficient method to solve constrained convex problems. In [@beck2003mirror], this method is shown to be equivalent to a projected subgradient method, using non-Euclidean projections. It plays an important role for some categories of constrained optimization problem; see e.g., [@bauschke2016descent] for recent developments and [@Dragomir19] for a surprising example.
Let us recall beforehand some definitions. Given a Legendre function $h$ with domain ${\mathrm{dom}\,}h$, define the [*Bregman distance*]{}[^5] associated to $h$ as $D_h(u,v)=h(u)-h(v)-\langle \nabla h(v),u-v\rangle$ where $u$ is in $ {\mathrm{dom}\,}h$ and $v$ is in the interior of ${\mathrm{dom}\,}h$.
Given a smooth convex function $f$ that we wish to minimize on $\overline{ {\mathrm{dom}\,}h}$, we consider the Bregman method $$x_{i+1}=\operatorname*{argmin}\left\{\langle \nabla f(x_i),u-x_i\rangle+\lambda D_h(u,x_i):u\in {\mathbb{R}}^2\right\},$$ where $x_0$ is in ${\mathrm{int}\,}{\mathrm{dom}\,}h$ and $\lambda>0$ is a step size. When the above iteration is well defined, e.g. when ${\mathrm{dom}\,}h$ is bounded, it writes: $$x_{i+1}=\nabla h^*\left(\nabla h(x_i)-\lambda \nabla f(x_i)\right).$$ In [@bauschke2016descent] the authors identified a generalized smoothness condition which confers good minimizing properties to the above method: $$\begin{aligned}
\label{hsmooth}Lh-f \mbox{ convex},\\ \lambda\in(0,L).\label{step}\end{aligned}$$
The corollary below shows that such an algorithm may not converge, even though we assume the cost to satisfy , the step to satisfy , and the Legendre function to have a compact domain.
There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, and $x_0$ in ${\mathbb{R}}^2$ such that the Bregman descent recursion $$\begin{aligned}
x_{i+1} = \nabla h^*\left( \nabla h(x_i) - c \right),
\end{aligned}$$ produces a bounded sequence $(x_i)_{i \in {\mathbb{N}}}$ which has at least two distinct accumulation points. \[cor:mirrorDescent\]
We fix $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$, $\theta \neq 0$, and consider $h$ constructed in Corollary \[cor:legendreOscillating\] and choose $c = (-1,0)$. In this case the Bregman descent recursion writes for all $i$ in ${\mathbb{N}}$, $$\begin{aligned}
\nabla h (x_{i+1}) - \nabla h(x_i) = -c
\end{aligned}$$ so that we actually have $\nabla h (x_{i}) - \nabla h(x_{0}) = \nabla h (x_{i}) = - i c$ and thus $$x_i = \nabla h^*(-ic) = \nabla h^*(i,0).$$ By Corollary \[cor:legendreOscillating\], we have for all $i \in {\mathbb{N}}$ that $\nabla h^*(i,0)$ proportional to $(\cos(\theta), (-1)^i \sin(\theta))$. Since the norm of the gradient of $h^*$ cannot vanish at infinitiy (no flat direction) and is bounded, this proves that the sequence $(x_i)_{i \in {\mathbb{N}}}$ has at least two accumulation points which is the desired result.
Central paths of Legendre barriers may not converge
---------------------------------------------------
Consider the problem $$\begin{aligned}
\min_{x \in D} \left\langle c, x\right\rangle
\label{eq:LP}\end{aligned}$$ where $D$ is a subset of ${\mathbb{R}}^2$ and $c$. Given a Legendre function $h$ on $D$, we introduce the $h$ central path through $$\begin{aligned}
x(r) = \operatorname*{argmin}\left\{ \left\langle c, x\right\rangle + r h(x):x\in{\mathbb{R}}^2\right\}
\label{eq:interiorLegendre}\end{aligned}$$ where $r>0$ is meant to tend to $0$. Central paths are one of the essential tools behind interior point methods, see e.g., [@NN; @Aus99] and references therein.
Note that the accumulation points of $x(r)$ as $r \to 0$, have to be in the the solution set of . It is even tempting to think that the convergence of the path to some specific minimizer could occur, as it is the case for many barriers, see e.g. [@Aus99]. We have however:
There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, such that the $h$ central path $r \mapsto x(r)$ has two distinct accumulation points. \[cor:interiorPoint\]
The optimality condition which characterizes $x(r)$ for any $r > 0$ writes, $$\begin{aligned}
x(r)= \nabla h^*\left( \frac{c}{r} \right),
\end{aligned}$$ and the construction is the same as in Corollary \[cor:mirrorDescent\].
Hessian Riemannian gradient dynamics may not converge
-----------------------------------------------------
The construction of this paragraph is similar to the two previous paragraphs. Consider a $C^k$ ($k\geq 2$) Legendre function $h \colon D \mapsto {\mathbb{R}}$ and the continuous time dynamics $$\begin{aligned}
\dot{x}(t) = - \nabla_H f(x(t)), \, t\geq 0,
\label{eq:hessianRiemannian}\end{aligned}$$ where $H = \nabla^2 h$ is the Hessian of $h$ and $\nabla_H f = H^{-1} \nabla f$ is the gradient of some differentiable function $f$ in the Riemannian metric induced by $H$ on ${\mathrm{int}\,}D$. Such dynamics were considered in [@BT03; @alvarez2004hessian].
We have the following result:
There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, and $x_0$ in ${\mathbb{R}}^2$ such that the solution to with $f=\langle c,\cdot\rangle$ has two distinct accumulation points. \[cor:hessianRiemannian\]
Equation may be rewritten $$\begin{aligned}
\frac{d}{dt} \nabla h(x(t))= - \nabla f(x(t)),
\end{aligned}$$ so choosing $c = (-1,0)$, we have for all $t \in {\mathbb{R}}$, $\nabla h(x(t)) = \nabla h(x(0)) + (t,0) = (t,0)$ and the construction is the same as in Corollary \[cor:mirrorDescent\].
Appendix
========
There exists a $C^\infty$ strictly increasing concave function $\phi\colon [0,1] \mapsto [0,1]$ such that $$\begin{aligned}
\phi(t) &= \sqrt{2t/3} \quad \forall t \leq 1/6\\
\phi(1) &= 1 \\
\phi'(1) &= 2/3\\
\phi^{(m)}(1) &= 0, \quad \forall m \geq 2
\end{aligned}$$ \[lem:interpolationAroundZero\]
Consider a $C^\infty$ function $g_0 \colon {\mathbb{R}}\mapsto [0,1]$ such that $g_0 = 1$ on $(-\infty,-1)$, $g_0 = 0$ on $(1, +\infty)$ (for example convoluting the step function with a smooth bump function). Set $g(t) = \frac{1}{2}\left( g_0(t) + 1 - g_0(-t) \right)$ we have that $g$ is $C^\infty$, $g = 1$ on $(-\infty,-1)$, $g = 0$ on $(1, +\infty)$ and $g(t) + g(-t) = 1$ for all $t$. We have $$\begin{aligned}
\int_{-1}^1 g(s) ds = 1\\
\int_{-1}^1 \left( \int_{-1}^t g(s)ds \right) dt = 1
\end{aligned}$$ Set $\phi_0 \colon [-3,3] \mapsto {\mathbb{R}}$, such that $$\begin{aligned}
\phi_0(t) = \int_{-3}^t \left( \int_{-3}^r g(s) ds\right)dr.
\end{aligned}$$ For all $r $ in $ [-3,3]$, we have $$\begin{aligned}
\int_{-3}^r g(s) ds =
\begin{cases}
r+3 & \text{ if } r \leq -1\\
2 + \int_{-1}^r g(s)ds & \text{ if } -1 \leq r \leq 1\\
3 & \text{ if } r \geq 1
\end{cases}
\end{aligned}$$ and thus $$\begin{aligned}
\phi_0(t) =
\begin{cases}
\frac{t^2}{2} - 9/2 + 3(t+3) & \text{ if } t \leq -1\\
2 + 2(t + 1) + \int_{-1}^t \left( \int_{-1}^r g(s)ds \right)dr& \text{ if } -1 \leq t \leq 1\\
6 + 3(t-1) & \text{ if } 1 \geq t
\end{cases}
\end{aligned}$$ and in particular $\phi_0(3) = 12$ and $\phi_0'(3) = 3$. Set $\phi_1(s) = \phi_0(6 s -3)/12$. $$\begin{aligned}
\phi_1(0) &= 0\\
\phi_1(t) &= \left(\frac{(6t-3)^2}{2} - 9/2 + 2(3t )\right)/12 = 3 t^2 / 2 = \text{ if } t \leq 1/3\\
\phi_1(1) &= 1\\
\phi_1'(1) &= 3/2.
\end{aligned}$$ $\phi_1$ is stricly increasing, let $\phi \colon [0,1] \mapsto [0,1]$ denote the inverse of $\phi_1$, we have $$\begin{aligned}
\phi(1) &= 1\\
\phi'(1) & = 2 / 3\\
\phi(t) &= \sqrt{2t/3} \text{ if } t\leq 1/6.
\end{aligned}$$
Consider any strictly increasing $C^k$ function $\phi \colon (0,2) \mapsto {\mathbb{R}}$ such that $\phi(1) = 1$ and $\phi^{(m)}(1) = 0$, $m = 2,\ldots k$. Then the function $$\begin{aligned}
G \colon (0,2) \times {\mathbb{R}}/ 2 \pi {\mathbb{Z}}& \mapsto {\mathbb{R}}^2 \\
(s,\theta) &\mapsto \phi(s) n(\theta)
\end{aligned}$$ is diffeomorphism which satisfies for any $m=1 \ldots,k$ and $l =2,\ldots, k$, $$\begin{aligned}
&\frac{\partial^m G}{\partial \theta^m}(1,\theta) = n^{(m)}(\theta)\\
&\frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (1,\theta) = \phi'(1)n^{(m)}(\theta) \\
&\frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_-,\theta) = 0.
\end{aligned}$$ \[lem:diffGauge2\]
[**: Combinatorial Arbogast-Faà di Bruno Formula (from [@ma2009higher]).**]{} Let $g \colon {\mathbb{R}}\mapsto {\mathbb{R}}$ and $f \colon {\mathbb{R}}^p \mapsto [0, +\infty)$ be $C^k$ functions. Then we have for any $m \leq k$ and any indices $i_1,\ldots,i_m \in \left\{ 1,\ldots, p \right\}$. $$\begin{aligned}
\frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} g \circ f(x) = \sum_{\pi \in {\mathcal{P}}} g^{(|\pi|)}(f(x)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l \in B}\partial x_{i_l}}(x),
\end{aligned}$$ where ${\mathcal{P}}$ denotes all partitions of $\left\{ 1,\ldots, m \right\}$, the product is over subsets of $\left\{ 1,\ldots,m \right\}$ given by the partition $\pi$ and $|\cdot|$ denotes the number of elements of a set. We rewrite this as follows $$\begin{aligned}
\frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} g \circ f(x) = \sum_{k = 1}^m\sum_{\pi \in {\mathcal{P}}_k} g^{(k)}(f(x)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x),
\end{aligned}$$ where ${\mathcal{P}}_k$ denotes all partitions of size $k$ of $\left\{ 1,\ldots, m \right\}$. \[lem:faaDiBruno\]
[From [@bolte2010characterization Lemma 45]]{} Let $h $ in $ C^0\left( (0,r_0],{\mathbb{R}}_+^* \right)$ be an increasing function. Then there exists a function $\psi $ in $ C^\infty({\mathbb{R}},{\mathbb{R}}_+)$ such that $\psi = 0$ on, ${\mathbb{R}}_-$ and $0 < \psi(s) \leq h(s)$ for any $s$ in $(0,r_0]$ and $\psi$ is increasing on ${\mathbb{R}}$ \[lem:CinfiniteLowerBound\]
Let $D \subset {\mathbb{R}}^p$ be a nonempty compact convex set and $f \colon D \mapsto {\mathbb{R}}$ convex, continuous on $D$ and $C^k$ on $D \setminus \operatorname*{argmin}_{D} f$. Assume further that $\operatorname*{argmin}_D f \subset \mathrm{int}(D)$, $k \geq 1$, with $\min_D f = 0$. Then there exists $\phi \colon {\mathbb{R}}\mapsto {\mathbb{R}}_+$, $C^k$, convex and increasing with positive derivative on $(0,+\infty)$, such that $\phi\circ f$ is convex and $C^k$ on $D$. \[lem:CkSmoothing\]
By a simple translation, we may assume that $\min_D f = 0$ and $\max_D f = 1$. Any convex function is locally Lipschitz continuous on the interior of its domain so that $f$ is globally Lipschitz continuous on $D$ and its gradient is bounded. Hence, $f^2$ is $C^1$ and convex on $D$. We now proceed by recursion. For any $m =1,\ldots, k$, we let $Q_m$ denote the $m$-order tensor of partial derivatives of order $m$. Fix $m$ in $\{1,\ldots,k\}$. Assume that $f$ is $C^m$ throughout $D$ while it is $C^{m+1}$ on $D \setminus \arg\min_D f$. Note that all the derivatives up to order $m$ are bounded. We wish to prove that $f$ is globally $C^{m+1}$.
Consider the increasing function $$\begin{aligned}
h \colon (0,1] &\mapsto {\mathbb{R}}_+^*\\
s &\mapsto \frac{s}{1 + \sup_{s \leq f(x) \leq 1}\|Q_{m+1}(x)\|_{\infty}}
\end{aligned}$$ and set $\psi$ as in Lemma \[lem:CinfiniteLowerBound\]. Recall that $\psi$ is $C^\infty$ and all its derivative vanish at $0$ and $\psi \leq h$ on $(0,1]$. Let $\phi$ denote the anti-derivative of $\psi$ such that $\phi(0) = 0$. $\phi$ is $C^\infty$ and convex increasing on ${\mathbb{R}}$ and, since its derivatives at $0$ vanish as well, one has, for any $q $ in $ {\mathbb{N}}$, $\phi^{(q)}(z) = o(z)$. Consider the function $\phi \circ f$. It is $C^m$ on $D$ and it has bounded derivatives up to order $m$. Furthermore, it is $C^{m+1}$ on $D \setminus \operatorname*{argmin}_D f$. Let $\bar{y} $ in $ \operatorname*{argmin}_D f$. If $\bar{y} $ in $ \mathrm{int}(\operatorname*{argmin}_D f)$, then $f$ and $\phi \,\circ f$ have derivatives of all order vanishing at $\bar{y}$. Assuming that $\bar{y} $ in $ \operatorname*{argmin}_D f\setminus \mathrm{int}(\operatorname*{argmin}_D f)$. By the induction assumption and Lemma \[lem:faaDiBruno\], we have for any indices $i_1,\ldots,i_m \in \left\{ 1,\ldots, p \right\}$ and any $h $ in $ {\mathbb{R}}^p$: $$\begin{aligned}
&\frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} (\phi \circ f)(\bar{y} + z) - \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} (\phi \circ f)(\bar{y}) \\
=\;& \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}}( \phi \circ f)(\bar{y} + z) \\
=\;&\sum_{q = 1}^{m}\sum_{\pi \in {\mathcal{P}}_q} \phi^{(q)}(f(\bar{y} + z)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(\bar{y} + z).
\end{aligned}$$ All the derivatives of $f$ are of order less or equal to $m$ and thus remain bounded as $z \to 0$. Further more $f$ is Lipschitz continuous on $D$ so that $f(\bar{y} + z) = O(\|z\|)$ near $0$, and, for any $q $ in $ {\mathbb{N}}$, $\phi^{(q)}(f(\bar{y} + z)) = o(\|z\|)$. Hence $\phi \circ f$ has derivative of order $m+1$ at $\bar{y}$ and it is $0$.
Since $\operatorname*{argmin}_D f \subset \mathrm{int}(D)$, we may consider any sequence of point $(y_{j})_{j \in {\mathbb{N}}}$ in $D \setminus \operatorname*{argmin}_D f$ converging to $\bar{y}$. By Lemma \[lem:faaDiBruno\], we have for any indices $i_1,\ldots,i_{m+1} \in \left\{ 1,\ldots, p \right\}$, and any $j $ in $ {\mathbb{N}}$, $$\begin{aligned}
\frac{\partial^{(m+1)}}{\prod_{l=1}^{m+1}\partial x_{i_l}} (\phi \circ f)(y_j) &= \phi'(f(y_j)) \frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j) + \sum_{q = 2}^{m+1}\sum_{\pi \in \Pi_q} \phi^{(q)}(f(y_j)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x)\\
&\leq h(f(y_j))\frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j) + \sum_{q = 2}^{m+1}\sum_{\pi \in \Pi_q} \phi^{(q)}(f(y_j)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x)\\
&= f(y_j) \frac{\frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j)}{1 + \sup_{f(y_j) \leq f(x) \leq 1}\|Q_{m+1}(x)\|_{\infty}} + O(f(y_j))\\
&= O(f(y_j)),
\end{aligned}$$ where the inequality follows from the construction of $\phi$. The third step follows using the definition of $h$ and the fact that, for any $q \geq 2$,
1. Each partition of $\left\{ 1,\ldots,m+1 \right\}$ of size $q$ contains subsets of size at most $m$. Thus in the product, the terms $\partial^{|B|} f$ correspond to bounded derivatives of $f$ by the induction hypothesis.
2. $\phi^{(q)}(a) = o(a)$ as $a \to 0$.
The last step stems from the fact that the ratio has asbolute value less than $1$. This shows that the derivatives of order $m+1$ of $\phi \circ f$ are decreasing to $0$ as $j \to \infty$ and $\phi \circ f$ is actually $C^{m+1}$ and convex on $D$. The result follows by induction up to $m = k$ and by the fact that a composition of increasing convex functions is increasing and convex.
Let $p \colon {\mathbb{R}}_+ \mapsto {\mathbb{R}}_+$ be concave increasing and $C^1$ with $p' \geq c$ for some $c > 0$. Assume that there exists $ A > 0$ such that for all $x $ in $ {\mathbb{R}}_+$ $$\begin{aligned}
p(x) - x p'(x) \leq A.
\end{aligned}$$ Then setting $a= A/c$, we have for all $x \geq a$, $$\begin{aligned}
p(x-a) - x p'(x-a) \leq 0
\end{aligned}$$ \[lem:techConcaveAsymptotic\]
For all $x \geq a$, we have $$\begin{aligned}
f(x-a) - (x-a)f'(x-a) \leq A,
\end{aligned}$$ hence $$\begin{aligned}
f(x-a) - xf'(x-a) \leq A - af'(x-a) \leq A - ac = 0.
\end{aligned}$$
[**Acknowledgements.**]{} The authors acknowledge the support of AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future – PIA3” program under the Grant agreement n°ANR-19-PI3A-0004, Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226, and ANR MasDol. J. Bolte acknowledges the support of ANR Chess, grant ANR-17-EURE-0010 and ANR OMS.
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[^1]: Toulouse School of Economics, Université Toulouse 1 Capitole, France.
[^2]: IRIT, ANITI, Université de Toulouse, CNRS. DEEL, IRT Saint Exupery, Toulouse, France.
[^3]: In the sense of sets inclusion the sequence being indexed on ${\mathbb{N}}$ or ${\mathbb{Z}}$
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[^5]: It is actually not a proper distance.
| ArXiv |
---
abstract: 'Along with increasingly popular virtual reality applications, the three-dimensional (3D) point cloud has become a fundamental data structure to characterize 3D objects and surroundings. To process 3D point clouds efficiently, a suitable model for the underlying structure and outlier noises is always critical. In this work, we propose a hypergraph-based new point cloud model that is amenable to efficient analysis and processing. We introduce tensor-based methods to estimate hypergraph spectrum components and frequency coefficients of point clouds in both ideal and noisy settings. We establish an analytical connection between hypergraph frequencies and structural features. We further evaluate the efficacy of hypergraph spectrum estimation in two common point cloud applications of sampling and denoising for which also we elaborate specific hypergraph filter design and spectral properties. The empirical performance demonstrates the strength of hypergraph signal processing as a tool in 3D point clouds and the underlying properties.'
author:
- 'Songyang Zhang, Shuguang Cui, , and Zhi Ding, '
title: Hypergraph Spectral Analysis and Processing in 3D Point Cloud
---
3D point clouds, hypergraph signal processing, hypergraph construction, denoising, sampling.
Introduction {#intro}
============
Recent developments in depth sensors and softwares make it easier to capture the features and create a three-dimensional (3D) model for an object and its surroundings[@c1]. In particular, with the low-cost scanners such as light detection and ranging (LIDAR) and Kinect, a new data structure known as the point cloud has achieved significant success in many areas, including virtual reality, geographic information system, reconstruction of art document and high-precision 3D maps for self-driving cars [@c2]. A point cloud consists of 3D coordinates with attributes such as color, temperature, texture, and depth [@c3]. Owing to the easy access to scanning sensors and the huge need in describing the 3D features, the use of point clouds has attracted significant attentions in areas of computer vision, virtual reality, and medical science. How to process the point clouds efficiently becomes an important topic of research in many 3D imaging and vision systems.
To analyze the features of point cloud, the first step is to construct an analytical model to represent the 3D structures. The literature provides several different models. In [@c4], the 3D space is partitioned into several boxes or voxels, and the point clouds are then discretized therein. One disadvantage of voxels is that a dense grid is required to achieve fine resolution, leading to spatial inefficiency [@c3]. A spatially efficient approach [@c5; @c6] is the octree representation of point clouds. An octree is a tree data structure in which each node has exactly eight children. It can partition a 3D space recursively, and represent the point clouds with partitioned boxes. Although efficient, octree suffers from discretization errors [@c3]. The bd-tree is another spatial decomposition technique and is robust in highly cluttered point cloud dataset. However, compared to octree structures, bd-trees are more difficult to update.
Recently, graphs and graph signal processing (GSP) have found applications in modeling point clouds. For example, the authors of [@c3] construct a graph based on pairwise point distances. Some other works, such as [@c8], construct graphs based on the $k$-nearest neighbors, where each vertex (point) has an edge connection to its $k$ nearest neighbors. There are several clear connections between graph features and point cloud characteristics. For example, the smoothness over a graph can describe the flatness of surfaces in point clouds. GSP-based tools such as filters and graph learning methods can process the point clouds and have shown great success because of the graph model’s ability to capture the underlying geometric structures. However, graph-based methods still face some challenges such as limited orders and measurement inefficiency. In a traditional graph, each edge can only connect two nodes, constraining graph-based models to describe only pairwise relationships. However, a multilateral relationship among multiple nodes is far more informative as in a point cloud model. For example, the points (nodes) on the same surface of a point cloud exhibit a strong multilateral relationship, which cannot be easily captured by an edge of a traditional graph. In fact, construction of an efficient graph for a given dataset is always an open question. Thus, studies on point clouds can benefit from more general and efficient models.
To develop an efficient model for point clouds, we explore a high-dimensional graph model, known as hypergraph [@c9]. Hypergraph can be a useful model in processing 3D point clouds. A hypergraph $\mathcal{H}=\{\mathcal{V},\mathcal{E}\}$ consists of a set of nodes $\mathcal{V}=\{\mathbf{v}_1, \dots,\mathbf{v}_K\}$ and a set of hyperedges $\mathcal{E}=\{\mathbf{e}_1, \dots,\mathbf{e}_K\}$. Each hyperedge in a hypergraph can connect more than two nodes. For example, a 3D shape together with its hypergraph model are shown as Fig. \[hyper\]. Obviously, a normal graph is a special case of hypergraph, where each hyperedge degrades to connect two nodes exactly. The hyperedge in a hypergraph can characterize the multilateral relationship among several related nodes (e.g., on a surface), thereby making hypergraph a natural and intuitive model for point clouds. Moreover, advances in hypergraph signal processing (HGSP) [@c9] are providing more hypergraph tools, such as HGSP-based filters and spectrum analysis, for effective point cloud processing.
However, processing the point clouds based on hypergraph still poses several challenges. Similar to GSP, the first problem lies in the construction of hypergraph for point clouds. The traditional hypergraph construction method for a general dataset relies on data structure. For example, in [@c11], a hypergraph model is constructed according to the sentence structure in natural language processing. The $k$-nearest neighbor model is another method to construct the hypergraph. In [@c9], a hypergraph can be formed from the feature distances for an animal dataset to achieve clustering. However, such distance-based or structure-based model may be rather lossy in information preservation. For example, the structure-based method may not preserve the correlation of some irregular structures, whereas the $k$-nearest neighbor method may narrowly emphasize the distance information. In addition to hypergraph construction, another issue in analyzing point cloud with hypergraph tools is the computation complexity of the spectrum space. In the HGSP framework, spectrum-based analysis plays an important role but needs to compute the spectrum space. Usually, the computation of hypergraph spectrum is based on orthogonal-CP decomposition, which incurs high-complexity when there are many nodes. Another challenge in point cloud processing is the effect of noise and outliers. Since a hypergraph model is constructed from observed data, noise can distort the hypergraph and degrade the performances of HGSP. Thus, mitigating noise effect and robustly estimating the hypergraph model for point clouds pose a significant challenge.
This work addresses the aforementioned problems. We propose novel spectrum-based hypergraph construction methods for both clean and noisy point clouds. For clean point clouds, we first estimate their spectrum components based on the hypergraph stationary process and optimally determine their frequency coefficients based on smoothness to recover the original hypergraph structure. For noisy point clouds, we introduce a method for joint hypergraph structure estimation and data denoising. We shall illustrate the effectiveness of the proposed hypergraph construction and spectrum estimation in two point clould applications: sampling and denoising. Our experimental results clearly establish a connection between hypergraph frequencies and point cloud features. The performance improvement in both applications demonstrates the strength and power of hypergraph in point cloud processing and the practical value of our estimation methods.
We organize the rest of the paper as follows. In Section \[pre\], we lay the foundation with respect to the preliminaries and notations of point clouds, tensor basics and hypergraph signal processing. Next, we propose means in estimating hypergraph spectrum for basic point clouds in Section \[h1\] and further develop means for hypergraph structure estimation of noisy point clouds in Section \[h2\]. With the proposed estimation methods, we study two important application scenarios and establish the effectiveness of hypergraph signal processing in Section \[appli\]. Finally, we present the conclusion and future directions in Section \[con\].
Preliminaries and Notations {#pre}
===========================
In this section, we cover basic background with respect to point cloud, tensor basics and hypergraph signal processing.
Point Clouds
------------
A point cloud is a set of 3D points obtained from sensors, where each point is attributed with coordinates and other features, like colors [@c10]. Since the coordinates are basic features of a point cloud, in this work, we mainly focus on gray-scale point clouds, where each node is characterized by its coordinates. We consider a matrix representation of the gray-scale point clouds, where a point cloud with $N$ nodes is denoted by a location matrix $$\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]=
\begin{bmatrix}
\mathbf{s}_1^T\\
\mathbf{s}_2^T\\
\ddots\\
\mathbf{s}_N^T
\end{bmatrix}\in\mathbb{R}^{N\times 3},$$ where $\mathbf{X}_i$ denotes a vector of the $i$th coordinates of all the points, and $\mathbf{s}_i$ is the three coordinates of $i$th point. With the information of coordinates, different models, such as graphs [@c3] and octrees [@c5], can be constructed to analyze the point clouds, for which we will discuss more in Section \[appli\].
Tensor Basics
-------------
Tensor is a high-dimensional generalization of matrix. A tensor can be interpreted as multi-dimensional arrays. The order of tensor is the number of indices to label the components of arrays [@c17]. For example, a scalar is a zeroth-order tensor; a vector is a first-order tensor; a matrix is a second-order tensor; and an $M$-dimensional array is an $M$th-order tensor [@c18]. In this work, an $M$th-order tensor is denoted by $\mathbf{A}\in\mathbb{R}^{I_1
\times I_2\times \cdots \times I_M}$, whose entry in position $(i_1,i_2,\cdots,i_M)$ is labeled as $a_{i_1\cdots i_M}$. Here, $I_k$ is the dimension of $k$th order.
Tensor outer product is a widely used operation to construct a higher-order tensor from lower-order tensors. The tensor outer product between an $M$th-order tensor $\mathbf{U}\in \mathbb{R}^{I_1\times I_2\times ...\times I_M }$ with entries $u_{i_1 ... i_M}$ and an $N$th-order tensor $\mathbf{V}\in \mathbb{R}^{J_1\times J_2\times ...\times J_N }$ with entries $v_{j_1 ... j_N}$ is denoted by $$\mathbf{W}=\mathbf{U} \circ \mathbf{V},$$ where the result $\mathbf{W}\in \mathbb{R}^{I_1\times I_2\times ...\times I_M \times J_1 \times J_2 \times ... \times J_N}$ is an $(M+N)$th-order tensor with entries $$w_{i_1 ... i_M j_1 ... j_N}= u_{i_1 ... i_M} \cdot v_{j_1 ... j_N}.$$
Hypergraph Signal Processing
----------------------------
Hypergraph signal processing (HGSP) is a tensor-based framework [@c9]. In the HGSP framework, a hypergraph with $N$ nodes and longest hyperedge connecting $M$ nodes, is represented by an $M$-th order $N$-dimension representing tensor $\mathbf{A}=(a_{i_1i_2\cdots i_M})\in\mathbb{R}^{N^M}$. The representing tensor can be adjacency tensor or Laplacian tensor in different purposes [@c16]. In this paper, we refer the adjacency tensor as the representing tensor, in which each entry $a_{i_1i_2\cdots i_M}$ indicates whether nodes $\{\mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_M\}$ are connected. The computation of the edge weight can be found in [@c9].
With the orthogonal-CP decomposition, the representing tensor can be decomposed via $$\label{decom}
\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{$M$ times}},$$ where $\mathbf{f}_r$’s are orthonormal basis called spectrum components and $\lambda_r$ are frequency coefficients related to the hypergraph frequency. All the spectrum components $\{\mathbf{f}_1,\cdots,\mathbf{f}_N\}$ construct the hypergraph spectral space. Each pair $(\mathbf{f}_r,\lambda_r)$ is called the spectral pair of the hypergraph.
Given an original signal $\mathbf{s}=[s_1\quad s_2\quad...\quad s_N]^{\mathrm{T}}$, the hypergraph signal is defined as the $(M-1)$ times tensor outer product of $\mathbf{s}$, i.e., $$\mathbf{s}^{[M-1]}=\underbrace{\mathbf{s\circ...\circ s}}_{\text{$M-1$ times}}.$$
The hypergraph frequency is ordered by the total variation of the spectrum component, which is defined as $$\mathbf{TV}(\mathbf{\mathbf{f}_r})=||\mathbf{f}_r-\frac{1}{\lambda_{max}}\mathbf{A}\mathbf{f}_r^{[M-1]}||_1,$$ where $\mathbf{A}\mathbf{f}_r^{[M-1]}$ is the contraction between representing tensor $\mathbf{A}$ and the hypergraph signal.
A spectrum component with larger total variation is a higher-frequency component, which indicates a faster propagation over the given hypergraph. Moreover, a supporting matrix $$\label{sup}
\mathbf{P_s}=\frac{1}{\lambda_{\max}}
\begin{bmatrix}
\mathbf{f}_1& \cdots& \mathbf{f}_N
\end{bmatrix}
\begin{bmatrix}
\lambda_1& & \\
&\ddots& \\
& &\lambda_N
\end{bmatrix}
\begin{bmatrix}
\mathbf{f}_1^{\mathrm{T}}\\
\vdots\\
\mathbf{f}_N^{\mathrm{T}}
\end{bmatrix},$$ can be defined to capture the overall spectral information of the hypergraph.
Instead of reviewing many properties of HGSP here, other aspects such as hypergraph Fourier transform, hypergraph filter design and sampling theory can be found in [@c9].
Hypergraph Spectrum Estimation for Point Clouds {#h1}
===============================================
To process the 3D point clouds, the first step is to construct an optimal hypergraph to model the point clouds. As we mentioned in the Section \[intro\], it is time-comsuming and inefficient to first construct a hypergraph structure before tensor decomposition to obtain the hypergraph spectrum. Instead, we propose to directly estimate the hypergraph spectral pairs based on the observed data, and then recover the original representing tensor with Eq. (\[decom\]). In this section, we first estimate the hypergraph spectrum components $\mathbf{f}_r$’s based on the hypergraph stationary process, and optimize the frequency coefficients $\lambda_r$’s based on the smoothness for original point clouds.
Estimation of Hypergraph Spectrum Components
--------------------------------------------
In this part, we propose a method to estimate the hypergraph spectral components based on the hypergraph stationary process.
### Hypergraph Stationary Process
Before providing details of the estimation, let us first introduce some new definitions and properties necessary for spectrum estimation.
Stationarity is a cornerstone property that facilities the analysis of random signals and observations in traditional signal processing [@c19]. It has equal importance in graph and hypergraph signal processing. Based on graph shifting introduced in [@c22], a definition of graph stationary process proposed in [@c19] can analyze the properties of the different observations of nodes, or the random signals over the graphs. Furthermore, [@c123] introduces a method to estimate the graph spectrum space and graph diffusion for multiple observations based on the graph stationary process. Similarly, the hypergraph stationary process can be defined to estimate hypergraph spectrum.
Now, let us introduce the definition of the hypergraph stationary process. In [@c9], a polynomial hypergraph filter based on supporting matrix is defined as $$\mathbf{s}'=\sum_{k=1}^{a}\alpha_k\mathbf{P}^k\mathbf{s},$$ where $\mathbf{P}=\lambda_{max}\mathbf{P_s}$.
Similarly, based on the supporting matrix, a $\tau$-step shifting operation is defined as $\mathbf{P}_{\tau}=\mathbf{P}^{\tau}$. Then, similar to the definition of the stationary process in traditional digital signal processing and graph signal processing, a strict-sense stationary process in HGSP can be defined as follows.
(Strict-Sense Stationary Process) A stochastic signal $\mathbf{x}\in\mathbb{R}^N$ is strict-sense stationary over the hypergraph with $\mathbf{P}_\tau$ if and only if $$\mathbf{x}\overset{d}{=}\mathbf{P}_{\tau}\mathbf{x}$$ holds for any $\tau$.
Since the strict-sense stationary is hard to achieve and analyze in the real datasets, we introduce the weak-sense stationary process similar to traditional digital signal processing.
(Weak-Sense Stationary Process) A stochastic signal $\mathbf{x}\in\mathbb{R}^N$ is weak-sense stationary over the hypergraph with $\mathbf{P}_\tau$ if and only if $$\label{mean}
\mathbb{E}[\mathbf{x}]=\mathbb{E}[\mathbf{P}_{\tau}\mathbf{x}]$$ and $$\label{time}
\mathbb{E}[(\mathbf{P}_{\tau_1}\mathbf{x})((\mathbf{P}^H)_{\tau_2}\mathbf{x})^H]=\mathbb{E}[(\mathbf{P}_{\tau_1+\tau}\mathbf{x})((\mathbf{P}^H)_{\tau_2-\tau}\mathbf{x})^H]$$ hold for any $\tau$, where $\mathbb{E}(\cdot)$ refers to the mean of observations and $(\cdot)^H$ is the Hermitian transpose.
From the definition of the weak-sense stationary process (WSS), Eq. (\[mean\]) implies that the mean function of the signal must be constant, which is the same condition as in traditional digital signal processing (DSP) [@c23]. From the definition of supporting matrix, the $(i,j)$-th entry of $\mathbf{P}$ is the same as the $(j,i)$-th entry of $\mathbf{P}^H$, which indicates that $\mathbf{P}^H$ is the shifting in the opposite direction of $\mathbf{P}$. Then, the condition in Eq. (\[time\]) indicates that the hypergraph covariance function $K_{\mathbf{xx}}({\tau_1},-\tau_2)=K_{\mathbf{xx}}({\tau_1}+\tau,\tau-\tau_2)=K_{\mathbf{xx}}({\tau_1}+\tau_2,0)$, which is also consistent with the definition in traditional DSP.
With the definition of the hypergraph stationary process, we have the following properties regarding the relationship between signals and hypergraph spectrum.
A stochastic signal $\mathbf{x}$ is WSS if and only if it has zero-mean and its covariance matrix has the same eigenvectors as the hypergraph spectrum basis, i.e., $$\label{s1}
\mathbb{E}[\mathbf{x}]=\mathbf{0}$$ and $$\label{s2}
\mathbb{E}[\mathbf{x}\mathbf{x}^H]=\mathbf{V}\Sigma_\mathbf{x}\mathbf{V}^{H},$$ where $\mathbf{V}=[\mathbf{f}_1,\mathbf{f}_2,\cdots,\mathbf{f}_N]\in\mathbb{R}^{N\times N}$ are the hypergraph spectrum.
Since the hypergraph spectrum basis are orthonormal, we have $\mathbf{VV}^T=\mathbf{I}$. Then, the $\tau$-step shifting based on supporting matrix can be calculated as $$\begin{aligned}
\mathbf{P}_\tau&=\underbrace{\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T\cdots\mathbf{V}\Lambda_\mathbf{P}\mathbf{V}^T}_{\tau\quad times}\\
&=\mathbf{V}\Lambda_\mathbf{P}^\tau\mathbf{V}^T.\label{poly}
\end{aligned}$$
Now, the Eq. (\[mean\]) can be written as $$\mathbb{E}[\mathbf{x}]=\mathbf{V}\Lambda_P^\tau\mathbf{V}^T\mathbb{E}[\mathbf{x}].$$ Since $\mathbf{V}\Lambda_P^\tau\mathbf{V}^T$ does not always equal to $\mathbf{I}$, Eq. (\[mean\]) holds for arbitrary supporting matrix and $\tau$ if and only if $\mathbb{E}[\mathbf{x}]=\mathbf{0}$.
Next we show the sufficiency and necessity of the condition in Eq. (\[s2\]). The condition in Eq. (\[time\]) can be written as $$\mathbf{P}_{\tau_1}\mathbb{E}[\mathbf{xx}^H]((\mathbf{P})^H_{\tau_2})^H=\mathbf{P}_{\tau_1+\tau}\mathbb{E}[\mathbf{xx}^H]((\mathbf{P})^H_{\tau_2-\tau})^H.$$ Considering Eq. (\[poly\]) and the fact that hypergraph spectrum is real [@c9], Eq. (\[time\]) is equivalent to $$\mathbf{V}\Lambda_\mathbf{P}^{\tau_1}\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}\Lambda_\mathbf{P}^{\tau_2}\mathbf{V}^H=\mathbf{V}\Lambda_\mathbf{P}^{\tau_1+\tau}\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}\Lambda_\mathbf{P}^{\tau_2-\tau} \mathbf{V}^H,$$ which can be written as $$\label{eig}
(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V})\Lambda_\mathbf{P}^{\tau}=\Lambda_\mathbf{P}^{\tau}(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V}).$$ If Eq. (\[eig\]) holds for arbitrary $\mathbf{P}$, $(\mathbf{V}^H\mathbb{E}[\mathbf{xx}^H]\mathbf{V})$ should be diagonal, which indicates $\mathbb{E}[\mathbf{x}\mathbf{x}^H]=\mathbf{V}\Sigma_\mathbf{x}\mathbf{V}^{H}$. Thus, the sufficiency of the condition is proved.
Similarly, we can apply Eq. (\[s2\]) on both sides of Eq. ($\ref{time}$), we can establish the necessity of the condition in Eq. (\[s2\]).
This theorem can be used to estimate the hypergraph spectrum, given multiple observations of several signal points.
### Estimation of Spectrum Components for Point Clouds
Now, we can use the property of stationary process to estimate the hypergraph spectrum of point clouds. The three coordinates of a point can be interpreted as three observations of the point from different angles, which describe the underlying multilateral relationship. Thus, we can assume that the point cloud signals follow the stationary process over the estimated underlying hypergraph structure. If the point cloud signals $\mathbf{s}$ follow the hypergraph stationarity, it should satisfy Eq. (\[s1\]) and Eq. (\[s2\]). Thus, a spectrum estimation method can be based on hypergraph staionarity. The details of the algorithm is described as follows.
: Point cloud dataset $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]\in\mathbb{R}^{N\times 3}$. Calculate the mean of each row in $\mathbf{s}$, i.e.,\
$\mathbf{\overline s}=(\mathbf{X_1+X_2+X_3})/3$; Normalize the original point cloud data as zero-mean in each row, i.e., $\mathbf{s}'=[\mathbf{X_1-{\overline s},X_2-{\overline s},X_3-{\overline s}}]$; Calculate the eigenvectors $\{\mathbf{f}_1,\cdots,\mathbf{f}_N\}$ for $R_{\mathbf{s}'}=\mathbf{s'}(\mathbf{s'}^T)$; : Hypergraph spectrum $\mathbf{V}=[\mathbf{f}_1,\cdots,\mathbf{f}_N]$.
With *Theorem 1*, we can directly obtain an estimation of the hypergraph spectrum based on the hypergraph stationarity. Note that, here, we assume all the observations are from a clean point cloud without noise. The case of noisy point clouds will be discussed later in Section \[h2\].
Estimation of Frequency Coefficients {#es}
------------------------------------
Next, we discuss how we estimate the hypergraph frequency coefficients with the spectrum components based on the hypergraph smoothness.
In real applications, the large-scale networks are usually sparse, which makes it meaningful to infer that most entries of the hypergraph representing tensor for real datasets are zero [@c24]. In addition, the smoothness of signals is a widely-used assumption when estimating the underlying structure of graphs and hypergraphs [@c25]. Thus, the estimation of the hypergraph representing tensor with known spectrum components for a given dataset $\mathbf{s}$ can be generally formulated as $$\begin{aligned}
&\min_{\mathbf{\boldsymbol{\lambda}}}\quad \alpha \mbox{Smooth}
(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)+\beta||\mathbf{A}||_T^2\label{e1}\\
s.t.\quad &\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}. \label{dec}\\
&\quad\mathbf{A}\in \mathcal{A}.\label{cs1}\\
&||\mathbf{A}||_T=\sqrt{\sum_{i_1,i_2,\cdots, i_M=1}^N a_{i_1i_2\cdots i_M}^2}.\label{t_norm}\end{aligned}$$
The constraint set $\mathcal{A}$ in (\[cs1\]) includes the prior information of the representing tensor. For example, if the representing tensor is the adjacency tensor, its entries should be non-negative. In the constraint of $(\ref{t_norm})$, $||\mathbf{A}||_T$ is the tensor norm which controls the sparsity of the hypergraph structure. The smoothness function Smooth$(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)$ can be designed for specific problems. Typical functions can be hypergraph Laplacian regularization, label ranking, and total variation [@c9]. For convenience, we use the quadratic-form total variation based on the supporting matrix to describe the hypergraph smoothness, i.e., $$\mathbf{TV}(\mathbf{s})=||\mathbf{s}-(1/\lambda_{max})\mathbf{P}\mathbf{s}||^2_2.$$ This form of smoothness function suggested in [@c9] can capture the differences between one node and its neighbors over hypergraph. Since the signals are smooth over the estimated hypergraph, observations are also smooth. Thus, the final smoothness function for point cloud $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]$ is $$\begin{aligned}
\mbox{Smooth}(\mathbf{s,\boldsymbol{\lambda}},\mathbf{f}_r)&=\sum_{i=1}^3||\mathbf{X}_i-\mathbf{P_sX}_i||^2_2\nonumber\\
&=\sum_{i=1}^3||\mathbf{X}_i-\sum_r \sigma_r(\mathbf{f}_r^T\mathbf{X}_i)\mathbf{f}_r||^2_2\nonumber\\
&=\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2,\label{sm1}\end{aligned}$$ where $\mathbf{W}_i=[(\mathbf{f}_1^T\mathbf{X}_i)\mathbf{f}_1\quad(\mathbf{f}_2^T\mathbf{X}_i)\mathbf{f}_2\quad\cdots\quad(\mathbf{f}_N^T\mathbf{X}_i)\mathbf{f}_N]$, $\sigma_r=\lambda_r/\lambda_{max}$ and $\boldsymbol{\sigma}=[\sigma_1\cdots\sigma_N]$.
Moreover, the tensor norm of a given hypergraph has the following property with the frequency coefficients.
Given a representing tensor $\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}$, the tensor norm $||\mathbf{A}||^2_T=\sum_{i_1,i_2,\cdots,i_M=1}^{N}a_{i_1i_2\cdots i_M}^2$ can be written in the form of frequency coefficients as $$||\mathbf{A}||^2_T=\sum_{r=1}^N \lambda_r^2=\boldsymbol{\lambda}^T\boldsymbol{\lambda},$$ where $\boldsymbol\lambda=[\lambda_1 \quad\lambda_2 \quad\cdots\quad\lambda_N]$.
Since $\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}}$, we have $$a_{i_1i_2\cdots i_M}=\sum_{r=1}^N \lambda_rf_{r,i_1}f_{r,i_2}\cdots f_{r,i_M},$$ where $f_{r,i}$ is the $i$th element of $\mathbf{f}_r$. Then, the tensor norm is $$\begin{aligned}
||\mathbf{A}||^2_T
%&=\sum_{i_1,i_2,\cdots,i_M}a_{i_1i_2\cdots i_M}^2\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}(\sum_{r=1}^N \lambda_rf_{r,i_1}f_{r,i_2}\cdots f_{r,i_M})^2\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}(\sum_{r=1}^N \lambda_rf_{r,i_1}\cdots f_{r,i_M})(\sum_{t=1}^N \lambda_tf_{t,i_1}\cdots f_{t,i_M})\nonumber\\
&=\sum_{i_1,i_2,\cdots,i_M}\sum_{r,t}\lambda_r\lambda_tf_{r,i_1}\cdots f_{r,i_M} f_{t,i_1}\cdots f_{t,i_M}\nonumber\\
&=\sum_{r,t}\lambda_r\lambda_t\sum_{i_1,i_2,\cdots,i_M=1}^N(f_{r,i_1}f_{t,i_1})\cdots(f_{r,i_M}f_{t,i_M})\nonumber\\
% &=\sum_{r,t}\lambda_r\lambda_t(\sum_{i=1}^Nf_{r,i}f_{t,i})^M\nonumber\\
&=\sum_{r,t}\lambda_r\lambda_t(\mathbf{f}_r^T\mathbf{f}_t)^M.
\end{aligned}$$ Since $\mathbf{f}_r$ is orthogonal, $\mathbf{f}_r^T\mathbf{f}_t=1$ holds if $r=t$; otherwise, $\mathbf{f}_r^T\mathbf{f}_t=0$. Thus, we obtain $||\mathbf{A}||^2_T=\sum_{r=1}^N \lambda_r^2$.
This property can help us build a connection from the tensor norm to the frequency coefficients directly.
Now, if we consider the representing tensor as the adjacency tensor and each hyperedge consists of three nodes since at least three nodes are required to construct a surface, we optimize the normalized frequency coefficients $\boldsymbol{\sigma}=\frac{1}{\lambda_{max}}\boldsymbol{\lambda}=[\sigma_1\quad
\sigma_2 \quad \cdots\quad \sigma_N]^T$ via $$\begin{aligned}
\label{target}
& \min_{\boldsymbol{\sigma}}\quad \alpha\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2+\beta{ \boldsymbol\sigma^T\boldsymbol\sigma}\\
\hspace{-3mm}s.\; t. \; &\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1,\label{non}\\
&\sum_{r=1}^N \sigma_r f_{r,i_1}f_{r,i_2}f_{r,i_3}\geq 0, \quad i_1,i_2,i_3=1,2,\cdots,N.\label{adj}\end{aligned}$$
The constraint (\[adj\]) limits the estimated representing tensor as the adjacency tensor. The constraint (\[non\]) is the nonnegative constraint on weight and the factor [@c26]. Clearly, the optimization is non-convex with the constraint $\max_i \sigma_{i}=1$. However, if the position of the maximal frequency is known, the optimization problem can be solved by tools such as cvx [@c27; @c28]. Thus, we can develop the following algorithm to estimate the frequency coefficients.
: Point cloud dataset $\mathbf{s}=[\mathbf{X_1,X_2,X_3}]\in\mathbb{R}^{N\times 3}$, hypergraph spectrum $\mathbf{V}=[\mathbf{f}_1,\cdots,\mathbf{f}_N]$. i=1,2,...,iter [**[do]{}**]{}: Set $\sigma_i=1$ as the maximal normalized eigenvalue. Solve the optimization problem in Eq. (\[target\]). Find the optimal $i$ to minimize the target function. The optimal coefficients is the solution of Eq. (\[target\]) correlated to the optimal $i$. : Frequency coefficients [$\boldsymbol\sigma$]{}.
Note that, since we consider clean point cloud without noise, we usually set parameter $\alpha\ll\beta$. Then, from the estimated spectrum pair $(\mathbf{f}_r,\sigma_r)$ under normalization, we can recover the original adjacency tensor as Eq. (\[dec\]). Hence, the hypergraph construction process for a clean point cloud can be summarized as Fig. \[fram1\]. The recovery of original adjacency tensor is not always necessary in practical applications since storing the representing tensor is less efficient than storing the spectrum pairs.
![Estimation of Hypergraph Spectral Pairs for Original Point Clouds[]{data-label="fram1"}](Framework1.jpg){width="3in"}
Joint Spectrum Estimation and Denoising {#h2}
=======================================
In practical 3D imaging, perturbations such as noises and outliers often exist when generate a point cloud of an unknown object. These noises may significantly affect the performance of point cloud processing since many existing algorithms require quality datasets [@c27]. Thus, denoising remains a vital issue in practical point cloud applications.
Usually, to denoise point sets with sharp features is difficult, especially when the noise is large, as such features are hard to distinguish from noise effect. Generally, smoothness-based methods are common. In [@c30], a method based on $L_0$ norm of differences between $k$-nearest neighbors is introduced. In [@c31], Laplacian regularization is used to describe smoothness and to denoise noisy point sets. Other works, such as [@c8; @c29], minimize the total variation over graphs to denoise the point sets. Although smoothness-based methods have achieved notable successes, how to interpret and define an effective smoothness function for a general point set remains open. Furthermore, for graph-based smoothness methods, the construction of graph model remains a critical problem, since traditional methods based on distance suffers from the imprecise location measurement. To this end, a more general definition of smoothness and a more efficient denoising method for arbitrary point clouds are highly desirable.
In this section, we introduce a joint method to simultaneously estimate the hypergraph structure and denoise noisy point clouds. In Section \[h1\], we already introduce an estimation method of spectral pair $(\mathbf{f}_r,\sigma_r)$ for clean point clouds. A similar construction process can be developed for the noisy point clouds. As the estimation of spectrum components only depends on the observed data, we need to denoise the noisy observations while optimizing the frequency coefficients. As already discussed, the problem of denoising a signal on a hypergraph can be written as a convex minimization problem with the constraints that denoised signals should be smooth over the hypergraph. Accordingly, the general process of hypergraph denoising and estimation can be summarized as the following steps:
- Step 1: Estimate the approximated hypergraph spectrum components from the observed noisy point clouds;
- Step 2: Jointly estimate frequency coefficients and denoise the noisy observations;
- Step 3: Update the noisy observations as denoised data and repeat Step 1 until enough iterations.
To estimate hypergraph spectral components of noisy data, the process is the same as Algorithm 1 based on hypergraph stationary process. To jointly estimate the frequency coefficients to recover the original underlying structure and to denoise the noisy point clouds, we propose the following objective. Given $N$ noisy points $\mathbf{s}=[\mathbf{X_1\quad X_2\quad X_3}]$, the joint estimation task can be formulated as
$$\begin{aligned}
\min_{\boldsymbol{\sigma},\mathbf{Y}}\quad& \sum_{i=1}^3[||\mathbf{X}_i-\mathbf{Y}_i||^2_2+\alpha ||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2]+\beta ||\mathbf{A}||_2^2\label{esfram}\\
s.t.\quad &\mathbf{A}=\sum_{r=1}^{N}\lambda_r\cdot\underbrace{\mathbf{f}_r\circ...\circ \mathbf{f}_r}_{\text{M times}} \in \mathcal{A},\nonumber\\
&\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1, \nonumber\\
&\mathbf{W}_i=[(\mathbf{f}_1^T\mathbf{X}_i)\mathbf{f}_1\quad(\mathbf{f}_2^T\mathbf{X}_i)\mathbf{f}_2\quad\cdots\quad(\mathbf{f}_N^T\mathbf{X}_i)\mathbf{f}_N].
\nonumber\end{aligned}$$
The resulting $\mathbf{Y=[Y_1\quad Y_2 \quad Y_3]}$ is the denoised point clouds, and $(\alpha, \beta)$ are two positive regularization parameters. The first part in Eq. (\[esfram\]) lets the denoised point cloud maintain the observed structural features. The second part is the smoothness function derived from Eq. (\[sm1\]) which adjusts positions of noisy points. The third part is the tensor norm regularization to control hypergraph sparsity.
The optimization problem of Eq. (\[esfram\]) is not convex in $\mathbf{Y}$ and $\boldsymbol{\sigma}$. Therefore, similar to [@c25], we split the problem into two subproblems. For each subproblem, we fix one variable set to solve the other one. Upon convergence, the solution corresponds to a local minimum and not necessarily a global minimum.
We first initialize $\mathbf{Y}$ as the observed signals $\mathbf{X}$ and solve the following problem similar to that in Section \[h1\]. $$\label{target1}
\min_{\boldsymbol{\sigma}} \alpha\sum_{i=1}^3||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2+\beta{ \boldsymbol\sigma^T\boldsymbol\sigma}$$ $$\begin{aligned}
s.t.\quad
&\;\;0\leq \sigma_r\leq \max_i \sigma_{i}=1, \nonumber\\
&\sum_{r=1}^N \sigma_r f_{r,i_1}f_{r,i_2}f_{r,i_3}\geq 0, \quad i_1,i_2,i_3=1,2,\cdots,N.\nonumber\end{aligned}$$ This problem can be solved similarly to the solution of clean point cloud with Algorithm 2.
Once the estimated frequency coefficients are found, we solve the subproblem of point cloud denoising $$\begin{aligned}
\label{ss2}
\min_\mathbf{Y} \sum_{i=1}^3[||\mathbf{X}_i-\mathbf{Y}_i||^2_2+\alpha ||\mathbf{X}_i-\mathbf{W}_i\boldsymbol{\sigma}||^2_2],\end{aligned}$$ whose close-form solution for each coordinate is $$\label{ss3}
\mathbf{Y}_i=[\mathbf{I}+\alpha(\mathbf{I-P_s})^T(\mathbf{I-P_s})]^{-1}\mathbf{X}_i.$$
Note that $\mathbf{P_s}$ is the supporting matrix. We then update the frequency components based on the denoised point clouds, and repeatedly carry out Step 1 to Step 3 until getting the final solution. In practice, we generally observe the convergence within only a few iterations. The complete algorithm is summarized in Algorithm 3 as shown in Fig. \[fram2\]. Unlike for clear point clouds, we emphasize more on the smoothness of signals over the hypergraph. The parameter $\alpha$ can be set larger than used when dealing with clean point clouds.
: Noisy observations of point clouds $\mathbf{s}=[\mathbf{X_1,X_2,X_3}]\in\mathbb{R}^{N\times 3}$. : Calculate the spectrum components $\mathbf{f}_r$’s from the observed point cloud $\mathbf{s}$ as Algorithm 1. i=1,2,...,iter [**[do]{}**]{}: Find the optimal $\boldsymbol{\sigma}$ for the first subproblem in Eq. (\[target1\]) with Algorithm 2. Solve the optimization problem in Eq. (\[ss2\]) with $\mathbf{Y}$ in Eq. (\[ss3\]). Update the observed signals as $\mathbf{Y}$ and recalculate the spectrum components $\mathbf{f}_r$’s. : Spectral pairs $(\mathbf{f}_r,\sigma_r)$’s, denoised point clouds $\mathbf{Y}$.
![Joint Hypergraph Estimation and Denoising for Noisy Point Cloud.[]{data-label="fram2"}](Framework2.jpg){width="3in"}
Application Examples {#appli}
====================
In this section, we examine two application examples to test the efficacy of the proposed method in estimating hypergraph structure for both clear and noisy point clouds.
Sampling
--------
Sampling is an important operation to facilitate analysis of very large point clouds. In this part, we consider different sampling strategies depending on different kinds of applications. Some interesting connections are found from the hypergraph frequency and point cloud features.
### Resampling using Harr-like Highpass Filtering
Filtering helps extract select features of a given dataset. In some applications such as boundary detection, accurate extraction of shape features of point clouds is important. Thus, an efficient sampling should retain the features of the original point cloud. In our estimation of hypergraph structure, smoothness is a significant feature to model point clouds. Ideally, smoothness over the original surface of a point cloud should correspond to smoothness over its hypergraph model. Therefore, we can also design a Harr-like high-pass filter to extract sharp features over the surfaces.
Let $\mathbf{I}$ be an identity matrix of appropriate size. Similar to that in GSP [@c3], a Haar-like high-pass filter is designed as $$\begin{aligned}
\mathbf{H}
&=\mathbf{I-P_s}\\
&=\mathbf{V}
\begin{bmatrix}
1-\sigma_1&0&\cdots&0\\
0&1-\sigma_2&\cdots&0\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&1-\sigma_N
\end{bmatrix}\mathbf{V}^T.\end{aligned}$$ The filtered signal is $$(\mathbf{Hs})_i=\mathbf{s}_i-\sum_j {P_s}_{(ij)}\mathbf{s}_j,$$ which reflects the differences between nodes and their neighbors over the hypergraph. Note that, the frequency coefficients together with their corresponding spectral components are ordered decreasingly here, i.e., $\sigma_{i}\geq\sigma_{i+1}$. From the definition of total variation, more smoothness corresponds to larger total variation. Thus, we can extract the sharp features over the point clouds by sampling the nodes with large value of $||\mathbf{s}_i-\sum_j {P_s}_{(ij)}\mathbf{s}_j||^2_2$.
To test this application, we estimate the spectral pairs for clean point clouds and filter the signals over several synthetic datasets. We randomly generate multiple points over the surfaces of basic graphics shown as Fig. \[ori\], and sample the point clouds using the high-pass filter (HPF) given in Fig. \[sam\]. From the test results, we can see that the sampled points of the surfaces in Fig. \[sur2\] mainly congregate near the corners and edges, which are the sharp parts of the point clouds. In addition, the sampled nodes for a cube shape are also crowded near edges and corners. On the other hand, the sampled nodes of a cylinder are mostly at the boundaries of the cylinder. Our test results show that the Harr-like HPF can extract sharp features from point cloud surfaces, which correspond to the least smooth parts of the estimated hypergraph. Moreover, since the total variation measures the order of frequency, sharp features over the point cloud correspond to high frequency components. Thus, the hypergraph model and the estimated spectral pairs are efficient when extracting features of 3D point clouds.
### Down-Sampling with Hypergraph Fourier Transform
Projecting signals into a suitable orthonormal basis is a widely-used sampling method [@c32]. The work of [@c9] develops a sampling theory based on hypergraph signal processing as follows:
- Step 1: Order the spectrum components from low frequency to high frequency based on their total variations.
- Step 2: Implement hypergraph Fourier tranform as $$\mathcal{F}(\mathbf{s})=[(\mathbf{f}_1^T\mathbf{s})^{M-1}\quad (\mathbf{f}_2^T\mathbf{s})^{M-1}\quad \cdots\quad (\mathbf{f}_N^T\mathbf{s})^{M-1}]^T.$$
- Step 3: Use $C$ transformed signal components in the hypergraph frequency domain to represent $N$ signals in the original vertex domain.
More specifically, for a $K$-bandlimitted hypergraph signal, a perfect recovery is available with $K$ samples in hypergraph frequency domain. Similarly, we can sample the point clouds based on the hypergraph Fourier transform. To test the performance of the sampled signals, we implement hypergraph Fourier transform (HGFT) on each coordinates of the point clouds, i.e., $\mathcal{F}(\mathbf{X}_i)$ for all $i$. Then, we take the first $C$ transformed signals in all coordinates. Finally, we implement the inverse hypergraph Fourier transform (iHGFT) to obtain the sampled shapes of the original point clouds. Note that, perfect recovery happens with $C$ samples, if $(\mathcal{F}(\mathbf{X}_i))_{j+C}=0$ for $i,j\in {\cal Z}^+$.
We test the recovered point clouds for animal point datasets [@c33; @c34; @c35; @c36] with the GSP-based methods. For the GSP-based method, we construct the a graph adjacency matrix $\mathbf{W}$ with Guassian model, i.e., $$\label{adjj}
W_{ij}=\left\{
\begin{aligned}
\exp\left(-\frac{||\mathbf{s}_i-\mathbf{s}_j||^2_2}{\delta^2}\right),&\quad||\mathbf{s}_i-\mathbf{s}_j||^2_2\leq t;\\
0,&\quad \mbox{otherwise},
\end{aligned}
\right.$$ where $\mathbf{s}_i$ is the coordinates of the $i$th node. Then, we sample the point clouds using the signals after the graph Fourier transform (GFT).
The test point cloud is shown as Fig. \[sam2\]. We first compare the mean squared error (MSE) between the recovered point clouds and original point clouds shown as Fig. \[sam3\]. From the experimental results, we can see that the HGSP-based method has smaller error than the GSP downsampling method, clearly indicating hypergraph to be a better model. However, sometimes, MSE alone cannot tell the true story in terms of the performance for the recovered point clouds. To explore more, we compare the recovered point clouds directly in Fig. \[sam4\]. From the experimental results, we can see that HGSP-based method captures the overall structure of the point clouds with very few samples, whereas the GSP-based method requires more samples to get sufficient details. The MSE of GSP mainly stems from some outliers when taking more than 90 percent of the samples. The experiments show that HGSP-based method is a better tool for applications which need to recover an overall shape of point clouds from limited data storage. Our test shows hypergraph to be a suitable model for point clouds, and the estimated hypergraph spectral pairs capture the point cloud characteristics very well.
Denoising
---------
From estimated hypergraph spectral pairs from noisy point clouds, the performance of denoising is an intuitive metric of how good the estimates are. There are multiple methods developed to denoise noisy point clouds. The authors of [@c8] proposed a graph-based method to denoise based on total variation (GSP-TV). This method constructs a graph based on observed coordinates first before solving the denoising optimization $$\min_{\mathbf{Y}} ||\mathbf{X}-\mathbf{Y}||_2^2+\alpha \mathbf{TV}(\mathbf{Y,W}),$$ where $\mathbf{X}$ is the observed coordinates, and $\mathbf{W}$ is the adjacency matrix. Here, the graph total variation $\mathbf{TV}(\mathbf{Y,W})$ is applied in describing the smoothness over the graphs. In addition to total variation, Laplacian regularization (LR) has also been used in denoising with a basic formulation $$\min_{\mathbf{Y}} ||\mathbf{X}-\mathbf{Y}||_2^2+\alpha ||\mathbf{Y}^T\mathbf{LY}||^2_2,$$ where $\mathbf{L}$ is the Laplacian matrix. Developed from traditional Laplacian regularization methods, a mesh Laplacian smooth (MLS) method is given in [@c37].
HGSP GSP(TV) MLS LR Noisy
---------------------------- ----------- --------- -------- -------- --------
Uniform$\sim$U(-0.03,0.03) **32.60** 45.94 56.63 48.86 63.84
Uniform$\sim$U(0.08,0.16) **98.36** 160.18 205.15 168.17 220.96
Guassian$\sim$N(0,0.08) **41.10** 42.36 49.41 64.00 76.54
Guassian$\sim$N(0.02,0.08) **73.43** 76.07 83.25 123.08 142.11
Impulse (p=0.08) **34.53** 45.45 50.89 40.53 60.5
: Error in Dfferent Kinds of Noise[]{data-label="t1"}
\
To validate the performance of our denoising method, we compare with the aforementioned traditional methods using the Standford bunny dataset with 3595 points and sampled bunny with 397 points shown as Fig. \[den\]. We compare different methods in the sampled bunny dataset adding zero-mean Guassian noise with variance $\sigma^2$, and zero-mean Uniform noise with the interval $B-A$, respectively. We use the error denoted by $$Error=\sum_{i=1}^N\sum_{j=1}^3 |X_{ji}-Y_{ji}|,$$ where $X_{ij}$ and $Y_{ji}$ are the $j$th coordinates of observed and denoised point $i$, respectively, to measure the performance. We repeat the test on 1000 randomly generated noisy data. The error between the original dataset and the denoised dataset is shown in Fig. \[den1\]. The error of the noisy point clouds before denoising is also given as a reference in Fig. \[den1\]. From the test results, we can see that the HGSP-based method can achieve the lowest error, which demonstrates the effectiveness of the proposed denoising methods and estimated spectral pairs. The comparison in other types of noise is shown in Table. \[t1\]. More specifically, the methods based on total variation, i.e, HGSP and GSP-TV, have better performance than the methods based on Laplacian regularization, which indicates the total variation has a more efficient representation of the surface smoothness. The denoised bunny with 3595 samples is shown in Fig. \[den11\], using our proposed method to denoise the noisy bunny. The successful recovery of the bunny point cloud presents a strong evidence that our estimated spectral pairs and denoising method are powerful tools in processing noisy datasets.
Conclusions and Future Directions {#con}
=================================
In this work, we develop HGSP tools for effectively processing 3D point clouds. We first introduce a novel method to estimate hypergraph spectral components and presented an optimization formulation to optimally select frequency coefficients to recover the optimal hypergraph structure. We develop a HGSP algorithm to jointly estimate hypergraph spectrum pairs and denoise noisy point clouds. To test the practicality and efficacy of our proposed hypergraph tools, we study two point cloud application examples. Our results illustrate significant performance improvements for both sampling and denoising applications. Moreover, we establish a clear connection between hypergraph frequency components and features on point-cloud surface that can be exploited in future studies.
Our work establish hypergraph signal processing as an efficient tool in tackling high-dimensional interactions among multiple nodes. In addition to sampling and denoising, HGSP can find good applications in many other aspects of point clouds through estimation of spectral components and frequency coefficients. One direction is the design of filters to analyze the spectral properties and surface features of 3D point clouds. Another interesting problem is the recovery of point clouds from low dimensional samples. Beyond point clouds, HGSP can also effectively handle datasets with other complex underlying structure.
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| ArXiv |
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abstract: 'We compute the number of coverings of ${{\mathbb{C}}}P^1\setminus\{0, 1, \infty\}$ with a given monodromy type over $\infty$ and given numbers of preimages of 0 and 1. We show that the generating function for these numbers enjoys several remarkable integrability properties: it obeys the Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and satisfies a topological recursion in the sense of Eynard-Orantin.'
address:
- |
Steklov Mathematical Institute\
8 Gubkin St.\
Moscow 119991 Russia
- |
St.Petersburg Department of the Steklov Mathematical Institute\
Fontanka 27\
St. Petersburg 191023, and Chebyshev Laboratory of St. Petersburg State University\
14th Line V.O. 29B\
St.Petersburg 199178 Russia
author:
- 'M. Kazarian, P. Zograf'
title: 'Virasoro constraints and topological recursion for Grothendieck’s dessin counting'
---
Introduction and preliminaries
==============================
Enumerative problems arising in various fields of mathematics, from combinatorics and representation theory to algebraic geometry and low-dimensional topology, often bear much in common. In many cases the generating functions associated with these problems exhibit similar behavior – in particular, they may satisfy
- Virasoro constraints,
- Evolution equations of the “cut-and-join” type,
- Integrable hierarchy (such as Kadomtsev-Petviashvili (KP), Korteveg-DeVries (KdV) or Toda equations),
- Topological recursion (also known as Eynard-Orantin recursion).
Simple Hurwitz numbers provide one of the best studied examples of such an enumerative problem – indeed, their generating function satisfies the celebrated cut-and-join equation [@GJ1], the Virasoro constraints (via the ELSV theorem [@ELSV] and the famous Mumford’s Grothendieck-Riemann-Roch formula [@M] it reduces to the Witten-Kontsevich potential), the KP hierarchy [@O], [@KL] or [@K], and the topological recursion [@EMS]. Other examples include the Witten-Kontsevich theory, Mirzakhani’s Weil-Petersson volumes, Gromov-Witten invariants of the complex projective line, invariants of knots, etc. (see [@EO1], [@EO2] for a review).
These remarkable integrability properties of generating functions usually result from matrix model reformulations of the corresponding counting problems. However, in this paper we show that for the enumeration of Grothendieck’s [*dessins d’enfants*]{} all these properties follow from pure combinatorics in a rather straightforward way.
The origin of Grothendieck’s theory of dessins d’enfants [@G] lies in the famous result by Belyi:
[(Belyi, [@B])]{} A smooth complex algebraic curve $C$ is defined over the field of algebraic numbers ${\overline{\mathbb{Q}}}$ if and only if there exist a non-constant meromorphic function $f$ on $C$ (or a holomorphic branched cover $f:C\to{\mathbb{C}P^1}$) that is ramified only over the points $0,1,\infty\in{\mathbb{C}P^1}$.
We call $(C,f)$, where $C$ is a smooth complex algebraic curve and $f$ is a meromorphic function on $C$ unramified over ${\mathbb{C}P^1}\setminus\{0,1,\infty\}$, a [*Belyi pair*]{}. For a Belyi pair $(C,f)$ denote by $g$ the genus of $C$ and by $d$ the degree of $f$. Consider the inverse image $f^{-1}([0,1])\subset C$ of the real line segment $[0,1]\subset{\mathbb{C}P^1}$. This is a connected bicolored graph with $d$ edges, whose vertices of two colors are the preimages of 0 and 1 respectively, and the ribbon graph structure is induced by the embedding $f^{-1}([0,1])\hookrightarrow C$. (Recall that a ribbon graph structure is given by prescribing a cyclic order of half-edges at each vertex of the graph.) The following is straightforward (cf. also [@LZ]):
\[Gr\][(Grothendieck, [@G])]{} There is a one-to-one correspondence between the isomorphism classes of Belyi pairs and connected bicolored ribbon graphs.
A connected bicolored ribbon graph representing a Belyi pair is called Grothendieck’s [*dessin d’enfant*]{}.[^1]
Let $(C,f)$ be a Belyi pair of genus $g$ and degree $d$, and let ${\Gamma}=f^{-1}([0,1])\hookrightarrow C$ be the corresponding dessin. Put $k=|f^{-1}(0)|,\;l=|f^{-1}(1)|$ and $m=|f^{-1}(\infty)|$, then we have $2g-2=d-(k+l+m)$. We assume that the poles of $f$ are labeled and denote the set of their orders by $\mu=(\mu_1,\ldots,\mu_m)$, so that $d=\sum_{i\geq 1}\mu_i$. The triple $(k,l,\mu)$ will be called here the [*type*]{} of the dessin ${\Gamma}$, and the set of all dessins of type $(k,l,\mu)$ will be denoted by ${\mathcal{D}}_{k,l;\mu}$.
Actually, instead of the dessin ${\Gamma}=f^{-1}([0,1])$ corresponding to a Belyi pair $(C,f)$ it is more convenient to consider the graph ${\Gamma}^*=\overline{f^{-1}(1/2+\sqrt{-1}{\mathbb{R}})}$ dual to $\Gamma$ (where the bar denotes the closure in $C$), see Fig. \[dual\]. The graph ${\Gamma}^*$ is connected, has $m$ ordered vertices of even degrees $2\mu_1,\ldots,2\mu_m$ at the poles of $f$ and inherits a natural ribbon graph structure. Moreover, the boundary components (faces) of ${\Gamma}^*$ are naturally colored: a face is colored in white (resp. in gray) if it contains a preimage of 0 (resp. 1), and every edge of ${\Gamma}^*$ belongs to precisely two boundary components of different color.
![Decomposition of ${\mathbb{C}P^1}$ into two 1-gons.[]{data-label="dual"}](dual){width="5cm"}
In this paper we are interested in the weighted count of labeled dessins d’enfants of a given type. Namely, define $$\begin{aligned}
N_{k,l}(\mu)=N_{k,l}(\mu_1,\ldots,\mu_m)=\sum_{{\Gamma}\in{\mathcal{D}}_{k,l,\mu}}\frac{1}{|{\rm Aut}_b {\Gamma}|}\;,\end{aligned}$$ where ${\rm Aut}_b {\Gamma}$ denotes the group of automorphisms of ${\Gamma}$ that preserve the boundary componentwise.[^2] Consider the total generating function $$\begin{aligned}
\label{gf}
F(s,u,v,p_1,p_2,\dots) = \sum_{k,l,m\geq 1}\frac{1}{m!}\sum_{\mu\in{\mathbb{Z}}_+^m} N_{k,l}(\mu) s^{d} u^k v^l\, p_{\mu_1}\ldots p_{\mu_m}\;,\end{aligned}$$ where the second sum is taken over all ordered sets $\mu=(\mu_1,\ldots,\mu_m)$ of positive integers, and $d=\sum_{i=1}^m \mu_i$.
The objective of this paper is to show that the generating function $F$ satisfies all four integrability properties listed at the beginning of this section – namely, Virasoro constraints, an evolution equation, the KP (Kadomtsev-Petviashvili) hierarchy and a topological recursion. We prove the Virasoro constraints by a bijective combinatorial argument and derive from them all other properties of $F$.[^3] As a result, we obtain a simpler version of the topological recursion in terms of homogeneous components of $F$. We also revisit the problem of enumeration of the ribbon graphs with a prescribed boundary type. Topological recursion for this problem was first established in [@EO2] (cf. also [@DMSS]). In this paper we give a different, more streamlined proof of it based on the Virasoro constraints and show that the corresponding generating function satisfies an evolution equation and the KP hierarchy as well. These (and other) examples convincingly demonstrate that Virasoro constraints imply topological recursion and are in fact equivalent to it.
Additionally, we show how our results can be applied to effectively enumerate orientable maps and hypermaps regardless of the boundary type. In particular, we present a very straightforward derivation of the famous Harer-Zagier recursion [@HZ] for the numbers of genus $g$ polygon gluings from the Walsh-Lehman formula [@WL] (a higher genus generalization of Tutte’s recursion [@T]).[^4]
Virasoro constraints
====================
Virasoro constraints for the numbers of dessins
-----------------------------------------------
For any integer $n\geq 0$ consider the differential operator $$\begin{gathered}
L_n=-\frac{n+1}{s}\frac{\partial}{\partial p_{n+1}}+(u+v)n\frac{\partial}{\partial p_{n}}
+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}\\
{}+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}+\delta_{0,n}uv\;.\label{V}\end{gathered}$$
A straightforward check shows that for any integer $m,n\geq 0$ $$\begin{aligned}
[L_m, L_n]=(m-n)L_{m+n}\,.\end{aligned}$$ In other words, the operators $L_n$ form (a half of) a representation of the Virasoro (or, rather, Witt) algebra.
The main technical statement of this section is the following
\[Virasoro\] The partition function $e^F=e^{F(s,u,v,p_1,p_2,\dots)}$ satisfies the infinite system of non-linear differential equations (Virasoro constraints) $$\begin{aligned}
\label{cons}
L_ne^F=0\;.\end{aligned}$$ The equations determine the partition function $e^F$ uniquely.
The Virasoro constraints can be re-written as follows: $$\begin{aligned}
\frac{n+1}{s}\frac{\partial F}{\partial p_{n+1}}&=
\sum_{j=1}^\infty p_j(n+j)\frac{\partial F}{\partial p_{n+j}}+(u+v)n\frac{\partial F}{\partial p_{n}}\nonumber\\
&+\sum_{i+j=n}ij\left(\frac{\partial^2 F}{\partial p_{i} \partial p_{j}}+
\frac{\partial F}{\partial p_{i}} \frac{\partial F}{\partial p_{j}}\right) + \delta_{0,n}uv\;.\label{vc}\end{aligned}$$ Eq. (\[vc\]) for $n+1=\mu_1$ can be further re-written as a recursion relation for the coefficients $N_{k,l}(\mu)$ of $F$: $$\begin{aligned}
\mu_1\,N_{k,l}(\mu_1,&\ldots,\mu_m)
=\sum_{j=2}^m (\mu_1+\mu_j-1)N_{k,l}(\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)\nonumber\\
&{}+(\mu_1-1)(N_{k-1,l}(\mu_1-1,\mu_2,\ldots,\mu_m)+N_{k,l-1}(\mu_1-1,\mu_2,\ldots,\mu_m))\nonumber\\
&{}+\sum_{i+j=\mu_1-1}ij \bigg(N_{k,l}(i,j,\mu_2,\ldots,\mu_m)\nonumber\\
&\qquad{}+\mathop{\sum_{k_1+k_2=k}}_{l_1+l_2=l}\quad\sum_{I\sqcup J=\{2,\ldots,m\}}
N_{k_1,l_1}(i,\mu_I)N_{k_2,l_2}(j,\mu_J)\bigg)\;,\label{vt}\end{aligned}$$ where $\mu_I=\mu_{i_1},\ldots,\mu_{i_k},\;I=\{i_1,\ldots,i_k\}$, and the hat means that the corresponding term is omitted.[^5] This recursion is valid for $\sum_{i=1}^m\mu_i>1$ and expresses the numbers $N_{k,l}(\mu)$ recursively in terms of $N_{1,1}(1)=1$.
We prove this recursion similar to [@WL] (cf. also [@DMSS], [@EO2], [@N]) by establishing a direct bijection between dessins counted in the left and right hand sides of . Here it is more convenient to deal with the dual graphs instead. Let ${\Gamma}^*$ be the ribbon graph dual to a dessin ${\Gamma}$ of type $(k,l,\mu)$. There are $2\mu_1$ ways to pick a half-edge incident to the first vertex of ${\Gamma}^*$. Following [@DMSS] we label this half-edge with an arrow (labeling of half-edges allow us to forget about nontrivial automorphisms). When ${\Gamma}$ varies over the set ${\mathcal{D}}_{k,l,\mu}$, this gives twice the number in the l.h.s. of .
Let us now express the same number in terms of dessins with one edge less. This can be done by contracting (or expanding) the labeled edges in the dual graphs in a way that preserves the proper coloring of faces. The following possibilities can occur:
(i) The labeled edge connects the first vertex with the $j$-th vertex, $j\neq 1$. Contracting this edge we get a ribbon graph with properly bicolored faces of type $(k,l,\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)$, see Fig. \[contract\]. Conversely, given a graph of type $(k,l,\mu_1+\mu_j-1,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)$, there are $2(\mu_1+\mu_j-1)$ ways to split its first vertex into two ones of degrees $2\mu_1$ and $2\mu_j$. Since $j$ can vary from 2 to $m$, this gives twice the first sum in the r.h.s. of .
![Contracting an edge with different endpoints.[]{data-label="contract"}](contract){width="9cm"}
(ii) The labeled edge forms a loop that bounds a white 1-gon, see Fig. \[loop\]. Contracting such a loop we reduce both $k$ and $\mu_1$ by 1, leaving $l$ and $\mu_j,\; j=2,\ldots,m,$ unchanged. Conversely, if we have a graph of type $(k-1,l,\mu_1-1,\mu_2,\ldots,\mu_m$, we can insert a loop into any of the $\mu_1-1$ gray sectors at the first vertex in order to get a graph of type $(k,l,\mu_1,\ldots,\mu_m)$, and 2 ways to label one of its half-edges. The case of a loop bounding a gray 1-gon can be treated verbatim, giving twice the second term in the r.h.s. of .
![Contracting a loop that bounds a 1-gon.[]{data-label="loop"}](loop){width="8cm"}
(iii) The labeled edge forms a loop whose half-edges are not adjacent relative to the cyclic order of half-edges at the first vertex. Contracting such a loop we split the first vertex into two ones, say, of degrees $2i$ and $2j$, where $i+j=\mu_1-1$, see Fig. \[split\]. Under this operation the graph may remain connected, or may split into two connected components. In the former case we get a graph of type $(k,l,i,j,\mu_2,\ldots,\mu_m)$. Reversing this operation, we join the first two vertices and add a loop. We can place the labeled half-edge of the loop in any of the $2i$ sectors at the first vertex, but its other half-edge can be placed only in one of $j$ sectors of different color at the second vertex (otherwise it will not be compatible with the face coloring). This gives us twice the third term in the r.h.s. of . The latter case when the graph becomes disconnected can be treated similarly.
![Contracting a loop that splits the vertex into two ones of degrees $2i$ and $2j$ with $i+j=\mu_1-1$.[]{data-label="split"}](split){width="8cm"}
The operations (i)–(iii) are reversible and compatible with the face coloring, thus establishing a required bijection. This proves the Virasoro constraints . It is also not hard to see that the Virasoro constraints determine the partition function $e^F$ uniquely, since they are equivalent to the recursion .
\[evo\] Put $$\begin{aligned}
\Lambda_1&=\sum_{i=2}^\infty (i-1)p_i\,\frac{\partial}{\partial p_{i-1}}\;,\nonumber\\
M_1&=\sum_{i=2}^\infty \sum_{j=1}^{i-1} \left((i-1)p_j p_{i-j}\,\frac{\partial}{\partial p_{i-1}}
+ j(i-j) p_{i+1}\,\frac{\partial^2}{\partial p_j \partial p_{i-j}}\right)\;.\end{aligned}$$ Then the partition function $e^F$ satisfies the evolution equation $$\begin{aligned}
\frac{\partial e^F}{\partial s}=((u+v)\Lambda_1+M_1+uvp_1)e^F\;,\label{eveq}\end{aligned}$$ and is uniquely determined by the initial condition $F\left|_{s=0}\right.=0$. In other words, $e^F$ is explicitly given by the formula $$\begin{aligned}
e^F=e^{s((u+v)\Lambda_1+M_1+uvp_1)}\,1\end{aligned}$$ were “1” stands for the constant function identically equal to 1.
Multiply the both sides of by $p_{n+1}$ and sum over $n$. We get $$\begin{aligned}
\sum_{n=0}^\infty p_{n+1}L_n
&=\sum_{n=0}^\infty p_{n+1}\left(-\frac{n+1}{s}\frac{\partial}{\partial p_{n+1}}+(u+v)n\frac{\partial}{\partial p_{n}}\right.\nonumber\\
&\hspace{0.4in}+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}
+\sum_{i+j=n}\left. ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}\right)+uvp_1\nonumber\\
&=-\frac{\partial}{\partial s}+(u+v)\Lambda_1+M_1+uvp_1\;,\end{aligned}$$ and the required statement immediately follows from .
A different proof of Corollary \[evo\] has recently appeared in [@Z].
\[ps\] Denote by $\psi$ the principal specialization of the partition function $e^F$: $$\begin{aligned}
\psi=\psi(s,t,u,v)=e^{F(s,u,v,p_1,p_2,\ldots)}\left|_{p_i=t^i}\right.\;,\end{aligned}$$ where $t$ is a new formal variable. It is not hard to check that $$\begin{aligned}
\Lambda_1 e^F\left|_{p_i=t^i}\right.=t\left(t\frac{d}{dt}\right)\psi\,, \qquad M_1 e^F\left|_{p_i=t^i}\right.=t\left(t\frac{d}{dt}\right)^2\psi\,.\end{aligned}$$ Then, with the help of the obvious identity $s\frac{\D\psi}{\D s}=t\frac{\D\psi}{\D t}$, the evolution equation translates into the following equation for the wave function $\psi$: $$\begin{aligned}
\frac{1}{st}\left(t\frac{d}{dt}\right)\psi=(u+v)\left(t\frac{d}{dt}\right)\psi+\left(t\frac{d}{dt}\right)^2\psi+uv\psi\;.\end{aligned}$$ It can be further rewritten as the Schrödinger equation $$\begin{aligned}
\label{qc}
t^2\,\frac{d^2\psi}{dt^2}+\left((u+v+1)\,t-\frac{1}{s}\right)\,\frac{d\psi}{dt}+uv\psi=0\;\end{aligned}$$ Eq. is often referred to as the [*quantum curve equation*]{} in the literature on topological recursions. Note that the coefficients of $\log\psi$ enumerate dessins with given numbers of white and black vertices, and a given number of edges regardless of genus (or the number of boundary components).
Another observation is that the generating function $F=F(s,u,v,p_1,p_2,\dots)$ satisfies an infinite system of non-linear partial differential equations called the KP (Kadomtsev-Petviashvili) hierarchy (this means that the numbers $N_{k,l}(\mu)$ additionally obey an infinite system of recursions). The KP hierarchy is one of the best studied completely integrable systems in mathematical physics. Below are the first few equations of the hierarchy: $$\label{KP}
\begin{aligned}
&F_{22}=-\frac12\,F_{11}^2+F_{31}-\frac1{12}\,F_{1111}\;,\\
&F_{32}=-F_{11}F_{21}+F_{41}-\frac16F_{2111}\;,\\
&F_{42}=-\frac12\,F_{21}^2-F_{11}F_{31}+F_{51}+\frac18\,F_{111}^2
+\frac1{12}\,F_{11}F_{1111}-\frac14\,F_{3111}+\frac1{120}\,F_{111111}\;,\\
&F_{33}=\frac13\,F_{11}^3-F_{21}^2-F_{11}F_{31}+F_{51}
+\frac14\,F_{111}^2+\frac13\,F_{11}F_{1111}-\frac13\,F_{3111}\\
&\hspace{3.5in}+\frac1{45}\,F_{111111}\;,
\end{aligned}$$ where the subscript $i$ stands for the partial derivative with respect to $p_i$.
The exponential $Z=e^F$ of any solution is called a [*tau function*]{} of the hierarchy. The space of solutions (or the space of tau functions) has a nice geometric interpretation as an infinite-dimensional Grassmannian (called the [*Sato Grassmannian*]{}), see [@MJD] or [@K] for details. In particular, the space of solutions is homogeneous: there is a Lie algebra $\widehat{\mathfrak{gl}(\infty)}$ that acts infinitesimally on the space of solutions, and the action of the corresponding Lie group is transitive.
\[tau\] The generating function $F=F(s,u,v,p_1,p_2,\dots)$ satisfies the infinite system of KP equations with respect to $p_1,p_2,\dots$ for any parameters $s,u,v$. Equivalently, the partition function $Z=e^F$ is a 3-parameter family of KP tau functions.
To begin with, we notice that $1$ is obviously a KP tau function. Then, since $p_1, \Lambda_1, M_1\in\widehat{\mathfrak{gl}(\infty)}$ (cf. [@K]), the linear combination $s((u+v)\Lambda_1+M_1 +uvp_1)$ also belongs to $\widehat{\mathfrak{gl}(\infty)}$ for any $s,u,v$. The exponential $e^{s((u+v)\Lambda_1+M_1+uvp_1)}$ therefore preserves the Sato Grassmannian and maps KP tau functions to KP tau functions. Thus, $e^F$ is a KP tau function as well, and $F$ is a solution to KP hierarchy.
Corollary \[tau\] was earlier proven in [@GJ2] by a different method. However, [@GJ2] contains no analogs of the Virasoro constraints or the evolution equation.
At the end of this subsection we will sketch how to enumerate dessins with $k$ white vertices, $l$ black vertices, $d$ edges and $m$ boundary components regardless of the partition $\mu=(\mu_1,\ldots,\mu_m)$. To these ends, consider the specialization operator $$\begin{aligned}
\theta: F\mapsto F|_{p_i=t, i=1,2,\ldots}\label{sp}\end{aligned}$$ and put $f=\theta(F)$. This specialization is more subtle than the one considered in Remark \[ps\], and the coefficients of $f$ do not mix dessins of different genera since by Euler’s formula $(k+l)-d+m=2-2g$. Expanding $f$ into a series in the variables $s$ and $t$, we recompose it as $$\begin{aligned}
f(s,t,u,v)=\sum_{g=0}^\infty\sum_{d=2g+1}^\infty \frac{f_{g,d}(t,u,v)}{d}\,s^d\;,\end{aligned}$$ where each coefficient $f_{g,d}(t,u,v)$ is a homogeneous polynomial in $t,u,v$ of degree $d+2-2g$ with integer coefficients.
Furthermore, using the Virasoro constraints (\[V\]) with $n=0,1,2$ and the homgeneity equation $$s\frac{\partial F}{\partial s}=\sum_{i=1}^\infty ip_i\frac{\partial F}{\partial p_i}\;,$$ we can express the specializations of partial derivatives of $F$ with respect to the variables $p_1,p_2,p_3$ in terms of $s$-derivatives of $f$. More precisely, a straightforward computation yields
We have $$\begin{aligned}
&\theta(F_{1})=s^2f'+suv, \\
&\theta(F_{11})=s^2(s^2f'+suv)', \\
&\theta(F_{1111})=s^2(s^2(s^2(s^2f'+suv)')')',\\
&2\theta(F_{2})=(1+s(u+v-t))(s^2f'+suv)-suv, \\
&3\theta(F_{3})=2s(u+v-t)\theta(F_{2})+(1-st)\theta(F_{1})+s\theta(F_{11})+s\theta(F_{1})^2-suv, \\
&\theta(F_{12})=s^2\theta(F_{2})', \\
&\theta(F_{13})=s^2\theta(F_{3})', \\
&2\theta(F_{22})=(1+s(u+v-t))\theta(F_{12})+3s\theta(F_{3})-2s\theta(F_{2}),\end{aligned}$$ where the subscript $i$ stands for the partial derivative with respect to $p_i$, and the prime $'$ denotes the derivative in $s$.
Applying the specialization operator $\theta$ to the first KP equation $$\theta(F_{22})=-\frac12\,\theta(F_{11})^2+\theta(F_{31})-\frac1{12}\,\theta(F_{1111})\,,$$ cf. (\[KP\]), and using the above formulas, we get an ordinary differential equation for $f$ as a function of $s$ that translates into the following quadratic recursion: $$\begin{aligned}
(d+1)f_{g,d}&=(2d-1)\,a\,f_{g,d-1}+(d-2)\,b\,f_{g,d-2}+(d-1)^2(d-2)f_{g-1,d-2}\nonumber\\
&+\sum_{i=0}^g\sum_{j=1}^{d-3}(4+6j)(d-2-j)f_{i,j}f_{g-i,d-2-j}\;,\label{ad}\end{aligned}$$ where $a=t+u+v,\; b=4(tu+tv+uv)-a^2.$ Starting with $f_{0,1}=tuv$, one can recursively compute the polynomials $f_{g,d}$ for all $g$ and $d$.
Finally, let us restrict ourselves to the case of dessins with one boundary component (or bicolored polygon gluings). Denote by $h_{g,d}$ the linear term in $f_{g,d}$ with respect to $t$. Then the recursion (\[ad\]) takes the form $$\begin{aligned}
(d+1)h_{g,d}&=(2d-1)(u+v)h_{g,d-1}-(d-2)(u-v)^2 h_{g,d-2}\\
&+(d-1)^2(d-2)h_{g-1,d-2}\;,\end{aligned}$$ and we reproduce the well-known result of [@A] on the enumeration of genus $g$ gluings of a bicolored $2d$-gon with given numbers of white and black vertices (cf. also [@J]).
Virasoro constraints for the numbers of ribbon graphs
-----------------------------------------------------
A closely related, but somewhat different enumerative problem was considered in [@WL]. Recall that a dessin d’enfant $f^{-1}([0,1])$ is a bicolored connected ribbon graph with vertices “colored" by either 0 or 1 depending on whether $f$ maps the vertex to 0 or 1 in ${{\mathbb{C}}}P^1$. One can similarly try to enumerate all (not necessarily bicolored) connected ribbon graphs, and this is the problem that was addressed in [@WL]. To make it precise, let us label the boundary components of a ribbon graph ${\Gamma}$ (or, equivalently, the vertices of the dual graph ${\Gamma}^*$) by integers from 1 to $m$, and let $\mu_1,\ldots,\mu_m$ be the lengths of the boundary components of ${\Gamma}$ (or the degrees of vertices of ${\Gamma}^*$).
Ribbon graphs can naturally be represented by dessins of a special type called [*clean dessins*]{} in [@DMSS]. Namely, color each vertex of a ribbon graph in white and place black vertices at the midpoints of edges. Such a dessin corresponds to a covering of ${\mathbb{C}P^1}$ of even degree $d$ with ramification of type $[2^{d/2}]$ over 1 and arbitrary ramification over 0 and $\infty$. As before, we put $k=|f^{-1}(0)|$ (the number of vertices of the ribbon graph ${\Gamma}$), $l=|f^{-1}(1)|=d/2$ (the number of edges of ${\Gamma}$), and $m=|f^{-1}(\infty)|$ (the number of boundary components of ${\Gamma}$). Clearly, we have $k-d/2+m=2-2g$.
Denote by $D_{g,m}(\mu)=D_{g,m}(\mu_1,\ldots,\mu_m)$ the number of genus $g$ ribbon graphs with $m$ labeled vertices of degrees $\mu_1,\ldots,\mu_m$ counted with weights $\frac{1}{|{\rm Aut}_v\,{\Gamma}|}$, where the automorphisms preserve each vertex of ${\Gamma}$ pointwise. Apparently, the same numbers enumerate pure dessins with $m$ labeled boundary components of lengths $(2\mu_1,\ldots,2\mu_m)$. The following recursion for $D_{g,m}(\mu)$ was derived in [@WL], Eq. (6):[^6] $$\begin{aligned}
\mu_1 D_{g,m}(\mu_1,\ldots,&\mu_m)
=\sum_{j=2}^m (\mu_1+\mu_j-2)D_{g,m-1}(\mu_1+\mu_j-2,\mu_2,\ldots,\widehat{\mu}_j,\ldots,\mu_m)\nonumber\\
&{}+2(\mu_1-2)D_{g,m}(\mu_1-2,\mu_2,\ldots,\mu_m)\nonumber\\
&{}+\sum_{i+j=\mu_1-2}ij \bigg(D_{g-1,m+1}(i,j,\mu_2,\ldots,\mu_m)\nonumber\\
&{}+\sum_{g_1+g_2=g}\quad\sum_{I\sqcup J=\{2,\ldots,m\}}
D_{g_1,|I|+1}(i,\mu_I)D_{g_2,|J|+1}(j,\mu_J)\bigg)\;.\label{wl}\end{aligned}$$ Recursion is valid for all $g\geq 0,\;m\geq 1$, and $\mu$ such that $d=\sum_{i=1}^m\mu_i>2$, whereas for $d=2$ the only nonzero numbers are $D_{0,1}(2)=1/2,\;D_{0,2}(1,1)=1$. Below we give a convenient interpretation of this recursion in terms of PDEs.
Similar to (\[gf\]), introduce the generating function $$\begin{aligned}
\label{gfM}
\tF(s,u,p_1,p_2,\ldots)
&=\sum_{g=0}^\infty\sum_{m=1}^\infty\frac{1}{m!} \sum_{\mu\in{{\mathbb{Z}}}_+^m} D_{g,m}(\mu)s^d u^k p_{\mu_1}\ldots p_{\mu_m}\;,\end{aligned}$$ where $d=\sum_{i=1}^m\mu_i$, $k=2-2g-m+d/2$, and $\mu=(\mu_1,\ldots,\mu_m)$ (compared to (\[gf\]), we omit here the trivial factor $v^{d/2}$ that carries no additional information in this case).
\[rg\] The generating function $\tF$ enjoys the following integrability properties:
(i) Let $$\begin{aligned}
\tL_n=-\frac{n+2}{s^2}\frac{\partial}{\partial p_{n+2}}
+2\,u\,n\frac{\partial}{\partial p_{n}}
+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}\\
{}+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_i \partial p_j}+\delta_{-1,n}u p_1+\delta_{0,n}u^2\;,\end{aligned}$$ where $n\geq -1$. Then the partition function $e^\tF$ satisfies the infinite system of PDE’s (“Virasoro constraints”) $$\begin{aligned}
\tL_n e^\tF=0\end{aligned}$$ that determine $\tF$ uniquely.
(ii) Put $$\begin{aligned}
&\Lambda_2=\sum_{i=3}^\infty (i-2)p_i\,\frac{\partial}{\partial p_{i-2}}+\frac12p_1^2\;,\nonumber\\
&M_2=\sum_{i=2}^\infty \sum_{j=1}^{i-1} \left((i-2)p_j p_{i-j}\,\frac{\partial}{\partial p_{i-2}}
+ j(i-j) p_{i+2}\,\frac{\partial^2}{\partial p_j \partial p_{i-j}}\right)\;.\end{aligned}$$ Then $e^\tF$ satisfies the evolution equation $$\begin{aligned}
\frac{1}{s}\frac{\partial e^\tF}{\partial s}=(2\,u\,\Lambda_2+M_2+u^2p_2)e^\tF\,,\end{aligned}$$ that, together with the initial condition $\tF|_{s=0}=0$, determines $\tF$ uniquely. In other words, $e^\tF$ is explicitly given by the formula $$\begin{aligned}
e^\tF=e^{\frac{s^2}{2}(2\,u\,\Lambda_2+M_2+u^2p_2)}\,1\;.\end{aligned}$$
(iii) The partition function $e^\tF$ is a tau function of the KP hierarchy, i.e. its logarithm $\tF(s,u,p_1,p_2,\ldots)$ satisfies for any $s$ and $u$.
Part (i) of the theorem is just a reformulation of the recursions (\[wl\]) for $\mu_1=n+2$. Note that the operators $\tL_n$ obey the commutation relations $[\tL_m,\tL_n]=(m-n)\tL_{m+n}$ for $m>n\geq -1$.
To prove (ii) we multiply $\tL_n$ by $p_{n+2}$ and sum over $n$: $$\begin{aligned}
\sum_{n=-1}^\infty p_{n+2}\tL_n
&=\sum_{n=-1}^\infty p_{n+2}\bigg(-\frac{n+2}{s}\frac{\partial}{\partial p_{n+2}}+2\,u\,n\frac{\partial}{\partial p_{n}}\nonumber\\
&\hspace{0.6in}
{}+\sum_{j=1}^\infty p_j(n+j)\frac{\partial}{\partial p_{n+j}}
+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_{i} \partial p_{j}}\bigg)+u\,p_1^2+u^2p_2\nonumber\\
&=-\frac{1}{s}\frac{\partial}{\partial s}+2\,u\,\Lambda_2+M_2+u^2 p_2\;.\label{evom}\end{aligned}$$
Part (iii) follows from the fact that $\Lambda_2, M_2$ and $p_2$ belong to $\widehat{\mathfrak{gl}(\infty)}$, cf. Corollary \[tau\].
Similar to Eq. we can write the quantum curve equation for the wave function $$\widetilde{\psi}=\widetilde{\psi}(s,t,u)=e^{\tF(s,u,p_1,p_2,\ldots)}\left|_{p_i=t^i}\right.\;,$$ that reads in this case as follows: $$t^2\,\frac{d^2\widetilde{\psi}}{dt^2}+\left(2(u+1)\,t-\frac{1}{s^2\,t}\right)\,\frac{d\widetilde{\psi}}{dt}+(u+u^2)\,\widetilde{\psi}=0\;.$$ Note that this equation differs from the one obtained in [@MS]. The reason for that is explained in Footnote \[M\] above. To fix that, put $Z(x,\hbar)=e^{-\frac{\log x}{\hbar}}\,\widetilde{\psi}|_{s=\hbar/x,t=1,u=1/\hbar}$. Then one has $$\left(\hbar^2\frac{d^2}{dx^2}+\hbar\frac{d}{dx}+1\right)Z(x,\hbar)=0$$ precisely like in [@MS].
To complete this section, we will show that the Walsh-Lehman formula (\[wl\]) implies the Harer-Zagier [@HZ] recursion for the numbers of orientable polygon gluings. We will follow the same lines as at the end of the previous subsection. To begin with, put $\tf=\theta(\tF)=\tF|_{p_i=t, i=1,2,\ldots}$, where the specialization operator $\theta$ is defined by Eq. (\[sp\]). The coefficients of $\tf$ enumerate ribbon graphs with given numbers of vertices, edges and boundary components and, therefore, do not mix graphs of different genera. Rearrange the series $f$ as follows: $$\begin{aligned}
\tf(s,t,u)=\sum_{g=0}^\infty\sum_{l=2g}^\infty \frac{\tf_{g,l}(t,u)}{2l}\,s^{2l}\;,\end{aligned}$$ where $l=d/2$, and each coefficient $\tf_{g,l}(t,u)$ is a homogeneous polynomial in $t,u$ of degree $l+2-2g$ with integer coefficients. Like in the case of dessins, using the Virasoro constraints of Theorem \[rg\] (i) with $n=-1,0,1$, we can express the specializations of partial derivatives of $\tF$ with respect to the variables $p_1,p_2,p_3$ in terms of $s$-derivatives of $\tf$. A straightforward computation yields
We have $$\begin{aligned}
&\theta(\tF_{1})=s^3\tf'+s^2tu, \\
&\theta(\tF_{11})=s^3\theta(\tF_{1})'-s^2\theta(\tF_{1})+s^2u, \\
&\theta(\tF_{111})=s^3\theta(\tF_{11})'-2s^2\theta(\tF_{11}),\\
&\theta(\tF_{1111})=s^3\theta(\tF_{111})'-3s^2\theta(\tF_{111}),\\
&2\theta(\tF_{2})=s^3\tf'+s^2u^2, \\
&4\theta(\tF_{22})=s^6\tf''+3s^5\tf'+2s^4u^2, \\
&\theta(\tF_{13})=s^5((2s^2u+1-s^2t)(s\tf')'+(2s^2u+2-s^2t)\tf')+s^6(2tu^2-t^2u)+3s^4u^2, \end{aligned}$$ where, as before, the subscript $i$ stands for the partial derivative with respect to $p_i$, and the prime $'$ denotes the derivative in $s$.
Applying the specialization operator $\theta$ to the first KP equation (\[KP\]) and using the above formulas, we get an ordinary differential equation for $\tf$ as a function of $s$ that translates into the following quadratic recursion: $$\begin{aligned}
(l+1)\tf_{g,l}&=2(2l-1)(t+u)\tf_{g,l-1}+(2l-1)(2l-3)(l-1)\tf_{g-1,l-2}\nonumber\\
&+3\sum_{i=0}^g\sum_{j=0}^{l-2}(2j+1)(2(l-2-j)+1)\tf_{i,j}\tf_{g-i,l-2-j}\;,\label{hz}\end{aligned}$$ where we put by definition $\tf_{0,0}=u$. This is essentially the formula from [@CC], but derived in a more straightforward way. Note that starting with $\tf_{0,1}=t^2u+tu^2$, one can recursively compute the polynomials $\tf_{g,l}$ for all $g$ and $l$.
To enumerate genus $g$ ribbon graphs with one boundary component (or $2l$-gon gluings), it is sufficient to consider the linear terms $\epsilon_{g,l}$ in $\tf_{g,l}$ with respect to $t$. Then Eq. (\[hz\]) turns into the famous Harer-Zagier recursion $$\begin{aligned}
(l+1)\epsilon_{g,l}=2(2l-1)u\epsilon_{g,l-1}+(2l-1)(2l-3)(l-1)\epsilon_{g-1,l-2}\;,\end{aligned}$$ cf. [@HZ].
Topological recursion
=====================
The generating function $F=\sum_{g,m}F_{g,m}$ of enumerating Belyi pairs $(C,f)$ (or Grothendieck’s dessins) can be naturally decomposed into components with fixed $g$ and $m$, where $g$ is the genus of $C$ and $m$ is the number of poles of $f$: $$\begin{aligned}
F_{g,m}(s,u,v,p_1,p_2,\dots)
= \frac{1}{m!}\sum_{\mu_1,\dots,\mu_m}\sum_{k+l=d-m+2-2g}
N_{k,l}(\mu) s^{d} u^k v^l\, p_{\mu_1}\ldots p_{\mu_m}\label{gm}
\end{aligned}$$ (here, as usual, $d=\sum\mu_i$). Another way to collect these numbers into a generating series is to use the *$m$-point correlation functions* $$W_{g,m}(x_1,\dots,x_m)
=\sum_{\mu_1,\dots,\mu_m}\sum_{k+l=d-m+2-2g} N_{k,l}(\mu)\,x_1^{\mu_1}\dots x_m^{\mu_m}.$$
*Topological recursion* (cf. [@EO2]) is an “ansatz” that allows to reconstruct the coefficients of certain generating series recursively in $g$ and $m$. Traditionally, it is formulated in terms of correlation functions or, rather, *differentials*
$$\begin{aligned}
w_{g,m}(x_1,\dots,x_m)&=\frac{\D^mW_{g,m}}{\D x_1\ldots \D x_m} \,dx_1\ldots dx_m\\
&=\sum_{\mu}\sum_{\substack{k,\,l\\k+l=d-m+2-2g}} N_{k,l}(\mu)\prod_{i=1}^m \mu_i x_i^{\mu_i-1} dx_i.\end{aligned}$$
We will present the topological recursion in terms of components $F_{g,m}$ of the generating function $F$. The advantage of this approach is that we need only one set of variables $p_i$ for all $g$ and $n$. The two approaches being equivalent, it proves out, however, that many properties of the recursion become more clear in terms of $p$-variables.
One of the nice features of topological recursion is that the generating functions $F_{g,m}$ become polynomilas under a linear change of variables $p_i$. The components $F_{g,m}$ of the total generating function $F$ are infinite formal series in $p_i$, and their polynomiality is far from being an immediate consequence of the Virasoro constraints for $F$. On the other hand, this polynomiality automatically follows from the equations of topological recursion. Another advantage of topological recursion is its universality. For a variety of enumerative problems it takes the same form, differing only in initial conditions.
Introduce the formal variables $x$ and $z$ related by $$z(x)=\sqrt{\frac{1-\beta s x}{1-\alpha s x}},\qquad
x(z)=\frac{z^2-1}{s \left(\alpha z^2-\beta \right)},
\label{eqzx}$$ where $$\a=(\sqrt{u}-\sqrt{v})^2,\qquad
\b=(\sqrt{u}+\sqrt{v})^2.
\label{eqabs}$$ We consider as a formal change of coordinates on the complex projective line ${\mathbb{C}P^1}$ near the point $x=0$ (resp., $z=1$) depending on the parameters $\a,\b,s$ (or $u,v,s$).
Put $$T_j(p)=\sum_{i=1}^\infty c_j^{(i)}p_i\,,
\label{eqT}$$ where the coefficients $c_j^{(i)}$ are defined by the relation $$\sum_{i=1}^\infty c_j^{(i)}x^i=(z(x))^{j}-1\,.$$
\[th2\] Let $F_{g,m}$ be the infinite series defined by . Then
(i) For each $g,m$ with $2g-2+m>0$ there exists a polynomial $G_{g,m}$ of the variables $t_j$, $j\in{\mathbb{Z}}_\odd$, such that $$F_{g,m}(p)=G_{g,m}(t)\bigm|_{t_j=T_j(p)}$$ (i.e. each $F_{g,m}$ is a polynomial in the linear functions $T_{\pm1},T_{\pm3},T_{\pm5},\dots$).
(ii) The polynomials $G_{g,m}$ can be recursively computed starting from $G_{0,3}$ and $G_{1,1}$, cf. Eqs. – below.
Let us now formulate the recursion for the polynomials $G_{g,m}$ precisely. This can be done in terms of the so-called *spectral curve*. In our case the spectral curve is the projective line ${\mathbb{C}P^1}$ equipped with the globally defined holomorphic involution $z\mapsto -z$ with respect to some affine coordinate $z$.
By a [*Laurent form*]{} we understand here a globally defined meromorphic 1-form on ${\mathbb{C}P^1}$ with poles only at $0$ and $\infty$. Denote by $L$ the space of odd Laurent forms relative to the involution $z\mapsto -z$. The forms $d(z^j)=j\,z^{j-1}\,dz$, $j\in{\mathbb{Z}}_\odd$, provide a convenient basis in $L$. Let $P_L$ denote the projector to the space $L$ in the space of all Laurent forms. For a Laurent form $\phi$ its projection $\psi=P_L(\phi)$ to $L$ is uniquely determined by the requirement that the form $\psi-\phi_\odd$ is regular at both $0$ and $\infty$, where $\phi_\odd(z)=\frac12(\phi(z)-\phi(-z))$ is the odd part of $\phi$. More explicitly, the action of $P_L$ is given by the formula $$\begin{aligned}
(P_L\phi)(z)&=\sum_{i=0}^\infty\res_{w=0}(\phi(w)w^{2i+1})\;z^{-2i-2}dz-
\sum_{i=0}^\infty\res_{w=\infty}(\phi(w)w^{-2i-1})\;z^{2i}dz\nonumber\\
&=\res_{w=0}\left(\phi(w)\frac{w\;dz}{z^2-w^2}\right)+\res_{w=\infty}\left(\phi(w)\frac{w\;dz}{z^2-w^2}\right)\;.
\label{res}\end{aligned}$$
Note that for the validity of this definition it will suffice to assume that $\phi$ is defined in a neighborhood of the points $0$ and $\infty$, or even that $\phi$ is a formal Laurent series at these points. On the other hand, the form $P_L\phi\in L$ is always globally defined on ${\mathbb{C}P^1}$.
In fact, the recursion applies not to the polynomials $G_{g,m}$ themselves, but to certain $1$-forms $U_{g,m}$. For the set of variables $z$ and $t=(t_{\pm 1},t_{\pm 3},\ldots)$ introduce the differential operator $$\d_{z,t}=\sum_{j\in{\mathbb{Z}}_\odd}jz^{j-1}\pd{}{t_j}\,.$$ For $2g-2+m>0$ put $$\begin{aligned}
U_{g,m}(z)&=\d_{z,t} G_{g,m}(z)=\sum_{j\in{\mathbb{Z}}_\odd}jz^{j-1}\pd{G_{g,m}}{t_j}\,.\end{aligned}$$ As we will see later, $U_{g,m}=U_{g,m}(z)dz$ is an odd Laurent form on ${\mathbb{C}P^1}$ that is polynomial in $t_j$.
\[remdelta\] The operator $\d_{z,t}$ written in terms of $x$ and $p$-variables becomes $$\d_{x,p}=\frac{dz}{dx}\sum_{i=1}^\infty ix^{i-1}\,\pd{}{p_i}\,,$$ where $x$ is related to $z$ by (\[eqzx\]) and $t_j=T_j(p)$, see . (For brevity we will omit the subscripts ‘$p$’ and ‘$t$’ by $\d$, always associating $p$- and $t$-variables with $x$ and $z$ respectively.) More precisely, assume that a function $f$ of $p$-variables can be expressed as a composition $f=h\circ T$, where $h$ is a function of $t$-variables and $T$ is the linear change . Then we have $\d_xf\,dx=(\d_zh)|_{t_k=T_k(p)}\,dz$. In particular, if $g$ is a polynomial in $t$-variables, then $\d_xfdx$ is a Laurent form in $z$ (with coefficients depending on $p_i$’s). Indeed, the operators on both sides satisfy the Leibnitz rule, and therefore it is sufficient to prove the equality for the case $h=t_k$, that is, $$\d_x T_k(p)dx=k z^{k-1} dz,\qquad z=z(x)\,,$$ which is essentially the definition of the linear functions $T_k(p)$, cf. (\[eqT\]).
In the unstable cases (i.e. when $2g-2+m\leq 0$) the definition of $U_{g,m}$ should be modified. Namely, we set $U_{0,1}=0$ and define $U_{0,2}$ by the following formal expansions $$\begin{aligned}
U_{0,2}(z)dz&=-\sum_{i=0}^\infty t_{-2i-1}z^{2i}dz=-(t_{-1}+t_{-3}z^2+\dots)\,dz,\quad z\to 0,\\
U_{0,2}(z)dz&=-\sum_{i=0}^\infty t_{2i+1}z^{-2i}d(z^{-1})=(t_1z^{-2}+t_3z^{-4}+\dots)\,dz,\quad z\to\infty,\\
\end{aligned}
\label{U02}$$
In general, the homogeneous degree $m$ polynomial $G_{g,m}$ can be recovered form the form $U_{g,m}$ by the Euler formula $$G_{g,m}=\frac{1}{m}\sum t_k\pd{G_{g,m}}{t_k}=\frac1m\,\O(U_{g,m},U_{0,2})\;,
\label{eq1}$$ where for odd forms $\phi$ and $\psi$ we set $$\O(\phi,\psi)=-\O(\psi,\phi)=\res_{z=0}\left(\phi\int\psi\right)+\res_{z=\infty}\left(\phi\int\psi\right)\;.$$
The last ingredient needed to write down the topological recursion is the form $$\label{eta}
\eta=\eta(z)dz=\frac{\s\,z^2dz}{(z^2-1)^2 (\a\,z^2 -\b)},\qquad
\s=(\a-\b)^2=16\,u\,v.$$ This form is odd and has the property that the dual vector field $$\frac{1}{\eta}=\frac{1}{\eta(z)}\,\frac{d}{dz}=\frac{\a\,z^4-(2\,\a+\b)\,z^2+\a +2\,\b -\b\,z^{-2}}{\s}\,\frac{d}{dz}$$ is meromorphic with poles of order $2$ at $z=0$ and $z=\infty$ and regular elsewhere in ${\mathbb{C}P^1}$.
The main recursive relation of this paper is $$U_{g,m}=P_L\Biggl(\frac{1}{2\eta}\Biggl(\d U_{g-1,m+1}+\sum_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}}U_{g_1,m_1}U_{g_2,m_2}\Biggr)\Biggr)
\label{eq2}$$ (here and below we tacitly assume that $\d f=\d_z f\,dz$). Note that almost all terms in the sum on the right hand side of (\[eq2\]) belong to $L$, so that $P_L$ is identical on these terms. Therefore, (\[eq2\]) can equivalently be rewritten as $$U_{g,m}=\frac{1}{2\eta}\Biggl(\d U_{g-1,m+1}+\mathop{\sum{}^*}_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}}
U_{g_1,m_1}U_{g_2,m_2}\Biggr)+P_L\left(\frac1\eta\,U_{g,m-1}U_{0,2}\right)\;,$$ where the star $^*$ by the summation sign means that the terms involving $U_{0,2}$ are excluded (recall that $U_{0,1}=0$ by assumption). This recursion relation is valid for all $g$ and $n$ with $2g-2+m>1$. Moreover, it applies for $(g,m)=(0,3)$ as well: $$U_{0,3}=P_L\left(\frac{U_{0,2}^2}{2\eta}\right).
\label{eq3}$$ In the case $(g,m)=(1,1)$ the formula is not applicable since $\d U_{0,2}$ is not defined. This is why we set by definition $$U_{1,1}=U_{1,1}(z)\,dz=P_L\left(\frac{1}{2\eta}\left(\frac{dz}{2z}\right)^2\right)=\frac{1}{2\eta}\left(\frac{dz}{2z}\right)^2
=\frac{1}{8z^2\eta(z)}\,dz\,.
\label{eq4}$$
Eqs. – concretize the second statement of Theorem \[th2\].
Below we list the polynomials $U_{g,m}$ and $G_{g,m}$ for small $g$ and $m$:
$$\begin{aligned}
\s\,U_{0,3}(z)&=\frac{\a}{2}\,t_1^2-\frac{\b}{2}\,t_{-1}^2\,z^{-2},\\[-12pt]\\
\s\,G_{0,3}&=\frac{\a}{6}\,t_1^3+\frac{\b}{6}\,t_{-1}^3,\\[-12pt]\\
\s\,U_{1,1}(z)&=\frac{\a}{8}\,z^2 -\frac{2\a+\b}{8}
+\frac{\a+2\b}{8}\,z^{-2}-\frac{\b}{8}\,z^{-4},\\[-12pt]\\
\s\,G_{1,1}&=\frac{\a}{24}\,t_3-\frac{2\a+\b}{8}\,t_1-\frac{\a+2\b}{8}\,t_{-1}+\frac{\b}{24}\,t_{-3},\\[-12pt]\\
\s^2\,U_{0,4}(z)&=\frac{\a^2}{2}t_1^3\,z^2
+\Bigl(\frac{\a^2}{2}\,t_3\, t_1^2-\frac{\a(2\a+\b)}{2}\,t_1^3-\frac{\a\b}{2}\,t_{-1}^2\, t_1\Bigr)\\
&+\Bigl(\frac{\b(\a+2\b)}{2}\,t_{-1}^3+\frac{\a\b}{2}\,t_1^2\, t_{-1}-\frac{\b^2}{2}\,t_{-3}\, t_{-1}^2\Bigr)\,z^{-2}
-\frac{\b^2}{2}\,t_{-1}^3\,z^{-4},\\[-12pt]\\
\s^2\,G_{0,4}&=\frac{\a^2}{6}\,t_1^3\, t_3-\frac{\a(2\a+\b)}{8}\,t_1^4-\frac{\a\b}{4}\,t_1^2\, t_{-1}^2
-\frac{\b(\a+2\b)}{8}\,t_{-1}^4+\frac{\b^2}{6}\,t_{-3}\,t_{-1}^3,\\[-12pt]\\
\s^2\,U_{1,2}(z)&=\frac{5\a^2}{8}\,t_1\,z^4+\Bigl(\frac{\a^2}{8}\,t_3-\frac{3\a(2\a+\b)}{4}\,t_1\Bigr)\,z^2\\
&+\Bigl(\frac{10\a^2+16\a\b+\b^2}{8}\,t_1+\frac{\a^2}{8}\,t_5+\frac{\a\b}{2}\,t_{-1}-\frac{\a(2\a+\b)}{4}\,t_3\Bigr)\\
&+\Bigl(-\frac{\a^2+16\a\b+10\b^21}{8}\,t_{-1}+\frac{\b(\a+2\b)}{4}\,t_{-3}-\frac{\a\b}{2}\,t_1-\frac{\b^2}{8}\,t_{-5}\Bigr)\,z^{-2}\\
&+\Bigl(\frac{3\b(\a+2\b)}{4}\,t_{-1}-\frac{\b^2}{8}\,t_{-3}\Bigr)\,z^{-4}
-\frac{5\b^2}{8}\,t_{-1}\,z^{-6},\\[-12pt]\\
\s^2\,G_{1,2}&=\frac{\a^2}{8}\,t_1\,t_5+\frac{\a^2}{48}\,t_3^2-\frac{\a(2\a+\b)}{4}\,t_1\,t_3+\frac{\a^2+16\a\b+10\b^2}{16}\,t_{-1}^2\\
&+\frac{\a\b}{2}\,t_{-1}\,t_1+\frac{10\a^2+16\a\b+\b^2}{16}\,t_1^2-\frac{\b(\a+2\b)}{4}\,t_{-3}\,t_{-1}\\
&+\frac{\b^2}{48}\,t_{-3}^2+\frac{\b^2}{8}\,t_{-5}\,t_{-1}\;,\end{aligned}$$
where $\a,\b,\s$ are given by , . This list can be continued further on.
Here we compare our form of the topological recursion with the one that can be found in the literature, see, e.g. [@EO1], [@EO2].
(i) Traditionally, the spectral curve comes with an embedding to (a compactification of) ${\mathbb{C}}^2$ by means of certain meromorphic functions $X,Y$ on ${\mathbb{C}P^1}$. These functions are chosen so that $X$ is even with respect to the involution $z\mapsto -z$, and $\eta=Y\,dX$. In our case we could have set, for example, $$X=\frac1x=s\,\frac{\a z^2-\b}{z^2-1},\qquad Y=-\frac{\a-\b}{2s}\,\frac{z}{\alpha z^2-\beta}.$$ The formulas of the topological recursion, however, involve the coordinates $X$ and $Y$ only in the combination $\eta=Y\,dX$.
(ii) The topological recursion is usually formulated in terms of $m$-point correlators. They are related to the homogeneous components $F_{g,m}$ of the generating function $F$ by the formulas $$\begin{aligned}
w_{g,m}(x_1,\dots,x_m)&=\d_{x_1}\ldots\d_{x_m} F_{g,m}\,dx_1\ldots dx_m\nonumber\\
&=\d_{x_2}\ldots\d_{x_m} U_{g,m}(x_1)\,dx_1\ldots dx_m\;,
\label{eqwgn}\end{aligned}$$ where $\d_{x_j}=\sum_{i=1}^\infty i\,x_j^{i-1}\,\pd{}{p_i}$. Via the change of variables , $w_{g,m}$ can be viewed as a meromorphic $m$-differential on $\left({\mathbb{C}P^1}\right)^m$ that is a Laurent form with respect to each of its arguments provided $2g-2+m>0$.
(iii) A version of holds also for $(g,m)=(0,2)$. Namely, $w_{0,2}(z_1,z_2)=\d_{z_2}U_{0,2}\,dz_2$ is the odd part of the *Bergman kernel* $B(z_1,z_2)=\frac{dz_1\,dz_2}{(z_1-z_2)^2}$: $$w_{0,2}(z_1,z_2)=\d_{z_2}U_{0,2}\,dz_2=\frac12\left(B(z_1,z_2)-B(z_1,-z_2)\right).$$ This can be interpreted as an equality of asymptotic expansions of the left and right hand sides at $z_1\to 0$ and $z_1\to\infty$, cf. : $$w_{0,2}(z_1,z_2)=
\begin{cases}-\sum_{i=0}^\infty d(z_2^{-2i-1})\;z_1^{2i}dz_1,&z_1\to 0\;,\medskip\\
-\sum_{i=0}^\infty d(z_2^{2i+1})\;z_1^{-2i}d(z_1^{-1}),& z_1\to\infty\;.\end{cases}$$
(iv) The projector $P_L$, see , is given by the contour integral $$\begin{aligned}
(P_L\phi)(z)\,dz=\frac{1}{2\pi\sqrt{-1}}\left(\int_{|w|=\epsilon}\frac{\phi(w)\,w\,dw}{z^2-w^2}
+\int_{|w|=1/\epsilon}\frac{\phi(w)\,w\,dw}{z^2-w^2}\right)\,dz\;,\end{aligned}$$ for small $\epsilon > 0$, where $$\frac{w\;dz}{z^2-w^2}=\frac12\int\limits_{-w}^w B(z,\cdot).$$ This explains the appearance of the Bergman kernel in the majority of expositions of the topological recursion.
(v) The above items (i)–(iv) demonstrate that the traditional form of the topological recursion in terms of the correlators $w_{g,n}$ is obtainable from by applying $\d_{z_2}\ldots\d_{z_n}$ to the both sides of it.
Proof of the topological recursion
==================================
Master Virasoro equation
------------------------
As we will see below, the topological recursion relations are just the equivalently reformulated Virasoro constraints. To begin with, let us collect the Virasoro constraints into a single equation by multiplying the $n$th equation by $x^n$ (where $x$ is a formal variable) and summing them up: $$\begin{gathered}
\sum_{n=0}^\infty x^n\Biggl(-\frac1s(n+1)\pd{F}{p_{n+1}}+(u+v)\,n\pd{F}{p_n}+
\sum_{j=1}^\infty p_j(n+j)\pd{F}{p_{n+j}}\Biggr.\\\Biggl.+
\sum_{i+j=n}ij\left(\pd{^2F}{p_i\D p_j}+\pd{F}{p_i}\pd{F}{p_j}\right)\Biggr)+uv=0.\end{gathered}$$ This equation can be simplified. Notice that $$\begin{aligned}
\sum_{n=0}^\infty&\; x^n\left(-\frac1s(n+1)\pd{F}{p_{n+1}}+(u+v)\,n\pd{F}{p_n}\right)
=\left(-\frac1s+(u+v)\,x\right)\d_x F\;,\\
\sum_{n=0}^\infty&\; x^n\sum_{i+j=n}ij\left(\pd{^2F}{p_i\D p_j}+\pd{F}{p_i}\pd{F}{p_j}\right)=
x^2\left(\d_x^2 F+(\d_x F)^2\right)\;,\end{aligned}$$ where, as in the previous section, $\d_x=\sum_{n=1}^\infty nx^{n-1}\pd{}{p_n}$ (cf. Remark \[remdelta\]). As for the third term in the sum, we use the identity $p_j=\d_y^{-1}(j\,y^{j-1})=\d_y^{-1}d_y(y^j)$, where $y$ is a new independent formal variable and $d_y=\frac{d}{dy}$, to re-write it as follows: $$\begin{aligned}
\sum_{n=0}^\infty\; x^n\sum_{j=1}^\infty& p_j(n+j)\pd{F}{p_{n+j}}\\
&=\sum_{n=0}^\infty\;\sum_{i+j=n} x^{i}\,p_j\,n\,\pd{F}{p_n}
=\d_y^{-1} d_y\left(\sum_{n=0}^\infty\;\sum_{i+j=n} x^{i}\,y^j\,n\,\pd{F}{p_n}\right)\\
&=\d_y^{-1} d_y\left(\sum_{k=0}^\infty \frac{x^{k+1}-y^{k+1}}{x-y}\, k\,\pd{F}{p_k}\right)
=\d_y^{-1} d_y\left(\frac{x^2{\d_x F}-y^2\d_y F}{x-y}\right)\end{aligned}$$ This yields the following *master Virasoro equation* that unifies all Eqs. : $$\begin{aligned}
\Bigl(-\frac1s+(u+v)\,x\Bigr)\d_x F&+x^2\left(\d_x^2 F+(\d_x F)^2\right)\nonumber\\
&+\d_y^{-1} d_y\left(\frac{x^2\d_x F-y^2\d_y F}{x-y}\right)+uv=0\;.
\label{mve}\end{aligned}$$
Unstable terms and the spectral curve
-------------------------------------
Our immediate goal is to extract the homogeneous terms in contributing to $\d_x F_{g,n}$ for fixed $g$ and $n$. We start with the unstable cases. For $g=0$ and $n=1$ we get $$x^2\,(\d_x F_{0,1})^2+\Bigl(-\frac1{s}+(u+v)\,x\Bigr)\d_x F_{0,1}+u v=0\,.\label{eqspectr}$$ Solving this equation for $x\,\d_x F_{0,1}$, choosing the proper root and expanding it into the Taylor series at $x=0$ we get $$\begin{aligned}
x\,\d_x F_{0,1}&=\frac{1}{2} \left(\frac{1}{s x}-u-v-\sqrt{\left(\frac{1}{s x}-u-v\right)^2-4 u v}\right)\\
&=u\,v\,s\,x+u\,v\,(u+v)\,s^2x^2+ u\,v\,\left(u^2+3\,u\,v+v^2\right)\,s^3x^3 +\dots\end{aligned}$$ and $$F_{0,1}=u\,v\,s\,p_1+\frac12u\,v\,(u+v)\,s^2p_2+\frac13u\,v\,\left(u^2+3\,u\,v+v^2\right)\,s^3p_3 +\dots.$$
The *spectral curve* is the Riemann surface of the algebraic function $x\,\d_x F_{0,1}$ in the $x$-variable.
In other words, the spectral curve is an algebraic curve such that $x$ and $x\d_x F_{0,1}$ are globally defined univalued meromorphic functions on it. In our case the spectral curve is given by . It is rational (admits a rational parametrization). Let $z$ be an affine coordinate on ${\mathbb{C}P^1}$. Its choice is not important, but, for convenience, we choose it in such a way that the two critical points of the function $x$ on ${\mathbb{C}P^1}$ are $z=0$ (with the critical value $x=\frac{1}{s\b}$) and $z=\infty$ (with the critical value $x=\infty$). The corresponding rational parametrization has the following form: $$\begin{aligned}
x&=\frac{z^2-1}{s\,(\a z^2-\b)},&
z&=\sqrt{\frac{1-\b s x}{1-\a s x}},\\
x\,\d_x F_{0,1}&=\sqrt{uv}\,\frac{1-z}{1+z},&\d_x F_{0,1}\frac{dx}{dz}&=\frac{8\,u\,v\,z}{(z+1)^2 \left(\alpha z^2-\beta \right)},\end{aligned}$$ where $\a,\b$ are related to $u,v$ by . All functions entering these equalities can be regarded either as rational functions in $z$-variable or as formal power expansions of these functions at $z=1$.
We continue with the terms with $g=0$ and $n=2$ in . We have $$\Bigl(-\frac1s+(u+v)x\Bigr)\d_x F_{0,2}+2\,x^2\,\d_x F_{0,1}\,\d_x F_{0,2}
+\d_y^{-1} d_y\left(\frac{x^2\,\d_x F_{0,1}-y^2\,\d_y F_{0,1}}{x-y}\right)
=0\,,$$ from where we get $$\d_y\,\d_x\,F_{0,2}=\frac{d_y\left(\frac{x^2\,\d_x\, F_{0,1}-y^2\,\d_y\, F_{0,1}}{x-y}\right)}
{\frac1s-(u+v)x-2\,x^2\,\d_x F_{0,1}}\;.$$ This equality uniquely determines $\d_y\d_x F_{0,2}dxdy$ as a meromorphic bidifferential on ${\mathbb{C}P^1}\times{\mathbb{C}P^1}$. Substituting the obtained above expressions for $x,\;dx/dz$ and $\d_x F_{0,1}$ into the last formula, after some miraculous cancellations we finally get $$\d_y\d_x F_{0,2}\,dx\,dy=\frac{dz\,dw}{(z+w)^2}=-B(z,-w),
\label{ddF02}$$ where $B(z,w)=\frac{dz\,dw}{(z-w)^2}$ is the Bergman kernel, and $w$ is related to $y$ by the same formulas that relate $z$ to $x$.
Rational recursion formula
--------------------------
Now we look at the homogeneous terms of genus $g$ and degree $m$ with $2g-2+m>1$ in . To begin with, let us extract the unstable terms from the expression $(\d F)^2$ in in order to re-group them with the other summands. Multiplying by $x^{-2}$ we get $$\begin{aligned}
&2\left(-\frac{1}{2\,s\,x^2}+\frac{u+v}{2\,x}+\d_x F_{0,1}\right)\,\d_x(F-F_{0,1}-F_{0,2})+\d_x^2 F\\
&\quad +\bigl(\d_x(F-F_{0,1}-F_{0,2})\bigr)^2+x^{-2}\d_y^{-1}\,d_y\,\left(\frac{x^2\,\d_x(F-F_{0,1})-y^2\,\d_y(F-F_{0,1})}{x-y}\right)\\
&\hspace{2.4in}+2\d_x(F-F_{0,1})\,\d_x F_{0,2}-(\d_x F_{0,2})^2=0.\end{aligned}$$ Let us rewrite the coefficients of this equation in the $z$-coordinate. It is convenient to put $$\eta=\eta(x)\,dx=-\left(-\frac{1}{2\,s\,x^2}+\frac{u+v}{2\,x}+\d_x F_{0,1}\right)\,dx\;,$$ so that in terms of the coordinate $z$ $$\eta=\eta(z)\,dz=\frac{(\a-\b)^2\,z^2}{(z^2-1)^2 (\a\,z^2 -\b)}\,dz\,.$$ Then, using the already known expressions for $x$, $\frac{dx}{dz}$, $\d_x F_{0,1}$, and $\d_y\d_x F_{0,2}$, we find that $$\begin{aligned}
\d_x F_{0,2}=-\d_y^{-1}\,d_y\left(\frac{1}{z+w}\right).
\label{delf}\end{aligned}$$ From the above identities we obtain the following equation for $\d_x F$: $$\begin{aligned}
\d_x(F-F_{0,1}-F_{0,2})&=\frac{1}{2\,\eta(x)}\Bigl(\d_x^2 F+(\d_x(F-F_{0,1}-F_{0,2}))^2-(\d_xF_{0,2})^2\Bigr)\nonumber\\
+\d_y^{-1}\,d_y&\left(\frac{w}{z^2-w^2}\left(\frac{\d_x(F-F_{0,1})}{\eta(x)}-\frac{\d_y (F-F_{0,1})}{\eta(y)}\right)\right)\,
\frac{dz}{dx}\;.\label{eqvirz}\end{aligned}$$ In particular, for the homogeneous components $U_{g,m}=\d_x F_{g,m}$ with $2g-2+m>1$ this equation reads $$\begin{gathered}
U_{g,m}(z)=\frac{1}{2\,\eta(z)}\Biggl(\d_z U_{g-1,m+1}(z)+\mathop{\sum{}^{\mathrlap{*}}}_{\substack{g_1+g_2=g,\\m_1+m_2=m+1}}
U_{g_1,m_1}(z)\,U_{g_2,m_2}(z)\Biggr)\\
{}+\d_w^{-1}\,d_w\left(\frac{w}{z^2-w^2}\left(\frac{U_{g,m-1}(z)}{\eta(z)}-\frac{U_{g,m-1}(w)}{\eta(w)}\right)\right)\;,
\label{eqrat}\end{gathered}$$ where the star $^*$ by the summation sign means that the unstable terms with $2g-2+m\leq 0$ are omitted. We refer to this equation as the *rational recursion formula* for the forms $U_{g,m}=U_{g,m}\,(z)dz$.
This equation allows to prove the polynomiality property for the forms $U_{g,m}$ by induction in $g$ and $m$. Indeed, assume that $U_{g',m'}$ is a Laurent form in $z$ with coefficients polynomially depending on $t_k=T_k(p)$, $k\in Z_\odd$, for all $(g',m')$ with $0<2g'-2+m'<2g-2+m$. Then the first summand in obviously also has this form. Let us examine the second summand. Note that the operator $\d_w^{-1}\,d_w$ is well defined on the space of odd Laurent polynomials in variable $w$, so let us check that this condition always holds. Indeed, the function $\frac{U_{g,m-1}(z)}{\eta(z)}$ is an even Laurent polynomial in $z$, therefore, it can be represented as a Laurent polynomial in $z^2$. Therefore, $\frac1{z^2-w^2}\left(\frac{U_{g,m-1}(z)}{\eta(z)}-\frac{U_{g,m-1}(w)}{\eta(w)}\right)$ is a Laurent polynomial in $z^2$ and $w^2$ regular at $z=\pm w$. Multiplying it by $w$ and applying $d_w$ we obtain an odd Laurent form in $w$. The polynomiality property for the forms $U_{g,m}$ follows now from Remark \[remdelta\].
Utilising Remark \[remdelta\] once again, we obtain the polynomiality property for $F_{g,m}$ with $2g-2+m>0$ as well. This proves the main assertion of Theorem \[th2\] (under the assumption that it is valid for the initial terms $F_{0,3}$ and $F_{1,1}$; this is checked below).
The residual formalism
----------------------
Since the both sides of belong to $L$, it is convenient to equate the projections of the terms entering this equality by applying $P_L$ to each of them. The terms of the first summand on the right hand side already belong to $L$, so $P_L$ is identical on them. Compute the image of the second summand on the right hand side under the projection $P_L$. The key observation is that $P_L$ can be applied to the two terms of this summand separately. In particular, the form $\frac{dz}{z^2-w^2}$ is regular both at $z=0$ and at $z=\infty$ so that it does not contribute to the image. It remains to compute the term $$\begin{aligned}
&P_L\!\left(\d_w^{-1}d_w\!\left(\frac{w}{z^2-w^2}\frac{U_{g,m-1}(z)}{\eta(z)}\right)dz\right)\\
&\hspace{2in}=P_L\!\left(\frac{U_{g,m-1}(z)}{\eta(z)}\,\d_w^{-1}\!\left(\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw\right)\right).\end{aligned}$$ The form $\frac{(z^2+w^2)}{(z^2-w^2)^2}\,dz\,dw$ is not Laurent so that $\d_w^{-1}$ is not applicable to it directly. However, what we actually need in order to apply $P_L$ is the expansion of this form at $z=0$ and $z=\infty$. The coefficients of these expansions are Laurent with respect to $w$: $$\begin{aligned}
&\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw=-\sum_{i=0}^\infty d(w^{-2i-1})z^{2i}dz=
-\d_w\sum_{i=0}^\infty t_{-2i-1}z^{2i}dz,\quad z\to 0,\\
&\frac{(z^2+w^2)}{(z^2-w^2)^2}dz\,dw=-\sum_{i=0}^\infty d(w^{2i+1})z^{-2i}d(z^{-1})\\
&\hspace{2.2in}=-\d_w\sum_{i=0}^\infty t_{2i+1}z^{-2i}d(z^{-1}),\quad z\to \infty.\end{aligned}$$ In other words, the form $\d_w^{-1}\Bigl(\frac{(z^2+w^2)}{(z^2-w^2)^2}\,dz\,dw\Bigr)$ is well defined in some neighborhoods of the points $z=0$ and $z=\infty$ and coincides with the form $U_{0,2}$ defined by . This proves the equality of the topological recursion.
Initial terms of the recursion
------------------------------
In order to finish the proof of Theorem \[th2\], it remains to check it for the initial terms of the recursion, that is, for $U_{1,1}=\d_x F_{1,1}\,dx$ and $U_{0,3}=\d_x F_{0,3}dx$.
For the case $g=m=1$, equating the corresponding terms in we get $$U_{1,1}=\frac{1}{2\eta(x)}\d_x^2F_{0,2}\,dx\,,$$ and using the explicit formula for $\d^2F_{0,2}$, we obtain the required formula for $U_{1,1}$.
For the case $g=0$, $m=3$ the computations are slightly more involved. Eq. uniquely determines the form $U_{0,3}$ as $$U_{0,3}=
\d_w^{-1}\, d_w\left(\frac{w}{z^2-w^2}\left(\frac{\d_x F_{0,2}(x)}{\eta(x)}-\frac{\d_y F_{0,2}(y)}{\eta(y)}\right)\right)\,dz
-\frac{(\d_x F_{0,2})^2}{2\,\eta(x)}\,dx\;,$$ where, as before, $y$ is related to $w$ by the same formulas that relate $x$ to $z$.
The form $\d_x F_{0,2}\,dx$ being known, cf. , we directly compute $$U_{0,3}=\frac{1}{32 u v}\left(\a\,t_1^2dz-\b t_{-1}^2\frac{dz}{z^2} \right)=P_L\left(\frac{(U_{0,2}(z))^2}{2\eta(z)}\,dz\right)$$ which agrees with . This completes the proof of Theorem \[th2\].
Topological recursion for ribbon graphs
=======================================
Here we discuss the toplogical recursion for the numbers $D_{g,m}$ of genus $g$ ribbon graphs with $m$ labeled boundary components of lengths $\mu_1,\ldots,\mu_m$, see Section 2. A topological recursion for these numbers was first obtained in [@EO1], Theorem 7.3 (it was later rediscovered in [@DMSS]). We present a simple proof of this recursion in terms of the homogeneous components $\tF_{g,m}$ of the generating function that follows directly from the Virasoro constraints, cf. Theorem \[rg\], (i). In fact, the argument is quite parallel to that of the case of dessins d’enfants. This is why we skip the details paying attention only to the differences between these two enumerative problems. More specifically, we have the same spectral curve ${\mathbb{C}}P^1$ equipped with an affine coordinate $z$ and the involution $z\mapsto-z$. What is different, is the choice of the local coordinate $x$ at the point $z=1$ and the form $\eta$.
Consider the linear functions $\tT_k(p)$ given by with $$\label{zxM}
z(x)=\sqrt{\frac{1+2\sqrt{u}\; s\, x}{1-2\sqrt{u}\; s\, x}}$$
\[th3\] Let $\tF_{g,m}$ be the infinite series defined by $$\begin{aligned}
\tF_{g,m}(s,u,p_1,p_2,\ldots)
&=\frac{1}{m!} \sum_{\mu\in{{\mathbb{Z}}}_+^m} D_{g,m}(\mu)s^d u^k p_{\mu_1}\ldots p_{\mu_m}\;.\end{aligned}$$ Then
(i) For each $g,m$ with $2g-2+m>0$ there exist a polynomial $\tG_{g,m}$ of the variables $t_j$, $j\in{\mathbb{Z}}_\odd$, such that $$\tF_{g,m}(p)=\tG_{g,m}(t)\bigm|_{t_j=\tT_j(p)}$$ (i.e. each $\tF_{g,m}$ is a polynomial in the linear functions $\tT_{\pm1},\tT_{\pm3},\tT_{\pm5},\ldots$).
(ii) The polynomials $\tG_{g,m}$ can be recursively computed, starting from $\tG_{0,3}$ and $\tG_{1,1}$, by the same Eqs. – with $\eta$ given by the formula $$\eta=\eta(z)\,dz=-\frac{16uz^2dz}{(1-z^2)^3}\;.$$
Like in the case of dessins, we start with the master Virasoro equation that readily follows from Theorem \[th2\], (i): $$\begin{aligned}
\Bigl(-\frac1{s^2}+2\,u\,x^2\Bigr)\,\d_x \tF&+x^3\Bigl(\d_x^2 \tF+(\d_x \tF)^2\Bigr)\\
&+\d_y^{-1}\,d_y\left(\frac{x^3\,\d_x \tF-y^3\,d_y \tF}{x-y}\right)+u^2\,x+u\,p_1=0\;.\end{aligned}$$ The spectral curve in this case (an analog of above) is given by the equation $$x^2\,(\d_x\tF_{0,1})^2-x\,\d_x\tF_{0,1}\Bigl(\frac{1}{s^2 x^2}-2 u\Bigr)+u^2=0\,.$$ Solving this equation for $x\,\d_x\tF_{0,1}$ we get the following rational parametrization of the spectral curve $$\begin{aligned}
&x(z)=\frac{z^2-1}{2s\sqrt{u}\;(z^2+1)}\;,\\
&x\,\d_x\tF_{0,1}=\frac{1-2\,s^2\,u\,x^2-\sqrt{1-4\,s^2\,u\,x^2}}{2s^2\,x^2}=u\left(\frac{1-z}{1+z}\right)^2\;,\end{aligned}$$ with the inverse change $z(x)$ given by . The genus $0$ two point correlator is the same as in the case of dessins: $$\d_y\d_x\tF_{0,2}\,dx\,dy=\frac{dz\,dw}{(z+w)^2},$$ and instead of the form $\eta$ we have $$\tilde\eta=\tilde\eta(x)dx=\left(-\d_x \tF_{0,1}-\frac{u}{x}+\frac{1}{2\,s^2\,x^3}\right)\,dx=-\frac{16\,u\,z^2}{(1-z^2)^3}\,dz\;.$$ Thus, the master Virasoro equation in $z$-coordinate acquires the same form and implies the topological recursion with $\eta$ replaced by $\tilde\eta$.
Below are the first few terms of the recursion: $$\begin{aligned}
U_{0,3}&=\frac{1}{32\,u}\left(t_1^2-t_{-1}^2z^{-2}\right)\,,\\[-12pt]\\
G_{0,3}&=\frac{1}{96\,u} \left(t_{1}^3+t_{-1}^3\right)\,,\\[-12pt]\\
U_{1,1}&=\frac{1}{128\,u}\left(z^2-3+3\,z^{-2}-z^{-4}\right)\,,\\[-12pt]\\
G_{1,1}&=\frac{1}{384\,u}\left(t_{3}-9 t_{1}-9 t_{-1}+t_{-3}\right)\,,\\[-12pt]\\
U_{0,4}&=\frac{1}{512\,u^2}\left(t_1^3 z^2+(t_3 t_1^2-3 t_1^3-t_{-1}^2 t_1)\right.\\[-12pt]\\
&\hspace{.9in}\left.+(t_1^2 t_{-1}+3 t_{-1}^3-t_{-3} t_{-1}^2)\,z^{-2}-t_{-1}^3\,z^{-4}\right)\,,\\[-12pt]\\
G_{0,4}&=\frac{1}{6144\,u^2}\left(4 t_1^3 t_3-9 t_1^4-6 t_1^2 t_{-1}^2-9 t_{-1}^4+4 t_{-3} t_{-1}^3\right)\,,\\[-12pt]\\
U_{1,2}&=\frac{1}{2048\,u^2}\left(5\,t_1\,z^4+(t_3-18 t_1)\,z^2+(4 t_{-1}+27 t_1-6 t_3+t_5)\right.\\[-12pt]\\
&\hspace{.3in}\left.+(-t_{-5}+6 t_{-3}-27 t_{-1}-4 t_1)\,z^{-2}+(18 t_{-1}-t_{-3})\,z^{-4}-5\,t_{-1}\,z^{-6}\right)\,,\\[-12pt]\\
G_{1,2}&=\frac{1}{12288\,u^2}\left(6 t_1 t_5+t_3^2-36 t_1 t_3+81 t_1^2+24 t_{-1} t_1\right.\\[-12pt]\\
&\hspace{2in}\left.+81 t_{-1}^2-36 t_{-1} t_{-3}+t_{-3}^2+6 t_{-5} t_{-1}\right)\,.\end{aligned}$$
[**Acknowledgments.**]{} The main results of the paper, Theorems 4 and 5, were obtained under support of the Russian Science Foundation grant 14-21-00035. The work of MK was additionally supported by the President of Russian Federation grant NSh-5138.2014.1 and by the RFBR grant 13-01-00383. PZ acknowledges hospitality of the Center for Quantum Geometry of Moduli Spaces at Aarhus University. We thank JSC “Gazprom Neft" for funding short-term visits of MK to the Chebyshev Laboratory at SPbSU. We are grateful to L. Chekhov, B. Eynard, P. Norbury and G. Schaeffer for useful discussions, and to the anonymous referee for correcting a few typos and suggesting several improvements in the text.
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[^1]: An important observation of Grothendieck that, by Belyi’s theorem, the absolute Galois group ${\rm Gal}({\overline{\mathbb{Q}}}/{\mathbb{Q}})$ naturally acts on dessins, lies beyond the scope of this paper; we refer the reader to [@LZ] for details.
[^2]: Equivalently, we can put $N_{k,l}(\mu)=\sum_{{\Gamma}\in{\mathcal{D}}_{k,l,\mu}}\frac{1}{|{\rm Aut}_v {\Gamma}^*|}\;,$ where ${\rm Aut}_v {\Gamma}^*$ is the group of automorphisms of the dual graph ${\Gamma}^*$ preserving each vertex pointwise. A closely related problem of the weighted count of unlabeled dessins ${\Gamma}$ with weights $\frac{1}{|{\rm Aut}\,{\Gamma}|}$ is equivalent to the above one. If one treats $\mu$ as the unordered partition $[1^{m_1}2^{m_2}\ldots]$, where $m_j=\#\{\mu_i=j\}$, then the corresponding number of dessins of type $(k,l,\mu)$ is equal to $\frac{1}{|{\rm Aut}\,\mu|}\,N_{k,l}(\mu)$ with $|{\rm Aut}\,\mu|=m_1!m_2!\ldots\;.$
[^3]: While this paper was in preparation, similar results were independently obtained by matrix integration methods in [@AC] and generalized further in [@AMMN].
[^4]: Recently we came across the paper [@CC] where this result was proven along similar lines. However, the authors of [@CC] do not explicitly use Virasoro constraints that considerably simplify and clarify the proof.
[^5]: Formula (\[vt\]) is a “bicolored" analogue of Tutte’s recursion, cf. [@T], Eq. 2.1, for $g=0$ and [@WL], Eq. (6), for any $g\geq 0$ (a more general form of Tutte’s recursion one can find, e. g., in [@EO2]).
[^6]: \[M\] This is (a specialization of) Tutte’s recursion for arbitrary $g$, cf. [@EO2]. This formula, undeservedly forgotten, was recently reproduced in [@DMSS], Eq. (3.15). Note that the second term in the r.h.s. of , corresponding to a loop bounding a 1-gon, was inadvertently omitted there. This required some “modification” of the numbers $D_{g,m}(\mu_1,\ldots,\mu_m)$ in [@DMSS].
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abstract: |
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is called irreducible if it lies in no proper parabolic subgroup of $G$. In this paper we complete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various $G$-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of $G$, with one exception.
A result of Liebeck and Testerman shows that each irreducible connected subgroup $X$ of $G$ has only finitely many overgroups and hence the overgroups of $X$ form a lattice. We provide tables that give representatives of each conjugacy class of connected overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of $G$: for example, when the characteristic is $2$, there exists a maximal connected subgroup of $G$ containing a conjugate of every irreducible subgroup $A_1$ of $G$.
address: 'School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK, and Heilbronn Institute for Mathematical Research, Bristol, UK'
author:
- 'Adam R. Thomas'
bibliography:
- 'biblio.bib'
title: The Irreducible Subgroups of Exceptional Algebraic Groups
---
[^1]
[^1]: The author is indebted to Prof. M. Liebeck for his help in producing this paper. He would also like to thank Dr A. Litterick and Dr T. Burness for their comments on previous versions of this paper. Finally, the author would like to thank the anonymous referee for their careful reading of this paper and many insightful comments and corrections.
| ArXiv |
---
abstract: 'We propose the generalized uncertainty principle (GUP) with an additional term of quadratic momentum motivated by string theory and black hole physics as a quantum mechanical framework for the minimal length uncertainty at the Planck scale. We demonstrate that the GUP parameter, $\beta_0$, could be best constrained by the the gravitational waves observations; GW170817 event. Also, we suggest another proposal based on the modified dispersion relations (MDRs) in order to calculate the difference between the group velocity of gravitons and that of photons. We conclude that the upper bound reads $\beta_0 \simeq 10^{60}$. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant $\Lambda$ (apparently manifesting gravitational influences on the vacuum energy density), known as [*catastrophe of non-gravitating vacuum*]{}, we suggest a possible solution for this long-standing physical problem, $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$.'
author:
- Abdel Magied Diab
- Abdel Nasser Tawfik
bibliography:
- 'upperBoundofGUPParameterV1.bib'
title: A Possible Solution of the Cosmological Constant Problem based on Minimal Length Uncertainty and GW170817 and PLANCK Observations
---
@pre@post
Introduction {#intro}
============
The cosmological constant, $\Lambda$, an essential ingredient of the theory of general relativity (GR) [@Einstein1917As], was guided by the idea that the evolution of the Universe should be static [@Tawfik:2011mw; @Tawfik:2008cd]. This model was subsequently refuted and accordingly the $\Lambda$-term was abandoned from the Einstein field equation (EFE), especially after the confirmation of the celebrated Hubble obervations in 1929 [@Hubble:1929ig], which also have verified the consequences of Friedmann solutions for EFE with vanishing $\Lambda$ [@Friedman:1922kd]. Nearly immediate after publishing GR, a matter-free solution for EFE with finite $\Lambda$-term was obtained by de Sitter [@deSitter:1917zz]. Later on when it has been realised that the Einstein [*static*]{} Universe was found unstable for small perturbations [@Mulryne:2005ef; @Wu:2009ah; @delCampo:2011mq], it was argued that the inclusion of the $\Lambda$-term remarkably contributes to the stability and simultaniously supports the expansion of the Universe, especially that the initial singularity of Friedmann-Lem$\hat{\mbox{a}}$itre-Robertson-Walker (FLRW) models could be improved, as well [@Weinberg1972AA; @Misner1984B]. Furthermore, the observations of type-Ia high redshift supernovae in late ninteeth of the last century [@Riess:1998cb; @Perlmutter:1998np] indicated that the expanding Universe is also accelerating, especially at a small $\Lambda$-value, which obviously contributes to the cosmic negative pressure [@Garriga:1999bf; @Martel:1997vi]. With this regard, we recall that the cosmological constant can be related to the vacuum energy density, $\rho$, as $\Lambda=8\pi G \rho/c^2$, where $c$ is the speed of light in vacuum and $G$ is the gravitational constant. In 2018, the PLANCK observations have provided us with a precise estimation of $\Lambda$, namely $\Lambda_{\mbox{Planck}} \simeq 10^{-47}$GeV$^4/\hbar^3 c^3$ [@Aghanim:2018eyx]. When comparing this tiny value with the theoretical estimation based on quantum field theory in weakly- or non-gravitating vacuum, $\Lambda_{\mbox{QFT}} \simeq 10^{74}$GeV$^4/\hbar^3 c^3$, there is, at least, a $121$-orders-of-magnitude-difference to be fixed [@Adler:1995vd; @Weinberg:1988cp; @Zeldovich:1968ehl].
The disagreement between both values is one of the greatest mysteries in physics and known as the cosmological constant problem or [*catastrophe of non-gravitating vacuum*]{}. Here, we present an attempt to solve this problem. To this end, we utilize the generalized uncertainty principle (GUP), which is an extended version of Heisenberg uncertainty principle (HUP), where a correction term encompassing the gravitational impacts is added, and thus an alternative quantum gravity approach emerges [@Tawfik:2014zca; @Tawfik:2015rva]. To summarize, the present attempt is motivated by the similarity of GUP (including non-gravitating and gravitating impacts on HUP) and the disagreement between theoretical and observed estimations for $\Lambda$ (manifesting gravitational influences on the vacuum energy density) and by the remarkable impacts of $\Lambda$ on early and late evolution of the Universe [@Tawfik:2019jsa; @Tawfik:2011mw; @Tawfik:2008cd]. So far, there are various quantum gravity approaches presenting quantum descriptions for different physical phenomena in presence of gravitational fields to be achnowledged, here [@Tawfik:2014zca; @Tawfik:2015rva].
The GUP offers a quantum mechanical framework for a potential minimal length uncertainty in terms of the Planck scale [@Tawfik:2017syy; @Tawfik:2016uhs; @Dahab:2014tda; @Ali:2013ma]. The minimal length uncertainty, as proposed by GUP, exhibits some features of the UV/IR correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj], which has been performed in viewpoint of local quantum field theory. Thus, it is argued that the UV/IR correspondence is relevant to revealing several aspects of short-distance physics, such as, the cosmological constant problem [@Weinberg:1988cp; @Banks:2000fe; @Cohen:1998zx; @ArkaniHamed:2000eg]. Therefore, a precise estimation of the minimal length uncertainty strongly depends on the proposed upper bound of the GUP parameter, $\beta_0$ [@Dahab:2014tda; @Tawfik:2013uza].
Various ratings for the upper bound of $\beta_0$ have been proposed, for example, by comparing quantum gravity corrections to various quantum phenomena with electroweak [@Das:2008kaa; @Das:2009hs] and astronomical [@Scardigli:2014qka; @Feng:2016tyt] observations. Accordingly, $\beta_0$ ranges between $10^{33}$ to $10^{78}$ [@Scardigli:2014qka; @Feng:2016tyt; @Walker:2018muw]. As a preamble of the present study, we present a novel estimation for $\beta_0$ from the binary neutron stars merger, the gravitational wave event GW170817 reported by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the Advanced Virgo collaborations [@TheLIGOScientific:2017qsa]. With this regard, there are different efforts based on the features of the UV/IR correspondence in order to interpret the $\Lambda$ problem [@Chang:2001bm; @Chang:2011jj; @Miao:2013wua; @Shababi:2017zrt; @Vagenas:2019wzd] with Liouville theorem in the classical limit [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct]. Having a novel estimation of $\beta_0$, a solution of the $\Lambda$ problem, [*catastrophe of non-gravitating vacuum*]{}, could be best proposed.
The present paper is organized as follows. Section \[MDRGUP\] reviews the basic concepts of the GUP approach with quadratic momentum. The associated modifications of the energy-momentum dispersion relations related to GR and rainbow gravity are also outlined in this section. In section \[GUPparameter\], we show that the dimensionless GUP parameter, $\beta_o$, could be, for instance, constrained to the gravitational wave event GW170817. Section \[LamdaProblem\] is devoted to calculating the vacuum energy density of states and shows how this contributes to understanding the cosmological constant problem with an quantum gravity approach, the GUP. The final conclusions are outlined in section \[conclusion\].
Generalized Uncertainty Principle and Modified Dispersion Relations \[MDRGUP\]
==============================================================================
Several approaches to the quantum gravity, such as GUP, predict a minimal length uncertainties that could be related to the Planck scale [@Tawfik:2015rva; @Tawfik:2014zca]. There were various laboratory experiments conducted to examine the GUP effects [@Bawaj:2014cda; @Marin:2013pga; @Pikovski:2011zk; @Khodadi:2018kqp]. In this section, we focus the discussion on GUP with a quadratic momentum uncertainty [@Tawfik:2015rva; @Tawfik:2014zca]. This version of GUP was obtained from black hole physics [@Gross:1987kza] and supported by [*gedanken*]{} experiments [@Maggiore:1993zu], which have been proposed Kempf, Mangano, and Mann (KMM), [@Kempf:1994su] x p , \[GUPuncertainty\] where $\Delta x$ and $\Delta p$ are the uncertainties in position and momentum, respectively. The GUP parameter can be exressed as $\beta = \beta_0 (\ell_p/\hbar)^2 = \beta_0/ (M_p c)^2$, where $\beta_0$ is a dimensionless parameter, $\ell_p=1.977 \times 10^{-16}~$GeV$^{-1}$ is the Planck length, and $M_p= 1.22 \times 10^{19}~$GeV$/c^2$ is the Planck mass. Equation (\[GUPuncertainty\]) implies the existence of a minimum length uncertainty, which is related to the Planck scale, $\Delta x_{\mbox{min}} \approx \hbar \sqrt{\beta} =\ell_p \sqrt{\beta_0}$. It should be noticed that the minimum length uncertainty exhibits features of the UV/IR correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj]. $\Delta x$ is obviously proportional to $\Delta p$, where large $\Delta p$ (UV) becomes proportional to large $\Delta x$ (IR). Equation (\[GUPuncertainty\]) is a noncommutative relation; $[\hat{x}_i,\; \hat{p}_j] = \delta_{ij} i \hbar [1+\beta p^2]$, where both position and momentum operators can be defined as \_i = \_[0i]{}, \_j= \_[0j]{} (1+p\^2), where $\hat{x}_{0i}$ and $\hat{p}_{0j}$ are corresponding operators obtained from the canonical commutation relations $[\hat{x}_{0i},\; \hat{p}_{0j}]=\delta_{ij} i \hbar,$ and $p^2= g_{ij} p^{0i} \; p^{0j}$.
We can now construct the modified dispersion relation (MDR) due to quadratic GUP. We start with the background metric in GR gravitational spacetime ds\^2 =g\_ dx\^ dx\^= g\_[00]{} c\^2 dt\^2 + g\_[ij]{} dx\^i dx\^j, with $g_{\mu \nu}$ is the Minkowski spacetime metric tensor $(-,+,+,+)$. Accordingly, the modified four-momentum squared is given by p\_p\^= g\_ p\^p\^&=& g\_[00]{} (p\^0)\^2 + g\_[ij]{} p\^[0i]{} p\^[0j]{} (1+p\^2)\
&=& -(p\^0)\^2 + p\^2 + 2 p\^2 p\^2. \[modifyMomentum\] Comparing this with the conventional dispersion relation, $p_\mu p^\mu = - m^2c^2$, the time component of the momentum can then be written as (p\^0)\^2 &=& m\^2c\^2 + p\^2 (1+p\^2). The energy of the particle $\omega$ can be defined as $\omega/c = - \zeta_\mu p^\mu = - g_{\mu \nu} \zeta^\mu p^\nu$, where the killing vector is given as $\zeta^\mu = (1,0,0,0)$. Therefore, the energy of the particle could be expressed as $\omega=-g_{00} c (p^0)=c (p^0)$ and the modified dispersion relation in GR gravity reads \^2 = m\^2 c\^4 + p\^2 c\^2 (1+ 2p\^2). \[MDRrel\] For $\beta \rightarrow 0$, the standard dispersion can be obtained.
The rainbow gravity generalizes the MDR in doubly special relativity to curved spacetime [@magueijo2004gravity], where the geometry spacetime is explored by a test particle with energy $\omega$ [@Magueijo:2001cr; @Magueijo:2002am], \^2 f\_1 ()\^2 - (pc)\^2 f\_2 ()\^2 = (mc\^2)\^2, where $\omega_p$ is the Planck energy and $f_1 (\omega/\omega_p)$ and $f_2 (\omega/\omega_p)$ are known as the rainbow functions which are model-depending. The rainbow functions can be defined as [@AmelinoCamelia:1996pj; @AmelinoCamelia:1997gz], f\_1 (/\_p) &=& 1, f\_2 (/\_p) = , \[Rainbowfuncs\] where $\eta$ and $n$ are free positive parameters. It was argued that for the logarithmic corrections of black hole entropy [@Tawfik:2015kga], the integer $n$ is limited as $n=1,2$ [@Gangopadhyay:2016rpl]. Therefore, it would be eligible to assume that $n=2$. Thus, the MDR for rainbow gravity with GUP can be written as, \^2 = . \[MDRrain\] Again, as $\beta \rightarrow 0$, Eq. (\[MDRrain\]) goes back to the standard dispersion relation.
We have constructed two different MDRs for quadratic GUP, namely Eqs (\[MDRrel\]) and (\[MDRrain\]) in GR and rainbow gravity, respectively. Bounds on GUP parameter from GW170817 shall be outlined in the section that follows.
Bounds on GUP parameter from GW170817 {#GUPparameter}
=====================================
Instead of violating Lorentz invariance [@Tawfik:2012hz], we intend to investigate the speed of the graviton from the GW170817 event. To this end, we use MDRs obtained from the quadratic GUP approaches, section \[MDRGUP\]. Thus, defining an upper bound on the dimensionless GUP parameter $\beta_0$ for given bounds on mass and energy of the graviton, where $m_g\lesssim 4.4 \times 10^{-22}~$eV$/c^2$ and $\omega = 8.5 \times 10^{-13}~$eV, respectively, plays an essential role. Assuming that the gravitational waves propagate as free waves, we could, therefore, determine the speed of the mediator, that of the graviton, from the group velocity of the accompanying wavefront, i.e. $v_g = \partial \omega/\partial p$, where $\omega$ and $p$ are the energy and momentum of the graviton, respectively [@Mirshekari:2011yq]. The idea is that the group velocity of the graviton can be simply deduced from the MDRs, Eqs. (\[MDRrel\]) for the GR gravity and (\[MDRrain\]) and the rainbow gravity, in presence and then in absence of the GUP impacts, which have been discussed in section \[MDRGUP\]. Accordingly, Eq. (\[MDRrel\]) implies that the group velocity reads v\_g = = (1+ 4 p\^2). \[vgMDR1\]
The unmodified momentum $p$ in terms of the modified parameters up to $\mathcal{O} (\beta)$, can be expressed as $p=a+b \beta$, where $a$ and $b$ are arbitrary parameters. By substituting this expression into Eq. (\[MDRrel\]), we find that $p^2= (\omega_g/c)^2 - m^2 c^2 $. Thus, Eq. (\[vgMDR1\]) can be rewritten as v\_g = c { \^[1/2]{} +4 \^[3/2]{} }, where $\omega_g$ is the energy of the graviton. It is obvious that for $\beta \rightarrow 0 $, i.e. in absence of GUP impacts, the group velocity reads v\_g = c . Then, the difference between the speed of photon (light) and that of graviton without GUP impacts is given as | v| = | c-v\_g| = c | ( )\^2 | 1.34 10\^[-19]{}c. \[vDr\] Although the small difference obtained, we are - in the gravitational waves epoch - technically able to measure even a such tiny difference! In light of this, we could use the results associated with the GW170817 event, such as the graviton velocity, in order to set an upper bound on the GUP parameter, $\beta_0$.
For a massless graviton, the difference between the speed of photons (light) and that of the gravitons in presence of the GUP impacts reads |v\_| &=&| 4| = | 4\_0 ()\^2| 1.95 10 \^[-80]{} \_0 c. \[vMDR\] Thus, the upper bound on the dimensionless parameter, $\beta_0$, of the quadratic GUP can be simply deduced from Eqs. (\[vDr\]) and (\[vMDR\]), \_0 8.89 10\^[60]{}. \[MDRbeta1\]
The group velocity of the graviton due to MDR and rainbow gravity when applying the quadratic GUP approach, Eq. (\[MDRrain\]), can be expressed as v\_g = = () . Similarly, one can for a massless graviton express the conventional momentum in terms of the GUP parameter. In order of $\mathcal{O}(\beta)$, we get c p = \_g . \[pcRainbow\] The unmodified momentum can be expressed in GUP-terms up to $\mathcal{O}(\beta)$; $p=a_0 + a_1 \beta$, where $a_0$ and $a_1$ are arbitrary parameters. Nevertheless, the investigation of the speed of the graviton from the GW150914 observations [@Abbott:2016blz] specifies the rainbow gravity parameter, $ \eta (\omega_g/\omega_p)^2\leq 3.3\times 10^{-21}$ [@Gwak:2016wmg]. Accordingly, Eq. (\[pcRainbow\]) can be reduced to $c p=\omega_g (1- \beta \omega_g^2/c^2)$ and the group velocity of the massless graviton becomes v\_g = c .
Then, the difference between the speed of photons and that of the gravitons reads |v\_| &=&| 5 | 2.43 10\^[-80]{} \_0 c. \[vgRainbow\] By comparing Eqs. (\[vgRainbow\]) and (\[vDr\]), the upper bound of the GUP parameter $\beta_0$ can be estimated as \_0 5.5 10\^[60]{}. \[Rainbowbeta1\]
It is obvious that both results, Eqs. (\[MDRbeta1\]) and (\[Rainbowbeta1\]), are very close to each other; $\beta_0 \lesssim 10^{60}$. The improved upper bound of $\beta_0$ is very similar to the ones reported in refs. [@Scardigli:2014qka; @Feng:2016tyt], which - as well - are depending on astronomical observations. The present results are based on mergers of spinning neutron stars. Thus, it is believed that more accurate observations, the more precise shall be $\beta_0$.
Having set a upper bound on the GUP parameter and counting on the spoken similarities between GUP and the catastrophe of non-gravitating vacuum, we can now propose a possible solution of the cosmological constant problem.
A Possible Solution of the Cosmological Constant Problem {#LamdaProblem}
========================================================
The cosmological constant can be given as $\Lambda = 3 H_0^2 \Omega_\Lambda$, where $H_0$ and $\Omega_\Lambda$ are the Hubble parameter and the dark energy density, respectively [@Carroll:2000fy]. On the other hand, the origin of the catastrophe of non-gravitating vacuum would be understood from the disproportion of the value of $\Lambda$ in the theoretical calculations, while this is apparently impacting the GW observations [@Sahni:2002kh]. From the most updated PLANCK observations, the values of $\Omega_\Lambda = 0.6889 \pm 0.0056$ and $H_0 = 67.66 \pm 0.42~$Km $\cdot$ s$^{-1}$ $\cdot$ Mpc$^{-1}$ [@Aghanim:2018eyx]. Then, the vacuum energy density &=& () \_= \_, \[VacuEnergy\] where the scale of the visible light, $\ell_0= c/H_0 \simeq 1.368 \times 10^{23}~$Km [@Aghanim:2018eyx]. Therefore, one can use Eq. (\[VacuEnergy\]) to esiamte the vacuum energy density in order of $10^{-47}~$GeV$^4/(\hbar^3c^3)$. In quantum field theory, the cosmological constant is to be calculated from sum over the vacuum fluctuation energies corresponding to all particle momentum states [@Carroll:2000fy]. For a massless particle, we obtain d\^3 (\_p /2) 9.6010\^[74]{} \^4/ (\^3 c\^3). \[QFTlamda\] This is clearly infinite integral. But, it is usually cut off, at the Planck scale, $\mu_p = \hbar/\ell_p$. We assume $\omega_p$ is the vacuum energy of quantum harmonic state $\hbar \omega_p = [p^2c^2+m_g^2c^4]^{1/2}$.
To propose a possible solution of the cosmological constant problem, it is initially needed to determine the number of states in the phase space volume taking into account GUP, Eq. (\[GUPuncertainty\]). An analogy can be found in Liouville theorem in the classical limit. We need to make sure that the size of each quantum mechanical state in phase space volume is depending on the modified momentum $p$, especially when taking GUP into consideration, Eq. (\[GUPuncertainty\]). In other words, the number of quantum states in the phase space volume is assumed not depending on time.
In the classical limit, the relation of the quantum commutation relations and the Poisson brackets is given as $[\hat{A}, \hat{B}] = i\hbar \{A, B\}$. Details on the Poisson bracket in D-Dimensions are outlined in appendix \[LiouvilleTheorem\]. Consequently, the modified density of states implies different implications on quantum field theory, such as, the cosmological constant problem.
In D-dimensional spherical coordinate systems, the density of states in momentum space is given as [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct] , where $V$ is the volume of space. It should be noticed that in quantum mechanics, the number of quantum stated per unit volume is given as $V/(2\pi \hbar)^D$. Therefore, for Liouville theorem, the weight factor in 3-D dimension reads [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct] (review appendix \[LiouvilleTheorem\]) . \[densityStates\] In quantum field theory, the modification in the quantum number of state of the phase space volume should have consequences on different quantum phenomena, such as, the cosmological constant problem and the black body radiation. At finite weight factor of GUP, the sum over all momentum states per unit volume of the phase space modifies the vacuum energy density. The cosmological constant, on the other hand, is determined by summing over the vacuum fluctuations, the energies, corresponding to a particular momentum state \_ (m) &=& d\^3 (p\^2) (\_p /2) = For a massless particle, the vacuum energy density, which is directly related to $\Lambda$, reads \_(m=0) &=& dp = = 1.78 10\^[-48]{} \^4/(\^3 c\^3). \[GUPLamda\] The agreement between the observed value of the cosmological constant, $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$, and our calculations based on quantum gravity approach, Eq. (\[GUPLamda\]), is very convincing. We conclude that the connection between the estimated upper bound on $\beta_0$, Eqs. (\[vgRainbow\]) and (\[vDr\]), from GW170817 event [@TheLIGOScientific:2017qsa] and the most updated observations of the PLANCK collaboration [@Aghanim:2018eyx] for the cosmological constant $\Lambda$, Eq. (\[QFTlamda\]), and our estimated value of $\Lambda(m=0)$, Eq. (\[GUPLamda\]), gives an interpretation for the cosmological constant problem in presence of the minimal length uncertainty.
Conclusions \[conclusion\]
===========================
In the present study, we have proposed the generalized uncertainty principle (GUP) with an addition term of quadratic momentum, from which we have driven the modified dispersion relations for GR and rainbow gravity, Eq. (\[MDRrel\]) and Eq. (\[MDRrain\]), respectively. Counting on the similarities between GUP (manifesting gravitational impacts on HUP) and the likely origin of the great discrepancy between the theoretical and observed values of the cosmological constant that in the gravitational impacts on the vacuum energy density, the present study suggests a possible solution for the long-standing cosmological constant problem ([*catastrophe of non-gravitating vacuum*]{}) that $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$.
We have assumed that the gravitational waves propagate as a free wave. Therefore, we could drive the group velocity in terms of the GUP parameter $\beta_0$ for GR and rainbow gravity, Eq. (\[MDRbeta1\]) and Eq. (\[Rainbowbeta1\]), respectively. Moreover, we have used recent results on gravitational waves, the binary neutron stars merger, GW170817 event, in order to determine the speed of the gravitons. Then, we have calculated the difference between the speed of gravitons and that of (photons) light, at finite and visnishing GUP parameter. We have shown that the upper bound on the dimensionless GUP parameter, $\beta \sim 10^{60}$, is merely constrained by such a speed difference. We have concluded that the speed of graviton is directly related to the GUP approach utilized in.
The cosmological constant problem, which is stemming from the large discrepancy between the QFT-based calculations and the cosmological observations, is tagged as $\Lambda_{QFT}/\Lambda_{exp} \sim 10^{121}$. This quite large ratio can be interpreted by features of the UV/IR correspondence and the impacts of gravity. For the earlier, the large $\Delta x$ (IR) corresponds to a large $\Delta p$ (UV) in scale of Planck momentum. For the later, the GUP approach, for instance, Eq. (\[GUPuncertainty\]), plays an essential role. We have assumed that in calculating the density of states where GUP approach is taken into account, a possible solution of the cosmological constant problem, Eq. (\[densityStates\]), can be proposed. At Planck scale, the resulting density of the states seems to impact the vacuum energy density of each quantum state, Eq. (\[GUPLamda\]). A refined value of the cosmological constant we have obtained for a novel upper bound on $\beta_0$, which - in turn - was determined from the GW170817 observations. Finally, the possible matching between the estimation of the upper bound on the GUP parameter deduced from the gravitational waves, GW170817 event, and the one estimated from the PLANCK 2018 observations seems to support the conclusion about the great importance of constructing a theory for quantum gravity. This likely helps in explaining various still-mysterious phenomena in physics.
Algebra of quantum mechanical commutators and Poisson brackets \[LiouvilleTheorem\]
===================================================================================
For a binary set of anticommutative functions on position and momentum, for instance, in D-dimensions, the Poisson bracket expresses their binary operation { F(x\_1, x\_D; p\_1, p\_D ), G(x\_1, x\_D; p\_1, p\_D )} &=&\
( - ) { x\_i, p\_j } &+& { x\_i, x\_j }. During a time duration, $\delta t$, the Hamilton’s equations of motion for position and momentum can be given as x\_i\^= x\_i + x\_i, p\_i\^= p\_i + p\_i, where, x\_i, &=& { x\_i, H} t = { x\_i, p\_j } + { x\_i, x\_j} ,\
p\_i, &=& { p\_i, H} t = - { x\_i, p\_j } , where $H\equiv H(x,p;t)$ is the Hamiltonian, itself.
The estimation of the change in the phase space volume during the time evolution requires to determine the Jacobain of the transformation from $(x_1, \cdots x_D;\; p_1, \cdots p_D)$ to $(x_1^\prime, \cdots x_D^\prime;\; p_1^\prime, \cdots p_D^\prime)$, i.e. d\^Dx\^ d\^D p\^= , where $\mathcal{J}$ is the Jacobain of the transformation, which can be expressed as &=& = 1 + ( + ) t. The general notations of position and momentum brackets lead to following algebraic relations {x\_i p\_i} = f\_[ij]{} (x, p), {x\_i, x\_j} = g\_[ij]{}(x,p), {p\_i,p\_j}= h\_[ij]{}(p). Thus, the Jacobain of the transformation is given as [@Fityo:2008zz] = \_[i=1]{}\^D f\_[ii]{}(x,p) = 1+ \_[i=1]{}\^D (f\_[ii]{} (x,p) - 1). \[jacobian\] Therefore the invariant phase space in D-dimension reads . Finally, the quantum density of states can be determined from .
| ArXiv |
---
abstract: 'The influence of a thermodynamic constraint on the critical finite-size scaling behavior of three-dimensional Ising and XY models is analyzed by Monte-Carlo simulations. Within the Ising universality class constraints lead to Fisher renormalized critical exponents, which modify the asymptotic form of the scaling arguments of the universal finite-size scaling functions. Within the XY universality class constraints lead to very slowly decaying corrections inside the scaling arguments, which are governed by the specific heat exponent $\alpha$. If the modification of the scaling arguments is properly taken into account in the scaling analysis of the data, finite-size scaling functions are obtained, which are [*independent*]{} of the constraint as anticipated by analytic theory.'
author:
- Michael Krech
title: 'Critical finite-size scaling with constraints:Fisher renormalization revisited'
---
Introduction
============
The theoretical investigation of classical spin systems has played a key role in the understanding of phase transitions, critical behavior, scaling, and universality [@Amit78; @Parisi88]. In particular, the classical Ising, the XY, and the Heisenberg model are the most relevant spin models in three dimensions. Each of these simple models represents a universality class which, apart from the spatial dimensionality and the range of the interactions, is characterized by the number of components of the order parameter , e.g, the magnetization in the case of ferromagnetic models. Real systems, however, suffer from various kinds of imperfections, e.g., lattice defects, impurities, or vacancies. In an experiment, which is designed to probe critical behavior as a function of temperature, the presence of, say, impurities on the lattice constitutes a thermodynamic constraint, because in a given sample the impurity concentration will remain constant during the temperature scans. According to the concepts of thermodynamics the impurity concentration $n_i$ can be written as the derivative of the grand canonical potential with respect to the chemical potential $\mu_i$ of the impurities, where other parameters like the temperature and the volume of the system are kept fixed. Now the question arises how the critical singularities in the grand canonical potential are affected when the thermodynamic ensemble is changed from ’fixed $\mu_i$’ to ’fixed $n_i$’, where the location of the critical temperature $T_c$ depends on the particular values of $\mu_i$ or $n_i$, respectively. The answer to this question has been given a long time ago by Michael Fisher [@Fisher68]. Provided, that the critical singularites have their usual form in the ’fixed $\mu_i$’ ensemble, then the constraint $n_i = const.$ amounts to a reparameterization of the reduced temperature $t = (T-T_c(n_i))/T_c(n_i)$ of the [*constrained*]{} system in terms of the reduced temperature $\tau = (T-T_c(\mu_i))/T_c(\mu_i)$ of the [*unconstrained*]{} system according to [@Fisher68] $$\label{ttau}
t = a \tau + b \tau |\tau|^{-\alpha} + \dots ,$$ where $a$ and $b$ are nonuniversal constants and the dots indicate higher order contributions. Apart from a linear term (\[ttau\]) contains a singular contribution which is characterized by the critical exponent $1 - \alpha$ of the entropy density . Which of the two terms in (\[ttau\]) is the leading one for $t, \tau \to 0$ depends on the sign of $\alpha$. Within the Ising universality class in $d = 3$ dimensions $\alpha \simeq 0.109$ [@LGZJ85] so that $|\tau| \sim |t|^{1/(1-\alpha)}$ to leading order and therefore the critical exponents $\beta$ (order parameter), $\gamma$ (susceptibility), and $\nu$ (correlation length) of the unconstrained system undergo ’Fisher renormalization’ in the constrained system according to [@Fisher68] $$\label{fren}
\beta \to \beta' = \beta / (1 - \alpha), \quad
\gamma \to \gamma' = \gamma / (1 - \alpha), \quad
\nu \to \nu' = \nu / (1 - \alpha) .$$ The specific heat exponent $\alpha$ requires a more careful analysis, because the specific heat is the temperature derivative of the entropy which in addition to the ’renormalization’ displayed in (\[fren\]) causes a sign change $$\label{frena}
\alpha \to \alpha' = -\alpha / (1 - \alpha) .$$ Note that analytic background contributions to the entropy of the unconstrained system become [*singular*]{} in the constrained system due to the singularity in the reparameterization given by (\[ttau\]).
Within the XY universality class in $d = 3$ the exponent $\alpha$ is negative [@LGZJ85], where probably the best current estimate $\alpha \simeq -0.013$ is obtained from an experiment on $^4$He near the superfluid transition [@LSNCI96]. For negative $\alpha$ the linear term on the r.h.s. of (\[ttau\]) is the dominating one for $\tau \to 0$. However, the XY universality class $\alpha$ is so small, that in practice the singular term in (\[ttau\]) can never be neglected. Instead, the singular contribution to (\[ttau\]) gives rise to very slowly decaying correction terms which must not be confused with Wegner corrections to scaling . These correction terms have to be considered in any scaling analysis in order to obtain correct values for the critical exponents.
If the system is finite, which is neccessarily the case for any Monte - Carlo simulation, all critical singularities are rounded, i.e., all quantities are analytic functions of the thermodynamic parameters [@Fisher71] so that a thermal singularity as shown in (\[ttau\]) does not occur. Critical finite-size rounding effects in, e.g., a cubic box $L^d$ are captured by [*universal*]{} finite-size scaling functions [@Fisher71; @Barber83] which restore all critical singularities in the limit $L \to \infty$. Following the line of argument in [@Fisher68], (\[ttau\]) then has to be replaced by $$\label{ttaufL}
t = a \tau + \tau |\tau|^{-\alpha} f(\tau L^{1/\nu}) ,$$ where $f(x)$ is the finite-size scaling function of the entropy density and $x = \tau L^{1/\nu}$ is a convenient choice of its scaling argument. For $\tau \to 0$ at finite $L$ the singular prefactor of $f(x)$ in (\[ttaufL\]) must be cancelled so that one has $f(x) = A |x|^{\alpha} +
\dots$ in the limit $x \to 0$, where $A$ is a nonuniversal constant such that $f(x)/A$ is a [*universal*]{} function of its argument. To leading order in $\tau$ the reparameterization of the reduced temperature $t$ of the constrained system is therefore [*linear*]{} in the reduced temperature $\tau$ of the unconstrained system and one finds $$\label{ttauL}
t = \tau (a + A L^{\alpha/\nu}).$$ According to (\[ttauL\]) the finite-size scaling argument $x$ in the constrained system is given by $$\label{x}
x = \tau L^{1/\nu} = t L^{1/\nu} / (a + A L^{\alpha/\nu}),$$ where the [*shape*]{} of the finite-size scaling functions is maintained [@VD], i.e., the presence of the constraint [*only*]{} affects the form of the scaling argument $x$. For $\alpha > 0$ (\[x\]) asymptotically reduces to $x = t L^{1/\nu'}/A$ for large $L$ in accordance with Fisher renormalization (see (\[fren\])). For $\alpha < 0$ (\[x\]) captures the aforementioned slowly decaying corrections to the asymptotic critical behavior in the XY universality class when a thermodynamic constraint is present. Note that $A > 0$ for the Ising universality class and that $A < 0$ for the XY universality class.
In the remainder of this paper a simple spin model is introduced which can be efficiently simulated with existing Monte - Carlo algorithms both with and without constraints in three dimensions. For the Ising and the XY version of the model finite-size scaling according to (\[x\]) is tested for the modulus of the order parameter, the susceptibility, and the specific heat.
Model and simulation method
===========================
The model system which is investigated here can be described as an $O(N-1)$ symmetric classical ’planar’ ferromagnet in a transverse magnetic field. The model Hamiltonian reads $$\label{H}
{\cal H} = -J \sum_{\langle i j \rangle}
\sum_{x=1}^{N-1} S_i^x S_j^x - h \sum_i S_i^N,$$ where $\langle i j \rangle$ denotes a nearest neighbor pair of spins on a simple cubic lattice in $d = 3$ dimensions. The lattice contains $L$ lattice sites in each direction and in order to avoid surface effects periodic boundary conditions are applied. Each spin ${\bf S}_i$ is a classical spin with $N$ components $\vec{S}_i = \left(S_i^1,S_i^2,
\dots,S_i^N\right)$ with the normalization $|\vec{S}_i| = 1$ for each lattice site $i$. The magnetic field $h$ in (\[H\]) only acts on the $N$-components of the spins which are not coupled by the exchange interaction $J$. From the symmetry of the Hamiltonian it is obvious, that the model belongs to the $O(N-1)$ universality class in $d = 3$, where nonuniversal quantities like the critical temperature $T_c = T_c(h)$ depend on the strength of the transverse field $h$. Note that $T_c(h)$ is symmetric around $h = 0$ and decreases with increasing $h$, because the spins become more and more aligned with the $N$-direction as $\pm h$ is increased and due to the normalization condition the typical interaction energy between pairs of spins is decreased.
The Hamiltonian given by (\[H\]) defines the unconstrained model. The constraint is imposed on the transverse magnetization $M$ in the form $$\label{Mconst}
M \equiv \sum_i S_i^N = const.,$$ where the Hamiltonian of the constrained model is given by $$\label{Hconst}
{\cal H}_M = -J \sum_{\langle i j \rangle}
\sum_{x=1}^{N-1} S_i^x S_j^x = {\cal H} + hM .$$ The critical temperature of the constrained model is a symmetric and monotonically decreasing function of the prescribed transverse magnetization $M$. The transverse field $h$ here plays the part of the chemical potential $\mu_i$ of impurities (see Sect. 1) and the transverse magnetization $M$ accordingly plays the part of the impurity concentration $n_i$. It is also possible to implement $O(N)$ spin models with impurities or vacancies with diffusion in order to mimic the situation discussed in [@Fisher68]. However, the fact that the Hamiltonians given by (\[H\]) and (\[Hconst\]) only require a single ’species’ with a single coupling constant leads to some simplifications in the algorithms. Note that the symmetric constrained model $(M = 0)$ becomes equivalent to the symmetric unconstrained model $(h = 0)$ for sufficiently large lattices. In particular, both versions of the symmetric model have the same $T_c$.
The Monte-Carlo algorithm is chosen as a hybrid scheme, where each hybrid Monte-Carlo step consists of 10 updates each of which can be one of the following: one Metropolis sweep of the whole lattice, one single cluster Wolff update [@Wolff89], or one overrelaxation update of the whole lattice [@CL94], where the latter can only be applied for $N \geq 3$. The Metropolis algorithm updates the lattice sequentially and works in the standard way for the unconstrained model. For the constrained model the constraint $M = const.$ is observed locally by applying a Kawasaki update dynamics for the $N^{th}$ components of the spins. For each lattice site $i$ a nearest neighbor site $j$ is chosen randomly and a random amount of the $N^{th}$ spin component is proposed for exchange such that $S_i^N + S_j^N$ remains constant. Then new spin components $(S_i^1,S_i^2,\dots ,S_i^{N-1})$ are proposed and the spin components $(S_j^1,S_j^2,\dots ,S_j^{N-1})$ are adjusted according to the spin normalization condition $|\vec{S}_i| =
|\vec{S}_j| = 1$. The local change $\Delta E$ of the configurational energy is calculated according to (\[Hconst\]). According to detailed balance the proposed update is accepted with probability $p(\beta \Delta E)$, where $\beta = 1/(k_B T)$. For our simulation we have chosen $p(x) = 1 / (\exp(x)
+ 1)$. Note that all updates must be proposed such that the new spin at lattice site $i$ is taken from the uniform distribution on the unit sphere in $N$ dimensions.
The Wolff algorithm also works the standard way [@Wolff89], except that [*only*]{} the first $N-1$ components of the spins are used for the cluster growth, i.e., (\[H\]) and (\[Hconst\]) are treated as planar ferromagnets. This means that a cluster update never changes the $N^{th}$ component of any spin so that the Wolff algorithm is nonergodic in this case. The cluster update is still a valid Monte-Carlo step in the sense that it fulfills detailed balance, however, in order to provide a valid Monte-Carlo algorithm it has to be used together with the Metropolis algorithm described above in a hybrid fashion. The use of Wolff updates allows us to take advantage of improved estimators [@Has90] for magnetic quantities.
The overrelaxation part of the algorithm performs a microcanonical update of the configuration in the following way. The local configurational energy has the functional form of a scalar product of the spins, where according to (\[H\]) and (\[Hconst\]) only the first $N-1$ components are involved. With respect to the sum of its nearest neighbor spins each spin has a transverse component in the $(S_i^1,S_i^2,\dots ,S_i^{N-1})$ plane which does not enter the scalar product. The overrelaxation algorithm scans the lattice sequentially, determines this transverse component for each lattice site and flips its sign. This overrelaxation algorithm is similar to the one used in [@CL94] and it quite efficiently decorrelates subsequent configurations over a wider range of temperatures around the critical point than the Wolff algorithm. However, overrelaxation can only be applied for $N \geq 3$. In the following only the cases $N = 2$ (transverse Ising) and $N = 3$ (transverse XY) are considered.
In a typical hybrid Monte-Carlo step we use three Metropolis [*(M)*]{}, seven single cluster Wolff [*(C)*]{} updates for $N = 2$ and three Metropolis, five single cluster Wolff, and two overrelaxation updates [*(O)*]{} for $N = 3$ in the critical region of the models. The inividual updates are mixed automatically in the program so that the update sequences [*(M C C M C C M C C C)*]{} for $N = 2$ and [*(M C C M O C M C C O)*]{} for $N = 3$ are generated as one hybrid Monte-Carlo step. The shift register generator R1279 given by the recursion relation $X_n = X_{n-p} \oplus
X_{n-q}$ for $(p,q) = (1279,1063)$ is used as the random number generator. Generators like this are known to cause systematic errors in combination with the Wolff algorithm [@cluerr]. However, for lags $(p,q)$ as large as the ones used here these errors will be far smaller than typical statistical errors. They are further reduced by the hybrid nature of our algorithm due to the presence of several Metropolis updates in one hybrid Monte-Carlo step [@AMFDPL].
The hybrid Monte-Carlo scheme described above is employed for lattice sizes $L$ between $L = 20$ and $L = 80$. For each system size and temperature we perform at least 10 blocks of $10^3$ hybrid steps for equilibration followed by $10^4$ hybrid steps for measurements. Each measurement block yields an estimate for all static quantities of interest and from these we obtain our final estimates and estimates of their statistical error following standard procedures. At the critical point (see below) two or three times as many updates have been performed. The integrated autocorrelation time of the hybrid algorithm is determined by the autocorrelation function of the energy or, equivalently, the modulus of the order parameter, which yield the slowest modes for the Wolff algorithm. The autocorrelation times are generally rather short, at the critical point they range from about 5 hybrid Monte-Carlo steps for $L = 20$ to about 10 hybrid Monte-Carlo steps for $L = 80$. The values for the equilibration and measurement periods given above thus translate to roughly 100 and 1000 autocorrelation times, respectively. In order to obtain the best statistics for magnetic quantities a measurement is made after every hybrid Monte-Carlo step. All error bars quoted in the following correspond to one standard deviation. The simulations have been performed on the DEC alpha AXP workstation cluster at the Physics Department and on HP RISC8000 workstations at the Computer Center of the RWTH Aachen.
Ising universality class
========================
For $N = 2$ (\[H\]) and (\[Hconst\]) describe a classical Ising model in a (fixed) transverse field or with fixed transverse magnetization, respectively. In the following we will only consider the constrained model with the symmetric constraint $M = 0$ and with the constraint $m \equiv M/L^3
= 1/\sqrt{2}$. The symmetrically constrained model does not show Fisher renormalization [@Fisher68] and we therefore use this case for tests of the algorithm and for the production of data representative of the Ising universality class in $d = 3$. The constraint $m = const. \neq 0$ breaks the $S_i^N \to -S_i^N$ symmetry of the model and Fisher renormatization should become visible within a certain temperature window around $T_c =
T_c(m)$. The width of this window is of course a nonuniversal property of the model and in particular one expects this window to widen as $m$ is increased. Due to the spin normalization condition $m$ cannot exceed unity and one therefore also expects, that critical behavior becomes very difficult to resolve numerically if $m$ is too close to its maximum value. Therefore, $m = 1/\sqrt{2}$ is chosen as a compromise between good resolution in the critical regime and a prominent Fisher renormalization effect.
The critical temperatures $T_c(m=0)$ and $T_c(m=1/\sqrt{2})$ are determined from temperature scans of the Binder cumulant ratio according to standard procedures [@CFL93]. We obtain the following reduced critical coupling constants $K_c(m) \equiv J / k_B T_c(m)$: $$\label{KcI}
K_c(0) = 0.41638 \pm 0.00005 \quad \mbox{and} \quad
K_c(1/\sqrt{2}) = 0.6371 \pm 0.0001 .$$ The corresponding estimates for the Binder cumulant ratio obtained for $m = 0$ and $m = 1/\sqrt{2}$ agree with previous estimates obtained for the Ising universality class within two standard deviations [@BLH95], where for the latter choice of $m$ Wegner corrections to scaling are considerable and must be subtracted in order to obtain a reliable estimate. In order to obtain an estimate for the exponent $\nu$ which enters the finite-size scaling argument according to (\[x\]) the cumulant $$\label{X}
X \equiv {\partial \over \partial T} \ln \langle \phi^2 \rangle
= {1 \over k_B T^2} \left(
{\langle \phi^2 {\cal H}_M \rangle \over \langle \phi^2 \rangle}
- \langle {\cal H}_M \rangle \right)$$ has been measured, where $\phi = L^{-3}\sum_i S_i^1$ is the order parameter. At the critical temperature $T_c(m)$ the scaling behavior $X \sim x/t$ is expected (see (\[x\])). Corresponding numerical results for $m = 0$ and $m = 1/\sqrt{2}$ are displayed in Fig.\[fig1\] on a double logarithmic scale.
![Cumulant $X$ at the critical point for $m = 0$ ($\times$) and $m = 1/\sqrt{2}$ (+). The solid and dashed lines display power law fits to the data for $30 \leq L \leq 70$ for $m = 0$ and $m =
1/\sqrt{2}$, respectively[]{data-label="fig1"}](fig1.eps){width="80.00000%"}
The data are compatible with simple power laws, where the exponents $\nu = 0.622 \pm 0.005$ $(m = 0)$ and $\nu' = 0.714 \pm 0.004$ $(m =
1/\sqrt{2})$ have been obtained. Compared to the best currently known estimate $\nu \simeq 0.630$ [@LGZJ85] the above estimate is too small and only agrees with the theoretical value within two standard deviations. A more thorough analysis shows that the discrepancy can be explained by a mismatch of the order $5 \times 10^{-5}$ between the actual critical temperature and the estimate used here (see (\[KcI\])), which on the other hand is of the same magnitude as the statistical error of $K_c(0)$. The agreement between the above estimate for $\nu' = \nu / (1-\alpha)$ and the theoretical value $\nu' \simeq 0.708$ [@LGZJ85] is better, however, it may again be affected by a mismatch between the actual value of $T_c(1/\sqrt{2})$ and the estimate used here. If the literature values for $\nu$ and $\alpha$ are substituted in (\[x\]), where $a$ and $A$ are used as fit parameters, $a/A \simeq 0.1$ is obtained which is small enough to be ignored in the scaling analysis (see below).
The finite-size scaling analysis has been performed for several thermodynamic quantities, in particular, the average modulus of the order parameter $\langle |\phi| \rangle$, the susceptibilities $$\label{Chi}
\chi_+ \equiv {L^3 \over k_B T} \langle \phi^2 \rangle , \quad
\chi_- \equiv {L^3 \over k_B T} \left( \langle \phi^2 \rangle
- \langle |\phi| \rangle^2 \right),$$ and the specific heat $C$. Data will only be shown for $\langle |\phi|\rangle$, $\chi_-$, and $C$, because the finite-size scaling functions for $\langle
|\phi| \rangle$ and $\chi_+$ are very similar. According to finite-size scaling theory it must be possible to callapse the data for all $m$ onto one and the same curve, where two nonuniversal scaling factors are required for each quantity. One scaling factor adjusts the magnitude of the scaling argument $x$ (see (\[x\])), the other adusts the absolute normalization of the quantity. Note that the former saling factor must be the same for [*all*]{} quantities. For $m = 0$ the scaling argument $x = t L^{1/\nu}$ is used, whereas for $m = 1/\sqrt{2}$ the choice $x = t L^{(1-\alpha)/\nu}
/ A$ has been made, where $A \simeq 1.1$ and the coefficient $a$ in (\[x\]) has been neglected. The exponents $\nu$ and $\alpha$ are taken from the literature [@LGZJ85].
![Scaling plot of $\langle |\phi| \rangle$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.007$ and 0.003. Statistical errors are much smaller than the symbol sizes[]{data-label="fig2"}](fig2.eps){width="80.00000%"}
The scaling plot of $\langle |\phi|
\rangle$ is shown in Fig.\[fig2\], where the abolute normalization of the data for $m = 1/\sqrt{2}$ can be adjusted to the $m = 0$ data by a scale factor of $\sim 0.7$ as one would expect from simple mean field arguments. As shown in Fig.\[fig2\], the espected data collapse can be reproduced rather well, where the literature value for the exponent $\beta / \nu =
0.5168$ [@LGZJ85] has been used. The same holds for the susceptibility $\chi_-$ which is displayed in Fig.\[fig3\], where $\gamma / \nu = 2 - \eta = 1.967$ is also taken form [@LGZJ85]. The absolute magnitudes of $\chi_-/L^{\gamma/\nu}$ for $m = 0$ and $m = 1/\sqrt{2}$ are different by a factor of about $0.5$ which is in accordance with simple mean field arguments. Note that the scaling function of $\chi_-$ has a maximum for $x < 0$ [@Dohm9596].
![Scaling plot of $\chi_-$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.007$ and 0.003. Statistical errors are much smaller than the symbol sizes[]{data-label="fig3"}](fig3.eps){width="80.00000%"}
The specific heat $C$ requires a somewhat different treatment due to the fact that unlike the other quantities presented so far the specific heat requires an [*additive*]{} renormalization within renormalized field theory [@Dohm9596]. For the data analysis this means that scaling can only be obtained after a suitable subtraction is applied to the specific heat. One option to obtain scaling is to subtract the bulk specific heat $C^0_b(t)$ at a reference reduced temperature $t = t_0$ which are given by $$\label{Cbt0}
C^0_b(t) = {A_\pm \over \alpha} |t|^{-\alpha} + B
\quad \mbox{and} \quad t_0 = (L/\xi^0_\pm)^{-1/\nu},$$ where $A_\pm$ and $B$ are nonuniversal constants and $\xi^0_\pm$ is the amplitude of the correlation length. The index $\pm$ refers to temperatures above or below $T_c(m)$, respectively. The reference reduced temperature chosen in (\[Cbt0\]) is positive and therefore only $A_+$ and $\xi^0_+$ are needed. Specifically, the choice $\nu = 0.630$, $\alpha = 0.109$ [@LGZJ85], $A_+ = 0.1552$, $B = -1.697$, and $\xi^0_+ = 0.495$ [@Dohm9596] guarantee scaling of the [*relative*]{} specific heat $\Delta C^0 \equiv C - C^0_b(t_0)$ for the Ising model in $d = 3$. Note that $\xi^0_+$ is measured in units of the lattice constant. For the data to be analyzed here the subtraction defined by (\[Cbt0\]) is only valid for the case $m = 0$, where $\Delta C^0$ scales as $L^{\alpha/\nu}$. For $m = 1/\sqrt{2}$ Fisher renormalization according to (\[ttau\]) must be applied to $C^0_b(t)$ in order to obtain the correct form $C^m_b(t)$ of the subtraction. The result is $$\label{Cbt0m}
C^m_b(t) = {A'_+ \over \alpha} |t|^{\alpha/(1-\alpha)} + B'
\quad \mbox{and} \quad t_0 = (L/\xi^0_\pm)^{(\alpha-1)/\nu},$$ where $A'_+ = -0.1728$, $B' = 1.598$, and $\xi^0_+ = 0.495$ is not changed. The scaling factor $L^{\alpha/\nu}$, which usually governs the finite-size scaling of the specific heat, is cancelled here, i.e., one expects data collapse for $\Delta C^0 / L^{\alpha/\nu}$ and $\Delta C^m \equiv C -
C^m_b(t_0)$ up to an overall scale factor of about $0.5$. The result of the data analysis is shown in Fig.\[fig4\].
![Relative specific heats $\Delta C^0 /
L^{\alpha/\nu}$ for $m = 0$ and $\Delta C^m$ for $m = 1/\sqrt{2}$ for $L =
30$, 40, 50, and 60. The reduced temperature $t$ has been varied between $-0.007$ and 0.003[]{data-label="fig4"}](fig4.eps){width="80.00000%"}
The data collapse reasonably well onto a single curve except near the maximum of the scaling function, where also the scatter of the individual data is substantial due to a few bad samples for $L = 50$ and 60. However, there are also systematic deviations from scaling in the data, because the maximum in the scaling functions for the $m = 0$ data is more pronounced than in the $m = 1/\sqrt{2}$ data. These deviations could be due to enhanced Wegner corrections to scaling for $m = 1/\sqrt{2}$ as compared to $m = 0$.
XY universality class
=====================
For $N = 3$ (\[H\]) and (\[Hconst\]) describe a classical XY model in a (fixed) transverse field or with fixed transverse magnetization, respectively. As for the case $N = 2$ we will only consider the constrained model with the symmetric constraint $m = 0$ and with the constraint $m =
1/\sqrt{2}$ in the following. The symmetrically constrained model is again used for algorithmic tests and data production for the XY universality class. The nonsymmetric constrained XY model does not show Fisher renormalization, however, according to (\[x\]) very slowly decaying corrections to the asymptotic critical behavior are expected, which will be discussed in the following. First, the cumulant $X$ defined by (\[X\]) is evaluated at the critical point, which is given by the reduced coupling constants $$\label{KcXY}
K_c(0) = 0.6444 \pm 0.0001 \quad \mbox{and} \quad
K_c(1/\sqrt{2}) = 1.1126 \pm 0.0003 ,$$ respectively. The result for $X$ is displayed in Fig.\[fig5\].
![Cumulant $X$ at the critical point for $m = 0$ (x) and $m = 1/\sqrt{2}$ (+). The solid and dashed lines display fits to the data for $30 \leq L \leq 80$ for $m = 0$ and $m =
1/\sqrt{2}$, respectively (see main text)[]{data-label="fig5"}](fig5.eps){width="80.00000%"}
For $m = 0$ the data can be fitted by a power law $\sim L^{1/\nu}$, where $\nu = 0.678 \pm 0.008$ is obtained which agrees with the best current estimate $\nu = 0.671$ [@LGZJ85]. For $m = 1/\sqrt{2}$ the data can also be fitted by a power law, however, the resulting exponent $\nu$ only has the meaning of an effective exponent which does not fit into the XY universality class. As shown in Fig.\[fig5\] the expression $x/t$ according to (\[x\]) also yields a very good representation of the data where the parameter $b$ in Fig.\[fig5\] is given by $b = A/a = -0.941$. The exponents used in the fit (XY universality class) are taken from [@LGZJ85]. The value of the Binder cumulant found here agrees with results reported in the literature for the standard (plane rotator) XY model [@GH93], however, Wegner corrections to scaling become quite substantial for $m = 1/\sqrt{2}$.
In the following scaling analysis the finite-size scaling argument for the case $m = 0$ takes its standard form $x = t L^{1/\nu}$ and for $m =
1/\sqrt{2}$ the combination $x = t L^{1/\nu} / (1+b L^{\alpha/\nu})$ for $b = -0.941$ takes care of the slowly decaying correction terms to the asymptotic critical behavior caused by the very small and negative value of $\alpha$ in the XY universality class. As in Sect. 3 we consider $\langle |\phi| \rangle$, $\chi_+$, $\chi_-$, and the specific heat $C$ in the scaling analysis. The scaling functions for $\langle
|\phi| \rangle$ and $\chi_+$ again look very similar so that we do not reproduce scaling plots for $\chi_+$ here. The result for $\langle |\phi|
\rangle$ is shown in Fig.\[fig6\], where the order parameter $\phi
\equiv L^{-3} \sum_i (S^1_i,S^2_i)$ has two components here.
![Scaling plot of $\langle |\phi| \rangle$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.005$ and 0.005. Statistical errors are much smaller than the symbol sizes[]{data-label="fig6"}](fig6.eps){width="80.00000%"}
The data collapse very well onto a single curve. Note that the absolute magnitudes of $\langle |\phi| \rangle$ for $m = 0$ and $m = 1/\sqrt{2}$ are again related by a factor of $\sim 0.7$ as suggested by mean-field arguments. The values for the critical exponents $\nu = 0.671$ and $\beta
= 0.347$ are taken from the literature [@LGZJ85]. The susceptibility $\chi_-$ can be treated essentially as described in Sect.3, where the exponent $\gamma / \nu = 2 - \eta = 1.965$ is taken from [@LGZJ85]. The result of the scaling analysis is displayed in Fig.\[fig7\].
![Scaling plot of $\chi_-$ for $L = 30$, 40, 50, and 60 for $m = 0$ and $m = 1/\sqrt{2}$. The reduced temperature $t$ has been varied between $-0.005$ and 0.005. Statistical errors are much smaller than the symbol sizes[]{data-label="fig7"}](fig7.eps){width="80.00000%"}
The data do not collapse as well as in Fig.\[fig3\]. Especially near the maximum of the scaling function the scatter of the data is substantially larger than in Fig.\[fig3\]. Slight systematic deviations from scaling for $m = 1/\sqrt{2}$ are observed which may again be due to enhanced Wegner corrections to scaling as compared to $m = 0$. Note that contrary to Fig.\[fig3\] the scaling function of $\chi_-$ has a maximum for $x > 0$ in the XY universality class.
The specific heat $C$ of the XY model also requires a subtraction before scaling is obtained [@CDE95]. The subtraction $C^0_b(t_0)$ is again used in the form given by (\[Cbt0\]), where $A_+ = 0.42$, $\alpha =
-0.013$, $B = -A_+/\alpha$, and $\xi^0_+ = 1.0$ are used here which differ somewhat from the choices made for the standard XY model in [@CDE95]. It turns out, that the quality of the data collapse for the relative specific heat $\Delta C^0 = C - C^0_b(t_0)$ for $m = 0$ is rather insensitive to the choice of $\xi^0_+$. The form of the subtraction $C^m_b(t_0)$ for $m =
1/\sqrt{2}$ requires a little analysis in order to include the slowly varying corrections to the asymptotic behavior coming from (\[ttau\]). One obtains the approximate form $$\label{Cbt0XY}
C^m_b(t_0) = {A_+ / \alpha \over 1 + c t_0^{-\alpha}} \left[ \left({t_0 \over
1 + c t_0^{-\alpha}}\right)^{-\alpha} - 1\right], \quad t_0 = L^{-1/\nu}
(1 + b L^{\alpha/\nu}),$$ where $b = -0.941$ (see Fig.\[fig5\]), $A_+ = 0.42$ as before, and $c
\simeq 2.0$ for optimal data collapse.
![Relative specific heats $\Delta C^{(0,m)} / \left[ L^{\alpha/\nu}
/ (1 + bL^{\alpha/\nu}) \right]$ with $b = 0$ for $m = 0$ and $b = -0.941$ for $m = 1/\sqrt{2}$ for $L = 30$, 40, 50, and 60. The reduced temperature $t$ has been varied between $-0.005$ and 0.005[]{data-label="fig8"}](fig8.eps){width="80.00000%"}
The resulting scaling plot of $\Delta C^0$ and $\Delta C^m \equiv C -
C^m_b(t_0)$ is displayed in Fig.\[fig8\]. The overall shape of the scaling function is similar to the one shown in Fig.\[fig4\]. However, the scatter of the data near the maximum is so strong, that data collapse cannot be obtained in this region. In part this deficiency in the data may be due to the presence of ’bad’ samples, e.g, for $L = 50$ and $m = 0$ at $t = -0.003$ and $t = -0.002$ and for $L = 60$ and $m = 1/\sqrt{2}$ at $t = -0.005$ and $t = -0.003$. Apart from that deviations from scaling as in Fig.\[fig4\] may be present which are due to an enhancement of Wegner corrections to scaling for $m = 1/\sqrt{2}$ as compared to the case $m = 0$. However, Figs.\[fig6\] - \[fig8\] confirm, that the choice of the scaling variable $x$ given by (\[x\]) captures the slowly decaying correction term inside the scaling argument in an appropiate way and that furthermore the finite-size scaling behavior of constrained models in the Ising and the XY universality class can be treated on the same footing.
Summary and conclusions
=======================
The influence of constraints on the critical finite-size scaling behavior of Ising and XY models has been investigated by Monte-Carlo simulations of $O(N-1)$ planar ferromagnets with fixed transverse magnetization. The theoretical idea that only the form of the scaling argument is modified, whereas the shape of the universal scaling functions remains unchanged is verified within the statistical uncertainty of the data for the modulus of the order parameter, the susceptibilites $\chi_+$ (not shown) and $\chi_-$, and the specific heat. The form of the scaling argument used here allows to deal with critical finite-size effects in constrained Ising and XY models on the same footing, where constrained Heisenberg models can be included as well. Within the Ising universality class the finite-size behavior is consistent with the Fisher renormalization of critical exponents. In the XY universality class slowly decaying corrections to the asymptotic critical behavior are generated which are captured systematically by the analytic form of the scaling argument. The treatment of these corrections within the XY universality class may serve as a paradigm for the finite-size scaling analysis of dynamic quantities, where the constraints imposed here reappear as conserved quantities which are statically or dynamically coupled to the order parameter. These corrections may also be important for the interpretation of spin dynamics data for planar ferromagnets, where the energy of the system is conserved during the simulated time evolution of spin models similar to the ones investigated here.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author gratefully acknowledges many helpful discussions with V. Dohm and financial support of this work through the Heisenberg program of the Deutsche Forschungsgemeinschaft.
[99]{}
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| ArXiv |
---
abstract: 'We admit that the vacuum is not empty but is filled with continuously appearing and disappearing virtual fermion pairs. We show that if we simply model the propagation of the photon in vacuum as a series of transient captures within the virtual pairs, we can derive the finite light velocity $c$ as the average delay on the photon propagation. We then show that the vacuum permittivity $\epsilon_0$ and permeability $\mu_0$ originate from the polarization and the magnetization of the virtual fermions pairs. Since the transit time of a photon is a statistical process within this model, we expect it to be fluctuating. We discuss experimental tests of this prediction. We also study vacuum saturation effects under high photon density conditions.'
address:
-
- 'LAL, Univ Paris-Sud, CNRS/IN2P3, Orsay, France.'
author:
- 'Marcel Urban[^1], François Couchot, Xavier Sarazin'
title: 'A mechanism giving a finite value to the speed of light, and some experimental consequences'
---
Introduction
============
The speed of light in vacuum $c$, the vacuum permittivity $\epsilon_0$ and the vacuum permeability $\mu_0$ are widely considered as being fundamental constants and their values, escaping any physical explanation, are commonly assumed to be invariant in space and time. In this paper, we propose a mechanism based upon a “natural” quantum vacuum description which leads to sensible estimations of these three electromagnetic constants, and we start drawing some consequences of this perspective.
The idea that the vacuum is a major partner of our world is not new. It plays a part for instance in the Lamb shift [@lamb], the variation of the fine structure constant with energy [@bare-charge], and the electron [@magnetic-moment] and muon [@Davier] anomalous magnetic moments. But these effects, coming from the so-called vacuum polarization, are second order corrections. Quantum Electrodynamics is a perturbative approach to the electromagnetic quantum vacuum. This paper is concerned with the description of the fundamental, non perturbed, vacuum state.
While we were writing this paper, Ref. [@Leuchs] proposed a similar approach to give a physical origin to $\epsilon_0$ and $\mu_0$. Although this derivation is different from the one we propose in this paper, the original idea is the same: [“The physical electromagnetic constants, whose numerical values are simply determined experimentally, could emerge naturally from the quantum theory” [@Leuchs]]{}. We do not know of any other paper proposing a direct derivation of $\epsilon_0$ and $\mu_0$ or giving a mechanism based upon the quantum vacuum leading to $c$.
The most important consequence of our model is that $c$, $\epsilon_0$ and $\mu_0$ are not fundamental constants but are observable parameters of the quantum vacuum: they can vary if the vacuum properties vary in space or in time.
The paper is organized as follows. First we describe our model of the quantum vacuum filled with virtual charged fermion pairs and we show that, by modeling the propagation of the photon in this vacuum as a series of interactions with virtual pairs, we can derive its velocity. Then we show how $\epsilon_0$ and $\mu_0$ might originate from the electric polarization and the magnetization of these virtual pairs. Finally we present two experimental consequences that could be at variance with the standard views and in particular we predict statistical fluctuations of the transit time of photons across a fixed vacuum path.
An effective description of quantum vacuum {#sec:model}
==========================================
The vacuum is assumed to be filled with virtual charged fermion pairs (particle-antiparticle). The other vacuum components are assumed not to be connected with light propagation (we do not consider intermediate bosons, nor supersymmetric particles). All known species of charged fermions are taken into account: the three families of charged leptons $e$, $\mu$ and $\tau$ and the three families of quarks ($u$, $d$), ($c$, $s$) and ($t$, $b$), including their three color states. This gives a total of $21$ pair species, noted $i$.
A virtual pair is assumed to be the product of the fusion of two virtual photons of the vacuum. Thus its total electric charge and total color are null, and we suppose also that the spins of the two fermions of a pair are antiparallel. The only quantity which is not conserved is therefore the energy and this is, of course, the reason for the limited lifetime of the pairs. We assume that first order properties can be deduced assuming that pairs are created with an average energy, not taking into account a full probability density of the pair kinetic energy. Likewise, we will neglect the total momentum of the pair.
We describe this vacuum in terms of five quantities for each pair species: average energy, lifetime, density, size of the pairs and cross section with photons.
We use the notation $Q_i = q_i/e$, where $q_i$ is the modulus of the $i$-kind fermion electric charge and $e$ the modulus of the electron charge.
The average energy $W_i$ of a pair is taken proportional to its rest mass energy $2W_i^0$, where $W_i^0$ is the fermion $i$ rest mass energy: $$\begin{aligned}
\label{eq:energy}
W_i\ = K_W\ 2 W_i^0 ,\end{aligned}$$ where $K_W$ is an unknown constant, assumed to be independent from the fermion type. We take $K_W$ as a free parameter, greater than unity. The value of $K_W$ could be calculated if we knew the energy spectrum of the virtual photons together with their probability to create virtual pairs.
The pair lifetime $\tau_i$ follows from the Heisenberg uncertainty principle $(W_i\tau_i=\hbar/2)$. So $$\begin{aligned}
\label{eq:tau}
\tau_i = \frac{1}{K_W}\frac{\hbar}{4 W_i^0} .\end{aligned}$$ We assume that the virtual pair densities $N_i$ are driven by the Pauli Exclusion Principle. Two pairs containing two identical virtual fermions in the same spin state cannot show up at the same time at the same place. However at a given location we may find 21 pairs since different fermions can superpose spatially. In solid state physics the successful determination of Fermi energies [@Kittel] implies that one electron spin state occupies a hyper volume $h^3$. We assume that concerning the Pauli principle, the virtual fermions are similar to the real ones. Noting $\Delta x_i$ the spacing between identical virtual $i-$type fermions and $p_i$ their average momentum, the one dimension hyper volume is $p_i\Delta x_i$ and dividing by $h$ should give the number of states which we take as one per spin degree of freedom. The relation between $p_i$ and $\Delta x_i$ reads $p_i\Delta x_i/h = 1$, or: $$\begin{aligned}
\label{eq:deltax}
\Delta x_i=\frac{2 \pi \hbar}{p_i}\, .\end{aligned}$$
We can express $\Delta x_i$ as a function of $W_i$ if we suppose the relativity to hold for the virtual pairs $$\begin{aligned}
\label{eq:dx}
\Delta x_i =\frac{2\pi \hbar c}{\sqrt{(W_i/2)^2-(W_i^0)^2}} = \frac{{2\pi\lambda_C}_i}{\sqrt{K_W^2-1}} ,\end{aligned}$$ where ${\lambda_C}_i$ is the Compton length associated to fermion $i$.
We write the density as $$\begin{aligned}
\label{eq:density}
N_i \approx \frac{1}{\Delta x_i^3} = \left(\frac{\sqrt{K_W^2-1}}{{2\pi\lambda_C}_i}\right)^3 .\end{aligned}$$
Each pair can only be produced in two fermion-antifermion spin combinations: up-down and down-up. We define $N_i$ as the density of pairs for a given spin combination. It is very sensitive to $K_W$, being zero for pairs having no internal kinetic energy.
The separation between the fermion and the antifermion in a pair is noted $\delta_i$. This parameter has to do with the physics of the virtual pairs. We assume it does not depend upon the fermion momentum. We will use the Compton wavelength of the fermion $\lambda_{C_i}$ as this scale: $$\begin{aligned}
\label{eq:compton-length}
\delta_i \approx {{\lambda}_C}_i .\end{aligned}$$
The interaction of a real photon with a virtual pair must not exchange energy or momentum with the vacuum. For instance, Compton scattering is not possible. To estimate this interaction probability, we start from the Thomson cross-section $\sigma_{Thomson}= {8 \pi}/{3}\ \alpha^2 {\lambda^2_{C}}_i$ which describes the interaction of a photon with a free electron. The factor $\alpha^2$ corresponds to the probability $\alpha$ that the photon is temporarily absorbed by the real electron times the probability $\alpha$ that the real electron releases the photon. However, in the case of the interaction of a photon with a virtual pair, the second $\alpha$ factor must be ignored since the photon is released with a probability equal to 1 as soon as the virtual pair disappears. Therefore the cross-section $\sigma$ for a real photon to interact and to be trapped by a virtual pair of fermions will be expressed as $$\begin{aligned}
\label{eq:sigma}
\sigma_i \approx \left(\frac{8 \pi}{3} \alpha\ Q_i^2 {\lambda^2_C}_i\right)\times 2 .\end{aligned}$$ The photon interacts equally with the fermion and the antifermion, which explains the factor $2$. A photon of helicity $1$ ($-1$ respectively) can interact only with a fermion or an antifermion with helicity $-1/2$ ($+1/2$ respectively) to flip temporarily its spin to helicity $+1/2$ ($-1/2$ respectively). During such a photon capture by a pair, both fermions are in the same helicity state and cannot couple to another incoming photon in the same helicity state as the first one.
Derivation of the light velocity in vacuum {#sec:speedoflight}
==========================================
We propose in this section a mechanism which leads to a **finite** speed of light. The propagation of the photon in vacuum is modeled as a series of interactions with the virtual fermions or antifermions present in the pairs. When a real photon propagates inside the vacuum, it interacts and is temporarily captured by a virtual pair during a time of the order of the lifetime $\tau_i$ of the virtual pair. As soon as the virtual pair disappears, it releases the photon to its initial energy and momentum state. The photon continues to propagate with a **bare** velocity $c_0$ which is assumed to be much greater than $c$. Then it interacts again with a virtual pair and so on. The delay on the photon propagation produced by these successive interactions implies that the velocity of light is finite.
The mean free path of the photon between two successive interactions with a $i-$type pair is: $$\begin{aligned}
\label{eq:freepath}
\Lambda_i = \frac{1}{\sigma_i N_i}\ ,\end{aligned}$$ where $\sigma_i$ is the cross-section for the photon capture by the virtual $i-$type pair and $N_i$ is the numerical density of virtual $i-$type pairs.
Travelling a distance $L$ in vacuum leads on average to $N_{stop,i}$ interactions on the $i-$kind pairs. One has: $$\begin{aligned}
\label{eq:Nstop}
N_{stop,i} = \frac{L}{\Lambda} = L{\sigma_i N_i}\ .\end{aligned}$$
Each kind of fermion pair contributes in reducing the speed of the photon. So, if the mean photon stop time on a $i-$type pair is $\tau_i$, the mean time $\overline{T}$ for a photon to cross a length $L$ is assumed to be: $$\begin{aligned}
\label{eq:Tbar}
\overline{T} = L/c_0 + \sum_{i}{N_{stop,i}\tau_i}\ .\end{aligned}$$
The bare velocity $c_0$ is the velocity of light in an **empty** vacuum with no virtual particles. We assume that $c_0$ is infinite, which is equivalent to say that the time does not flow in an empty vacuum (with a null zero point energy). There are no “natural” time or distance scales in such an empty vacuum, whereas the $\tau_i$ and $\Delta x_i$ scales allow to build a speed scale. So, the total delay reduces to: $$\begin{aligned}
\label{eq:Tbar2}
\overline{T} = \sum_{i}{N_{stop,i}\tau_i}\ .\end{aligned}$$ So, a photon, although propagating at the speed of light, is at any time resting on one fermion pair.
Using Eq. (\[eq:Nstop\]), we obtain the photon velocity $\tilde{c}$ as a function of three parameters of the vacuum model: $$\begin{aligned}
\label{eq:c-1}
\tilde{c} = \frac{L}{\overline{T} }= \frac{1}{\sum_{i}{\sigma_i N_i \tau_i}} .\end{aligned}$$
We notice that the cross-section $\sigma_i$ in (\[eq:sigma\]) does not depend upon the energy of the photon. It implies that the vacuum is not dispersive as it is experimentally observed. Using Eq. (\[eq:tau\]), (\[eq:density\]) and (\[eq:sigma\]), we get the final expression: $$\begin{aligned}
\label{eq:c-2}
\tilde{c} = \frac{K_W}{\left(K_W^2-1\right)^{3/2}}\ \frac{\ 6\pi^2}{\alpha \hbar \sum_{i}{Q_i^2/({\lambda_C}_iW_i^0)}} .\end{aligned}$$
${\lambda_C}_iW_i^0/\hbar$ is equal to the speed of light: $$\begin{aligned}
\label{eq:lcwi}
{{\lambda_C}_iW_i^0}/{\hbar}=\frac{\hbar}{m_i c}\ m_i c^2 \frac{1}{\hbar} = c .\end{aligned}$$ So $$\begin{aligned}
\label{eq:c-3}
\tilde{c} = \frac{K_W}{\left(K_W^2-1\right)^{3/2}}\ \frac{\ 6\pi^2}{\alpha \sum_{i}{Q_i^2}}\ c .\end{aligned}$$
The photon velocity depends only on the electrical charge units $Q_i$ of the virtual charged fermions present in vacuum. It depends neither upon their masses, nor upon the vacuum energy density.
The sum in Eq. (\[eq:c-3\]) is taken over all pair types. Within a generation the absolute values of the electric charges are 1, 2/3 and 1/3 in units of the positron charge. Thus for one generation the sum writes $(1+3 \times(4/9+1/9))$. The factor 3 is the number of colours. Each generation contributes equally, hence for the three families of the standard model: $$\begin{aligned}
\label{eq:sommeq2}
\sum_{i}{Q_i^2} = 8 .\end{aligned}$$
One obtains $$\begin{aligned}
\label{eq:c-4}
\tilde{c} = \frac{K_W}{(K_W^2-1)^{3/2}}\frac{3\pi^2} {4 \alpha}\ c .\end{aligned}$$
The calculated light velocity $\tilde{c}$ is equal to the observed value $c$ when $$\begin{aligned}
\label{eq:cadoublev}
\frac{K_W}{(K_W^2-1)^{3/2}}=\frac{4\alpha}{3\pi^2} ,\end{aligned}$$ which is obtained for $K_W \approx 31.9\,$, greater than one as required.
The average speed of the photon in our medium being $c$, the photon propagates, on average, along the light cone. As such, the effective average speed of the photon is independent of the inertial frame as demanded by relativity. This mechanism relies on the notion of an absolute frame for the vacuum at rest. It satisfies special relativity only in the Lorentz-Poincaré sense.
Derivation of the vacuum permittivity {#sec:permittivity}
=====================================
Consider a parallel-plate capacitor with a gas inside. When the pressure of the gas decreases the capacitance decreases too until there is no more molecules in between the plates. The strange thing is that the capacitance is not zero when we hit the vacuum. In fact the capacitance has a very sizeable value as if the vacuum were a usual material body. The dielectric constant of a medium is coming from the existence of opposite electric charges that can be separated under the influence of an applied electric field $\vec{E}$. Furthermore the opposite charges separation stays finite because they are bound in a molecule. These opposite translations result in opposite charges appearing on the dielectric surfaces in regard of the metallic plates. This leads to a decrease of the effective charge, which implies a decrease of the voltage across the dielectric slab and finally to an increase of the capacitance. In our model of the vacuum the virtual pairs are the pairs of opposite charges and the separation stays finite because the electric field acts only during the lifetime of the pairs. In an absolute **empty** vacuum the induced charges would be null because there would be no charges to be separated and the capacitance of our parallel-plate capacitor would go to zero when we would remove all molecules of the gas. We will see in this section that introducing our vacuum filled by virtual fermions will cause its electric charges to be separated and to appear at the level of $5.10^7$ electron charges per $m^2$ under an electric stress $E = 1\ V/m$.
We assume that every fermion-antifermion virtual pair of the $i$-kind bears a mean electric dipole $d_i$ given by: $$\begin{aligned}
\label{eq:elecdipole}
\vec{d_i} = Q_i e \vec{\delta_i} .\end{aligned}$$ where $\delta_i$ is the average size of the pairs. If no external electric field is present, the dipoles point randomly in any direction and their resulting average field is zero. We propose to give a physical interpretation of the observed vacuum permittivity $\epsilon_0$ as originating from the mean polarization of these virtual fermions pairs in presence of an external electric field $\vec{E}$. This polarization would show up due to the dipole lifetime dependence on the electrostatic coupling energy of the dipole to the field. In a field homogeneous at the $\delta_i$ scale, this energy is $d_i E \cos \theta$ where $\theta$ is the angle between the virtual dipole and the electric field $\vec{E}$. The electric field modifies the pair lifetimes according to their orientation: $$\begin{aligned}
\label{eq:taudipel}
\tau_i(\theta)= \frac{\hbar/2} {W_i - d_i E \cos \theta} .\end{aligned}$$
Since it costs less energy to produce such an elementary dipole aligned with the field, this configuration lasts a bit longer than the others, leading to an average dipole different from zero. This average dipole $\langle D_i \rangle$ is aligned with the electric field $\vec{E}$. Its value is obtained by integration over $\theta$ with a weight proportional to the pair lifetime: $$\begin{aligned}
\label{eq:D}
\langle D_i \rangle = \frac{\int_0^{\pi} d_i\ \cos\theta\ \tau_i(\theta)\ 2\pi \sin\theta\ d\theta}{\int_0^{\pi} \tau_i(\theta)\ 2\pi \sin\theta\ d\theta} .\end{aligned}$$
To first order in $E$, one gets: $$\begin{aligned}
\label{eq:polar}
\langle D_i \rangle = d_i \frac{d_i E}{3 W_i} = {{ d_i^2}\over {3W_i}} E .\end{aligned}$$
We estimate the permittivity $\tilde{\epsilon}_{0,i}$ due to $i$-type fermions using the relation $P_i=\tilde{\epsilon}_{0,i}E$, where the polarization $P_i$ is equal to the dipole density $P_i=2 N_i \langle D_i \rangle$, since the two spin combinations contribute. Thus: $$\begin{aligned}
\label{eq:epsi}
\tilde{\epsilon}_{0,i} =2 N_i \frac{\langle D_i \rangle}{E} =2 N_i \frac{d_i^2}{3W_i} =2 N_i e^2 \frac{Q_i^2 \delta_i^2}{3W_i} .\end{aligned}$$
Each species of fermions increases the induced polarization and therefore the vacuum permittivity. By summing over all pair species, one gets the estimation of the vacuum permittivity: $$\begin{aligned}
\label{eq:epsi0}
\tilde{\epsilon}_{0} = e^2 \sum_{i}{2 N_i Q_i^2 \frac{\delta_i^2}{3W_i}} .\end{aligned}$$
We can write that permittivity as a function of our units, using Eq. (\[eq:energy\]), (\[eq:density\]), (\[eq:compton-length\]) and (\[eq:lcwi\]): $$\begin{aligned}
\label{eq:epsi0bis}
\tilde{\epsilon}_{0} = \frac{(K_W^2-1)^{3/2}}{K_W}\frac{e^2}{24 \pi^3\hbar c} \sum_{i}{Q_i^2} .\end{aligned}$$
The sum is again taken over all pair types. From Eq. (\[eq:sommeq2\]) one gets: $$\begin{aligned}
\label{eq:permittivity}
\tilde{\epsilon}_0 = \frac{(K_W^2-1)^{3/2}}{K_W} \frac{e^2}{3\pi^3 \hbar c}\, .\end{aligned}$$
And, from Eq. (\[eq:cadoublev\]) one gets: $$\begin{aligned}
\label{eq:permittivity}
\tilde{\epsilon}_0 = {\left(\frac{4\alpha}{3\pi^2}\right)}^{-1} \frac{e^2}{3\pi^3 \hbar c}=\frac{e^2}{4\pi\hbar c\alpha} =8.85\, 10^{-12} F/m\end{aligned}$$
It is remarkable that Eq. (\[eq:cadoublev\]) obtained from the derivation of the speed of light leads to a calculated permittivity $\tilde{\epsilon}_0$ exactly equal to the observed value of $\epsilon_0$.
Derivation of the vacuum permeability {#sec:permeability}
=====================================
The vacuum acts as a highly paramagnetic substance. When a torus of a material is energized through a winding carrying a current $I$, there is a resulting magnetic flux density $B$ which is expressed as: $$\begin{aligned}
\label{eq:mu-1}
B = \mu_0 n I + \mu_0 M .\end{aligned}$$ where $n$ is the number of turns per unit of length, $nI$ is the magnetic intensity in $A/m$ and $M$ is the corresponding magnetization induced in the material and is the sum of the induced magnetic moments divided by the corresponding volume. In an experiment where the current $I$ is kept a constant and where we lower the quantity of matter in the torus, $B$ decreases. As we remove all matter, $B$ gets to a non zero value: $B = \mu_0 n I$ showing experimentally that the vacuum is paramagnetic with a vacuum permeability $\mu_0 = 4\pi\ 10^{-7} {N/A^2}$.
We propose to give a physical interpretation to the observed vacuum permeability as originating from the magnetization of the charged virtual fermions pairs under a magnetic stress, following the same procedure as in the former section.
Each charged virtual fermion carries a magnetic moment proportional to the Bohr magneton: $$\begin{aligned}
\label{eq:magneton}
\mu_i = \frac{eQ_i\ c{\lambda_C}_i}{2} .\end{aligned}$$
Since the total spin of the pair is zero, and since fermion and antifermion have opposite charges, each pair carries twice the magnetic moment of one fermion. The coupling energy of a $i$-kind pair to an external magnetic field $\vec{B}$ is then $-2 \mu_i B \cos \theta$ where $\theta$ is the angle between the magnetic moment and the magnetic field $\vec{B}$. The pair lifetime is therefore a function of the orientation of its magnetic moment with respect to the applied magnetic field: $$\begin{aligned}
\label{eq:taumag}
\tau_i(\theta)= \frac{\hbar/2}{W_i - 2 \mu_i B \cos \theta} .\end{aligned}$$
As in the electrostatic case, pairs with a dipole moment aligned with the field last a bit longer than anti-aligned pairs. This leads to a non zero average magnetic moment $<\mathcal{M}_i>$ for the pair, aligned with the field and given, to first order in $B$, by: $$\begin{aligned}
\label{eq:magnet}
<\mathcal{M}_i> = \frac{4\mu_i^2}{3W_i} B .\end{aligned}$$
The volume magnetic moment is $M_i = {2 N_i <\mathcal{M}_i>}$, since one takes into account the two spin states per cell.
The contribution $\tilde{\mu}_{0,i}$ of the $i$-type fermions to the vacuum permeability is given by $ B=\tilde{\mu}_{0,i}M_i $ or ${1}/{\tilde{\mu}_{0,i}}={M_i}/{B}$.
This leads to the estimation of the vacuum permeability $$\begin{aligned}
\label{eq:permeability-1}
\frac{1}{\tilde{\mu}_0}=\sum_{i}{\frac{M_i}{B}} = \sum_{i}{\frac{8 N_i\mu_i^2}{3W_i}}= c^2 e^2\sum_{i}{\frac{2 N_iQ_i^2{\lambda^2_C}_i}{3W_i}} .\end{aligned}$$
Using Eq. (\[eq:energy\]), (\[eq:density\]) and (\[eq:lcwi\]) and summing over all pair types, one obtains $$\begin{aligned}
\label{eq:permeability-3}
\tilde{\mu}_0 = \frac{K_W}{(K_W^2-1)^{3/2}} \frac{24\pi^3\hbar}{c\,e^2 \sum_{i}{Q_i^2}}= \frac{K_W}{(K_W^2-1)^{3/2}} \frac{3\pi^3 \hbar}{ c\,e^2}\, .\end{aligned}$$
Using the $K_W$ value constrained by the calculus of $c$ (\[eq:cadoublev\]), we end up with: $$\begin{aligned}
\label{eq:permeability-4}
\tilde{\mu}_0 = \frac{4\pi\alpha}{3} \frac{3\ \hbar}{c\,e^2} =\frac{4\pi\alpha\hbar}{c\,e^2} = 4\pi 10^{-7}N/A^2 .\end{aligned}$$
It is again remarkable that Eq. (\[eq:cadoublev\]) obtained from the derivation of the speed of light leads to a calculated permeability $\tilde{\mu}_0$ equal to the right $\mu_0$ value.
We notice that the permeability and the permittivity do not depend upon the masses of the fermions, as in Ref. [@Leuchs]. The electric charges and the number of species are the only important parameters. This is at variance with the common idea that the energy density of the vacuum is the dominant factor [@Latorre].
This expression, combined with the expression (\[eq:permittivity\]) of the calculated permittivity, verifies the Maxwell relation, typical of wave propagation, $ \tilde{\epsilon}_0 \tilde{\mu}_0 = 1/c^2$, although our mechanism for a finite $c$ is purely corpuscular.
A generalized model
====================
We have shown that our model of vacuum leads to coherent calculated values of $c$, $\epsilon_0$ and $\mu_0$ equal to the observed values if we assume that the virtual fermion pairs are produced with an average energy which is about 30 times their rest mass.
This solution corresponds to some **natural** hypotheses for the density and the size of the virtual pair, the cross-section with real photons and their capture time. We can generalize the model by introducing the free parameters $K_N$, $K_{\delta}$, $K_{\sigma}$ and $K_\tau$ in the expressions (\[eq:density\]), (\[eq:compton-length\]), (\[eq:sigma\]) and (\[eq:Tbar\]) of the physical quantities: $$\begin{aligned}
\label{eq:1}
N_i = K_N \left( \frac{\sqrt{K_W^2-1}}{2\pi{\lambda_C}_i} \right)^3\end{aligned}$$ $$\begin{aligned}
\label{eq:2}
\delta_i = K_{\delta}\, {\lambda_C}_i\end{aligned}$$ $$\begin{aligned}
\label{eq:3}
\sigma_i = K_{\sigma} \frac{16 \pi}{3} \alpha\, Q_i^2 {\lambda^2_C}_i\end{aligned}$$ $$\begin{aligned}
\label{eq:4}
\overline{T} = \sum_{i}{N_{stop,i}K_\tau\tau_i}\ .\end{aligned}$$ $K_N$, $K_{\delta}$, $K_{\sigma}$ and $K_\tau$ are assumed to be universal factors independent of the fermion species. Their values are expected to stay close to $1$. Among the model parameters, $K_W$ is the only unconstrained unknown.
The general solutions of the calculation of $c$ (\[eq:c-4\]), $\epsilon_0$ (\[eq:permittivity\]) and $\mu_0$ (\[eq:permeability-4\]) read now: $$\begin{aligned}
\label{eq:5}
\tilde{c} = \frac{1}{K_N K_{\sigma}K_\tau } \frac{K_W}{(K_W^2-1)^{3/2}}\frac{3\pi^2} {4 \alpha}\ c\, ,\end{aligned}$$ $$\begin{aligned}
\label{eq:6}
\tilde{\epsilon}_0 = K_N K_{\delta}^2 \frac{(K_W^2-1)^{3/2}}{K_W} \frac{e^2}{3\pi^3 \hbar c}\, ,\end{aligned}$$ $$\begin{aligned}
\label{eq:7}
\tilde{\mu}_0 = \frac{1}{K_N} \frac{K_W}{(K_W^2-1)^{3/2}} \frac{3\pi^3 \hbar}{c\,e^2}\, ,\end{aligned}$$ from which one can get, for instance:
- first $\tilde{\epsilon}_0\tilde{\mu}_0=K_{\delta}^2/c^2$ which implies $K_{\delta} = 1$ ,
- then either $\tilde{\epsilon}_0$ or $\tilde{\mu}_0$ fixes $\frac{1}{K_N} \frac{K_W}{(K_W^2-1)^{3/2}} = \frac{4 \alpha}{3\pi^2}\ $
- which applied to $\tilde{c}$ gives $K_{\sigma}K_\tau = 1$.
So, the model parameters satisfy: $$\begin{aligned}
\label{eq:9}
K_{\delta} = 1\ , K_{\sigma}K_\tau = 1\ ,
\frac{1}{K_N} \frac{K_W}{(K_W^2-1)^{3/2}} = \frac{4\alpha}{3\pi^2}\ .
$$
$K_{\delta}$ is precisely constrained to its first guess value. More relations or observables are required to extract the other quantities and check this vacuum model. A measurement of the expected fluctuations of the speed of light, and a measurement of speed of light variations with light intensity, as discussed in the following sections, could bring such relations.
Transit time fluctuations {#sec:prediction transit}
=========================
Prediction
----------
Quantum gravity theories including stochastic fluctuations of the metric of compactified dimensions, predict a fluctuation $\sigma_t$ of the propagation time of photons [@Yu-Ford]. However observable effects are expected to be too small to be experimentally tested. It has been also recently predicted that the non commutative geometry at the Planck scale should produce a spatially coherent space-time jitter [@Hogan].
In our model we also expect fluctuations of the speed of light $c$. Indeed in the mechanism proposed here $c$ is due to the effect of successive interactions and transient captures of the photon with the virtual particles in the vacuum. Thus statistical fluctuations of $c$ are expected, due to the statistical fluctuations of the number of interactions $N_{stop}$ of the photon with the virtual pairs and to the capture time fluctuations.
The propagation time of a photon which crosses a distance $L$ of vacuum is $$\begin{aligned}
\label{eq:transit}
t = \sum_{i,k}{ t_{i,k}} ,\end{aligned}$$ where $t_{i,k}$ is the duration of the $k^{th}$ interaction on an $i$-kind pair. As in section \[sec:speedoflight\], let $ N_{stop,i}$ be the mean number of such interactions. The variance of $t$ due to the statistical fluctuations of $N_{stop,i}$ is: $$\begin{aligned}
\label{eq:sig1}
\sigma_{t,N}^2 = \sum_{i}{N_{stop,i} K_\tau^2\tau_i^2} .\end{aligned}$$ The photon may arrive on a virtual pair any time between its birth and its death. If we assume a flat probability distribution between $0$ and $\tau_i$, the mean value of $t_{i,k}$ is $\tau_i/2$, so one has $K_\tau=1/2$. The variance of the stop time is $(K_\tau\tau_i)^2/3$: $$\begin{aligned}
\label{eq:sig2}
\sigma_{t,\tau}^2 = \sum_{i}{N_{stop,i} \frac{(K_\tau\tau_i)^2}{3}} .\end{aligned}$$ Then $$\begin{aligned}
\label{eq:fluctu}
\sigma_t^2 = \sum_{i}{N_{stop,i} (K_\tau\tau_i)^2(1+\frac{1}{3}})= \frac{4\,K_\tau^2}{3}\sum_{i}{N_{stop,i} \tau_i^2}.\end{aligned}$$ And, using Eq. (\[eq:Nstop\]): $$\begin{aligned}
\label{eq:fluctuation-0}
\sigma_t^2 = \frac{4 K_\tau^2L}{3} \sum_{i}{\sigma_i N_i \tau_i^2} . \end{aligned}$$ Once reduced, the current term of the sum is proportional to ${\lambda_C}_i$. Therefore the fluctuations of the propagation time are dominated by virtual $e^+e^-$ pairs. Neglecting the other fermion species, and using $\sigma_e N_eK_\tau\tau_e=1/(8c)$, one gets : $$\begin{aligned}
\label{eq:formulesigmat2}
\sigma_t^2 = \frac{K_\tau\tau_eL}{6c}= \frac{K_\tau{\lambda_C}_eL}{24 K_W c^2} .\end{aligned}$$ So $$\begin{aligned}
\label{eq:fluctuation}
{\sigma_t} = \sqrt{\frac{L}{c}}\sqrt{\frac{{\lambda_C}_e}{c}}\sqrt{\frac{K_\tau}{K_W}}\frac{1}{\sqrt{24}} .\end{aligned}$$
For the simple solution of the vacuum model where $K_W=31.9$ and $K_\tau=1/2$ the predicted fluctuation is: $$\begin{aligned}
\label{eq:fluctuation-2}
\sigma_t \approx 53 \ as\ m^{-1/2} \sqrt{L(m)} .\end{aligned}$$ This corresponds for instance on a $1\ m$ long travel, to an average of $8.\, 10^{13} $ stops by $e^+e^-$ pairs during which the photon stays on average $5\, 10^{-24} s$ (it spends $7/8$ of its time trapped on other species pairs). Fluctuations of both quantities lead to this $50\ as$ expected dispersion on the photon transit time which represents a $1.5\,10^{-8}$ relative fluctuation over a meter.
This prediction must be modulated by the remaining degree of freedom on $K_N$ or $K_W$, but the mechanism would loose its physical basis if $\sigma_t$ would not have that order of magnitude.
A positive measurement of $\sigma_t$, apart from being a true revolution, would tighten our understanding of the fundamental constants in the vacuum, by fixing the ratio $K_\tau/K_W$.
The experimental way to test fluctuations is to measure a possible time broadening of a light pulse travelling a distance $L$ of vacuum. This may be done using observations of brief astrophysical events, or dedicated laboratory experiments.
Constraints from astrophysical observations
-------------------------------------------
The very bright GRB 090510, detected by the Fermi Gamma-ray Space Telescope [@Abdo], at MeV and GeV energy scale, presents short spikes in the $8~keV - 5~MeV$ energy range, with the narrowest widths of the order of $10\,ms$. Observation of the optical after glow, a few days later by ground based spectroscopic telescopes gives a common redshift of $z = 0.9$. This corresponds to a distance, using standard cosmological parameters, of about $2\, 10^{26} m$. Translated into our model, this sets a limit of about $0.7\, fs\, m^{-1/2}$ on $c$ fluctuations. It is important to notice that there is no expected dispersion of the bursts in the interstellar medium at this energy scale.
If we move six orders of magnitude down in distances we arrive to kpc and pulsars. Short microbursts contained in main pulses from the Crab pulsar have been recently observed at the Arecibo Observatory telescope at 5 GHz [@Crab-pulsar-2010]. The frequency-dependent delay caused by dispersive propagation through the interstellar plasma is corrected using a coherent dispersion removal technique. The mean time width of these microbursts after dedispersion is about 1 $\mu$s, much larger than the expected broadening caused by interstellar scattering. If this unknown broadening would not be correlated to the emission properties, it could come from $c$ fluctuations of about $0.2\, fs\, m^{-1/2}$.
In these observations of the Crab pulsar, some very sporadic pulses with a duration of less than $1 ns$ have been observed at 9 GHz [@Crab-pulsar-2007]. This is 3 orders of magnitude smaller than the usual pulses. These nanoshots can occasionally be extremely intense, exceeding $2\, MJy$, and have an unresolved duration of less than $0.4\, ns$ which corresponds to a light-travel size $c\delta t \approx 12\, cm$. From this the implied brightness temperature is $2\ 10^{41} K$. Alternatively we might assume the emitting structure is moving outward with a Lorentz factor $\gamma_b \approx 10^2 - 10^3$. In that case, the size estimate increases to $10^3 - 10^5\, cm$, and the brightness temperature decreases to $10^{35} - 10^{37}\, K$. We recall that the Compton temperature is $10^{12}\, K$ and that the Planck temperature is $10^{32}\, K$ so the phenomenon, if real, would be way beyond known physics. We emphasize also two features. Firstly, these nanoshots are contained in a single time bin (2 ns at 5 GHz and 0.4 ns at 9 GHz) corresponding to a time width less than $2/\sqrt{12} \approx 0.6\, ns$ at $5~\,GHz$ and $0.4/\sqrt{12} \approx 0.1\,ns$ at $9~\,GHz$, below the expected broadening caused by interstellar scattering. Secondly, their frequency distributions appear to be almost monoenergetic and very unusual, since the shorter the pulse the narrower its reconstructed energy spectrum.
Constraints from Earth bound experiments
----------------------------------------
The very fact that the predicted statistical fluctuations should go like the square root of the distance implies the exciting idea that experiments on Earth do compete with astrophysical constraints, since going from the kpc down to a few hundred meters, which means a distance reduction by a $10^{17}$ factor, we expect fluctuations in the $fs$ range.
An experimental setup using femtosecond lasers pulses sent to a rather long multi-pass cavity equipped with metallic mirrors could be able to detect such a phenomenon.
The attosecond laser pulse generation and characterization, by itself, might allow already to set the best limit on $\sigma_t$. This limit would be of the order of our predicted value in the simplest version of the model [@atto-1]. However, for the time being, the unambiguous measurement of the pulse time spread is available only for $fs$ pulses through the correlation of the short pulses in a non linear crystal.
Modification of the light speed in extremely intense light pulses {#sec:vacmod}
=================================================================
The vacuum, considered as a peculiar medium, should be able to undergo changes. This is suspected to be the case in the Casimir effect which predicts a pressure to be present between electrically neutral conducting surfaces [@Casimir]. This force has been observed in the last decade by several experiments [@Casimir-exp-1] and is interpreted as arising from the modification of the zero-point energies of the vacuum due to the presence of material boundaries.
The vacuum can also be seen as the triggering actor in the spontaneous decay of excited atomic states through a virtual photon stimulating the emission [@Purcell]. In that particular case experimentalists were able to change the vacuum, producing a huge increase [@Goy] or a decrease [@Hulet] of the spontaneous emission rate by a modification of the virtual photon density.
This model predicts that the local vacuum is also modified by a light beam because of photon capture by virtual pairs, which in a sense pumps the vacuum.
Let us apply the mechanism exposed in section \[sec:speedoflight\] to the propagation of a pulse when photon densities are not negligibly small compared to the $e^+e^-$ pair density.
If the pulse is fully circularly polarized, all its photons bear the same helicity. So, a photon cached by a pair makes it transparent to the other incoming photons, till it jumps on another one.
If the photon density is $N_\gamma$, the fraction of $i-$type species masked this way is, to first order in $N_\gamma$, equal to: $$\begin{aligned}
\label{eq:masked}
\Delta N_i/N_i=N_\gamma K_\sigma \sigma_i c K_\tau\tau_i .\end{aligned}$$ So, from (\[eq:9\]) $$\begin{aligned}
\label{eq:masked-2}
\Delta N_i=N_\gamma N_i \sigma_i c\tau_i .\end{aligned}$$ The remaining densities available to interact with photons are $N_i-\Delta N_i$. So the speed of light is given by : $$\begin{aligned}
\label{eq:cstar}
\tilde{c}^* = \frac{1}{\sum_{i}{\sigma_i (N_i-\Delta N_i)\tau_i}} =\frac{1}{\sum_{i}{\sigma_i N_i
\tau_i(1-N_\gamma\sigma_i c\tau_i )}} .\end{aligned}$$
$\sigma_i \tau_i$ being proportional to ${{\lambda^3_C}_i}$, we keep only the $e^+e^-$ contribution in the corrective term. Using (\[eq:c-1\]), it comes $$\begin{aligned}
\label{eq:cstar-2}
\tilde{c}^* =\frac{ c}{1- N_\gamma N_e\sigma_e^2 c^2\tau_e^2}\, .\end{aligned}$$
Noticing that $N_e\sigma_e \tau_e = 1/8c$, this reduces to: $$\begin{aligned}
\label{eq:cstar-2}
\tilde{c}^* =\frac{ c}{1-N_\gamma/(64 N_e)}\, .\end{aligned}$$
So, one ends up with: $$\begin{aligned}
\label{dcc}
\frac{\delta{c}}{c} =\frac{N_\gamma}{64 N_e}\, ,\end{aligned}$$ which shows that $c$ would be an increasing function of the photon densities. This anti Kerr effect is directly related to the $e^+e^-$ pair density. One can express it as a function of $K_W$, using (\[eq:1\]) and (\[eq:9\]): $$\begin{aligned}
\label{dcckw}
\frac{\delta{c}}{c} =\frac{N_\gamma\pi\alpha{{\lambda^3_C}_e}}{6K_W}\, .\end{aligned}$$
This prediction could in principle be tested in a dedicated laboratory experiment where one huge intensity pump pulse would be used to stress the vacuum and change the transit time of a small probe pulse going in the same direction and having the same circular polarization (or going in the opposite direction with the opposite polarization). Other helicity combinations would give no effect on the transit time.
Let us convert Eq. \[dcckw\] into numbers. Using $K_W=31.9$, $N_e$ amounts to: $$\begin{aligned}
\label{eq:densnum-e+e-}
N_e \approx 2\ 10^{39}\ e^+e^-/m^3 \,.\end{aligned}$$
Now, the photon density in a pulse of power $P$, frequency $\omega$ and section $S$ is: $$\begin{aligned}
\label{eq:densphot}
N_\gamma = \frac{P}{\hbar\omega S c} = \frac{P\lambda}{2\pi\hbar S c^2}\,.\end{aligned}$$ A petawatt source at $\lambda\approx .5\,\mu m$, such as the one of Ref. [@Bayramian], allows to reach focused irradiances of the order of $P/S=10^{23}W/cm^2$ in a few $\lambda^3$ volume. This means photons densities in the range: $$\begin{aligned}
N_\gamma = \frac{10^{27}\ .5 10^{-6}}{6.6\, 10^{-34}\ 9\, 10^{16}}= 10^{37}\ \gamma/ m^{3} \,,\label{eq:densphot}\end{aligned}$$ leading to: $$\begin{aligned}
\label{eq:ratio}
N_\gamma/N_e \approx 5\,10^{-3}\,,\end{aligned}$$ and a relative effect on $c$ of $8\,10^{-5}$ over a short length. This would affect the time transit of a probe pulse focused through the same volume. Creating a $1\ fs$ advance on that probe pulse would need a common travel with such a pump pulse over a distance of $4\ mm$. This seems a difficult challenge, but we think that this matter deserves a specific study to build a real proposal for testing this prediction.
Conclusions
===========
We describe the ground state of the unperturbed vacuum as containing a finite density of charged virtual fermions. Within this framework, the finite speed of light is due to successive transient captures of the photon by these virtual particles. $\epsilon_0$ and $\mu_0$ originate also simply from the electric polarization and from the magnetization of these virtual particles when the vacuum is stressed by an electrostatic or a magnetostatic field respectively. Our calculated values for $c$, $\epsilon_0$ and $\mu_0$ are equal to the measured values when the virtual fermion pairs are produced with an average energy of about $30$ times their rest mass. This model is self consistent and it proposes a quantum origin to the three electromagnetic constants. The propagation of a photon being a statistical process, we predict fluctuations of the speed of light. It is shown that this could be within the grasp of nowadays experimental techniques and we plan to assemble such an experiment. Another prediction is a light propagation faster than $c$ in a high density photon beam.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank N. Bhat, J.P. Chambaret , I. Cognard, J. Haïssinski, P. Indelicato, J. Kaplan, P. Wolf and F. Zomer for fruitful discussions, and J. Degert, E. Freysz, J. Oberl' e and M. Tondusson for their collaboration on the experimental aspects. This work has benefited from a GRAM[^2] funding.
[00]{} W.E. Lamb and R.C. Retherford, Phys. Rev. 72 (1947) 241-243. I. Levine et al., Phys. Rev. Lett. 78 (1997) 424-427. J. Schwinger, Phys. Rev. 73 (1948) 416-417. M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur.Phys.J. C71 (2011) 1515. Ch. Kittel, Elementary Solid State Physics, John Wiley & Sons (1962) 120. G. Leuchs, A.S. Villar and L.L. Sanchez-Soto, Appl. Phys. B 100 (2010) 9-13. Yu H. and Ford L.H., Phys. Lett. B 496 (2000) 107-112. C. J. Hogan, ArXiv.org 1002.4880 (2011). Abdo A.A. et al., Nature 462 (2009) 331-334. Crossley J.H. et al., The Astrophysical Journal 722 (2010) 1908-1920. Hankins T.H. and Eilek J.A., The Astrophysical Journal 670 (2007) 693-701. H. Casimir, Phys. Rev., 73 (1948) 360. S.K. Lamoreaux, Rep. Prog. Phys. 68 (2005) 201. E.M. Purcell, Phys. Rev. 69 (1946) 681. P. Goy, J.M. Raymond, M. Gross and S. Haroche Phys. Rev. Lett. 50 (1983) 1903. R. G. Hulet, E.S. Hilfer and D. Kleppner, Phys. Rev. Lett. 55 (1985) 2137. J. I. Latorre et al., Nuclear Physics B437 (1995) 60. E. Goulielmakis et al., Science 320, 1614 (2008). A. Bayramian et al., JOSA B 25 - Issue 7 (2008) B57-B61.
[^1]: [email protected]
[^2]: CNRS INSU/INP program with CNES & ONERA participations (Action Sp' ecifique “Gravitation, Références, Astronomie, Métrologie”)
| ArXiv |
---
abstract: 'Accurate depth estimation from images is a fundamental task in many applications including scene understanding and reconstruction. Existing solutions for depth estimation often produce blurry approximations of low resolution. This paper presents a convolutional neural network for computing a high-resolution depth map given a single RGB image with the help of transfer learning. Following a standard encoder-decoder architecture, we leverage features extracted using high performing pre-trained networks when initializing our encoder along with augmentation and training strategies that lead to more accurate results. We show how, even for a very simple decoder, our method is able to achieve detailed high-resolution depth maps. Our network, with fewer parameters and training iterations, outperforms state-of-the-art on two datasets and also produces qualitatively better results that capture object boundaries more faithfully. Code and corresponding pre-trained weights are made publicly available[^1].'
author:
- |
Ibraheem Alhashim\
KAUST\
[[email protected]]{}
- |
Peter Wonka\
KAUST\
[[email protected]]{}
bibliography:
- 'zbib.bib'
title: High Quality Monocular Depth Estimation via Transfer Learning
---
Introduction
============
Depth estimation from 2D images is a fundamental task in many applications including scene understanding and reconstruction [@Lee2011; @moreno2007active; @Hazirbas2016FuseNetID]. Having a dense depth map of the real-world can be very useful in applications including navigation and scene understanding, augmented reality [@Lee2011], image refocusing [@moreno2007active], and segmentation [@Hazirbas2016FuseNetID]. Recent developments in depth estimation are focusing on using convolutional neural networks (CNNs) to perform 2D to 3D reconstruction. While the performance of these methods has been steadily increasing, there are still major problems in both the quality and the resolution of these estimated depth maps. Recent applications in augmented reality, synthetic depth-of-field, and other image effects [@Hedman2018; @Cao2018; @Wang2018] require fast computation of high resolution 3D reconstructions in order to be applicable. For such applications, it is critical to faithfully reconstruct discontinuity in the depth maps and avoid the large perturbations that are often present in depth estimations computed using current CNNs.
![**Comparison of estimated depth maps:** input RGB images, ground truth depth maps, our estimated depth maps, state-of-the-art results of [@Fu2018DeepOR].[]{data-label="fig:teaser"}](teaser){width="\linewidth"}
Based on our experimental analysis of existing architectures and training strategies [@Eigen2014; @Li2015; @Laina2016; @Xu2017; @Fu2018DeepOR] we set out with the design goal to develop a simpler architecture that makes training and future modifications easier. Despite, or maybe even due to its simplicity, our architecture produces depth map estimates of higher accuracy and significantly higher visual quality than those generated by existing methods (see Fig. \[fig:teaser\]). To achieve this, we rely on transfer learning were we repurpose high performing pre-trained networks that are originally designed for image classification as our deep features encoder. A key advantage of such a transfer learning-based approach is that it allows for a more modular architecture where future advances in one domain are easily transferred to the depth estimation problem.
{width="\linewidth"}
#### Contributions:
Our contributions are threefold. First, we propose a simple transfer learning-based network architecture that produces depth estimations of higher accuracy and quality. The resulting depth maps capture object boundaries more faithfully than those generated by existing methods with fewer parameters and less training iterations. Second, we define a corresponding loss function, learning strategy, and simple data augmentation policy that enable faster learning. Third, we propose a new testing dataset of photo-realistic synthetic indoor scenes, with perfect ground truth, to better evaluate the generalization performance of depth estimating CNNs.
We perform different experiments on several datasets to evaluate the performance and quality of our depth estimating network. The results show that our approach not only outperforms the state-of-the-art and produces high quality depth maps on standard depth estimation datasets, but it also results in the best generalization performance when applied to a novel dataset.
Related Work
============
The problem of 3D scene reconstruction from RGB images is an ill-posed problem. Issues such as lack of scene coverage, scale ambiguities, translucent or reflective materials all contribute to ambiguous cases where geometry cannot be derived from appearance. In practice, the more successful approaches for capturing a scene’s depth rely on hardware assistance, e.g. using laser or IR-based sensors, or require a large number of views captured using high quality cameras followed by a long and expensive offline reconstruction process. Recently, methods that rely on CNNs are able to produce reasonable depth maps from a single or couple of RGB input images at real-time speeds. In the following, we look into some of the works that are relevant to the problem of depth estimation and 3D reconstruction from RGB input images. More specifically, we look into recent solutions that depend on deep neural networks.
#### Monocular depth estimation
has been considered by many CNN methods where they formulate the problem as a regression of the depth map from a single RGB image [@Eigen2014; @Laina2016; @Xu2017; @Hao2018DetailPD; @Xu2018StructuredAG; @Fu2018DeepOR]. While the performance of these methods have been increasing steadily, general problems in both the quality and resolution of the estimated depth maps leave a lot of room for improvement. Our main focus in this paper is to push towards generating higher quality depth maps with more accurate boundaries using standard neural network architectures. Our preliminary results do indicate that improvements on the state-of-the-art are possible to achieve by leveraging existing simple architectures that perform well on other computer vision tasks.
#### Multi-view
stereo reconstruction using CNN algorithms have been recently proposed [@Huang2018DeepMVSLM]. Prior work considered the subproblem that looks at image pairs [@Ummenhofer2017], or three consecutive frames [@Godard2018DiggingIS]. Joint key-frame based dense camera tracking and depth map estimation was presented by [@Zhou2018DeepTAMDT]. In this work, we seek to push the performance for single image depth estimation. We suspect that the features extracted by monocular depth estimators could also help derive better multi-view stereo reconstruction methods.
#### Transfer learning
approaches have been shown to be very helpful in many different contexts. In recent work, Zamir et al. investigated the efficiency of transfer learning between different tasks [@Zamir2018TaskonomyDT], many of which were are related to 3D reconstruction. Our method is heavily based on the idea of transfer learning where we make use of image encoders originally designed for the problem of image classification [@huang2017densely]. We found that using such encoders that do not aggressively downsample the spatial resolution of the input tend to produce sharper depth estimations especially with the presence of skip connections.
#### Encoder-decoder
networks have made significant contributions in many vision related problems such as image segmentation [@Ronneberger2015u], optical flow estimation [@Dosovitskiy2015], and image restoration [@LehtinenMHLKAA18]. In recent years, the use of such architectures have shown great success both in the supervised and the unsupervised setting of the depth estimation problem [@Godard2017; @Ummenhofer2017; @Huang2018DeepMVSLM; @Zhou2018DeepTAMDT]. Such methods typically use one or more encoder-decoder network as a sub part of their larger network. In this work, we employ a single straightforward encoder-decoder architecture with skip connections (see Fig. \[fig:network\_overview\]). Our results indicate that it is possible to achieve state-of-the-art high quality depth maps using a simple encoder-decoder architecture.
Proposed Method {#sec:method}
===============
In this section, we describe our method for estimating a depth map from a single RGB image. We first describe the employed encoder-decoder architecture. We then discuss our observations on the complexity of both encoder and decoder and its relation to performance. Next, we propose an appropriate loss function for the given task. Finally, we describe efficient augmentation policies that help the training process significantly.
Network Architecture
--------------------
#### Architecture.
Fig. \[fig:network\_overview\] shows an overview of our encoder-decoder network for depth estimation. For our *encoder*, the input RGB image is encoded into a feature vector using the DenseNet-169 [@huang2017densely] network pretrained on ImageNet [@Deng2009]. This vector is then fed to a successive series of up-sampling layers [@LehtinenMHLKAA18], in order to construct the final depth map at half the input resolution. These upsampling layers and their associated skip-connections form our *decoder*. Our decoder does not contain any Batch Normalization [@Ioffe2015BNA] or other advanced layers recommended in recent state-of-the-art methods [@Fu2018DeepOR; @Hao2018DetailPD]. Further details about the architecture and its layers along with their exact shapes are described in the appendix.
#### Complexity and performance.
The high performance of our surprisingly simple architecture gives rise to questions about which components contribute the most towards achieving these quality depth maps. We have experimented with different state-of-the-art encoders [@Bianco2018], of more or less complexity than that of DenseNet-169, and we also looked at different decoder types [@Laina2016; @Wojna2017TheDI]. What we experimentally found is that, in the setting of an encoder-decoder architecture for depth estimation, recent trends of having convolutional blocks exhibiting more complexity do not necessarily help the performance. This leads us to advocate for a more thorough investigation when adopting such complex components and architectures. Our experiments show that a simple decoder made of a $2\times$ bilinear upsampling step followed by two standard convolutional layers performs very well.
{width="\linewidth"}
Learning and Inference
----------------------
#### Loss Function.
A standard loss function for depth regression problems considers the difference between the ground-truth depth map $y$ and the prediction of the depth regression network $\hat{y}$ [@Eigen2014]. Different considerations regarding the loss function can have a significant effect on the training speed and the overall depth estimation performance. Many variations on the loss function employed for optimizing the neural network can be found in the depth estimation literature [@Eigen2014; @Laina2016; @Ummenhofer2017; @Fu2018DeepOR]. In our method, we seek to define a loss function that balances between reconstructing depth images by minimizing the difference of the depth values while also penalizing distortions of high frequency details in the image domain of the depth map. These details typically correspond to the boundaries of objects in the scene.
For training our network, we define the loss $L$ between $y$ and $\hat{y}$ as the weighted sum of three loss functions: $$L(y,\hat{y}) = \lambda L_{depth}(y,\hat{y}) + L_{grad}(y,\hat{y}) + L_{SSIM}(y,\hat{y}).$$
The first loss term $L_{depth}$ is the point-wise L1 loss defined on the depth values: $$L_{depth}(y,\hat{y}) = \frac{1}{n} \sum_{p}^{n} \lvert y_p -\hat{y}_p \rvert.$$
The second loss term $L_{grad}$ is the L1 loss defined over the image gradient $\boldsymbol{g}$ of the depth image: $$L_{grad}(y,\hat{y}) = \frac{1}{n} \sum_{p}^{n} \lvert \boldsymbol{g_\mathrm{x}}(y_p,\hat{y}_p) \rvert + \lvert \boldsymbol{g_\mathrm{y}}(y_p,\hat{y}_p) \rvert$$ where $\boldsymbol{g_\mathrm{x}}$ and $\boldsymbol{g_\mathrm{y}}$, respectively, compute the differences in the $\mathrm{x}$ and $\mathrm{y}$ components for the depth image gradients of $y$ and $\hat{y}$.
Lastly, $L_{SSIM}$ uses the Structural Similarity (SSIM) [@Wang2004SSIM] term which is a commonly-used metric for image reconstruction tasks. It has been recently shown to be a good loss term for depth estimating CNNs [@Godard2017]. Since SSIM has an upper bound of one, we define it as a loss $L_{SSIM}$ as follows: $$L_{SSIM}(y,\hat{y}) = \frac{1 - SSIM(y,\hat{y})}{2}.$$ Note that we only define one weight parameter $\lambda$ for the loss term $L_{depth}$. We empirically found and set $\lambda=0.1$ as a reasonable weight for this term.
An inherit problem with such loss terms is that they tend to be larger when the ground-truth depth values are bigger. In order to compensate for this issue, we consider the reciprocal of the depth [@Ummenhofer2017; @Huang2018DeepMVSLM] where for the original depth map $y_{orig}$ we define the target depth map $y$ as $y = m / y_{orig}$ where $m$ is the maximum depth in the scene (e.g. $m=10$ meters for the NYU Depth v2 dataset). Other methods consider transforming the depth values and computing the loss in the log space [@Eigen2014; @Ummenhofer2017].
#### Augmentation Policy.
Data augmentation, by geometric and photo-metric transformations, is a standard practice to reduce over-fitting leading to better generalization performance [@krizhevsky2012imagenet]. Since our network is designed to estimate depth maps of an entire image, not all geometric transformations would be appropriate since distortions in the image domain do not always have meaningful geometric interpretations on the ground-truth depth. Applying a vertical flip to an image capturing an indoor scene may not contribute to the learning of expected statistical properties (e.g. geometry of the floors and ceilings). Therefore, we only consider horizontal flipping (i.e. mirroring) of images at a probability of $0.5$. Image rotation is another useful augmentation strategy, however, since it introduces invalid data for the corresponding ground-truth depth we do not include it. For photo-metric transformations we found that applying different color channel permutations, e.g. swapping the red and green channels on the input, results in increased performance while also being extremely efficient. We set the probability for this color channel augmentation to $0.25$. Finding improved data augmentation policies and their probability values for the problem of depth estimation is an interesting topic for future work [@Cubuk2018AutoAugmentLA].
Experimental Results
====================
In this section we describe our experimental results and compare the performance of our network to existing state-of-the-art methods. Furthermore, we perform ablation studies to analyze the influence of the different parts of our proposed method. Finally, we compare our results on a newly proposed dataset of high quality depth maps in order to better test the generalization and robustness of our trained model.
Datasets
--------
#### NYU Depth v2
is a dataset that provides images and depth maps for different indoor scenes captured at a resolution of $640\times480$ [@Silberman2012]. The dataset contains 120K training samples and 654 testing samples [@Eigen2014]. We train our method on a 50K subset. Missing depth values are filled using the inpainting method of [@Levin2004]. The depth maps have an upper bound of 10 meters. Our network produces predictions at half the input resolution, i.e. a resolution of $320\times240$. For training, we take the input images at their original resolution and downsample the ground truth depths to $320\times240$. Note that we do not crop any of the input image-depth map pairs even though they contain missing pixels due to a distortion correction preprocessing. During test time, we compute the depth map prediction of the full test image and then upsample it by $2\times$ to match the ground truth resolution and evaluate on the pre-defined center cropping by Eigen et al. [@Eigen2014]. At test time, we compute the final output by taking the average of an image’s prediction and the prediction of its mirror image.
#### KITTI
is a dataset that provides stereo images and corresponding 3D laser scans of outdoor scenes captured using equipment mounted on a moving vehicle [@geiger2013vision]. The RGB images have a resolution of around $1241\times376$ while the corresponding depth maps are of very low density with lots of missing data. We train our method on a subset of around 26K images, from the left view, corresponding to scenes not included in the 697 test set specified by [@Eigen2014]. Missing depth values are filled using the inpainting method mentioned earlier. The depth maps have an upper bound of 80 meters. Our encoder’s architecture expects image dimensions to be divisible by 32 [@huang2017densely], therefore, we upsample images bilinearly to $1280\times384$ during training. During testing, we first scale the input image to the expected resolution and then upsample the output depth image from $624\times192$ to the original input resolution. The final output is computed by taking the average of an image’s prediction and the prediction of its mirror image.
{width="\linewidth"}
Method $\delta_{1}\uparrow$ $\delta_{2}\uparrow$ $\delta_{3}\uparrow$ rel$\downarrow$ rms$\downarrow$ $log_{10}\downarrow$
------------------------------- ---------------------- ---------------------- ---------------------- ----------------- ----------------- ----------------------
Eigen et al. [@Eigen2014] 0.769 0.950 0.988 0.158 0.641 -
Laina et al. [@Laina2016] 0.811 0.953 0.988 0.127 0.573 0.055
MS-CRF [@Xu2017] 0.811 0.954 0.987 0.121 0.586 0.052
Hao et al. [@Hao2018DetailPD] 0.841 0.966 0.991 0.127 0.555 0.053
Fu et al. [@Fu2018DeepOR] 0.828 0.965 0.992 **0.115** 0.509 **0.051**
Ours **0.846** **0.974** **0.994** 0.123 **0.465** 0.053
Ours (scaled) **0.895** **0.980** **0.996** **0.103** **0.390** **0.043**
Method $\delta_{1}\uparrow$ $\delta_{2}\uparrow$ $\delta_{3}\uparrow$ rel$\downarrow$ sq. rel$\downarrow$ rms$\downarrow$ $log_{10}\downarrow$
------------------------------------- ---------------------- ---------------------- ---------------------- ----------------- --------------------- ----------------- ----------------------
Eigen et al. [@Eigen2014] 0.692 0.899 0.967 0.190 1.515 7.156 0.270
Godard et al. [@Godard2017] 0.861 0.949 0.976 0.114 0.898 4.935 0.206
Kuznietsov et al. [@Kuznietsov2017] 0.862 0.960 0.986 0.113 0.741 4.621 0.189
Fu et al. [@Fu2018DeepOR] **0.932** **0.984** **0.994** **0.072** **0.307** **2.727** **0.120**
Ours
Implementation Details
----------------------
We implemented our proposed depth estimation network using TensorFlow [@tensorflow2015-whitepaper] and trained on four NVIDIA TITAN Xp GPUs with 12GB memory. Our encoder is a DenseNet-169 [@huang2017densely] pretrained on ImageNet [@Deng2009]. The weights for the decoder are randomly initialized following [@glorot2010understanding]. In all experiments, we used the ADAM [@jlb2015adam] optimizer with learning rate $0.0001$ and parameter values $\beta_1=0.9$, $\beta_2=0.999$. The batch size is set to 8. The total number of trainable parameters for the entire network is approximately 42.6M parameters. Training is performed for 1M iterations for NYU Depth v2, needing 20 hours to finish. Training for the KITTI dataset is performed for 300K iterations, needing 9 hours to train.
Evaluation
----------
#### Quantitative evaluation.
We quantitatively compare our method against state-of-the-art using the standard six metrics used in prior work [@Eigen2014]. These error metrics are defined as:
- average relative error (rel): $\frac{1}{n}\sum_p^n \frac{\lvert y_p-\hat{y}_p \rvert}{y}$;
- root mean squared error (rms): $\sqrt{\frac{1}{n}\sum_p^n (y_p-\hat{y}_p)^2)}$;
- average ($\log_{10}$) error: $\frac{1}{n}\sum_p^n \lvert \log_{10}(y_p)-\log_{10}(\hat{y}_p) \rvert$;
- threshold accuracy ($\delta_i$): $\%$ of $y_p$ s.t. $\text{max}(\frac{y_p}{\hat{y}_p},\frac{\hat{y}_p}{y_p}) = \delta < thr$ for $thr=1.25,1.25^2,1.25^3$;
where $y_p$ is a pixel in depth image $y$, $\hat{y}_p$ is a pixel in the predicted depth image $\hat{y}$, and $n$ is the total number of pixels for each depth image.
#### Qualitative results.
We conduct three experiments to approximately evaluate the quality of the results using three measures on the NYU Depth v2 test set. The first measure is a perception-based qualitative metric that measures the quality of the results by looking at the similarity of the resulting depth maps in image space. We do so by rendering a gray scale visualization of the ground truth and that of the predicted depth map and then we compute the mean structural similarity term (mSSIM) of the entire test dataset $\frac{1}{T} \sum_i^T {SSIM}(y_i,\hat{y}_i)$. The second measure considers the edges formed in the depth map. For each sample, we compute the gradient magnitude image of both the ground truth and the predicted depth image, using the Sobel gradient operator [@Sobel1968], and then threshold this image at values greater than 0.5 and compute the F1 score averaged across the set. The third measure is the mean cosine distance between normal maps extracted from the depth images of the ground truth and the predicted depths also averaged across the set. Fig. \[fig:qualitative\] shows visualizations of some of these measures.
Fig. \[fig:gallery\] shows a gallery of depth estimation results that are predicated using our method along with a comparison to those generated by the state-of-the-art. As can be seen, our approach produces depth estimations at higher quality where depth edges better match those of the ground truth and with significantly fewer artifacts.
Method mSSIM$\uparrow$ F1$\uparrow$ mne$\downarrow$
--------------------------- ----------------- -------------- -----------------
Laina et al. [@Laina2016] 0.957 0.395 0.698
Fu et al. [@Fu2018DeepOR] 0.949 0.351 0.730
**Ours** **0.968** **0.519** **0.636**
: **Qualitative evaluation.** For the NYU Depth v2 testing set, we compute three measures that reflect the quality of the depth maps generated by different methods. The measures are: mean SSIM of the depth maps, mean F1 score of the edge maps, and mean of the surface normal errors. Higher values indicate better quality for the first two measures while lower values are better for the third. []{data-label="tab:2"}
Comparing Performance
---------------------
In Tab. \[tab:1\], the performance of our depth estimating network is compared to the state-of-the-art on the NYU Depth v2 dataset. As can be seen, our model achieves state-of-the-art on all but two quantitative metrics. Our model is able to outperform the existing state-of-the-art [@Fu2018DeepOR] while requiring fewer parameters, 42.6M vs 110M, fewer number of training iterations, 1M vs 3M, and with fewer input training data, 50K samples vs 120K samples. A typical source of error for single image depth estimation networks is the estimated absolute scale of the scene. The last row in Tab. \[tab:1\] shows that when accounting for this error, by multiplying the predicted depths by a scalar that matches the median with the ground truth [@Zhou2017], we are able to achieve with a good margin state-of-the-art for the NYU Depth v2 dataset on all metrics. The results in Tab. \[tab:2\] show that for the same dataset our method outperforms state-of-the-art on our defined quality approximating measures. We conduct these experiments for methods with published pre-trained models and code.
In Tab. \[tab:kitti\], the performance of our network is compared to the state-of-the-art on the KITTI dataset. Our method is the second best on all the standard metrics. We suspect that one reason our method does not outperform the state-of-the-art on this particular dataset is due to the nature of the provided depth maps. Since our loss function is designed to not only consider point-wise differences but also optimize for edges and appearance preservation by looking at regions around each point, the learning process does not converge well for very sparse depth images. Fig. \[fig:kitti\] clearly shows that while quantitatively our method might not be the best, the quality of the produced depth maps is much better than those produced by the state-of-the-art.
Ablation Studies
----------------
We perform ablation studies to analyze the details of our proposed architecture. Fig. \[fig:ablation\] shows a representative look into the testing performance, in terms of validation loss, when changing some parts of our standard model or modifying our training strategy. Note that we performed these tests on a smaller subset of the NYU Depth v2 dataset.
#### Encoder depth.
In this experiment we substitute the pretrained DenseNet-169 with a denser encoder, namely the DenseNet-201. In Fig. \[fig:ablation\] (red), we can see the validation loss is lower than that of our standard model. The big caveat, though, is that the number of parameters in the network grows by more than $2\times$. When considering using DenseNet-201 as our encoder, we found that the gains in performance did not justify the slow learning time and the extra GPU memory required.
#### Decoder depth.
In this experiment we apply a depth reducing convolution such that the features feeding into the decoder are half what they are in the standard DenseNet-169. In Fig. \[fig:ablation\] (blue), we see a reduction in the performance and overall instability. Since these experiments are not representative of a full training session the performance difference in halving the features might not be as visible as we have observed when running full training session.
#### Color Augmentation.
In this experiment, we turn off our color channel swapping-based data augmentation. In Fig. \[fig:ablation\] (green), we can see a significant reduction as the model tends to quickly falls into overfitting to the training data. We think this simple data augmentation and its significant effect on the neural network is an interesting topic for future work.
Generalizing to Other Datasets
------------------------------
To illustrate how well our method generalizes to other datasets, we propose a new dataset of photo-realistic indoor scenes with nearly perfect ground truth depths. These scenes are collected from the Unreal marketplace community [@UnrealMarket2018]. We refer to this dataset as **Unreal-1k**. It is a random sampling of 1000 images with their corresponding depth maps selected from renderings of 32 virtual scenes using the Unreal Engine. Further details about this dataset can be found in the appendix. We compare our NYU Depth v2 trained model to two supervised methods that are also trained on the same dataset. For inference, we use the public implementations for each method. The hope of this experiment is to demonstrate how well do models trained on one dataset perform when presented with data sampled from a different distribution (i.e. synthetic vs. real, perfect depth capturing vs. a Kinect, etc.).
Tab. \[tab:1\] shows quantitative comparisons in terms of the average errors over the entire Unreal-1k dataset. As can be seen, our method outperforms the other two methods. We also compute the qualitative measure mSSIM described earlier. Fig. \[fig:unreal\] presents a visual comparison of the different predicted depth maps against the ground truth.
![**Ablation Studies.** Three variations on our standard model are considered. *DenseNet-201* (red) refers to a deeper version of the encoder. The *half decoder* variation (blue) represents the model with only half the features coming out of the last layer in the encoder. Lastly, we consider the performance when disabling the *color-swapping* data augmentations (green). []{data-label="fig:ablation"}](ablation){width="\linewidth"}
Method $\delta_{1}\uparrow$ $\delta_{2}\uparrow$ $\delta_{3}\uparrow$ rel$\downarrow$ rms$\downarrow$ $log_{10}\downarrow$ mSSIM$\uparrow$
--------------------------- ---------------------- ---------------------- ---------------------- ----------------- ----------------- ---------------------- -----------------
Laina et al. [@Laina2016] 0.526 0.786 0.896 0.311 1.049 0.130 0.903
Fu et al. [@Fu2018DeepOR] **0.545** 0.794 0.898 0.313 1.040 0.128 0.895
**Ours** 0.544 **0.803** **0.904** **0.301** **1.030** **0.125** **0.910**
{width="\linewidth"}
{width="0.98\linewidth"}
Conclusion
==========
In this work, we proposed a convolutional neural network for depth map estimation for single RGB images by leveraging recent advances in network architecture and the availability of high performance pre-trained models. We show that having a well constructed encoder, that is initialized with meaningful weights, can outperform state-of-the-art methods that rely on either expensive multistage depth estimation networks or require designing and combining multiple feature encoding layers. Our method achieves state-of-the-art performance on the NYU Depth v2 dataset and our proposed Unreal-1K dataset. Our aim in this work is to push towards generating higher quality depth maps that capture object boundaries more faithfully, and we have shown that this is indeed possible using an existing architectures. Following our simple architecture, one avenue for future work is to substitute the proposed encoder with a more compact one in order to enable quality depth map estimation on embedded devices. We believe their are still many possible cases of leveraging standard encoder-decoder models alongside transfer learning for high quality depth estimation. Many questions on the limits of our proposed network and identifying more clearly the effect on performance and contribution of different encoders, augmentations, and learning strategies are all interesting to purse for future work.
Appendix
========
Network Architecture
--------------------
Tab. \[table:1\] shows the structure of our encoder-decoder with skip connections network. Our encoder is based on the DenseNet-169 [@huang2017densely] network where we remove the top layers that are related to the original ImageNet classification task. For our decoder, we start with a $1 \times 1$ convolutional layer with the same number of output channels as the output of our truncated encoder. We then successively add upsampling blocks each composed of a $2\times$ bilinear upsampling followed by two $3 \times 3$ convolutional layers with output filters set to half the number of inputs filters, and were the first convolutional layer of the two is applied on the concatenation of the output of the previous layer and the pooling layer from the encoder having the same spatial dimension. Each upsampling block, except for the last one, is followed by a leaky ReLU activation function [@Maas13LeRELU] with parameter $\alpha=0.2$. The input images are represented by their original colors in the range $[0,1]$ without any input data normalization. Target depth maps are clipped to the range $[0.4, 10]$ in meters.
The Unreal-1K Dataset
---------------------
We propose a new dataset of photo-realistic synthetic indoor scenes having near perfect ground truth depth maps. The scenes cover categories including living areas, kitchens, and offices all of which have realistic material and different lighting scenarios. These scenes, 32 scenes in total, are collected from the Unreal marketplace community [@UnrealMarket2018]. For each scene we select around 40 objects of interest and we fly a virtual camera around the object and capture images and their corresponding depth maps of resolution $640 \times 480$. In all, we collected more than 20K images from which we randomly choose 1K images as our testing dataset Unreal-1k. Fig. \[fig:append-fig1\] shows example images from this dataset along with depth estimations using various methods.
Additional Ablation Studies {#sec:sup_ablation}
---------------------------
![**Visual comparison of estimated depth maps on the Unreal-1K dataset:** input RGB images, ground truth depth maps, results using Laina et al. [@Laina2016], our estimated depth maps, results of Fu et al. [@Fu2018DeepOR]. Note that, for better visualization, we normalize each depth map with respect to the range of its corresponding ground truth.[]{data-label="fig:append-fig1"}](append_unreal.pdf){width="\linewidth"}
We perform additional ablation studies to analyze more details of our proposed architecture. Fig. \[fig:sup\_ablation\] shows a representative look into the testing performance, in terms of validation loss, when changing some parts of our standard model. The training in these experiments is performed on the NYU Depth v2 dataset [@Silberman2012] for 750K iterations (15 epochs).
#### Pre-trained model.
In this experiment, we examine the effect of using an encoder that is initialized using random weights as opposed to being pre-trained on ImageNet which is what we use in our proposed standard model. In Fig. \[fig:sup\_ablation\] (purple), we can see the validation loss is greatly increased when training from scratch. This further validates that the performance of our depth estimation is positively impacted by transfer learning.
#### Skip connections.
In this experiment, we examine the effect of removing the skip connections between layers of the encoder and decoder. In Fig. \[fig:sup\_ablation\] (green), we can see the validation loss is decreased, compared to our proposed standard model, resulting in worse depth estimation performance.
#### Batch size.
In this experiment, we look at different values for the batch size and its effect on performance. In Fig. \[fig:sup\_ablation\] (red and blue), we can see the validation loss for batch sizes 2 and 16 compared to our standard model (orange) with batch size 8. Setting the batch size to 8 results in the best performance out of the three values while also training for a reasonable amount of time.
![**Additional ablation studies.** Four variations on our standard model are considered. The horizontal axis represents the number of training iterations (in epochs). The vertical axis represents the average loss of the validation set at each epoch. Please see Sec. \[sec:sup\_ablation\] for more details. []{data-label="fig:sup_ablation"}](sup_ablation){width="\linewidth"}
[^1]: https://github.com/ialhashim/DenseDepth
| ArXiv |
---
author:
- 'Taiya <span style="font-variant:small-caps;">Munenaka</span> and Hirohiko <span style="font-variant:small-caps;">Sato</span>[^1]'
title: ' A Novel Pyrochlore Ruthenate: Ca$_{2}$Ru$_{2}$O$_{7}$ '
---
Frustration results in many types of unexpected phenomena. Among three-dimensional frustrated systems, the pyrochlore lattice is particularly interesting for its strong frustration originating from a purely geometric reason. A typcial pyrochlore oxide has the composition A$_{2}$B$_{2}$O$_{7}$. In this system, the B sites (and also the A sites) form a three-dimensional network based on the B$_{4}$ tetrahedron. From another point of view, we can regard the pyrochlore lattice as a three-dimensional version of a Kagomé lattice. Therefore, perfect geometric frustration is inherent in this structure, and many interesting phenomena emerge. For example, Y$_{2}$Mo$_{2}$O$_{7}$ exhibits spin-glass behavior,[@gingras97; @gardner99; @miyoshi00] demonstrating that the geometric frustration due to the antiferromagnetic pyrochlore lattice itself is responsible for the glassy state, even if there is no structural disorder. On the other hand, the nearest-neighbor interaction is ferromagnetic in Ho$_{2}$Ti$_{2}$O$_{7}$ and Dy$_{2}$Ti$_{2}$O$_{7}$. In this case, single-ion magnetic anisotropy causes another type of frustration and consequently, “spin ice” behavior appears.[@harris97; @ramirez99; @higashinaka05]
Conductive pyrochlores are also remarkable systems. For Nd$_{2}$Mo$_{2}$O$_{7}$, there was the epoch-making interpretation that the anomalous Hall effect detects the chirality of Nd magnetic moments.[@taguchi03] A theoretical study proved that the Berry phase plays an important role in systems with a chiral spin arrangement.[@onoda03] Tl$_{2}$Mn$_{2}$O$_{7}$ has metallic conductivity and undergoes a ferromagnetic transition. Near the transition temperature, a giant magnetoresistance appears[@shimakawa96]. In Cd$_{2}$Re$_{2}$O$_{7}$, a superconducting transition was discovered at 1.5 K.[@sakai01; @hanawa01] Furthermore, $\beta$-type pyrochlore osmates, AOs$_{2}$O$_{6}$ (A = K, Rb, Cs), also exhibit superconductivity with a relatively high $T_{c}$ (9.7 K for A = K).[@yonezawa04; @yonezawa04b; @hiroi04; @hiroi05]
While searching for new materials with exotic electronic states, we have become interested in pyrochlore ruthenates. Because ruthenium 4$d$-orbitals have a character intermediate between localized and itinerant orbitals, a variety of electronic phases appear. In particular, the discovery of spin-triplet superconductivity in Sr$_{2}$RuO$_{4}$[@maeno94; @ishida98] has aroused the interest of many material scientists.
Ruthenates with pyrochlore structures have also been actively investigated. Bi$_{2}$Ru$_{2}$O$_{7}$ and Pb$_{2}$Ru$_{2}$O$_{6.5}$ are metallic with Pauli paramagnetism,[@longo69; @cox83; @hsu88] whereas Ln$_{2}$Ru$_{2}$O$_{7}$ and Y$_{2}$Ru$_{2}$O$_{7}$ are insulators with localized magnetic moments.[@aleonard62; @subramanian83; @yoshii99; @ito00] Tl$_{2}$Ru$_{2}$O$_{7}$ undergoes a metal-insulator transition.[@takeda98] These observations reveal that pyrochlore ruthenates display a variety of electronic phases, widely distributed over the Mott boundary. Their electronic states are very sensitive to the Ru-O distance or the Ru-O-Ru bond angle, which is related to the radius of the cations on the A site. In addition to controlling the band width by changing the cation radius, filling control of the $4d$-band also seems important in searching for novel electronic phases. However, there have been few trials[@yoshii99] on controlling the band filling of pyrochlore ruthenates. This is probably because the cation on the A site is trivalent in most stable pyrochlore ruthenates. Apart from Cd$_{2}$Ru$_{2}$O$_{7}$[@wang98], there are no reports on stoichiometric pyrochlores composed of only Ru$^{5+}$. In the present study, we succeeded in synthesizing a new pyrochlore ruthenate with Ru$^{5+}$, Ca$_{2}$Ru$_{2}$O$_{7}$, by maintaining a high-oxidization atmosphere.
Single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$ were synthesized by a hydrothermal method. A mixture of RuO$_{2}$ (40 mg), obtained by oxidizing Ru metal (Furuya Metals, 99.99% purity), CaO (34 mg, Soekawa Chemical, 99.99% purity), and 0.3 ml of 30% H$_{2}$O$_{2}$ solution was encapsulated in a gold tube. Then, it was kept in an autoclave under 150 MPa hydrostatic pressure at 600$^{\circ}$C for 3 days. The chemical composition was determined using an energy dispersive X-ray spectrometer (EDS) installed on a scanning electron microscope. The crystal structure was analyzed using a single crystal and an imaging-plate X-ray diffractometer (Rigaku, R-Axis RAPID), in which Mo-K$\alpha$ radiation was generated using an X-ray tube and monochromized using graphite. We also used a powder X-ray diffractometer to check whether there was any contamination due to impurity phases. The magnetic susceptibility between 2 and 400 K was measured using a superconducting quantum-interference-device magnetometer. In the measurement, approximately 10 mg of nonoriented single crystals were wrapped in a piece of aluminum foil. The resistivity was measured by a DC four-wire method on an array of single crystals, connected with each other, in a closed-cycle helium refrigerator whose temperature range was between 5.5 and 300 K. The array was composed of four single crystals, and we attached the voltage leads to the same crystal located at the center. Therefore, we consider that the observed resistivity approximately reflects the behavior of a single crystal.
The obtained materials were black crystals with an octahedral shape. Observations using a microscope did not detect any other type of crystal as shown in Fig. \[fig1\](a). An EDS analysis showed that the atomic compositional ratio of Ca and Ru is almost 1:1. The single-crystal X-ray diffraction revealed an F-type cubic unit cell with $a = 10.197$ Å, which is very close to 10.143 Å for Y$_{2}$Ru$_{2}$O$_{7}$[@kennedy95]. This strongly suggested that the our material has a pyrochlore structure. A further structural refinement was carried out by observing about 2000 reflections at room temperature. The results are summarized in Table \[table1\],
------- ---------- ----------- ------- ------- ----------
Atom Position $x$ $y$ $z$ $B_{eq}$
Ru(1) 16$c$ 0 0 0 0.602(4)
Ca(1) 16$d$ 0.5 0.5 0.5 2.587(9)
O(1) 48$f$ 0.3219(1) 0.125 0.125 1.48(2)
O(2) 8$b$ 0.375 0.375 0.375 3.65(3)
------- ---------- ----------- ------- ------- ----------
: Fractional atomic coordinates and equivalent isotropic displacement parameters (Å$^{2}$) for Ca$_{2}$Ru$_{2}$O$_{7}$. The lattice symmetry and the space group are *cubic* and $Fd\bar{3}m$ (\#227), respectively. The lattice parameters are $a = 10.197(2)$ Å, $V = 1060.4(3)$ Å$^3$ and $Z=8$. The final reliability factor is $R(F) = 2.7 \%$ for 1888 observed reflections.[]{data-label="table1"}
![(a) Micrograph of single crystals of Ca$_{2}$Ru$_{2}$O$_{7}$. The typical dimensions of the crystals are $0.1 \times 0.1 \times 0.1$ mm$^{3}$. (b) X-ray powder pattern of batch used for magnetic measurement. The whole batch was ground before measurement. The weak peaks at 28.1$^{\circ}$ and at 54.3$^{\circ}$ are from traces of RuO$_{2}$.[]{data-label="fig1"}](fig1.eps){width="1.0\linewidth"}
and they coincide with those for a pyrochlore structure. The analysis did not detect any clear evidence that the composition deviates from the ideal pyrochlore, although the temperature factors of Ca(1) and O(2) seem unexpectedly large. We also analyzed the powder X-ray diffraction pattern. As shown in Fig. \[fig1\](b), almost all of the peaks are in agreement with those of a pyrochlore lattice, except for two weak peaks from the traces of RuO$_{2}$. To our best knowledge, there have been no reports on the existence of the Ca$_{2}$Ru$_{2}$O$_{7}$ phase despite many studies on pyrochlore ruthenates. There has only been a study on the mixed crystal Y$_{2-x}$Ca$_{x}$$_{2}$Ru$_{2}$O$_{7}$,[@yoshii99] although the upper limit of $x$ was 0.6. The reason why Ca$_{2}$Ru$_{2}$O$_{7}$ has not been successfully obtained is probably that a strong oxidization atmosphere is necessary for maintaining the Ru$^{5+}$ valence at high temperatures. We suppose that a hydrothermal reaction using a strong oxidant, H$_{2}$O$_{2}$, is advantageous for realizing this condition.
Figure \[fig2\](a) shows the temperature dependence of the magnetic susceptibility.
![(a) Magnetic susceptibility of Ca$_{2}$Ru$_{2}$O$_{7}$ at 0.1 T. The magnetization becomes irreversible below 23 K. The inset shows a comparison of the low-temperature susceptibilities under several magnetic fields. (b) Time dependence of magnetization. The sample was cooled under the ZFC condition down to 15 K. After waiting for 1800 s, a magnetic field of 0.1 T was applied. Then, the magnetization was recorded as a function of time.[]{data-label="fig2"}](fig2.eps){width="0.8\linewidth"}
The susceptibility above 30 K is almost perfectly reproduced using the function $$\label{eq1}
\chi=\frac{C}{T-\Theta} + \chi_{0},$$ where $C$, $\Theta$ and $\chi_{0}$ are the Curie constant, the Weiss temperature and a constant term independent of temperature, respectively. The best-fitted values are $C = 1.65 \times 10^{-2}$ emu$\cdot$K/mol Ru, $\Theta = -4.3$ K and $\chi_{0} = 5.95 \times 10^{-4} $ emu/mol Ru. Assuming a simple ionic model, the oxidation number of ruthenium is 5+. Therefore, there are three 4$d$ electrons per Ru atom, and each Ru atom has a $S=3/2$ spin. In this case, we can expect $\mu_{\mbox{\scriptsize eff}}= 3.87$ $\mu_{\mbox{\scriptsize B}}$. However, the observed $\mu_{\mbox{\scriptsize eff}}$ is only 0.36 $\mu_{\mbox{\scriptsize B}}$, smaller by one order of magnitude. On the other hand, the value of $\chi_{0}$ is comparable to that of typical Pauli paramagnetism in highly correlated oxides, which is consistent with the resistivity result, as shown below.
The most striking feature is the appearance of a sharp transition with an irreversibility between the field-cooled (FC) curve and the zero-field-cooled (ZFC) curve below 23 K. The ZFC curve exhibits a sharp cusp and decreases with decreasing temperature. On the other hand, the FC curve below 23 K is almost constant. This is a typical behavior of a spin glass. Under a higher magnetic field, the cusp becomes less sharp, and the onset temperature of the irreversibility becomes lower as shown in the inset. The time dependence of the magnetization below the glass-transition temperature is shown in Fig. \[fig2\](b). The magnetization exhibits an aging phenomenon, clearly demonstrating another characteristic of a spin-glass state.
Figure \[fig3\] shows the temperature dependence of the resistivity of Ca$_{2}$Ru$_{2}$O$_{7}$. The resistivity at room temperature is 2$\times$$10^{-3}$ $\Omega$cm, as large as a typical value for metallic, highly correlated oxides.
![Temperature dependence of resistivity of Ca$_{2}$Ru$_{2}$O$_{7}$ single crystal. The resistivity is also plotted as a function of $\log T$ in the inset.[]{data-label="fig3"}](fig3.eps){width="1.0\linewidth"}
The behavior is not that of a simple metal, because the temperature coefficient is negative over the whole temperature range measured between 5.5 and 300 K. In the lowest temperature region, the resistivity does not appear to saturate but continues to increase with decreasing temperature. However, the rise in the resistivity at low temperatures is less steep; the ratio, $\rho_{\mbox{\scriptsize 5.5K}} / \rho_{\mbox{\scriptsize 295K}}$, is only about 2. Clearly, an activation function, $\rho \propto \exp (E_{\mbox{\scriptsize A}}/k_{\mbox{\scriptsize B}}T)$, does not reproduce the observation, indicating that no gap is formed at the Fermi level. Therefore, the electronic state is basically metallic, and the $\chi_{0}$ term in the susceptibility can be interpreted in terms of Pauli paramagnetism. We also attempted to fit the data with a variable-range hopping funcition, $\rho \propto \exp \{ (E_{\mbox{\scriptsize A}}/k_{\mbox{\scriptsize B}}T)^{(1/(d+1))} \}$, but were unsuccessful. On the other hand, it seems that a $\rho$ vs $\log T$ plot begins to saturate at low temperatures, as shown in the inset. The resistivity shows no anomaly at the glass-transition temperature, $T_{g}$. This is not unusual for a conductive spin-glass system.[@taniguchi04]
It is known that Y$_{2}$Ru$_{2}$O$_{7}$ is an insulator but becomes metallic when Bi is substituted for Y.[@yoshii99] This is explained in terms of a Mott transition. The Ru-O-Ru bond angles are 129$^{\circ}$ and 139$^{\circ}$ in Y$_{2}$Ru$_{2}$O$_{7}$ and Bi$_{2}$Ru$_{2}$O$_{7}$, respectively.[@ishii00] The larger angle gives a broader band width, resulting in a metallic state. Ca substitution was also attempted by the same author. However, the upper limit of $x$ was 0.6 in Y$_{2-x}$Ca$_{x}$$_{2}$Ru$_{2}$O$_{7}$, and the electronic state was still that of an insulator. In Ca$_{2}$Ru$_{2}$O$_{7}$, the Ru-O-Ru bond angle is 135.72$^{\circ}$, larger than that in Y$_{2}$Ru$_{2}$O$_{7}$. This may be one of the reasons why Ca$_{2}$Ru$_{2}$O$_{7}$ is metallic. Furthermore, the change in the band filling is also important. Note that the isovalent Cd$_{2}$Ru$_{2}$O$_{7}$ is also metallic.[@wang98]
We now explain the origin of the localized spins, all of which undergo the spin-glass transition. One might interpret the data to conclude that Ca$_{2}$Ru$_{2}$O$_{7}$ itself exhibits only Pauli paramagnetism, and that the coexistence of a small amount of another phase contributes to the spin-glass behavior. However, our sample is an ensemble of small single crystals with well-defined shapes, and the powder X-ray diffraction demonstrated that there is no contamination, as shown in Fig. \[fig1\], except for a very small amount of RuO$_{2}$, which is known to exhibit Pauli paramagnetism.[@ryden70] We measured the magnetic susceptibility for three different batches. All the results quantitatively agreed; the ratio of the Curie-Weiss contribution and the Pauli paramagnetism is always the same. If the Curie-Weiss contribution were from an extrinsic origin, this ratio would vary from batch to batch. Therefore, it is most likely that the spin-glass behavior is an intrinsic feature of our phase.
Next, we clarify whether our pyrochlore includes some magnetic impurity atoms in the crystal lattice, because an inclusion of only 0.4 % of Fe$^{3+}$ ions may cause the observed magnetic moments. We can exclude this possibility because we used reagents of more than 99.99 % purity. In the Ru powder used in this study, the content of magnetic elements such as Fe was less than 14 ppm. In the CaO powder, it was less than 1 ppm. The H$_{2}$O$_{2}$ solution used was almost free of magnetic impurities.
Whatever the origin of the magnetic moments, it is unlikely that only nearest-neighbor interactions cause the spin-glass state, because the glass temperature is as high as 23 K despite the very low spin concentration. Therefore, there should be Ruderman-Kittel-Kasuya-Yoshida interactions mediated by the conduction electrons. In such a mechanism, the exchange interactions can be ferromagnetic or antiferromagnetic. The coexistence of these interactions reduces the absolute value of the Weiss temperature. In our crystal, the factor $T_{\mbox{\scriptsize g}} / |\Theta|$ is as large as 5.4. This is in contrast with the insulating spin-glass pyrochlore, Y$_{2}$Mo$_{2}$O$_{7}$, in which $T_{\mbox{\scriptsize g}} / |\Theta|$ is as small as 0.11, showing that geometric frustration is predominant in this molybdate.
Although our X-ray structural analysis did not detect any clear evidence, we cannot yet exclude the possibility of a deviation from ideal Ca$_{2}$Ru$_{2}$O$_{7}$ composition. If there are oxygen defects, for example, some of the Ru atoms are deoxidized into Ru$^{4+}$, possessing an excess electron. If these excess electrons behave as localized moments, they can cause a Curie-Weiss magnetism in addition to the Pauli paramagnetism coming from the conduction electrons. Another possibility is that the conduction electrons themselves have weak magnetic polarizations delocalized over many atoms, similar to those in stoner glass[@hertz79] and spin-density glass.[@sachdev95] In any case, there remains a fundamental question: why did the Ru 4$d$ electrons become partially magnetic despite the metallic conduction? Note that many perovskite-structure metallic ruthenates exhibit anomolous magnetism instead of ordinary Pauli paramagnetism. For example, CaRuO$_{3}$, which has metallic conduction, exhibits Curie-Weiss magnetic susceptibility with weak irreversibility.[@felner00] Therefore, the metallic state in Ca$_{2}$Ru$_{2}$O$_{7}$ might be so fragile that a small purturbation causes magnetic moments. This is in contrast with the study on Bi$_{2-x}$$M_{x}$Ru$_{2}$O$_{7}$ ($M=$ Mn, Fe, Co, Ni and Cu), where the substitution with magnetic atoms $M$ does not affect the Pauli paramagnetic state of the Ru-O sublattice of Bi$_{2}$Ru$_{2}$O$_{7}$.[@haas02]
In conclusion, we have discovered a novel pyrochlore ruthenate, Ca$_{2}$Ru$_{2}$O$_{7}$. The magnetic susceptibility exhibits the behavior of a spin glass in addition to Pauli paramagnetism, although the effective magnetic moment of the spin glass was smaller than expected by one order of magnitude. The resistivity of $2 \times 10^{-3}$ $\Omega$cm at room temperature suggests a metallic electronic state, but the $\rho$ vs $T$ curve has negative slope over the whole temperature region, indicating that the conduction electrons are strongly scattered by the magnetic moments. In future studies, we intended to first verify whether the spin-glass behavior is intrinsic in Ca$_{2}$Ru$_{2}$O$_{7}$. Detailed measurements of the transport properties under a magnetic field will be useful, because they may detect chirality in the spin-glass state.[@taniguchi04] NMR or neutron-scattering measurements are necessary to clarify the origin of the magnetic moments and the mechanism of the spin glass. Investigation of the electronic phase diagram by doping with Y will also be important.
This study was supported by Grants-in-Aid for Scientific Research No. 15750127 and No. 17540341 from the Ministry of Education, Culture, Sports, Science and Technology.
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[^1]: E-mail address: [email protected]
| ArXiv |
---
abstract: 'A fuzzy expert system (FES) for the prediction of prostate cancer (PC) is prescribed in this article. Age, prostate-specific antigen (PSA), prostate volume (PV) and $\%$ Free PSA ($\%$FPSA) are fed as inputs into the FES and prostate cancer risk (PCR) is obtained as the output. Using knowledge based rules in Mamdani type inference method the output is calculated. If PCR $\ge 50\%$, then the patient shall be advised to go for a biopsy test for confirmation. The efficacy of the designed FES is tested against a clinical data set. The true prediction for all the patients turns out to be $68.91\%$ whereas only for positive biopsy cases it rises to $73.77\%$. This simple yet effective FES can be used as supportive tool for decision making in medical diagnosis.'
author:
- 'Juthika Mahanta$^{1,}$[^1]'
- 'Subhasis Panda$^{2,}$[^2]'
bibliography:
- 'biblio.bib'
title: Fuzzy expert system for prediction of prostate cancer
---
Introduction {#intro}
============
Artificial intelligence (AI) is defined as the intelligence processed by machines. With the advancement in the computer system, machines exhibit tasks which normally need human intelligence. Development of AI techniques has revolutionized many areas like robotics, transportation, education, marketing etc. including medical diagnosis and health care. Medical diagnosis deals with the analysis of complex medical data. The primary job in medical diagnosis is to reach to a decision using expert’s logical reasoning. Handling large complex data and many uncertainties make this job very difficult. AI appears to be very handy for this job. AI in medical diagnosis has added expert human reasoning in simulation of computer-aided diagnosis process. There are different AI methods used in medical diagnosis, fuzzy logic is one of the most popular one. AI is the technique of mimicking human intelligence by the help of advance computer systems. Human brain takes natural languages as inputs which aren’t feasible to be represented by Boolean logic (either true or false). So, there is a requirement of representation outside these two possibilities. Fuzzy logic exactly does that. Fuzzy logic appears closer to the way human brain works. Therefore, in AI, fuzzy logic shows the sign of being a natural choice.\
Uncertainties and imprecision are connected with every aspect of our day to day life activities. Specifically in medical diagnosis domain, one encounters a lot of uncertainty and vagueness. It becomes very difficult to identify a particular disease from the said symptoms of the patients, as it contains lot of approximate and inaccurate information. On the other hand, a particular symptom can possibly lead to many different disjoint diseases, whereas for the same disease, the symptom may manifest itself in completely different ways from person to person. There are inherent uncertainties in the process of decision making in medical diagnosis, even for an expert too. To tackle these inexact, linguistic inputs, fuzzy logic based expert system is turned out to be very useful. The concepts of fuzzy set and fuzzy logic were introduced by Prof. L. A. Zadeh [@zadeh]. In contrast to binary logic, fuzzy logic deals with multi-valued logic which is a mathematical tool to represent the real world effectively. Due to its usefulness and simplicity, fuzzy logic has drawn huge attention of interdisciplinary researchers round the globe.\
Fuzzy logic based expert systems are widely used in many areas of medical diagnosis and decision-making process. Particularly, in the area of prostate cancer, very few literature are available [@saritas; @benecchi; @yuksel; @seker; @kar; @castanho; @abbod_review; @lorenz]. These literature address the problem in different angles and also use fuzzy logic in disjoint ways. Some researchers have used hybrid system to treat this problem. We focus our attention to fuzzy logic based expert system to predict the prostate cancer risk. From careful study of the literature, we found that $\%$FPSA is a very crucial parameter, along with age, PSA and PV for early detection of PC. Therefore, we formulate a fuzzy expert system by taking care of all these inputs.\
The paper is organized as follows. Section \[fes\] describes the fuzzy expert system, where all the inputs and the output are discussed. In section \[res\], we apply our FES to a medical data set and discuss our findings. Finally we summarize and conclude in section \[con\].
Fuzzy Expert System (FES) {#fes}
=========================
An expert system which uses fuzzy logic instead of Boolean logic is called fuzzy expert system (FES). A FES is a form of artificial intelligence which deals with membership functions and some prescribed rule base to evaluate a set of data. Fuzziness is introduced to the crisp inputs of a FES by means of suitable membership functions. Once membership functions are defined for all input variables, then they are fed to a particular inference method for further action. Here, we have used Mamdani (max-min) inference method which is most popular in literature. Rules of the FES developed here are of IF-THEN form. Mamdani type inference method results in fuzzy sets as output. For a given set of input values, some relevant rules will be fired to produce a fuzzy output in Mamdani type inference method. Fuzzy output is defuzzified using different techniques to obtain crisp output. Centroid method is used for defuzzification in our FES. General structure of a FES is shown in below figure \[fig:0\].
![General architecture of a FES.[]{data-label="fig:0"}](fes_block.eps){width="\textwidth"}
We have used medical data of the patients as given in reference [@saritas]. Depending on the data set, the range of different inputs are determined. In the following two subsections, we discuss about different inputs and the output of the FES.
Input variables
---------------
### Age
Age of man is an important parameter to calculate the risk factor for prostate cancer. For a man having no family history of PC, the chance of getting it increases after the age of 50. This number changes from race to race. However, two out of three PCs are diagnosed in men at the age of 65 or above. The input variable “age" is represented by four fuzzy sets, namely, “very young", “young", “middle age" and “old". First and fourth fuzzy sets are represented by trapezoidal membership functions whereas for second and third we have used triangular membership functions. Table \[tab:1\] lists crisp sets and the corresponding fuzzy sets for the input “age". The membership functions for the same are plotted in figure \[fig:1\].
[lll]{} Input variable & Crisp set & Fuzzy set\
Age (year) & 0-30 & Very young\
& 20-50 & Young\
& 30-60 & Middle age\
& 40-100 & Old\
![Membership functions for “age".[]{data-label="fig:1"}](age.eps){width="\textwidth"}
### Prostate-specific Antigen (PSA)
PSA has altered drastically the management of prostate cancer in men. The PSA test for blood can provide early stage detection of PC [@brawer99]. PSA is a protein secreted by the prostate gland which helps to keep the semen in liquid form. Some parts of this protein will pass into blood which give rise to the increase in normal PSA level. Elevation in PSA level in blood depends up on the health of prostate gland and the age of the person. A healthy prostate will release less PSA in blood compared to a cancerous gland. So, a rise in PSA level over normal range could be a possible indicator of PC. Although, elevated PSA level may be caused due to other factors like acute bacterial prostatitis, enlargement of prostate and other urinary retention. The measurement of PSA is expressed as nanograms per milliliter of blood. The normal range of PSA number can be age specific and also race specific. The input variable “PSA" is represented by five fuzzy sets, e.g., “very low", “low", “middle", “high" and “very high". For the first and fifth sets we have used trapezoidal membership functions while for the rest, triangular membership functions are used. In table \[tab:2\], we have shown the crisp sets and the corresponding fuzzy sets. The plot of the membership functions for the input “PSA" is displayed in figure \[fig:2\].
[lll]{} Input variable & Crisp set & Fuzzy set\
PSA (ng/ml) & 0-4 & Very low\
& 2-8 & Low\
& 4-12 & Middle\
& 8-16 & High\
& 12-50 & Very high\
![Membership functions for “PSA".[]{data-label="fig:2"}](psa.eps){width="\textwidth"}
### Prostate Volume (PV)
A healthy human male’s prostate is marginally larger than a walnut. It is a crucial parameter for early detection of PC. There is a characteristic pattern in the growth of prostate with age. That pattern can change from race to race. With the increase in the prostate volume there is a possibility of sampling error in systematic sextant needle biopsy. It is wise to use prostate volume as a factor while determining the necessity to repeat biopsy with initial negative result [@brawer_pv]. Prostate is mainly divided in four zones in pathological terminology and total prostate volume as well as transition zone volume are measured in ultrasound. According to transrectal ultrasound (TRUS) guidance [@zhang], prostate width ($W$) (maximal transverse diameter) is estimated on an axial image. Prostate length ($L$) (longitudinal diameter, the distance between proximal external sphincter and urinary bladder) and height ($H$) (maximal antero-posterior diameter) are measured on a mid-sagittal image [@bangma]. The total prostate volume (TPV) is calculated using the prolate elliptical formula, TPV = $\frac{\pi} {6} \times W \times L \times H$. The transition zone volume can be calculated using the same formula by measuring the required dimensions from the ultrasound. The length, width and height changes with the age. The rate of change for the length is significant compared to other two dimensions as the man approaches the age 60. So, based on the above information we have divided the input “prostate volume" in four fuzzy sets such as “small", “middle", “big" and “very big". Trapezoidal and triangular membership functions are used to represent them. Crisp sets, fuzzy sets and the membership functions are listed in table \[tab:3\] and figure \[fig:3\] respectively for the input variable “prostate volume".
[lll]{} Input variable & Crisp set & Fuzzy set\
PV (ml) & 0-60 & Small\
& 30-120 & Middle\
& 60-200 & Big\
& 180-300 & Very big\
![Membership functions for “PV".[]{data-label="fig:3"}](pv.eps){width="\textwidth"}
### Percentage of Free PSA ($\%$FPSA)
PSA is a protein which exists in different forms in serum. PSA circulates through body via bound to some other proteins or in unbound form. Free PSA test measure the ratio of the unbound PSA to bound PSA whereas normal PSA test measure the total PSA (both bound and unbound) level in blood [@labmed; @ito]. $\%$FPSA is calculated as $\frac{{\rm Free ~ PSA}}{{\rm Total PSA}} \times 100\%$. As we already discussed that PSA level may rise not only due to cancerous prostate but because of many other reasons. Therefore, for the early detection of the PC, which is potentially in its curable stage, requires a lower cutoff for PSA level, gives rise to avoidable biopsies. It is also observed that the men with PC are likely to have $\%$FPSA lower than those of benign disease [@labmed; @catalona]. So, along with an elevated PSA level, $\%$FPSA cutoff will be a good indicator for the early stage detection. Based on this we have categorized the input variable “$\%$FPSA" in three fuzzy sets, namely, “low", “middle" and “high". For their representation we have used trapezoidal and triangular membership functions. Crisp sets, fuzzy sets and the membership functions are shown in table \[tab:4\] and figure \[fig:4\].
[lll]{} Input variable & Crisp set & Fuzzy set\
$\%$FPSA & 0-11 & Low\
& 9-21 & Middle\
& 18-100 & High\
![Membership functions for “$\%$FPSA".[]{data-label="fig:4"}](fpsa.eps){width="\textwidth"}
Output Variable
---------------
### Prostate Cancer Risk (PCR)
The output of the FES is PCR. Numerical value of it will help us to identify the benignant one or the malignant one by considering PSA, age, PV and $\%$FPSA of the patient as input variables for the FES. If the PCR $\%$ value is $\ge 50$, then we can anticipate that the patient has a high chance of having PC. Therefore, he will be recommended for biopsy test for confirmation. The positive biopsy result confirms our prediction whereas negative result contradicts. The case of PCR $\%$ value $<50$ can be understood in similar fashion. The output variable, PCR is categorized into three fuzzy sets, namely, “low", “middle" and “high". Trapezoidal and triangular membership functions are used to represent them. The table \[tab:5\] and the figure \[fig:5\] are displaying these fuzzy sets and their membership functions respectively.
[lll]{} Output variable & Crisp set & Fuzzy set\
Prostate Cancer Risk ($\%$) & 0-30 & Low\
& 10-50 & Middle\
& 45-100 & High\
![Membership functions for “PCR".[]{data-label="fig:5"}](pcr.eps){width="\textwidth"}
Fuzzy Rule Base
---------------
In this FES, four input variables are age, PSA, PV and $\%$FPSA which are represented by four, five, four and three membership functions respectively. So, our FES has total $4\times5\times4\times3=240$ rules to estimate PCR as the output, which is also characterized by three membership functions. Some of the selected rules are displayed below.
``` {fontsize="\footnotesize"}
1. If (Age is vy) and (PSA is vl) and (PV is small) and (%FPSA is low) then (PCR is low) (1)
8. If (Age is vy) and (PSA is vl) and (PV is big) and (%FPSA is mid) then (PCR is low) (1)
65. If (Age is yo) and (PSA is vl) and (PV is middle) and (%FPSA is mid) then (PCR is low) (1)
97. If (Age is yo) and (PSA is hi) and (PV is small) and (%FPSA is low) then (PCR is high) (1)
130. If (Age is ma) and (PSA is vl) and (PV is verybig) and (%FPSA is low) then (PCR is low) (1)
172. If (Age is ma) and (PSA is vh) and (PV is middle) and (%FPSA is low) then (PCR is high) (1)
196. If (Age is ol) and (PSA is lo) and (PV is middle) and (%FPSA is low) then (PCR is low) (1)
240. If (Age is ol) and (PSA is vh) and (PV is verybig) and (%FPSA is high) then (PCR is mid) (1)
```
For example, rule $1$ can be elucidated as, if age of the patient is very young, PSA is very low, PV is small and $\%$FPSA is low then PCR for that patient is low. The wieght of this rule is maximum which is specified by number $1$ in the parentheses after the rule. Other rules can be spelled out in similar manner. Surface plot gives a visualization of the rules applied in FES. Since, our FES has four inputs and one output, it is not possible to plot the dependence of the output against all inputs. So, in surface plot we have plotted the output against any two input variables by keeping some reference values for the rest two inputs. Fig \[fig:7\] displays two such surface plots where relevant parameters are mentioned in the caption. For example, PCR is high (very high) for $\%$FPSA value less than $20\%$ ($10\%$) irrespective of the patient’s age.
![Surface plot of four-input FES (left) by taking two variables as inputs and PCR as output while keeping reference values for other two inputs. For the figure on the left, age & $\%$FPSA are inputs and reference values for PV & PSA are $100$ and $6$ respectively. The figure on the right, we have taken PSA & age as inputs while fixing reference values for PV and $\%$FPSA at 100 and $15$ respectively. []{data-label="fig:7"}](surf1.eps "fig:"){width="63.00000%"} ![Surface plot of four-input FES (left) by taking two variables as inputs and PCR as output while keeping reference values for other two inputs. For the figure on the left, age & $\%$FPSA are inputs and reference values for PV & PSA are $100$ and $6$ respectively. The figure on the right, we have taken PSA & age as inputs while fixing reference values for PV and $\%$FPSA at 100 and $15$ respectively. []{data-label="fig:7"}](surf2.eps "fig:"){width="60.00000%"}
Defuzzification & Mamdani Inference Engine
------------------------------------------
Defuzzification is the process of obtaining a precise quantity from a fuzzy set. In this FES, we have employed centroid method to defuzzify which is given by $${\bar z}=\frac{\int \mu_C(z)\cdot z~ dz}{\int \mu_C(z)~dz}.$$ For a given set of input variables, viz. age=$56$, PSA=$9.05$, PV=$39$ and $\%$FPSA=$8.51$, we calculate the corresponding degrees of membership by identifying the proper fuzzy sets for each input variable. This particular choice of inputs will fire, say $k$ number of rules. Truth degree ($\alpha_i$) of the $i^{\rm th}$ firing rule is determined by taking min of corresponding membership values of each input variable. Thereafter, max of all $\alpha_i$ will be the membership value of PCR. This is how Mamdani max-min inference technique is used. Then applying the centroid method, we obtain crisp value of PCR. Steps are shown below for the above mentioned input data.
1. Age = $56$, $\mu_{\rm ma}(56)=0.2$ and $\mu_{\rm ol}(56)=0.533$.
2. PSA = $9.05$, $\mu_{\rm mi}(9.05)=0.7375$ and $\mu_{\rm hi}(9.05)=0.2625$.
3. PV = $39$, $\mu_{\rm sl}(39)=0.7$ and $\mu_{\rm mi}(39)=0.3$.
4. $\%$FPSA = $8.51$, $\mu_{\rm lo}(8.51)=0.83$.
For the above set of input data, eight rules will come into action and yield
1. $\alpha_{145}={\rm min}(0.2,0.7375,0.7,0.83)=0.2$,
2. $\alpha_{148}={\rm min}(0.2,0.7375,0.3,0.83)=0.2$,
3. $\alpha_{157}={\rm min}(0.2,0.2625,0.7,0.83)=0.2$,
4. $\alpha_{160}={\rm min}(0.2,0.2625,0.3,0.83)=0.2$,
5. $\alpha_{205}={\rm min}(0.533,0.7375,0.7,0.83)=0.533$,
6. $\alpha_{208}={\rm min}(0.533,0.7375,0.3,0.83)=0.3$,
7. $\alpha_{217}={\rm min}(0.533,0.2625,0.7,0.83)=0.2625$,
8. $\alpha_{220}={\rm min}(0.533,0.2625,0.3,0.83)=0.2625$,
which further provides
&= [max]{}(\_[145]{},\_[148]{},\_[157]{},\_[160]{},\_[205]{},\_[208]{},\_[217]{},\_[220]{}),\
&= [max]{}(0.2,0.2,0.2,0.2,0.533,0.3,0.2625,0.2625)= 0.533.
Using $\alpha = 0.533$, and applying centroid method we get the PCR is $74.01\%$, which is much greater than our cutoff value of $50\%$. Therefore, in this case, the patient should be advised to go for a biopsy to confirm. We have used fuzzy logic toolbox of Matlab software for calculation of the entire set of medical data of 119 patients. Block diagram of FES is portrayed in the figure \[fig:6\].
![FES structure[]{data-label="fig:6"}](mod_block.eps){width="120.00000%"}
Results and Discussion {#res}
======================
We have applied our proposed FES to analyze the data (given in reference [@saritas]) presented in table \[tab:6\]. Here, the range of different inputs are chosen depending upon the minimum and the maximum values of the respective input variables. For example, for the input “PV", the maximum value is $235$ ml and the minimum value is $15$ ml. Therefore, we have chosen the range for “PV" to be $0-300$ ml. Feeding the four inputs into the FES, we obtain PCR as the output. If the value of PCR is greater or equal to $50\%$, then the patient should be advised to go for a biopsy test to confirm whether the prostate problem is of benignant or malignant type. We have compared outcomes of our FES with the results of biopsy. Out of total $119$ patients, $61$ patients had positive biopsy results, while the rest $58$ patients had negative results. The true prediction by our FES is $68.91\%$. The proposed FES has correctly predicted the positive biopsy results for $45$ patients ($73.77\%$) and the negative biopsy results for $37$ patients ($63.79\%$). True prediction percentage of our FES is better than that of the existing systems for prediction in literature. Saritas [*et. al*]{} [@saritas] has claimed $64.71\%$ true prediction for the same set of data, while online calculator for PC prediction and FPSA/PSA ratio provide only $62.18\%$ and $60.50\%$ respectively. Inclusion of one parameter has definitely increased the number of rules, but the prediction of the designed FES has become more accurate.
Summary and Conclusions {#con}
=======================
We have developed a fuzzy rule based expert system to predict the chances of having prostate cancer (PC) from a given set of input parameters. For this FES, we used age, PSA, PV and $\%$FPSA as the inputs to calculate prostate cancer risk (PCR) as the output. Existing literature on FES for PC, has mainly dealt with first three of the above mentioned inputs. The cutoff of $\%$FPSA is also a very crucial parameter along the PSA value, as it can reduce avoidable biopsy tests significantly. Most important advantages, apart from the cost of biopsy, the patient who do not need biopsy will be free from uneasiness and worries of the procedure, as well as the possible aftermath medical complications [@labmed]. Therefore, inclusion of this parameter in our FES was essential and it has altered the outcome appreciably. For a case study of $119$ patients, our FES has correctly predicted for $82$ patients ($68.91\%$). It is to be noted that, true prediction percentage for positive biopsy cases is excellent ($73.77\%$) whereas for the negative biopsy cases, it deviates slightly. The salient feature of our FES is that, the true prediction for positive biopsy is much higher compared to negative biopsy cases. This may cause an unnecessary biopsy test (along the with the pain associated with it) but it will certainly save a life. One should keep in mind that the FES designed here is not to confirm whether a person is having PC or not, but to assist a doctor to take a decision whether to go for biopsy or not. Since, four inputs are considered for this FES, number of rules has increased which makes it little complicated to start with. Once the burden of deciding these rules based on experts’ opinion are done, it does the prediction efficaciously. On the other hand, the behaviour of different bio-markers varies drastically from race to race, from demographic region to region, family history, life style, food habits and also depends on many more hidden variables. So, modification in cutoff values has to be done accordingly to use the FES. Hybridizing our FES with other AI techniques and incorporating more experts’ opinions may lead to an improved result.
Acknowledgements {#acknowledgements .unnumbered}
================
Authors would like to thank Dr. Bidesh Karmakar and Dr. Aniruddha Dewri for the useful discussions made with them during the manuscript preparation.
[\*[12]{}c]{}
\
Age & PSA & PV & & Biopsy & & Age & PSA & PV & & Biopsy &\
(year) & (ng/ml) & (ml) & & result & & (year) & (ng/ml) & (ml) & & result &\
[[** – continued from previous page**]{}]{}\
Age & PSA & PV & & Biopsy & & Age & PSA & PV & & Biopsy &\
(year) & (ng/ml) & (ml) & & result & & (year) & (ng/ml) & (ml) & & result &\
\
44 & 7.6 & 38 & 10.53 & Negative & 45.42 & 67 & 15.93 & 69 & 6.09 & Positive & 74.82\
51 & 6.76 & 15 & 4.14 & Positive & 57.78 & 67 & 28 & 47 & 15.00 & Positive & 74.15\
51 & 44 & 83 & 31.82 & Positive & 30.00 & 68 & 5.09 & 47 & 2.36 & Negative & 42.96\
53 & 4.5 & 39 & 18.89 & Negative & 19.96 & 68 & 5.51 & 45 & 11.25 & Negative & 21.28\
53 & 5.83 & 25 & 6.86 & Negative & 53.32 & 68 & 7.2 & 33 & 3.61 & Positive & 67.21\
53 & 8.34 & 25 & 7.43 & Negative & 73.83 & 68 & 9.25 & 91 & 3.57 & Positive & 74.03\
54 & 5.62 & 28 & 14.95 & Negative & 21.96 & 68 & 12.1 & 61 & 16.12 & Negative & 74.25\
54 & 17.3 & 90 & 27.46 & Negative & 30.00 & 68 & 23.7 & 109 & 10.04 & Positive & 73.55\
54 & 17.3 & 45 & 8.90 & Positive & 73.91 & 68 & 140 & 117 & 14.29 & Positive & 74.80\
55 & 10.51 & 54 & 22.45 & Negative & 23.57 & 68 & 140 & 54 & 3.29 & Positive & 74.70\
56 & 8.9 & 26 & 34.16 & Negative & 18.80 & 69 & 8.8 & 34 & 8.98 & Positive & 74.40\
56 & 9.05 & 39 & 8.51 & Positive & 74.07 & 69 & 11.06 & 38 & 29.84 & Negative & 26.64\
56 & 16 & 146 & 8.44 & Negative & 74.07 & 69 & 15.31 & 74 & 30.57 & Positive & 30.00\
57 & 12.56 & 52 & 65.84 & Negative & 30.00 & 69 & 61 & 46 & 9.93 & Negative & 73.65\
58 & 4.48 & 67.5 & 16.07 & Negative & 16.09 & 69 & 70.56 & 45 & 6.02 & Positive & 73.98\
58 & 4.62 & 48 & 11.04 & Negative & 17.62 & 69 & 146 & 29 & 7.33 & Positive & 75.10\
58 & 5.2 & 58 & 23.46 & Negative & 12.80 & 70 & 5.39 & 120 & 19.11 & Negative & 23.58\
58 & 16.39 & 27 & 92.07 & Negative & 30.00 & 70 & 5.39 & 42 & 12.80 & Negative & 19.91\
59 & 0.28 & 168 & 42.86 & Negative & 13.30 & 70 & 13 & 40 & 15.46 & Negative & 74.39\
59 & 8.36 & 55 & 7.54 & Positive & 74.31 & 70 & 13.95 & 119 & 13.76 & Negative & 74.02\
59 & 18.2 & 77 & 17.75 & Negative & 73.76 & 70 & 19.2 & 44 & 10.10 & Positive & 73.49\
59 & 19.48 & 79 & 25.00 & Positive & 30.00 & 70 & 21.94 & 29 & 7.11 & Positive & 75.17\
59 & 22.51 & 42 & 7.02 & Negative & 74.22 & 70 & 27.7 & 63 & 8.99 & Negative & 74.40\
59 & 22.65 & 66 & 10.82 & Negative & 73.88 & 71 & 6.08 & 48 & 21.38 & Positive & 13.52\
60 & 6.58 & 65 & 14.74 & Negative & 24.82 & 71 & 12.64 & 50 & 7.99 & Positive & 74.39\
60 & 10.6 & 30 & 16.79 & Positive & 61.27 & 71 & 22 & 57 & 12.00 & Positive & 74.59\
60 & 11.45 & 46 & 19.48 & Negative & 53.47 & 72 & 6.64 & 32 & 27.41 & Negative & 12.43\
60 & 14.79 & 38 & 6.90 & Positive & 74.39 & 72 & 13.31 & 33 & 3.83 & Positive & 74.40\
60 & 15.51 & 35 & 21.02 & Negative & 30.00 & 72 & 13.31 & 33 & 3.76 & Positive & 74.40\
61 & 4.6 & 37 & 10.87 & Negative & 25.02 & 72 & 20 & 48 & 7.90 & Positive & 74.22\
61 & 10.33 & 62 & 25.36 & Negative & 23.78 & 72 & 46 & 36 & 10.70 & Positive & 73.81\
61 & 10.36 & 35 & 19.79 & Negative & 47.88 & 72 & 77 & 48 & 8.31 & Positive & 74.22\
61 & 10.59 & 56 & 17.00 & Positive & 60.72 & 73 & 4.65 & 41 & 41.94 & Negative & 12.52\
61 & 18.3 & 62 & 6.99 & Positive & 74.46 & 73 & 7.25 & 19 & 5.52 & Negative & 67.73\
62 & 6.12 & 52 & 24.18 & Negative & 12.86 & 73 & 7.6 & 74 & 31.32 & Positive & 12.09\
62 & 6.2 & 25 & 4.35 & Positive & 56.05 & 73 & 19 & 90 & 6.84 & Positive & 73.98\
62 & 8.37 & 43 & 11.23 & Negative & 40.23 & 73 & 29.52 & 91 & 9.82 & Negative & 73.73\
62 & 8.79 & 45 & 10.92 & Positive & 48.39 & 73 & 47.4 & 87 & 15.89 & Positive & 74.11\
62 & 20 & 53 & 5.20 & Positive & 74.55 & 74 & 12.52 & 27 & 11.82 & Negative & 74.47\
62 & 51.74 & 29 & 6.80 & Positive & 74.55 & 74 & 150 & 54 & 16.67 & Positive & 74.09\
63 & 8.8 & 31 & 22.50 & Positive & 19.22 & 75 & 4.61 & 16 & 17.57 & Positive & 18.39\
64 & 5.7 & 36 & 29.82 & Negative & 12.71 & 75 & 10 & 34 & 7.60 & Positive & 73.98\
64 & 6.96 & 45 & 9.20 & Negative & 60.28 & 76 & 9.81 & 56 & 37.41 & Negative & 21.46\
64 & 8 & 40 & 7.50 & Positive & 74.39 & 76 & 13.61 & 61 & 19.91 & Positive & 52.54\
64 & 11.08 & 26 & 10.11 & Negative & 59.05 & 76 & 13.83 & 54 & 19.96 & Positive & 51.76\
64 & 16.28 & 21 & 6.94 & Positive & 74.70 & 76 & 21 & 86 & 5.43 & Positive & 74.15\
65 & 4.39 & 30 & 21.64 & Negative & 13.42 & 77 & 10 & 60 & 6.00 & Positive & 73.98\
65 & 5.15 & 47 & 15.73 & Negative & 19.46 & 77 & 12.05 & 28 & 27.05 & Positive & 30.00\
65 & 7.61 & 23 & 5.78 & Positive & 70.95 & 77 & 56 & 51 & 7.34 & Positive & 74.46\
65 & 7.82 & 75 & 22.76 & Negative & 13.04 & 78 & 4.5 & 180 & 20.44 & Negative & 18.05\
65 & 8.33 & 32 & 14.53 & Positive & 38.08 & 78 & 26.1 & 46 & 8.62 & Negative & 74.07\
66 & 4.38 & 33 & 23.52 & Negative & 12.78 & 78 & 26.13 & 235 & 8.27 & Negative & 74.96\
66 & 6.72 & 61 & 13.84 & Positive & 25.38 & 78 & 31.6 & 57 & 8.86 & Negative & 74.50\
66 & 7.65 & 89 & 23.66 & Negative & 12.90 & 79 & 17.1 & 41 & 7.60 & Negative & 74.31\
66 & 9 & 74 & 18.89 & Positive & 53.84 & 80 & 69.51 & 28 & 28.77 & Positive & 30.00\
66 & 9.86 & 49 & 23.83 & Negative & 21.69 & 81 & 4.5 & 28 & 21.56 & Positive & 13.45\
67 & 4.39 & 28 & 0.91 & Negative & 23.99 & 81 & 68.36 & 52 & 35.27 & Positive & 30.00\
67 & 5.65 & 24 & 10.27 & Positive & 46.65 & 88 & 10.4 & 32 & 7.50 & Positive & 74.22\
67 & 6.24 & 65 & 21.96 & Negative & 13.31 & & & & & &\
67 & 8.2 & 36 & 20.37 & Positive & 31.78 & & & & & &\
67 & 9.68 & 41 & 7.44 & Positive & 74.18 & & & & & &\
[^1]: [email protected]
[^2]: [email protected]
| ArXiv |
---
abstract: 'Electroweak, gluons, and gravity fields arise as gauge fields from probabilities of physical events.'
author:
- |
G. Quznetsov\
[email protected], [email protected]
date: 'August 17, 2007'
title: |
All four forces without superstrings.\
(Other game in town)
---
Introduction
============
As is obvious, the strings theory arrives at it’s logical finish [@Sch]. But it is possible that Nature is done more simply and more naturally than that. In this article I’m propose the deduction of the electroweak, gluons, and gravity forces from the representation of physical events’ probabilities by spinors as “other game in town” [@Sch1].
Electroweak fields
==================
Let $\left\langle \rho \left( \underline{x}\right) ,j_1\left( \underline{x}%
\right) ,j_2\left( \underline{x}\right) ,j_3\left( \underline{x}\right)
\right\rangle =\left\langle \rho \left(t,\mathbf{x}\right) ,\mathbf{j}%
\left( t,\mathbf{x}\right) \right\rangle $ be a probability density 3+1-vector of any physical event [@Q0].
Complex functions $\varphi _1\left( \underline{x}%
\right) $, $\varphi _2\left( \underline{x}\right) $, $\varphi _3\left(
\underline{x}\right) $, $\varphi _4\left( \underline{x}\right) $ exist [@Q1] such that
$$\begin{aligned}
\rho &=&\sum_{s=1}^4\varphi _s^{*}\varphi _s\mbox{,}
\label{j} \\
\frac{j_{\alpha}}{\mathrm{c}} &=&-\sum_{k=1}^4\sum_{s=1}^4%
\varphi _s^{*}\beta _{s,k}^{\left[ \alpha \right] }\varphi _k \nonumber\end{aligned}$$
for every such density vector. Here $\alpha \in \left\{ 1,2,3\right\} $ and $\beta ^{\left[ \alpha \right]
}$ - are diagonal elements of the light Clifford’s pentad [@Q2].
If
$$\varphi =\left[
\begin{array}{c}
\varphi _1 \\
\varphi _2 \\
\varphi _3 \\
\varphi _4
\end{array}
\right]$$
then [@Q4]
$$\frac 1{\mathrm{c}}\partial _t\varphi +\left( \mathrm{i}\Theta _0+\mathrm{i}%
\Upsilon _0\gamma ^{\left[ 5\right] }\right) \varphi =\left(
\begin{array}{c}
\beta ^{\left[ 1\right] }\partial _1+\mathrm{i}\Theta _1\beta ^{\left[
1\right] }+\mathrm{i}\Upsilon _1\beta ^{\left[ 1\right] }\gamma ^{\left[
5\right] }+ \\
+\beta ^{\left[ 2\right] }\partial _2+\mathrm{i}\Theta _2\beta ^{\left[
2\right] }+\mathrm{i}\Upsilon _2\beta ^{\left[ 2\right] }\gamma ^{\left[
5\right] }+ \\
+\beta ^{\left[ 3\right] }\partial _3+\mathrm{i}\Theta _3\beta ^{\left[
3\right] }+\mathrm{i}\Upsilon _3\beta ^{\left[ 3\right] }\gamma ^{\left[
5\right] }+ \\
+\mathrm{i}M_0\gamma ^{\left[ 0\right] }+\mathrm{i}M_4\beta ^{\left[
4\right] }- \\
-\mathrm{i}M_{\zeta ,0}\gamma _\zeta ^{[0]}+\mathrm{i}M_{\zeta ,4}\zeta
^{[4]}- \\
-\mathrm{i}M_{\eta ,0}\gamma _\eta ^{[0]}-\mathrm{i}M_{\eta ,4}\eta ^{[4]}+
\\
+\mathrm{i}M_{\theta ,0}\gamma _\theta ^{[0]}+\mathrm{i}M_{\theta ,4}\theta
^{[4]}
\end{array}
\right) \varphi \mbox{.} \label{ham0}$$
with real $\Theta _k$, $\Upsilon _k$, $M_0$, $M_4$, $M_{\zeta ,0}$, $%
M_{\zeta ,4}$, $M_{\eta ,0}$, $M_{\eta ,4}$, $M_{\theta ,0}$, $M_{\theta ,4}$ and $$\gamma ^{\left[ 5\right] }\stackrel{def}{=}\left[
\begin{array}{cc}
1_2 & 0_2 \\
0_2 & -1_2
\end{array}
\right] \mbox{,} \label{g5}$$
here $\gamma ^{\left[ 0\right] }$, $\beta ^{\left[ 4\right] }$ are antidiagonal elements of the light Clifford’s pentad, and $\gamma _\zeta
^{[0]}$, $\zeta ^{[4]}$, $\gamma _\eta ^{[0]}$, $\eta ^{[4]}$, $\gamma
_\theta ^{[0]}$, $\theta ^{[4]}$ are antidiagonal elements of colored Clifford’s pentads [@Q3].
If $M_{\zeta ,0}=0$, $M_{\zeta ,4}=0$, $M_{\eta ,0}=0$, $M_{\eta ,4}=0$, $%
M_{\theta ,0}=0$, $M_{\theta ,4}=0$ then the Dirac lepton moving equation is derived from (\[ham0\]):
$$\left( \mathrm{i}\frac 1{\mathrm{c}}\partial _t-\Theta _0-\Upsilon _0\gamma
^{\left[ 5\right] }\right) \varphi =\sum_{k=1}^3\left( \beta ^{\left[
k\right] }\left( \mathrm{i}\partial _k-\Theta _k-\Upsilon _k\gamma ^{\left[
5\right] }\right) -m\gamma \right) \varphi \label{eq3}$$
with $m=\sqrt{M_0^2+M_4^2}$ and $\gamma =\left( \frac{M_0}{\sqrt{M_0^2+M_4^2}%
}\gamma ^{\left[ 0\right] }+\frac{M_4}{\sqrt{M_0^2+M_4^2}}\beta ^{\left[
4\right] }\right) $.
Let $x_4$, $x_5$ be some real variables such that
$$-\frac {\pi\mathrm{c}}{\mathrm{h}}\leq x_5\leq \frac {\pi\mathrm{c}}{\mathrm{%
h}},-\frac {\pi\mathrm{c}} {\mathrm{h}}\leq x_4\leq \frac {\pi\mathrm{c}}{%
\mathrm{h}}\mbox{.}$$
and let
$$\begin{aligned}
\widetilde{\varphi }\left( t,x_1,x_2,x_3,x_5,x_4\right) \stackrel{def}{=}%
\varphi \left( t,x_1,x_2,x_3\right) \cdot \nonumber \\
\cdot \left( \exp \left(\mathrm{i}\left( x_5M_0\left( t,x_1,x_2,x_3\right)
+x_4M_4\left( t,x_1,x_2,x_3\right) \right) \right) \right) \mbox{.}
\nonumber\end{aligned}$$
In this case $\widetilde{\varphi }$ obeys to the following moving equation:
$$\left( \sum_{s=0}^3\beta ^{\left[ s\right] }\left( \mathrm{i}\partial
_s-\Theta _s-\Upsilon _s\gamma ^{\left[ 5\right] }\right) -\gamma ^{\left[
0\right] }\mathrm{i}\partial _5-\beta ^{\left[ 4\right] }\mathrm{i}\partial
_4\right) \widetilde{\varphi }=0$$
(here $\beta ^{\left[ 0\right] }=-1$).
This equation can be transformated to the following form [@Q5]:
$$\left( \sum_{s=0}^3\beta ^{\left[ s\right] }\left( \mathrm{i}\partial
_s+F_s+0.5g_1YB_s\right) -\gamma ^{\left[ 0\right] }\mathrm{i}\partial
_5-\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \widetilde{\varphi }%
=0 \label{eq1}$$
with real $F_s$, $B_s$, real positive $g_1$, and with the charge matrix[^1] :
$$Y\stackrel{def}{=}-\left[
\begin{array}{cc}
1_2 & 0_2 \\
0_2 & 2\cdot 1_2
\end{array}
\right] \mbox{.} \label{chrg}$$
If $\chi \left( t,x_1,x_2,x_3\right) $ is a real function and:
$$\widetilde{U}\left( \chi \right) \stackrel{def}{=}\left[
\begin{array}{cc}
\exp \left( \mathrm{i}\frac \chi 2\right) 1_2 & 0_2 \\
0_2 & \exp \left( \mathrm{i}\chi \right) 1_2
\end{array}
\right] \mbox{.} \label{ux}$$
then equation (\[eq1\]) is invariant for the following transformations:
$$\begin{aligned}
\ &&x_4\rightarrow x_4^{\prime }=x_4\cos \frac \chi 2-x_5\sin \frac \chi 2%
\mbox{;} \nonumber \\
\ &&x_5\rightarrow x_5^{\prime }=x_5\cos \frac \chi 2+x_4\sin \frac \chi 2%
\mbox{;} \nonumber \\
\ &&x_\mu \rightarrow x_\mu ^{\prime }=x_\mu \mbox{ for
}\mu \in \left\{ 0,1,2,3\right\} \mbox{;} \label{T} \\
\ &&\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }=%
\widetilde{U}\widetilde{\varphi }\mbox{,} \nonumber \\
\ &&B_\mu \rightarrow B_\mu ^{\prime }=B_\mu -\frac 1{g_1}\partial _\mu \chi %
\mbox{,} \nonumber \\
\ &&F_\mu \rightarrow F_\mu ^{\prime }=\widetilde{U}F_s\widetilde{U}%
^{\dagger }\mbox{.} \nonumber\end{aligned}$$
Therefore, $B_\mu $ are components of the Standard Model gauge field $B$.
Further $\Im _{\mathbf{J}}$ be [@DVB], [@AV] the space spanned of the following basis [@Q7]:
$$\mathbf{J}\stackrel{def}{=}\left\langle
\begin{array}{c}
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( s_0x_4\right) \right) \epsilon _1,\frac{\mathrm{h}}{2\pi
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
s_0x_4\right) \right) \epsilon _2, \\
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( s_0x_4\right) \right) \epsilon _3,\frac{\mathrm{h}}{2\pi
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
s_0x_4\right) \right) \epsilon _4, \\
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( n_0x_5\right) \right) \epsilon _1,\frac{\mathrm{h}}{2\pi
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
n_0x_5\right) \right) \epsilon _2, \\
\frac{\mathrm{h}}{2\pi \mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{%
\mathrm{c}}\left( n_0x_5\right) \right) \epsilon _3,\frac{\mathrm{h}}{2\pi
\mathrm{c}}\exp \left( -\mathrm{i}\frac{\mathrm{h}}{\mathrm{c}}\left(
n_0x_5\right) \right) \epsilon _4
\end{array}
\right\rangle \label{TJ}$$
with some integer numbers $s_0$ and $n_0$ and with
$$\epsilon _1=\left[
\begin{array}{c}
1 \\
0 \\
0 \\
0
\end{array}
\right] ,.\epsilon _2=\left[
\begin{array}{c}
0 \\
1 \\
0 \\
0
\end{array}
\right] ,.\epsilon _3=\left[
\begin{array}{c}
0 \\
0 \\
1 \\
0
\end{array}
\right] ,.\epsilon _4=\left[
\begin{array}{c}
0 \\
0 \\
0 \\
1
\end{array}
\right] \mbox{.}$$
Farther $U$ be any linear transformation of space $\Im _{\mathbf{J}}$ such that for every $\widetilde{\varphi }$: if $\widetilde{\varphi }\in \Im _{%
\mathbf{J}}$ then:
$$\begin{array}{c}
\left( U\widetilde{\varphi }\right) ^{\dagger }\left( U\widetilde{\varphi }%
\right) =\rho \mbox{,} \\
\left( U\widetilde{\varphi }\right) ^{\dagger }\beta ^{\left[ s\right]
}\left( U\widetilde{\varphi }\right) =-\frac{j_{,s}}{\mathrm{c}}
\end{array}
\label{un1}$$
for $s\in \left\{ 1,2,3\right\} $.
Matrix $U$ factorized as the following [@Q8]:
$$U=\exp \left( \mathrm{i}\varsigma \right) \widetilde{U}U^{\left( -\right)
}U^{\left( +\right) }$$
with real $\varsigma $ and with
$$U^{\left( +\right) }\stackrel{def}{=}\left[
\begin{array}{cccc}
1_2 & 0_2 & 0_2 & 0_2 \\
0_2 & \left( u+\mathrm{i}v\right) 1_2 & 0_2 & \left( k+\mathrm{i}s\right) 1_2
\\
0_2 & 0_2 & 1_2 & 0_2 \\
0_2 & \left( -k+\mathrm{i}s\right) 1_2 & 0_2 & \left( u-\mathrm{i}v\right)
1_2
\end{array}
\right] \label{upls}$$
and
$$U^{\left( -\right) }\stackrel{def}{=}\left[
\begin{array}{cccc}
\left( a+\mathrm{i}b\right) 1_2 & 0_2 & \left( c+\mathrm{i}q\right) 1_2 & 0_2
\\
0_2 & 1_2 & 0_2 & 0_2 \\
\left( -c+\mathrm{i}q\right) 1_2 & 0_2 & \left( a-\mathrm{i}b\right) 1_2 &
0_2 \\
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \label{llll}$$
with real $a$, $b$, $c$, $q$, $u$, $v$, $k$, $s$.
Matrix $U^{\left( +\right) }$ refers to antiparticles [@Q9]. And transformation $U^{\left( -\right) }$ reduces equation (\[eq1\]) to the following shape [@Q10]:
$$\left(
\begin{array}{c}
\sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\mathrm{i}\left( \partial _\mu -%
\mathrm{i}0.5g_1B_\mu Y-\mathrm{i}\frac 12g_2W_\mu -\mathrm{i}F_\mu \right)
\\
+\gamma ^{\left[ 0\right] }\mathrm{i}\partial _5+\beta ^{\left[ 4\right] }%
\mathrm{i}\partial _4
\end{array}
\right) \widetilde{\varphi }=0\mbox{.} \label{hW}$$
with real positive $g_2$ and with
$$W_\mu =\left[
\begin{array}{cccc}
W_{0,\mu }1_2 & 0_2 & \left( W_{1,\mu }-\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2
\\
0_2 & 0_2 & 0_2 & 0_2 \\
\left( W_{1,\mu }+\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 & -W_{0,\mu }1_2 &
0_2 \\
0_2 & 0_2 & 0_2 & 0_2
\end{array}
\right]$$
with real $W_{0,\mu }$, $W_{1,\mu }$ and $W_{2,\mu }$ .
Equation (\[hW\]) is invariant under the following transformation [@Q12]:
$$\begin{aligned}
\ &&\varphi \rightarrow \ \varphi ^{\prime }=U\varphi \mbox{,} \nonumber \\
\ &&x_4\rightarrow x_4^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_4+\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_5\mbox{,}
\nonumber \\
\ &&x_5\rightarrow x_5^{\prime }=\left( \ell _{\circ }+\ell _{*}\right)
ax_5-\left( \ell _{\circ }-\ell _{*}\right) \sqrt{1-a^2}x_4\mbox{,}
\label{tt2} \\
\ &&x_\mu \rightarrow x_\mu ^{\prime }=x_\mu \mbox{,
for }\mu \in \left\{ 0,1,2,3\right\} \mbox{,} \nonumber\\
\ &&B_\mu \rightarrow B_\mu ^{\prime }=B_\mu \mbox{,}\nonumber \\
\ &&W_\mu \rightarrow W_\mu ^{\prime }=UW_\mu U^{\dagger }-\frac{2\mathrm{i}%
}{g_2}\left( \partial _\mu U\right) U^{\dagger }\nonumber\end{aligned}$$
with [@Q11]
$$\begin{aligned}
&&\ \ \ell _{\circ }\stackrel{def}{=}\frac 1{2\sqrt{\left( 1-a^2\right) }%
}\left[
\begin{array}{cc}
\left( b+\sqrt{\left( 1-a^{\prime 2}\right) }\right) 1_4 & \left( q-\mathrm{i%
}c\right) 1_4 \\
\left( q+\mathrm{i}c\right) 1_4 & \left( \sqrt{\left( 1-a^2\right) }%
-b\right) 1_4
\end{array}
\right] \mbox{,} \\
&&\ \ \ell _{*}\stackrel{def}{=}\frac 1{2\sqrt{\left( 1-a^2\right) }}\left[
\begin{array}{cc}
\left( \sqrt{\left( 1-a^2\right) }-b\right) 1_4 & \left( -q+\mathrm{i}%
c\right) 1_4 \\
\left( -q-\mathrm{i}c\right) 1_4 & \left( b+\sqrt{\left( 1-a^2\right) }%
\right) 1_4
\end{array}
\right] \mbox{.}\end{aligned}$$
Hence $W_\mu $ behaves as components of the weak field $W$ of Standard Model.
Field $W_{0,\mu }$ obeys the following equation [@Q13]:
$$\left( -\frac 1{\mathrm{c}^2}\partial _t^2+\sum_{s=1}^3\partial _s^2\right)
W_{0,\mu }=g_2^2\left( \widetilde{W}_0^2-\widetilde{W}_1^2-\widetilde{W}_2^2-%
\widetilde{W}_3^2\right) W_{0,\mu }+\Lambda \label{eq2}$$
with
$$\widetilde{W}_\nu =\left[
\begin{array}{c}
W_{0,\nu } \\
W_{1,\nu } \\
W_{2,\nu }
\end{array}
\right]$$
and $\Lambda $ is the action of other components of field $W$ on $W_{0,\mu }$.
Equation (\[eq2\]) looks like to the Klein-Gordon equation of field $%
W_{0,\mu }$ with mass
$$m=\frac{\mathrm{h}}{\mathrm{c}}g_2\sqrt{\widetilde{W}_0^2-\sum_{s=1}^3%
\widetilde{W}_s^2} \label{z10}$$
and with additional terms of the $W_{0,\mu }$ interactions with others components of $\widetilde{W}$. Fields $W_{1,\mu }$ and $W_{2,\mu }$ have similar equations.
The ”mass” (\[z10\]) is invariant [@Q14] under the Lorentz transformations
$$\widetilde{W}_0^{\prime }\stackrel{def}{=}\frac{\widetilde{W}_0-\frac v{%
\mathrm{c}}\widetilde{W}_k}{\sqrt{1-\left( \frac v{\mathrm{c}}\right) ^2}}%
\mbox{, }\widetilde{W}_k^{\prime }\stackrel{def}{=}\frac{\widetilde{W}%
_k-\frac v{\mathrm{c}}\widetilde{W}_0}{\sqrt{1-\left( \frac v{\mathrm{c}%
}\right) ^2}}\mbox{, }\widetilde{W}_k^{\prime }\stackrel{def}{=}\widetilde{W}%
_k\mbox{, if }s\neq k \mbox{,}$$
is invariant under the turns of the $\left\langle \widetilde{W}_1,\widetilde{%
W}_2,\widetilde{W}_3\right\rangle $ space
$$\left\{
\begin{array}{c}
\widetilde{W}_r^{\prime }=\widetilde{W}_r\cos \lambda -\widetilde{W}_s\sin
\lambda \mbox{.} \\
\widetilde{W}_s^{\prime }=\widetilde{W}_r\sin \lambda +\widetilde{W}_s\cos
\lambda \mbox{;}
\end{array}
\right|$$
and invariant under a global weak isospin transformation $U$:
$$W_\nu \rightarrow W_\nu ^{\prime }=U^{\left( -\right) }W_\nu U^{\left(
-\right) \dagger }\mbox{,}$$
but is not invariant for a local transformation $U^{\left( -\right) }$. But local transformations for $W_{0,\mu }$, $W_{1,\mu }$ and $W_{2,\mu }$ is insignificant since all three particles are very short-lived.
That is the form (\[z10\]) varies in space, but locally acts like a mass - i.e. it does not allow to particles of this field to behave as a massless ones.
If
$$\begin{array}{c}
Z_\mu \stackrel{def}{=}\left( W_{0,\mu }\cos \alpha -B_\mu \sin \alpha
\right) \mbox{,} \\
A_\mu \stackrel{def}{=}\left( B_\mu \cos \alpha +W_{0,\mu }\sin \alpha
\right)
\end{array}$$
with
$$\alpha \stackrel{def}{=}\arctan \frac{g_1}{g_2}$$
then masses of $Z$ and $W$ fulfill to the following ratio [@Q15]:
$$m_Z=\frac{m_W}{\cos \alpha }\mbox{.}$$
If
$$\mathrm{e}\stackrel{def}{=}\frac{g_1g_2}{\sqrt{g_1^2+g_2^2}}\mbox{,}$$ and
$$\widehat{Z}_\mu \stackrel{def}{=}Z_\mu \frac 1{\sqrt{g_2^2+g_1^2}}\left[
\begin{array}{cccc}
\left( g_2^2+g_1^2\right) 1_2 & 0_2 & 0_2 & 0_2 \\
0_2 & 2g_1^21_2 & 0_2 & 0_2 \\
0_2 & 0_2 & \left( g_2^2-g_1^2\right) 1_2 & 0_2 \\
0_2 & 0_2 & 0_2 & 2g_1^21_2
\end{array}
\right] \mbox{,}$$
$$\widehat{W}_\mu \stackrel{def}{=}g_2\left[
\begin{array}{cccc}
0_2 & 0_2 & \left( W_{1,\mu }-\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 \\
0_2 & 0_2 & 0_2 & 0_2 \\
\left( W_{1,\mu }+\mathrm{i}W_{2,\mu }\right) 1_2 & 0_2 & 0_2 & 0_2 \\
0_2 & 0_2 & 0_2 & 0_2\cdot 1_2
\end{array}
\right] \mbox{,}$$
$$\widehat{A}_\mu \stackrel{def}{=}A_\mu \left[
\begin{array}{cccc}
0_2 & 0_2 & 0_2 & 0_2 \\
0_2 & 1_2 & 0_2 & 0_2 \\
0_2 & 0_2 & 1_2 & 0_2 \\
0_2 & 0_2 & 0_2 & 1_2
\end{array}
\right] \mbox{.}$$
then equation (\[hW\]) has got the following form [@Q16]:
$$\left( \sum_{\mu =0}^3\beta ^{\left[ \mu \right] }\mathrm{i}\left( \partial
_\mu +\mathrm{i}e\widehat{A}_\mu -\mathrm{i}0.5\left( \widehat{Z}_\mu +%
\widehat{W}_\mu \right) \right) +\gamma ^{\left[ 0\right] }\mathrm{i}%
\partial _5+\beta ^{\left[ 4\right] }\mathrm{i}\partial _4\right) \widetilde{%
\varphi }=0. \label{AZW}$$
Here the vector field $A_\mu $ is* the electromagnetic potential*, and $\left( \widehat{Z}_\mu +\widehat{W}_\mu \right) $ is *the weak potential*.
Gluons fields
=============
If
$$U_{1,2}\left( \alpha \right) =\cos \frac \alpha 2\cdot 1_4-\sin \frac \alpha
2\cdot \beta ^{\left[ 1\right] }\beta ^{\left[ 2\right] }$$
then equation (\[eq3\]) is invariant under the following turning transformations [@Q17]:
$$\begin{array}{c}
\widetilde{\varphi }\rightarrow \widetilde{\varphi }^{\prime }=U_{1,2}\left(
\alpha \right) \widetilde{\varphi }\mbox{,} \\
x_0\rightarrow x_0^{\prime }=x_0\mbox{,} \\
x_1\rightarrow x_1^{\prime }=x_1\cos \alpha -x_2\sin \alpha \mbox{,} \\
x_2\rightarrow x_2^{\prime }=x_1\sin \alpha +x_2\cos \alpha \mbox{,} \\
x_3\rightarrow x_3^{\prime }=x_3\mbox{,} \\
\Theta _0\rightarrow \Theta _0^{\prime }=\Theta _0\mbox{,} \\
\Theta _1\rightarrow \Theta _1^{\prime }=\Theta _1\cos \alpha -\Theta _2\sin
\alpha \mbox{,} \\
\Theta _2\rightarrow \Theta _2^{\prime }=\Theta _1\sin \alpha +\Theta _2\cos
\alpha \mbox{,} \\
\Theta _3\rightarrow \Theta _3^{\prime }=\Theta _3\mbox{,} \\
\Upsilon _0\rightarrow \Upsilon _0^{\prime }=\Upsilon _0\mbox{,} \\
\Upsilon _1\rightarrow \Upsilon _1^{\prime }=\Upsilon _1\cos \alpha
-\Upsilon _2\sin \alpha \mbox{,} \\
\Upsilon _2\rightarrow \Upsilon _2^{\prime }=\Upsilon _1\sin \alpha
+\Upsilon _2\cos \alpha \mbox{,} \\
\Upsilon _3\rightarrow \Upsilon _3^{\prime }=\Upsilon _3\mbox{,} \\
M_0\rightarrow M_0^{\prime }=M_0\mbox{,} \\
M_4\rightarrow M_4^{\prime }=M_4\mbox{:}
\end{array}
\label{T12}$$
and is invariant under all other turnings of Cartesian system [@Q100].
But under such rotation the mass members of colored pentads of equation (\[ham0\]) are interfused [@Q18]:
$$\begin{aligned}
M_{\zeta ,0}^{\prime }\cos \alpha +M_{\eta ,0}^{\prime }\sin \alpha
&=&M_{\zeta ,0}\mbox{,} \\
M_{\zeta ,4}^{\prime }\cos \alpha +M_{\eta ,4}^{\prime }\sin \alpha
&=&M_{\zeta ,4}\mbox{,} \\
M_{\eta ,0}^{\prime }\cos \alpha -M_{\zeta ,0}^{\prime }\sin \alpha
&=&M_{\eta ,0}\mbox{.} \\
M_{\eta ,4}^{\prime }\cos \alpha -M_{\zeta ,4}^{\prime }\sin \alpha
&=&M_{\eta ,4}\mbox{,} \\
M_{\theta ,0}^{\prime } &=&M_{\theta ,0}\mbox{,} \\
M_{\theta ,4}^{\prime } &=&M_{\theta ,4}\mbox{.}\end{aligned}$$
The mass members of colored pentads are interfused under other Cartesian rotations, too.
Therefore the chromatic triplet elements can not be separated in space. These elements must be *confined* into identical place (*confinement*).
Each chromatic pentad contains two mass elements. Hence, every family contains two sorts of the chromatic particles of tree colors. These particles are quarks.
The hamiltonian of form:
$$\begin{array}{c}
\widehat{H}_{clr}\stackrel{def}{=}\sum_{s=1}^3\beta ^{\left[ s\right]
}\left( \mathrm{i}\partial _s+F_s\right) + \\
+M_{\zeta ,0}\gamma _\zeta ^{[0]}-M_{\zeta ,4}\zeta ^{[4]}+M_{\eta ,0}\gamma
_\eta ^{[0]}+M_{\eta ,4}\eta ^{[4]}-M_{\theta ,0}\gamma _\theta
^{[0]}-M_{\theta ,4}\theta ^{[4]}\mbox{.}
\end{array}
\label{Hclr}$$
can be formulated as the following:
$$\widehat{H}_{clr}=\sum_{s=1}^33\lfloor \widehat{\beta }^{\left[ s\right]
}\left( \mathrm{i}\partial _s+F_s\right) +3\lfloor \widehat{M}\mbox{,}
\label{Hcl}$$
here $\lfloor $ is the left bracket of the product K-matricies designation [@Q19], $\widehat{M}$ is a K-matrix such that:
$$\widehat{M}\stackrel{def}{=}\left\|
\begin{array}{ccc}
\left( M_{\zeta ,0}-\widehat{\mathrm{i}}M_{\zeta ,4}\right) \gamma _\zeta
^{[0]} & 0_4 & 0_4 \\
0_4 & \left( M_{\eta ,0}+\widehat{\mathrm{i}}M_{\eta ,4}\right) \gamma _\eta
^{[0]} & 0_4 \\
0_4 & 0_4 & \left( -M_{\theta ,0}-\widehat{\mathrm{i}}M_{\theta ,4}\right)
\gamma _\theta ^{[0]}
\end{array}
\right\| \label{Mx}$$
with
$$\widehat{\mathrm{i}}\stackrel{def}{=}\mathrm{i}\gamma ^{\left[ 5\right]
}=\left[
\begin{array}{cccc}
\mathrm{i} & 0 & 0 & 0 \\
0 & \mathrm{i} & 0 & 0 \\
0 & 0 & -\mathrm{i} & 0 \\
0 & 0 & 0 & -\mathrm{i}
\end{array}
\right] \mbox{,}$$
$$\widehat{\beta }^{\left[ 1\right] }\stackrel{def}{=}-\left\|
\begin{array}{ccc}
\zeta ^{[1]} & 0_4 & 0_4 \\
0_4 & \eta ^{[1]} & 0_4 \\
0_4 & 0_4 & \theta ^{[1]}
\end{array}
\right\| \mbox{,}$$
$$\widehat{\beta }^{\left[ 2\right] }\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & \gamma _\zeta ^{[0]}\eta ^{[2]}\gamma _\eta ^{[0]} & 0_4 \\
0_4 & 0_4 & \gamma _\eta ^{[0]}\theta ^{[2]}\gamma _\theta ^{[0]} \\
\gamma _\theta ^{[0]}\zeta ^{[2]}\gamma _\zeta ^{[0]} & 0_4 & 0_4
\end{array}
\right\| \mbox{,}$$
$$\widehat{\beta }^{\left[ 3\right] }\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 0_4 & \gamma _\zeta ^{[0]}\theta ^{[3]}\gamma _\theta ^{[0]} \\
\gamma _\eta ^{[0]}\zeta ^{[3]}\gamma _\zeta ^{[0]} & 0_4 & 0_4 \\
0_4 & \gamma _\theta ^{[0]}\eta ^{[3]}\gamma _\eta ^{[0]} & 0_4
\end{array}
\right\| \mbox{.}$$
Equation
$$\widehat{H}_{clr}\widetilde{\varphi }=\sum_{s=1}^33\lfloor \widehat{\beta }%
^{\left[ s\right] }\left( \mathrm{i}\partial _s+F_s\right) \widetilde{%
\varphi }\rceil +3\lfloor \widehat{M}\widetilde{\varphi }\rceil \mbox{.}
\label{Hclr2}$$
is invariant [@Q20] under the following transformation:
$$\begin{aligned}
\widetilde{\varphi } &\rightarrow &\widetilde{\varphi }^{\prime }\stackrel{%
def}{=}\lfloor U\widetilde{\varphi }\rceil \mbox{,} \label{TG} \\
\widehat{M} &\rightarrow &\widehat{M}^{\prime }\stackrel{def}{=}\lfloor U%
\widehat{M}U^{\dagger }\rceil \mbox{,} \nonumber \\
\widehat{\beta }^{\left[ s\right] } &\rightarrow &\widehat{\beta }^{\left[
s\right] \prime }\stackrel{def}{=}\lceil U\widehat{\beta }^{\left[ s\right]
}U^{\dagger }\rceil \mbox{,} \nonumber \\
F_s &\rightarrow &F_s^{\prime }\stackrel{def}{=}\left( F_s-\lceil \left(
\mathrm{i}\partial _sU\right) U^{\dagger }\rceil \right) \nonumber\end{aligned}$$
with an unitary 3$\times $3 K-matrix $U$.
There:
$$\lfloor \left( \mathrm{i}\partial _sU\right) U^{\dagger }\rceil =\mathrm{i}%
\widehat{\mathrm{i}}\sum_{r=1}^8\lambda _rg\frac 12\gamma ^{\left[ 5\right]
}G_s^r$$
with $G_s^r$ are real, $g$ is a real positive and:
$$U=\exp \left( \widehat{\mathrm{i}}\sum_{r=1}^8\lambda _r\alpha _r\right)$$
with
$$\lambda _1\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 1_4 & 0_4 \\
1_4 & 0_4 & 0_4 \\
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _2\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & -\widehat{\mathrm{i}} & 0_4 \\
\widehat{\mathrm{i}} & 0_4 & 0_4 \\
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{,}\lambda _3\stackrel{def}{=}\left\|
\begin{array}{ccc}
1_4 & 0_4 & 0_4 \\
0_4 & -1_4 & 0_4 \\
0_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{,}$$
$$\lambda _4\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 0_4 & 1_4 \\
0_4 & 0_4 & 0_4 \\
1_4 & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _5\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 0_4 & -\widehat{\mathrm{i}} \\
0_4 & 0_4 & 0_4 \\
\widehat{\mathrm{i}} & 0_4 & 0_4
\end{array}
\right\| \mbox{, }\lambda _6\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 0_4 & 0_4 \\
0_4 & 0_4 & 1_4 \\
0_4 & 1_4 & 0_4
\end{array}
\right\| \mbox{,}$$
$$\lambda _7\stackrel{def}{=}\left\|
\begin{array}{ccc}
0_4 & 0_4 & 0_4 \\
0_4 & 0_4 & -\widehat{\mathrm{i}} \\
0_4 & \widehat{\mathrm{i}} & 0_4
\end{array}
\right\| \mbox{, }\lambda _8\stackrel{def}{=}\frac 1{\sqrt{3}}\left\|
\begin{array}{ccc}
1_4 & 0_4 & 0_4 \\
0_4 & 1_4 & 0_4 \\
0_4 & 0_4 & -2\cdot 1_4
\end{array}
\right\| \mbox{.}$$
Here $G_s^r$ are *the gluon fields*.
Gravity fields
==============
Equation (\[ham0\]) can be reformed as the following [@Q21]:
$$\left(
\begin{array}{c}
\beta ^{\left[ 0\right] }\left( \partial _0+\mathrm{i}\Theta _0+\mathrm{i}%
\Upsilon _0\gamma ^{\left[ 5\right] }-\mathrm{i}M_0\gamma ^{\left[ 0\right]
}-\mathrm{i}M_4\beta ^{\left[ 4\right] }\right) \\
+\beta ^{\left[ 1\right] }\left( \partial _1+\mathrm{i}\Theta _1+\mathrm{i}%
\Upsilon _1\gamma ^{\left[ 5\right] }+M_{\zeta ,0}\beta ^{\left[ 4\right]
}-M_{\zeta ,4}\gamma ^{\left[ 0\right] }\right) \\
+\beta ^{\left[ 2\right] }\left( \partial _2+\mathrm{i}\Theta _2+\mathrm{i}%
\Upsilon _2\gamma ^{\left[ 5\right] }+M_{\eta ,0}\beta ^{\left[ 4\right]
}+M_{\eta ,4}\gamma ^{\left[ 0\right] }\right) \\
+\beta ^{\left[ 3\right] }\left( \partial _3+\mathrm{i}\Theta _3+\mathrm{i}%
\Upsilon _3\gamma ^{\left[ 5\right] }-M_{\theta ,0}\beta ^{\left[ 4\right]
}-M_{\theta ,4}\gamma ^{\left[ 0\right] }\right)
\end{array}
\right) \varphi =0\mbox{.}$$
Therefore the leptons mass members $M_0$, $M_4$ and the quarks mass members $%
M_{\zeta ,0}$, $M_{\zeta ,4}$, $M_{\eta ,0}$, $M_{\eta ,4}$, $M_{\theta ,0}$, $M_{\theta ,4}$ form a field
$$\mathcal{G}\left( M_0,M_4,M_{\zeta ,0},M_{\zeta ,4},M_{\eta ,0},M_{\eta
,4},M_{\theta ,0},M_{\theta ,4}\right)$$
which acts as other gauge fields $\Theta $ and $\Upsilon \gamma ^{\left[
5\right] }$. Since $\mathcal{G}$ is defined by values of masses then this field is *the gravity field*.
Conclusion
==========
Thus, all four forces arise from the representation of physical events’ probabilities by spinors. And all physics events are interpreted by these four forces. Superstrings are not necessary.
A researcher obtains from Nature only probabilities of physical events. But these probabilities give only the known by this time particles (leptons, quarks) and the known by this time gauge fields (electroweak, gluons, gravity) [@Q]. Therefore, if anybody shall assert that he found a super or he found a Higgs then this assertion is false because all these events[^2] can be interpreted by leptons, quarks, and by electroweak, gluons, gravity fields.
[99]{} Bert Schroer, String theory and the crisis in particle physics,\
arXiv:physics/0603112v4 \[physics.soc-ph\] 5 Oct 2006
Idem, p.29
Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p. 38; G. Quznetsov, *Logical Foundation of Theoretical Physics*, Nova Sci. Publ. Inc., NY, (2006), p. 80
Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p.39-41
Idem., p.38 (formula (2.1))
Idem., p.61 (formula (2.27))
Idem, p.39 (formulas 2.2, 2.3, 2.5, 2.7, 2.9)
Idem, p.62
Idem, p.64 (formula 2.34)
Dvoeglazov, V. V., Additional Equations Derived from the Ryder Postulates in the (1/2,0)+(0,1/2) Representation of the Lorentz Group. hep-th/9906083. *Int. J. Theor. Phys.* **37** (1998) 1909. *Helv. Phys. Acta* **70** (1997) 677. *Fizika B* **6** (1997) 75; *Int. J. Theor. Phys.* **34** (1995) 2467.* Nuovo Cimento* **108** A (1995) 1467. *Nuovo Cimento* **111** B (1996) 483. *Int. J. Theor. Phys.* **36** (1997) 635.
Ahluwalia, D. V., (j,0)+(0,j) Covariant spinors and causal propagators based on Weinberg formalism. nucl-th/9905047. *Int. J. Mod. Phys.* *A* **11** (1996) 1855.
Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007), p.95
Idem, p.96-98
Idem, p.91-94
Idem, p.129
Idem, p.121
Idem, p.98
Idem, p.127
Idem, p.127-128
Idem, p.129
Idem, p.131
Idem, p.137
Idem, p.136
Idem, p.138
Idem, p.140-146
Idem, p.150-152
Idem, p.155
Gunn Quznetsov, *Probabilistic Treatment of Gauge Theories*, in series Contemporary Fundamental Physics, ed. V. Dvoeglazov, Nova Sci. Publ. Inc., NY (2007); G. Quznetsov, *Logical Foundation of Theoretical Physics*, Nova Sci. Publ. Inc., NY, (2006)
[^1]: I denote $n\times n$ matrices:
$$1_n\stackrel{def}{=}\left[
\begin{array}{cccc}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots \\
0 & 0 & \cdots & 1
\end{array}
\right] ,0_n\stackrel{def}{=}\left[
\begin{array}{cccc}
0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 \\
\cdots & \cdots & \cdots & \cdots \\
0 & 0 & \cdots & 0
\end{array}
\right] \mbox{.}$$
[^2]: and a dark energy and a dark matter, may be, too
| ArXiv |
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1.8truecm
Quiver Grassmannians and Auslander varieties
for wild algebras.
Claus Michael Ringel
Let $k$ be an algebraically closed field and $\Lambda$ a finite-dimensional $k$-algebra. Given a $\Lambda$-module $M$, the set $\Bbb G_{\bold e}(M)$ of all submodules of $M$ with dimension vector $\bold e$ is called a quiver Grassmannian. If $D,Y$ are $\Lambda$-modules, then we consider $\Hom(D,Y)$ as a $\Gamma(D)$-module, where $\Gamma(D) =
\End(D)^\op$, and the Auslander varieties for $\Lambda$ are the quiver Grassmannians of the form $\Bbb G_{\bold e}\Hom(D,Y)$. Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs in this way. There is a tendency to relate this fact to the wildness of quiver representations and the aim of this note is to clarify these thoughts: We show that for an algebra $\Lambda$ which is (controlled) wild, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian.
[**1. Introduction.**]{} Let $k$ be an algebraically closed field and $\Lambda$ a finite-dimensional $k$-algebra. A [*dimension vector*]{} $\bold d$ for $\Lambda$ is a function defined on the set of isomorphism classes of simple $\Lambda$-modules $S$ with values $d_S$ being non-negative integers. If $M$ is a $\Lambda$-module, its dimension vector $\bdim M$ attaches to the simple module $S$ the Jordan-Hölder multiplicity $(\bdim M)_S = [M:S].$
Given a $\Lambda$-module $M$, the set $\Bbb G_{\bold e}(M)$ of all submodules of $M$ with dimension vector $\bold e$ is called a quiver Grassmannian. Quiver Grassmannians are projective varieties and every projective variety occurs in this way (see the Appendix). If $D,Y$ are $\Lambda$-modules, then we consider $\Hom(D,Y)$ as a $\Gamma(D)$-module, where $\Gamma(D) =
\End(D)^\op$. The easiest way to define the Auslander varieties for $\Lambda$ is to say that they are just the quiver Grassmannians $\Bbb G_{\bold e}\Hom(D,Y)$ (here, we rely on the Auslander bijections; the proper definition of the Auslander varieties would have to refer to right equivalence classes of right $D$-determined maps ending in $Y$, see \[Ri\]). The Auslander varieties are part of Auslander’s approach of describing the global directedness of the category $\mod\Lambda$. Let as add that the quiver Grassmannians for $\Lambda$ are special Auslander varieties, namely the Auslander varieties $\Bbb G_{\bold e}\Hom(D,Y)$ with $D = \Lambda$.
According to Drozd \[D1\], any finite dimensional $k$-algebra is either tame or wild (note that there are few tame algebras, most of the algebras are wild; for example, the path algebra of a connected quiver is tame only in case we deal with a Dynkin or an extended Dynkin quiver). It has been conjectured that wild algebras are actually controlled wild (the definition will be recalled in section 2). A proof of this conjecture has been announced by Drozd \[D2\] in 2007, but apparently it has not yet been published. We show that for a fixed (controlled) wild algebra $\Lambda$, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian. We denote by $\mod\Lambda$ the category of all (finite-dimensional left) $\Lambda$-modules. Let $\rad$ be the radical of $\mod\Lambda$, this is the ideal generated by all non-invertible maps between indecomposable modules. If $\Cal C$ is a collection of object of $\mod \Lambda$, we denote by $\add \Cal C$ the closure under direct sums and direct summands. For every pair $X,Y$ of $\Lambda$-modules, $\Hom(X,\Cal C,Y)$ denotes the subgroup of $\Hom(X,Y)$ given by the maps $X \to Y$ which factor through a module in $\add\Cal C$. Here is now the definition. The algebra $\Lambda$ is said to be [*controlled wild*]{} provided for any finite-dimensional $k$-algebra $\Gamma$, there is an exact embedding functor $F\:\mod \Gamma \to \mod \Lambda$ and a full subcategory $\Cal C$ of $\mod \Lambda$ (called the [*control class*]{}) such that for all $\Gamma$-modules $X,Y$, the subgroup $\Hom(FX,\Cal C,FY)$ is contained in $\rad (FX,FY)$ and we have $$\Hom(FX,FY) = F\Hom(X,Y) \oplus \Hom(FX,\Cal C,FY).$$
In order to check that $\Lambda$ is controlled wild, it is sufficient to exhibit such a functor $F$ for just one suitable algebra $\Gamma$, for example for the 3-Kronecker algebra (this is the path algebra of the quiver with two vertices, say $a$ and $b,$ and three arrows $b \to a$).
We also mention that $\Lambda$ is said to be [*strictly wild*]{} provided for any finite-dimensional $k$-algebra $\Gamma$, there is a full exact embedding functor $F\:\mod \Gamma \to \mod \Lambda$ (thus, strictly wild algebras are controlled wild and we can take as control class $\Cal C$ the zero subcategory). The 3-Kronecker algebra is a typical strictly wild algebra. The special case of a strictly wild algebras has been considered already in \[Ri\]. The proof of Proposition 1 will be given in this section. We start with the following Lemma. Proof: Let $X = \bigoplus X_i$. It is sufficient to show that $\Hom(X,C,X) = \Hom(X,\Cal C,X)$ for some module $C\in \add \Cal C$. Since the subgroups $\Hom(X,C,X)$ with $C \in \add\Cal C$ are subspaces of the finite-dimensional vector space $\Hom(X,X)$, there is $C\in \add\Cal C$ such that $\Hom(X,C,X)$ is of maximal dimension. Let $C'\in \add\Cal C$. Then also $C\oplus C'$ belongs to $\add\Cal C$ and we have $\Hom(X,C,X) \subseteq
\Hom(X,C\oplus C',X)$. The maximality of the dimension of $\Hom(X,C,X)$ implies that $\Hom(X,C,X) =
\Hom(X,C\oplus C',X),$ and thus $\Hom(X,C',X) \subseteq \Hom(X,C,X)$. But $\Hom(X,\Cal C,X) = \bigcup_{C'} \Hom(X,C',X)$.
Proof. Let $U$ be an element of $\Bbb G_{\bold g+\bold c} N$. We want to show that $U \supseteq ReN$. Given dimension vectors $\bold d, \bold d'$ for $\Lambda$, one writes $\bold d'\le \bold d$ provided $\bold d-\bold d'$ has non-negative coefficients. Since $\bdim U = \bold g+\bold c,$ we have $\bdim U \ge \bold c.$ Let $S$ be a simple $R$-module with $eS \neq 0$. Then $$[U:S] = (\bdim U)_S \ge \bold c_S = (\bdim \Lambda eN)_S = [\Lambda eN:S],$$ and therefore $eN \subseteq U$, thus also $\Lambda eN \subseteq U.$
[**Proof of proposition 1.**]{} Let $V$ be a projective variety. There is a finite-dimensional algebra $\Gamma$, a $\Gamma$-module $M$ and a dimension vector $\bold g$ for $\Gamma$ such that $\Bbb G_{\bold g}M$ is of the form $V$ (see the Appendix). Since $\Lambda$ is controlled wild, there is a controlled embedding $F$ of $\mod
\Gamma$ into $\mod\Lambda$, say with control class $\Cal C$. Let $G = F({}_\Gamma\Gamma)$ and $Y = F(M).$ According to Lemma 1, there is $C\in \add\Cal C$ such that $\Hom(G,C,G) = \Hom(G,\Cal C,G)$ and $\Hom(G,C,Y) = \Hom(G,\Cal C,Y)$. Let $D = G\oplus C$ and $R = \End(D)^\op.$ Let $e_G$ be the projection of $D$ onto $G$ with kernel $C$ and $e = e_C$ the projection of $D$ onto $C$ with kernel $G$, both $e_G, e_C$ considered as elements of $R$. Note that $$\align
\Hom(D,D) &= \Hom(G\oplus C,G\oplus C) = \Hom(G,G)\oplus \Hom(G,C) \oplus \Hom(C,G) \oplus \Hom(C,C)\cr
&= F(\Hom(\Gamma,\Gamma))
\oplus \Hom(G\oplus C,\Cal C,G\oplus C) \oplus \Hom(G,C) \oplus \Hom(C,G) \oplus \Hom(C,C),
\endalign$$ and $$e\Hom(D,D)e = \Hom(G\oplus C,\Cal C,G\oplus C) \oplus \Hom(G,C) \oplus \Hom(C,G) \oplus \Hom(C,C).$$ It follows that the map $\gamma \mapsto F(\gamma)\in
e_GRe_G$ yields an isomorphism $\Gamma \to R/ReR.$ Also, we are interested in the $R$-module $N = \Hom(G\oplus C,Y).$ Here, we have $$\align
N = \Hom(G\oplus C,Y) &= \Hom(G\oplus 0,Y) \oplus \Hom(0\oplus C,Y) \cr
&= F\Hom(\Gamma,M) \oplus \Hom(G\oplus 0,C,Y) \oplus
\Hom(0\oplus C,Y).
\endalign$$ If we multiply $N$ with the element $e = e_C\in R$, we obtain $$eN = \Hom(0\oplus C,Y),$$ thus $$ReN =
R\Hom(0\oplus C,Y) = \Hom(G\oplus 0,C,Y) \oplus
\Hom(0\oplus C,Y).$$ This shows that $N/ReN$ is canonically isomorphic to $F\Hom(\Gamma,M)$ as an $R$-module. Of course, these modules are annihilated by $e$, thus they are $R/ReR$-modules and as we know $R/ReR = \Gamma.$
It remains to apply Lemma 2. [**4. Quiver Grassmannians.**]{} Proof. Let $\Lambda$ be any local radical square zero $k$-algebra of dimension at least 4 (thus $\Lambda = k[T_1,\dots,T_n]/(T_1,\dots,T_n)^2$ with $n \ge 3$). It is well-known (and easy to see) that such an algebra is controlled wild. Let $M$ be a $\Lambda$-module. The Grothendieck group $K_0(\Lambda)$ is free of rank one, thus the quiver Grassmannians are of the form $\Bbb G_i(M)$ with $i$ a non-negative integer (the elements of $\Bbb G_i(M)$ are the submodules of $M$ of dimension $i$). In order to determine the possible varieties $\Bbb G_i(M)$, we can assume that $i
\le\dim \soc M$. Namely, if $i > \dim\soc M,$ then we consider the dual module $M^*$ and the quiver Grassmannian $\Bbb G_{d-i}(M^*)$, where $d = \dim M = \dim M^*$. On the one hand, the varieties $G_i(M)$ and $G_{d-i}(M^*)$ are obviously isomorphic, on the other hand we have $d-i < \dim M - \dim\soc M \le \dim M - \dim\rad M =
\dim \top M = \dim\soc(M^*)$, here we have used that $\rad M \subseteq \soc M$.
Thus, let $i \le s = \dim\soc M.$ The submodules $U$ of $\soc M$ of dimension $i$ are just the subspaces of $\soc M$ of dimension $i$ considered as a $k$-space, thus they form the usual Grassmannian $\Bbb G_i(\soc M) = \Bbb G_i(k^s)$, in particular, this is an irreducible (and rational) variety. Now let $U$ be a submodule of $M$ of dimension $i$ which is not contained in $\soc M$. We claim that there is a projective line in $\Bbb G_i(M)$ which contains both $U$ and a submodule $U'$ of $\soc M$. Namely, let $b_1,\dots, b_t$ be a basis of $\soc U$, and extend it to a basis $b_1,\dots,b_i$ of $U$. Now $b_1,\dots, b_t$ are linearly independent elements of the socle of $M$. Since $i \le \dim\soc M$, there is an $i$-dimensional subspace $U'$ inside $\soc M$ which contains $\soc U' = U \cap \soc M.$ We can extend the basis $b_1,\dots, b_t$ of $\soc U$ to a basis of $U'$, say $b_1,\dots, b_t, b'_{t+1},\dots, b'_i$, For $\lambda = (\lambda_0:\lambda_1) \in \Bbb P^1$, define $U_\lambda$ as the subspace of $M$ with basis the elements $b_1,\dots, b_t$ as well as the elements $\lambda_0b_j+\lambda_1b'_j$ where $t < j \le i.$ Of course, $U_\lambda$ is a submodule of $M$. On the one hand, we have $U_{(1:0)} = U$, on the other hand, $U_{(0:1)} = U'$ is an $i$-dimensional submodule of $M$ which lies inside the socle of $M$.
Since the Grassmannian $\Bbb G_i(\soc M)$ is (rational and) connected and for any element $U\in
\Bbb G_i(M)$ there is a $\Bbb P^1$-family of submodules which contains $U$ and an element in $\Bbb G_i(\soc M)$, it follows that also $\Bbb G_i(M)$ is connected (even rationally connected, see \[Ha\]).
We have shown that any projective variety occurs as an Auslander variety for any (controlled) wild algebra. It seems that the Auslander varieties for the tame algebras are quite restrictive — is there a special property which all have? Such a result would provide a characterization of the tame-wild dichotomy in terms of Auslander varieties.
We have shown that dealing with local algebras with radical square zero, all quiver Grassmannians are rationally connected. Are there further properties which they share? On the other hand, the quiver Grassmannians for wild algebras should be of quite a general nature. Is there a class of varieties which can be realized as quiver Grassmannians for all wild algebras, but not for tame algebras?
We say that a $k$-module $M$ is a [*brick*]{} provided $\End(M) = k.$ Let us outline a proof, following Van den Bergh \[L\]. Let $V$ be a projective variety. We can assume $V$ is a closed subset of the projective space $\Bbb P^n$, defined by the vanishing of homogeneous polynomials $f_1,\dots,f_m$ of degree 2. Let $\Delta$ be the Beilinson quiver with 3 vertices, say $a,b,c$, with $n+1$ arrows $b\to a$ labeled $x_0,\dots,x_n$ as well as $n+1$ arrows $c\to b$, also labeled $x_0,\dots,x_n$. The path algebra of $\Delta$ with all the relations $x_ix_j = x_jx_i$ (whenever this makes sense) is called the Beilinson algebra. Let $\Lambda$ be the factor algebra of this Beilinson algebra taking the elements $f_1,\dots,f_m$ as additional relations (obviously, these elements may be considered as linear combinations of paths of length 2 in the quiver $\Delta$). Take $M = I_{\Lambda}(a),$ the indecomposable injective $\Lambda$-module corresponding to the vertex $a$, and take $\bold e = (1,1,1)$. Note that the elements of $\Bbb G_{\bold e}M$ are just all the serial $\Lambda$-modules, one from each isomorphism class (we call a module $X$ [*serial*]{}, provided it has a unique composition series). Here are some remarks on the history: The title of the appendix is also the title of a recent paper \[Re\] by Reineke, who answered in this way a question by Keller. The 2-page paper attracted a lot of interest, see for example blogs by Le Bruyn \[L\] (with the proof by Van den Bergh presented above) and by Baez \[Bz\]. Actually, the construction given in the proof of Van den Bergh is much older, it has been used before by several mathematicians dealing with related problems and may, of course, be traced back to Beilinson \[Be\].
The quiver Grassmannians play an important role in the representation theoretical approach to cluster algebras. Here one deals with the quiver Grassmannians $\Bbb G_{\bold e}M$, where $M$ is a quiver representation without self-extensions. It has been asserted by Caldero and Reineke \[CR\] that the quiver Grassmannians $\Bbb G_{\bold e}M$, where $M$ is a quiver representation without self-extensions, are very special: [*if $\Bbb G_{\bold e}M$ is non-empty, then the Euler characteristic of $\Bbb G_{\bold e}M$ is positive*]{}; a complete proof was given by Nakajima \[N\] and Qin \[Q\]. Note that an indecomposable quiver representation without self-extensions is a brick. If we consider the bricks $M$ constructed by Van den Bergh as representations of the quiver $\Delta$, then such a $k\Delta$-module will have self-extensions, but it seems to be remarkable to observe that $M$ has no self-extensions when it is considered as a $k\Delta/\Ann(M)$-module, where $\Ann(M)$ is the annihilator of $M$ in $k\Delta$ (in contrast, the examples constructed by Reineke are usually faithful quiver representations). Namely, we have $k\Delta/\Ann(M) = \Lambda$, and by construction, $M$ is an injective $\Lambda$-module. We should recall that for any ring $R$, an $R$-module is said to be [*quasi-injective*]{} provided for any submodule $U$ of $M$ any map $U \to M$ can be extended to an endomorphism of $M$; for an artinian ring, a module $M$ is quasi-injective if and only if $M$ considered as an $R/\Ann(M)$-module is injective.
There is a tendency to relate the fact that every variety is a quiver Grassmannian to the tame-wild dichotomy as established by Drozd \[D1\]. For example, Baez \[B\] writes that one may suppose that this is [*just another indication of the ‘wildness’ of quiver representations once we leave the safe waters of Gabriel’s theorem.*]{} The aim of this note was to clarify these thoughts.
Some mathematicians (see \[L\],\[V\]) refer in this context to “Murphy’s law”: [*Anything that can go wrong, will go wrong*]{} (as formulated in the Wikipedia \[W\]), or: [*Anything that can happen, will happen.*]{} But one should be aware that Murphy’s law may be a challenging assertion in daily life, but it is just a tautology when we consider mathematical questions. Indeed, in mathematics, if we know (that means: if we can prove) that something [**does not**]{} happen, then of course we have a proof that it [**cannot**]{} happen. [**References.**]{}
[\[Bz\]]{} Baez, J.: The n-category cafe: Quivering with Excitement. Blog May 4, 2012. http://golem.ph.utexas.edu/category/2012/05/quivering$\_$with$\_$excitement.html
[\[Be\]]{} Beilinson, A.A.: Coherent sheaves on $\Bbb P^n$ and problems in linear algebra. Funktsional. Anal. i Prilozhen 12.3 (1978), 68-29.
[\[CR\]]{} Caldero, Ph., Reineke, M.: On the quiver Grassmannian in the acyclic csae. J. Pure Appl. Algebra 212 (2008), 2369-2380. arXiv:math/0611074
[\[D1\]]{} Drozd, Yu.A.: Tame and wild matrix problems, in: Representation Theory II, Lecture Notes in Math., Vol. 832 (1980), 242-258
[\[D2\]]{} Drozd, Yu.A.: Wild Algebras are Controlled Wild. Presentation ICRA XII, (2007). http://icra12.mat.uni.torun.pl/lectures/Drozd-Torun.pdf
[\[Ha\]]{}Harris, J.: Lectures on rationally connected varieties. http://mat.uab.es/kock/RLN/rcv.pdf.
[\[L\]]{} Le Bruyn, L.: Quiver Grassmannians can be anything. Blog, May 2, 2012. http://www.neverendingbooks.org/quiver-grassmannians-can-be-anything.
[\[N\]]{} Nakajima, H.; Quiver varieties and cluster algebras, http://arxiv.org/abs/0905.0002v5.
[\[Q\]]{} Qin, F.: Quantum Cluster Variables via Serre Polynomials. http://arxiv.org/abs/1004.4171.
[\[Re\]]{} Reineke, M.: Every projective variety is a quiver Grassmannian. Algebras and Representation Theory (to appear). arXiv:1204.5730.
[\[Ri\]]{}Ringel, C. M.: The Auslander bijections: How morphisms are determined by modules. arXiv:1301.1251.
[\[V\]]{} Vakil, R.: Murphy’s Law in algebraic geometry: Badly-behaved deformation spaces. Invent. math. 164 (2006), 569-590.
[\[A1\]]{}Auslander, M.: Functors and morphisms determined by objects. In: Representation Theory of Algebras. Lecture Notes in Pure Appl. Math. 37. Marcel Dekker, New York (1978), 1-244. Also in: Selected Works of Maurice Auslander, Amer. Math. Soc. (1999).
[\[A2\]]{}Auslander, M.: Applications of morphisms determined by objects. In: Representation Theory of Algebras. Lecture Notes in Pure Appl. Math. 37. Marcel Dekker, New York (1978), 245-327. Also in: Selected Works of Maurice Auslander, Amer. Math. Soc. (1999).
[\[ARS\]]{}Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36. Cambridge University Press. 1997.
blogs: a large part of the contributions - ignorant and superficial.
| ArXiv |
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abstract: '[Dans cet article je donnerai une nouvelle démonstration courte et directe pour le Théorème des Nombres Premiers. C’est vrai que ce théorème a été complétement démontré au début du 20ème siecle mais la démonstration était basé sur des résultats élémentaires (théorème de **Chebyshev**) et aussi analytiques compliqués (théorème de **Ikehrara**), mais ici j’ai pas utilisé le théorème de Chebyshev ainsi que j’ai remplacé et j’ai généralisé le théorème de **Ikehara** grâce à la notion des fonctions à variation bornée qui est ancienne mais récent dans la théorie analytique des nombres.]{}'
author:
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`[email protected]`\
`[email protected]`
title: |
<span style="font-variant:small-caps;">Tauberian Theorem of Laplace Transformation\
And\
Application of Prime Number Theorem</span>
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Préliminaires
=============
Les fonctions à variation bornée
--------------------------------
Les fonctions à variation bornée joue un rôle très important dans la théorie de l’intégration au sens de **Stieltjes**, ici on va s’interésser à les fonctions à variation bornée sur $\R^+$ à valeurs complexes. Soit $x$ un réel positif et soit $(x_k)_{k=0,\cdots, n}$ une suite finie et strictement croissante des réels de l’intervalle $[0,x]$ tels que $0=x_0 < x_1<x_2< \cdots < x_n = x$ est une subdivision de l’intervalle $[0,x]$, on note $\Sigma$ pour cette subdivion et $\mathcal{S}([0,x])$ l’ensemble de toutes les subdivions possibles de $[0,x]$. *La fonction variation totale* d’une fonction complexe définie sur $\R^+$, notée $T_f$, est la fonction définie par [$$T_f(x) := \sup_{\Sigma \in \mathcal{S}([0,x])} \sum_{k=1}^n|f(x_k) - f(x_{k-1})|\label{e1}$$]{}
Il est bien clair que la fonction $T_f$ est une fonction croissante sur $\R^+$, par conséquent si $T_f$ est majorée sur $\R^+$ alors on dira que $f$ est à *variation bornée* sur $\R^+$ et on note $$V(f) = \lim_{x\to + \infty} T_f(x) \in \R^+$$ pour *la variation totale* de la fonction $f$.
- Toute fonction $g$ de classe $\mathcal{C}^1$ sur $\R^+$ à valeurs complexes telle que $g' \in L^1(\R^+)$ est à variation bornée, en effet, pour une subdivision $\Sigma: 0=x_0 < x_1<\cdots < x_n=x$ et puisque $g$ est continue sur chaque intervalle $[x_{i-1},x_i]$ (pour $i=1,\cdots n$) et dérivable sur leurs interieurs topologique alors d’après le théorème des accroissements finis il existe des $c_i$ dans $]x_{i-1},x_i[$ tels que $$|g(x_i) - g(x_{i-1})|=|g'(c_i)||x_{i} - x_{i-1}|$$
D’où $$T_g(x) = \sup_{\Sigma \in \mathcal{S}([0,x])} \sum_{k=1}^n|g'(c_i)||x_{i} - x_{i-1}|$$
Or cette somme est une somme de **Darboux** ce qu’on peut déduire grâce à l’intégrale de **Riemann** que $$T_g(x) = \int_0^x|g'(t)| dt$$
Donc $$V(g) = \int_{0}^{+\infty}|g'(t)| dt$$ qui est finie puisque $g' \in L^1(\R^+)$, alors $g$ est à variation bornée sur $\R^+$.
- toute fonction à variation bornée sur $\R^+$ est bornée sur $\R^+$, en effet, soit $f$ une fonction à variation bornée sur $\R^+$ alors pour un $x\geq 0$ $$\begin{aligned}
|f(x) - f(0)| &= \left|\sum_{k=1}^n f(x_i) - f(x_{i-1})\right| \\ &\leq \sum_{k=1}^n|f(x_i) - f(x_{i-1})| \\ &\leq T_f(x) \\ &\leq V(f) <+\infty \end{aligned}$$
Alors $f$ est bornée sur $\R^+$.
\[prop1\]
On dit qu’une fonction $f$ définie de $\R^+$ à valeurs complexes admet une limite à gauche en $x \in \R^+$, notée $f(x^-)$ si à tout $\eps >0$ on peut associer un $0 \leq \delta < x$ tel que $$a < t < x \Longrightarrow |f(t) - f(x^-)| < \eps$$
Et en plus si $f(x^-) = f(x)$ on dit que $f$ est continue à gauche en $x$.
On note pour $\vb$ la classe des fonctions, définies de $\R^+$ à valeurs complexes, à variation bornée, continues à gauche en tout point de $\R^+$ et qui s’annullent en $0$.
Intégrale de Lebesgue-Stieltjes
-------------------------------
Le théorème 8.14 page 156 du livre \[Rud\] a établi le lien entre la théorie de la mesure et la théorie des fonctions à variation bornée. Donc d’après le même théorème, soit $f \in \vb$ alors il existe une unique mesure complexe de Borel $\mu_f$ telle que [$$f(x) = \mu_f([0,x[), \qquad \forall x\geq 0 \label{e2}$$]{}
Et en plus pour tout $x \in \R^+$ on a [$$T_f(x)=|\mu_f|([0,x[)$$]{}
Où $|\mu_f|$ est une mesure positive de Borel, dite *la variation totale de la mesure complexe $\mu_f$*, qui est *finie* d’après le théorème 6.4 page 114 de \[2\].
- On peut facilement montrer que $|\mu_f|$ est finie autant que $f\in \vb$, en effet, soit $f\in \vb$ alors $$\begin{aligned}
|\mu_f|(\R^+) &= \lim_{x\to +\infty} |\mu_f|([0,x[) \\ &= \lim_{x\to +\infty}T_f(x) \\&= V(f) <+ \infty\end{aligned}$$
- D’une autre part, si $f$ est à valeurs dans $\R$ alors $\mu_f$ est dite *une mesure signée* alors de la même manière on démontre que cette mesure est finie.
- Soit $f\in \vb$, si $y>x$ alors $$\begin{aligned}
f(y) - f(x) &= \mu_f([0,y[) - \mu_f([0,x[)\\ &= \mu_f([x,y[) \end{aligned}$$
Donc $$\mu_f(\{x\}) = f(x^+) - f(x)$$
D’où $f$ est continue en $x$ si et seulement si $$\mu_f(\{x\}) = 0$$
Le théorème de **Radon-Nikodym**, voir le théorème 6.12 page 120 de \[2\], assure que pour toute mesure complexe $\mu$ il existe une fonction mesurable complexe $h$ de module égal à $1$ telle que $$d\mu = h d|\mu| .$$
Ainsi, on déduit que pour toute fonction $g:\R^+ \longrightarrow \C$ mesurable et bornée sur $\R^+$ on a $g \in L_{\mu_f}^1(\R^+)$ où $f\in \vb$. En effet: $$\begin{aligned}
\left| \int_{\R^+} g d\mu_f \right| &\leq \int_{\R^+} |g| d|\mu_f| \\ &\leq \|g\|_{\infty} |\mu_f|(\R^+)\\ &< +\infty \end{aligned}$$
Où $$\|g\|_{\infty} = \sup_{x \in \R^+}|g(x)|.$$
Maintenant, d’après le théorème 6.1.4 du livre \[1\] on constate que pour $f \in \vb$ on a[$$\int_0^x df(t) = \mu_f([0,x[), \qquad x\geq 0 \label{e4}$$]{}
Soient donc $f\in \vb$ et $g:\R^+ \longrightarrow \C$ une fonction de classe $\mathcal{C}^1$ sur $\R^+$ telle que $g' \in L^1(\R^+)$, alors d’après le théorème 6.2.2 (grâce au résultat \[e4\]) du même livre on démontre que [$$\int_0^{+\infty} g(t) df(t) = \mu_{fg}(\R^+) - \int_0^{+\infty} f(t)g'(t) dt \label{e5}$$]{}
la mesure complexe $\mu_{fg}$ a bien un sens, en effet d’après les propriétés \[prop1\] on démontre que $g$ est à variation bornée or le produit de deux éléments de $\vb$ est un élément de $\vb$ alors $fg \in \vb$ (car $fg$ est à variation bornée et continue à gauche à chaque point de $\R^+$ et $(fg)(0) =0$), et en plus $$\mu_{fg}([0,x[) = f(x)g(x), \qquad \forall x\in \R^+$$
Et $$|\mu_{fg}(\R^+)| \leq |\mu_{fg}|(\R^+) = \lim_{x\to + \infty} T_{fg}(x)< + \infty$$
Théorème Tauberien de la transformation de Laplace complexe
===========================================================
Dans tout ce qui suit $s = \sigma + it$ où $\sigma , t \in \R$ est un nombre complexe et $\rho$ est une fonction de la classe $\vb^* := \{f \in \vb , \quad \ \Im(f) = 0 \}$. Ainsi $\C_*^+$ est l’ensemble des nombres complexes de partie réelle strictement positive.
On définit la transformation de Laplace-Stieltjes de la fonction $\rho$ par $$\L_{\rho}^*(s) = \int_0^{+\infty} e^{-sx}d\rho(x), \qquad \sigma > 0$$
il est bien clair d’après ce qui précéde, puisque $x\mapsto e^{-sx}$ est continue et bornée sur $\R^+$ pour tout $\sigma > 0$, que la fonction $\L_{\rho}^*$ est bien définie.
******
On pose pour tout $(x,s) \in \R^+ \times \C_+^*$ $$\phi(x,s) = e^{-sx}$$
Alors on a
- Pour tout $x\geq 0$ la fonction $s\mapsto \phi(x,s)$ est continue en $0^+$.
- Pour tout $\sigma > 0$ la fonction $x \mapsto \phi(x,s)$ est continue donc mésurable sur $\R^+$.
- Pour tout $\sigma > 0$ et pour $d\rho$-presque tout $x \in \R^+$ on a $$|\phi(x,s)|\leq 1$$
Où $1 \in L_{d\rho}^1(\R^+)$ car $$\int_{\R^+} d\rho(x) = \mu_{\rho}(\R^+) <+ \infty$$
Alors la fonction $x \mapsto \phi(x,s)$ est $d\rho$-intégrable sur $\R^+$ et la fonction $\L_{\rho}^*$ est est continue en $0^+$. Donc $$\begin{aligned}
\lim_{s \to 0^+} \L_{\rho}^*(s) &= \L_{\rho}^*(0) \\ &= \int_{\R^+}d\rho (x) \\ &= \lim_{x\to + \infty} \mu_{\rho}([0,x[) \qquad \text{d'après \ref{e4}} \\ &= \lim_{+ \infty} \rho \qquad \qquad \qquad \quad \! \! \text{d'après \ref{e2}} \end{aligned}$$
$\blacksquare$
Maintenant on définit la transformation de Laplace complexe d’une fonction $\rho \in \vb^*$ par $$\L_{\rho}(s) = \int_{0}^{+ \infty} \rho(x)e^{-sx}dx, \qquad \sigma > 0 .$$
La fonction $\L_{\rho}$ est bien définie, en effet puisque $\rho \in \vb^*$ alors $\rho$ est bornée sur $\R^+$ en plus $$\left|\int_0^{+ \infty} \rho(x)e^{-sx}dx \right| \leq \| \rho\|_{\infty} \int_0^{+ \infty} e^{- \sigma x} dx = \frac{\|\rho\|_{\infty}}{\sigma} < +\infty$$
******
Soit $s\in \C_+^*$, d’après l’équation \[e5\] on a $$\L_{\rho}^*(s) = \mu_{\rho e^{-s \cdot}}(\R^+) + s \int_0^{+\infty} \rho(x) e^{-sx} dx$$
Or $$|\mu_{\rho e^{-s \cdot}}(\R^+)| = |\mu_{\rho}(\R^+)|\lim_{x \to + \infty} e^{-\sigma x} = 0$$
Donc $$\L_{\rho}^*(s) = s \L_{\rho}(s)$$
Passons à la limite $s \to 0^+$ on a d’après le <span style="font-variant:small-caps;">Lemme</span> \[L1\] $$\lim_{x\to + \infty} \rho(x) =Res(\L_{\rho},0)$$
$\blacksquare$
D’une manière générale, soit $\alpha$ un réel positif alors il est clair, d’après ce qui précéde, que pour tout $\rho \in \vb^*$ on a $\varrho(x) = \rho(x) e^{-\alpha x}$ est un élément de $\vb^*$. Ainsi on déduit le résultat suivant:
******
Soit $\sigma > \alpha$, alors $$\L_{\rho}(s) = \int_0^{+\infty} \rho(x)e^{-sx}dx = \int_{0}^{+\infty} \rho(x) e^{-\alpha x} e^{-(s-\alpha)x}dx$$
On pose $$\varrho(x) = \rho(x)e^{- \alpha x}, \qquad \forall x \geq 0 .$$
Alors $$\L_{\rho}(s) = \int_0^{+ \infty} \varrho(x)e^{-(s-\alpha)x}dx = \L_{\varrho}(s - \alpha)$$
Donc $$(s-\alpha) \L_{\rho}(s) = (s-\alpha)\L_{\varrho}(s- \alpha) = z \L_{\varrho}(z)$$
D’où quand $s \to \alpha$ on aura $z \to 0$ et d’après le <span style="font-variant:small-caps;">Théorème</span> \[T1\] on a $$\lim_{x \to + \infty} \varrho(x) = Res(\L_{\varrho}(z),z=0) = Res(\L_{\rho},\alpha)$$
Alors $$\rho(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\alpha) e^{\alpha x} .$$
$\blacksquare$
Théorème des Nombres Premiers (nouvelle démonstration)
======================================================
Soit $\chi: \N^* \longrightarrow \R^+$ une fonction arithmétique positive, on pose pour tout $x \in (1,+\infty)$ $$f(x) = \sum_{1 \leq n < x}\chi(n) \qquad \text{et} \qquad f(1) =0 .$$
Il est clair que la fonction $f$ est croissante sur $[1,+\infty)$ et continue à gauche en tout point de $[1,+\infty)$. Ainsi, les points de discontinuité de $f$ sont des éléments de $\N^*$. Si $f$ est continue en $x\in \N^*$ alors on aura $$f(x^+) = f(x)$$
Donc $$\begin{aligned}
0 &= f(x^+) - f(x) \\ &= \sum_{x \leq n < x^+}\chi(n) \\ &= \chi(x) \end{aligned}$$
Alors [$$f \ \text{est \ continue \ en } \ x \in \N^* \Longleftrightarrow \chi(x) = 0 \label{R1}$$]{}
Soit maintenant $(a_k)_{k \in \N}$ une suite croissante des points de discontinuité de la fonction $f$ sur $[1,+\infty)$ alors $f$ est constante sur chaque intervalle $I_k=(a_{k-1},a_{k}]$ (où $k \in \N^*$). En effet, soit $k\in \N^*$ s’il existe $n \in I_k$ tel que $f$ est continue en $n$ alors d’après \[R1\] $\chi(n) = 0$ ainsi et d’une manière générale soit $(\beta_i)_{i \in \N^*}$ une suite strictement croissante des entiers de $\overset{\circ}{I_k}$ (l’interieur de $I_k$), alors $f$ est continue en chaque $\beta_i$ d’où $\chi(\beta_i)=0$ pour tout $i=1,2,\cdots$ et en conséquent pour tout $x \in (a_{k-1},a_k]$ on a $f(x) = f(a_{k-1}^+)$ ($k \in \N^*$).
Soient $\alpha >1$ un réel et $\rho$ la fonction définie sur $\R^+$ par $$\rho(x) = f\left( e^{x} \right) e^{-\alpha x} .$$
Soit $k$ un entier strictement positif on note $ (\lambda_k)_{k \in \N}$ pour la suite croissante des points de discontinuité de la fonction $\rho$ sur $\R^+$ ($\lambda_k = \log a_k \in \log\N^*$). Alors la fonction $\rho$ est décroissante sur chaque intervalle $J_k = (\lambda_{k-1},\lambda_k]$, en effet: soient $x,y \in J_k$ tels que $x>y$, donc puisque $f$ est constante ($\equiv c_k$) sur $J_k$ alors $\rho(x) - \rho(y) = c_k \left(e^{-\alpha x} - e^{-\alpha y}\right) < 0$ d’où $\rho$ est strictement décroissante sur $J_k$ pour tout $k \in \N^*$. D’une autre part, pour tout $k \in \N^*$ $$\rho(\lambda_k^+) > \rho(\lambda_k).$$
En effet, puisque la fonction $x \mapsto e^{- \alpha x}$ est continue sur $\R^+$ alors $e^{-\alpha \lambda_k^+} = e^{- \alpha \lambda_k}$ donc $$\rho(\lambda_k^+)-\rho(\lambda_k) = (f(a_k^+) - f(a_k))e^{- \alpha \lambda_k}$$ et puisque $f$ est discontinue en $a_k$ et croissante sur $[1,+\infty)$ alors $f(a_k^+) > f(a_k)$. D’où $$\rho(\lambda_k^+) > \rho(\lambda_k).$$
******
Soit $x \in \R^+$, on pose $0=x_0 < x_1 < \cdots < x_n = x$ une subdivision de l’intervalle $[0,x]$ et on note pour $m$ le plus grand entier naturel non nul tel que $\lambda_{m-1} < x \leq \lambda_m$ où les $(\lambda_k)_{k \in \N}$ sont les points de discontinuité de la fonction $\rho$ définis précédamment, alors $$\sum_{i=1}^n |\rho(x_i) - \rho(x_{i-1})| = \sum_{k=0}^m\sum_{\underset{x_i \in J_k}{i=1}}^n|\rho(x_i) - \rho(x_{i-1})|$$ où $(J_k)_{k \in \N^*}$ sont les intervalles $(\lambda_{k-1},\lambda_k]$ et $J_0=[0,\lambda_0]$, et on note bien que $\displaystyle \cup_{k=0}^mJ_k =[0, \lambda_m] $ donc puisque $\rho$ est strictement décroissante sur chaque $J_k$ alors
$$\begin{aligned}
\sum_{k=0}^m\sum_{\underset{x_i \in J_k}{i=1}}^n|\rho(x_i) - \rho(x_{i-1})| & \leq \rho(0) - \rho(\lambda_0) + \sum_{k=1}^m\left(\rho(\lambda_{k-1}^+) - \rho(\lambda_k)\right) - (\rho(x) - \rho(\lambda_m)) \\ &=-\rho(\lambda_0)+\rho(\lambda_0^+)-\rho(\lambda_1)+\rho(\lambda_1^+)+ \cdots -\rho(\lambda_m) - \rho(x) + \rho(\lambda_m)\\ &= - \rho(x) + \sum_{k=0}^{m-1}\left(\rho(\lambda_k^+) -\rho(\lambda_k)\right) \\ &= - \rho(x) + \sum_{k=0}^{m-1} \frac{f(a_k^+) - f(a_k)}{a_k^{\alpha}} \\ &= -\rho(x) + \sum_{k=0}^{m-1}\frac{\chi(a_k)}{a_k^{\alpha}}\end{aligned}$$
Où les $(a_k)_{k \in \N}$ sont les points de discontinuité de la fonction $f$ et qui sont des éléments de $\N^*$. Donc $$\sum_{k=0}^{m-1}\frac{\chi(a_k)}{a_k^{\alpha}} \leq \sum_{1\leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}}$$
D’où $$\sum_{i=1}^n |\rho(x_i) - \rho(x_{i-1})| \leq - \rho(x) + \sum_{1\leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}}.$$
Alors $$T_{\rho}(x) \leq - \rho(x) + \sum_{1 \leq \ell <e^x}\frac{\chi(\ell)}{\ell^{\alpha}} .$$
Or puisque $\rho$ est une fonction positive alors $$T_{\rho}(x) \leq \sum_{1 \leq \ell < e^x} \frac{\chi(\ell)}{\ell^{\alpha}} .$$
Donc $$\text{la série} \ \sum_{n \geq 1} \frac{\chi(n)}{n^{\alpha}} \ \text{converge} \ \Longrightarrow \rho \in \vb^* .$$
$\blacksquare$
Sans perte de généralité le résultat est vrai pour toute fonction arithmétique $\chi:\N^* \longrightarrow \R$ croissante. Dans ce cas, le <span style="font-variant:small-caps;">Lemme</span> \[L1\] peut être reformulé: $$\text{la série} \ \sum_{n \geq 1} \frac{\chi(n)}{n^{\alpha}} \ \text{est absolument convergente} \Longrightarrow \rho \in \vb^*.$$ où $\alpha > 1$.
Maintenant on arrive au résultat le plus important dans cette section:
******
Soit $\alpha > 1$ un réel tel que la série du terme générale $\frac{\chi(n)}{n^{\alpha}}$ est convergente alors, d’après le <span style="font-variant:small-caps;">Lemme</span> \[L1\], la fonction $\rho(x) = f(e^x)e^{-\alpha x}$ est un élément de $\vb^*$. Or d’après le <span style="font-variant:small-caps;">Corollaire</span> \[C1\] on déduit que $$\rho(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\beta) e^{\beta x} .$$
Ce qui est $$f(e^x) \underset{x \to + \infty}{\sim}Res(\L_{\rho},\beta) e^{(\alpha + \beta)x} .$$
D’où $$f(x) \underset{x \to + \infty}{\sim} Res(\L_{\rho},\beta) x^{\alpha + \beta} .$$
Ce qu’il fallait démontrer.
$\blacksquare$
On rappelle que la fonction $\Lambda$ de **Von Mangoldt** est une fonction arithmétique définie sur $\N^*$ par $$\Lambda(n) := \begin{cases} \log p \quad \text{si} \ n=p^k , \quad k \in \N^*,p\in \P \\ \\ \quad 0 \qquad \text{sinon} \end{cases}.$$
La fonction définie pour tout $x \in [1,+ \infty)$ tel que $x \neq p^k$ où $k\in \N^*$ et $p\in \P$ par $ \displaystyle \psi(x) = \sum_{n < x} \Lambda(n)$ est dite la fonction de **Chebyshev**, ainsi pour démontrer le théorème des nombres premiers il faut et il suffit de démontrer que $$\psi(x) \underset{x\to + \infty}{\sim} x .$$
Il existe une forte relation entre la fonction $\zeta$ de **Riemann** et la fonction $\psi$, en effet $$-\frac{\zeta'(s)}{\zeta(s)} = \sum_{n \geq 1} \frac{\Lambda(n)}{n^s} = s \int_0^{+ \infty} \psi(e^x)e^{-sx}dx, \qquad \forall \sigma > 1.$$
On rappelle aussi que la fonction $\zeta$ est holomorphe sur $\{\sigma \geq 1\}$ sauf au $s=1$ qui est le seul pôle simple de la fonction $\zeta$, ainsi d’après **Hadamard** et **De La Vallée Poussin** la fonction $\zeta$ ne s’annulle en aucun point du demi-plan $\{\sigma \geq 1 \}$. Alors on déduit que la fonction $-\frac{\zeta'}{\zeta}$ est holomorphe sur $\{\sigma \geq 1 \}$ sauf au point $s=1$ qui est le seul pôle simple de résidu égal à $1$.
Et on a le **Théorème des Nombres Premiers**:
******
Soit $\alpha > 1$ un réel donné, on pose $$\rho(x) = \psi(e^x)e^{-\alpha x}, \qquad \forall x \in \R^+.$$
Alors puisque la fonction $- \frac{\zeta'}{\zeta}$ est holomorphe sur $\{\sigma > 1\}$ alors la série su terme général $\frac{\Lambda(n)}{n^{\alpha}}$ est convergente pour tout $\alpha > 1$.
D’une autre part, soit $\sigma > 1$ alors
$$\begin{aligned}
\L_{\rho}(s) &= \int_0^{+ \infty}\rho(x)e^{-sx}dx \\ &= \int_0^{+\infty} \psi(e^x)e^{-\alpha x} e^{-sx}dx \\ &= \int_0^{+ \infty} \psi(e^x)e^{-(s+\alpha)x}dx \\&= - \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)} \end{aligned}$$
Alors puisque la fonction $s\mapsto - \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)}$ est holomorphe sur $\{\sigma \geq 1-\alpha\}$ sauf au point $s=1-\alpha$ qui est le seul pôle simple de cette fonction où $$Res\left(- \frac{\zeta'(s+\alpha)}{(s+\alpha)\zeta(s+\alpha)},1-\alpha \right)=1$$
Alors d’après le <span style="font-variant:small-caps;">Théorème</span> \[T2\] on a $$\psi(x) \underset{x \to + \infty}{\sim}x^{\alpha + 1 - \alpha}$$
C’est à dire $$\psi(x) \underset{x \to + \infty}{\sim} x .$$
Ce qu’il fallait démontrer.
$\blacksquare$
[plain]{}
M.Carter et B. Van Brunt, *The Lebesgue-Stieltjes Integral* a practical introduction, Springer (2000). E.C. Titchmarsh, *The Theory of The Riemann Zeta-Function* 2nd ed, revised by D. R. Heath-Brown, Oxford University Press (1986). Walter Rudin, *Analyse réelle et complexe*, Troisième tirage MASSON Paris New York Barcelone Milan 1980.
| ArXiv |
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abstract: 'We study the irreducible components of the moduli space of instanton sheaves on ${{\mathbb{P}^{3}}}$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${{\mathcal T}}(d)$ of stable sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=dt$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${{\mathcal T}}(d)$ for $d\le4$.'
address:
- |
IMECC - UNICAMP\
Departamento de Matemática\
Rua Sérgio Buarque de Holanda, 651\
13083-970 Campinas-SP, Brazil
- 'Institute of Mathematics of the Romanian Academy, Calea Grivitei 21, Bucharest 010702, Romania'
- |
Department of Mathematics\
National Research University Higher School of Economics\
6 Usacheva Street\
119048 Moscow, Russia
author:
- Marcos Jardim
- Mario Maican
- 'Alexander S. Tikhomirov'
title: Moduli spaces of rank 2 instanton sheaves on the projective space
---
Introduction
============
Instanton bundles on ${\mathbb{C}}{{\mathbb{P}^{3}}}$ were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970’s as the holomorphic counterparts, via twistor theory, to anti-self-dual connections with finite energy (instantons) on the 4-dimensional round sphere $S^4$. To be more precise, an *instanton bundle of charge $n$* is a $\mu$-stable rank 2 bundle $E$ on ${{\mathbb{P}^{3}}}$ with $c_1(E)=0$ and $c_2(E)=n$ satisfying the cohomological condition $h^1(E(-2))=0$; equivalently, an instanton bundle of charge $n$ is a locally free sheaf which arises as cohomology of a linear monad of the form $$\label{instanton monad}
0 \to n\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(-1) \longrightarrow (2+2n)\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \longrightarrow n\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(1) \to 0 .$$
The moduli space ${{\mathcal I}}(n)$ of such objects has been thoroughly studied in the past 35 years by various authors and it is now known to be an irreducible [@T1; @T2], nonsingular [@JV] affine [@CO] variety of dimension $8n-3$.
The closure of ${{\mathcal I}}(n)$ within the moduli space ${{\mathcal M}}(n)$ of semistable rank 2 sheaves with Chern classes $c_1=0$, $c_2=n$ and $c_3=0$ contains non locally free sheaves which also arise as cohomology of monads of the form (\[instanton monad\]). Such *instanton sheaves* can alternatively be defined as rank 2 torsion free sheaves satisfying the cohomological conditions $$h^0(E(-1)) = h^1(E(-2)) = h^2(E(-2)) = h^3(E(-3)) = 0.$$ We prove that such sheaves are always stable, see Theorem \[instanton stability\] below, so they admit a moduli space ${{\mathcal L}}(n)$ regarded as an open subset of ${{\mathcal M}}(n)$ which, of course, contains ${{\mathcal I}}(n)$.
The spaces ${{\mathcal L}}(1)$ and ${{\mathcal L}}(2)$ were essentially known to be irreducible, see details in the first few paragraphs of Section \[L(n)\] below. However, ${{\mathcal L}}(3)$ was observed to have at least two irreducible components [@JMT1 Remark 8.6], while several new components of ${{\mathcal L}}(n)$ were constructed in [@JMT2].
The main goal of this paper is to characterize the irreducible components of ${{\mathcal L}}(3)$ and ${{\mathcal L}}(4)$. We prove:
\[mthm1\]
- ${{\mathcal L}}(3)$ is a connected quasi-projective variety consisting of exactly two irreducible components each of dimension 21;
- ${{\mathcal L}}(4)$ is a connected quasi-projective variety consisting of exactly four irreducible components, three of dimension 29 and one of dimension 32.
For every instanton sheaf $E$, the quotient $E^{\vee\vee}/E$ is a semistable sheaf with Hilbert polynomial $d\cdot(t+2)$ (see Section \[SIS\] below), therefore an essential ingredient for the proof of Main Theorem \[mthm1\] is the study of the moduli space ${{\mathcal T}}(d)$ of semistable sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=d\cdot t$. Since these spaces are also interesting in their own right, we prove:
\[mthm2\]
- ${{\mathcal T}}(1)$ is an irreducible projective variety of dimension 5;
- ${{\mathcal T}}(2)$ is a connected projective variety consisting of exactly two irreducible components of dimension 8;
- ${{\mathcal T}}(3)$ is a connected projective variety consisting of exactly four irreducible components, two of dimension 12 and two of dimension 13.
- ${{\mathcal T}}(4)$ is a connected projective variety consisting of exactly eight irreducible components, four of dimension 16, two of dimension 17, one of dimension 18 and one of dimension 20.
We also give a precise description of a generic point in each of the irreducible components mentioned in the statement of the theorem, see Section \[1d chi=0\].
[**Acknowledgements.**]{} MJ is partially supported by the CNPq grant number 303332/2014-0, and the FAPESP grants number 2014/14743-8 and 2016/03759-6; this work was completed during a visit to the University of Edinburgh, and he is grateful for its hospitality. MJ also thanks Daniele Faenzi and Simone Marchesi for their help in the proof of Theorem \[instanton stability\] below. AST was supported by a subsidy to the HSE from the Government of the Russian Federation for the implementation of the Global Competitiveness Program. AST also acknowledges the support from the Max Planck Institute for Mathematics in Bonn, where this work was finished during the winter of 2017.
Stability of instanton sheaves {#SIS}
==============================
Recall from [@J-i] that a torsion free sheaf $E$ on ${{\mathbb{P}^{3}}}$ is called an *instanton sheaf* if $c_1(E)=0$ and the following cohomological conditions hold $$h^0(E(-1))=h^1(E(-2))=h^{2}(E(-2))=h^3(E(-3))=0.$$ The integer $n:=-\chi(E(-1))$ is called the charge of $E$; it is easy to check that $n=h^1(E(-1))=c_2(E)$, and that $c_3(E)=0$. The trivial sheaf $r\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$ of rank $r$ is considered as an instanton sheaf of charge zero. In this paper, we will only be interested in rank 2 instanton sheaves.
Recall that the singular locus ${\rm Sing}(G)$ of a coherent sheaf $G$ on a nonsingular projective variety $X$ is given by $${\rm Sing}(G) := \{ x\in X ~|~ G_x ~~\text{is not free over}~~ \mathcal{O}_{X,x} \} ,$$ where $G_x$ denotes the stalk of $G$ at a point $x$ and $\mathcal{O}_{X,x}$ is its local ring. The following result, proved in [@JG Main Theorem], provides a key piece of information regarding the singular loci of rank 2 instanton sheaves.
\[jg-thm\] If $E$ is a non locally free instanton sheaf of rank $2$ on ${{\mathbb{P}^{3}}}$, then
- its singular locus has pure dimension $1$;
- $E^{\vee\vee}$ is a (possibly trivial) locally free instanton sheaf.
\[remark 4\] In fact, the quotient sheaf $Q_E:=E^{\vee\vee}/E$ is a *rank 0 instanton sheaf*, in the sense of [@hauzer Section 6.1]; see also [@JG Section 3.2]. More precisely, a *rank $0$ instanton sheaf* is a coherent sheaf $Q$ on ${{\mathbb{P}^{3}}}$ such that $h^0(Q(-2))=h^1(Q(-2))=0$; the integer $d:=h^0(Q(-1))$ is called the *degree* of $Q$.
The Hilbert polynomial of a rank 2 instanton sheaf $E$ (in fact, of any coherent sheaf on ${{\mathbb{P}^{3}}}$ of rank 2 with $c_1=0$, $c_2=n$ and $c_3=0$) is given by $$\label{pe(t)}
P_E(t) = \frac{1}{3}(t+3)\cdot(t+2)\cdot(t+1) - n\cdot(t+2) = 2\cdot\chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - n\cdot(t+2) .$$ Let $n':=c_2(E^{\vee\vee})\ge 0$; it follows from the standard sequence $$\label{std dual sqc}
0 \to E \to E^{\vee\vee} \to Q_E \to 0$$ that $$P_{Q_E}(t)=d\cdot(t+2) ~~ {\rm where} ~~ d:=n-n' ~.$$ Note that the $d=n-n'$ is precisely the multiplicity of $Q_E$ as a rank 0 instanton sheaf.
Rank 0 instanton sheaves can be characterized in the following way.
\[generic\_case\] Every rank 0 instanton sheaf $Q$ admits a resolution of the form $$\label{generic_resolution}
0 {\longrightarrow}d\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(-1) {\longrightarrow}2d\cdot{{\mathcal O}_{\mathbb{P}^{3}}} {\longrightarrow}d\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(1) {\longrightarrow}Q {\longrightarrow}0.$$
Consider the Beilinson spectral sequence from [@choi_chung_maican Section 6], applied to the sheaf $Q':=Q(-2)$. We have $\H^0(Q') = 0$, hence also $\H^0(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = 0$ and $\H^0(Q'(-1)) = 0$. We adopt the notations of [@choi_chung_maican Section 6]. Since $\ker({\varphi}_5)/{{\mathcal Im}}({\varphi}_4) = 0$, we deduce that $\H^0(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = 0$. Thus, the bottom row of the $E^1$-term of the spectral sequence vanishes. Since ${\varphi}_7$ is an isomorphism, we deduce that ${\varphi}_1$ is injective. Since ${\varphi}_8$ is injective, we deduce that $\ker({\varphi}_2) = {{\mathcal Im}}({\varphi}_1)$. The top row of the $E^1$-term of the spectral sequence yields the resolution $$0 {\longrightarrow}\H^1(Q'(-1)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-3) \overset{{\varphi}_1}{{\longrightarrow}}
\H^1(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-2) \overset{{\varphi}_2}{{\longrightarrow}}$$ $$\overset{{\varphi}_2}{{\longrightarrow}} \H^1(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-1) {\longrightarrow}Q' {\longrightarrow}0.$$ We have $$\chi(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = -d, \qquad \chi(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = -2d, \qquad \chi(Q'(-1)) = -d,$$ hence $${\operatorname{h}}^1(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = d, \qquad {\operatorname{h}}^1(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = 2d, \qquad {\operatorname{h}}^1(Q'(-1)) = d.$$ The above exact sequence yields (\[generic\_resolution\]).
Let now examine the stability properties of instanton sheaves.
\[instanton stability\] Every nontrivial rank 2 instanton sheaf $E$ is stable. In addition, a nontrivial instanton sheaf $E$ is $\mu$-stable if and only if its double dual $E^{\vee\vee}$ is nontrivial.
Since rank 2 instanton sheaves have no global sections [@J-i Prop. 11], every nontrivial locally free rank 2 instanton sheaf is $\mu$-stable; therefore, if $E^{\vee\vee}$ is nontrivial, then $E$ is also $\mu$-stable. Conversely, if $E$ is $\mu$-stable, then so is $E^{\vee\vee}$, hence it must be nontrivial.
Therefore, in order to prove the first claim of the Theorem, it is enough to consider *quasi-trivial instanton sheaves*, i.e. rank 2 instanton sheaves $E$ with $E^{\vee\vee}\simeq 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$; note that the multiplicity of $Q_E$ is exactly $n=c_2(E)$.
Since $E$ has no global sections, it can only be destabilized by the ideal sheaf ${I_{C/{{\mathbb{P}^{3}}}}}$ of a subscheme $C\subset{{\mathbb{P}^{3}}}$. Moreover, we can assume that the quotient sheaf $E/{I_{C/{{\mathbb{P}^{3}}}}}$ is torsion free, thus it is also the ideal sheaf ${I_{D/{{\mathbb{P}^{3}}}}}$ of another subscheme $D\subset{{\mathbb{P}^{3}}}$. We obtain two exact sequences $$\xymatrix{
& 0 \ar[d] & & & \\
& {I_{C/{{\mathbb{P}^{3}}}}}\ar[d] & & & \\
0 \ar[r] & E \ar[d]\ar[r] & 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \ar[r] & Q_E \ar[r] & 0 \\
& {I_{D/{{\mathbb{P}^{3}}}}}\ar[d] & & & \\
& 0 & & &
}$$
Taking the double dual of the top vertical morphisms we obtain, using also the Snake Lemma, the following commutative diagram $$\label{big diag}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & {I_{C/{{\mathbb{P}^{3}}}}}\ar[d]\ar[r] & {{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & {{\mathcal O}_{C}}\ar[d]\ar[r] & 0 \\
0 \ar[r] & E \ar[d]\ar[r] & 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & Q_E \ar[d]\ar[r] & 0 \\
0 \ar[r] & {I_{D/{{\mathbb{P}^{3}}}}}\ar[d]\ar[r] & {{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & {{\mathcal O}_{D}}\ar[d]\ar[r] & 0 \\
& 0 & 0 & 0 &
}$$ Since $h^0(Q_E(-2))=0$, then also $h^0({{\mathcal O}_{C}}(-2))=0$, hence $C$ must have pure dimension 1. Moreover, note also that $h^1(Q_E(-2))=0$ implies $h^1({{\mathcal O}_{D}}(-2))=0$.
We show that $\dim D=0$. Indeed, assume that $D$ has dimension 1. Let $U$ be the maximal 0-dimensional subsheaf of ${{\mathcal O}_{D}}$, and set ${\mathcal O}_{D'}:={{\mathcal O}_{D}}/U$; clearly, $D'$ has pure dimension 1. Next, let $D'':=D'_{\rm red}$ be the underlying reduced scheme. We end up with two exact sequences $$0 \to U \to {{\mathcal O}_{D}}\to {\mathcal O}_{D'} \to 0 ~~{\rm and}~~$$ $$0 \to T \to {\mathcal O}_{D'} \to {\mathcal O}_{D''} \to 0,$$ so that the vanishing of $h^1({{\mathcal O}_{D}}(-2))$ forces $h^1({\mathcal O}_{D''}(-2))=0$.
Still, $D''$ may be reducible, so let $D'' := D''_1 \cup\dots\cup D''_p$ be its decomposition into irreducible components. For each index $j=1,\dots,p$ we obtain a sequence: $$0 \to S_j \to {\mathcal O}_{D''} \to {\mathcal O}_{D''_j} \to 0,$$ thus also $h^1({\mathcal O}_{D''_j}(-2))=0$. Let $d_j$ and $p_j$ denote the degree and arithmetic genus of $D''_j$, respectively. It follows that $$0\le h^0({\mathcal O}_{D''_j}(-2))=\chi({\mathcal O}_{D''_j}(-2))=-2d_j+1-p_j$$ hence $p_j\le-2d_j+1\le-1$, which is imposible for a reduced and irreducible curve.
Now let $\delta=h^0({{\mathcal O}_{D}})$ be the length of $D$; since $\deg(C)=n$, we have $$P_{{I_{C/{{\mathbb{P}^{3}}}}}}(t) = \chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - \chi({{\mathcal O}_{C}}(t)) = \chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - nt + (\delta-2n) .$$ Comparing with equation (\[pe(t)\]), we have $$\label{pe-pic}
\frac{P_E(t)}{2} - P_{{I_{C/{{\mathbb{P}^{3}}}}}}(t) = \frac{n}{2}t + n - \delta ,$$ which is positive for $n$ sufficiently large, and $E$ contains no destabilizing subsheaves.
As a consequence of the proof above, we also obtain the following interesting fact.
Every rank 2 quasi-trivial instanton on ${{\mathbb{P}^{3}}}$ is an extention of the ideal of a 0-dimensional scheme $D$ by the ideal of a pure 1-dimensional scheme containing $D$.
On the other hand, it is easy to check that every rank 0 instanton sheaf is semistable.
\[rk 0 semistable\] Every rank 0 instanton sheaf is semistable.
Let $Z$ be a rank 0 instanton sheaf, and let $T$ be a subsheaf of $Z$ with Hilbert polynomial $P_Z(t)=a\cdot t+\chi(Z)$. Since $h^0(Z(-2))=0$, then $h^0(T(-2))=0$ and $-2a+\chi(Z)=-h^1(T(-2))\leq0$. It follows that $$\frac{\chi(Z)}{a} \leq 2 = \frac{\chi(Q)}{d} . \qedhere$$
Clearly, not every rank 0 instanton sheaf is stable: if $Q_1$ and $Q_2$ are rank 0 instanton sheaves, then so is any extention of $Q_1$ by $Q_2$, and this cannot possibly be stable.
Conversely, there are semistable sheaves with Hilbert polynomial $dt+2d$ which are not rank 0 instanton sheaves: just consider $Q:={{\mathcal O}}_\Sigma(2)$ for an elliptic curve $\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$, so that $h^0(Q(-2))\ne0$.
Moduli space of instanton sheaves {#L(n)}
=================================
Let ${{\mathcal L}}(n)$ denote the open subscheme of the Maruyama moduli space ${{\mathcal M}}(n)$ of semistable rank 2 torsion free sheaves with Chern classes $c_1=0$, $c_2=n$ and $c_3=0$ consisting of instanton sheaves of charge $n$. Let also ${{\mathcal I}}(n)$ denote the open subscheme of ${{\mathcal M}}(n)$ consisting of locally free instanton sheaves. Finally, let $\overline{{{\mathcal L}}(n)}$ and $\overline{{{\mathcal I}}(n)}$ denote the closures within ${{\mathcal M}}(n)$ of ${{\mathcal L}}(n)$ and ${{\mathcal I}}(n)$, respectively. We also consider the set ${{\mathcal I}}^{0}(n):=\overline{{{\mathcal I}}(n)}\cap {{\mathcal L}}(n)$, which consists of those instanton sheaves which either are locally free, or can be deformed into locally free ones.
It was shown in [@T1; @T2] that ${{\mathcal I}}(n)$ is irreducible for every $n>0$; its closure $\overline{{{\mathcal I}}(n)}$ is called the *instanton component* of ${{\mathcal M}}(n)$. However, the same is not true for ${{\mathcal L}}(n)$ as soon as $n\ge 3$. Indeed, it is well known that $$\overline{{{\mathcal I}}(1)}={{\mathcal L}}(1)={{\mathcal M}}(1)\simeq {{\mathbb{P}^{5}}} ,$$ see for instance [@JMT2 Section 6].
The case $n=2$ has also been understood.
\[L(2)\] $\overline{{{\mathcal L}}(2)} = \overline{{{\mathcal I}}(2)}$.
In particular, ${{\mathcal L}}(2)$ possesses a single irreducible component of dimension 13.
Le Potier showed in [@LeP] that ${{\mathcal M}}(2)$ has exactly 3 irreducible components; according to the description of these components provided in [@JMT2 Section 6], only the instanton component $\overline{{{\mathcal I}}(2)}$ contains instanton sheaves.
Let us now describe the irreducible components of ${{\mathcal L}}(n)$ for $n\ge 3$ introduced in [@JMT2 Section 3].
Let $\Sigma$ be an irreducible, nonsingular, complete intersection curve in ${{\mathbb{P}^{3}}}$, given as the intersection of a surface of degree $d_1$ with a surface of degree $d_2$, with $1\le d_1\le d_2$; denote by $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ the inclusion morphism. Let also $L\in\operatorname{{Pic}}^{g-1}(\Sigma)$ such that $h^0(\Sigma,L)=h^1(\Sigma,L)=0$. Given a (possibly trivial) locally free instanton sheaf $F$ of charge $c\ge0$ and an epimorphism $\varphi:E{\twoheadrightarrow}(\iota_*L)(2)$, the kernel $F:=\ker\varphi$ is an instanton sheaf of charge $c+d_1d_2$. Thus we may consider the set $$\label{c-comp's}
{{\mathcal C}}(d_1,d_2,c) := \left\{ [E]\in{{\mathcal M}}(c+d_1d_2) ~|~ E^{\vee\vee}\in{{\mathcal I}}(c) ~,~
E^{\vee\vee}/E \simeq (\iota_*L)(2) \right\}$$ as a subvariety of ${{\mathcal M}}(c+d_1d_2)$. The following result is proved in [@JMT2], cf. Theorems 15, 17 and 23.
\[dim C thm\] For each $c\ge0$ and $1\le d_1\le d_2$ such that $(d_1,d_2)\ne(1,1),(1,2)$, $\overline{{{\mathcal C}}(d_1,d_2,c)}$ is an irreducible component of ${{\mathcal M}}(c+d_1d_2)$ of dimension $$\label{dim C}
\dim \overline{{{\mathcal C}}(d_1,d_2,c)} = 8c-3 + \frac{1}{2}d_1d_2(d_1+d_2+4) + h,$$ where $$h = \left\{ \begin{array}{l}
2{{d_1+3}\choose{3}} - 4 , ~~{\rm if}~~ d_1=d_2 \\ ~~ \\
{{d_1+3}\choose{3}} + {{d_2+3}\choose{3}} - {{d_2-d_1+3}\choose{3}} - 2, ~~{\rm if}~~ d_1<d_2
\end{array} \right.$$ In addition, $\overline{{{\mathcal C}}(d_1,d_2,c)}\cap{{\mathcal I}}^{0}(c+d_1d_2)\ne
\emptyset$.
We do not know whether the families ${{\mathcal C}}(d_1,d_2,c)$ exhaust all components of ${{\mathcal L}}(n)$, though we prove that this holds for $n=3,4$ in Sections \[L(3) section\] and \[L(4) section\] below, respectively.
However, we remark that the previous result allows for a partial count of the number of components of ${{\mathcal L}}(n)$. Indeed, let $\tau(n)$ denote the number of irreducible components of the union $$\overline{{{\mathcal I}}(n)} \bigcup \left( \bigcup_{d_1d_2+c=n} \overline{{{\mathcal C}}(d_1,d_2,c)} \right).$$ To estimate $\tau(n)$, we must count the different ways in which an integer $n\ge3$ can be written as $n=d_1d_2+c$ with $c\ge0$, and $1\le d_1\le d_2$ excluding the pairs $(d_1,d_2)=(1,1),(1,2)$. Consider the function $$\delta(p) = \left\{ \begin{array}{l}
\frac{1}{2} ( d(p) + 1 ), \mbox{ if } p \mbox{ is a perfect square} \\ ~~ \\
\frac{1}{2} d(p), \mbox{ otherwise}
\end{array} \right.$$ where $d(p)$ is the *divisor function*, i.e. the number of divisors of a positive integer $p$, including $p$ itself. Note that $\delta(p)$ is the number of different ways in which we can write $p$ as a product $d_1d_2$ with $1\le d_1\le d_2$. Adding the instanton component, we have that the number of irreducible components of ${{\mathcal L}}_0(n)$ is given by: $$\label{ell0}
\tau(n) = 1 + \sum_{p=3}^{n} \delta(p) =
\frac{1}{2}\left( \sum_{p=3}^{n} d(p) + \left\lfloor \sqrt{n} \right\rfloor + 1 \right),$$ since $\left\lfloor\sqrt{n}\right\rfloor - 1$ accounts for the number of perfect squares between 3 and $n$.
Let $l(n)$ be the number of irreducible components of the moduli space of instanton sheaves of charge $n$. Then, for $n$ sufficiently large $l(n)> \frac{1}{2} n\cdot\log(n)$.
Determining the asymptotic behaviour of the sum of divisors function is a relevant problem in Number Theory called the *Dirichlet divisor problem*; indeed, it is known that $$\sum_{p=1}^{n} d(p) = n\cdot\log(n) + (2\gamma - 1)n + O(n^\theta),$$ where $\gamma$ denotes the Euler–Mascheroni constant, and $1/4\le \theta\le 131/416$, cf. [@Huxley]. Comparing with equation (\[ell0\]), we easily obtain the desired estimate.
Also relevant for us is a class of instanton sheaves studied in [@JMT1]; more precisely, for $n>0$ and each $m=1,\dots,n$, consider the subset ${{\mathcal D}}(m,n)$ of ${{\mathcal M}}(n)$ consisting of the isomorphism classes $[E]$ of the sheaves $E$ obtained in this way: $${{\mathcal D}}(m,n) :=
\{ [E]\in{{\mathcal M}}(n) ~|~ [E^{\vee\vee}]\in{{\mathcal I}}(n-m), ~~
\Gamma=\mathrm{Supp}(E^{\vee\vee}/E)\in {{\mathcal R}}^*_{0}(m)_{E^{\vee\vee}}~,$$ $$~~{\rm and}~~
E^{\vee\vee}/E\simeq{{\mathcal O}}_\Gamma(2m-1) \},$$ where the space ${{\mathcal R}}^*_{0}(m)_{E^{{{\scriptscriptstyle \operatorname{D}}}{{\scriptscriptstyle \operatorname{D}}}}}$ is decribed as follows: first, let ${{\mathcal R}}^*_{0}(m)$ denote the space of nonsingular rational curves $\Gamma\hookrightarrow{{\mathbb{P}^{3}}}$ of degree $m$ whose normal bundle $N_{\Gamma/{{\mathbb{P}^{3}}}}$ is given by $2\cdot{{\mathcal O}}_\Gamma((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$; then, for any instanton sheaf $F$ we set $${{\mathcal R}}^*_{0}(m)_F := \{\:\Gamma\in {{\mathcal R}}^*_{0}(m)~|~ F|_\Gamma\simeq2\cdot{{\mathcal O}}_\Gamma\ \}.$$ One can show that for every rank 2 instanton sheaf $F$, the space ${{\mathcal R}}^*_{0}(m)_F$ is a nonempty open subset of ${{\mathcal R}}^*_{0}(m)$, cf. [@JMT1 Lemma 6.2].
Let $\overline{{{\mathcal D}}(m,n)}$ denote the closure of ${{\mathcal D}}(m,n)$ within ${{\mathcal M}}(n)$. Note that since $E^{\vee\vee}$ is a locally free instanton sheaf of charge $n-m$, and ${{\mathcal O}}_\Gamma(2m-1)$ is a rank 0 instanton sheaf of degree $m$, then $E$ is an instanton sheaf of charge $n$, so that ${{\mathcal D}}(m,n)\subset{{\mathcal L}}(n)$. In fact, it is shown in [@JMT2 Theorem 7.8] that $\overline{{{\mathcal D}}(m,n)}\subset{{\mathcal I}}^0(n)$. In addition, we prove:
\[disjoint rat curves\] Let $\Gamma_1,\dots,\Gamma_r$ be disjoint, smooth irreducible rational curves in ${{\mathbb{P}^{3}}}$ of degrees $m_1,\dots,m_r$, respectively; set $Q:=\bigoplus_{j=1}^{r}{{\mathcal O}}_{\Gamma_j}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$. If $F$ is a locally free instanton sheaf of charge $c$ such that $F|_{\Gamma_j}\simeq2\cdot{{\mathcal O}}_{\Gamma_j}$ for each $j=1,\dots,r$, and $\varphi:F{\twoheadrightarrow}Q(2)$ is an epimorphism, then $[\ker\varphi]\in{{\mathcal I}}^0(c+m_1+\cdots+m_r)$.
The proof of the previous proposition requires the following technical lemma, proved in [@JMT1 Lemma 7.1].
\[F,G\] Let $C$ be a smooth irreducible curve with a marked point $0$, and set $\mathbf{B}:=C \times{{\mathbb{P}^{3}}}$. Let $\mathbf{F}$ and $\mathbf{G}$ be $\mathcal{O}_{\mathbf{B}}$-sheaves, flat over $C $ and such that $\mathbf{F}$ is locally free along $\mathrm{Supp}(\mathbf{G})$. Denote $$\mathbf{G}_t:=\mathbf{G}|_{\{t\}\times{{\mathbb{P}^{3}}}} ~~{\rm and}~~
\mathbf{F}_t=\mathbf{F}|_{\{t\}\times{{\mathbb{P}^{3}}}}\ \ {\rm for}\ t\in C .$$ Assume that, for each $t\in C$, $$\label{vanish Hi}
H^i({{\mathcal H}{\it om}}(\mathbf{F}_t,\mathbf{G}_t))=0,\ \ \ i\ge1.$$ If $s:\mathbf{F}_0\to\mathbf{G}_0$ is an epimorphism, then, after possibly shrinking $C$, $s$ extends to an epimorhism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$.
We argue by induction on $r$; the case $r=1$ is just the aforementioned result, namely [@JMT2 Theorem 7.8].
Let $Q':=\bigoplus_{j=1}^{r-1}{{\mathcal O}}_{\Gamma_j}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $Q=Q'\oplus{{\mathcal O}}_{\Gamma_r}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$. Let $E:=\ker\varphi$, and let $E'$ denote the kernel of the composition $F\stackrel{\varphi}{{\twoheadrightarrow}} Q(2) {\twoheadrightarrow}Q'(2) $. We obtain the following exact sequence: $$0 \to E \to E' \stackrel{\varphi'}{\rightarrow} {{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}}) \to 0.$$ By the induction hypothesis, $[E']\in{{\mathcal I}}^0(c+m_1+\cdots+m_{r-1})$, thus one can find an affine open subset $0\in U\subset\mathbb{A}^1$ and a coherent sheaf $\mathbf{E}$ on ${{\mathbb{P}^{3}}}\times U$, flat over $U$, such that $\mathbf{E}_0=E'$ and $\mathbf{E}_t$ is a locally free instanton sheaf of charge $c+m_1+\cdots+m_{r-1}$ satisfying $\mathbf{E}_t|_{\Gamma_r}\simeq 2\cdot{{\mathcal O}}_{\Gamma_r}$ for every $t\in U\setminus\{0\}$. Setting $\mathbf{G}=:\pi^*Q'$ where $\pi:{{\mathbb{P}^{3}}}\times U\to U$ is the projection onto the first factor, note that $$H^i({{\mathcal H}{\it om}}(\mathbf{E}_t,\mathbf{G}_t)) = H^i(2\cdot{{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})) = 0 ~~{\rm for}~~ i\ge1 ~~{\rm and}~~ t\in U.$$ This claim is clear for $t\ne0$; when $t=0$, simply observe that the sequence $0\to E'\to F \to Q'(2)\to 0$ implies that $E'_{\Gamma_r}\simeq F_{\Gamma_r}$, since the support of $Q'$ is disjoint from $\Gamma_r$.
By Lemma \[F,G\], there exists an epimorphism $\mathbf{s}:\mathbf{E}\twoheadrightarrow\mathbf{G}$ extending $\varphi':E'\to {{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $[\ker\mathbf{s}_t]\in{{\mathcal D}}(m_r,c+m_1+\cdots+m_r)$, by construction. It then follows that $[E]\in\overline{{{\mathcal D}}(m_r,c+m_1+\cdots+m_r)}$, hence, by [@JMT2 Theorem 7.8], $[E]\in{{\mathcal I}}^0(c+m_1+\cdots+m_r)$, as desired.
Next, we consider the following situation: let $\Sigma$ be an irreducible, nonsingular, complete intersection curve in ${{\mathbb{P}^{3}}}$, given as the intersection surfaces of degrees $d_1$ and $d_2$, with $1\le d_1\le d_2$ and $(d_1,d_2)\ne(1,1),(1,2)$, and let $\Gamma$ be a smooth irreducible rational curve in ${{\mathbb{P}^{3}}}$ of degree $m$ disjoint from $\Sigma$. Set $Q:=L\oplus{{\mathcal O}}_{\Gamma}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$ for some $L\in\operatorname{{Pic}}^{g-1}(\Sigma)$ such that $h^0(\Sigma,L)=h^1(\Sigma,L)=0$, where $g$ is the genus of $\Sigma$.
\[disjoint curves\] If $F$ is a locally free instanton sheaf of charge $c$ such that $F|_{\Gamma}\simeq2\cdot{{\mathcal O}}_{\Gamma}$, and $H^1(F^\vee|_{\Gamma}\otimes L(2))=0$. If $\varphi:F{\twoheadrightarrow}Q(2)$ is an epimorphism, then $[\ker\varphi]\in\overline{{{\mathcal C}}(d_1,d_2,c+m)}$.
The idea is the same as in the proof of Proposition \[disjoint rat curves\]. Let $E'$ be the kernel of the composition $F\stackrel{\varphi}{{\twoheadrightarrow}}Q(2){\twoheadrightarrow}{{\mathcal O}}_{\Gamma}((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $E:=\ker\varphi$ and $E'$ are related via the following exact sequence: $$0 \to E \to E' \stackrel{\varphi'}{\rightarrow} L(2) \to 0.$$ By [@JMT2 Theorem 7.8], one can find an affine open subset $0\in U\subset\mathbb{A}^1$ and a coherent sheaf $\mathbf{E}$ on ${{\mathbb{P}^{3}}}\times U$, flat over $U$, such that $\mathbf{E}_0=E'$ and $\mathbf{E}_t$ is a locally free instanton sheaf of charge $c+m$ for every $t\in U\setminus\{0\}$.
Setting $\mathbf{G}:=\pi^*L(2)$, we must, in order to apply Lemma \[F,G\], check that $$H^i({{\mathcal H}{\it om}}(E_t,G_t))=0 ~~{\rm for}~~ i\ge1 ~~{\rm and}~~ t\in U.$$ Indeed, since $\dim G_t=1$, it is enough to show that $H^1({{\mathcal H}{\it om}}(E_t,G_t))=0$. Note that $${{\mathcal H}{\it om}}(E_0,G_0) = {{\mathcal H}{\it om}}(E',L(2)) \simeq {{\mathcal H}{\it om}}(F,L(2)) \simeq F^\vee|_{\Sigma}\otimes L(2) ,$$ where the middle isomorphism follows from applying the functor ${{\mathcal H}{\it om}}(\cdot,L(2))$ to the sequence $$0\to E' \to F \to {{\mathcal O}}_{\Gamma}((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}}) \to0,$$ also exploring the fact that $\Sigma$ and $\Gamma$ are disjoint. It follows that $H^1({{\mathcal H}{\it om}}(E_0,G_0))=H^1(F^\vee|_{\Sigma}\otimes L(2))=0$ by hypothesis. By semicontinuity of $h^1({{\mathcal H}{\it om}}(E_t,G_t))$, we can shrink $U$ to another affine open subset $U'\subset\mathbb{A}^1$, if necessary, to guarantee that $H^1({{\mathcal H}{\it om}}(E_t,G_t))=0$ for every $t\in U'$.
By Lemma \[F,G\], there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending $\varphi':E'\to L(2)$, so that $[\ker\mathbf{s}_t]\in{{\mathcal C}}(d_1,d_2,c+m)$, by construction. Since $E\simeq \ker\mathbf{s}_0$, it follows that $[E]\in\overline{{{\mathcal C}}(d_1,d_2,c+m)}$.
Moduli of sheaves of dimension one and Euler characteristic zero {#1d chi=0}
================================================================
Given two integers $d$ and $\chi$, $d\ge1$, let ${{\mathcal T}}(d,\chi)$ be the moduli space of semistable coherent sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=d\cdot t+\chi$. In this section, we focus on the space ${{\mathcal T}}(d):={{\mathcal T}}(d,0)$.
Apart from its intrinsic interest, the space ${{\mathcal T}}(d)$ is also relevant for the study of instanton sheaves, and the description of ${{\mathcal T}}(d)$ for $d\le4$ provided in this section will be a key ingredient for the proof of the Main Theorem \[mthm1\].
In addition, let ${{\mathcal Z}}(d)$ denote the set of rank 0 instanton sheaves of degree $d$ *modulo S-equivalence* (which makes sense, since, by Lemma \[rk 0 semistable\], every rank 0 instanton sheaf is semistable). After a twist by ${{\mathcal O}_{\mathbb{P}^{3}}}(-2)$, ${{\mathcal Z}}(d)$ can be regarded as an open subscheme of the moduli space ${{\mathcal T}}(d)$ consisting of those sheaves $Q$ satisfying $h^0(Q)=0$.
The space ${{\mathcal T}}(d)$ has several distinguished subsets, which we now describe.
First, let ${{\mathcal P}}_d \subset {{\mathcal T}}(d)$ be the subset of planar sheaves; it is a fiber bundle over $({{\mathbb P}}^3)^*$ with fiber being the moduli space of semistable coherent sheaves on ${{\mathbb{P}^{2}}}$ with Hilbert polynomial $P=d\cdot t$. In view of [@lepotier Theorem 1.1], ${{\mathcal P}}_d$ is a projective irreducible variety of dimension $d^2+4$. In particular, ${{\mathcal P}}_d$ is closed.
Next, consider the subsets ${{\mathcal R}}^o_d,~ {{\mathcal E}}^o_d \subset {{\mathcal T}}(d)$ of sheaves supported on smooth rational curves of degree $d$, respectively, on smooth elliptic curves of degree $d$. Let ${{\mathcal R}}_d$ and ${{\mathcal E}}_d$ denote their closures.
Given a partition $(d_1,\ldots,d_s)$ of $d$ such that $d_1 \ge \cdots \ge d_s$, we denote by ${{\mathcal T}}_{d_1, \ldots, d_s} \subset {{\mathcal T}}(d)$ the locally closed subset of points of the form $$\label{polystable}
[Q_1 \oplus \cdots \oplus Q_s],$$ where $Q_i$ gives a stable point in ${{\mathcal T}}(d_i)$; in particular, ${{\mathcal T}}_d$ is the open subset of stable points in ${{\mathcal T}}(d)$. Let ${{\mathcal T}}^o_{d_1, \ldots, d_s} \subset {{\mathcal T}}_{d_1, \ldots, d_s}$ be the open dense subset given by the condition that ${\operatorname{supp}}(Q_i)$ be mutually disjoint. Clearly, each irreducible component of ${{\mathcal T}}^o_{d_1, \ldots, d_s}$ is an open dense subset of an irreducible component of ${{\mathcal T}}_d$. Hence irreducible components of $\overline{{{\mathcal T}}}_{d_1, \ldots, d_s}$ are also irreducible components of ${{\mathcal T}}(d)$. On the other hand, each point of ${{\mathcal T}}(d)$ is an $S$-equivalence class of a polystable (e. g. stable) sheaf of the form (\[polystable\]). Hence, the following result follows.
\[irred comp\] (i) All irreducible components of ${{\mathcal T}}(d)$ are exhausted by the irreducible components of the union $$\label{union}
\underset{(d_1,...,d_s)}{\bigcup }\overline{{{\mathcal T}}}_{d_1, \ldots, d_s},$$ this union being taken over all the partitions $(d_1,...,d_s)$ of $d$.\
(ii) For a given partition $(d_1,...,d_s)$ of $d$, each irreducible component of $\overline{{{\mathcal T}}}_{d_1, \ldots, d_s}$ is birational to a symmetric product $$({{\mathcal X}}_1 \times \cdots \times {{\mathcal X}}_s)/\Sigma$$ of irreducible components ${{\mathcal X}}_i$ of ${{\mathcal T}}_{d_i}$, where $\Sigma$ is the subgroup of the full symmetric group $\Sigma_s$ of degree $s$ generated by the transpositions $(i, j)$ for which $d_i = d_j$ and ${{\mathcal X}}_i = {{\mathcal X}}_j$.
We have only to prove statement (ii). Indeed, let $\Sigma' \subset \Sigma_s$ be the subgroup generated by the transpositions $(i, i+1)$ for which $d_i = d_{i+1}$. We have a bijective morphism $$({{\mathcal T}}_{d_1} \times \cdots \times {{\mathcal T}}_{d_s})/\Sigma' {\longrightarrow}{{\mathcal T}}_{d_1, \ldots, d_s}, \qquad ([Q_1], \ldots, [Q_s]) \longmapsto [Q_1 \oplus \cdots \oplus Q_s],$$ which is an isomorphism over ${{\mathcal T}}^o_{d_1, \ldots, d_s}$, because over this set we can construct local inverse maps. Whence, the statement (ii) follows.
\[irred of Td\] Lemma \[irred comp\] implies that the problem of finding the irreducible components of ${{\mathcal T}}(d)$ is reduced to the problem of finding the irreducible components of ${{\mathcal T}}_2, \ldots, {{\mathcal T}}_d$.
It also follows from Lemma \[irred comp\] that the number of irreducible components of ${{\mathcal T}}(d)$ is at least as large as the number of partitions of $d$, usualy denoted $p(d)$. A well-known formula by Hardy and Ramanujan gives the following asymptotic expression $$p(d) \sim \frac{1}{4\sqrt{3}\cdot d}\exp\left(\pi\sqrt{\frac{2d}{3}}\right).$$ Therefore, the number of irreducible components of ${{\mathcal T}}(d)$ grows at least exponentially on $\sqrt{d}$. However, as we shall see below in the cases $d=3$ and $d=4$, $p(d)$ is just a rough underestimate of the number of irreducible components of ${{\mathcal T}}(d)$.
Given a coherent sheaf $Q$ on ${{\mathbb{P}^{3}}}$, we define $Q^{{\scriptscriptstyle \operatorname{D}}}:= {\mathcal Ext}^c (Q, \omega_{{{\mathbb{P}^{3}}}})$, where $c = \operatorname{codim} (Q)$. We will use below the following general result regarding stable sheaves in ${{\mathcal T}}(d)$.
\[trivial\_lemma\] Assume that ${{\mathcal F}}$ gives a stable point in ${{\mathcal T}}(d)$ and that $P \in {\operatorname{supp}}({{\mathcal F}})$ is a closed point. Then there are exact sequences $$\label{trivial_extension}
0 {\longrightarrow}{{\mathcal E}}{\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ and $$\label{trivial_sequence}
0 {\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ for some sheaves ${{\mathcal E}}\in {{\mathcal T}}(d,-1)$ and ${{\mathcal G}}\in {{\mathcal T}}(d,+1)$.
Choose a surjective morphism ${{\mathcal F}}\to {{\mathbb C}}_P$ and denote its kernel by ${{\mathcal E}}$. Since ${{\mathcal F}}$ is stable, ${{\mathcal E}}$ is semi-stable, so we have sequence (\[trivial\_extension\]). According to [@rendiconti Theorem 13], the dual sheaf ${{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}$ gives a stable point in ${{\mathcal T}}(d)$. Thus, we have an exact sequence $$0 {\longrightarrow}{{\mathcal E}}_1 {\longrightarrow}{{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ with ${{\mathcal E}}_1 \in {{\mathcal T}}(d,-1)$. According to [@rendiconti Remark 4], ${{\mathcal F}}$ is reflexive. According to [@rendiconti Theorem 13], the sheaf ${{\mathcal G}}= {{\mathcal E}}_1^{{\scriptscriptstyle \operatorname{D}}}$ gives a point in ${{\mathcal T}}(d,1)$. Since ${{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}$ is pure, we can apply [@huybrechts_lehn Proposition 1.1.10] to deduce that ${\mathcal Ext}^3({{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}, \omega_{{{\mathbb P}}^3}) = 0$. The long exact sequence of extension sheaves associated to the above exact sequence yields (\[trivial\_sequence\]).
The goal of this section is to describe the irreducible components of ${{\mathcal T}}(d)$ for $d\le4$. According to [@dedicata], for ${{\mathcal F}}\in {{\mathcal T}}(d)$ we have the following cohomological conditions $$\begin{aligned}
& {\operatorname{h}}^0 ({{\mathcal F}}) = 0 \qquad \text{if $d = 1$ or $2$}, \\
& {\operatorname{h}}^0 ({{\mathcal F}}) \le 1 \qquad \text{if $d = 3$ or $4$}.
$$
Moduli of sheaves of degree 1 and 2
-----------------------------------
The case $d=1$ is easy: clearly, ${{\mathcal T}}(1)\simeq {{\mathcal R}}_1$, being isomorphic to the Grassmanian of lines in ${{\mathbb{P}^{3}}}$.
The moduli space ${{\mathcal T}}(1)$ is an irreducible projective variety of dimension 4.
In addition, it is easy to see that ${{\mathcal Z}}(1)={{\mathcal T}}(1)$.
\[components\_2\] The moduli space ${{\mathcal T}}(2)$ is connected, and has two irreducible components, each of dimension 8: ${{\mathcal P}}_2$ (which coincides with ${{\mathcal R}}_2$) and $\overline{{{\mathcal T}}}_{1, 1}$.
If ${{\mathcal F}}\in {{\mathcal T}}_2$, then we have the exact sequence (\[trivial\_sequence\]) in which ${{\mathcal G}}\in {{\mathcal T}}(2,1)$. Thus, ${{\mathcal G}}$ is the structure sheaf of a conic curve, hence ${{\mathcal G}}$ is planar, and hence ${{\mathcal F}}$ is planar. We conclude that $ {{\mathcal T}}(2) = {{\mathcal P}}_2 \cup \overline{{{\mathcal T}}}_{1,1}$. The intersection ${{\mathcal P}}_2 \cap \overline{{{\mathcal T}}}_{1,1}$ consists of those points of the form $[{{\mathcal O}}_{\ell_1}(-1)\oplus{{\mathcal O}}_{\ell_2}(-1)]$ where $\ell_1$ and $\ell_2$ are two intersecting (and possibly coincident) lines.
Note also that ${{\mathcal Z}}(2)={{\mathcal T}}(2)$; the fact that ${{\mathcal Z}}(2)$ consists of two irreducible components of dimension 8 should be compared with [@hauzer Corollary 6.12], where Hauzer and Langer prove that the moduli space of *framed* rank 0 instanton sheaves also consists of two irreducible components of dimension 8.
Moduli of sheaves of degree 3
-----------------------------
\[components\_3\] The moduli space ${{\mathcal T}}(3)$ has four irreducible components ${{\mathcal P}}_3$, ${{\mathcal R}}_3$, $\overline{{{\mathcal T}}}_{2,1}$ and $\overline{{{\mathcal T}}}_{1,1,1}$, of dimension 13, 13, 12, respectively, 12. If ${{\mathcal F}}\in {{\mathcal T}}_3$ and $\H^0({{\mathcal F}}) \neq 0$, then ${{\mathcal F}}$ is the structure sheaf of a planar cubic curve.
By Proposition \[components\_2\] we have $\overline{{{\mathcal T}}}_2={{\mathcal P}}_2$, so that in view of Lemma \[irred comp\] we already obtain the irreducible components $\overline{{{\mathcal T}}}_{2,1}$ and $\overline{{{\mathcal T}}}_{1,1,1}$ of ${{\mathcal T}}(3)$. Therefore, by Remark \[irred of Td\], we only have to find the irreducible components of ${{\mathcal T}}_3$.
Thus, given ${{\mathcal F}}\in {{\mathcal T}}_3$, take a point $P \in {\operatorname{supp}}({{\mathcal F}})$. We then have the exact sequence (\[trivial\_sequence\]) for ${{\mathcal G}}\in {{\mathcal T}}(3,1)$. According to [@freiermuth_trautmann Theorem 1.1], ${{\mathcal T}}(3,1)$ has two irreducible components: the subset ${{\mathcal P}}$ of planar sheaves and the subset ${{\mathcal R}}$ that is the closure of the set of structure sheaves of twisted cubics. Moreover, all sheaves in ${{\mathcal R}}\setminus {{\mathcal P}}$ are structure sheaves of curves $R \subset {{\mathbb P}}^3$ of degree $3$ and arithmetic genus zero. If ${{\mathcal G}}$ is planar, then ${{\mathcal F}}$ is planar. If ${{\mathcal G}}= {{\mathcal O}}_R$, then $R = {\operatorname{supp}}({{\mathcal F}})$, where the scheme-theoretic support ${\operatorname{supp}}({{\mathcal F}})$ of the sheaf ${{\mathcal F}}$ is defined by the 0-th Fitting ideal ${{\mathcal F}}itt^0({{\mathcal F}})$: ${{\mathcal I}}_{R/{{\mathbb P}}^3}={{\mathcal F}}itt^0({{\mathcal F}})$. The morphism $$\rho \colon {{\mathcal T}}_3 \setminus {{\mathcal P}}_3 {\longrightarrow}{{\mathcal R}}\setminus {{\mathcal P}}, \qquad \rho ([{{\mathcal F}}]) = [{{\mathcal O}}_{{\operatorname{supp}}({{\mathcal F}})}],$$ is injective. Indeed, if $\rho ([{{\mathcal F}}_1]) = \rho ([{{\mathcal F}}_2])$, then ${\operatorname{supp}}({{\mathcal F}}_1) = {\operatorname{supp}}({{\mathcal F}}_2) = R$. Choose a point $P \in R$. We have exact sequences $$0 {\longrightarrow}{{\mathcal F}}_1 {\longrightarrow}{{\mathcal G}}_1 {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0, \qquad
0 {\longrightarrow}{{\mathcal F}}_2 {\longrightarrow}{{\mathcal G}}_2 {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0,$$ with ${{\mathcal G}}_1, {{\mathcal G}}_2 \in{{\mathcal T}}(3,1)$. Clearly, ${{\mathcal G}}_1$ and ${{\mathcal G}}_2$ are both isomorphic to ${{\mathcal O}}_R$, hence ${{\mathcal F}}_1$ and ${{\mathcal F}}_2$ are both isomorphic to the ideal sheaf ${{\mathcal I}}_{P, R}$ of $P$ in $R$. The image of $\rho$ is a constructible set of the irreducible variety ${{\mathcal R}}\setminus {{\mathcal P}}$ and contains an open subset of ${{\mathcal R}}\setminus {{\mathcal P}}$, namely the subset given by the condition that $R$ be irreducible. Indeed, if $R$ is irreducible, then it is easy to check that ${{\mathcal I}}_{P, R}$ is stable; we have $\rho ([{{\mathcal I}}_{P, R}]) = [{{\mathcal O}}_R]$. We deduce that ${{\mathcal T}}_3 \setminus {{\mathcal P}}_3$ is irreducible. It follows that ${{\mathcal R}}_3^o$ is dense in ${{\mathcal T}}_3 \setminus {{\mathcal P}}_3$. Thus, ${{\mathcal T}}_3$ has two irreducible components, hence ${{\mathcal T}}(4)$ has the four irreducible components announced in the proposition.
Assume now that $\H^0({{\mathcal F}}) \neq 0$. Then ${{\mathcal F}}$ cannot be isomorphic to ${{\mathcal I}}_{P, R}$, hence ${{\mathcal F}}$ is planar. Take a non-zero morphism ${{\mathcal O}}\to {{\mathcal F}}$. This morphism factors through an injective morphism ${{\mathcal O}}_C \to {{\mathcal F}}$, where $C$ is a planar curve. The semi-stability of ${{\mathcal F}}$ implies that $C$ is a cubic. Comparing Hilbert polynomials, we see that ${{\mathcal O}}_C \to {{\mathcal F}}$ is an isomorphism.
Moduli of sheaves of degree 4
-----------------------------
\[components\_4\] The moduli space ${{\mathcal T}}(4)$ has eight irreducible components: ${{\mathcal P}}_4$, ${{\mathcal E}}_4$, ${{\mathcal R}}_4$, $\overline{{{\mathcal T}}}_{2,2}$, $\overline{{{\mathcal T}}}_{2,1,1}$, $\overline{{{\mathcal T}}}_{1,1,1,1}$ and two irreducible components of ${{\mathcal T}}_{3,1}$ that are birational to ${{\mathcal P}}_3 \times {{\mathcal T}}_1$, respectively, to ${{\mathcal R}}_3 \times {{\mathcal T}}_1$. Their dimensions are, respectively, $20$, $18$, $16$, $16$, $16$, $16$, $17$, $17$.
By Propositions \[components\_2\] and \[components\_3\] and Lemma \[irred comp\] we already have 5 irreducible components of ${{\mathcal T}}(4)$ which are $\overline{{{\mathcal T}}}_{2,2}$, $\overline{{{\mathcal T}}}_{2,1,1}$, $\overline{{{\mathcal T}}}_{1,1,1,1}$ and two irreducible components of ${{\mathcal T}}_{3,1}$ that are birational to ${{\mathcal P}}_3 \times {{\mathcal T}}_1$, respectively, to ${{\mathcal R}}_3 \times {{\mathcal T}}_1$. Therefore by Remark \[irred of Td\] we have only to find irreducible components of ${{\mathcal T}}_4$. Thus, given ${{\mathcal F}}\in {{\mathcal T}}_4$, take a point $P \in {\operatorname{supp}}({{\mathcal F}})$. We then have the exact sequence (\[trivial\_sequence\]) for ${{\mathcal G}}\in {{\mathcal T}}(4,1)$. According to [@choi_chung_maican Theorem 4.12], ${{\mathcal T}}(4,1)$ has three irreducible components: the subset ${{\mathcal P}}$ of planar sheaves, the subset ${{\mathcal R}}$ that is the closure of the set of structure sheaves of rational quartic curves, and the set ${{\mathcal E}}$ that is the closure of the set of sheaves of the form ${{\mathcal O}}_E(P')$, where $E$ is a smooth elliptic quartic curve and $P' \in E$. If ${{\mathcal G}}\in {{\mathcal P}}$, then ${{\mathcal F}}\in {{\mathcal P}}_4$. The sheaves in ${{\mathcal R}}\setminus ({{\mathcal P}}\cup {{\mathcal E}})$ are structure sheaves of quartic curves of arithmetic genus zero. The sheaves in ${{\mathcal E}}\setminus {{\mathcal P}}$ are supported on quartic curves of arithmetic genus $1$. Let ${{\mathcal T}}_{4, {\operatorname{rat}}} \subset {{\mathcal T}}_4$ be the subset of sheaves whose support is a quartic curve of arithmetic genus zero. As in Proposition \[components\_3\], we can construct an injective dominant morphism $$\rho \colon {{\mathcal T}}_{4, {\operatorname{rat}}} {\longrightarrow}{{\mathcal R}}\setminus ({{\mathcal P}}\cup {{\mathcal E}}), \qquad \rho [{{\mathcal F}}] = [{{\mathcal O}}_{{\operatorname{supp}}({{\mathcal F}})}].$$ It follows that ${{\mathcal T}}_{4, {\operatorname{rat}}}$ is irreducible, hence ${{\mathcal T}}_{4, {\operatorname{rat}}} \subset {{\mathcal R}}_4$. To finish the proof of the proposition we need to show that ${{\mathcal T}}_4 \setminus ({{\mathcal P}}_4 \cup {{\mathcal T}}_{4, {\operatorname{rat}}})$ is contained in ${{\mathcal E}}_4$.
According to [@space_quartics], discussion after Proposition 8, the sheaves ${{\mathcal G}}$ in ${{\mathcal E}}\setminus {{\mathcal P}}$ are of two kinds:
1. ${{\mathcal O}}_E(P')$ for a curve $E$ of arithmetic genus $1$ given by an ideal of the form $(q_1, q_2)$, where $q_1$, $q_2$ are quadratic forms, and $P' \in E$. Notice that ${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathbb C}}_{P'}, {{\mathcal O}}_E) {{\settowidth{\rrrr}{$\scriptstyle{xx}$}
\xrightarrow{\makebox[\rrrr]{$\scriptstyle{}$}}
\hspace{-0.5\rrrr }\hspace{-1.1 em}
\raisebox{- 0.5 ex}{$\sim$}\hspace{0.7\rrrr }
}}{{\mathbb C}}$, so the notation ${{\mathcal O}}_E(P')$ is justified. Also note that ${{\mathcal O}}_E$ is stable;
2. non-planar extensions of the form $$0 {\longrightarrow}{{\mathcal O}}_L(-1) {\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathcal C}}{\longrightarrow}0,$$ where $L$ is a line and ${{\mathcal C}}$ gives a point in ${{\mathcal T}}_H(3,1)$ for a plane $H$ possibly containing $L$. (Here and below we use the notation ${{\mathcal T}}_S(d,\chi)$ for the moduli space of 1-dimensional sheaves on a given surface $S$ in ${{\mathbb P}}^3$ with Hilbert polynomial $P(t)=dt+\chi$. We also set ${{\mathcal T}}_S(d):={{\mathcal T}}_S(d,0)$.)
\
*Claim 1*. Case (ii) is unfeasible.
\
Assume, firstly, that $P \in H$. Tensoring (\[trivial\_sequence\]) with ${{\mathcal O}}_H$, we get the exact sequence $${{\mathcal F}}_{| H} {\longrightarrow}{{\mathcal G}}_{| H} \overset{\alpha}{{\longrightarrow}} {{\mathcal C}}_P {\longrightarrow}0.$$ Thus, ${{\mathcal Ker}}(\alpha)$ is a quotient sheaf of ${{\mathcal F}}$ of slope zero. This contradicts the stability of ${{\mathcal F}}$. Assume, secondly, that $P \notin H$. According to [@space_quartics Sequence (10)], we have an exact sequence $$0 {\longrightarrow}{{\mathcal E}}{\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathcal O}}_L {\longrightarrow}0$$ for some sheaf ${{\mathcal E}}\in {{\mathcal T}}_H (3)$. The composite map ${{\mathcal E}}\to {{\mathcal G}}\to {{\mathcal C}}_P$ is zero, hence ${{\mathcal E}}$ is a subsheaf of ${{\mathcal F}}$. This contradicts the stability of ${{\mathcal F}}$ and proves Claim 1.
\
It remains to deal with the sheaves from (i). We have one of the following possibilities:
1. $E$ is contained in a smooth quadric surface $S$;
2. $E$ is contained in an irreducible cone $\Sigma$ but not in a smooth quadric surface;
3. $\operatorname{span}\{ q_1, q_2 \}$ contains only reducible quadratic forms and $q_1$, $q_2$ have no common factor.
\
*Claim 2*. In case (a), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Notice that ${{\mathcal F}}\in {{\mathcal T}}_S(4)$. According to [@ballico_huh Proposition 7], ${{\mathcal T}}_S(4)$ has five disjoint irreducible components ${{\mathcal T}}_S (p, q, 4)$, where $(p, q)$ is the type of the support of the 1-dimensional sheaf w.r.t. $\mathrm{Pic}S$. Clearly, ${{\mathcal F}}\in {{\mathcal T}}_S (2, 2, 4)$. Thus, ${{\mathcal F}}$ is a limit of sheaves in ${{\mathcal T}}_S (2, 2, 4)$ supported on smooth curves of type $(2, 2)$, hence ${{\mathcal F}}\in {{\mathcal E}}_4$.
\
It remains to deal with cases (b) and (c). Next we reduce further to the case when $P = P'$. Notice that, if $P \neq P'$, then ${{\mathcal F}}\simeq {{\mathcal O}}_E(P') {\otimes}({{\mathcal O}}_E(P))^{{\scriptscriptstyle \operatorname{D}}}$, hence the notation ${{\mathcal F}}= {{\mathcal O}}_E(P' - P)$ is justified.
\
*Claim 3*. Assume that ${{\mathcal F}}= {{\mathcal O}}_E(P' - P)$ for an elliptic quartic curve $E$ and distinct closed points $P', P \in E$. Then ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Let $Z_1, \ldots, Z_m$, denote the irreducible components of $E$. Fix $i, j \in \{ 1, \ldots, m \}$. Consider the locally closed subset ${{\mathcal X}}\subset {{\mathcal E}}\times {{\mathcal E}}$ of pairs $([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)])$, where $E'$ is a quartic curve of arithmetic genus $1$ whose ideal is generated by two quadratic polynomials, and $P_1$, $P_2$ are distinct points on $E'$ such that $P_1 \notin \cup_{k \neq i} Z_k$, $P_2 \notin \cup_{k \neq j} Z_k$. Consider the morphisms $$\xi \colon {{\mathcal X}}{\longrightarrow}{{\mathcal T}}(4), \qquad ([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)]) \longmapsto [{{\mathcal O}}_{E'}(P_1 - P_2)],$$ $$\sigma \colon {{\mathcal X}}{\longrightarrow}{\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t), \qquad ([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)]) \longmapsto E',$$ where ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ is the Hilbert scheme of subschemes of ${{\mathbb P}}^3$ with Hilbert polynomial $P(t)=4t$. According to [@chen_nollet Examples 2.8 and 4.8], ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ consists of two irreducible components, denoted ${\mathbf H}_1$ and ${\mathbf H}_2$. The generic member of ${\mathbf H}_1$ is a smooth elliptic quartic curve. The generic member of ${\mathbf H}_2$ is the disjoint union of a planar quartic curve and two isolated points. Note that ${\mathbf H}_2$ lies in the closed subset $\{ E' \mid \ {\operatorname{h}}^0({{\mathcal O}}_{E'}) \ge 3 \}$. Since $E$ lies in the complement of this subset, we deduce that $E \in {\mathbf H}_1$. It follows that there exists an irreducible quasi-projective curve $\Gamma \subset {\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ containing $E$, such that $\Gamma \setminus \{ E \}$ consists of smooth elliptic quartic curves (see the proof of [@space_quartics Proposition 12]). The fibers of the map $\sigma^{-1} (\Gamma) \to \Gamma$ are irreducible of dimension $2$. By [@shafarevich Theorem 8, p. 77], we deduce that $\sigma^{-1}(\Gamma)$ is irreducible. Thus, $\xi (\sigma^{-1}(\Gamma))$ is irreducible. This set contains $[{{\mathcal O}}_{E}(P'-P)]$ for $P' \in Z_i \setminus \cup_{k \neq i} Z_k$, $P \in Z_j \setminus \cup_{k \neq j} Z_k$. The generic member of $\xi (\sigma^{-1}(\Gamma))$ is a sheaf supported on a smooth elliptic quartic curve. We conclude that $[{{\mathcal O}}_E(P' - P)] \in {{\mathcal E}}_4$. Since $i$ and $j$ are arbitrary, the result is true for all $P'$, $P$ closed points on $E$.
\
*Claim 4*. In case (c), $E$ is a quadruple line supported on a line $L$. More precisely, there are three distinct planes $H$, $H'$, $H''$ containing $L$, such that $E = (H \cup H') \cap (2H'')$.
\
The claim will follow if we can show that there are linearly independent forms $u, v \in V^*$ such that $q_1, q_2 \in {{\mathbb C}}[u, v]$. Indeed, in this case $(q_1, q_2)$ has the normal form $(uv, (u+v)^2)$. We argue by contradiction. Assume that $q_1 = XY$ and $q_2 = Zl$. Consider first the case when $l = aX + bY + cZ$. We will find $\lambda \in {{\mathbb C}}$ such that $f = XY + \lambda Z l$ is irreducible, which is equivalent to saying that $$\frac{\partial f}{\partial X} = Y + a\lambda Z, \qquad \frac{\partial f}{\partial Y} = X + b\lambda Z, \qquad \frac{\partial f}{\partial Z} = \lambda (aX + bY + 2cZ)$$ have no common zero, or, equivalently, $$\left|
{\begin{array}}{ccc}
0 & 1 & a\lambda \\
1 & 0 & b\lambda \\
a\lambda & b\lambda & 2c\lambda
{\end{array}}\right| \neq 0.$$ We have reduced to the inequality $2 a b \lambda^2 - 2c\lambda \neq 0$. If $c \neq 0$ we can find a solution. If $c = 0$, then $a b \neq 0$, otherwise $q_1$ and $q_2$ would have a common factor, and we can choose any $\lambda \in {{\mathbb C}}^*$. Assume now that $l = aX + bY + cZ + dW$ with $d \neq 0$. Note that $f = XY + \lambda Z l$ is irreducible if its image in $${{\mathbb C}}[X, Y, Z, W]/\langle (c-1)Z + dW \rangle \simeq {{\mathbb C}}[X, Y, Z]$$ is irreducible. The above isomorphism sends $f$ to $XY + \lambda Z (aX + bY + Z)$ which brings us to the case examined above.
\
*Claim 5*. In case (c), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
We have ${{{\mathcal O}}_E}_{|H} \simeq {{\mathcal O}}_C$, ${{{\mathcal O}}_E}_{|H'} \simeq {{\mathcal O}}_{C'}$ for conic curves $C$ and $C'$ supported on $L$. The kernel of the map ${{\mathcal O}}_E \to {{\mathcal O}}_C$ has Hilbert polynomial $2t-1$ and is stable, because ${{\mathcal O}}_E$ is stable, hence it is isomorphic to ${{\mathcal O}}_{C'}(-1)$. We have a commutative diagram $$\xymatrix
{
0 \ar[r] & {{\mathcal O}}_E \ar[r] \ar[d] & {{\mathcal O}}_E(P') \ar[r] \ar[d] & {{\mathbb C}}_{P'} \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & {{\mathcal O}}_C \ar[r] & {{\mathcal O}}_E(P')_{|H} \ar[r] & {{\mathbb C}}_{P'} \ar[r] & 0
}$$ in which the second row is obtained by restricting the first row to $H$. Applying the snake lemma, we obtain the first row of the following exact commutative diagram: $$\xymatrix
{
0 \ar[r] & {{\mathcal O}}_{C'}(-1) \ar[r] & {{\mathcal O}}_E(P') \ar[r] \ar[d] & {{\mathcal O}}_E(P')_{|H} \ar[r] \ar[d]^-{\alpha} & 0 \\
& & {{\mathbb C}}_P \ar@{=}[r] & {{\mathbb C}}_P
}$$ Applying the snake lemma to this diagram, we get the exact sequence $$0 {\longrightarrow}{{\mathcal O}}_{C'}(-1) {\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathcal Ker}}(\alpha) {\longrightarrow}0.$$ Note that ${{\mathcal Ker}}(\alpha)$ has Hilbert polynomial $2t+1$ and is semi-stable, being a quotient of the stable sheaf ${{\mathcal F}}$. It follows that ${{\mathcal Ker}}(\alpha) \simeq {{\mathcal O}}_C$. Thus, ${{\mathcal F}}$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \big)^{{\scriptstyle \operatorname{s}}}$ of stable non-split extensions of ${{\mathcal O}}_C$ by ${{\mathcal O}}_{C'}(-1)$.
Consider the family of planes $H''_t$, $t \in {{\mathbb P}}^1 \setminus \{ 0, \infty \}$, containing $L$ and different from $H$ and $H'$. Denote $E_t = (H \cup H') \cap (2 H''_t)$. We have a two-dimensional family of semi-stable sheaves $$\{ {{\mathcal O}}_{E_t}(P' - P'') \mid t \in {{\mathbb P}}^1 \setminus \{ 0, \infty \}, \ P'' \in L \setminus \{ P' \} \} \subset {{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big).$$ This family is dense in the right-hand-side because ${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}} ({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \simeq {{\mathbb C}}^3$. To prove this we use the standard exact sequence obtained from Thomas’ spectral sequence $$\begin{gathered}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{{\mathcal O}}_C}_{| H'}, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}} ({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \to \\ {\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'}), {{\mathcal O}}_{C'}(-1))
\to {\operatorname{Ext}}^2_{{{\mathcal O}}_{H'}} ({{{\mathcal O}}_C}_{| H'}, {{\mathcal O}}_{C'}(-1)),\end{gathered}$$ see also [@choi_chung_maican Lemma 4.2]. Note that ${{{\mathcal O}}_C}_{| H'} \simeq {{\mathcal O}}_L$. Using Serre duality we obtain the isomorphisms $$\begin{aligned}
{\operatorname{Ext}}^2_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_L, {{\mathcal O}}_{C'}(-1)) & \simeq {\operatorname{Hom}}_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3))^* = 0, \\
{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_L, {{\mathcal O}}_{C'}(-1)) & \simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3))^* \simeq {{\mathbb C}}^2.\end{aligned}$$ The last isomorphism follows from the long exact sequence of extension sheaves $$\begin{gathered}
0 = {\operatorname{Hom}}({{\mathcal O}}_{H'}(-1), {{\mathcal O}}_L(-3)) {\longrightarrow}{\operatorname{Hom}}({{\mathcal O}}_{H'}(-3), {{\mathcal O}}_L(-3)) \simeq \H^0 ({{\mathcal O}}_L) \simeq {{\mathbb C}}\\
{\longrightarrow}{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3)) {\longrightarrow}\\
{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{H'}(-1), {{\mathcal O}}_L(-3)) \simeq \H^1({{\mathcal O}}_L(-2)) \simeq {{\mathbb C}}{\longrightarrow}{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{H'}(-3), {{\mathcal O}}_L(-3)) = 0\end{gathered}$$ derived from the short exact sequence $$0 {\longrightarrow}{{\mathcal O}}_{H'}(-3) {\longrightarrow}{{\mathcal O}}_{H'}(-1) {\longrightarrow}{{\mathcal O}}_{C'}(-1) {\longrightarrow}0.$$ Choose linear forms $u$ and $u'$ defining $H$ and $H'$. Restricting the standard resolution $$0 {\longrightarrow}{{\mathcal O}}(-3) \xrightarrow{\tiny \left[ \!\! {\begin{array}}{l} -u \\ (u')^2 {\end{array}}\!\!\! \right]} {{\mathcal O}}(-2) \oplus {{\mathcal O}}(-1)
\xrightarrow{\tiny \left[ \!\! {\begin{array}}{cc} (u')^2 & \!\! u {\end{array}}\!\! \right]} {{\mathcal O}}{\longrightarrow}{{\mathcal O}}_C {\longrightarrow}0$$ to $H'$, we see that ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'})$ is isomorphic to the cohomology of the complex $${{\mathcal O}}_{H'}(-3) \xrightarrow{\tiny \left[ \!\! {\begin{array}}{l} - u_{| H'} \\ \phantom{-} 0 {\end{array}}\!\! \right]} {{\mathcal O}}_{H'}(-2) \oplus {{\mathcal O}}_{H'}(-1)
\xrightarrow{\tiny \left[ \!\! {\begin{array}}{cc} 0 & \!\! u_{| H'} {\end{array}}\!\! \right]} {{\mathcal O}}_{H'}$$ that is, to ${{\mathcal O}}_L(-2)$. Using the fact that ${{\mathcal O}}_{C'}(-1)$ and ${{\mathcal O}}_L(-2)$ are reflexive, we have the isomorphisms $${\operatorname{Hom}}({{\mathcal O}}_L(-2), {{\mathcal O}}_{C'}(-1)) \simeq {\operatorname{Hom}}({{\mathcal O}}_{C'}(-1)^{{\scriptscriptstyle \operatorname{D}}}, {{\mathcal O}}_L(-2)^{{\scriptscriptstyle \operatorname{D}}}) \simeq {\operatorname{Hom}}({{\mathcal O}}_{C'}, {{\mathcal O}}_L) \simeq {{\mathbb C}}.$$ The above discussion shows that $[{{\mathcal F}}]$ is a limit of points in ${\operatorname{M}}_{{{\mathbb P}}^3}(4m)$ of the form $[{{\mathcal O}}_{E_t}(P' - P'')]$, with $P' \neq P''$. Claim 5 now follows from Claim 3.
\
It remains to consider sheaves ${{\mathcal F}}$ given by sequence (\[trivial\_sequence\]) in which ${{\mathcal G}}= {{\mathcal O}}_E(P)$ and $E$ is as at (b). We reduce further to the case when $E$ has no regular points.
\
*Claim 6*. Assume that $E$ has a regular point. Then ${{\mathcal F}}\simeq {{\mathcal O}}_E$, hence ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
The proof of the claim is obvious because $P$ in sequence (\[trivial\_sequence\]) can be chosen arbitrarily on $E$. We choose $P \in \operatorname{reg}(E)$. The kernel of the map ${{\mathcal O}}_E(P) \to {{\mathbb C}}_P$ is ${{\mathcal O}}_E$. Note that $E$ belongs to the irreducible component ${\mathbf H}_1$ of ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$, hence it is the limit of smooth elliptic quartic curves.
\
*Claim 7*. Let $E \subset {{\mathbb P}}^3$ be a quartic curve of arithmetic genus $1$ which is contained in an irreducible cone $\Sigma$, but not in a smooth quadric surface. Assume that $E$ has no regular points. Then we have one of the following two possibilities:
1. $E = \Sigma \cap (H \cup H')$, where $H$, $H'$ are distinct planes each intersecting $\Sigma$ along a double line;
2. $E = \Sigma \cap (2H)$, where $H$ is a plane intersecting $\Sigma$ along a double line.
\
To fix notations assume that $\Sigma$ has vertex $O$ and base a conic curve $\Gamma$ contained in a plane $\Pi$. Assume first that $E = \Sigma \cap \Sigma'$ for $\Sigma'$ another irreducible cone. If $\Sigma$ and $\Sigma'$ have distinct vertices, then $E$ has regular points. Thus, $\Sigma'$ has vertex $O$ and base an irreducible conic curve $\Gamma'$ contained in $\Pi$. Since $E$ has no regular points, $\Gamma \cap \Gamma'$ is the union of two double points $Q_1$ and $Q_2$. Now $E$ is the cone with vertex $O$ and base $Q_1 \cup Q_2$, so $E$ is as at (b1).
Assume next that $E = \Sigma \cap (H \cup H')$ for distinct planes $H$ and $H'$. If $H$ or $H'$ does not contain $O$, then $E$ has regular points. If $H$ or $H'$ is not tangent to $\Gamma$, then $E$ has regular points. We deduce that $E$ is as in (b1).
Assume, finally, that $E = \Sigma \cap (2H)$ for a double plane $2H$. If $O \notin H$, then it can be shown that $E$ is contained in a smooth quadric surface. Indeed, assume that $\Sigma$ has equation $X^2 + Y^2 + Z^2 = 0$ and $H$ has equation $W=0$. Then $E$ is contained in the smooth quadric surface with equation $X^2 + Y^2 + Z^2 + W^2 = 0$. Thus, $O \in H$. If $\Gamma \cap H$ is the union of two distinct points, then $\Gamma \cap (2H)$ is the union of two double points $Q_1$ and $Q_2$ and $E$ is as in (b1). If $\Gamma \cap H$ is a double point, then $E$ is as in (b2).
\
*Claim 8*. In case (b1), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
We have ${{\mathcal O}}_{E | H} \simeq {{\mathcal O}}_C$, ${{\mathcal O}}_{E | H'} \simeq {{\mathcal O}}_{C'}$ for conic curves $C$, $C'$ supported on lines $L$, respectively, $L'$. Assume that $P \in L$ and choose a point $P' \in L$ not necessarily distinct from $P$. Let ${{\mathcal F}}' \in {{\mathcal T}}_4$ be given by the exact sequence $$0 {\longrightarrow}{{\mathcal F}}' {\longrightarrow}{{\mathcal O}}_E(P') {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0.$$ As in the first paragraph in the proof of Claim 5, we see that ${{\mathcal F}}'$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big)^{{\scriptstyle \operatorname{s}}}$. We have $\dim {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \le 2$. Indeed, start with the exact sequence $$\begin{aligned}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)) & \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \to \\ & \to {\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'}), {{\mathcal O}}_{C'}(-1)).\end{aligned}$$ The group on the second line vanishes because ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'})$ is supported on $O$ while ${{\mathcal O}}_{C'}(-1)$ has no zero-dimensional torsion. It follows that $${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)).$$ The sheaf ${{\mathcal O}}_{C | H'}$ is the structure sheaf of a double point supported on $O$, hence we have the exact sequence $${{\mathbb C}}\simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathbb C}}_O, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathbb C}}_O, {{\mathcal O}}_{C'}(-1)) \simeq {{\mathbb C}}$$ from which we get our estimate on the dimension of the middle vector space.
The one-dimensional family ${{\mathcal O}}_E(P' - P)$, $P' \in L \setminus \{ P \}$, is therefore dense in ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big)^{{\scriptstyle \operatorname{s}}}$, hence, in view of Claim 3, ${{\mathcal F}}$ is a limit of sheaves in ${{\mathcal E}}_4$. We conclude that ${{\mathcal F}}\in {{\mathcal E}}_4$.
\
*Claim 9*. In case (b2), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Let $L$ be the reduced support of $\Sigma \cap H$. We have ${{\mathcal O}}_{E | H} \simeq {{\mathcal O}}_C$ for a conic curve supported on $L$. Choose a point $P' \in L$ not necessarily distinct from $P$ and let ${{\mathcal F}}' \in {{\mathcal T}}_4$ be given by the exact sequence $$0 {\longrightarrow}{{\mathcal F}}' {\longrightarrow}{{\mathcal O}}_E(P') {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0.$$ As in the first paragraph of the proof of Claim 5, we see that ${{\mathcal F}}'$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_C(-1))\big)^{{\scriptstyle \operatorname{s}}}$. We have $\dim {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) = 5$. This follows from the exact sequence $$\begin{gathered}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to \\
{\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H), {{\mathcal O}}_C(-1)) \to {\operatorname{Ext}}^2_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)).\end{gathered}$$
From Serre duality we get $${\operatorname{Ext}}^2_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \simeq {\operatorname{Hom}}_{{{\mathcal O}}_H}({{\mathcal O}}_C(-1), {{\mathcal O}}_C(-3))^* \simeq \H^0({{\mathcal O}}_C(-2))^* = 0.$$ We have ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H) \simeq {{\mathcal O}}_C(-1)$ hence ${\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H), {{\mathcal O}}_C(-1)) \simeq {{\mathbb C}}$. Applying the functor ${\operatorname{Hom}}(\cdot,{{\mathcal O}}_C(-1))$ to the short exact sequence $$0 {\longrightarrow}{{\mathcal O}}_H(-2) {\longrightarrow}{{\mathcal O}}_H {\longrightarrow}{{\mathcal O}}_C {\longrightarrow}0$$ we obtain the following exact sequence, $$\begin{gathered}
0 \to {\operatorname{Hom}}({{\mathcal O}}_H(-2), {{\mathcal O}}_C(-1)) \simeq \H^0({{\mathcal O}}_C(1)) \simeq {{\mathbb C}}^3 \\
\to {\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to \\
{\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_H, {{\mathcal O}}_C(-1)) \simeq \H^1({{\mathcal O}}_C(-1)) \simeq {{\mathbb C}}\to 0,\end{gathered}$$ since ${\operatorname{Hom}}({{\mathcal O}}_H, {{\mathcal O}}_C(-1)) \simeq \H^0({{\mathcal O}}_C(-1)) = 0$, and ${\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_H(-2), {{\mathcal O}}_C(-1)) \simeq \H^1({{\mathcal O}}_C(1)) = 0$.
Denote $Q = L \cap \Pi$. We have a three-dimensional family $\Gamma_t$ of irreducible conic curves in $\Pi$ that contain $Q$ and are tangent to $H$. Let $\Sigma_t$ be the cone with vertex $O$ and base $\Gamma_t$. Put $E_t = \Sigma_t \cap (2H)$. The four-dimensional family ${{\mathcal O}}_{E_t}(P' - P)$, $P' \in L \setminus \{ P \}$ is dense in ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_C(-1))\big)^{{\scriptstyle \operatorname{s}}}$, hence, in view of Claim 3, ${{\mathcal F}}$ is the limit of sheaves in ${{\mathcal E}}_4$. We conclude that ${{\mathcal F}}\in {{\mathcal E}}_4$.
The proof of Main Theorem \[mthm2\] is finally complete.
Components and connectedness of ${{\mathcal L}}(3)$ {#L(3) section}
===================================================
We are now ready to prove that the moduli space of rank 2 instanton sheaves of charge 3 on ${{\mathbb{P}^{3}}}$ is connected and has precisely two irreducible components. Indeed, the two components of $\overline{{{\mathcal L}}(3)}$ have already been identified above:
- $\overline{{{\mathcal I}}(3)}$, whose generic point corresponds to locally free instanton sheaves.
- $\overline{{{\mathcal C}}(1,3,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,3,0)}
0 \to E \to 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}} \to \iota_*L(2) \to 0$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane cubic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$.
Both components have dimension 21; this is classical result for the component ${{\mathcal I}}^{0}(3)$, while the dimension of ${{\mathcal C}}(1,3,0)$ is given by Theorem \[dim C thm\]. In addition, this same result also guarantees that the union ${{\mathcal I}}^{0}(3)\cup {{\mathcal C}}(1,3,0)$ is connected.
Therefore, our task is to prove that $\overline{{{\mathcal L}}(3)}$ has no other irreducible components, i.e. that every instanton sheaf of charge 3 can be deformed either into a locally free instanton sheaf, or into an instanton sheaf given by a sequence of the form (\[C(1,3,0)\]).
So let $E$ be a non locally free instanton sheaf of charge 3, and let $Q_E:=E^{\vee\vee}/E$ be the corresponding rank 0 instanton sheaf; let $d_E$ denote the degree of $Q_E$. There are three possibilities to consider: $d_E=1$, $d_E=2$ and $d_E=3$.
The first possibility is easy to deal with: if $d_E=1$, then $Q_E={{\mathcal O}}_\ell(1)$, where $\ell\hookrightarrow{{\mathbb{P}^{3}}}$ is a line in ${{\mathbb{P}^{3}}}$. It follows that $E$ fits into an exact sequence of the form $$0 \to E \to F \to {{\mathcal O}}_\ell(1) \to 0 ~,$$ where $F$ is a locally free instanton sheaf of charge 2. However, [@JMT1 Proposition 7.2] ensures that $E$ can be deformed in a (’t Hooft) locally free instanton sheaf of charge 3. In other words, if $d_E=1$, then $E$ lies within ${{\mathcal I}}^0(3)$.
Now, if $d_E=2$, then, since $Q_E$ is semistable and by Proposition \[components\_2\] above, one can find an affine open subset $0\in U\subset{{\mathbb A}}^1$ and a coherent sheaf $\mathbf{G}$ on ${{\mathbb{P}^{3}}}\times U$ such that $G_0=Q_E$ and, for $u\ne 0$,
- either $G_u={{\mathcal O}}_\Gamma(3\mathrm{pt})$, where $\Gamma$ is a nonsingular conic in ${{\mathbb{P}^{3}}}$;
- or $G_u={{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)$ where $\ell_1$ and $\ell_2$ are skew lines in ${{\mathbb{P}^{3}}}$.
Since $d_E=2$, $E^{\vee\vee}$ is a locally free instanton sheaf of charge 1 (a.k.a. a null-correlation bundle); we set $N:=E^{\vee\vee}$ denote such sheaf. Take $\mathbf{F}:=\pi^*N$, where $\pi:{{\mathbb{P}^{3}}}\times U \to {{\mathbb{P}^{3}}}$ is the projection onto the first factor. Let $s:N\twoheadrightarrow Q_E$ be the epimorphism given by the standard sequence (\[std dual sqc\]). For every $u\in U$, the sheaf ${{\mathcal H}{\it om}}(F_u,G_u)\simeq N\otimes G_u$ is supported in dimension 1, thus clearly $H^i({{\mathcal H}{\it om}}(F_u,G_u))=0$ for $i=2,3$. For $u\ne 0$ we can, after possibly shrinking $U$, assume that either $N|_\Gamma\simeq 2\cdot{{\mathcal O}}_\Gamma$ or $N|_{\ell_1}\simeq 2\cdot{{\mathcal O}}_{\ell_1}$ and $N|_{\ell_2}\simeq 2\cdot{{\mathcal O}}_{\ell_2}$; in both situations, it is easy to check that $H^1({{\mathcal H}{\it om}}(F_u,G_u))=0$. Finally, for $u=0$, we twist the resolution of $Q_E$ $$0 \to 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}}(-1) \stackrel{\alpha}{\longrightarrow} 4\cdot {{\mathcal O}_{\mathbb{P}^{3}}} \stackrel{\beta}{\longrightarrow} 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}}(1) \to Q_E \to 0$$ by $N$ and check that $H^1({{\mathcal H}{\it om}}(F_0,G_0))=H^1(N\otimes Q_E)\simeq H^2(N\otimes B)=0$, where $B:=\operatorname{im}\beta$.
Therefore, it follows from Lemma \[F,G\] that there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending $s:N\twoheadrightarrow Q_E$. Let $\mathbf{E}:=\ker\mathbf{s}$; clearly, $E_0:=\mathbf{E}|_{\{0\}\times{{\mathbb{P}^{3}}}}=E$. For $u\ne 0$, $E_u$ fits into the exact sequence $$0 \to E_u \to N \to G_u \to 0.$$
In the case [**(i)**]{} described above, $G_u$ lies within ${{\mathcal D}}(2,3)$ for $u\ne0$, hence $E=E_0$ lies within $\overline{{{\mathcal D}}(2,3)}$, which is contained in $\overline{{{\mathcal I}}(3)}$ by [@JMT1 Theorem 7.8]. In other words, $E_0$ can be deformed into a locally free instanton sheaf of charge 3, thus it lies within ${{\mathcal I}}^0(3)$.
In the case [**(ii)**]{}, Propostion \[disjoint rat curves\] also implies that $[E_0]\in{{\mathcal I}}^0(3)$.
An argument similar to the one used in the proof of [@JMT1 Proposition 7.2] works to show that $E$ can be deformed into a locally free (’t Hooft) instanton sheaf.
Summing up, we conclude that if $d_E=2$, then $E$ lies within ${{\mathcal I}}^0(3)$.
Finally, consider $d_E=3$, so that $E^{\vee\vee}=2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$. Since $Q_E$ is semistable, it follows from Proposition \[components\_3\] that one can find an affine open subset $0\in U\subset{{\mathbb A}}^1$ and a coherent sheaf $\mathbf{G}$ on ${{\mathbb{P}^{3}}}\times U$ such that $G_0=Q_E$ and, for $u\ne 0$,
- either $G_u={{\mathcal O}}_\Delta(5\mathrm{pt})$, where $\Delta$ is a nonsingular twisted cubic in ${{\mathbb{P}^{3}}}$;
- or $G_u={{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus
{{\mathcal O}}_{\ell}(1)$, where $\Gamma$ is a nonsingular conic and $\ell$ is a line disjoint from $\ell$;
- or $G_u={{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)\oplus{{\mathcal O}}_{\ell_3}(1)$ where $\ell_j$ are 3 skew lines in ${{\mathbb{P}^{3}}}$;
- or $G_u=L(2)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$.
Now set $\mathbf{F}:2\cdot\pi^*{{\mathcal O}_{\mathbb{P}^{3}}}$. Note that $H^i({{\mathcal H}{\it om}}(F_u,G_u))=H^i(2\cdot G_u)$, and this vanishes for $i=1,2,3$ in all of the four cases outlined above for $u\ne0$; for $u=0$, $H^i(G_0)=H^i(Q_E)$ and this vanishes by dimension of $Q_E$ when $i=2,3$, and by the vanishing of $h^1(Q_E(-2))$ when $i=1$.
We complete the argument as before; again, it follows from Lemma \[F,G\] that there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending the epimorphism $s:2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\twoheadrightarrow Q_E$ obtained from the standard sequence (\[std dual sqc\]) for $E$. Let $\mathbf{E}:=\ker\mathbf{s}$; clearly, $E_0:=\mathbf{E}|_{\{0\}\times{{\mathbb{P}^{3}}}}=E$. For $u\ne 0$, $E_u$ fits into the exact sequence $$0 \to E_u \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \to G_u \to 0.$$
In the cases [**(i)**]{} through [**(iii)**]{}, we know from [@JMT1 Theorem 7.8] and Proposition \[disjoint rat curves\] above that $[E_0]\in\overline{{{\mathcal D}}(3,3)}$, thus also $[E_0]\in{{\mathcal I}}^0(3)$.
In the case [**(iv)**]{}, $E_u$ lies within ${{\mathcal C}}(1,3,0)$, by definition. This completes the proof of the first part of Main Theorem \[mthm1\].
Components and connectedness of ${{\mathcal L}}(4)$ {#L(4) section}
===================================================
In this section we prove the second part of Main Theorem \[mthm1\], i.e. we enumerate the irreducible components of ${{\mathcal L}}(4)$ and show that ${{\mathcal L}}(4)$ is connected. Note that we already know from Theorem \[dim C thm\] four irreducible components of $\overline{{{\mathcal L}}(4)}$:
- $\overline{{{\mathcal I}}(4)}$, whose generic point corresponds to locally free instanton sheaves;
- $\overline{{{\mathcal C}}(1,3,1)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,3,1)}
0 \to E \to N\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane cubic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$;
- $\overline{{{\mathcal C}}(2,2,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(2,2,0)}
0 \to E \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular elliptic space quartic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$;
- $\overline{{{\mathcal C}}(1,4,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,4,0)}
0 \to E \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane quartic $\Sigma$, and $L\in{\rm Pic}^2(\Sigma)$ is such that $h^0(\Sigma,L)=0$.
The first three components have dimension 29, and the last one has dimension 32; this is a classical result for the component ${{\mathcal I}}^0(4)$, while the dimensions of ${{\mathcal C}}(1,3,1)$, ${{\mathcal C}}(2,2,0)$ and ${{\mathcal C}}(1,4,0)$ are given by Theorem \[dim C thm\] above. Furthermore, [@JMT2 Theorem 23] implies that each of the last three components intersects ${{\mathcal I}}^0(4)$. Thus the union of these four components is connected.
To finish the proof of the second part of Main Theorem \[mthm1\], it is again enough to show that there are no other components in ${{\mathcal L}}(4)$, except for those described above. The argument here is the same as before, exploring Theorem \[jg-thm\], Remark \[remark 4\] and Proposition \[components\_4\].
Take any $[E]\in{{\mathcal L}}(4)$ and consider the triple (\[std dual sqc\]). Then, in view of Theorem \[jg-thm\] and Remark \[remark 4\], $Q_E$ is a rank 0 instanton sheaf of multiplicity $1\le d_E\le 4$, and $E^{\vee\vee}$ is an instanton bundle of charge $4-d_E$. Consider the possible cases for $d_E$.
[$\mathbf{d_E=1.}$]{} As in the similar case in Section \[L(3) section\], $Q_E={{\mathcal O}}_{\ell}(1)$ where $l$ is a line in ${{\mathbb{P}^{3}}}$. Respectively, $[E^{\vee\vee}]\in{{\mathcal I}}(3)$. Deforming $\ell$ in ${{\mathbb{P}^{3}}}$ we may assume that $E^{\vee\vee}|_{\ell}\simeq2\cdot{{\mathcal O}}_{\ell}$, so that $[E]\in{{\mathcal D}}(1,4)$. Therefore, $[E]\in{{\mathcal I}}^0(4)$.
[$\mathbf{d_E=2.}$]{} As in the similar case in Section \[L(3) section\], $Q_E$ can be deformed in a flat family either into a sheaf ${{\mathcal O}}_\Gamma(3\mathrm{pt})$, where $\Gamma$ is a nonsingular conic in ${{\mathbb{P}^{3}}}$, or into a sheaf ${{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)$ where $\ell_1$ and $\ell_2$ are skew lines in ${{\mathbb{P}^{3}}}$. Respectively, $[E^{\vee\vee}]\in{{\mathcal I}}(2)$. Now the same argument as in Section \[L(3) section\] shows that $[E]\in{{\mathcal I}}^0(4)$.
[$\mathbf{d_E=3.}$]{} Then $E^{\vee\vee}$ is a null-correlation bundle and, as in the case $d_E=3$ of Section \[L(3) section\], the sheaf $Q_E$ deforms in a flat family to one of the sheaves:
- $L(2)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$.
- ${{\mathcal O}}_\Delta(5\mathrm{pt})$, where $\Delta$ is a nonsingular twisted cubic in ${{\mathbb{P}^{3}}}$;
- ${{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus{{\mathcal O}}_{\ell}(1)$, where $\Gamma$ is a nonsingular conic and $\ell$ is a line disjoint from $\ell$;
- ${{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)\oplus
{{\mathcal O}}_{\ell_3}(1)$ where $\ell_j$ are 3 skew lines in ${{\mathbb{P}^{3}}}$.
By definition, $[E]\in\overline{{{\mathcal C}}(1,3,1)}$ in case [**(i)**]{}. The same argument as in Section \[L(3) section\], based on [@JMT1 Theorem 7.8] and Proposition \[disjoint rat curves\], shows that $[E]\in\overline{{{\mathcal I}}(4)}$ in cases [**(ii)**]{} through [**(iv)**]{}.
$\mathbf{d_E=4.}$ Then $E^{\vee\vee}\simeq2\cdot{{\mathcal O}}_{{{\mathbb{P}^{3}}}}$ and, according to Proposition \[components\_4\], the sheaf $Q_E$ deforms in a flat family to one of the sheaves:
- $L(2)$, where $L\in\operatorname{{Pic}}^2(\Sigma)$, for some nonsingular plane quartic $\Sigma$ in ${{\mathbb{P}^{3}}}$, and $L$ satisfies an open condition $h^1(L)=0$;
- $L(2)$, where $0\ne L\in\operatorname{{Pic}}^0(\Delta)$, for some nonsingular space elliptic quartic $\Delta$ in ${{\mathbb{P}^{3}}}$;
- ${{\mathcal O}}_{\Delta}(7\mathrm{pt})$ for some nonsingular rational space quartic $\Delta$ in ${{\mathbb{P}^{3}}}$;
- $L(2)\oplus{{\mathcal O}}_{\ell}(1)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$ and a line $\ell$ disjoint from $\Sigma$;
- ${{\mathcal O}}_\Delta(5\mathrm{pt})\oplus{{\mathcal O}}_{\ell}(1)$, where $\Delta$ is a nonsingular twisted cubic and $\ell$ is a line disjoint from $\Delta$;
- ${{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus{{\mathcal O}}_{\ell_1}(1)
\oplus{{\mathcal O}}_{\ell_2}(1)$, where $\Gamma$ is a nonsingular conic and $\ell_1,\ \ell_2$ are two skew lines disjoint from $\Gamma$;
- ${{\mathcal O}}_{\ell}(1)\oplus{{\mathcal O}}_{\ell_2}(1)
\oplus{{\mathcal O}}_{\ell_3}(1)\oplus{{\mathcal O}}_{\ell_4}(1)$, where $\ell_1,\ \ell_2,\ \ell_3,\ \ell_4$ are four disjoint lines in ${{\mathbb{P}^{3}}}$.
In case [**(i)**]{}, since, in the notation of Lemma $\ref{F,G}$, $F_0=2\cdot{{\mathcal O}}_{{{\mathbb{P}^{3}}}}$ and $G_0=L$, hence $H^i({{\mathcal H}{\it om}}(F_0,G_0))=0,\ \ \ i\ge1,$ and therefore the condition (\[vanish Hi\]) is satisfied by the semicontinuity, so that the deformation argument as above shows that $E\in\overline{{{\mathcal C}}(1,4,0)}$.
In case [**(ii)**]{}, by the same reason, $[E]\in\overline{{{\mathcal L}}(2,2,0)}$.
In case [**(iii)**]{}, a similar argument shows that $[E]\in\overline{{{\mathcal D}}(4,4)}$, and therefore $[E]\in\overline{{{\mathcal I}}(4)}$.
In case [**(iv)**]{}, Proposition \[disjoint curves\] above guarantees that $[E]\in\overline{{{\mathcal L}}(1,3,1)}$.
In the cases remaining, [**(v)**]{} through [**(vii)**]{}, as in cases [**(ii)**]{} and [**(iii)**]{} for $d_E=3$ above, we again obtain $[E]\in\overline{{{\mathcal I}}(4)}$.
Main Theorem \[mthm1\] is finally proved.
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| ArXiv |
---
author:
- 'N. G. Guseva'
- 'P. Papaderos'
- 'H. T. Meyer'
- 'Y. I. Izotov'
- 'K. J. Fricke'
date: 'Received ; Accepted'
title: 'An investigation of the luminosity-metallicity relation for a large sample of low-metallicity emission-line galaxies [^1], [^2]'
---
[We present 8.2m VLT spectroscopic observations of 28 H [[ii]{}]{} regions in 16 emission-line galaxies and 3.6m ESO telescope spectroscopic observations of 38 H [[ii]{}]{} regions in 28 emission-line galaxies. These emission-line galaxies were selected mainly from the Data Release 6 (DR6) of the Sloan Digital Sky Survey (SDSS) as metal-deficient galaxy candidates. ]{} [We collect photometric and high-quality spectroscopic data for a large uniform sample of star forming galaxies including new observations. Our aim is to study the luminosity-metallicity ($L-Z$) relation for nearby galaxies, especially at its low-metallicity end and compare it with that for higher-redshift galaxies. ]{} [Physical conditions and element abundances in the new sample are derived with the $T_{\rm e}$-method, excluding six H [[ii]{}]{} regions from the VLT observations and nearly two third of the H [[ii]{}]{} regions from the 3.6m observations. Element abundances for the latter galaxies were derived with the semiempirical strong-line method. ]{} [ From our new observations we find that the oxygen abundance in 61 out of the 66 H [[ii]{}]{} regions of our sample ranges from 12 + log O/H = 7.05 to 8.22. Our sample includes 27 new galaxies with 12 + log O/H $<$ 7.6 which qualify as extremely metal-poor star-forming galaxies (XBCDs). Among them are 10 H [[ii]{}]{} regions with 12 + log O/H $<$ 7.3. The new sample is combined with a further 93 low-metallicity galaxies with accurate oxygen abundance determinations from our previous studies, yielding in total a high-quality spectroscopic data set of 154 H [[ii]{}]{} regions. 9000 more galaxies with oxygen abundances, based mainly on the $T_{\rm e}$-method, are compiled from the SDSS. Photometric data for all galaxies of our combined sample are taken from the SDSS database while distances are from the NED. Our data set spans a range of 8 mag with respect to its absolute magnitude in SDSS $g$ (–12 $\ga M_g \ga$ –20) and nearly 2 dex in its oxygen abundance (7.0$\la$12 + log O/H$\la$8.8), allowing us to probe the $L-Z$ relation in the nearby universe down to the lowest currently studied metallicity level. The $L-Z$ relation established on the basis of the present sample is consistent with previous ones obtained for emission-line galaxies. ]{}
Introduction \[intro\]
======================
It was shown more than 20 years ago that low-luminosity dwarf galaxies have systematically lower metallicities compared to more luminous galaxies [@Lequeux1979; @Skillman1989; @RicherMcC1995]. This dependence, initially obtained for irregular galaxies, was later confirmed for galaxies of different morphological types [e.g. @Vila1992; @KobylZarit1999; @MelbourneSalzer2002; @Lee2004; @Pil2004; @Lee45mu2006].
The differences between giant and dwarf galaxies are usually attributed to different chemical evolution of galaxies with different masses [e.g. @Lequeux1979; @Tremonti2004; @Lee45mu2006; @Ellison2008; @Gavilan2009]. Thus, more efficient mechanisms seem to be at work in massive galaxies converting gas into stars and/or less efficient ones ejecting enriched matter into the galactic halo or even into the intergalactic medium. While the mass of a galaxy is one of the key physical parameters governing galaxy evolution, its determination is not easy and somewhat uncertain. Therefore, very often the luminosity, which is directly derived from observations, is used instead of the mass. In addition, some authors also use other global characteristics of a galaxy such as Hubble morphological type, rotation velocity, the gas mass fraction, surface brightness of the galaxy, to study correlations between metallicity and macroscopic properties of a galaxy [e.g. @Tremonti2004; @Pil2004].
Metallicity reflects the level of the gas astration in the galaxy. Hence, the metallicity of a galaxy depends strongly on its evolutionary state, specifically, on the fraction of the gas converted into stars. The metallicity in emission-line galaxies is defined in terms of the relative abundance of oxygen to hydrogen (usually 12 + log O/H) in the interstellar medium (ISM). Different mechanisms were considered in chemical evolution models to account for the low metallicity of dwarf galaxies, mainly 1) enriched galactic wind outflow which expells the newly synthesized heavy elements from the galaxy, resulting in slowing enrichment of the galaxy ISM; 2) inflow of metal-poor intergalactic gas and its mixing with the galaxy ISM which results in decreasing ISM metallicity, and 3) the burst character of star formation with a very low level of astration between the bursts. In principle, chemical evolution models could predict the slope and scatter of the mass-metallicity $M-Z$ (and luminosity-metallicity $L-Z$) relations over a large range in mass (luminosity) and metallicity invoking the mechanisms mentioned above.
Usually, $L-Z$ relations are based on optical observations of nearby galaxies. However, it was shown in recent studies that the near infrared (NIR) range could be more promising for such studies. @Saviane2008 collected abundances obtained by means of the temperature-sensitive method and NIR luminosities for a sample of dwarf irregular galaxies with –20 $<$ $M_H$ $<$ –13, located in nearby groups of galaxies. They obtained a tight $M-Z$ relation with a low scatter of 0.11 dex around its linear fit. @Salzer05 [see also @Vaduvescu07] noted that the NIR luminosities are more fundamental than the $B$-band ones, since they are largely free of absorption effects and are more directly related to the stellar mass of the galaxy than optical luminosities. Nevertheless, this statement is correct only for galaxies with low and moderate SF activity. In emission-line galaxies with high star formation rate (SFR), such as blue compact dwarf (BCD) galaxies, the young, low mass-to-light ($M/L$) ratio stellar component may provide up to $\sim$50% of the total $K$ band emission [@Noeske03]. Additionally, in such systems the contribution of ionized gas to the total luminosity could be high [see e.g. @I97b; @P98; @P02], especially in the NIR range [see e.g. @Vanzi00; @Smith2009], and should be taken into account.
Recently, studies of the $L-Z$ relation were extended to larger volumes by including moderate- and high-redshift galaxies [@KobylZarit1999; @Contini2002; @Maier2004]. Variations of the $L-Z$ relation with redshift can provide a means to study the galaxy evolution with look-back time [see, e.g., @Kobulniky2003]. It was established in this study that the slopes and zero points of the $L-Z$ relation evolve smoothly with redshift. Its large dispersion has been attributed to galaxy evolution effects. However, these results and their comparison with those for nearby galaxies should be considered with caution. The high-redshift samples are biased by different selection criteria and metallicity calibrations as compared to the local galaxies. They consist on average of more luminous and higher metallicity galaxies. Star-forming dwarf galaxies in the relatively high-redshift (up to $z$ $\sim$ 1) samples are rare because of their intrinsic faintness. Moreover, due to the weakness of the \[O [iii]{}\]$\lambda$4363 emission line in the spectra of these galaxies, their abundance determinations are more uncertain and could lead to a large scatter in the $L-Z$ diagrams. This fact could be the reason for a larger scatter of high-redshift dwarf galaxies if the direct $T_{\rm e}$-method is used instead of the empirical R$_{23}$ one [e.g., @Kakazu07].
In summary, it is difficult to obtain reliable metallicities over a large luminosity range in a homogeneous manner, i.e. employing a unique technique (e.g. the direct $T_{\rm e}$-method), even for nearby galaxies. Therefore, different methods for abundance determination are applied for galaxies of different types. The direct method is mainly used for nearby low-metallicity galaxies, while various empirical methods are used for nearby high-metallicity galaxies and for almost all high-redshift galaxies. The variety of methods results in significant differences in the $L-Z$ relations obtained with the direct $T_{\rm e}$-method and those based on strong emission line ratio calibrations, such as $R_{23}$, $P$-method, N2 and O3N2 methods. These differences were reported by many authors [e.g., @Pil2004; @Shi2005; @Hoyos2005; @Kakazu07].
Large surveys, such as the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and Sloan Digital Sky Survey (SDSS), provide rich data sets for statistically improved studies of the $L-Z$ relation. For example, @Lamareille2004 using more than 6000 spectra of SF galaxies at $z$ $<$ 0.15 from the 2dFGRS have obtained an $L-Z$ relation that is much steeper than that for nearby irregulars and spiral galaxies. @Tremonti2004 studied the mass-metallicity ($M-Z$) relation for 53000 SF galaxies within $z$ $\sim$ 0.2 extracted from SDSS, using their stellar continuum and line fitting method. This method is applicable because the bulk of their emission-line galaxies show weak emission lines and strong stellar absorption features, and therefore the contribution of gaseous emission to the galaxy luminosity is low. The @Tremonti2004 $M-Z$ relation is relatively steep but it flattens for massive galaxies at masses above 10$^{10}$ M$_{\odot}$. On the contrary, @MelbourneSalzer2002 using 519 emission-line galaxies from the KPNO International Spectroscopic Survey (KISS) found that the slope of the $L-Z$ relation for luminous galaxies is steeper than that for dwarf galaxies. Nevertheless, @Pil2004 have compared the $L-Z$ relation based on more than 1000 published spectra of H [ii]{} regions in spiral galaxies to that for irregular galaxies. They found that the slope of the relation for spirals is slightly shallower than the one for irregular galaxies. Furthermore, using 72 star-forming galaxies, @Shi2005 have also shown that the slope of the $L-Z$ relation for luminous galaxies is slightly shallower than that for dwarf galaxies.
Is the slope of the $L-Z$ relation invariant for galaxies of different type, such as local dwarf and spiral galaxies and high-redshift galaxies? If differences in the $L-Z$ relations for intermediate- and high-redshift and local ones are present, they may yield important constraints on the Star Formation History (SFH) of galaxies. For this, an as accurate as possible $L-Z$ relation, based on homogeneous high-quality photometric and spectroscopic data is required for galaxies that covers a large range in metallicity and luminosity. In particular, probing the slope of the $L-Z$ relation in its low-metallicity end, i.e. in the range expected for unevolved low-mass galaxies in the faraway universe, is much needed. For this purpose, in this paper we focus our study on the lowest-metallicity galaxy candidates selected from large spectroscopic surveys, using deep follow-up spectroscopic observations.
Specifically, the objective of the work is to study the $L-Z$ relation for a large uniform sample of emission-line galaxies in the local Universe for which the element abundances are obtained with high precision. The main feature of our sample is that it is one of the richest currently available at the low-metallicity end. For the galaxy selection we used different surveys such as 2dFGRS, SDSS and others. Most of our sample galaxies currently undergo strong episodes of star formation (SF).
We performed 3.6m ESO spectroscopic observations of a sample of 38 H [ii]{} regions in 28 emission-line galaxies and 8.2m VLT spectroscopic observations of a sample of 28 H [ii]{} regions in 16 emission-line galaxies. We supplement our new data with our previous data collected from the MMT observations [@IT2007], and from the 3.6m ESO observations [@BJlarge2007; @Pap2008] of the low-metallicity emission-line galaxies selected from the SDSS, with the sample used by @IT04a to study the helium abundance in low-metallicity BCDs (henceforth referred to as the HeBCD sample) and with the MMT sample used by @TI2005 to study high-ionization emission lines in low-metallicity BCDs. Our MMT, 3.6m ESO and HeBCD low-metallicity galaxies were selected from different surveys [a more complete description of the MMT, 3.6m ESO and HeBCD subsamples can be found in @IT2007; @BJlarge2007; @Pap2008; @IT04a]. During past years we selected from the SDSS and performed follow-up spectroscopic observations with large telescopes of (i) BCDs with strong ongoing SF, i.e. galaxies with high EW(H$\beta$), blue colours, high ionisation parameter, and (ii) low-metallicity galaxies in a relatively quiescent phase of SF, i.e. galaxies with low EW(H$\beta$), low ionisation parameter or older starburst age. For this, we selected galaxies with weak or not detected \[O [iii]{}\]$\lambda$4363 emission line and with \[O [iii]{}\]$\lambda$4959/H$\beta$ $\la$ 1 and \[N [ii]{}\]$\lambda$6583/H$\beta$ $\la$ 0.05 [@IPGFT2006; @IT2007].
SDSS is an excellent source of both photometric and spectroscopic data for more than one million galaxies in its Data Release 7 (DR7) [@A09]. Despite that, our stringent selection criteria resulted in a very small sample of extremely metal-deficient emission-line galaxies with reliable abundance determinations. This sample is supplemented for the purpose of comparison by a sample of $\sim$ 9000 SDSS emission-line galaxies (SDSS sample) over a larger range of metallicities. The oxygen abundances for the galaxies from the SDSS sample are obtained using the direct $T_{\rm e}$-method. In addition, only high-quality spectra of SDSS galaxies with the non-detected \[O [iii]{}\]$\lambda$4363 emission line are included, for which oxygen abundances are derived by a semiempirical method [@IT2007].
Thus, we construct a large homogeneous sample with uniform selection criteria, uniform data reduction methods, and uniform techniques for the element abundance determinations. The apparent $g$ magnitudes for our entire data set are taken from the SDSS. They were used to derive absolute $g$ magnitudes which were corrected for the Galactic extinction and Virgo cluster infall, except for the comparison SDSS sample galaxies. For the latter galaxies the absolute magnitude was derived from the observed redshift, adopting a Hubble constant of $H_0$ = 75 km s$^{-1}$ Mpc$^{-1}$.
The paper is organized as follows. Observations and data reduction are described in Sect. 2. Physical conditions and element abundances in the galaxies from the new observations are presented in Sect. 3. We discuss the properties of the $L-Z$ relation in Sect. 4 and summarise our conclusions in Sect. 5.
Observations and data reduction
===============================
The new spectra of the 3.6m ESO sample were obtained on 14 - 16 September, 2007 with the spectrograph EFOSC2. The grism Gr\#7 and a long slit with the width of 12 were used yielding a wavelength coverage of $\lambda$$\lambda$3400–5200Å. The long slit was centered on the brightest part of each galaxy and simultaneously on another H [ii]{} regions, whenever present. The name of each galaxy with its different H [ii]{} regions, the coordinates R.A., Dec. (J2000.0), date of observation, exposure time, number of exposures for each observation, average airmass and seeing are given in Table \[obs36\]. All spectra were obtained at low airmass or with the slit oriented along the parallactic angle, so no corrections for atmospheric refraction have been applied.
The new VLT spectra were obtained during several runs in October - December, 2006 and January, 2007 with the spectrograph FORS2 mounted at the ESO VLT UT2. The observing conditions were photometric during the nights with seeing $<$ 15. Several observations were performed under excellent seeing conditions ($<$ 08). The grisms 600B ($\lambda$$\lambda$$\sim$3400–6200) and 600RI and filter GG435 ($\lambda$$\lambda$$\sim$5400–8620) for the blue and red parts of the spectrum, respectively, were used. A 1$\times$360 long slit was centered on the brightest H [ii]{} regions of each galaxy. In Table \[obsVLT\], the same parameters as in Table \[obs36\] are given for the VLT observations. Note that for each galaxy the first and the second lines are related to the observations in the blue and red ranges, respectively. Again, as for the EFOSC2 spectra, the observations were obtained at low airmass, and no corrections for atmospheric refraction were applied.
The data were reduced with the IRAF[^3] software package. This included bias–subtraction, flat–field correction, cosmic-ray removal, wavelength calibration, night sky background subtraction, correction for atmospheric extinction and absolute flux calibration of the two–dimensional spectrum. The spectra were also corrected for interstellar extinction using the reddening curve of @W58. One-dimensional spectra of one or several H [ii]{} regions in each galaxy were extracted from two-dimensional observed spectra. The flux- and redshift-calibrated one–dimensional EFOSC2 3.6m spectra of the H [ii]{} regions are shown in Fig. \[fig1\] for all galaxies given in Table \[obs36\]. One-dimensional VLT spectra are shown in Fig. \[fig2\] for 28 objects listed in Table \[obsVLT\]. For VLT spectra of the four background galaxies and H [ii]{} region No.2 in the galaxy J2354-0004 without a detectable \[O [iii]{}\]$\lambda$4363$\AA$ emission line, no abundance determination has been done.
Emission-line fluxes were measured using Gaussian profile fitting. The errors of the line fluxes were calculated from the photon statistics in the non-flux-calibrated spectra. They have been propagated in the calculations of the elemental abundance errors. The quality of the VLT data reduction could be verified by a comparison of He [i]{} $\lambda$5876 emission line fluxes measured in the blue and red spectra of the same object. We found that the fluxes of the He [i]{} $\lambda$5876 emission line in spectra of bright objects differ by no more than 1-2% indicating an accuracy in the flux calibration at the same level. For faint objects the difference between the flux of the He [i]{} $\lambda$5876 emission line in the blue and red spectra is higher, $\sim$5 – 10%, and is comparable to the statistical errors listed in Table \[t4\_1\_VLT\].
The extinction coefficient $C$(H$\beta$) and equivalent widths of the hydrogen absorption lines EW(abs) are calculated simultaneously, minimizing the deviations of corrected fluxes $I(\lambda)$/$I$(H$\beta$) of all hydrogen Balmer lines from their theoretical recombination values as $$\begin{aligned}
\frac{I(\lambda)}{I({\rm H}\beta)} & = &\frac{F(\lambda)}{F({\rm H}\beta)}
\frac{EW(\lambda)+|EW(abs)|}{EW(\lambda)}\frac{EW({\rm H}\beta)}{EW({\rm H}\beta)+|EW(abs)|} \\
& \times & 10^{C({\rm H}\beta)f(\lambda)}.\end{aligned}$$ Here $f$($\lambda$) is the reddening function normalized to the value at the wavelength of the H$\beta$ line, $F$($\lambda$)/$F$(H$\beta$) are the observed hydrogen Balmer emission line fluxes relative to that of H$\beta$, EW($\lambda$) and EW(H$\beta$) the equivalent widths of emission lines, and EW(abs) the equivalent widths of hydrogen absorption lines which we assumed to be the same for all hydrogen lines. For $f$($\lambda$) we adopted the reddening law by @W58. The extinction-corrected fluxes of emission lines other than hydrogen ones are derived from equation $$\begin{aligned}
\frac{I(\lambda)}{I({\rm H}\beta)} & = &\frac{F(\lambda)}{F({\rm H}\beta)}
\times 10^{C({\rm H}\beta)f(\lambda)} .\end{aligned}$$
The extinction-corrected emission line fluxes $I$($\lambda$) relative to the H$\beta$ fluxes multiplied by 100, the extinction coefficients $C$(H$\beta$), the equivalent widths EW(H$\beta$), the observed H$\beta$ fluxes $F$(H$\beta$) (in units 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$), and the equivalent widths of the hydrogen absorption lines are listed in Table \[t3\_1\_36\] (3.6m ESO observations) and in Table \[t4\_1\_VLT\] (VLT observations). $C$(H$\beta$) and EW(abs) are set to zero in Tables \[t3\_1\_36\] and \[t4\_1\_VLT\] if we do not have enough observational data or their values are negative.
Physical conditions and element abundances
==========================================
The electron temperature $T_{\rm e}$, the ionic and total heavy element abundances were derived following @Iz06. In particular, for the ions O$^{2+}$, Ne$^{2+}$ and Ar$^{3+}$ we adopt the temperature $T_{\rm e}$(O [iii]{}) directly derived from the \[O [iii]{}\] $\lambda$4363/($\lambda$4959 + $\lambda$5007) emission-line ratio. For $T_{\rm e}$(O [ii]{}) and $T_{\rm e}$(S [iii]{}) we use the relation between the electron temperatures $T_{\rm e}$(O [iii]{}) and the temperatures characteristic for ions O$^{+}$ and S$^{2+}$ obtained by @Iz06 from the H [ii]{} photoionization models based on recent stellar atmosphere models and improved atomic data [@Stasin2003].
We use $T_{\rm e}$(O [ii]{}) for the calculation of O$^{+}$, N$^{+}$, S$^{+}$ and Fe$^{2+}$ abundances and $T_{\rm e}$(S [iii]{}) for the calculation of S$^{2+}$, Cl$^{2+}$ and Ar$^{2+}$ abundances. The electron number densities for some H [ii]{} regions were obtained from the \[S [ii]{}\] $\lambda$6717/$\lambda$6731 emission line ratio. These lines were not observed or not measured in the remaining H [ii]{} regions. For the abundance determination in those H [ii]{} regions we adopt $N_{\rm e}$ = 10 cm$^{-3}$. The precise value of the electron number density makes little difference in the derived abundances since in the low-density limit which holds for the H [ii]{} regions considered here, the element abundances do not depend sensitively on $N_{\rm e}$. The electron temperatures $T_{\rm e}$(O [iii]{}), $T_{\rm e}$(O [ii]{}), the ionization correction factors ($ICF$s), the ionic and total O and Ne abundances are given in Table \[t5\_1\_36\] for 3.6m observations. The electron temperatures $T_{\rm e}$(O [iii]{}), $T_{\rm e}$(O [ii]{}), $T_{\rm e}$(S [iii]{}), electron number density $N_{\rm e}$(\[S [ii]{}\]), the ionization correction factors ($ICF$s), the ionic and total O, N, Ne, S, Cl, Ar and Fe abundances are given in Table \[t6\_1\_VLT\] for VLT observations.
The oxygen abundances 12 + log O/H in 61 H [ii]{} regions out of 66 obtained from the new 3.6m ESO and VLT observations range from 7.05 to 8.22. Among them, 27 H [ii]{} regions with 12 + log O/H $<$ 7.6 are found, including 10 H [ii]{} regions with 12 + log O/H $<$ 7.3. The combined sample consisting of the new observations, 43 BCDs from the HeBCD sample, 30 galaxies from our previous 3.6m ESO observations and 20 galaxies from the MMT observations yields a data set of 154 H [ii]{} regions. For comparison, we also use $\sim$9000 SDSS emission-line galaxies with the \[O [iii]{}\] $\lambda$4363 emission line detected at least at the 1$\sigma$ level, allowing abundance determination by the direct $T_{\rm e}$-method. In addition, SDSS galaxies with high-quality spectra where the \[O [iii]{}\]$\lambda$4363 emission line was not detected are used. In the latter case, the oxygen abundances were derived by the semiempirical method. SDSS galaxies from the comparison sample mostly populate the high-metallicity, high-luminosity ranges, as compared to the galaxies from our combined sample of low-metallicity emission-line galaxies (Figs. \[fig5\] - \[fig7\]). The considered galaxies span two dex in gas-phase oxygen abundance, from 12 + log O/H $\sim$ 7.0 through $\sim$ 9.0.
We use SDSS $g$ magnitudes for the determination of the absolute magnitude $M_g$ of all galaxies from our samples, while usually $B$ magnitudes and $M_B$ are considered in the literature. However, @Pap2008 have shown that for regions with ongoing bursts of star formation, which is the case for our sample galaxies, the $B$–$g$ colour index is of the order of 0.1 mag only and $<$0.3 mag during the first few Gyrs of galactic evolution. Therefore, we do not transform $M_g$ to $M_B$ and directly compare $M_g$’s for the galaxies from our samples with $M_B$’s for the galaxies available from the literature. The use of the SDSS $g$-band photometry for all our samples allows us to investigate the $L-Z$ relation over the $M_g$ range from –21 mag to the faintest magnitude of $\sim$ –12 mag at the low-metallicity end.
Results
=======
Luminosity-metallicity relation
-------------------------------
In order to illustrate the main properties of our sample we plot (a) the reddening parameter $C$(H$\beta$) obtained from the Balmer decrement and (b) the logarithm of the H$\beta$ equivalent width (Fig. \[fig3\]) and the logarithm of the H$\beta$ line luminosity (in erg s$^{-1}$) (Fig. \[fig4\]) as a function of absolute magnitude $M_g$. The new 3.6m telescope and VLT data are shown by open circles and stars, respectively. The metal-poor galaxies collected from previous 3.6m ESO observations are shown by filled triangles [@BJlarge2007; @Pap2008]. Filled circles denote the data from the HeBCD sample collected by @ING2004a and @IT04a. The MMT data [@IT2007] are shown by large filled circles. The comparison SDSS sample is represented by asterisks. From the latter sample H [ii]{} regions in nearby spiral galaxies are excluded, as are faint SDSS galaxies with $m_g$ $>$ 18, the nearest SDSS galaxies with the redshift $z$ $<$ 0.004 and all SDSS galaxies with $\sigma$\[$I(4363)$\]/$I(4363)$ $>$ 0.25, totaling 443 SDSS galaxies from the comparison sample.
Our sample does not show any trend with absolute magnitude of either $C$(H$\beta$) or EW(H$\beta$), contrary to what was obtained by @Salzer05 for the KISS sample. The extinction in our sample galaxies is low. Only a few galaxies have $C$(H$\beta$) $>$ 0.4. The range of EW(H$\beta$) $\sim$ 0 – 300 $\AA$ for the galaxies from our sample is similar to that for the KISS sample [@Salzer05] but it is higher than that for the high-redshift galaxies of @Kobulniky2003 where EW(H$\beta$) $\leq$ 60 $\AA$.
The logarithm of the H$\beta$ luminosity log $L$(${\rm{H}\beta}$) of our galaxies ranges from 36 to 42 (Fig. \[fig4\]). For comparison, the galaxies from the KISS sample by @Salzer05 and intermediate-redshift galaxies by @Kobulniky2003 have log $L$(${\rm{H}\beta}$) $\sim$ 39 – 43 and 39 – 42, respectively, i.e. low-luminosity galaxies are lacking.
In Fig. \[fig5\] we show the oxygen abundance - absolute magnitude $M_g$ relation for the galaxies with oxygen abundances calculated mainly with the $T_{\rm e}$-method. In this Figure, the same samples and symbols as in Fig \[fig3\] are used. The region denoted as “branch” is populated mainly by galaxies with relatively high redshifts ($z$ $>$ 0.02) and oxygen abundances derived by the $T_{\rm e}$-method. Note that selection effects could be present for “branch” high-redshift galaxies which are predominantly distant spirals. In these galaxies we select mainly low metallicity 2 regions with a detectable \[O [iii]{}\]$\lambda$4363 line while the abundance gradient is present in spirals. The dotted line is a mean least-squares fit to all our data and the solid line is a mean least-squares fit to our data excluding “branch” galaxies with M$_g$ $<$ –18.4 and systems with an oxygen abundance in the range 8.0 – 8.3. The dashed line is a mean least-squares fit to the local dwarf irregular galaxies by @Skillman1989. Our sample (including the SDSS subsample) shows the familiar trend of increasing metallicity with increasing luminosity. A linear least square fit to all data yields the relation
$${\rm 12+log(O/H)} = (5.706\pm 0.199) - (0.134\pm 0.012) {\rm M}_{g}$$
(dotted line in Fig. \[fig5\]). Excluding ”branch” galaxies we obtain the relation $${\rm 12+log(O/H)} = (5.076\pm 0.320) - (0.174\pm 0.200) {\rm M}_{g}$$ (solid line in Fig. \[fig5\]). We note that the Skillman et al. and Richer & McCall fits do not extend over the metallicity range of the present data. Therefore, we extrapolate the former fit in Fig. \[fig5\] (dashed line) to higher metallicities. Skillman’s and our fits are obviosuly very similar. The slopes of our $L-Z$ relation of 0.134 (0.174) are very close to the slope of 0.153 by @Skillman1989 and to the slope of 0.147 by @RicherMcC1995.
Our sample is well populated in the low-luminosity range, while less than 10 galaxies from the KISS sample [@Salzer05] which were used for the study of the $L-Z$ relation are fainter than $M_B$ = –15, and none of them has an oxygen abundance less than 7.6. Our sample, excluding the SDSS subsample, has a lower dispersion around the dotted line compared to all our data and shows a shift to lower metallicities or/and higher luminosities. This likely can be attributed to our selection criteria which are optimized for the search for very metal-poor emission-line galaxies. Additionally, our sample galaxies display significant to strong ongoing SF giving rise to a large contribution from young stars and ionized gas to the total light of the galaxy. @P96 [see also @P02], using surface brightness profile decomposition to separate the star-forming component from the underlying host galaxy of BCDs, found that SF regions provide on average 50% of the total $B$-band emission within the 25 $B$ [mag/$\sq\arcsec$]{} isophote, with several examples of more intense starbursts whose flux contribution exceeds 70%. As a result, a shift of BCDs by a $\Delta\,M \sim$–0.75 mag with respect to the relatively quiescent dIrr population is to be expected in Fig. \[fig5\] (see also Fig. \[fig9\]). A similar offset to lower metallicities or/and higher luminosities has been found by @Kakazu07 for their intermediate-redshift low-metallicity emission-line galaxies with strong SF activity. The mass estimate of the galaxy is less sensitive to the presence of star-forming regions as compared to its luminosity. This was demonstrated by @Ellison2008 who found that galaxies in close pairs show enhanced SF activity as compared to a control sample of isolated galaxies. At the same time galaxies in close pairs show a smaller offset in the mass-metallicity relation as compared to the luminosity-metallicity relation. Thus, the offset in Fig. \[fig5\] indicates that both higher luminosities and lower metallicities may contribute to the shift in the luminosity-metallicity diagram of our sample galaxies relative to more quiescent dIrrs.
In Fig. \[fig6\] we demonstrate that the region of “branch” galaxies is populated mainly by relatively high-redshift systems. The sample is the same as in Fig. \[fig5\] but in the left panel only SDSS galaxies with oxygen abundances derived with the $T_{\rm e}$-method are shown and in the right panel only relatively high-redshift galaxies with $z$ $>$ 0.02 are selected.
The location of the galaxies on the luminosity-metallicity diagram is also sensitive to the method used for the abundance determination. In order to illustrate its effect on the observed $L-Z$ relation, we compare in Fig. \[fig7\] the oxygen abundance of SDSS sample galaxies (dots) obtained with the direct $T_{\rm e}$-method (left panel) and with the semiempirical strong-line method (right panel). The abundances for other galaxies in Fig. \[fig7\] are derived with the $T_{\rm e}$-method. In this Figure we show the larger control sample of the SDSS ($N$=7964) as compared to Fig. \[fig5\]. Only H [ii]{} regions in nearby spiral galaxies and from the nearest SDSS galaxies with redshifts $z$ $<$ 0.004 were excluded from the $\sim$ 9000 SDSS sources while faint galaxies with $m_g$ $>$ 18 are included. Symbols in Fig. \[fig7\] are the same as in Fig. \[fig3\] except for SDSS galaxies which are shown by dots. The dotted line is a mean least-squares fit to all our data from Fig. \[fig5\], while the solid line is a mean least-squares fit to the same data excluding “branch” galaxies.
It can be seen from Fig. \[fig7\] that the oxygen abundance of a given galaxy obtained by different methods could differ by $\sim$0.3–0.5 dex, especially for luminous galaxies. This figure illustrates clearly above 12 + log O/H $\sim$ 8.5 and $M_g$ $<$ –19 - –20 significant discrepancies between oxygen abundances obtained from the $T_{\rm e}$-method and empirical methods. @Stas2002 emphasized that, at high metallicity, the $T_{\rm e}$ derived from \[O [iii]{}\] $\lambda$4363 would largely overestimate the temperature of the O$^{++}$ zone (and largely underestimate the metallicity) because cooling is dominated by the \[O [iii]{}\] $\lambda$52 $\mu$m and \[O [iii]{}\] $\lambda$88 $\mu$m lines. At the same time @Pil2007 demonstrated that there is an observational limit of the highest possible metallicities near 12 + log O/H $\sim$ 8.95. This maximum value was determined in the centers of the most luminous (–22.3 $\la$ $M_B$ $\la$ –20.3) galaxies using the semiempirical ff-method [@Pil2006]. Thus, although the main mechanisms determining the electron temperature in H [ii]{} nebulae have been known for a long time, there are still important unsolved problems.
The contribution of star-forming regions to the light of the galaxy can be quantified by the equivalent width EW(H$\beta$) of the H$\beta$ emission line which in turn depends on the age of the burst of star formation. In Fig. \[fig8\] we show the same samples as in Fig. \[fig5\] except for the SDSS galaxies now being split into two subsamples. In the left panel only those with high equivalent widths EW(H$\beta$) $>$ 80$\AA$ are shown while in the right panel only SDSS galaxies with low equivalent widths EW(H$\beta$) $<$ 20$\AA$ are plotted. The dotted line in the left and right panels is a mean least-squares fit to all our data shown in Fig. \[fig5\], while the solid line is a mean least-squares fit to the same data excluding “branch” galaxies. There is a clear difference between the two subsamples of the SDSS galaxies by $\sim$ 0.4 dex in oxygen abundance or, equivalently, by $\sim$ 3 mag in absolute magnitude. SDSS galaxies with EW(H$\beta$) $>$ 80$\AA$ nicely follow the relation for our dwarf low-metallicity emission-line galaxies shown as reference objects by filled and open circles, stars, filled triangles and large filled circles. On the other hand, the SDSS galaxies with EW(H$\beta$) $<$ 20$\AA$ are located systematically above the low-metallicity galaxies. We propose two possible explanations for such a difference between the two subsamples of SDSS galaxies: 1) the emission of the SDSS galaxies with high EW(H$\beta$) is dominated by star-forming regions, therefore they have higher luminosities compared to galaxies in a relatively quiescent stage; 2) SDSS galaxies with low EW(H$\beta$) are the ones with higher astration level, therefore they are more chemically evolved systems with higher oxygen abundances. Perhaps both of these explanations are tenable, accounting for the observed differences between SDSS galaxies with high and low EW(H$\beta$). Thus, the lowest-metallicity SDSS galaxies are found predominantly among galaxies with high EW(H$\beta$). On the other hand, no extremely low-metallicity SDSS galaxies are found among systems with EW(H$\beta$) $<$ 20$\AA$. Thus, mixing of the SDSS galaxies with EW(H$\beta$) $<$ 20$\AA$ and $>$ 80$\AA$ results in a significant increase of the dispersion of the luminosity-metallicity diagram.
The redshift of the galaxy could also play a role. In Fig. \[fig5\] the bulk of the galaxies with 12+log(O/H)=8.0 – 8.3 and absolute magnitudes between –19 and –21 mag (denoted as “branch” galaxies) is represented by higher-redshift systems as compared to other galaxies from the SDSS and a correction for redshift is required. Since “branch” galaxies are blue, a correction for redshift for systems with weak emission lines would increase their brightness by $\sim$ 0.1 - 0.3 mag. This will not be enough to remove the offset between “branch” galaxies and lower-redshift galaxies in Fig. \[fig5\]. The situation is more complicated for “branch” galaxies with strong emission lines since their effect on the apparent magnitudes of a galaxy in standard passbands will significantly depend on redshift [see e.g. @Z08]. Because of these reasons, we decided not to take into account corrections for redshift.
Comparison of our sample with other data
----------------------------------------
In Fig. \[fig9\] we compare our $L-Z$ relation with other published data for galaxies of different types. In this Figure, all of our galaxies from Fig. \[fig5\], including those from the comparison SDSS sample, are shown by small filled circles. Some well known metal-poor galaxies are depicted by large filled circles and are labelled. Their absolute magnitudes $M_B$ are taken from @Kewley2007. For comparison, 23 KISS emission-line galaxies by @Lee2004 are displayed with large open double circles. The abundances for these galaxies are derived with the $T_{\rm e}$-method, the $B$-band magnitudes are from @Salzer1989 and @GildePaz2003. With open double squares we show 25 nearby dIrrs with the 4.5$\mu$m [*Spitzer*]{} luminosities and compiled O/H abundances derived with the $T_{\rm e}$-method [@Lee45mu2006]. With large open circles and large crosses we respectively show 20 irregular galaxies from @Skillman1989 and 21 dwarf irregular galaxies from @RicherMcC1995 for which oxygen abundances are obtained mainly with the $R_{23}$ empirical method, and for a few objects only with the $T_{\rm e}$-method. The thick solid line is a mean least-squares fit to all our data. The thin solid line is a least-squares fit to the data by @RicherMcC1995 while the dotted line is a mean least-squares fit to the data by @Skillman1989. The dashed line is the luminosity-metallicity relation for local metal-poor BCDs obtained by @KunthOstlin2000.
Data for intermediate- and high-redshift galaxies are also shown. The most distant ($z$ $<$ 1) extremely metal-poor galaxies (XMPGs) [@Kakazu07] with the oxygen abundances derived with the empirical method are shown with filled squares, while relatively metal-poor luminous galaxies at $z$ $\sim$ 0.7 [@Hoyos2005] (O/H derived with the $T_{\rm e}$-method) with filled triangles. The remaining samples in Fig. \[fig9\] are the following: a) the large open circles correspond to the $z$ = 3.36 lensed galaxy [@Villar2004] and to the average position of luminous Lyman-break galaxies at redshifts $z$ $\sim$ 2.5 [@KobKoo2000] (O/H derived with the $R_{23}$ method); b) small open circles stand for 66 Canada-France Redshift Survey (CFRS) galaxies by @Lilly2003 in the redshift range of $\sim$ 0.5 – 1.0 (O/H derived with the $R_{23}$ empirical method); c) asterisks are for 204 GOODS-N (Great Observatories Origins Deep Survey - North) emission-line galaxies in the range of redshifts 0.3 $<$ $z$ $<$ 1.0 [@KobKew04] (O/H is derived with the $R_{23}$ empirical method); d) small open rombs indicate 64 emission-line field galaxies from the Deep Extragalactic Evolutionary Probe Groth Strip Survey (DGSS) in the redshift range of $\sim$ 0.3 – 0.8 [@Kobulniky2003] (O/H derived with the $R_{23}$ empirical method); e) open squares are for the gamma-ray burst (GRB) hosts by @Kewley2007. Small open squares are for galaxies with O/H derived with the empirical method [@KewleyDopita2002] and large open squares for the galaxies with O/H derived with the $T_{\rm e}$-method [@Kewley2007]; f) filled stars denote the 14 star-forming emission-line galaxies at intermediate redshifts (0.11 $<$ $z$ $<$ 0.5) by @KobylZarit1999 (O/H derived with the empirical $R_{23}$ method); g) open filled triangles are for 29 distant 15$\mu$m-selected luminous infrared galaxies (LIRGs) at $z$ $\sim$ 0.3 – 0.8 taken from the sample of @Liang2004 (O/H derived with the empirical method); and, finally, h) the large dotted rectangle depicts the position of Lyman break galaxies [LBGs, @Pettini2001] on the $L-Z$ diagram.
The location of our galaxies on the luminosity-metallicity diagram is similar to that obtained previously for local emission-line galaxies but is shifted to higher luminosities and/or lower metallicities compared to that obtained for quiescent irregular dwarf galaxies. For comparison, @Lee2004 have also demonstrated that their 54 H [ii]{} KISS galaxies with O/H derived with the $T_{\rm e}$-method follow the $L-Z$ relation with a slope similar to that for a more quiescent dIrrs but are shifted to higher brightness by 0.8 magnitudes. Furthermore, they have shown that H [ii]{} galaxies with disturbed irregular outer isophotes (likely due to the interaction) are shifted to a more luminous and/or more metal-poor region in the $L-Z$ diagram as compared to morphologically more regular galaxies. Note that their samples of H [ii]{} galaxies and of dIrrs are in the same luminosity range as our sample. @Pap2008 also note that in contrast to the majority ($>$90%) of BCDs, the extremely metal-poor SF dwarfs reveal more irregular and bluer hosts.
Thus, the difference in the zero point between our $L-Z$ relation for low-metallicity galaxies and for other galaxies seems to be primarily due to the differences in the intrinsic properties of the galaxies selected for different samples with various selection criteria.
A key question is whether a unique $L-Z$ relation does exist for galaxies of different types. The assessment of this issue is complicated by offsets of high-redshift galaxies with different look-back-times. In this context, @Kobulniky2003 have shown that both the slopes and zero points of the $L-Z$ relation exhibit a smooth evolution with redshift. A possible universal $L-Z$ relation for galaxies is also blurred by the fact that metallicity determinations of various galaxy samples, differing in their EW(H$\beta$), absolute magnitude and redshift, do not employ a unique technique. More specifically, several authors emphasize the presence of a well-known shift between the O/H ratio obtained by the direct $T_{\rm e}$-method and empirical strong-line methods. Oxygen abundances obtained by empirical methods are by 0.1 –0.25 dex [@Shi2005] and even by up to 0.6 dex [@Hoyos2005] higher than those obtained with the $T_{\rm e}$-method. For our sample we obtained an offset of $\sim$0.3–0.5 dex.
It can be seen from Fig. \[fig9\] that the high-redshift galaxies with an oxygen abundance derived by the $T_{\rm e}$-method have a shallower slope compared to local galaxies. On the other hand, oxygen abundances of high-redshift galaxies obtained with the $R_{23}$ empirical strong-line method [data in Fig. \[fig9\] by @Lilly2003; @KobKew04; @Liang2004] are higher and follow the relation for high-metallicity SDSS galaxies in Fig. \[fig7\]b despite the fact that oxygen abundances for the latter galaxies were calculated with the different semi-empirical strong-line method. Because of this agreement we decided not to re-calculate oxygen abundances of high-redshift galaxies with the semi-empirical method and adopted O/H values from the literature. Keeping in mind the systematic differences between oxygen abundances derived with the empirical and the $T_{\rm e}$-methods, it might be worth considering a decrease in oxygen abundance by $\sim$ 0.2 – 0.6 dex for all high-redshift galaxies with O/H derived with the empirical method. In that case, the position of high-redshift galaxies on the $L-Z$ diagram would be consistent with that of the “branch” galaxies. Such considerations add further support to the results obtained by @Pil2004 and @Shi2005 that the more luminous galaxies have a slope of the $L-Z$ relation more shallow than that of the dwarf galaxies.
We presume that our $L-Z$ relation could be useful as a local reference for studies of this relation for other types of local galaxies and/or of high-redshift galaxies.
Summary
=======
We present VLT spectroscopic observations of a new sample of 28 H [ii]{} regions from 16 emission-line galaxies and ESO 3.6m telescope spectroscopic observations of a new sample of 38 H [ii]{} regions from 28 emission-line galaxies. These galaxies have mainly been selected from the Data Release 6 (DR6) of the Sloan Digital Sky Survey (SDSS) as low-metallicity galaxy candidates.
Physical conditions and element abundances are derived with the $T_{\rm e}$-method for 38 H [ii]{} regions observed with the 3.6m telescope and for 23 H [ii]{} regions observed with the VLT.
From our new observations we find that the oxygen abundance in 61 out of the 66 observed H [ii]{} in our sample ranges from 12 + log O/H = 7.05 to 8.22. The oxygen abundance in 27 H [ii]{} regions is 12 + log O/H $<$ 7.6 and among them 10 H [ii]{} regions have an oxygen abundance less than 7.3.
This new data in combination with objects from our previous studies constitute a large uniform sample of 154 H [ii]{} regions with high-quality spectroscopic data which are used to study the luminosity-metallicity ($L-Z$) relation for the local galaxies with emphasis on its low-metallicity end.
As a comparison sample we use $\sim$ 9000 SDSS emission-line galaxies with higher oxygen abundances which are also obtained mainly by the direct $T_{\rm e}$-method. For all of our sample galaxies the $g$ magnitudes are taken from the SDSS while the distances are from the NED. The entire sample spans nearly two orders of magnitude with respect to its gas-phase metallicity, from 12 + log O/H $\sim$ 7.0 to $\sim$ 8.8, and covers an absolute magnitude range from $M_g$ $\sim$ –12 to $\sim$ –20.
We find that the metallicity-luminosity relation for our galaxies is consistent with previous ones obtained for objects of similar type. The local $L-Z$ relation obtained with high-quality spectroscopic data is useful for predictions of galaxy evolution models.
N. G. G. and Y. I. I. thank the Max Planck Institute for Radioastronomy (MPIfR) for hospitality, and acknowledge support through DFG grant No. Fr 325/57-1. P. P. thanks the Department of Astronomy and Space Physics at Uppsala University for its warm hospitality. K. J. Fricke thanks the MPIfR for Visiting Contracts during 2008 and 2009. This research was partially funded by project grant AYA2007-67965-C03-02 of the Spanish Ministerio de Ciencia e Innovacion. We acknowledge the work of the Sloan Digital Sky Survey (SDSS) team. Funding for the SDSS has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the US Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http:// www.sdss.org/.
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[lcclcccc]{}
Galaxy & R.A., Dec. & Date & Exp. time & Airmass & seeing & redshift\
& (J2000) &(2007) & (sec) & & &\
J0015+0104 & [ 00$^{\rm h}$15$^{\rm m}$20$^{\rm s}$.7 ]{}, [$01$$^{\circ}$04$^{\rm '}$37$^{\rm ''}$]{}& 14 Sep & 4$\,\times\,$900 & 1.207 & 174 & 0.00689 $\pm$ 0.00021\
J0016+0108 & 00 16 28. 3, 01 08 02 & 15 Sep & 4$\,\times\,$675 & 1.306 & 190 & 0.01035 $\pm$ 0.00006\
J0029-0108 & 00 29 04. 7, –01 08 26 & 16 Sep & 4$\,\times\,$450 & 1.614 & 188 & 0.01313 $\pm$ 0.00006\
J0029-0025 & 00 29 49. 5, –00 25 40 & 14 Sep & 4$\,\times\,$900 & 1.142 & 172 & 0.01440 $\pm$ 0.00013\
0057-0022 & 00 57 12. 6, –00 21 58 & 16 Sep & 4$\,\times\,$900 & 1.525 & 147 & 0.00956 $\pm$ 0.00023\
J0107+0001 & 01 07 50. 8, 00 01 28 & 16 Sep & 3$\,\times\,$267 & 1.188 & 141 & 0.01835 $\pm$ 0.00012\
J0109+0107 & 01 09 08. 0, 01 07 16 & 15 Sep & 4$\,\times\,$900 & 1.174 & 154 & 0.00396 $\pm$ 0.00013\
J0126-0038 No.1 & 01 26 46. 1, –00 38 39 & 16 Sep & 4$\,\times\,$900 & 1.332 & 136 & 0.00632 $\pm$ 0.00010\
J0126-0038 No.2 & 01 26 46. 1, –00 38 39 & 16 Sep & 4$\,\times\,$900 & 1.332 & 136 & 0.00642 $\pm$ 0.00016\
J0135-0023 & 01 35 44. 0, –00 23 17 & 14 Sep & 4$\,\times\,$900 & 1.142 & 233 & 0.01708 $\pm$ 0.00006\
J0213-0002 No.1 & 02 13 57. 7, –00 02 56 & 16 Sep & 4$\,\times\,$900 & 1.252 & 152 & 0.03640 $\pm$ 0.00008\
J0213-0002 No.2 & 02 13 57. 7, –00 02 56 & 16 Sep & 4$\,\times\,$900 & 1.252 & 152 & 0.03635 $\pm$ 0.00011\
J0216+0115 No.1 & 02 16 29. 3, –01 15 21 & 15 Sep & 4$\,\times\,$900 & 1.223 & 202 & 0.00939 $\pm$ 0.00004\
J0216+0115 No.2 & 02 16 29. 3, 01 15 21 & 15 Sep & 4$\,\times\,$900 & 1.223 & 202 & 0.00940 $\pm$ 0.00004\
096632 & 02 51 47. 5, –30 06 32 & 15 Sep & 4$\,\times\,$900 & 1.400 & 228 & 0.00354 $\pm$ 0.00009\
J0252+0017 & 02 52 16. 8, 00 17 41 & 16 Sep & 3$\,\times\,$800 & 1.165 & 138 & 0.00527 $\pm$ 0.00011\
J0256+0036 & 02 56 28. 3, 00 36 28 & 14 Sep & 4$\,\times\,$900 & 1.150 & 247 & 0.00919 $\pm$ 0.00013\
J0303-0109 No.1 & 03 03 31. 3, –01 09 47 & 14 Sep & 2$\,\times\,$800 & 1.170 & 243 & 0.03055 $\pm$ 0.00011\
J0303-0109 No.2 & 03 03 31. 3, –01 09 47 & 14 Sep & 2$\,\times\,$800 & 1.170 & 243 & 0.03039 $\pm$ 0.00017\
J0341-0026 No.1 & 03 41 18. 1, -00 26 28& 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03080 $\pm$ 0.00020\
J0341-0026 No.2 & 03 41 18. 1, –00 26 28 & 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03045 $\pm$ 0.00008\
J0341-0026 No.3 & 03 41 18. 1, –00 26 28 & 16 Sep & 3$\,\times\,$800 & 1.198 & 146 & 0.03047 $\pm$ 0.00008\
G1815456-670126 & 18 15 46. 5, –67 01 23 & 14 Sep & 4$\,\times\,$900 & 1.280 & 200 & 0.01131 $\pm$ 0.00008\
G2052078-691229 No.1 & 20 52 07. 1, –69 12 30 & 16 Sep & 3$\,\times\,$800 & 1.328 & 150 & 0.00212 $\pm$ 0.00008\
G2052078-691229 No.2 & 20 52 07. 1, –69 12 30 & 16 Sep & 3$\,\times\,$800 & 1.328 & 150 & 0.00203 $\pm$ 0.00011\
J2053+0039 & 20 53 12. 6, 00 39 15 & 15 Sep & 4$\,\times\,$525 & 1.219 & 155 & 0.01328 $\pm$ 0.00007\
J2105+0032 No.1 & 21 05 08. 6, 00 32 23 & 14 Sep & 3$\,\times\,$800 & 1.408 & 145 & 0.01431 $\pm$ 0.00003\
J2105+0032 No.2 & 21 05 08. 6, 00 32 23 & 14 Sep & 3$\,\times\,$800 & 1.408 & 145 & 0.01436 $\pm$ 0.00013\
J2112-0016 No.1 & 21 12 00. 8, –00 16 49 & 15 Sep & 3$\,\times\,$800 & 1.441 & 132 & 0.01195 $\pm$ 0.00015\
J2112-0016 No.2 & 21 12 00. 8, –00 16 49 & 15 Sep & 3$\,\times\,$800 & 1.441 & 132 & 0.01215 $\pm$ 0.00021\
J2119-0732 & 21 19 42. 4, –07 32 24 & 14 Sep & 3$\,\times\,$800 & 1.130 & 178 & 0.00966 $\pm$ 0.00006\
J2120-0058 & 21 20 25. 9, –00 58 27 & 15 Sep & 4$\,\times\,$900 & 1.172 & 154 & 0.01979 $\pm$ 0.00004\
J2150+0033 & 21 50 32. 0, 00 33 05 & 15 Sep & 3$\,\times\,$800 & 1.155 & 232 & 0.01508 $\pm$ 0.00003\
G2155572-394614 & 21 55 57. 9, –39 46 14 & 15 Sep & 3$\,\times\,$900 & 1.241 & 190 & 0.00740 $\pm$ 0.00008\
J2227-0939 & 22 27 30. 7, –09 39 54 & 14 Sep & 3$\,\times\,$800 & 1.060 & 214 & 0.00528 $\pm$ 0.00007\
PHL 293B & 22 30 36. 8, –00 06 37 & 14 Sep & 4$\,\times\,$900 & 1.230 & 135 & 0.00537 $\pm$ 0.00004\
J2310-0109 No.1 & 23 10 42. 0, –01 09 48 & 16 Sep & 3$\,\times\,$800 & 1.471 & 145 & 0.01254 $\pm$ 0.00008\
J2310-0109 No.2 & 23 10 42. 0, –01 09 48 & 16 Sep & 3$\,\times\,$800 & 1.471 & 145 & 0.01232 $\pm$ 0.00009\
[lcclcccc]{}
Galaxy & R.A., Dec. & Date & Exp. time & Airmass & seeing & redshift\
& (J2000) & & (sec) & & &\
J0004+0025 No.1 & [ 00$^{\rm h}$04$^{\rm m}$21$^{\rm s}$.6 ]{}, [$00$$^{\circ}$25$^{\rm '}$36$^{\rm ''}$]{} & 20.10.2006& 2$\,\times\,$750& 1.246&060 & 0.01269 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.181 & 037 & 0.01269 $\pm$ 0.00004\
J0004+0025 No.2 & 00 04 21. 6, 00 25 36 & 20.10.2006 &2$\,\times\,$750 & 1.246 & 060 & 0.01266 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.181 & 037 & 0.01266 $\pm$ 0.00003\
J0014-0044 No.1 & 00 14 28. 8, –00 44 44 & 19.11.2006 &2$\,\times\,$750 & 1.158 & 122 & 0.01361 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.217 & 093 & 0.01361 $\pm$ 0.00004\
J0014-0044 No.2 & 00 14 28. 8, –00 44 44 & 19.11.2006 &2$\,\times\,$750 & 1.158 & 122 & 0.01379 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.217 & 093 & 0.01379 $\pm$ 0.00002\
J0202-0047 & 02 02 38. 0, –00 47 44 & 13.01.2007 &2$\,\times\,$750 & 1.211 & 185 & 0.03371 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.294 & 163 & 0.03371 $\pm$ 0.00003\
J0301-0059 No.1 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03841 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03841 $\pm$ 0.00000\
J0301-0059 No.2 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03822 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03822 $\pm$ 0.00002\
J0301-0059 No.3 & 03 01 26. 3, –00 59 26 & 13.12.2006 &2$\,\times\,$750 & 1.140 & 146 & 0.03807 $\pm$ 0.00008\
& & &2$\,\times\,$540 & 1.191 & 138 & 0.03807 $\pm$ 0.00008\
J0315-0024 No.1 & 03 15 59. 9, –00 24 26 & 26.11.2006 &2$\,\times\,$1500 & 1.145 & 132 & 0.02247 $\pm$ 0.00005\
& & &2$\,\times\,$1080 & 1.197 & 108 & 0.02247 $\pm$ 0.00005\
J0315-0024 No.2 & 03 15 59. 9, –00 24 26 & 26.11.2006 &2$\,\times\,$1500 & 1.145 & 132 & 0.02261 $\pm$ 0.00006\
& & &2$\,\times\,$1080 & 1.197 & 108 & 0.02261 $\pm$ 0.00006\
J0338+0013 (BG) & 03 38 12. 2, 00 13 13 & 25.11.2006 &2$\,\times\,$750 & 1.207 & 076 & 0.39695 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.284 & 070 & 0.39695 $\pm$ 0.00000\
J0338+0013 & 03 38 12. 2, 00 13 13 & 25.11.2006 &2$\,\times\,$750 & 1.207 & 076 & 0.04266 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.284 & 070 & 0.04266 $\pm$ 0.00002\
G0405204-364859 No.1 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00281 $\pm$ 0.00004\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00281 $\pm$ 0.00004\
G0405204-364859 No.2 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00275 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00275 $\pm$ 0.00003\
G0405204-364859 No.3 & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.00276 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.00276 $\pm$ 0.00003\
G0405204-364859 (BG) & 04 05 18. 6, –36 48 49 & 15.11.2006 &2$\,\times\,$750 & 1.033 & 111 & 0.17236 $\pm$ 0.00000\
& & &2$\,\times\,$540 & 1.054 & 100 & 0.17236 $\pm$ 0.00000\
J0519+0007 & 05 19 02. 7, 00 07 29 & 16.11.2006 &2$\,\times\,$750 & 1.220 & 103 & 0.04438 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.166 & 127 & 0.04438 $\pm$ 0.00002\
J2104-0035 No.1 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00469 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00469 $\pm$ 0.00002\
J2104-0035 No.2 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00469 $\pm$ 0.00001\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00469 $\pm$ 0.00001\
J2104-0035 No.3 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00471 $\pm$ 0.00005\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00471 $\pm$ 0.00005\
J2104-0035 No.4 & 21 04 55. 3, –00 35 22 & 12.10.2006 &2$\,\times\,$750 & 1.099 & 148 & 0.00472 $\pm$ 0.00003\
& & &2$\,\times\,$540 & 1.096 & 140 & 0.00472 $\pm$ 0.00003\
J2302+0049 No.1 & 23 02 10. 0, 00 49 39 & 12.10.2006 &2$\,\times\,$750 & 1.170 & 118 & 0.03312 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.132 & 123 & 0.03312 $\pm$ 0.00002\
J2302+0049 No.2 & 23 02 10. 0, 00 49 39 & 12.10.2006 &2$\,\times\,$750 & 1.170 & 118 & 0.03311 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.132 & 123 & 0.03311 $\pm$ 0.00002\
J2324-0006 & 23 24 21. 3, –00 06 29 & 12.10.2006 &2$\,\times\,$750 & 1.102 & 155 & 0.00896 $\pm$ 0.00002\
& & &2$\,\times\,$540 & 1.119 & 153 & 0.00896 $\pm$ 0.00002\
J2354-0005 No.1 & 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.00771 $\pm$ 0.00003\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.00771 $\pm$ 0.00003\
J2354-0005 No.2 & 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.00798 $\pm$ 0.00004\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.00798 $\pm$ 0.00004\
J2354-0005 (BG1)& 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.16534 $\pm$ 0.00000\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.16534 $\pm$ 0.00000\
J2354-0005 (BG2)& 23 54 37. 3, –00 05 02 & 15.10.2006 &2$\,\times\,$1500 & 1.502 & 074 & 0.16520 $\pm$ 0.00000\
& & &2$\,\times\,$1080 & 1.362 & 077 & 0.16520 $\pm$ 0.00000\
$^a$first line for each galaxy is related to the observation in the blue range and second line to the one in the red range.
[lrrrrrrrr]{}
\
\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&\
& [J0015$+$0104 ]{}&[J0016$+$0108 ]{}& [J0029$-$0108 ]{}&[J0029$-$0025 ]{}& [J0057$-$0022 ]{}&[J0107$+$0001 ]{}& [J0109$+$0107 ]{}&[J0126$-$0038 ]{}\
& & & & & & & & No.1\
3727 \[O [ii]{}\] & 118.7 $\pm$ 5.7 & 221.2 $\pm$ 7.8 & 171.6 $\pm$ 30.8 & 130.3 $\pm$ 7.8 & 219.7 $\pm$ 5.4 & 162.3 $\pm$ 13.6 & 209.3 $\pm$ 4.3 & 238.1 $\pm$ 4.1\
3868 \[Ne [iii]{}\] & ... & ... & ... & ... & 31.0 $\pm$ 2.0 & ... & 30.1 $\pm$ 1.0 & 38.5 $\pm$ 0.9\
3889 He [i]{} + H8 & ... & ... & ... & ... & 17.3 $\pm$ 1.8 & ... & 20.6 $\pm$ 1.6 & 17.2 $\pm$ 0.9\
3968 \[Ne [iii]{}\] + H7 & ... & ... & ... & ... & 16.1 $\pm$ 1.9 & ... & 25.6 $\pm$ 1.5 & 22.2 $\pm$ 0.9\
4101 H$\delta$ & ... & ... & ... & ... & 24.9 $\pm$ 1.9 & ... & 24.9 $\pm$ 1.3 & 24.8 $\pm$ 0.9\
4340 H$\gamma$ & 49.3 $\pm$ 3.5 & 47.0 $\pm$ 3.4 & ... & 43.9 $\pm$ 4.8 & 47.2 $\pm$ 1.9 & ... & 46.9 $\pm$ 1.4 & 46.9 $\pm$ 1.0\
4363 \[O [iii]{}\] & ... & ... & ... & ... & 5.1 $\pm$ 0.1 & ... & 3.7 $\pm$ 0.4 & 6.9 $\pm$ 0.4\
4471 He [i]{} & ... & ... & ... & ... & ... & ... & 2.6 $\pm$ 0.4 & 2.4 $\pm$ 0.3\
4658 \[Fe [iii]{}\] & ... & ... & ... & ... & ... & ... & ... & 1.5 $\pm$ 0.3\
4686 He [ii]{} & ... & ... & ... & ... & ... & ... & ... & 1.7 $\pm$ 0.3\
4861 H$\beta$ & 100.0 $\pm$ 4.9 & 100.0 $\pm$ 4.2 & 100.0 $\pm$ 15.0 & 100.0 $\pm$ 6.2 & 100.0 $\pm$ 2.6 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 2.1 & 100.0 $\pm$ 1.7\
4959 \[O [iii]{}\] & 20.1 $\pm$ 2.0 & 61.1 $\pm$ 2.8 & 48.1 $\pm$ 8.8 & 38.8 $\pm$ 3.2 & 76.1 $\pm$ 2.0 & 31.0 $\pm$ 3.3 & 102.5 $\pm$ 2.1 & 139.7 $\pm$ 2.3\
5007 \[O [iii]{}\] & 57.8 $\pm$ 2.8 & 190.6 $\pm$ 6.1 & 144.3 $\pm$ 22.7 & 168.6 $\pm$ 7.9 & 219.4 $\pm$ 4.8 & 93.1 $\pm$ 7.9 & 305.1 $\pm$ 5.6 & 409.4 $\pm$ 6.4\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.245 ]{}& [ 0.000 ]{}& [ 0.120 ]{}& [ 0.410 ]{}\
$F$(H$\beta$)$^b$ & [ 2.77 ]{}& [ 4.66 ]{}& [ 1.19 ]{}& [ 2.16 ]{}& [ 12.99 ]{}& [ 2.88 ]{}& [ 18.18 ]{}& [ 75.47 ]{}\
EW(H$\beta$) ($\AA$) & [ 57.5 ]{}& [ 20.4 ]{}& [ 6.8 ]{}& [ 53.8 ]{}& [ 44.6 ]{}& [ 87.7 ]{}& [ 50.1 ]{}& [ 48.7 ]{}\
EW(abs) ($\AA$)$^a$ & [ 0.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 0.25 ]{}& [ 2.00 ]{}& [ 2.15 ]{}& [ 0.70 ]{}\
&\
& [J0126$-$0038 ]{}&[J0135$-$0023 ]{}& [J0213$-$0002 ]{}&[J0213$-$0002 ]{}& [J0216$+$0115 ]{}&[J0216$+$0115 ]{}& [096632 ]{}&[J0252$+$0017 ]{}\
&No.2 & &No.1 &No.2 &No.1 &No.2 & &\
3727 \[O [ii]{}\] & 224.9 $\pm$ 6.3 & 177.9 $\pm$ 10.0 & 182.2 $\pm$ 9.8 & 291.0 $\pm$ 16.6 & 225.2 $\pm$ 8.7 & 236.5 $\pm$ 12.4 & 306.5 $\pm$ 6.9 & 257.0 $\pm$ 26.1\
3868 \[Ne [iii]{}\] & ... & ... & ... & ... & ... & ... & 8.4 $\pm$ 0.9 & ... \
4101 H$\delta$ & 24.8 $\pm$ 2.2 & ... & ... & ... & ... & ... & 29.7 $\pm$ 1.7 & ... \
4340 H$\gamma$ & 44.8 $\pm$ 2.2 & 44.0 $\pm$ 5.8 & 47.1 $\pm$ 5.0 & 47.3 $\pm$ 7.3 & 49.3 $\pm$ 3.9 & ... & 48.1 $\pm$ 1.8 & ... \
4861 H$\beta$ & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 6.7 & 100.0 $\pm$ 6.6 & 100.0 $\pm$ 7.6 & 100.0 $\pm$ 4.6 & 100.0 $\pm$ 10.0 & 100.0 $\pm$ 2.4 & 100.0 $\pm$ 10.0\
4959 \[O [iii]{}\] & 59.3 $\pm$ 1.9 & 43.5 $\pm$ 3.5 & 97.0 $\pm$ 5.8 & 115.6 $\pm$ 7.2 & 83.8 $\pm$ 3.8 & 42.1 $\pm$ 3.9 & 35.2 $\pm$ 1.2 & 71.7 $\pm$ 8.5\
5007 \[O [iii]{}\] & 170.7 $\pm$ 4.3 & 155.9 $\pm$ 7.9 & 279.2 $\pm$ 13.1 & 332.9 $\pm$ 16.7 & 245.6 $\pm$ 8.3 & 126.9 $\pm$ 6.7 & 100.3 $\pm$ 2.3 & 270.5 $\pm$ 24.1\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}\
$F$(H$\beta$)$^b$ & [ 7.47 ]{}& [ 2.87 ]{}& [ 2.54 ]{}& [ 1.73 ]{}& [ 3.11 ]{}& [ 2.26 ]{}& [ 10.20 ]{}& [ 1.69 ]{}\
EW(H$\beta$) ($\AA$) & [ 45.6 ]{}& [ 14.0 ]{}& [ 52.1 ]{}& [ 36.7 ]{}& [ 17.2 ]{}& [ 24.3 ]{}& [ 48.3 ]{}& [ 11.1 ]{}\
EW(abs) ($\AA$)$^a$ & [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}\
&\
& [J0256$+$0036 ]{}&[J0303$-$0109 ]{}& [J0303$-$0109 ]{}&[J0341$-$0026 ]{}& [J0341$-$0026 ]{}&[J0341$-$0026 ]{}& [G1815-6701 ]{}&[G2052-6912 ]{}\
& &[ No.1 ]{}& [ No.2]{}&[ No.1 ]{}& [ No.2]{}&[ No.3 ]{}& &[ No.1 ]{}\
3727 \[O [ii]{}\] & 190.3 $\pm$ 5.1 & 137.9 $\pm$ 4.5 & 170.4 $\pm$ 22.5 & 227.5 $\pm$ 13.6 & 136.4 $\pm$ 5.9 & 87.5 $\pm$ 5.0 & 263.8 $\pm$ 4.4 & 137.6 $\pm$ 2.1\
3750 H12 & ... & ... & ... & ... & ... & ... & ... & 3.3 $\pm$ 0.2\
3771 H11 & ... & ... & ... & ... & ... & ... & ... & 3.8 $\pm$ 0.2\
3798 H10 & ... & ... & ... & ... & ... & ... & ... & 5.1 $\pm$ 0.2\
3835 H9 & ... & ... & ... & ... & ... & ... & ... & 7.1 $\pm$ 0.2\
3868 \[Ne [iii]{}\] & 31.7 $\pm$ 1.7 & 42.9 $\pm$ 2.2 & ... & ... & ... & ... & 52.3 $\pm$ 1.1 & 37.2 $\pm$ 0.6\
3889 He [i]{} + H8 & ... & 30.6 $\pm$ 2.8 & ... & ... & ... & ... & 17.5 $\pm$ 0.8 & 19.3 $\pm$ 0.4\
3968 \[Ne [iii]{}\] + H7 & ... & 36.4 $\pm$ 3.0 & ... & ... & ... & ... & 25.7 $\pm$ 0.8 & 27.2 $\pm$ 0.5\
4026 He [i]{} & ... & ... & ... & ... & ... & ... & ... & 1.5 $\pm$ 0.1\
4068 \[S [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 1.3 $\pm$ 0.1\
4101 H$\delta$ & ... & 33.8 $\pm$ 2.7 & ... & ... & ... & ... & 23.7 $\pm$ 0.8 & 25.9 $\pm$ 0.4\
4340 H$\gamma$ & 47.8 $\pm$ 2.2 & 47.0 $\pm$ 2.5 & ... & ... & 45.4 $\pm$ 4.4 & 50.0 $\pm$ 4.7 & 46.9 $\pm$ 1.0 & 46.6 $\pm$ 0.7\
4363 \[O [iii]{}\] & ... & 8.2 $\pm$ 1.0 & ... & ... & ... & ... & 8.7 $\pm$ 0.4 & 4.1 $\pm$ 0.1\
4471 He [i]{} & ... & ... & ... & ... & ... & ... & 3.5 $\pm$ 0.4 & 3.9 $\pm$ 0.1\
4658 \[Fe [iii]{}\] & ... & ... & ... & ... & ... & ... & 1.7 $\pm$ 0.3 & 0.3 $\pm$ 0.1\
4686 He [ii]{} & ... & ... & ... & ... & ... & ... & 1.6 $\pm$ 0.4 & 0.3 $\pm$ 0.1\
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & ... & ... & ... & ... & 0.7 $\pm$ 0.1\
4861 H$\beta$ & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 3.0 & 100.0 $\pm$ 13.0 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 4.7 & 100.0 $\pm$ 5.2 & 100.0 $\pm$ 1.7 & 100.0 $\pm$ 1.5\
4921 He [i]{} & ... & ... & ... & 18.9 $\pm$ 2.8 & ... & ... & ... & 0.9 $\pm$ 0.1\
4959 \[O [iii]{}\] & 110.1 $\pm$ 3.0 & 130.6 $\pm$ 3.5 & 26.6 $\pm$ 5.1 & 68.0 $\pm$ 4.9 & 77.0 $\pm$ 3.5 & 65.3 $\pm$ 3.3 & 171.5 $\pm$ 2.7 & 160.5 $\pm$ 2.3\
5007 \[O [iii]{}\] & 316.2 $\pm$ 7.3 & 368.4 $\pm$ 8.9 & 79.9 $\pm$ 11.6 & 253.6 $\pm$ 13.9 & 243.1 $\pm$ 8.4 & 186.1 $\pm$ 7.6 & 508.6 $\pm$ 7.9 & 481.2 $\pm$ 7.0\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.515 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.575 ]{}& [ 0.390 ]{}\
$F$(H$\beta$)$^b$ & [ 9.25 ]{}& [ 11.86 ]{}& [ 1.71 ]{}& [ 4.49 ]{}& [ 4.49 ]{}& [ 3.54 ]{}& [ 66.43 ]{}& [448.40 ]{}\
EW(H$\beta$) ($\AA$) & [ 26.2 ]{}& [ 95.0 ]{}& [ 21.2 ]{}& [ 27.5 ]{}& [ 85.2 ]{}& [ 145.8 ]{}& [ 35.9 ]{}& [ 221.4 ]{}\
EW(abs) ($\AA$)$^a$ & [ 0.00 ]{}& [ 1.50 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.40 ]{}& [ 0.75 ]{}\
&\
& [G2052-6912]{}&[J2053+0039 ]{}& [J2105+0032 ]{}&[J2105+0032 ]{}& [J2112-0016 ]{}&[J2112-0016 ]{}& [J2119-0732 ]{}&[J2120-0058 ]{}\
& [No.2]{}&[ ]{}& [ No.1]{}&[ No.2 ]{}& [ No.1]{}&[ No.2 ]{}& &[ ]{}\
3727 \[O [ii]{}\] & 104.5 $\pm$ 1.6 & 85.8 $\pm$ 8.2 & 109.0 $\pm$ 8.3 & 187.0 $\pm$ 10.8 & 106.6 $\pm$ 2.1 & 107.6 $\pm$ 4.6 & 161.7 $\pm$ 4.1 & 252.5 $\pm$ 5.3\
3750 H12 & 3.5 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3771 H11 & 4.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3798 H10 & 5.2 $\pm$ 0.2 & ... & ... & ... & ... & ... & ... & ... \
3820 He [i]{} & 1.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
3835 H9 & 7.2 $\pm$ 0.2 & ... & ... & ... & ... & ... & ... & ... \
3868 \[Ne [iii]{}\] & 47.9 $\pm$ 0.7 & ... & ... & ... & 42.7 $\pm$ 1.0 & 25.7 $\pm$ 2.0 & 38.5 $\pm$ 1.7 & 39.9 $\pm$ 1.4\
3889 He [i]{} + H8 & 18.1 $\pm$ 0.3 & ... & ... & ... & 18.4 $\pm$ 0.8 & 18.0 $\pm$ 3.1 & 24.5 $\pm$ 2.0 & 22.5 $\pm$ 1.7\
3968 \[Ne [iii]{}\] + H7 & 30.9 $\pm$ 0.5 & ... & ... & ... & 27.9 $\pm$ 1.0 & 37.0 $\pm$ 3.3 & ... & 20.4 $\pm$ 1.7\
4026 He [i]{} & 1.7 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4068 \[S [ii]{}\] & 1.0 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4101 H$\delta$ & 25.9 $\pm$ 0.4 & ... & ... & ... & 25.8 $\pm$ 1.0 & 41.1 $\pm$ 2.4 & 26.8 $\pm$ 1.8 & 27.5 $\pm$ 1.5\
4340 H$\gamma$ & 46.7 $\pm$ 0.7 & ... & ... & ... & 48.7 $\pm$ 1.1 & 46.9 $\pm$ 3.0 & 46.7 $\pm$ 1.8 & 47.2 $\pm$ 1.6\
4363 \[O [iii]{}\] & 6.7 $\pm$ 0.1 & ... & ... & ... & 5.3 $\pm$ 0.4 & 5.5 $\pm$ 0.9 & ... & 7.8 $\pm$ 0.8\
4387 He [i]{} & 0.6 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4471 He [i]{} & 4.1 $\pm$ 0.1 & ... & ... & ... & 3.7 $\pm$ 0.3 & ... & ... & ... \
4658 \[Fe [iii]{}\] & 0.2 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & 1.4 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4740 \[Ar [iv]{}\] & 0.6 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4861 H$\beta$ & 100.0 $\pm$ 1.5 & 100.0 $\pm$ 9.0 & 100.0 $\pm$ 8.0 & 100.0 $\pm$ 6.0 & 100.0 $\pm$ 1.8 & 100.0 $\pm$ 3.5 & 100.0 $\pm$ 2.6 & 100.0 $\pm$ 2.2\
4921 He [i]{} & 1.1 $\pm$ 0.1 & ... & ... & ... & ... & ... & ... & ... \
4959 \[O [iii]{}\] & 207.9 $\pm$ 3.0 & 75.2 $\pm$ 7.1 & 71.2 $\pm$ 5.3 & 53.4 $\pm$ 4.0 & 189.9 $\pm$ 3.2 & 152.6 $\pm$ 4.7 & 137.3 $\pm$ 3.1 & 102.4 $\pm$ 2.1\
5007 \[O [iii]{}\] & 623.9 $\pm$ 9.0 & 216.5 $\pm$ 14.3 & 263.8 $\pm$ 14.4 & 165.6 $\pm$ 8.1 & 565.0 $\pm$ 9.1 & 454.1 $\pm$ 12.5 & 413.1 $\pm$ 8.6 & 297.6 $\pm$ 5.6\
\
$C$(H$\beta$)$^a$ & [ 0.495 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.245 ]{}& [ 0.095 ]{}& [ 0.415 ]{}\
$F$(H$\beta$)$^b$ & [766.20 ]{}& [ 1.58 ]{}& [ 2.57 ]{}& [ 2.36 ]{}& [ 49.75 ]{}& [ 7.37 ]{}& [ 16.82 ]{}& [ 15.15 ]{}\
EW(H$\beta$) ($\AA$) & [ 395.9 ]{}& [ 24.4 ]{}& [ 18.2 ]{}& [ 121.7 ]{}& [ 95.2 ]{}& [ 176.1 ]{}& [ 52.8 ]{}& [ 40.1 ]{}\
EW(abs) ($\AA$)$^a$ & [ 1.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 0.00 ]{}& [ 4.15 ]{}& [ 2.00 ]{}& [ 0.90 ]{}\
&\
& [J2150+0033 ]{}&[G2155-3946 ]{}& [J2227-0939 ]{}&[PHL 293B ]{}& [J2310-0109 ]{}&[J2310-0109 ]{}\
& &[ ]{}& [ ]{}&[ ]{}& [ No.1]{}&[ No.2 ]{}& [ ]{}&[ ]{}\
3727 \[O [ii]{}\] & 190.0 $\pm$ 7.3 & 323.4 $\pm$ 10.2 & 197.0 $\pm$ 7.0 & 65.7 $\pm$ 1.1 & 128.0 $\pm$ 4.0 & 177.4 $\pm$ 6.0\
3750 H12 & ... & ... & ... & 5.0 $\pm$ 0.4 & ... & ... \
3771 H11 & ... & ... & ... & 6.4 $\pm$ 0.4 & ... & ... \
3798 H10 & ... & ... & ... & 8.2 $\pm$ 0.4 & ... & ... \
3835 H9 & ... & ... & ... & 8.4 $\pm$ 0.4 & ... & ... \
3868 \[Ne [iii]{}\] & ... & 16.6 $\pm$ 1.9 & 35.0 $\pm$ 3.0 & 41.0 $\pm$ 0.7 & 20.9 $\pm$ 1.4 & 27.8 $\pm$ 1.9\
3889 He [i]{} + H8 & ... & ... & 23.4 $\pm$ 7.2 & 21.4 $\pm$ 0.5 & ... & ... \
3968 \[Ne [iii]{}\] + H7 & ... & ... & 26.9 $\pm$ 4.9 & 28.9 $\pm$ 0.6 & ... & ... \
4101 H$\delta$ & ... & 22.8 $\pm$ 4.6 & 28.5 $\pm$ 4.4 & 25.7 $\pm$ 0.5 & ... & ... \
4340 H$\gamma$ & 39.9 $\pm$ 3.4 & 47.1 $\pm$ 3.0 & 47.3 $\pm$ 3.4 & 47.4 $\pm$ 0.8 & 38.3 $\pm$ 2.7 & 41.5 $\pm$ 3.4\
4363 \[O [iii]{}\] & 7.3 $\pm$ 1.3 & ... & ... & 12.3 $\pm$ 0.3 & ... & 8.7 $\pm$ 1.3\
4471 He [i]{} & ... & ... & ... & 3.5 $\pm$ 0.2 & ... & ... \
4686 He [ii]{} & ... & ... & ... & 1.4 $\pm$ 0.1 & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & 1.8 $\pm$ 0.1 & ... & ... \
4740 \[Ar [iv]{}\] & ... & ... & ... & 1.1 $\pm$ 0.1 & ... & ... \
4861 H$\beta$ & 100.0 $\pm$ 4.6 & 100.0 $\pm$ 3.8 & 100.0 $\pm$ 3.9 & 100.0 $\pm$ 1.5 & 100.0 $\pm$ 3.3 & 100.0 $\pm$ 3.8\
4959 \[O [iii]{}\] & 86.9 $\pm$ 3.5 & 59.3 $\pm$ 2.4 & 128.2 $\pm$ 4.3 & 157.1 $\pm$ 2.4 & 122.1 $\pm$ 3.4 & 115.5 $\pm$ 3.7\
5007 \[O [iii]{}\] & 255.9 $\pm$ 8.4 & 170.9 $\pm$ 5.1 & 373.0 $\pm$ 10.9 & 466.7 $\pm$ 7.0 & 369.4 $\pm$ 9.0 & 332.8 $\pm$ 9.2\
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}& [ 0.000 ]{}& [ 0.000 ]{}& [ 0.305 ]{}& [ 0.000 ]{}& [ 0.000 ]{}\
$F$(H$\beta$)$^b$ & [ 4.59 ]{}& [ 10.53 ]{}& [ 21.39 ]{}& [107.50 ]{}& [ 14.00 ]{}& [ 9.26 ]{}\
EW(H$\beta$) ($\AA$) & [ 30.0 ]{}& [ 13.8 ]{}& [ 20.3 ]{}& [ 117.2 ]{}& [ 26.1 ]{}& [ 30.5 ]{}\
EW(abs) ($\AA$)$^a$ & [ 2.00 ]{}& [ 2.00 ]{}& [ 2.00 ]{}& [ 1.50 ]{}& [ 2.00 ]{}& [ 2.00 ]{}\
$^a$ zero value is assumed if a negative value is derived.
$^b$ in units of 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$.
[lrrrrrrrr]{}
\
\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{} &[$I$($\lambda$)/$I$(H$\beta$)]{}\
&\
& [J0004$+$0025 ]{}&[J0004$+$0025 ]{}& [J0014$-$0044 ]{}&[J0014$-$0044 ]{}& [J0202$-$0047 ]{}&[J0301$-$0059 ]{}& [J0301$-$0059 ]{}&[J0301$-$0059 ]{}\
& No.1 & No.2 & No.1 & No.2 & & No.1 & No.2 & No.3\
3727 \[O [ii]{}\] & 214.05 $\pm$ 11.27 &253.26 $\pm$ 13.99 & 99.16 $\pm$ 1.73 &298.77 $\pm$ 9.30 &191.81 $\pm$ 3.50 &331.24 $\pm$ 6.29 &304.43 $\pm$ 6.86 &330.51 $\pm$ 21.26\
3750 H12 & ... & ... & 4.57 $\pm$ 0.56 & ... & ... & ... & ... & ... \
3771 H11 & ... & ... & 5.06 $\pm$ 0.52 & ... & 4.22 $\pm$ 0.66 & ... & ... & ... \
3798 H10 & ... & ... & 7.06 $\pm$ 0.52 & ... & 4.83 $\pm$ 0.67 & ... & ... & ... \
3835 H9 & ... & ... & 8.66 $\pm$ 0.46 & ... & 5.85 $\pm$ 0.58 & ... & ... & ... \
3868 \[Ne [iii]{}\] & ... & 33.25 $\pm$ 4.96 & 50.11 $\pm$ 0.85 & ... & 41.34 $\pm$ 0.92 & 24.06 $\pm$ 0.98 & 23.23 $\pm$ 1.72 & ... \
3889 He [i]{} + H8 & ... & ... & 21.72 $\pm$ 0.55 & ... & 22.44 $\pm$ 0.72 & 22.59 $\pm$ 1.80 & ... & ... \
3968 \[Ne [iii]{}\] + H7 & ... & ... & 32.07 $\pm$ 0.64 & ... & 28.21 $\pm$ 0.78 & 25.37 $\pm$ 1.37 & ... & ... \
4026 He [i]{} & ... & ... & 1.10 $\pm$ 0.11 & ... & ... & ... & ... & ... \
4101 H$\delta$ & ... & ... & 25.84 $\pm$ 0.51 & 28.41 $\pm$ 2.21 & 25.71 $\pm$ 0.70 & 28.75 $\pm$ 1.16 & 28.14 $\pm$ 2.37 & ... \
4340 H$\gamma$ & 47.25 $\pm$ 3.35 & 47.74 $\pm$ 3.21 & 46.79 $\pm$ 0.77 & 51.88 $\pm$ 1.98 & 48.61 $\pm$ 0.93 & 46.88 $\pm$ 1.15 & 45.15 $\pm$ 1.68 & ... \
4363 \[O [iii]{}\] & ... & ... & 8.87 $\pm$ 0.18 & ... & 8.97 $\pm$ 0.37 & 3.46 $\pm$ 0.42 & 4.43 $\pm$ 0.92 & ... \
4387 He [i]{} & ... & ... & 0.57 $\pm$ 0.08 & ... & ... & ... & ... & ... \
4471 He [i]{} & ... & ... & 3.83 $\pm$ 0.12 & ... & 3.49 $\pm$ 0.26 & ... & ... & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & 1.08 $\pm$ 0.09 & ... & ... & ... & ... & ... \
4740 \[Ar [iv]{}\] & ... & ... & 0.84 $\pm$ 0.07 & ... & ... & ... & ... & ... \
4861 H$\beta$ & 100.00 $\pm$ 3.77 &100.00 $\pm$ 3.98 &100.00 $\pm$ 1.49 &100.00 $\pm$ 2.52 &100.00 $\pm$ 1.60 &100.00 $\pm$ 1.81 &100.00 $\pm$ 2.17 &100.00 $\pm$ 17.01\
4921 He [i]{} & ... & ... & 0.94 $\pm$ 0.06 & ... & ... & ... & ... & ... \
4959 \[O [iii]{}\] & 37.84 $\pm$ 1.98 & 62.06 $\pm$ 2.64 &207.93 $\pm$ 3.04 & 59.47 $\pm$ 1.64 &161.71 $\pm$ 2.50 & 91.37 $\pm$ 1.62 & 90.51 $\pm$ 1.93 & 53.80 $\pm$ 4.75\
4988 \[Fe [iii]{}\] & ... & ... & 0.44 $\pm$ 0.01 & ... & ... & ... & ... & ... \
5007 \[O [iii]{}\] & 108.99 $\pm$ 3.45 &183.91 $\pm$ 5.64 &621.54 $\pm$ 9.04 &175.13 $\pm$ 3.94 &495.48 $\pm$ 7.53 &275.10 $\pm$ 4.60 &267.82 $\pm$ 5.17 &154.51 $\pm$ 8.64\
5015 He [i]{} & ... & ... & 0.90 $\pm$ 0.03 & ... & 2.20 $\pm$ 0.15 & ... & ... & ... \
5518 \[Cl [iii]{}\] & ... & ... & 0.38 $\pm$ 0.06 & ... & ... & ... & ... & ... \
5538 \[Cl [iii]{}\] & ... & ... & 0.61 $\pm$ 0.10 & ... & ... & ... & ... & ... \
5876 He [i]{} & ... & ... & 10.92 $\pm$ 0.19 & 13.81 $\pm$ 0.29 & 10.96 $\pm$ 0.27 & 9.94 $\pm$ 0.61 & 11.70 $\pm$ 0.82 & ... \
6300 \[O [i]{}\] & ... & ... & 1.92 $\pm$ 0.06 & ... & 3.11 $\pm$ 0.13 & 9.14 $\pm$ 0.52 & 9.57 $\pm$ 0.60 & ... \
6312 \[S [iii]{}\] & ... & ... & 1.73 $\pm$ 0.06 & ... & 1.74 $\pm$ 0.11 & 0.97 $\pm$ 0.02 & ... & ... \
6363 \[O [i]{}\] & ... & ... & 0.70 $\pm$ 0.05 & ... & 1.42 $\pm$ 0.02 & 2.73 $\pm$ 0.40 & 2.72 $\pm$ 0.05 & ... \
6548 \[N [ii]{}\] & ... & ... & 2.16 $\pm$ 0.06 & 10.39 $\pm$ 0.88 & 3.86 $\pm$ 0.13 & 6.66 $\pm$ 0.44 & 7.77 $\pm$ 0.53 & ... \
6563 H$\alpha$ & 269.07 $\pm$ 8.14 &266.58 $\pm$ 8.43 &283.02 $\pm$ 4.47 &288.13 $\pm$ 6.59 &281.45 $\pm$ 4.63 &284.02 $\pm$ 5.14 &281.73 $\pm$ 5.83 &288.79 $\pm$ 19.92\
6583 \[N [ii]{}\] & 15.55 $\pm$ 1.51 & 12.75 $\pm$ 1.07 & 5.73 $\pm$ 0.11 & 22.66 $\pm$ 0.99 & 9.59 $\pm$ 0.21 & 20.47 $\pm$ 0.54 & 26.57 $\pm$ 0.81 & 33.85 $\pm$ 3.43\
6678 He [i]{} & ... & ... & 3.05 $\pm$ 0.07 & ... & 2.69 $\pm$ 0.11 & 1.83 $\pm$ 0.35 & ... & ... \
6717 \[S [ii]{}\] & 32.18 $\pm$ 1.51 & 27.90 $\pm$ 1.41 & 8.04 $\pm$ 0.14 & 31.47 $\pm$ 1.01 & 16.63 $\pm$ 0.32 & 42.25 $\pm$ 0.88 & 47.03 $\pm$ 1.11 & 63.87 $\pm$ 4.85\
6731 \[S [ii]{}\] & 25.79 $\pm$ 1.51 & 22.66 $\pm$ 1.32 & 5.78 $\pm$ 0.11 & 24.87 $\pm$ 0.91 & 12.38 $\pm$ 0.26 & 29.45 $\pm$ 0.68 & 32.66 $\pm$ 0.86 & 44.23 $\pm$ 4.15\
7065 He [i]{} & ... & ... & 2.45 $\pm$ 0.06 & ... & 1.66 $\pm$ 0.09 & 1.58 $\pm$ 0.26 & ... & ... \
7136 \[Ar [iii]{}\] & ... & ... & 6.61 $\pm$ 0.12 & ... & 8.26 $\pm$ 0.19 & 6.60 $\pm$ 0.41 & 7.65 $\pm$ 0.41 & ... \
7281 He [i]{} & ... & ... & 0.51 $\pm$ 0.03 & ... & 0.92 $\pm$ 0.08 & ... & ... & ... \
7320 \[O [ii]{}\] & ... & ... & 1.51 $\pm$ 0.05 & ... & 2.16 $\pm$ 0.09 & ... & ... & ... \
7330 \[O [ii]{}\] & ... & ... & 1.36 $\pm$ 0.05 & ... & 1.85 $\pm$ 0.07 & ... & ... & ... \
\
$C$(H$\beta$)$^a$ & [ 0.000 ]{}&[ 0.000 ]{}&[ 0.210 ]{}&[ 0.030 ]{}&[ 0.305 ]{}&[ 0.215 ]{}&[ 0.185 ]{}&[ 0.110 ]{}\
$F$(H$\beta$)$^b$ & [ 0.76 ]{}&[ 0.80 ]{}&[ 20.90 ]{}&[ 0.81 ]{}&[ 5.47 ]{}&[ 5.40 ]{}&[ 3.59 ]{}&[ 0.38 ]{}\
EW(H$\beta$) ($\AA$) & [ 22.5 ]{}&[ 21.4 ]{}&[ 275.7 ]{}&[ 74.2 ]{}&[ 132.6 ]{}&[ 28.3 ]{}&[ 13.6 ]{}&[ 8.0 ]{}\
EW(abs) ($\AA$) & [ 2.00 ]{}&[ 2.00 ]{}&[ 2.20 ]{}&[ 2.00 ]{}&[ 0.50 ]{}&[ 1.95 ]{}&[ 1.75 ]{}&[ 2.00 ]{}\
&\
& [J0315$-$0024 ]{}&[J0315$-$0024 ]{}& [J0338$+$0013 ]{}&[G0405-3648]{}& [G0405-3648]{}&[G0405-3648]{}& [J0519$+$0007 ]{}&[J2104$-$0035 ]{}\
& No.1 & No.2 & & No.1 &No.2 & No.3 & & No.1\
3727 \[O [ii]{}\] & 174.28 $\pm$ 5.19 & 190.40 $\pm$ 9.12 & 60.18 $\pm$ 1.06 & 129.50 $\pm$ 5.31 & 160.45 $\pm$ 3.75 & 154.37 $\pm$ 3.90 & 27.98 $\pm$ 0.47 & 27.37 $\pm$ 0.97\
3750 H12 & ... & ... & 3.59 $\pm$ 0.30 & ... & ... & ... & 3.13 $\pm$ 0.18 & 4.11 $\pm$ 0.75\
3771 H11 & ... & ... & 3.80 $\pm$ 0.29 & ... & ... & ... & 4.07 $\pm$ 0.18 & 5.22 $\pm$ 0.73\
3798 H10 & ... & ... & 6.86 $\pm$ 0.30 & ... & ... & ... & 5.75 $\pm$ 0.18 & 6.80 $\pm$ 0.66\
3820 He [i]{} & ... & ... & 2.00 $\pm$ 0.18 & ... & ... & ... & 0.74 $\pm$ 0.09 & ... \
3835 H9 & ... & ... & 8.51 $\pm$ 0.28 & ... & ... & ... & 7.26 $\pm$ 0.19 & 8.01 $\pm$ 0.61\
3868 \[Ne [iii]{}\] & 8.88 $\pm$ 1.88 & ... & 41.54 $\pm$ 0.71 & 9.66 $\pm$ 1.46 & 11.59 $\pm$ 1.16 & 14.90 $\pm$ 1.09 & 35.86 $\pm$ 0.56 & 24.32 $\pm$ 0.53\
3889 He [i]{} + H8 & 20.76 $\pm$ 2.11 & ... & 18.64 $\pm$ 0.41 & 18.38 $\pm$ 3.37 & 18.70 $\pm$ 1.09 & 20.05 $\pm$ 1.24 & 17.21 $\pm$ 0.31 & 22.61 $\pm$ 0.66\
3968 \[Ne [iii]{}\] + H7 & 16.88 $\pm$ 1.83 & ... & 28.87 $\pm$ 0.53 & 17.38 $\pm$ 2.88 & 15.71 $\pm$ 1.02 & 20.71 $\pm$ 1.21 & 26.96 $\pm$ 0.44 & 23.31 $\pm$ 0.65\
4026 He [i]{} & ... & ... & 2.82 $\pm$ 0.17 & ... & ... & ... & 1.55 $\pm$ 0.08 & 2.12 $\pm$ 0.18\
4068 \[S [ii]{}\] & ... & ... & 1.16 $\pm$ 0.02 & ... & ... & ... & 0.65 $\pm$ 0.06 & ... \
4101 H$\delta$ & 21.87 $\pm$ 1.58 & ... & 26.20 $\pm$ 0.47 & 25.79 $\pm$ 1.85 & 21.62 $\pm$ 0.82 & 24.52 $\pm$ 0.99 & 25.67 $\pm$ 0.41 & 25.83 $\pm$ 0.63\
4227 \[Fe [v]{}\] & ... & ... & ... & ... & ... & ... & 0.86 $\pm$ 0.14 & ... \
4340 H$\gamma$ & 49.15 $\pm$ 1.56 & 50.73 $\pm$ 4.71 & 48.06 $\pm$ 0.76 & 47.97 $\pm$ 1.55 & 45.48 $\pm$ 1.00 & 49.12 $\pm$ 1.13 & 47.62 $\pm$ 0.71 & 48.28 $\pm$ 0.82\
4363 \[O [iii]{}\] & 2.04 $\pm$ 0.74 & ... & 14.66 $\pm$ 0.27 & 4.02 $\pm$ 0.76 & 4.03 $\pm$ 0.36 & 5.12 $\pm$ 0.50 & 15.04 $\pm$ 0.24 & 9.87 $\pm$ 0.20\
4387 He [i]{} & ... & ... & ... & ... & ... & ... & 0.37 $\pm$ 0.05 & ... \
4471 He [i]{} & ... & ... & 3.70 $\pm$ 0.13 & 2.18 $\pm$ 0.72 & 3.54 $\pm$ 0.43 & 2.63 $\pm$ 0.40 & 3.60 $\pm$ 0.08 & 3.65 $\pm$ 0.13\
4658 \[Fe [iii]{}\] & ... & ... & 0.49 $\pm$ 0.06 & ... & ... & ... & 0.49 $\pm$ 0.04 & ... \
4686 He [ii]{} & ... & ... & 1.15 $\pm$ 0.08 & ... & ... & ... & 1.52 $\pm$ 0.05 & 0.65 $\pm$ 0.10\
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & 2.35 $\pm$ 0.09 & ... & ... & ... & 2.33 $\pm$ 0.06 & 1.52 $\pm$ 0.09\
4740 \[Ar [iv]{}\] & ... & ... & 1.40 $\pm$ 0.07 & ... & ... & ... & 1.51 $\pm$ 0.06 & 0.74 $\pm$ 0.10\
4861 H$\beta$ & 100.00 $\pm$ 1.95 & 100.00 $\pm$ 4.69 & 100.00 $\pm$ 1.48 & 100.00 $\pm$ 2.03 & 100.00 $\pm$ 1.70 & 100.00 $\pm$ 1.77 & 100.00 $\pm$ 1.44 & 100.00 $\pm$ 1.50\
4921 He [i]{} & ... & ... & 1.27 $\pm$ 0.08 & ... & ... & ... & 0.98 $\pm$ 0.05 & 1.11 $\pm$ 0.08\
4959 \[O [iii]{}\] & 53.68 $\pm$ 1.19 & 51.95 $\pm$ 2.12 & 176.71 $\pm$ 2.58 & 70.62 $\pm$ 1.51 & 48.62 $\pm$ 0.91 & 45.69 $\pm$ 0.93 & 148.33 $\pm$ 2.13 & 96.10 $\pm$ 1.41\
4988 \[Fe [iii]{}\] & ... & ... & 0.54 $\pm$ 0.05 & ... & 1.96 $\pm$ 0.31 & ... & 0.65 $\pm$ 0.04 & 0.75 $\pm$ 0.13\
5007 \[O [iii]{}\] & 162.82 $\pm$ 2.95 & 151.44 $\pm$ 4.43 & 530.20 $\pm$ 7.71 & 206.51 $\pm$ 3.80 & 140.34 $\pm$ 2.29 & 137.62 $\pm$ 2.33 & 439.62 $\pm$ 6.30 & 288.91 $\pm$ 4.20\
5015 He [i]{} & ... & ... & 2.59 $\pm$ 0.08 & ... & ... & ... & 0.97 $\pm$ 0.04 & 1.64 $\pm$ 0.18\
5755 \[N [ii]{}\] & ... & ... & ... & ... & ... & ... & 0.15 $\pm$ 0.02 & ... \
5876 He [i]{} & 9.03 $\pm$ 0.55 & ... & 11.14 $\pm$ 0.21 & 8.41 $\pm$ 0.58 & 9.77 $\pm$ 0.48 & 5.72 $\pm$ 0.29 & 11.60 $\pm$ 0.18 & 9.38 $\pm$ 0.19\
6300 \[O [i]{}\] & ... & ... & 1.72 $\pm$ 0.07 & ... & 4.36 $\pm$ 0.36 & 2.75 $\pm$ 0.25 & 1.07 $\pm$ 0.03 & 0.49 $\pm$ 0.06\
6312 \[S [iii]{}\] & ... & ... & 1.01 $\pm$ 0.07 & ... & ... & ... & 0.78 $\pm$ 0.02 & 0.66 $\pm$ 0.06\
6363 \[O [i]{}\] & ... & ... & 0.53 $\pm$ 0.05 & ... & ... & ... & 0.37 $\pm$ 0.02 & ... \
6548 \[N [ii]{}\] & ... & ... & ... & ... & ... & ... & 1.17 $\pm$ 0.03 & ... \
6563 H$\alpha$ & 283.73 $\pm$ 5.33 & 288.03 $\pm$ 8.76 & 277.02 $\pm$ 4.37 & 279.40 $\pm$ 5.45 & 275.30 $\pm$ 4.73 & 272.50 $\pm$ 4.83 & 274.85 $\pm$ 4.27 & 274.51 $\pm$ 4.32\
6583 \[N [ii]{}\] & 6.88 $\pm$ 0.30 & 11.83 $\pm$ 1.44 & 1.64 $\pm$ 0.06 & 5.41 $\pm$ 0.37 & 6.48 $\pm$ 0.31 & 5.14 $\pm$ 0.23 & 3.10 $\pm$ 0.05 & 0.93 $\pm$ 0.06\
6678 He [i]{} & 1.89 $\pm$ 0.04 & ... & 2.69 $\pm$ 0.08 & ... & 2.31 $\pm$ 0.24 & 2.64 $\pm$ 0.22 & 2.95 $\pm$ 0.05 & 2.50 $\pm$ 0.07\
6717 \[S [ii]{}\] & 8.51 $\pm$ 0.36 & 13.97 $\pm$ 1.68 & 4.65 $\pm$ 0.10 & 13.04 $\pm$ 0.55 & 16.04 $\pm$ 0.40 & 14.92 $\pm$ 0.39 & 2.36 $\pm$ 0.04 & 2.10 $\pm$ 0.06\
6731 \[S [ii]{}\] & 9.99 $\pm$ 0.33 & 9.02 $\pm$ 0.97 & 3.55 $\pm$ 0.09 & 8.37 $\pm$ 0.42 & 12.20 $\pm$ 0.35 & 9.95 $\pm$ 0.30 & 2.19 $\pm$ 0.04 & 1.57 $\pm$ 0.06\
7065 He [i]{} & ... & ... & 4.94 $\pm$ 0.10 & 2.53 $\pm$ 0.33 & 1.35 $\pm$ 0.19 & 1.90 $\pm$ 0.19 & 6.54 $\pm$ 0.11 & 2.30 $\pm$ 0.06\
7136 \[Ar [iii]{}\] & ... & ... & 3.14 $\pm$ 0.08 & 3.69 $\pm$ 0.30 & 2.82 $\pm$ 0.20 & 2.33 $\pm$ 0.16 & 2.36 $\pm$ 0.04 & 1.25 $\pm$ 0.05\
7281 He [i]{} & ... & ... & ... & ... & ... & ... & ... & 0.72 $\pm$ 0.03\
7320 \[O [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 0.49 $\pm$ 0.03\
7330 \[O [ii]{}\] & ... & ... & ... & ... & ... & ... & ... & 0.25 $\pm$ 0.03\
\
$C$(H$\beta$)$^a$ & [ 0.135 ]{}& [ 0.015 ]{}& [ 0.245 ]{}& [ 0.005 ]{}& [ 0.000 ]{}& [ 0.075 ]{}& [ 0.285 ]{}& [ 0.235 ]{}\
$F$(H$\beta$)$^b$ & [ 1.98 ]{}& [ 0.42 ]{}& [ 15.97 ]{}& [ 3.05 ]{}& [ 4.21 ]{}& [ 3.55 ]{}& [ 81.87 ]{}& [ 19.30 ]{}\
EW(H$\beta$) ($\AA$) & [ 36.1 ]{}& [ 23.1 ]{}& [ 220.6 ]{}& [ 18.3 ]{}& [ 36.5 ]{}& [ 38.7 ]{}& [ 241.8 ]{}& [ 213.6 ]{}\
EW(abs) ($\AA$) & [ 1.15 ]{}& [ 2.00 ]{}& [ 0.80 ]{}& [ 1.25 ]{}& [ 0.65 ]{}& [ 1.30 ]{}& [ 0.65 ]{}& [ 0.60 ]{}\
&\
& [J2104$-$0035 ]{}&[J2104$-$0035 ]{}& [J2104$-$0035 ]{}&[J2302$+$0049 ]{}& [J2302$+$0049 ]{}&[J2324$-$0006 ]{}& [J2354$-$0004 ]{}\
& No.2 & No.3 &No.4 & No.1 &No.2 & & No.1\
3727 \[O [ii]{}\] & 114.48 $\pm$ 11.06 & 108.68 $\pm$ 4.99 & 88.02 $\pm$ 3.86 & 59.42 $\pm$ 1.00 & 190.27 $\pm$ 4.15 & 152.34 $\pm$ 2.43 & 138.52 $\pm$ 11.46\
3750 H12 & ... & ... & ... & 3.18 $\pm$ 0.25 & ... & 3.14 $\pm$ 0.33 & ... \
3771 H11 & ... & ... & ... & 4.04 $\pm$ 0.23 & ... & 3.83 $\pm$ 0.33 & ... \
3798 H10 & ... & ... & ... & 6.34 $\pm$ 0.24 & ... & 6.29 $\pm$ 0.32 & ... \
3820 He [i]{} & ... & ... & ... & 1.21 $\pm$ 0.14 & ... & ... & ... \
3835 H9 & ... & ... & ... & 7.43 $\pm$ 0.24 & ... & 7.53 $\pm$ 0.31 & ... \
3868 \[Ne [iii]{}\] & ... & ... & ... & 45.23 $\pm$ 0.72 & 40.05 $\pm$ 1.31 & 47.39 $\pm$ 0.75 & ... \
3889 He [i]{} + H8 & ... & 16.40 $\pm$ 1.55 & 22.24 $\pm$ 1.54 & 20.52 $\pm$ 0.39 & 22.92 $\pm$ 1.37 & 21.00 $\pm$ 0.42 & ... \
3968 \[Ne [iii]{}\] + H7 & ... & 17.76 $\pm$ 1.60 & 14.37 $\pm$ 1.37 & 29.75 $\pm$ 0.50 & 25.90 $\pm$ 1.29 & 31.37 $\pm$ 0.54 & ... \
4026 He [i]{} & ... & ... & ... & 1.64 $\pm$ 0.10 & ... & 1.84 $\pm$ 0.11 & ... \
4068 \[S [ii]{}\] & ... & ... & ... & ... & ... & 1.19 $\pm$ 0.09 & ... \
4076 \[S [ii]{}\] & ... & ... & ... & ... & ... & 0.41 $\pm$ 0.08 & ... \
4101 H$\delta$ & ... & 24.53 $\pm$ 1.22 & 24.61 $\pm$ 1.15 & 26.13 $\pm$ 0.43 & 30.30 $\pm$ 1.24 & 26.61 $\pm$ 0.45 & 25.66 $\pm$ 2.60\
4340 H$\gamma$ & 51.10 $\pm$ 3.95 & 49.03 $\pm$ 1.34 & 48.00 $\pm$ 1.33 & 47.28 $\pm$ 0.72 & 46.12 $\pm$ 1.24 & 47.16 $\pm$ 0.71 & 48.20 $\pm$ 2.01\
4363 \[O [iii]{}\] & ... & 3.59 $\pm$ 0.65 & 1.61 $\pm$ 0.69 & 14.90 $\pm$ 0.24 & 6.28 $\pm$ 0.54 & 9.47 $\pm$ 0.16 & 4.81 $\pm$ 1.57\
4387 He [i]{} & ... & ... & ... & ... & ... & 0.31 $\pm$ 0.08 & ... \
4471 He [i]{} & ... & ... & ... & 3.55 $\pm$ 0.09 & ... & 3.70 $\pm$ 0.08 & ... \
4658 \[Fe [iii]{}\] & ... & ... & ... & 0.49 $\pm$ 0.08 & ... & 0.59 $\pm$ 0.05 & ... \
4686 He [ii]{} & ... & ... & ... & 2.40 $\pm$ 0.08 & ... & 0.76 $\pm$ 0.05 & ... \
4711 \[Ar [iv]{}\] + He [i]{} & ... & ... & ... & 2.41 $\pm$ 0.08 & ... & 1.10 $\pm$ 0.05 & ... \
4740 \[Ar [iv]{}\] & ... & ... & ... & 1.41 $\pm$ 0.07 & ... & 0.42 $\pm$ 0.04 & ... \
4861 H$\beta$ & 100.00 $\pm$ 3.67 & 100.00 $\pm$ 2.01 & 100.00 $\pm$ 1.95 & 100.00 $\pm$ 1.45 & 100.00 $\pm$ 1.93 & 100.00 $\pm$ 1.44 & 100.00 $\pm$ 2.60\
4921 He [i]{} & ... & ... & ... & 0.89 $\pm$ 0.05 & ... & 1.02 $\pm$ 0.05 & ... \
4959 \[O [iii]{}\] & 18.99 $\pm$ 1.81 & 32.09 $\pm$ 0.90 & 18.23 $\pm$ 0.62 & 198.50 $\pm$ 2.86 & 105.26 $\pm$ 1.96 & 182.82 $\pm$ 2.62 & 59.53 $\pm$ 1.82\
4988 \[Fe [iii]{}\] & ... & ... & ... & 0.75 $\pm$ 0.05 & 3.20 $\pm$ 0.39 & 0.87 $\pm$ 0.04 & ... \
5007 \[O [iii]{}\] & 59.43 $\pm$ 2.53 & 94.72 $\pm$ 1.89 & 60.24 $\pm$ 1.23 & 567.92 $\pm$ 7.98 & 315.09 $\pm$ 5.54 & 543.75 $\pm$ 7.78 & 166.78 $\pm$ 3.88\
5015 He [i]{} & ... & ... & ... & 1.78 $\pm$ 0.06 & ... & 0.96 $\pm$ 0.05 & ... \
5518 \[Cl [iii]{}\] & ... & ... & ... & ... & ... & 0.25 $\pm$ 0.04 & ... \
5538 \[Cl [iii]{}\] & ... & ... & ... & ... & ... & 0.22 $\pm$ 0.03 & ... \
5876 He [i]{} & 5.84 $\pm$ 0.74 & 9.56 $\pm$ 0.57 & 9.39 $\pm$ 0.52 & 10.00 $\pm$ 0.18 & 8.03 $\pm$ 0.45 & 10.10 $\pm$ 0.16 & 9.03 $\pm$ 0.67\
6300 \[O [i]{}\] & ... & ... & ... & 1.30 $\pm$ 0.05 & ... & 2.69 $\pm$ 0.05 & ... \
6312 \[S [iii]{}\] & ... & ... & ... & 1.21 $\pm$ 0.06 & ... & 1.66 $\pm$ 0.04 & ... \
6363 \[O [i]{}\] & ... & ... & ... & 0.48 $\pm$ 0.04 & ... & 0.97 $\pm$ 0.03 & ... \
6548 \[N [ii]{}\] & ... & ... & ... & ... & ... & 1.46 $\pm$ 0.04 & ... \
6563 H$\alpha$ & 288.62 $\pm$ 8.99 & 273.23 $\pm$ 5.26 & 276.97 $\pm$ 5.20 & 277.79 $\pm$ 4.33 & 280.05 $\pm$ 5.31 & 281.56 $\pm$ 4.37 & 275.84 $\pm$ 6.36\
6583 \[N [ii]{}\] & 4.92 $\pm$ 0.56 & 4.28 $\pm$ 0.38 & 3.93 $\pm$ 0.26 & 1.70 $\pm$ 0.05 & 6.61 $\pm$ 0.45 & 4.94 $\pm$ 0.09 & 4.85 $\pm$ 0.46\
6678 He [i]{} & ... & 3.66 $\pm$ 0.33 & 2.53 $\pm$ 0.22 & 2.18 $\pm$ 0.05 & 2.49 $\pm$ 0.20 & 2.91 $\pm$ 0.06 & ... \
6717 \[S [ii]{}\] & 12.66 $\pm$ 1.06 & 8.30 $\pm$ 0.40 & 8.87 $\pm$ 0.34 & 4.47 $\pm$ 0.08 & 16.91 $\pm$ 0.51 & 11.75 $\pm$ 0.19 & 12.16 $\pm$ 0.69\
6731 \[S [ii]{}\] & 10.50 $\pm$ 1.04 & 4.19 $\pm$ 0.29 & 5.96 $\pm$ 0.35 & 3.48 $\pm$ 0.07 & 11.47 $\pm$ 0.39 & 8.79 $\pm$ 0.14 & 6.55 $\pm$ 0.56\
7065 He [i]{} & ... & ... & ... & 2.18 $\pm$ 0.05 & ... & 2.43 $\pm$ 0.05 & ... \
7136 \[Ar [iii]{}\] & ... & ... & ... & 3.41 $\pm$ 0.07 & 4.56 $\pm$ 0.09 & 5.80 $\pm$ 0.10 & ... \
7281 He [i]{} & ... & ... & ... & 0.52 $\pm$ 0.03 & ... & 0.59 $\pm$ 0.02 & ... \
7320 \[O [ii]{}\] & ... & ... & ... & 0.65 $\pm$ 0.03 & ... & 2.06 $\pm$ 0.04 & ... \
7330 \[O [ii]{}\] & ... & ... & ... & 0.55 $\pm$ 0.03 & ... & 1.67 $\pm$ 0.03 & ... \
\
$C$(H$\beta$)$^a$ & [ 0.015 ]{}& [ 0.225 ]{}& [ 0.205 ]{}& [ 0.260 ]{}& [ 0.195 ]{}& [ 0.250 ]{}& [ 0.080 ]{}\
$F$(H$\beta$)$^b$ & [ 1.10 ]{}& [ 1.81 ]{}& [ 2.02 ]{}& [ 32.24 ]{}& [ 2.59 ]{}& [ 55.62 ]{}& [ 1.05 ]{}\
EW(H$\beta$) ($\AA$) & [ 9.6 ]{}& [ 36.3 ]{}& [ 84.2 ]{}& [ 199.4 ]{}& [ 41.9 ]{}& [ 219.5 ]{}& [ 25.0 ]{}\
EW(abs) ($\AA$) & [ 2.00 ]{}& [ 0.65 ]{}& [ 0.85 ]{}& [ 0.10 ]{}& [ 2.55 ]{}& [ 0.15 ]{}& [ 0.90 ]{}\
$^a$ zero value is assumed if a negative value is derived.
$^b$ in units of 10$^{-16}$ erg s$^{-1}$ cm$^{-2}$.
[lrrrrrrrr]{}
\
\
&\
Property & [J0015+0104 ]{} & [J0016+0108 ]{} & [J0029-0108 ]{} & [J0029-0025 ]{} & [J0057-0022 ]{} & [J0107+0001 ]{} & [J0109+0107 ]{} & [J0126-0038 ]{}\
& & & & & & & & No.1\
$T_{\rm e}$(O [iii]{}) (K) & $20000 \pm1020 $ & $17023 \pm1008 $ & $18580 \pm1188 $ & $19025 \pm1020 $ & $16258 \pm 215 $ & $20000 \pm1054 $ & $12337 \pm 558 $ & $14070 \pm 351 $\
$T_{\rm e}$(O [ii]{}) (K) & $16284 \pm1293 $ & $15379 \pm1217 $ & $15961 \pm1470 $ & $16083 \pm1273 $ & $15006 \pm 180 $ & $16284 \pm1336 $ & $12196 \pm 517 $ & $13624 \pm 313 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.084 \pm 0.017$ & $ 0.186 \pm 0.038$ & $ 0.129 \pm 0.037$ & $ 0.096 \pm 0.019$ & $ 0.199 \pm 0.008$ & $ 0.115 \pm 0.025$ & $ 0.381 \pm 0.052$ & $ 0.294 \pm 0.020$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.033 \pm 0.004$ & $ 0.153 \pm 0.022$ & $ 0.096 \pm 0.019$ & $ 0.099 \pm 0.013$ & $ 0.200 \pm 0.007$ & $ 0.053 \pm 0.007$ & $ 0.571 \pm 0.076$ & $ 0.534 \pm 0.036$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & ... & ... & ... & ... & $ 2.421 \pm 0.457$\
O/H ($\times$10$^4$) & $ 0.117 \pm 0.017$ & $ 0.339 \pm 0.044$ & $ 0.225 \pm 0.042$ & $ 0.194 \pm 0.023$ & $ 0.399 \pm 0.011$ & $ 0.168 \pm 0.026$ & $ 0.952 \pm 0.092$ & $ 0.852 \pm 0.042$\
12 + log(O/H) & $ 7.070 \pm 0.063$ & $ 7.530 \pm 0.056$ & $ 7.353 \pm 0.080$ & $ 7.289 \pm 0.052$ & $ 7.601 \pm 0.012$ & $ 7.226 \pm 0.066$ & $ 7.979 \pm 0.042$ & $ 7.931 \pm 0.021$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & ... & $ 0.660 \pm 0.048$ & ... & $ 1.498 \pm 0.220$ & $ 1.247 \pm 0.092$\
ICF & ... & ... & ... & ... & 1.213 & ... & 1.155 & 1.141\
log(Ne/O) & ... & ... & ... & ... & $-0.697 \pm 0.053$ & ... & $-0.740 \pm 0.101$ & $-0.777 \pm 0.049$\
&\
Property& [J0126-0038 ]{} & [J0135-0023 ]{} & [J0213-0002 ]{} & [J0213-0002 ]{} & [J0216+0115 ]{} & [J0216+0115 ]{} & [096632 ]{} & [J0252+0017 ]{}\
& No.2 & & No.1 &No.2 &No.1 &No.2 & &\
$T_{\rm e}$(O [iii]{}) (K) & $17255 \pm1005 $ & $18365 \pm1022 $ & $16036 \pm1017 $ & $14366 \pm1019 $ & $16076 \pm1009 $ & $17938 \pm1023 $ & $17425 \pm1005 $ & $15617 \pm1064 $\
$T_{\rm e}$(O [ii]{}) (K) & $15481 \pm1218 $ & $15895 \pm1260 $ & $14888 \pm1209 $ & $13839 \pm1182 $ & $14910 \pm1200 $ & $15750 \pm1253 $ & $15552 \pm1221 $ & $14650 \pm1256 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.185 \pm 0.037$ & $ 0.135 \pm 0.028$ & $ 0.169 \pm 0.037$ & $ 0.341 \pm 0.084$ & $ 0.208 \pm 0.045$ & $ 0.185 \pm 0.038$ & $ 0.249 \pm 0.049$ & $ 0.251 \pm 0.063$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.136 \pm 0.019$ & $ 0.102 \pm 0.014$ & $ 0.263 \pm 0.043$ & $ 0.413 \pm 0.080$ & $ 0.229 \pm 0.036$ & $ 0.091 \pm 0.013$ & $ 0.078 \pm 0.011$ & $ 0.255 \pm 0.048$\
O/H ($\times$10$^4$) & $ 0.321 \pm 0.042$ & $ 0.238 \pm 0.031$ & $ 0.432 \pm 0.057$ & $ 0.754 \pm 0.116$ & $ 0.437 \pm 0.058$ & $ 0.276 \pm 0.040$ & $ 0.327 \pm 0.051$ & $ 0.506 \pm 0.079$\
12 + log(O/H) & $ 7.506 \pm 0.057$ & $ 7.376 \pm 0.056$ & $ 7.636 \pm 0.057$ & $ 7.877 \pm 0.067$ & $ 7.641 \pm 0.057$ & $ 7.441 \pm 0.063$ & $ 7.515 \pm 0.067$ & $ 7.704 \pm 0.068$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & ... & ... & ... & $ 0.148 \pm 0.026$ & ... \
ICF & ... & ... & ... & ... & ... & ... & 1.408 & ... \
log(Ne/O) & ... & ... & ... & ... & ... & ... & $-1.195 \pm 0.233$ & ... \
&\
Property& [J0256+0036 ]{} & [J0303-0109 ]{} & [J0303-0109 ]{} & [J0341-0026 ]{} & [J0341-0026 ]{} & [J0341-0026 ]{} & [G1815-6701 ]{} & [G2052-6912]{}\
& &No.1 &No.2 &No.1 &No.2 &No.3 & & No.1\
$T_{\rm e}$(O [iii]{}) (K) & $15446 \pm1004 $ & $15850 \pm 960 $ & $20000 \pm1143 $ & $16136 \pm1023 $ & $17233 \pm1010 $ & $19005 \pm1014 $ & $14202 \pm 308 $ & $10991 \pm 110 $\
$T_{\rm e}$(O [ii]{}) (K) & $14548 \pm1183 $ & $14784 \pm 816 $ & $16284 \pm1449 $ & $14942 \pm1218 $ & $15472 \pm1223 $ & $16078 \pm1264 $ & $13721 \pm 273 $ & $11007 \pm 105 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.190 \pm 0.042$ & $ 0.131 \pm 0.020$ & $ 0.121 \pm 0.030$ & $ 0.209 \pm 0.046$ & $ 0.113 \pm 0.023$ & $ 0.064 \pm 0.013$ & $ 0.318 \pm 0.019$ & $ 0.370 \pm 0.014$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.327 \pm 0.055$ & $ 0.359 \pm 0.055$ & $ 0.046 \pm 0.008$ & $ 0.222 \pm 0.036$ & $ 0.189 \pm 0.027$ & $ 0.120 \pm 0.015$ & $ 0.645 \pm 0.038$ & $ 1.281 \pm 0.043$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & ... & ... & ... & $ 1.764 \pm 0.426$ & $ 0.611 \pm 0.106$\
O/H ($\times$10$^4$) & $ 0.517 \pm 0.069$ & $ 0.490 \pm 0.059$ & $ 0.166 \pm 0.031$ & $ 0.430 \pm 0.059$ & $ 0.302 \pm 0.036$ & $ 0.184 \pm 0.020$ & $ 0.981 \pm 0.043$ & $ 1.657 \pm 0.045$\
12 + log(O/H) & $ 7.713 \pm 0.058$ & $ 7.690 \pm 0.052$ & $ 7.221 \pm 0.082$ & $ 7.634 \pm 0.059$ & $ 7.480 \pm 0.051$ & $ 7.265 \pm 0.047$ & $ 7.992 \pm 0.019$ & $ 8.219 \pm 0.012$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 0.780 \pm 0.138$ & $ 0.981 \pm 0.159$ & ... & ... & ... & ... & $ 1.645 \pm 0.105$ & $ 2.810 \pm 0.109$\
ICF & 1.138 & 1.092 & ... & ... & ... & ... & 1.126 & 1.125\
log(Ne/O) & $-0.765 \pm 0.121$ & $-0.660 \pm 0.102$ & ... & ... & ... & ... & $-0.724 \pm 0.042$ & $-0.720 \pm 0.023$\
&\
Property& [G2052-6912 ]{} & [J2053+0039 ]{} & [J2105+0032 ]{} & [J2105+0032 ]{} & [J2112-0016 ]{} & [J2112-0016 ]{} & [J2119-0732 ]{} & [J2120-0058 ]{}\
&No.2 & &No.1 &No.2 &No.1 &No.2 & &\
$T_{\rm e}$(O [iii]{}) (K) & $11865 \pm 89 $ & $18364 \pm1039 $ & $17397 \pm1027 $ & $17930 \pm1021 $ & $11339 \pm 320 $ & $12417 \pm 771 $ & $14591 \pm1003 $ & $17281 \pm 908 $\
$T_{\rm e}$(O [ii]{}) (K) & $11756 \pm 83 $ & $15894 \pm1282 $ & $15541 \pm1247 $ & $15747 \pm1251 $ & $11239 \pm 302 $ & $12268 \pm 713 $ & $13996 \pm1167 $ & $15492 \pm 736 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.218 \pm 0.006$ & $ 0.065 \pm 0.014$ & $ 0.089 \pm 0.019$ & $ 0.146 \pm 0.030$ & $ 0.264 \pm 0.025$ & $ 0.192 \pm 0.036$ & $ 0.183 \pm 0.043$ & $ 0.207 \pm 0.025$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 1.309 \pm 0.033$ & $ 0.150 \pm 0.021$ & $ 0.194 \pm 0.029$ & $ 0.119 \pm 0.017$ & $ 1.366 \pm 0.120$ & $ 0.834 \pm 0.151$ & $ 0.487 \pm 0.088$ & $ 0.235 \pm 0.030$\
O/H ($\times$10$^4$) & $ 1.527 \pm 0.034$ & $ 0.215 \pm 0.026$ & $ 0.283 \pm 0.035$ & $ 0.265 \pm 0.034$ & $ 1.630 \pm 0.123$ & $ 1.026 \pm 0.156$ & $ 0.669 \pm 0.098$ & $ 0.442 \pm 0.039$\
12 + log(O/H) & $ 8.184 \pm 0.009$ & $ 7.332 \pm 0.052$ & $ 7.451 \pm 0.053$ & $ 7.423 \pm 0.056$ & $ 8.212 \pm 0.033$ & $ 8.011 \pm 0.066$ & $ 7.826 \pm 0.064$ & $ 7.646 \pm 0.038$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 2.728 \pm 0.079$ & ... & ... & ... & $ 2.871 \pm 0.284$ & $ 1.249 \pm 0.261$ & $ 1.116 \pm 0.214$ & $ 0.723 \pm 0.093$\
ICF & 1.064 & ... & ... & ... & 1.079 & 1.062 & 1.094 & 1.194\
log(Ne/O) & $-0.721 \pm 0.017$ & ... & ... & ... & $-0.721 \pm 0.058$ & $-0.888 \pm 0.124$ & $-0.739 \pm 0.123$ & $-0.710 \pm 0.096$\
&\
Property& [J2150+0033 ]{} & [G2155-3946 ]{} & [J2227-0939 ]{} & [PHL 293B ]{} & [J2310-0109 No.1 ]{} & [J2310-0109 No.2 ]{}\
& & & & & No.1 &No.2\
$T_{\rm e}$(O [iii]{}) (K) & $16314 \pm1008 $ & $16087 \pm1008 $ & $14707 \pm1007 $ & $17410 \pm 228 $ & $15417 \pm1005 $ & $15359 \pm1006 $\
$T_{\rm e}$(O [ii]{}) (K) & $15036 \pm 845 $ & $14915 \pm1199 $ & $14075 \pm1173 $ & $15546 \pm 184 $ & $14531 \pm1183 $ & $14495 \pm 867 $\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.171 \pm 0.026$ & $ 0.298 \pm 0.064$ & $ 0.219 \pm 0.051$ & $ 0.053 \pm 0.002$ & $ 0.128 \pm 0.028$ & $ 0.179 \pm 0.029$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.230 \pm 0.036$ & $ 0.160 \pm 0.025$ & $ 0.434 \pm 0.079$ & $ 0.360 \pm 0.012$ & $ 0.379 \pm 0.063$ & $ 0.348 \pm 0.059$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 0.655 \pm 0.065$ & ... & ... \
O/H ($\times$10$^4$) & $ 0.401 \pm 0.044$ & $ 0.458 \pm 0.068$ & $ 0.653 \pm 0.094$ & $ 0.420 \pm 0.012$ & $ 0.507 \pm 0.069$ & $ 0.527 \pm 0.066$\
12 + log(O/H) & $ 7.603 \pm 0.048$ & $ 7.661 \pm 0.065$ & $ 7.815 \pm 0.062$ & $ 7.624 \pm 0.013$ & $ 7.705 \pm 0.060$ & $ 7.722 \pm 0.054$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & $ 0.363 \pm 0.070$ & $ 0.993 \pm 0.202$ & $ 0.729 \pm 0.026$ & $ 0.517 \pm 0.094$ & $ 0.695 \pm 0.127$\
ICF & ... & 1.319 & 1.122 & 1.046 & 1.086 & 1.124\
log(Ne/O) & ... & $-0.980 \pm 0.193$ & $-0.768 \pm 0.134$ & $-0.742 \pm 0.021$ & $-0.956 \pm 0.114$ & $-0.829 \pm 0.120$\
[lrrrrrrrr]{}
\
\
&\
Property& [J0004$+$0025 ]{} & [J0004$+$0025 ]{} & [J0014$-$0044 ]{} & [J0014$-$0044 ]{} & [J0202$-$0047 ]{} & [J0301$-$0059 ]{} & [J0301$-$0059 ]{} & [J0301$-$0059 ]{}\
&No.1 &No.2 &No.1 &No.2 & &No.1 &No.2 &No.3\
$T_{\rm e}$(O [iii]{}) (K) & $18630 \pm1019 $ & $16701 \pm1016 $ & $13195 \pm 123 $ & $16308 \pm1006 $ & $14607 \pm 277 $ & $12582 \pm 585 $ & $13999 \pm1236 $ & $16248 \pm1033 $\
$T_{\rm e}$(O [ii]{}) (K) & $15975 \pm1263 $ & $15229 \pm1221 $ & $12939 \pm 112 $ & $15033 \pm1201 $ & $14006 \pm 243 $ & $12416 \pm 539 $ & $13571 \pm1104 $ & $15001 \pm1232 $\
$T_{\rm e}$(S [iii]{}) (K) & $17163 \pm 846 $ & $15562 \pm 844 $ & $13067 \pm 102 $ & $15236 \pm 835 $ & $14307 \pm 230 $ & $12499 \pm 485 $ & $13785 \pm1026 $ & $15186 \pm 857 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 193 \pm 128$ & $ 213 \pm 132$ & $ 33 \pm 28$ & $ 167 \pm 78$ & $ 78 \pm 38$ & $ 10 \pm 10$ & $ 10 \pm 10$ & $ 10 \pm 10$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.164 \pm 0.033$ & $ 0.224 \pm 0.048$ & $ 0.146 \pm 0.005$ & $ 0.274 \pm 0.057$ & $ 0.218 \pm 0.011$ & $ 0.565 \pm 0.078$ & $ 0.381 \pm 0.089$ & $ 0.300 \pm 0.067$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.073 \pm 0.010$ & $ 0.156 \pm 0.024$ & $ 0.960 \pm 0.027$ & $ 0.158 \pm 0.024$ & $ 0.580 \pm 0.030$ & $ 0.485 \pm 0.065$ & $ 0.353 \pm 0.084$ & $ 0.141 \pm 0.023$\
O/H ($\times$10$^4$) & $ 0.237 \pm 0.034$ & $ 0.380 \pm 0.053$ & $ 1.106 \pm 0.028$ & $ 0.431 \pm 0.062$ & $ 0.798 \pm 0.032$ & $ 1.050 \pm 0.101$ & $ 0.734 \pm 0.123$ & $ 0.441 \pm 0.071$\
12 + log(O/H) & $ 7.374 \pm 0.063$ & $ 7.580 \pm 0.061$ & $ 8.044 \pm 0.011$ & $ 7.635 \pm 0.063$ & $ 7.902 \pm 0.017$ & $ 8.021 \pm 0.042$ & $ 7.866 \pm 0.073$ & $ 7.644 \pm 0.070$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 1.015 \pm 0.155$ & $ 0.913 \pm 0.140$ & $ 0.573 \pm 0.013$ & $ 1.663 \pm 0.240$ & $ 0.811 \pm 0.029$ & $ 2.245 \pm 0.202$ & $ 2.399 \pm 0.373$ & $ 2.491 \pm 0.409$\
ICF & 1.377 & 1.675 & 7.041 & 1.534 & 3.647 & 1.892 & 1.930 & 1.408\
log(N/O) & $-1.229 \pm 0.094$ & $-1.396 \pm 0.091$ & $-1.438 \pm 0.015$ & $-1.228 \pm 0.090$ & $-1.431 \pm 0.023$ & $-1.393 \pm 0.057$ & $-1.200 \pm 0.099$ & $-1.099 \pm 0.102$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & $ 0.659 \pm 0.138$ & $ 1.989 \pm 0.064$ & ... & $ 1.196 \pm 0.067$ & $ 1.118 \pm 0.167$ & $ 0.764 \pm 0.198$ & ... \
ICF & ... & 1.273 & 1.047 & ... & 1.094 & 1.268 & 1.226 & ... \
log(Ne/O) & ... & $-0.656 \pm 0.184$ & $-0.725 \pm 0.019$ & ... & $-0.785 \pm 0.035$ & $-0.870 \pm 0.119$ & $-0.894 \pm 0.205$ & ... \
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.510 \pm 0.059$ & $ 0.484 \pm 0.059$ & $ 0.176 \pm 0.003$ & $ 0.549 \pm 0.065$ & $ 0.318 \pm 0.010$ & $ 0.988 \pm 0.076$ & $ 0.920 \pm 0.121$ & $ 1.038 \pm 0.138$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 1.393 \pm 0.056$ & ... & $ 1.036 \pm 0.079$ & $ 0.919 \pm 0.113$ & ... & ... \
ICF & ... & ... & 1.724 & ... & 1.142 & 0.897 & ... & ... \
log(S/O) & ... & ... & $-1.612 \pm 0.019$ & ... & $-1.712 \pm 0.031$ & $-1.788 \pm 0.052$ & ... & ... \
\
Cl$^{++}$/H$^+$ ($\times$10$^8$) & ... & ... & $ 3.474 \pm 0.410$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 1.243 & ... & ... & ... & ... & ... \
log(Cl/O) & ... & ... & $-3.408 \pm 0.052$ & ... & ... & ... & ... & ... \
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 3.416 \pm 0.075$ & ... & $ 3.606 \pm 0.116$ & $ 3.719 \pm 0.336$ & $ 3.569 \pm 0.440$ & ... \
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 1.213 \pm 0.103$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 1.122 & ... & 1.141 & 1.097 & 1.043 & ... \
log(Ar/O) & ... & ... & $-2.460 \pm 0.020$ & ... & $-2.287 \pm 0.022$ & $-2.410 \pm 0.057$ & $-2.295 \pm 0.090$ & ... \
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & ... & ... & ... & ... & ... & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & $ 0.125 \pm 0.003$ & ... & ... & ... & ... & ... \
ICF & ... & ... & 10.398 & ... & ... & ... & ... & ... \
log(Fe/O) ($\lambda$4658) & ... & ... & ... & ... & ... & ... & ... & ... \
log(Fe/O) (4988) & ... & ... & $-1.932 \pm 0.015$ & ... & ... & ... & ... & ... \
&\
Property& [J0315$-$0024 ]{} & [J0315$-$0024 ]{} & [J0338$+$0013 ]{} & [G0405-3648]{} & [G0405-3648]{} & [G0405-3648]{} & [J0519$+$0007 ]{} & [J2104$-$0035 ]{}\
&No.1 &No.2 & &No.1 &No.2 &No.3 & &No.1\
$T_{\rm e}$(O [iii]{}) (K) & $12550 \pm1733 $ & $18111 \pm1012 $ & $17882 \pm 208 $ & $15013 \pm1293 $ & $18169 \pm 908 $ & $21295 \pm1349 $ & $20143 \pm 238 $ & $20194 \pm 294 $\
$T_{\rm e}$(O [ii]{}) (K) & $12387 \pm1597 $ & $15811 \pm1243 $ & $15729 \pm 165 $ & $14278 \pm1125 $ & $15830 \pm 712 $ & $16407 \pm 902 $ & $16306 \pm 171 $ & $16313 \pm 210 $\
$T_{\rm e}$(S [iii]{}) (K) & $12468 \pm1438 $ & $16732 \pm 840 $ & $16542 \pm 173 $ & $14645 \pm1073 $ & $16780 \pm 753 $ & $19375 \pm1119 $ & $18419 \pm 198 $ & $18461 \pm 244 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 1099 \pm 206$ & $ 10 \pm 10$ & $ 116 \pm 50$ & $ 10 \pm 10$ & $ 108 \pm 57$ & $ 10 \pm 10$ & $ 483 \pm 60$ & $ 84 \pm 74$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.339 \pm 0.138$ & $ 0.147 \pm 0.030$ & $ 0.048 \pm 0.002$ & $ 0.137 \pm 0.030$ & $ 0.125 \pm 0.014$ & $ 0.107 \pm 0.014$ & $ 0.021 \pm 0.001$ & $ 0.019 \pm 0.001$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.289 \pm 0.115$ & $ 0.108 \pm 0.014$ & $ 0.385 \pm 0.012$ & $ 0.228 \pm 0.051$ & $ 0.099 \pm 0.012$ & $ 0.069 \pm 0.010$ & $ 0.248 \pm 0.007$ & $ 0.162 \pm 0.006$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 0.531 \pm 0.042$ & ... & ... & ... & $ 0.444 \pm 0.021$ & $ 0.126 \pm 0.019$\
O/H ($\times$10$^4$) & $ 0.628 \pm 0.179$ & $ 0.255 \pm 0.033$ & $ 0.438 \pm 0.012$ & $ 0.365 \pm 0.059$ & $ 0.224 \pm 0.018$ & $ 0.176 \pm 0.017$ & $ 0.273 \pm 0.007$ & $ 0.183 \pm 0.006$\
12 + log(O/H) & $ 7.798 \pm 0.124$ & $ 7.406 \pm 0.056$ & $ 7.641 \pm 0.012$ & $ 7.562 \pm 0.070$ & $ 7.351 \pm 0.036$ & $ 7.246 \pm 0.043$ & $ 7.437 \pm 0.012$ & $ 7.261 \pm 0.014$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 0.768 \pm 0.202$ & $ 0.786 \pm 0.128$ & $ 0.110 \pm 0.004$ & $ 0.440 \pm 0.067$ & $ 0.430 \pm 0.036$ & $ 0.319 \pm 0.031$ & $ 0.195 \pm 0.004$ & $ 0.058 \pm 0.003$\
ICF & 1.847 & 1.713 & 8.642 & 2.686 & 1.785 & 1.616 & 12.223 & 8.833\
log(N/O) & $-1.646 \pm 0.169$ & $-1.277 \pm 0.091$ & $-1.662 \pm 0.019$ & $-1.490 \pm 0.096$ & $-1.465 \pm 0.051$ & $-1.535 \pm 0.061$ & $-1.059 \pm 0.015$ & $-1.551 \pm 0.029$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & $ 0.416 \pm 0.199$ & ... & $ 0.689 \pm 0.022$ & $ 0.258 \pm 0.071$ & $ 0.185 \pm 0.028$ & $ 0.164 \pm 0.024$ & $ 0.447 \pm 0.013$ & $ 0.301 \pm 0.011$\
ICF & 1.240 & ... & 1.040 & 1.143 & 1.251 & 1.286 & 1.031 & 1.037\
log(Ne/O) & $-1.085 \pm 0.385$ & ... & $-0.786 \pm 0.019$ & $-1.094 \pm 0.181$ & $-0.987 \pm 0.123$ & $-0.923 \pm 0.133$ & $-0.774 \pm 0.018$ & $-0.766 \pm 0.022$\
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.291 \pm 0.066$ & $ 0.202 \pm 0.028$ & $ 0.073 \pm 0.002$ & $ 0.225 \pm 0.028$ & $ 0.250 \pm 0.016$ & $ 0.205 \pm 0.016$ & $ 0.040 \pm 0.001$ & $ 0.031 \pm 0.001$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & $ 0.387 \pm 0.027$ & ... & ... & ... & $ 0.223 \pm 0.009$ & $ 0.189 \pm 0.019$\
ICF & ... & ... & 1.947 & ... & ... & ... & 2.539 & 1.978\
log(S/O) & ... & ... & $-1.689 \pm 0.028$ & ... & ... & ... & $-1.612 \pm 0.018$ & $-1.623 \pm 0.040$\
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 1.068 \pm 0.028$ & $ 1.544 \pm 0.201$ & $ 0.937 \pm 0.080$ & $ 0.618 \pm 0.052$ & $ 0.682 \pm 0.014$ & $ 0.357 \pm 0.015$\
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & $ 0.919 \pm 0.051$ & ... & ... & ... & $ 0.762 \pm 0.032$ & $ 0.371 \pm 0.049$\
ICF & ... & ... & 1.867 & 1.058 & 1.048 & 1.060 & 2.455 & 1.898\
log(Ar/O) & ... & ... & $-2.342 \pm 0.027$ & $-2.349 \pm 0.090$ & $-2.358 \pm 0.051$ & $-2.430 \pm 0.056$ & $-2.213 \pm 0.025$ & $-2.431 \pm 0.064$\
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & $ 0.083 \pm 0.010$ & ... & ... & ... & $ 0.076 \pm 0.007$ & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & $ 0.091 \pm 0.009$ & ... & $ 0.327 \pm 0.059$ & ... & $ 0.101 \pm 0.007$ & $ 0.117 \pm 0.021$\
ICF & ... & ... & 12.634 & ... & 2.360 & ... & 18.224 & 12.932\
log(Fe/O) ($\lambda$4658) & ... & ... & $-1.623 \pm 0.054$ & ... & ... & ... & $-1.298 \pm 0.041$ & ... \
log(Fe/O) (4988) & ... & ... & $-1.579 \pm 0.043$ & ... & $-1.464 \pm 0.086$ & ... & $-1.170 \pm 0.031$ & $-1.081 \pm 0.079$\
&\
Property& [J2104$-$0035 ]{} & [J2104$-$0035 ]{} & [J2104$-$0035 ]{} & [J2302$+$0049 ]{} & [J2302$+$0049 ]{} & [J2324$-$0006 ]{} & [J2354$-$0004 ]{}\
&No.2 &No.3 & No.4 &No.1 &No.2 & &No.1\
$T_{\rm e}$(O [iii]{}) (K) & $20000 \pm1061 $ & $21471 \pm2559 $ & $17794 \pm4091 $ & $17281 \pm 179 $ & $15203 \pm 616 $ & $14313 \pm 128 $ & $18120 \pm3255 $\
$T_{\rm e}$(O [ii]{}) (K) & $16284 \pm1345 $ & $16411 \pm1693 $ & $15697 \pm3257 $ & $15492 \pm 145 $ & $14399 \pm 533 $ & $13801 \pm 113 $ & $15814 \pm2559 $\
$T_{\rm e}$(S [iii]{}) (K) & $18300 \pm 880 $ & $19521 \pm2124 $ & $16469 \pm3396 $ & $16043 \pm 148 $ & $14801 \pm 511 $ & $14057 \pm 106 $ & $16740 \pm2702 $\
$N_{\rm e}$(S [ii]{}) (cm$^{-3}$) & $ 255 \pm 248$ & $ 10 \pm 10$ & $ 10 \pm 10$ & $ 143 \pm 44$ & $ 10 \pm 10$ & $ 84 \pm 31$ & $ 10 \pm 10$\
\
O$^+$/H$^+$ ($\times$10$^4$) & $ 0.083 \pm 0.018$ & $ 0.075 \pm 0.019$ & $ 0.070 \pm 0.036$ & $ 0.050 \pm 0.001$ & $ 0.196 \pm 0.020$ & $ 0.182 \pm 0.005$ & $ 0.107 \pm 0.044$\
O$^{++}$/H$^+$ ($\times$10$^4$) & $ 0.034 \pm 0.004$ & $ 0.047 \pm 0.012$ & $ 0.043 \pm 0.024$ & $ 0.450 \pm 0.012$ & $ 0.335 \pm 0.035$ & $ 0.675 \pm 0.018$ & $ 0.120 \pm 0.051$\
O$^{+++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 1.342 \pm 0.062$ & ... & $ 0.715 \pm 0.051$ & ... \
O/H ($\times$10$^4$) & $ 0.117 \pm 0.019$ & $ 0.122 \pm 0.023$ & $ 0.113 \pm 0.043$ & $ 0.513 \pm 0.012$ & $ 0.531 \pm 0.041$ & $ 0.864 \pm 0.019$ & $ 0.227 \pm 0.067$\
12 + log(O/H) & $ 7.068 \pm 0.070$ & $ 7.088 \pm 0.080$ & $ 7.052 \pm 0.166$ & $ 7.710 \pm 0.010$ & $ 7.725 \pm 0.033$ & $ 7.937 \pm 0.009$ & $ 7.355 \pm 0.128$\
\
N$^+$/H$^+$ ($\times$10$^6$) & $ 0.310 \pm 0.051$ & $ 0.266 \pm 0.049$ & $ 0.265 \pm 0.095$ & $ 0.117 \pm 0.003$ & $ 0.528 \pm 0.045$ & $ 0.431 \pm 0.009$ & $ 0.322 \pm 0.092$\
ICF & 1.323 & 1.592 & 1.587 & 9.717 & 2.732 & 4.662 & 2.133\
log(N/O) & $-1.455 \pm 0.103$ & $-1.462 \pm 0.114$ & $-1.428 \pm 0.230$ & $-1.653 \pm 0.017$ & $-1.566 \pm 0.049$ & $-1.633 \pm 0.013$ & $-1.518 \pm 0.177$\
\
Ne$^{++}$/H$^+$ ($\times$10$^5$) & ... & ... & ... & $ 0.819 \pm 0.024$ & $ 1.030 \pm 0.115$ & $ 1.457 \pm 0.043$ & ... \
ICF & ... & ... & ... & 1.040 & 1.139 & 1.072 & ... \
log(Ne/O) & ... & ... & ... & $-0.780 \pm 0.017$ & $-0.656 \pm 0.075$ & $-0.743 \pm 0.018$ & ... \
\
S$^{+}$/H$^+$ ($\times$10$^6$) & $ 0.198 \pm 0.026$ & $ 0.103 \pm 0.015$ & $ 0.132 \pm 0.038$ & $ 0.073 \pm 0.001$ & $ 0.293 \pm 0.018$ & $ 0.232 \pm 0.004$ & $ 0.164 \pm 0.037$\
S$^{++}$/H$^+$ ($\times$10$^6$) & ... & ... & ... & $ 0.508 \pm 0.027$ & ... & $ 1.045 \pm 0.032$ & ... \
ICF & ... & ... & ... & 2.124 & ... & 1.300 & ... \
log(S/O) & ... & ... & ... & $-1.619 \pm 0.023$ & ... & $-1.716 \pm 0.014$ & ... \
\
Cl$^{++}$/H$^+$ ($\times$10$^8$) & ... & ... & ... & ... & ... & $ 1.385 \pm 0.150$ & ... \
ICF & ... & ... & ... & ... & ... & 1.193 & ... \
log(Cl/O) & ... & ... & ... & ... & ... & $-3.719 \pm 0.048$ & ... \
\
Ar$^{++}$/H$^+$ ($\times$10$^7$) & ... & ... & ... & $ 1.222 \pm 0.027$ & $ 1.869 \pm 0.095$ & $ 2.613 \pm 0.053$ & ... \
Ar$^{+++}$/H$^+$ ($\times$10$^7$) & ... & ... & ... & $ 1.004 \pm 0.053$ & ... & $ 0.487 \pm 0.052$ & ... \
ICF & ... & ... & ... & 2.041 & 1.061 & 1.265 & ... \
log(Ar/O) & ... & ... & ... & $-2.313 \pm 0.024$ & $-2.428 \pm 0.040$ & $-2.417 \pm 0.015$ & ... \
\
Fe$^{++}$/H$^+$($\times$10$^6$)(4658) & ... & ... & ... & $ 0.086 \pm 0.014$ & ... & $ 0.139 \pm 0.012$ & ... \
Fe$^{++}$/H$^+$($\times$10$^6$)(4988) & ... & ... & ... & $ 0.131 \pm 0.010$ & $ 0.673 \pm 0.096$ & $ 0.204 \pm 0.010$ & ... \
ICF & ... & ... & ... & 14.309 & 3.639 & 6.495 & ... \
log(Fe/O) ($\lambda$4658) & ... & ... & ... & $-1.618 \pm 0.069$ & ... & $-1.982 \pm 0.039$ & ... \
log(Fe/O) (4988) & ... & ... & ... & $-1.437 \pm 0.034$ & $-1.336 \pm 0.071$ & $-1.815 \pm 0.024$ & ... \
[^1]: Based on observations collected at the European Southern Observatory, Chile, VLT and 3.6m telescopes.
[^2]: Tables 1 - 6 and Figures 1 - 2 are only available in electronic form in the online edition.
[^3]: IRAF is the Image Reduction and Analysis Facility distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation (NSF).
| ArXiv |
---
author:
- Atsuhisa Ota
- and Masahide Yamaguchi
bibliography:
- 'bib.bib'
title: Secondary isocurvature perturbations from acoustic reheating
---
Introduction
============
The conserved quantities on superhorizon scales play an important role in inflationary Universe because they connect the primordial perturbations generated during inflation with those at the late Universe. Even though most of characteristic signals of the early universe are washed away due to the thermalization processes, they keep the statistics of the primordial density fluctuations, which enables us to reveal the details of inflationary models. We usually evaluate such quantities when they exit horizons during inflation and consider them as initial conditions of the hot Big Bang universe. The curvature perturbation on the uniform density slice $\zeta$ is one of typical examples of such conserved quantities [@Malik:2003mv; @Lyth:2003im; @Lyth:2004gb]. Suppose the total energy momentum tensor is conserved and we drop the gradient terms, it is well-known that $\zeta$ is conserved even at nonlinear order when there are no non adiabatic pressure perturbations. We can also define the curvature perturbations $\zeta_{\alpha}$ on $\alpha$-fluid uniform density slice, where $\alpha=\nu,~b,~c$ represents neutrino, baryon, or cold dark matter (CDM) while $\gamma$ the photon fluid. Then, the isocurvature perturbations are introduced as $S_{\alpha \gamma}\equiv 3(\zeta_{\alpha}-\zeta_{\gamma})$. It should be noticed that $S_{\alpha \gamma}$ are also conserved at the leading order of the gradient expansion if the energy momentum tensors of $\alpha$- and $\gamma$-fluids are conserved, respectively. The conservation law of the total energy momentum tensor is universal so that the conservation laws of the curvature perturbations have been also considered to be robust as long as the other conditions are satisfied.
In this paper, we revisit the above two assumptions for the conservation laws of $\zeta_{\alpha}$: ignoring the gradient terms and the conservation laws of the energy momentum tensors. First, we point out that, at nonlinear order, we cannot justify to drop the gradient terms even when we consider the long wavelength modes; convolutions in Fourier space can pick up products of short wavelength modes, which might be significant. As a result, the total curvature perturbations might not be conserved at nonlinear order even without non-adiabatic pressure perturbations. We then newly introduce a second order conserved quantity in the presence of gradient terms. Second, we discuss energy transfer among components, that still conserves the total energy momentum but violates each one. This would lead to the evolution of superhorizon isocurvature perturbations. The typical example of the above process is acoustic reheating of the photon-baryon plasma [@Jeong:2014gna; @Nakama:2014vla; @Naruko:2015pva]. The short wavelength temperature fluctuations of the cosmic microwave background (CMB) are significantly damping due to imperfectness of the photon-baryon fluid, which produces the second order entropy production and the second order energy transfer between the photons and the baryons. These processes actually happen inside each diffusion scale; the secondary effects fluctuate on scales larger than those of corse graining. The distant patches are not necessarily reheated homogeneously if there exist three or four-point correlations of primordial density perturbations a priori [@Naruko:2015pva]. They are comparable to the non gradient terms at second order because the convolutions pick heat conduction and shear viscosity on small scales up. We investigate these diffusion effects in detail by employing the nonlinear cosmological perturbation theory, which enables us to follow the evolution of the photon distribution function directly.
We organize this paper as follows. First of all, we explain our set up for the second order perturbation theory in section \[nonlinearpert\]. Then, we discuss the non conservation of the curvature perturbations in the presence of gradient terms and introduce a new conserved quantity in section \[EMTevol\]. Section \[evophoton\] is devoted to describe the actual time evolution of the photon baryon plasma due to the weak Compton scattering. We comment on several definitions for the isocurvature perturbations during non-equilibrium periods in section \[sec:entropy\]. In the final section, we summarize our conclusions and describe future prospects related to the present results.
Set up for second order perturbation theory {#nonlinearpert}
===========================================
We need to perturb both the gravity and the matter sectors up to nonlinear order. Here, let us first define the nonlinear metric perturbations.
The metric perturbations {#section:metric}
------------------------
We start with writing the spacetime metric in the following 3+1 form: $$\begin{aligned}
ds^2&=-\mathcal N^2 d\eta^2 + \gamma_{ij}(\beta^i d\eta + dx^i)(\beta^j d\eta + dx^j)\notag \\
&=(-\mathcal N^2+\beta_k\beta^k) d\eta^2 +2\beta_i dx^i d\eta+ \gamma_{ij} dx^i dx^j.\label{def:metric}\end{aligned}$$ In other words, each component can be written as $$\begin{aligned}
g_{\mu\nu}&=\left(\begin{array}{cc}-\mathcal N^2+\beta_k\beta^k & \beta_j \\\beta_i & \gamma_{ij}\end{array}\right),\label{def:comp}\end{aligned}$$ where $\mathcal N$ and $\beta_i$ are the lapse and the shift, respectively. $\gamma_{ij}$ is the spatial metric. Let us consider nonlinear scalar perturbations introduced as $$\begin{aligned}
\mathcal N^2&=a^2 e^{2{A}},\\
\beta_i&=a^2e^{{D}}\partial_i e^{B},\label{def:shift}\\
\gamma_{ij}&=a^2e^{2{D}}\delta_{ij}\label{def:gamma},\end{aligned}$$ where $a$ is the scale factor, and we have fixed only the spacial coordinate by vanishing the anisotropic part of $\gamma_{ij}$. The nonlinear metric perturbations can be expanded as $X \equiv \sum_{n=1}
X^{(n)}$ for $X=A,B$ and $D$ with $n$ being the order in primordial perturbations. Note that the conformal Newtonian, the uniform density, the spatially flat and the velocity orthogonal isotropic gauges (comoving gauge) are mutually transformed by changing only the time slice. Here, we ignore the vector and the tensor perturbations for simplicity. This would be justified if the primordial vector perturbations and the primordial tensor ones are subdominant compared to the second order scalar ones. We include the curvature perturbation $D$ in Eq. (\[def:shift\]) to simplify the inverse matrix in the following discussions. The inverse matrixes for the induced metric and the shift vector are written as $$\begin{aligned}
\gamma^{ij}&=a^{-2}e^{-2{D}}\delta_{ij},\\
\beta^i&=e^{-{D}}\partial_ie^{B}.\end{aligned}$$ Then, we obtain $$\begin{aligned}
\beta^k\beta_k&=a^2\partial e^{{B}}\partial e^{{B}},\\
-\mathcal N^2+\beta_k\beta^k&=-a^2e^{2{A}}+a^2\partial e^{{B}}\partial e^{{B}},\label{def:lapse}\end{aligned}$$ where we write as $\partial X\partial Y\equiv \partial_i X\partial_i Y$ for notational simplicity. Eqs. (\[def:shift\]), (\[def:gamma\]) and (\[def:lapse\]) yield $$\begin{aligned}
g_{00}&=-a^2e^{2{A}}+a^2e^{2{B}}(\partial {{B}})^2,\\
g_{0i}&=a^2e^{{D}+{B}}\partial_i {B},\\
g_{ij}&=a^2e^{2{D}}\delta_{ij}.\end{aligned}$$ The inverse matrix of Eq. (\[def:comp\]) is well known: $$\begin{aligned}
g^{\mu\nu}&=\left(\begin{array}{cc}-\frac{1}{\mathcal N^2} & \frac{\beta^j}{\mathcal N^2} \\\frac{\beta^i}{\mathcal N^2} & \gamma^{ij}-\frac{\beta^i\beta^j}{\mathcal N^2}\end{array}\right).\end{aligned}$$ Then, each component of the inverse matrix can be obtained as
$$\begin{aligned}
g^{00}&=-a^{-2}e^{-2{A}},\\
g^{0i}&=a^{-2}e^{-2{A}-{D}+{B}}\partial_i{B},\\
g^{ij}&=a^{-2}e^{-2{D}}\delta^{ij}-a^{-2}e^{-2{A}-2{D}+2{B}}\partial_i{{B}}\partial_j{{B}}.\end{aligned}$$
The determinant of $g_{\mu\nu}$ can be also evaluated as $$\begin{aligned}
\sqrt{-g}=\mathcal N\sqrt{\gamma}=a^4 e^{{A}+3{D}}.\label{def:det}\end{aligned}$$
The Christoffel symbols at second order
---------------------------------------
Here and hereafter we consider only the perturbations up to second order. Up to second order, each component of the metric tensor can be rewritten as
$$\begin{aligned}
g_{00}&=-a^2e^{2{A}}+a^2(\partial {{B}})^2,\\
g_{0i}&=a^2e^{{D}+{B}}\partial_i {B},\\
g_{ij}&=a^2e^{2{D}}\delta_{ij},\end{aligned}$$
and the inverse matrix components are $$\begin{aligned}
g^{00}&=-a^{-2}e^{-2{A}},\\
g^{0i}&=a^{-2}e^{-2{A}-{D}+{B}}\partial_i{B},\\
g^{ij}&=a^{-2}e^{-2{D}}\delta^{ij}-a^{-2}\partial_i{{B}}\partial_j{{B}}.\end{aligned}$$ Let us evaluate the Christoffel symbol $$\begin{aligned}
\Gamma^\mu{}_{\nu\rho}\equiv \frac12g^{\mu\alpha}\left(\partial_\rho g_{\alpha\nu}+\partial_\nu g_{\alpha\rho}-\partial_\alpha g_{\nu\rho} \right).\end{aligned}$$ Each component of the symbols can be calculated as $$\begin{aligned}
\Gamma^0{}_{00}=&\mathcal H+{A}'+\mathcal H (\partial {B})^2+ \partial A\partial B,\\
\Gamma^0{}_{0i}=&\partial_i{A}+e^{-2{A}+{D}+B}(\mathcal H +{D}' )\partial_i {B}-\frac12\partial_i(\partial{B})^2,\\
\Gamma^0{}_{ij}=&
\frac12 \left[\partial_i {B}\partial_j{D}
+ \partial_j {B}\partial_i{D}\right]
-e^{-2{A}+{D}+{B}}\partial_i\partial_j{B}
\notag \\
&-\partial_i{B}\partial_j{B}
+e^{-2{A}+2{D}}\delta_{ij}\left[
\mathcal H+{D}'-\partial B\partial D\right],\\
\Gamma^{i}{}_{00}=&
e^{-{D}+{B}}(\mathcal H\partial_iB + \partial_iB')
+(-A'+D'+B')\partial_i B\notag\\
&+e^{-2{D}+2{A}}\partial_i{A}-\frac12\partial_i (\partial{B})^2,\\
\Gamma^i{}_{0j}=&(\mathcal H+{D}')\delta_{ij}-\partial_i {B}\partial_j {A}-\mathcal H\partial_i {B}\partial_j {B}\notag \\
& -\frac12(\partial_i{D}\partial_j {B}-\partial_j{D}\partial_i {B}) ,\\
\Gamma^i{}_{jk}=&-\partial_i{D}\delta_{jk}+\partial_k{D}\delta_{ij}+\partial_j{D}\delta_{ik}+(\partial_i{B})\partial_j\partial_k{B}\notag \\
&
-e^{-2{A}+{D}+{B}}(\mathcal H+{D}')\delta_{jk}\partial_i{B}.\end{aligned}$$
Conserved quantity at second order {#EMTevol}
==================================
In this section we show the conservation laws of the curvature perturbations and discuss the gradient corrections by full consideration of second order perturbation theory.
Divergence of the energy momentum tensor
----------------------------------------
Let $T^{(\alpha)\mu\nu}$ be energy momentum tensors of $\alpha$-fluid. Assuming the conservation of the energy momentum tensor for each fluid component $$\begin{aligned}
\nabla_\mu T^{(\alpha)\mu\nu}=0,\label{cons:colless}\end{aligned}$$ the curvature perturbations on $\alpha$-fluid uniform density slice $$\begin{aligned}
\zeta_\alpha\equiv D+\frac13\int^{\rho(\eta,\mathbf x)}_{\rho_{\rm rf}(\eta)} \frac{d \rho_\alpha}{\rho_\alpha+P_\alpha},\label{defzetatotal}\end{aligned}$$ are conserved as long as non-adiabatic pressure perturbations and the gradient terms are negligible [@Lyth:2004gb]. Let us first take a closer look at the above theorem. In this section, we do not specify a fluid component explicitly and drop the symbols from expressions. The time component of the covariant divergence can be given as $$\begin{aligned}
\nabla_\mu T^\mu{}_0=&\partial_\mu T^\mu{}_0 +T^\mu{}_0 \partial_\mu \ln\sqrt{-g}-\Gamma^\alpha{}_{\mu 0}T^\mu{}_\alpha.\label{tenkaicovemt}\end{aligned}$$ Note that only a spatial gradient term in $$\begin{aligned}
\partial_\mu T^\mu{}_0=\partial_0 T^0{}_0 + \partial_i T^i{}_0,\label{partderivs}\end{aligned}$$ is negligible on superhorizon scales. The other gradient terms arising in products of the linear perturbations cannot be dropped without their concrete evaluations since they may have significant contributions on small scales through convolutions in Fourier space. On the other hand, from Eq. (\[def:det\]), the second term in Eq. (\[tenkaicovemt\]) can be easily evaluated as $$\begin{aligned}
T^\mu{}_0 \partial_\mu \ln\sqrt{-g}=(4\mathcal H+{A}'+3{D}')T^0{}_0+T^i{}_0\partial_i({A}+3{D}).\label{volumefac}\end{aligned}$$ The term with the Christoffel symbol in Eq. (\[tenkaicovemt\]) is decomposed into 4 parts: $$\begin{aligned}
\Gamma^\alpha{}_{\mu 0}T^\mu{}_\alpha&=\Gamma^0{}_{0 0}T^0{}_0+\Gamma^0{}_{i 0}T^i{}_0+\Gamma^i{}_{0 0}T^0{}_i+\Gamma^i{}_{j 0}T^j{}_i.\end{aligned}$$ Each part can be easily calculated as $$\begin{aligned}
\Gamma^0{}_{0 0}T^0{}_0=&(\mathcal H+{A}')T^0{}_0+\mathcal H(\partial{B})^2T^0{}_0+(\partial A\partial B) T^0{}_0, \\
\Gamma^0{}_{i 0}T^i{}_0=&\mathcal H \partial_i{B} T^i{}_{0}+\partial_i{A} T^i{}_{0}, \\
\Gamma^i{}_{0 0}T^0{}_i=&
\mathcal H\partial_i BT^0{}_i+ \partial_i B'T^0{}_i + \partial_iAT^0{}_i\\
\Gamma^i{}_{j 0}T^j{}_i=&
3P(\mathcal H +{D}') -P(\partial{B}\partial{A})-P\mathcal H(\partial {B})^2,\label{Gamma0ijsubs}\end{aligned}$$ where we have decomposed $T^i{}_j$ into the trace part (that is, the pressure part) and the traceless part (the anisotropic pressure part), $$\begin{aligned}
T^i{}_j = P \delta^i{}_j + \widetilde{T}^i{}_j\end{aligned}$$ with $\widetilde{T}^i{}_i = 0$. Note that the anisotropic pressure is at least first order quantity, which would be included in the cubic order terms above; therefore, only the isotropic pressure arises in Eq. (\[Gamma0ijsubs\]). At linear order, the following relation is useful: $$\begin{aligned}
T^i{}_0+T^0{}_i=-\partial_iB(T^0{}_0 - P).\label{christoffelpart}\end{aligned}$$ Then, using Eqs. (\[partderivs\]), (\[volumefac\]) and (\[christoffelpart\]), we finally obtain
$$\begin{aligned}
\nabla_\mu T^\mu{}_0=&\partial_\mu T^\mu{}_0+3(\mathcal H+{D}')(T^0{}_0-P)-(T^0{}_0-P)\partial{B}\partial({A}+3D)-T^0{}_i\partial_i(A+3D+B').
\label{emtdiv0com}\end{aligned}$$
In most of the previous literatures where perfect fluid approximations are assumed, the gradient terms are automatically dropped. On the other hand, in our case, only the second term in Eq. (\[partderivs\]) is negligible, and products of the linear perturbations cannot be necessarily dropped. Let us introduce the energy density $\rho$ and the momentum transfer $q$ as $$\begin{aligned}
\rho&\equiv -T^0{}_0,\\
\partial_i q&\equiv \frac{ T^0{}_i}{\rho+P}.\end{aligned}$$ Then, (\[emtdiv0com\]) can be recast into $$\begin{aligned}
-\frac{1}{3(\rho+P)}\nabla_\mu T^\mu{}_0&=\mathcal H + D'+\frac{\rho'}{3(\rho+P)}-\frac13\partial{B}\partial({A}+3D)
+\frac13\partial q\partial(A+3D+B').
\label{emtdiv0com3}\end{aligned}$$
Note that we have not taken the specific time slice other than the spacial coordinate; therefore Eq. (\[emtdiv0com\]) is useful for conformal Newtonian ($B=0$), uniform density ($\delta \rho=0$), spatially flat ($D=0$) or velocity orthogonal isotropic gauges ($q=0$), respectively.\
Gradient corrections
--------------------
We are now ready to discuss the superhorizon conserved quantities in the presence of gradient terms. From Eqs. (\[cons:colless\]), (\[defzetatotal\]), and (\[emtdiv0com3\]), we immediately obtain
$$\begin{aligned}
\zeta_\alpha'=&\frac13\partial{B}\partial({A}+3D)
-\frac13\partial q_\alpha \partial(A+3D+B').
\label{non:cons:zeta}\end{aligned}$$
Eq. (\[non:cons:zeta\]) apparently shows that $\zeta_{\alpha}$ is not conserved in the presence of second order gradient terms. Note that we cannot simply ignore the RHS even for long wavelength modes as we already mentioned.\
As explained in section \[section:metric\], the spacial coordinate is already fixed; the residual linear gauge freedom is given by a shift of the time coordinate $$\begin{aligned}
\eta &\to \eta+\alpha.\label{alphadefgauge}\end{aligned}$$ Here, it should be noticed that the source term is composed of the products of linear perturbations; therefore, we only consider the linear gauge transformation here. In response to the above transformation, the metric perturbations obey the following transformation laws [@Ma:1995ey]: $$\begin{aligned}
A&= \tilde A-\alpha'-\mathcal H\alpha,\\
B&= \tilde B+ \alpha,\\
D&= \tilde D-\mathcal H\alpha.\end{aligned}$$ On the other hand, the energy density, the pressure and the momentum transfer transform as $$\begin{aligned}
\delta \rho &= \delta\tilde \rho -\alpha\rho^{(0)}{}',\\
\delta P &= \delta\tilde P -\alpha P^{(0)}{}',\\
q &= \tilde q + \alpha.\label{gt:q}\end{aligned}$$ Then, we find $$\begin{aligned}
A+3D+B'&=\tilde A+3\tilde D+\tilde B'-4\mathcal H \alpha.\label{gauge:a3dbp}\end{aligned}$$ Eqs. (\[non:cons:zeta\]) and (\[gauge:a3dbp\]) motivate us to move on to the gauge which satisfies the following relation: $$\begin{aligned}
A+3D+B'=0.\label{mygauge}\end{aligned}$$ This condition is useful since the fluid components and metric perturbations decouple in the covariant derivative of the energy momentum tensor, and gauge fixing is complete from Eq. (\[gauge:a3dbp\]). In this gauge, we find following quantities are conserved: $$\begin{aligned}
\xi_{\alpha} \equiv D + \frac16\partial{B}\partial B+\frac13\int^{\rho(\eta,\mathbf x)}_{\rho_{\rm rf}(\eta)} \frac{d \rho_\alpha}{\rho_\alpha+P_\alpha}.\label{cons:grad:second}\end{aligned}$$ Note that $\xi_{\alpha}\to \zeta_{\alpha}$ if we ignore the gradient term. We define the isocurvature perturbations in terms of $\xi_{\alpha}$ in the similar way: $$\begin{aligned}
S_{\alpha \gamma}=3(\xi_{\alpha}-\xi_{\gamma}),\label{def:iso:xi}\end{aligned}$$ which are also conserved if the energy momentum tensors are conserved and non-adiabatic pressure perturbations are absent. Thus the curvature perturbations on the uniform density slice are no more conserved in the presence of gradient terms. Instead, we introduced another conserved quantity $\xi$ at second order. $\xi$ is no more the curvature perturbation on the uniform density slice since we moved to another specific time slicing. In the next section, we consider the time evolution of $\xi$ in the presence of a collision process.
Energy transfer and time evolution of the isocurvature perturbations {#evophoton}
====================================================================
The local Minkowski frame for collision processes
-------------------------------------------------
Here, we discuss the collision processes for the weak Compton scattering, which are described by the quantum electrodynamics (QED) in the local Minkowski coordinate. To relate the local frame with the global one defined in Eq. (\[def:metric\]), let us consider the following coordinate transformations [@Pitrou:2007jy; @Naruko:2013aaa]:
$$\begin{aligned}
g_{\mu\nu}=\eta_{\bar\alpha\bar\beta}e^{\bar\alpha}{}_\mu e^{\bar\beta}{}_\nu,\end{aligned}$$
where each vierbein is defined as $$\begin{aligned}
e^{\bar 0}{}_0&=ae^A,\\
e^{\bar 0}{}_i&=0,\\
e^{\bar a}{}_0&=ae^B\partial_{\bar a}B,\\
e^{\bar a}{}_i&=ae^{D}\delta_{\bar ai}.\end{aligned}$$ For the inverse matrix, the coordinate transformation becomes $$\begin{aligned}
g^{\mu\nu}=e^\mu{}_{\bar\alpha} e^\nu{}_{\bar\beta}\eta^{\bar\alpha\bar\beta},\end{aligned}$$ where we have introduced $$\begin{aligned}
e^{0}{}_{\bar 0}&=a^{-1}e^{-A},\\
e^{0}{}_{\bar a}&=0,\\
e^{i}{}_{\bar 0}&=-a^{-1}e^{-A-D+B}\partial_{i}B,\\
e^{i}{}_{\bar a}&=a^{-1}e^{-D}\delta_{i\bar a}.\end{aligned}$$
Next, let us consider the physical momentum $\tilde p_{\bar\alpha}$ of a particle in the local Minkowski frame. The momentum satisfies $$\begin{aligned}
\tilde p_{\bar\alpha}\tilde p^{\bar\alpha}=\eta^{\bar\alpha\bar\beta}\tilde p_{\bar\alpha}\tilde p_{\bar\beta}=\eta_{\bar\alpha\bar\beta}\tilde p^{\bar\alpha}\tilde p^{\bar\beta}=-m^2,\end{aligned}$$ where $m$ is the mass of the particle. The evolution of the photon momentum in the expanding universe is written as $$\begin{aligned}
\tilde p^{\bar\alpha}\propto \frac{1}{a}.\end{aligned}$$ Then, it would be more convenient to introduce the comoving momentum so as to subtract the background spacetime evolution. For this purpose, we define the comoving momentum of the conformal flat coordinate as $$\begin{aligned}
p^{\bar\alpha}\equiv a \tilde p^{\bar\alpha}.\end{aligned}$$ The energy and the spacial direction of the photon are also introduced as $$\begin{aligned}
p&\equiv p^{\bar 0},\label{pdef}\\
n^{\bar a}&\equiv \frac{p^{\bar a}}{p}.\end{aligned}$$ Then we can write the conjugate momentum, $P^\mu=e^\mu{}_{\bar\alpha}\tilde p^{\bar\alpha}$, associated with the spacial coordinate by using $p$ and $n^i$ as $$\begin{aligned}
P^0&=\frac{\tilde p^{\bar 0}}{ae^A}=\frac{p}{a^2e^A},\label{P0def}\\
P^i&=\frac{p}{a^2e^D}(n^i-e^{B-A}\partial_i B),\\
P_0&=-pe^{A}(1- e^{B-A} n\partial {B}).\end{aligned}$$ 0 Now let us consider the time derivative of $p$ that will arise in the Boltzmann equation. First, we straightforwardly write the time derivative of Eq. (\[P0def\]): $$\begin{aligned}
\frac{d}{d\eta}\ln P^0=\frac{d\ln p}{d\eta} -\frac{dA}{d\eta}-2\mathcal H.\label{logP0prime}\end{aligned}$$ Combining Eq. (\[logP0prime\]) with the zeroth component of the geodesic equation $$\begin{aligned}
\frac{1}{P^0}\frac{dP^0}{d\eta}=-\Gamma^0{}_{\alpha\beta}\frac{P^\alpha P^\beta}{{(P^0)}^2},\end{aligned}$$ we obtain $$\begin{aligned}
\frac{d\ln p}{d\eta}=&-\frac{\partial D}{\partial \eta}-e^{A-D}(n\partial)A+e^{B-D}(n\partial)^2B\notag \\
&+\partial A\partial B+\partial B\partial D-(n\partial B)(n\partial D)+(n\partial B)^2.\label{logpbibun}\end{aligned}$$
Time evolution of the photon energy momentum tensor
---------------------------------------------------
In order to elucidate a concrete collision process, we start with constructing the photon energy momentum tensor from the phase space distribution function $f_{\gamma}$: $$\begin{aligned}
T^{(\gamma)\mu\nu} &\equiv 2\int \frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha P^\alpha)\theta(P^0)2P^\mu P^{\nu} f_\gamma, \label{def:EMT}\end{aligned}$$ where $\theta$ is a step function, $P$’s in this expression are conjugate momenta $P_\mu$, and $\alpha$ implies a fluid component. Then the covariant derivative of Eq. (\[def:EMT\]) is given by $$\begin{aligned}
\nabla_\mu T^{(\gamma)\mu}{}_{\nu}=2 \int \frac{d^3 P}{\sqrt{-g}(2\pi)^3P^0} P_{\nu} \frac{df_\gamma}{d\lambda},\label{div:emt:f}\end{aligned}$$ where $\lambda$ is an affine parameter and $P^0=d\eta/d\lambda$. Under the non canonical coordinate transformation $P_i \to p^{\bar a}$ $$\begin{aligned}
P_i=g_{ij} e^{j}{}_{\bar a}\frac{p^{\bar a}}{\bar a},\end{aligned}$$ the Jacobian is transformed as $$\begin{aligned}
|g_{ij} e^{j}{}_{\bar a}a^{-1}|=e^{3D}.\end{aligned}$$ Then, the three dimensional volume element in momentum space can be expressed as $$\begin{aligned}
d^3P\equiv
dP_1dP_2dP_3 = e^{3D}p^2 dp d\mathbf n,\end{aligned}$$ in terms of the momentum in the local conformal Minkowski frame. Using the above expression, Eq. (\[div:emt:f\]) yields
$$\begin{aligned}
\nabla_\mu T^{(\gamma)\mu}{}_{0}=-\frac{2}{a^{4}} \int \frac{p^{2} dp d\mathbf n}{(2\pi)^3} p(1 - n\partial {B}+\cdots ) \frac{df_\gamma}{d\eta},\label{div:emt:f:loc}\end{aligned}$$
where dots imply second order corrections. The integrand of Eq. (\[div:emt:f:loc\]) is directly related to the collision process through the Boltzmann equation:
$$\begin{aligned}
\frac{df_\gamma}{d\eta}=\mathcal C[f_{\gamma},\cdots],\end{aligned}$$
where the dots imply the distribution functions of the fluids which interact with the photons. When we consider the weak Compton scattering up to second order, a solution to the above Boltzmann equation can written as the superposition of a local blackbody and the spectral $y$ distortion. In this case, the collision term can be decomposed into the following form [@Ota:2016esq] $$\begin{aligned}
\mathcal C[f]=\mathcal A \mathcal G(p) +\mathcal B\mathcal Y(p),\label{col:exp}\end{aligned}$$ where we have also introduced $$\begin{aligned}
\mathcal G(p)&\equiv \left(-p\frac{\partial }{\partial p}\right)f^{(0)}(p),\\
\mathcal Y(p)&\equiv \left(-p\frac{\partial }{\partial p}\right)^{2}f^{(0)}(p)- 3\mathcal G(p),\end{aligned}$$ with $f^{(0)}(p) \equiv (e^{p/T_{\rm rf}}-1)^{-1}.$ $p$ is the local frame comoving momentum defined in Eq. (\[pdef\]), and $T_{\rm rf}$ is a (constant) comoving temperature of reference blackbody whose number density and energy density are defined as $$\begin{aligned}
N_{\gamma \rm rf}&=2\int\frac{p^{2}dp}{2\pi^{2}}f^{(0)},\\
\rho_{\gamma \rm rf}&=2\int\frac{p^{2}dp}{2\pi^{2}}pf^{(0)}.\end{aligned}$$
We can show that the isotropic component of $\mathcal A$ is zero from the fact that the weak Compton scattering does not change the number of photons. Here we introduce the following number density flux $$\begin{aligned}
N^\mu_\gamma\equiv 2\int \frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha P^\alpha)\theta(P^0)2P^\mu f_\gamma. \label{def:Nflux}\end{aligned}$$ The covariant derivative of the number flux can be calculated as $$\begin{aligned}
\nabla_\mu N_\gamma^\mu=2\int \frac{d^3P}{\sqrt{-g}(2\pi)^3P^0}\frac{d f_\gamma}{d\lambda}.\label{tochu:covN}\end{aligned}$$ Then, substituting Eqs. (\[col:exp\]) into (\[tochu:covN\]), we obtain $$\begin{aligned}
\nabla_\mu N_\gamma^\mu=3N_{\gamma \rm rf}\frac{1}{e^{A}a^4}\int \frac{d\mathbf n}{4\pi}\mathcal A=0,\end{aligned}$$ where we have used $$\begin{aligned}
2\int\frac{p^{2}dp }{2\pi^{2}}\mathcal G &= 3N_{\gamma \rm rf}, \\
2\int\frac{p^{2}dp }{2\pi^{2}}\mathcal Y &= 0.\end{aligned}$$ On the other hand, the dipole component of $\mathcal A$ is not zero. In our notation, the dipole component of $\mathcal A$ and the monopole component of $\mathcal B$ are written as [@Pitrou:2009bc; @Naruko:2013aaa; @Chluba:2012gq; @Ota:2016esq] $$\begin{aligned}
\int \frac{d\mathbf n}{4\pi}\mathbf n\mathcal A &=\frac 13n_{\rm e} \sigma_{\rm T}a\hat \partial (v+3i\Theta_{1})+\cdots \label{dip:A} \\
\int \frac{d \mathbf n}{4\pi}\mathcal B&=\frac13n_{\rm e} \sigma_{\rm T}a\hat \partial v\hat\partial (v+3i\Theta_{1})\label{Monopo:B},\end{aligned}$$ where the dots represent the second order corrections, and $\hat \partial$ corresponds to $i\mathbf k/|\mathbf k|$ in Fourier space [^1]. $v=|\mathbf v|$ is the magnitude of the velocity of the baryon fluid, and $\Theta_{1}$ is the dipole component of the photon temperature perturbations. $n_{\rm e}$ is the electron density, $\sigma_{\rm T}$ is the Thomson scattering cross section, and $a$ is a scale factor. Using Eqs. (\[div:emt:f:loc\]), (\[col:exp\]), (\[dip:A\]) and (\[Monopo:B\]), we find $$\begin{aligned}
\nabla_\mu T^{(\gamma )\mu}{}_{0}=-
\frac{4}{3a^{4}}\rho_{\gamma,{\rm rf}}n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),\label{res:cons:emt}\end{aligned}$$ where we have used $$\begin{aligned}
2\int\frac{p^{2}dp p}{2\pi^{2}}\mathcal G&=4\rho_{\gamma \rm rf},\\
2\int\frac{p^{2}dp p}{2\pi^{2}}\mathcal Y&=4\rho_{\gamma \rm rf}.\end{aligned}$$
We are now ready to discuss the superhorizon evolution of the isocurvature perturbations in the presence of heat conduction between electrons and photons. From Eqs. (\[emtdiv0com3\]), (\[cons:grad:second\]), and (\[res:cons:emt\]), we find $$\begin{aligned}
\xi_\gamma'=& \frac{1}{3} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),
\label{emtdiv0com4}\\
\xi_b'=&
-\frac{1}{3R} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),
\label{emtdiv0com5}\\
\xi_c'=&0,
\label{emtdiv0com6}\end{aligned}$$ where $R=3\rho_{b}/4\rho_{\gamma}=3a\rho_{b,{\rm rf}}/4\rho_{\gamma,{\rm rf}}$, and we used Eq. (\[emtdiv0com3\]) for the baryon fluid with $$\begin{aligned}
\nabla_\mu T^{(\gamma )\mu}{}_{0}+ \nabla_\mu T^{(b)\mu}{}_{0}=0.\end{aligned}$$ Then time derivatives of the isocurvature perturbations defined with Eq. (\[def:iso:xi\]) become $$\begin{aligned}
S_{b\gamma }'&=-\frac{(1+R)}{R} n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}),\label{Sgabbabprime}\\
S_{c\gamma }'&=- n_{\rm e} \sigma_{\rm T}a(\hat\partial v-\partial B)\hat\partial (v+3i\Theta_{1}).\label{Sgabbabprimec}\end{aligned}$$ These expressions imply that the heat conduction from electron fluid is responsible for the change of the total photon energy while the friction heat from the intrinsic photon shear viscosity $\Theta_2$ is not. This is because the friction heat from the photon anisotropic stress does not increase the net energy in a photon system as long as we deal with background and perturbations as a whole system. Some confusion may occur if one separates the background and perturbations as done in the previous literatures, in which energy transfers from perturbations to the background are discussed. In response to Eq. (\[res:cons:emt\]), the energy momentum conservation for baryons should be also broken while those of the total fluids and the other dark sectors remain conserved. Note that these expressions are independent of the gauge choice (\[mygauge\]) since Eq. (\[gt:q\]) for the baryons and the photons are written as $$\begin{aligned}
v&\to \tilde v=v+k\alpha,\\
\Theta_{1}&\to \tilde \Theta_{1}=\Theta_{1}+\frac{ik}{3}\alpha.\end{aligned}$$\
Role of the primordial non Gaussianity
--------------------------------------
Eqs. (\[Sgabbabprime\]) and (\[Sgabbabprimec\]) imply that the observed isocurvature perturbations are superposition of the primordial isocurvature and the secondary isocurvature. Suppose we only have the adiabatic perturbations at the beginning, the Fourier space isocurvature perturbations are simply given as $$\begin{aligned}
S_{\alpha \gamma,\mathbf k} &= \int \frac{d^{3}k_{1}d^{3}{k_{2}}}{(2\pi)^{6}}(2\pi)^{3}\delta^{(3)}(\mathbf k_{1}+\mathbf k_{2}-\mathbf k) \mathcal S_{\alpha}(\mathbf k_{1},\mathbf k_{2})\zeta_{\mathbf k_{1}}\zeta_{\mathbf k_{2}},\label{So:def}\end{aligned}$$ Here, the transfer functions in Fourier space are introduced as $$\begin{aligned}
\mathcal S_{\alpha }(\mathbf k_{1},\mathbf k_{2})= \hat k_{1}\cdot \hat k_{2} \int d\eta w_{\alpha} n_{\rm e}\sigma_{\rm T}a[v(k_{1})-k_{1} B(k_{1})][v(k_{2})+3i\Theta_{1}(k_{2})],\end{aligned}$$ where $w_{b}=(1+R)/R$, $w_{c}=1$. On the other hand, the statistics of the adiabatic perturbations in the Fourier spaces are written as $$\begin{aligned}
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\rangle
&=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^2 \mathbf k_i\right]P_\zeta(k_1),\label{power:zeta}\\
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\rangle
&=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^3 \mathbf k_i\right]B_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3),\label{bis:zeta}\\
\langle \zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4}\rangle&=(2\pi)^3\delta^{(3)}\left[\sum_{i=1}^4 \mathbf k_i\right]T_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4).\label{tri:zeta}\end{aligned}$$ Then, the cross correlations with the adiabatic perturbations and the auto correlations become
$$\begin{aligned}
\langle S_{\alpha\gamma ,\mathbf k}\zeta_{\mathbf k'} \rangle &=(2\pi)^{3}\delta(\mathbf k+\mathbf k') P_{\alpha \zeta}(\mathbf k),\\
\langle S_{\alpha \gamma ,\mathbf k}S_{\beta \gamma\mathbf k'} \rangle &=(2\pi)^{3}\delta(\mathbf k+\mathbf k') P_{\alpha \beta}(\mathbf k),\end{aligned}$$
where the powerspectra are calculated as $$\begin{aligned}
P_{\alpha \zeta}&=\int \frac{d^{3}k_{1}}{(2\pi)^{3}} \mathcal S_{\alpha}(\mathbf k_{1},\mathbf k- \mathbf k_{1}) B_{\zeta}(\mathbf k_{1},\mathbf k-\mathbf k_{1},\mathbf k),\\
P_{\alpha \beta}&=\prod_{i=\alpha,\beta}\left[\int \frac{d^{3}k^{(i)}_{1}}{(2\pi)^{3}}\mathcal S_{i}(\mathbf k^{(i)}_{1},\mathbf k- \mathbf k^{(i)}_{1}) \right] T_{\zeta}(\mathbf k^{(\alpha)}_{1},\mathbf k-\mathbf k^{(\alpha)}_{1},\mathbf k^{(\beta)}_{1},\mathbf k-\mathbf k^{(\beta)}_{1}).\end{aligned}$$ The scale dependences of the secondary powerspectra depend on the shape of the primordial non Gaussianity. As an example, consider the local forms of bispectra and trispectra: $$\begin{aligned}
&B_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3)=\frac65f^{\rm loc.}_{\rm NL}\left[P_{\zeta}(k_1)P_{\zeta}(k_2) + (\text{2 perms.}) \right],\label{def:fnl}\\
&T_\zeta(\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4)=\tau^{\rm loc.}_{\rm NL}\left[P_{\zeta}(k_1)P_{\zeta}(k_2)P_{\zeta}(|\mathbf k_1+\mathbf k_3|) + (\text{11 perms.}) \right]\label{def:tnl},\end{aligned}$$ where we have omitted terms proportional to $g^{\rm loc.}_{\rm NL}$ for simplicity. Then the dominant contributions become
$$\begin{aligned}
P_{\alpha \zeta}&\approx \frac{12}{5}f^{\rm loc.}_{\rm NL}P_{\zeta}(k)\times \int \frac{d^{3}k_{1}}{(2\pi)^{3}} \mathcal S_{\alpha}(\mathbf k_{1},-\mathbf k_{1})P_{\zeta}(k_{1}),\\
P_{\alpha \beta}&\approx 4\tau^{\rm loc.}_{\rm NL}P_{\zeta}(k) \times \prod_{i=\alpha,\beta}\left[\int \frac{d^{3}k^{(i)}_{1}}{(2\pi)^{3}}S_{i}(\mathbf k^{(i)}_{1},-\mathbf k^{(i)}_{1})P_{\zeta}(k^{(i)}_{1}) \right].\end{aligned}$$
0 In Eq. (\[So:def\]), the relative velocity $v+3i\Theta_{1}$ is strongly suppressed during tight coupling approximation, but we have a significant numerical factor coming from $n_{\rm e} \sigma_{\rm T}a$. As a result, the transfer function (\[So:def\]) becomes order of unity if there is no special cancellation. Therefore, the cross- and the auto correlation functions of the secondary isocurvature perturbations are expected to be $P_{\alpha \zeta}\approx f^{\rm loc.}_{\rm NL}\times 10^{-18}$ and $P_{\alpha \beta}\approx \tau^{\rm loc.}_{\rm NL}\times 10^{-27}$, respectively. Thus, the secondary effect on the isocurvature perturbations are negligible if we do not have significant local type primordial non Gaussianity. Thus, the powerspectra of the secondary isocurvature perturbations are the same form with the linear isocurvature powerspectrum. The disconnected part of the trispectrum leads to the following contribution for $\mathbf k\neq 0$ and $\mathbf k'\neq 0$:
$$\begin{aligned}
P^{(d)}_{\alpha \beta}&\approx \int \frac{d^{3}k_{1}}{(2\pi)^{3}} \prod_{i=\alpha,\beta}\ \mathcal S_{i}(\eta^{i},\mathbf k_{1},-\mathbf k_{1}) P_{\zeta}(k_{1})P_{\zeta}(k_{1}).\end{aligned}$$
Then we obtain $P^{(d)}_{\alpha \beta}\approx{\rm const.}$ for the disconnected trispectrum. This suggests the spectral index is 4, and the powerspectrum is mainly enhanced on scales where the physical process occurs. In other words, the Gaussian fluctuations cannot produce the superhorizon isocurvature modes.
0
Generation of Entropy perturbations {#sec:entropy}
===================================
Besides the conserved quantity (\[cons:grad:second\]), one may wonder if we could introduce the similar quantities by using the entropy flux. In this section, we introduce the secondary entropy perturbations, which are not identified with the isocurvature perturbations if we consider non equilibrium universe during recombination.
Entropy flux non conservation
-----------------------------
Suppose the universe is out of equilibrium states, the standard thermodynamic relation among the entropy density, the energy density and pressure is not applicable. Instead, we introduce the (Shannon) entropy flux, which is defined in terms of a logarithm of the number of states [@Khatri:2012rt], $$\begin{aligned}
S^\mu_\gamma&\equiv 2\int
\frac{d^4P}{\sqrt{-g}(2\pi)^4}2\pi\delta(P_\alpha
P^\alpha)\theta(P^0)2 P^\mu \mathcal F \label{def:shannon},\\
\mathcal F&\equiv \left[(f_\gamma+1)\ln(f_\gamma+1)-f_\gamma\ln f_\gamma \right].\label{def:calF}\end{aligned}$$ Note that this definition reproduces the thermodynamic entropy density for the Planck distribution. The covariant divergence of this entropy flux can be calculated as $$\begin{aligned}
\nabla_\mu S^\mu_\gamma =2\int \frac{d^3P}{\sqrt{-g}(2\pi)^3P^0}\frac{d
\mathcal F}{d\lambda}.\label{tochu:divS}\end{aligned}$$ A solution to the Boltzmann equation with the weak Compton collision process can be written as a superposition of the local blackbody and the spectral $y$ distortion up to the second order in the primordial fluctuations [@Pitrou:2009bc; @Naruko:2013aaa]. Such an ansatz can be expanded as follows: $$\begin{aligned}
f_{\gamma}=&f^{(0)}(p)+\left[\Theta +\frac32\Theta^2\right] \mathcal G(p) + \left[\frac12\Theta^2+y\right]\mathcal Y(p),\label{def:non-thermal_anz}\end{aligned}$$ where $\Theta=\Theta^{(1)}+\Theta^{(2)}$ and $y=y^{(2)}$ are the temperature perturbation and spectral $y$ distortion, respectively. Then, Eq. (\[tochu:divS\]) vanishes at zeroth and first orders of the perturbations, but there exist non-zero contributions at second order, which is manifest from the following expression, $$\begin{aligned}
\frac{1}{P^0}\frac{d \mathcal F}{d\lambda} &=
\frac{d \mathcal F}{d\eta}=\frac{p}{T_{\rm rf}}\left[ \left(1-\Theta\right)\mathcal A\mathcal G+\mathcal B\mathcal Y\right].\label{calc:F2}\end{aligned}$$ Here we have replaced the Liouville term with the collision terms by using the Boltzmann equation. Using the Boltzmann equation for the $y$ distortion [@Pitrou:2009bc; @Naruko:2013aaa; @Chluba:2012gq; @Ota:2016esq], $$\begin{aligned}
y'&=\mathcal B-\Theta\mathcal A,
\label{y:eq}\end{aligned}$$ with Eqs. (\[tochu:divS\]), and (\[calc:F2\]), we find $$\begin{aligned}
\nabla_\mu S^\mu_\gamma= \frac{4\pi^2}{15 a}\left(\frac{T_{\rm rf}}{a}\right)^3 y'_0.\label{eq:14}\end{aligned}$$ Thus, entropy increases with the generation of the spectral $y$ distortion. The physical entropy density can be defined as $S_\gamma\equiv - n_{\mu} S^{\mu}_\gamma$ with $ n_{\mu}\equiv
{\nabla_\mu \eta}(-\nabla_\nu\eta \nabla^\nu\eta)^{-\frac12}$ being the normalized 1-form orthogonal to a constant $\eta$ hypersurface. One may wonder if Eq. (\[eq:14\]) can also be derived from the standard thermodynamic relation, $$\begin{aligned}
\frac{dS_\gamma}{dt}=\frac{1}{T}\frac{dQ}{dt},\label{defthent}\end{aligned}$$ where $Q$ is thermodynamical heat. What we found is not a reinterpretation of this relation because we identify “heat” for the photon baryon fluid in the presence of non-equilibrium effect; thermodynamic arguments are not applicable to. Thus, the generation of $y$ distortion is not directly identified with the entropy perturbation production without a kinetic description based on the Boltzmann equation.\
Entropy perturbations at second order
-------------------------------------
We are now ready to introduce a quantity $$\begin{aligned}
\zeta^{(S)}_\gamma\equiv D+\frac{A}{3}+\frac13\ln \left(
\frac{S^0}{S^0_{\rm rf}}\right),\label{zeta_S}\end{aligned}$$ where $S^{0}_{\rm rf}= 4\pi^2T_{\rm rf}^3/(45a^4)$. This quantity is conserved as long as entropy flux conserves at leading order of the gradient expansion. Let us check this statement by considering the covariant derivative of the entropy flux: $$\begin{aligned}
\nabla_\mu S^\mu_\gamma=\partial_\mu S^{\mu}+S^\mu\partial_\mu\ln \sqrt{-g}.\end{aligned}$$ Dropping a gradient term $\partial_i S^i$, we find $$\begin{aligned}
\zeta_\gamma ^{(S)}{}'= -\frac{S^i}{3S^0}\partial_i(A+3D)+y'_0,\end{aligned}$$ where we have used Eq. (\[eq:14\]). The first term represents a volume effect, which is manifest only when we take into account the next leading order of the gradient expansion. $\zeta^{(S)}_{\gamma}$ is conserved even at second order when the scattering is negligible, but only if we move on to $A+3D=0$ gauge, where the volume element does not fluctuate. However, note that gauge is not completely fixed on this slice. The second term arises as a result of the entropy production, which, in this paper, we should keep since the imperfectness of a fluid on subhorizon scales could be non negligible due to convolutions.\
The entropy density is not necessarily proportional to the number density if both of them are evaluated for a non-equilibrium state. In our case, its discrepancy is expressed in terms of $y$ distortion, which characterizes the deviation from the thermodynamic system. The curvature perturbations on the uniform number density slice can be also defined through the same procedures with the entropy: $$\begin{aligned}
\zeta^{(N)}_\gamma\equiv D+\frac{A}{3}+\frac13\ln \left(
\frac{N^0}{N^0_{\rm rf}}\right),\end{aligned}$$ where $N^{0}_{\rm rf}= 2\zeta(3) T_{\rm rf}^3/(\pi^2 a^4)$. Using the number flux conservation laws and dropping $\partial_i N^i$, we find $$\begin{aligned}
\zeta_\gamma ^{(N)}{}'=-\frac{N^i}{3N^0}\partial_i(A+3D).\end{aligned}$$ Thus $\zeta_\gamma ^{(N)}$ is also a conserved quantity if we have the number conservation law and take the leading order of the gradient expansion. Note that $\zeta^{(N)}_\gamma$ is also conserved in $A+3D=0$ gauge even at second order without truncating the higher order gradient corrections.\
Now let us consider the following isocurvature perturbations: $$\begin{aligned}
S^{(NS)}_{\alpha \gamma} \equiv \zeta^{(N)}_{\alpha} - \zeta^{(S)}_{\gamma}.\label{def:iso:NS}\end{aligned}$$ This is a covariant extension of $\delta \left(N_{\alpha}/S_{\gamma}\right)$ at nonlinear order. It should be noticed that the following relation $$\begin{aligned}
\frac{N^i}{N^0}=\frac{S^i}{S^0},\end{aligned}$$ applies at linear order even for the present case since the spectral distortion is a second order effect. Then we obtain $$\begin{aligned}
S^{(NS)'}_{\alpha \gamma} = -y_{0}'.\end{aligned}$$ Thus, the entropy perturbations are also conserved quantity in the presence of gradient terms if the photon entropy flux and $\alpha$-fluid number density flux are conserved. We may also consider isocurvature perturbations defined as $$\begin{aligned}
S^{(N)}_{\alpha \gamma}\equiv 3(\zeta^{(N)}_\alpha-\zeta^{(N)}_\gamma),\label{def:photoniso:nn}\end{aligned}$$ which are conserved if each number density flux are conserved.
The above discordance between Eqs. (\[def:iso:NS\]) and (\[def:photoniso:nn\]) motivates us to newly define the *photon isocurvature perturbation* as a fluctuation of a fraction between the photon number density and the photon entropy density $$\begin{aligned}
S^{(NS)}_{\gamma\gamma}=-3y_0.\end{aligned}$$ This is nothing but the spectral $y$ distortion. For the chemical equilibrium period in the early universe where $y$ distortion is erased, it is obvious that $S^{(N-S)}_{\alpha\gamma}=S^{(N)}_{\alpha\gamma}=S_{\alpha\gamma}$ due to thermodynamic relations.\
Thus, Eqs. (\[def:iso:NS\]) and (\[def:photoniso:nn\]) can be also defined as superhorizon conserved quantities without scattering processes. However, in contrast to the conservation laws of energy momentum tensor, the conservation laws for the number flux and the entropy flux are not necessarily established in the whole cosmic history. Therefore, Eq. (\[def:iso:xi\]) is much more important than the others.
Conclusions {#conclusion}
===========
In this paper, we revisited the two assumptions for the conservation laws of the superhorizon isocurvature perturbations: the negligibility of the gradient terms and the energy conservation laws for the component fluids. We pointed out that the second order gradient terms are not necessarily dropped even if we consider the long wavelength modes. Then, we have introduced new second order quantities, which are conserved even in the presence of gradient terms if there are no non-adiabatic pressure perturbations. It should be noticed that they coincide with the curvature perturbations on the uniform density slice only when we can ignore the gradient terms. The total energy momentum tensor is always conserved, but that for each component fluid is not necessarily conserved. As such an example, we discuss the weak Compton scattering that transfers the energy between the photons and baryons. We found that the secondary isocurvature perturbations are generated due to this energy transfer. The powerspectra of secondary isocurvature perturbations become scale invariant if we consider the local form of the primordial tri- and bispectrum. On the other hand, the disconnected part of the trispectrum only produces the isocurvature perturbations on scales where the actual physical process occurs. We also commented on the entropy perturbations, which are usually equivalent to the isocurvature perturbations in thermal equilibrium states. However, in our case, we cannot identify these two quantities when the universe is dominated by the weak Compton scattering and is not in thermal equilibrium. We found that the entropy perturbations can be understood in terms of the spectral $y$ distortion, which is a non thermal deviation from the blackbody spectrum produced in the weak Compton scattering dominated universe.
The new quantity $\xi$ we have introduced in this paper is still gauge dependent. However, it should be noticed that we can always define the gauge invariant quantities recursively even at nonlinear order as pointed out in Ref. [@Nakamura:2014kza]. Using this formalism, the gauge invariant expressions for $\xi$ would be investigated in future works. Though we only consider the weak Compton scattering, it would also be interesting if we consider the similar heat conduction from the other species such as neutrinos in the earlier epoch. This would lead to a new constraint on curvature perturbations with extremely short wavelength though it requires explicit evaluation for each scattering process, which is left for our future works. So far, we have discussed the late epoch when the universe is in neither kinetic nor chemical equilibrium. In the early epoch, the full considerations of the Compton collision terms are necessary. When there exist relativistic electrons that can sufficiently transfer the photon energy, local kinetic equilibrium is expected. In this case, the $y$ distortion may be transformed into the $\mu$ distortion, which is defined as chemical potential of a Bose distribution function. In the earlier epoch, the number changing process such as the double Compton effects, Bremsstrahlung or pair annihilation are also non-negligible. They adjust the number density and erase the spectral distortions so as to realize chemical equilibrium. Referring to Eqs. (\[tochu:covN\]) and (\[div:emt:f\]), such violation of photon number density conservation would break photon energy conservation as well. Then, secondary isocurvature perturbations might be additionally generated on superhorizon scales, but further study is necessary to make a clearer statement.
We would like to thank Misao Sasaki and Atsushi Naruko for useful discussion on conservation of isocurvature perturbations on superhorizon scales. The authors are grateful to Kouji Nakamura and Karim Malik for helpful discussions. We also would like to thank Rampei Kimura for careful reading of our manuscript. This work was supported in part by JSPS Grant-in-Aid for PD Fellows (A.O.), JSPS Grant-in-Aid for Scientific Research Nos. 25287054 (M.Y.) and 26610062 (M.Y.), MEXT KAKENHI for Scientific Research on Innovative Areas “Cosmic Acceleration” No. 15H05888 (M.Y.).
[^1]: In Ref. [@Ota:2016esq], the angular dependence was not properly treated, and $\hat \partial$ was dropped.
| ArXiv |
---
abstract: 'We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for $\mathcal{C}^{1,1}$-smooth convex domains in $\mathbb{C}^{2}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in $\Bbb C^n$ are given.'
address:
- |
Institute of Mathematics and Informatics\
Bulgarian Academy of Sciences\
Acad. G. Bonchev 8, 1113 Sofia, BulgariaFaculty of Information Sciences\
State University of Library Studies and Information Technologies\
Shipchenski prohod 69A, 1574 Sofia, Bulgaria
- |
Université de Toulouse\
UPS, INSA, UT1, UTM\
Institut de Mathématiques de Toulouse\
F-31062 Toulouse, France
- |
Institute of Mathematics, Faculty of Mathematics and Computer Science\
Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
author:
- Nikolai Nikolov
- 'Pascal J. Thomas'
- Maria Trybuła
title: 'Gromov (non-)hyperbolicity of certain domains in $\mathbb{C}^{n}$'
---
Introduction and statements
===========================
In [@Gromov], Gromov introduced the notion of almost hyperbolic space. He discovered that “negatively curved” space equipped with some distance share many properties with the prototype, even though the distance does not come from a Riemannian metric. This gave the impulse to intensive research to find new interesting classes of spaces which are hyperbolic in that sense. In this paper we are mainly interested in investigating this concept with respect to the Kobayashi distance of convex domains. One may suspect that it is a restriction to consider only the Kobayashi metric. Actually, because the Kobayashi distance of a ($\Bbb C$-)convex domain containing no complex lines, as well as of a bounded strictly pseudoconvex domain, is bilipschitz equivalent to the (inner) Carathéodory and Bergman distances (see [@NPZ Theorem 12] and [@Nikolov Proposition 4]), it does not matter which one we choose (see below). Recall that a set $E$ in $\Bbb C^n$ is called *$\Bbb C$-convex* if any intersection of $E$ with a complex line $l$ and its complement in $l$ are both connected in $l$ (cf. [@APS]).
The notion of a bilipschitz equivalence has the following generalization.
Let $(X_1,d_1)$ and $(X_2,d_2)$ be two metric spaces. Then a map $\varphi:X_1\to X_2$ is said to be a *quasi-isometry* if there are constants $c_1,c_2 >0$ such that for any $x,y\in X_1$, $$c_1^{-1} d_1(x,y) - c_2 \le d_2\left( \varphi(x), \varphi(y) \right) \le c_1 d_1(x,y) + c_2.$$ Two distances $d_1,d_2$ on a set $X$ are said to be quasi-isometrically equivalent if the identity map is a quasi-isometry from $(X,d_1)$ to $(X,d_2)$.
Gromov hyperbolicity is well-known to be invariant under bijective quasi-isometries of path metric spaces (cf. [@Jesus Theorems 3.18, 3.20]).
\[defghyp\] Let $(D,d)$ be a metric space. Given points $x,y,z\in D,$ the *Gromov product* is $$(x,y)_{z}=d(x,z)+d(z,y)-d(x,y).$$ Let $$S_d(p,q,x,w)= \min\{(p,x)_{w},(x,q)_{w}\}-(p,q)_{w}.$$ $(D,d)$ is *Gromov hyperbolic* if $$\sup_{p,q,x,w\in D} S_d(p,q,x,w) <\infty.$$ If $S_d(p,q,x,w)\le 2\delta$, then $(D,d)$ is called *$\delta$-hyperbolic*.
We refer to [@Jesus] for other characterizations of Gromov hyperbolicity, especially for path metric spaces. We chose this one because it does not use geodesics explicitly.
$(D,d)$ is a *path metric space* if, for any two points $x,y\in D$ and any number $\varepsilon >0,$ there exists a rectifiable path joining $x$ and $y$ with length at most $d(x,y)+\varepsilon.$ Then the distance $d$ is called *intrinsic*.
From now on, let $D$ be a domain in $\mathbb{C}^{n}$.
Denote by $c_D$ and $l_{D}$ the Carathéodory distance and the Lempert function of $D$: $$c_{D}(z,w)=\sup\{\tanh^{-1}|f(w)|:f\in\mathcal{O}(D,\Bbb D), f(z)=0\},$$ $$l_{D}(z,w)=\inf\{\tanh^{-1}|\alpha|:\exists\varphi\in\mathcal{O}(\mathbb{D},D)
\hbox{ with }\varphi(0)=z,\varphi(\alpha)=w\},$$ where $\mathbb{D}$ is the unit disc. The Kobayashi distance $k_{D}$ is the largest pseudodistance not exceeding $l_{D}.$ The inner Carathéodory distance $c_D^i$ is the inner pseudodistance associated to $c_D.$ So, $c_D\le c_D^i\le k_D\le l_D.$ By Lempert’s seminal paper [@Lempert], we have equalities above if $D$ is convex (or bounded, $\mathcal{C}^2$-smooth and $\Bbb C$-convex).
An important property of $k_{D}$ is that it is the integrated form of the Kobayashi metric $\kappa_{D}$ of $D,$ i.e. $$\begin{gathered}
k_{D}(z,w)=\inf\{\int_0^1\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt:\\
\gamma:[0,1]\to D\textup{ is a smooth curve with }\gamma(0)=z\textup{ and }\gamma(1)=w\},\end{gathered}$$ where $$\kappa_{D}(z;X)=\inf\{|\alpha|:\exists\varphi\in\mathcal{O}(\mathbb{D},D)
\hbox{ with }\varphi(0)=z,\ \alpha\varphi'(0)=X\},$$ $z,w\in D,\ X\in\mathbb{C}^{n}.$
We refer to [@Jarnicki] for basic properties of the invariants defined here and of the Bergman distance $b_D.$
We shall say that $D$ is Gromov *$s$-hyperbolic* if $(D,s_D)$ is Gromov hyperbolic with respect to the distance $s$ (this should not be confused with $\delta$-hyperbolicity for some constant $\delta>0$).
The first result concerning Gromov $k$-hyperbolicity for domains in ${\Bbb C}^n$ was given by Balogh and Bonk [@Bonk] who gave both positive and negative examples. They proved that any bounded strictly pseudoconvex domain is Gromov $k$-hyperbolic [@Bonk Theorem 1.4]. They also showed that the Cartesian product of bounded strictly pseudoconvex domains is not Gromov $k$-hyperbolic [@Bonk Proposition 5.6] which is a special case of a general situation mentioned in many places, but without proof (cf. [@Gaussier]).
\[I\] Assume that $(X_1,d_1)$ is a path metric space with $d_1$ unbounded and $(X_2,d_2)$ a metric space with unbounded $d_2$. Let $d=\max\{d_1,d_2\}$. Then $(X_1\times X_2,d)$ is not Gromov hyperbolic.
\[I\]
The next proposition is more general than the previous one. However its proof uses Proposition \[I\].
\[II\] Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces, such that one of them is a path metric space. Let $d=\max\{d_1,d_2\}$. Then $(X_1\times X_2,d)$ is Gromov hyperbolic if and only if one of the factors is Gromov hyperbolic and the metric of the second one is bounded (in particular, it is also Gromov hyperbolic).
Moreover, the proof of Proposition \[I\] and Remark 1 (following this proof) show that the path property in Proposition \[II\] can be replaced the following.
A metric space $(Y,d)$ admits the *weak midpoints property* if either $d$ is bounded or there exist sequences $(x_k),(y_k),(z_k)\subset Y$ such that $d(x_k,y_k)\to\infty$ and $$\label{wmp}
\frac{d(x_k,z_k)}{d(x_k,y_k)}\to\frac{1}{2},\ \frac{d(y_k,z_k)}{d(x_k,y_k)}\to\frac{1}{2}.$$
\[III\] Let $D_1$ and $D_2$ be Kobayashi hyperbolic domains (i.e. $k_{D_1}$ and $k_{D_2}$ are distances) admitting non-constant bounded holomorphic functions (for example, bounded domains). Then $D_1\times D_2$ is not Gromov $k$-hyperbolic.
To see this, it is enough to observe that if a domain $G$ in $\Bbb C^n$ admits a non-constant bounded holomorphic function $f$ and $|f(z_j)|\to\sup_G|f|,$ then $k_G(z,z_j)\ge c_G(z,z_j)\to\infty.$
Note also that Proposition \[I\] implies that if $D_1$ and $D_2$ are planar domains with complements containing more than one point (i.e. they are Kobayashi hyperbolic), then $D_1\times D_2$ is not Gromov $k$-hyperbolic (use that $k_{D_k}(z,z_j)\to\infty$ as $z_j\to\partial
D_k,$ $k=1,2$).
As an immediate consequence we obtain that the polydisc is not Gromov $k$-hyperbolic. Moreover, even its “symmetrized” counterpart is not.
\[G\_n\] $\mathbb{G}_{n}$ is not Gromov $c$- nor $k$-hyperbolic for $n\geq 2$.
For the convenience of the reader, recall that the [*symmetrized polydisc*]{} $\mathbb{G}_{n},$ which is of great relevance due to its properties and role (cf. [@AY], [@Costara]), is the image of the holomorphic map $$\pi :\mathbb{D}^{n}\rightarrow\mathbb{C}^{n},\ \pi=(\pi_1,\ldots,\pi_n),$$ $$\pi_k(z_1,\ldots,z_n)=\sum_{1\leq j_1<\ldots<j_k\leq n}z_{j_1}\ldots z_{j_k},\ z_1,\ldots,z_n\in\mathbb{D},\ 1\leq k\leq n,$$ which is proper from $\mathbb{D}^{n}$ to $\mathbb{G}_{n}$.
Another interesting domain, the [*tetrablock*]{} (cf. [@AWY]), fails to be Gromov $k$-hyperbolic, too. Let $$\varphi:\mathcal{R}_{II}\rightarrow \mathbb{C}^{3},\ \varphi(z_{11},z_{22},z)=(z_{11},z_{22},z_{11}z_{22}-z^{2}),$$ where $\mathcal{R}_{II}$ denotes the classical Cartan domain of the second type (in $\mathbb{C}^{3}$), i.e. $$\mathcal{R}_{II} =\{\widetilde{z}\in\mathcal{M}_{2\times 2}(\mathbb{C}): \widetilde{z}=\widetilde{z}^{t},\ \lVert \widetilde{z}\rVert<1\},$$ where $\lVert \cdot \rVert$ is the operator norm and $\mathcal{M}_{2\times 2}(\mathbb{C})$ denotes the space of $2 \times 2$ complex matrices (we identify a point $(z_{11},z_{22},z)\in\mathbb{C}^{3}$ with a $2\times 2$ symmetric matrix $ \left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right) ). $ Then $\varphi$ is a proper holomorphic map and $\varphi(\mathcal{R}_{II})=\mathbb{E}$ is a domain, called the tetrablock.
\[tetrablock\] $\mathbb{E}$ is not Gromov $k$-hyperbolic.
Since $\Bbb G_2$ and $\mathbb{E}$ are bounded $\mathbb{C}$-convex domains (see [@G2 Theorem 1 (i)] and [@Zwonek Corollary 4.2]), it follows that they are not Gromov $c^i$- nor $b$-hyperbolic either.
Buckley in [@Buckley], claimed that it is because of the flatness of the boundary rather than the lack of smoothness that Gromov hyperbolicity fails. Recently, Gaussier and Seshadri have provided a proof of that conjecture. More precisely, their main result in [@Gaussier Theorem 1.1] states that any bounded convex domain in $\mathbb{C}^{n}$ whose boundary is $\mathcal{C}^{\infty}$-smooth and contains an analytic disc, is not Gromov $k$-hyperbolic. Lemma 5.4 in their proof used the $\mathcal{C}^{\infty}$ assumption in an essential way. Our aim is to prove this result in a shorter way in $\mathbb{C}^{2}$, assuming only $\mathcal{C}^{1,1}$-smoothness. Moreover, the proofs of the facts we use are more elementary.
\[Gaussier\] Let $D$ be a convex domain in $\mathbb{C}^{2}$ containing no complex lines.[^1] Assume that $\partial D$ is $\mathcal{C}^{1,1}$-smooth and contains an analytic disc. Then $D$ is not Gromov $k$-hyperbolic.
Besides, we give a partial answer to the question raised in [@Bonk].
\[Pascal\] Let $D$ be a $\mathcal{C}^{1,1}$-smooth convex bounded domain in $\mathbb{C}^{2}$ admitting a defining function of the form $\varrho (z)=-\Re z_{1}+\psi (|z_{2}|)$ near the origin, where $\psi$ is a $\mathcal{C}^{1,1}$-smooth nonnegative convex function near $0$ satisfying $\psi (0)=0,$ and $$\label{infinite type}
\limsup _{x\rightarrow 0}\frac{\log \psi (|x|)}{\log |x|}=\infty.\footnote{If $\psi$ is $\mathcal{C}^{\infty},$ then $0$ is of infinite type if and only if condition (\ref{infinite type}) holds.}$$ Then $D$ is not Gromov $k$-hyperbolic.
Finally, note that there is no connection between Gromov hyperbolicity and pseudoconvexity. Indeed, take any strictly pseudoconvex domain $G.$ As we have already mentioned, $G$ is Gromov $k$-hyperbolic, and $k_G$ and $c_G$ are bilipschitz equivalent. Hence $G$ is Gromov $c$-hyperbolic, too. Assume that, respectively, $A\Subset G$ and $B$ is a relatively closed subset of $G$ such that $G\setminus A$ is a domain and that $B$ is negligible with respect to the $(2n - 2)$-dimensional Hausdorff measure. Then $G\setminus A$ is Gromov $c$-hyperbolic and $G\setminus B$ is Gromov $k$-hyperbolic, since $$c_{G\setminus A} = c_{G}|_{(G\setminus A)\times (G\setminus A)}$$ (by the Hartogs extension theorem) and $$k_{G\setminus B} = k_{G}|_{(G\setminus B)\times (G\setminus B)}$$ (cf. [@Jarnicki Theorem 3.4.2]).
However, the example with $G\setminus B$ does not have a smooth boundary. The next proposition yields, in particular, a family of non-pseudoconvex domains with smooth boundaries which are Gromov $k$-hyperbolic.
\[ball\] Let $G$ be a bounded domain in $\Bbb C^n\ (n\geq 2).$ Assume that $D\Subset G$ is a $\mathcal{C}^{2}$-smooth domain in $\Bbb C^n$ and its Levi form has at least one positive eigenvalue at each boundary point. Then $G\setminus D$ is a domain such that $k_{G\setminus D}$ is quasi-isometrically equivalent to $k_{G}|_{(G\setminus D)\times (G\setminus D)}.$[^2]
In particular, if $G$ is Gromov $k$-hyperbolic, then so is $G\setminus\overline{D}.$
\[str\_psc\] If $D\Subset G$ are strictly pseudoconvex domains in $\Bbb C^n,$ then $G\setminus\overline{D}$ is a Gromov $k$-hyperbolic domain.
The estimates that we use in the proof of Proposition 5 do not hold for the planar annulus $\mathbb{A}_r=\{z\in\mathbb{C}: r^{-1}<|z|<r\}$ ($r>1$). However, any finitely connected proper planar domain is Gromov $k$-hyperbolic (cf. [@RT Proposition 3.2]).
\[compact\] Let $G$ be a bounded domain in $\Bbb C^n\ (n\geq 2).$ Assume that $K$ is compact subset of $G$ such that through any point $z\in\Bbb C^n\setminus K$ passes a complex line disjoint from $K.$ Then $G\setminus D$ is a domain such that $k_{G\setminus D}$ is quasi-isometrically equivalent to $k_{G}|_{(G\setminus D)\times (G\setminus D)}$.[^3]
In particular, if $G$ is Gromov $k$-hyperbolic, then so is $G\setminus K.$
Note that we may take $K$ to be any compact (${\Bbb C}$-)convex set, since any compact or open ${\Bbb C}$-convex set $E$ in $\Bbb C^n$ is *linearly convex*, i.e. through any point in $\Bbb C^n\setminus E$ passes a complex line disjoint from $E$ (cf. [@APS Theorem 2.3.9]).
Throughout the paper $d_{D}$ denotes the (Euclidean) distance to $\partial D.$ A point $z\in\mathbb{C}^{n}$ we write as $(z_{1},\ldots,z_{n}),\ z_{j}\in\mathbb{C}.$
An appendix at the end of the paper includes some of the estimates for the Kobayashi distance and metric used in the proofs.
Proofs
======
[*Proof of Proposition \[I\].*]{} Assume that $(X,d)$ is $\frac{\delta}{2}$-hyperbolic. Put $k=3+\delta$. Then there are points $y_1,y_2\in X_2$ such that $d_2(y_1,y_2)=2s\geq 2k$. Choose points $x_1,x_2^\ast\in X_1$ with $d_1(x_1,x_2^\ast)\geq 2s$. By the path property of $X_1$, there is a $d_1$-continuous curve $\gamma:[0,1]\to X_1$ joining the points $x_1$ and $x_2^\ast$ such that $L_{d_1}(\gamma)<d_1(x_1,x_2^\ast)+1$. Note that $t\to
d_1(x_1,\gamma(t))$ is continuous. Hence there is a smallest number $t_0$ such that $d_1(x_1,\gamma(t_0))=2s$. Set $x_2=\gamma(t_0)$.
Now $L(\gamma|_{[0,t_0]}) \ge d_1(x_1,x_2)=2s$, and $$L(\gamma|_{[0,t_0]}) = L(\gamma) - L(\gamma|_{[t_0,1]})
\le d_1(x_1,x_2^\ast)+1 - d_1(x_2,x_2^\ast) \le d_1(x_1,x_2) +1.$$ Let $t_1$ be the smallest number in $[0,t_0]$ such that $d_1(x_1,\gamma(t_1))=s$. Set $x_3=\gamma(t_1)$. Then $$d_1(x_2,x_3) \ge d_1(x_1,x_2) - d_1(x_1,x_3) = s, \mbox{ and }$$ $$d_1(x_2,x_3) = L(\gamma|_{[0,t_1]}) = L(\gamma|_{[0,t_0]}) - L(\gamma|_{[t_1,t_0]})
\le 2s+1 - d_1(x_1,x_2) = s+1.$$ Hence, $s= d_1(x_1,x_3)\leq
d_1(x_3,x_2)<s+1$.
Now define the following points in $X_1 \times X_2$: $x=(x_1,y_1)$, $y=(x_2,y_1)$, $w=(x_3,y_1)$, and $z=(x_3,y_2)$. Then $d(z,w)=d(z,x)=d(z,y)=2s$ and $(x,y)_w\leq 1$, $(x,z)_w= d(x,w)=s$, $(y,z)_w=d(y,w)\ge s-1$. By the assumption of $\frac{\delta}{2}$-hyperbolicity we reach the inequalities $$1\geq (x,y)_w\geq \min\{(y,z)_w,(x,z)_w\}-\delta\geq
s-1-\delta\ge 2$$ which is a contradiction.
An essential ingredient in the proof of Proposition \[I\] is the existence of points $x_{1},x_{2},x_{3}$ such the triangle inequality is a near-equality, namely $(x_1, x_2)_{x_3} \le 1$. The condition is equivalent to $(x_1, x_2)_{x_3} = o(d(x_1,x_2))$, and $\left| d(x_1,x_3) - d(x_2,x_3)\right| = o(d(x_1,x_2))$.
Using this weaker hypothesis and following the steps of the above proof, setting $2s =d(x_1,x_2)$ as before, we find $$o(s)\geq (x,y)_w\geq s - o(s) -\delta,$$ leading to a contradiction when $s\to\infty$. Similar changes can be made in the proof below.
[*Proof of Proposition \[II\].*]{} Let first $(X_1,d_1)$ be $2\delta$-hyperbolic and $d_2\le 2c.$ Since $d\le d_1+2c,$ it follows that $$(x_1,y_1)_{w_1}-2c\le (x,y)_w\le(x_1,y_1)_{w_1}+4c$$ and then $(X,d)$ is $(\delta+3c)$-hyperbolic.
Assume now that $(X,d)$ is $\delta$-hyperbolic. Following the proof of Proposition \[I\], we deduce that one of the distances is bounded, say $d_2\le 2c.$ Then we get as above that $(X_1,d_1)$ is $(\delta+3c)$-hyperbolic.
[*Proof of Proposition \[G\_n\].*]{} Let $a\in\Bbb D,$ $p_a=\pi (a,\ldots,a),$ $q_a=\pi (a,\ldots,a,-a)$ and $m_a=\pi (a,\ldots,a,0).$ We shall show that $$S_{c_{\Bbb G_n}}(p_a,q_a,m_a,0)\to\infty\mbox{ as }|a|\to 1.$$ It follows exactly in the same way that $S_{k_{\Bbb G_n}}(p_a,q_a,m_a,0)\to\infty\mbox{ as }|a|\to 1.$ So, $\Bbb G_n$ is not Gromov $c$- nor $k$-hyperbolic for $n\geq 2.$
The holomorphic contractibility implies that $$c_{\Bbb G_n}(p_a,q_a) \ge c_{\Bbb D}(a^n,-a^n)=2c_{\Bbb D}(a^n,0),$$ $$\max\{c_{\Bbb G_n}(p_a,0), c_{\Bbb G_n}(q_a,0),c_{\Bbb G_n}(p_a,m_a),c_{\Bbb G_n}(q_a,m_a)\}\le c_{\Bbb D}(a^n,0).$$ Thus $$S_{c_{\Bbb G_n}}(p_a,q_a,m_a,0)\ge c_{\Bbb G_n}(m_a,0)+2c_{\Bbb D}(a^n,0)-2c_{\Bbb D}(a,0).$$ Since $$2c_{\Bbb D}(a,0)-2c_{\Bbb D}(a,0)\to\log n\mbox{ as }|a|\to 1,$$ it remains to see that $c_{\Bbb G_n}(m_a,0)\to\infty$ as $|a|\to 1.$ This follows by the fact that any point $b\in\Bbb G_n$ is a weak peak point, i.e. there exists $f_b\in\mathcal O(\Bbb G_n,\Bbb D)$ such that $|f_b(z)|\to 1$ as $z\to b$ (a consequence of [@Costara Corollary 3.2]).
[*Proof of Proposition \[tetrablock\].*]{} Let $a\in (0,1),$ and put $P_a=\varphi(\textup{diag}(a,a)),\ Q_a=\varphi(\textup{diag}(a,-a)).$ Recall that $\Phi_a(Z)=(Z-a\textup{I})(\textup{I}-aZ)^{-1}$ is an automorphism of $\mathcal{R}_{II}.$ Direct computations show that $$\varphi\circ\Phi_a (\left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right))=\varphi\circ\Phi_a (\left( \begin{array}{ll}
z_{11} & -z \\
-z & z_{22}
\end{array} \right)),$$ whenever $\left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right)\in\mathcal{R}_{II}.$ Thus, $\Phi_a$ induces an automorphism $\widetilde{\Phi}_{a}$ of $\mathbb{E}.$ It follows from this and [@AWY Corollary 3.7] that $$\begin{gathered}
k_{\mathbb{E}}(0,(a,b,p))=\tanh^{-1}\max\Big{\{}\frac{|a-\overline{b}p|+|ab-p|}{1-|b|^{2}},\frac{|b-\overline{a}p|+|ab-p|}{1-|a|^{2}}\Big{\},}\\
(a,b,p)\in\mathbb{E},\end{gathered}$$ $$2k_{\mathbb{E}}(P_a,0),\ 2k_{\mathbb{E}}(Q_a,0),\ k_{\mathbb{E}}(P_a,Q_a)
=-\log d_{\mathbb{D}}(a) +\textup{O}(1).$$ Observe that if $f(\lambda)=(0,\lambda,0),$, then $g_a=\widetilde{\Phi}_{-a}\circ f$ is a complex geodesic for $k_{\Bbb E}$ with $P_a=g_a(0)$, $Q_a=g\left(-\frac{2a}{1+a^2}\right)$. Note that the Kobayashi middle point $R_{a}$ of $g_{a}|_{[-\frac{2a}{1+a^2},0]}$ tends to the boundary; more precisely, $$R_{a}=g_{a}(-a)\rightarrow \textup{diag}(1,0) \textup{ as }a\rightarrow 1.$$ Consequently, $S_{k_{\Bbb E}}(P_a,Q_a,R_a,0)$ is comparable with $k_{\mathbb{E}}(R_a,0).$ By Proposition A1(b) (see Appendix), $k_{\mathbb{E}}(R_a,0)\to\infty$ as $a\to 1,$ which finishes the proof.
[*Proof of Theorem \[Gaussier\].*]{} Since $\partial D$ contains an analytic disc, it is well known that it contains an affine disc (cf. [@NPZ Proposition 7]). We assume that this disc has center $0$ and lies in $\{z_{1}=0\},$ and that $D\subset\{\Re z_{1}>0\}.$
We can find an $r>0$ such that for any $\delta >0$ small enough there exist two discs $\Delta (\tilde{p}_{\delta},r)$ and $\Delta (\tilde{q}_{\delta},r)$ in $D_{\delta}=D\cap\{z_{1}=\delta\}$ which touch $\partial{D}$ at two points $\hat{p}_{\delta}$ and $\hat{q}_{\delta}$ with $\lVert \hat{p}_{\delta}-\hat{q}_{\delta}\rVert >5r.$
We identify $\partial D\cap \{ z_{1}=0\}$ with a closed, bounded, convex subset of $\mathbb C$, which is the closure of its interior. Call this interior $D_0.$
There exists $\zeta_0 \in D_0$ such that $d_{D_0}(\zeta_0)=\max_{\zeta\in D_0}d_{D_0}(\zeta).$ Then $$M= \left\{ p \in \partial D_0 : |p- \zeta_0 | = \min_{\xi \in \partial D_0} |\xi- \zeta_0 |
\right\}$$ is a not empty set which cannot be contained in any half plane $$H_\theta = \{ \zeta :\\ \Re [(\zeta-\zeta_0)e^{-i\theta}] <0\}:$$ if it were, one could find $\varepsilon>0$ such that $d_{D_0}( \zeta_0 + \varepsilon e^{i\theta})
> d_{D_0}(\zeta_0)$. So there are $\hat p \neq \hat q
\in M$ such that $\arg( (\hat p -\zeta_0) (\hat q-\zeta_0)^{-1} ) \ge 2\pi/3$. Take $r\in (0, \frac{\sqrt 3}{5+\sqrt 3}|\hat p -\zeta_0|)$, $\tilde{p} = \zeta_0 + ( 1 - r|\hat p -\zeta_0|^{-1} ) (\hat p -\zeta_0)$, and $\tilde{q}$ chosen likewise. Then the discs $\Delta (\tilde{p},r)\subset D_{0}$ and $\Delta (\tilde{q},r)\subset D_{0}$ are tangent to $\partial D_{0}$ at $\hat{p}$ and $\hat{q}$.
Now we want to move these discs inside $D.$ By $\mathcal{C}^{1,1}$-smoothness of $D,$ we can move them (in $\mathbb{C}^{2}$) along the vector $(1,0)$ inside $D,$ i.e. $\Delta (\tilde{p},r),\,\Delta (\tilde{q},r)\subset D\cap\{z_{1}=\delta\}=D_{\delta},$ for $0<\delta <\delta_{0}.$ If they do not touch $\partial D_{\delta},$ then shift them (separately at every sublevel set) to the boundary but leaving their centers on the real line passing through $\tilde{p}+(\delta,0)$ and $\tilde{q}+(\delta,0).$ Denote new discs by $\Delta (\tilde{p}_{\delta},r),\,\Delta (\tilde{q}_{\delta},r),$ and by $\hat{p}_{\delta},\,\hat{q}_{\delta}$ points of contact of those discs with $\partial D_{\delta}.$
Choose now a point $a=(\delta_{0},0)\in D\textup{ (}\delta_{0}>0\textup{)}$ and consider the cone with vertex at $a$ and base $\partial D\cap\{z_{1}=0\}.$ Denote by $G_{\delta}$ the intersection of this cone and $\{z_{1}=\delta\}.$ For any $\delta>0$ small enough the line segment with ends at $\tilde{p}_{\delta}$ and $\hat{p}_{\delta}$ intersects $\partial G_{\delta},$ say at $p_{\delta}.$ Define $q_{\delta}$ in a similar way.
Set $\tilde{s}_{\delta}=\frac{ \tilde{p}_{\delta}+\tilde{q}_{\delta}}{2}.$ We shall show that $S_{k_D}(p_{\delta},q_{\delta}, \tilde{s}_{\delta}, a) \rightarrow \infty$ as $\delta\rightarrow 0.$ For this we will see that $(p_{\delta},\tilde{s}_{\delta})_{a}-(p_{\delta},q_{\delta})_{a}\rightarrow \infty$ as $\delta\rightarrow 0.$ It will follow in the same way that $(q_{\delta},\tilde{s}_{\delta})_{a}-(p_{\delta},q_{\delta})_{a}\rightarrow \infty.$
It is enough to prove that $$\label{1}
k_{D}(q_{\delta},a)-k_{D}(\tilde{s}_{\delta},a)<c_{1}$$ and $$\label{2}
k_{D}(p_{\delta},q_{\delta})-k_{D}(p_{\delta},\tilde{s}_{\delta})\rightarrow \infty.$$ Here and below $c_{1},c_{2},\ldots$ denote some positive constants which are independent of $\delta.$
For (\[1\]), observe that, by Propositions A1(a) and A2 (see Appendix), $$\label{Kobayashi estimates}
k_{D}(\tilde{s}_{\delta},a)\geq\frac{1}{2}\log\frac{d_{D}(a)}{d_{D}(\tilde{s}_{\delta})}\textup{ and }2k_{D}(q_{\delta},a)\leq -\log d_{D}(q_{\delta})+c_{2}.$$ It remains to use that $d_{D}(\tilde{s}_{\delta})=d_{D}(q_{\delta})$ for any $\delta >0$ small enough.
To prove (\[2\]), denote by $F_{\delta}$ the convex hull of $\Delta (\tilde{p}_{\delta},r)$ and $\Delta (\tilde{s}_{\delta},r).$ Then by inclusion $k_{D}(p_{\delta},\tilde{s}_{\delta})\leq k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta})$.
$k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta}) < -\frac{1}{2}\log d^{\prime}_{D}(p_{\delta})+c_{3},$ where $d^{\prime}_{D}$ is the distance to $\partial D$ in the $z_{2}$-direction.
[*Proof.*]{} For $\delta>0$ small enough we have that $$d^{\prime}_{D}(p_{\delta}) = d_{D_\delta}(p_{\delta}) =
d_{F_\delta}(p_{\delta}) = d_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta})$$ because the closest point on $\partial D_\delta$ belongs to $\partial \Delta (\tilde{p}_{\delta},r)$. Now $k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta}) \le
k_{F_{\delta}}(p_{\delta},\tilde{p}_{\delta}) + k_{F_{\delta}}(\tilde p_{\delta},\tilde{s}_{\delta}) $.
Since $\Delta (\tilde{p}_{\delta},r)\subset F_{\delta}$, $$\begin{gathered}
k_{F_{\delta}}(p_{\delta},\tilde{p}_{\delta})
\le k_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta},\tilde{p}_{\delta})
= \frac{1}{2}\log \frac{1+\frac{|p_{\delta}-\tilde{p}_{\delta}|}{r} }{1-\frac{|p_{\delta}-\tilde{p}_{\delta}|}{r}}
\\
\le - \frac{1}{2} \log d_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta}) +\frac12 \log (2r)
= -\frac{1}{2}\log d^{\prime}_{D}(p_{\delta})+ \frac12 \log (2r) .\end{gathered}$$ On the other hand, by using a finite chain of discs of radius $r$ with centers on the line segment from $\tilde{p}_{\delta}$ to $\tilde{s}_{\delta}$, we obtain that $$k_{F_{\delta}}(\tilde p_{\delta},\tilde{s}_{\delta}) \le
4 \frac{|\tilde p_{\delta}-\tilde{s}_{\delta}|}{r} \le C(r).\qed$$
Now, we shall show that $$\label{3}
2k_{D}(p_{\delta},q_{\delta})>-\log d^{\prime}_{D}(p_{\delta})-\log d^{\prime}_{D}(q_{\delta})-c_{4},$$ which implies (\[2\]), because $d^{\prime}_{D}(q_{\delta})\rightarrow 0$ as $\delta\rightarrow 0.$
Since the Kobayashi distance is intrinsic, we may find a point $m_{\delta}\in D$ such that $$\lVert p_{\delta} - m_{\delta}\rVert=\lVert q_{\delta}-m_{\delta}\rVert\geq\frac{\lVert p_{\delta}-q_{\delta}\rVert}{2},$$ $$k_{D}(p_{\delta},q_{\delta})>
k_{D}(p_{\delta},m_{\delta})+k_{D}(m_{\delta},q_{\delta})-1.$$
Let $\check{p}_{\delta}\in\partial D$ be the closest point to $p_{\delta}$ in the direction of the complex line through $p_{\delta}$ and $m_{\delta}.$
Recall that $d_D'$ is the distance to $\partial D$ in the $z_{2}$-direction and $d_D(p_\delta)$ is attained in $z_{1}$-direction for any $\delta>0$ small enough. This means that the standard basis is adapted to the local geometry of $\partial D$ near $p_\delta$, and more precisely, if $X=(X_1,X_2)\in\Bbb C^2$ is a unit vector, [@NPZ (4)] states in this case that there exists a constant $C$ such that $$\frac{1}{d_D(p_\delta,X)}\le \frac{|X_1|}{d_{D}(p_\delta)} +
\frac{|X_2|}{d'_{D}(p_\delta)}\le\frac{C}{d_D(p_\delta,X)},$$ where $d_D(\cdot;X)$ is the distance to $\partial D$ in direction $X.$ Since $d'_{D}\ge d_D$, we obtain $$d_D(p_\delta;X)\le c_5d_D'(p_\delta).$$
Let $X= \frac{m_\delta-p_{\delta}}{\|m_\delta-p_{\delta}\|}.$ Then $\| p_{\delta}-\check{p}_{\delta}\| = d_X (p_{\delta})$ and thus $$\label{4}
\lVert p_{\delta}-\check{p}_{\delta}\rVert< c_{5}d^{\prime}_{D}(p_{\delta}).$$
By convexity, $D$ is on the one of the sides, say $H_{\delta},$ of the real tangent plane to $\partial D$ at $\check{p} _{\delta}.$ Since $\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{d_{H_\delta}(m_{\delta})}
=\frac{\lVert p_{\delta}-\check{p}_{\delta}\rVert}{d_{H_\delta}(p_{\delta})}$, it follows by that $$2k_{D}(p_{\delta},m_{\delta})
\geq 2k_{H_{\delta}}(p_{\delta},m_{\delta})
\geq \log \frac{d_{H_\delta}(m_{\delta})}{d_{H_\delta}(p_{\delta})}
=
\log\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}.$$ Applying the triangle inequality and , we get that $$\begin{gathered}
\log\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}\ge
\log\frac{\lVert m_\delta -p_\delta\rVert -\lVert p_\delta -\check{p}_\delta\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}\ge\\
\log\left(\frac{r}{2\lVert p_{\delta}-\check{p}_{\delta}\rVert}-1\right)\ge\log\frac{r}{2c_5 d^{\prime}(p_\delta)}\,-\,1,\end{gathered}$$ for any $\delta >0$ small enough. So $2k_{D}(p_{\delta},m_{\delta})>-\log d^{\prime}_{D}(p_{\delta})-c_{6}.$ Similarly, $2k_{D}(q_{\delta},m_{\delta})>-\log d^{\prime}_{D}(q_{\delta})-c_{6},$ which implies (\[3\]), and completes the proof.
All the above arguments hold in $\mathbb{C}^{n}$, $n\ge 3$, except (\[4\]).
[*Proof of Theorem \[Pascal\].*]{} Since the case when $\psi(z_{0}) =0$ for some $z_{0}\not= 0,$ is covered by Proposition \[Gaussier\], we may assume that $\psi^{-1}\{0\}=\{0\}$. Also assume $p=(1,0)\in D.$
Let $\alpha (x),$ small enough, an increasing function such that for any $x>0,\ \psi^{\prime}(x)\geq\psi^{\prime}((1-\alpha (x))x)\geq\frac{1}{2}\psi' (x)$. We choose, for $x>0,\ q(x)=(\psi(x),0),\ r(x)=(\psi (x),-(1-\alpha (x))x),\ s(x)=(\psi (x),(1-\alpha (x))x).$
We claim that for $x$ small enough:
1. $d_{D}(q)=\psi (x),$\[A\]
2. $\frac{\alpha (x)}{4}x\psi^{\prime}(x)\leq d_{D}(s),d_{D}(r)\leq \alpha (x)x\psi^{\prime}(x),$\[B\]
3. the functions $k_{D}(s,q)+\frac{1}{2}\log\alpha (x)$ and $k_{D}(r,q)+\frac{1}{2}\log\alpha (x)$ are bounded,\[C\]
4. the function $k_{D}(r,s)+\log\alpha (x)$ is bounded.\[D\]
Before we proceed to prove the claims we make some general observation about infinite order of vanishing.
\[psipsi\] For any $\varepsilon>0$ and $A>0$, there exists $x \in (0,\varepsilon)$ such that $\frac{x \psi'(x) }{\psi(x)} >A$.
Suppose instead that there exist $\varepsilon>0$ and $A>0$ such that $ \frac{x \psi'(x) }{\psi(x)} \le A$ for $0<x\le \varepsilon$. Then $$\frac{d}{dx} \left( \log \psi(x)\right) \le \frac{A}x, \quad 0<x\le \varepsilon,$$ so $\log(\psi(\varepsilon)) - \log (\psi(x)) \le A \left( \log \varepsilon - \log x \right)$, i.e. $$\psi(x) \ge \frac{\psi(\varepsilon)}{\varepsilon^{A}} x^{ A}, \quad 0<x\le \varepsilon,$$ which means that at the point $0$ there is finite order of contact with the tangent hyperplane, a contradiction.
Assume the claims for a while, and observe that for any $x$ verifying the conclusion of Lemma \[psipsi\] we have $$(r,p)_{q}-(r,s)_{q}, \quad (p,s)_{q}-(r,s)_{q}\geq -\frac{1}{2}\log\frac{\psi (x)}{x\psi^{\prime}(x)}+C_{1}.$$ Since the above quantity can be made arbitrarily large, it finishes the proof.
It remains to prove (\[A\])-(\[D\]).
(\[A\]) is clear. Next, since $(\psi((1-\alpha (x))x),(1-\alpha (x))x)\in D,\ d_{D}(s)\leq \psi (x)-\psi ((1-\alpha (x))x))\leq \alpha (x)x\psi^{\prime}(x)$ by convexity. Let $L$ be the real line through $(\psi((1-\alpha (x))x),(1-\alpha (x))x)$ and $(\psi (x),x).$ Its slope is less than $\psi^{\prime}(x),$ so $d_{D}(s)\geq \textup{dist}\,(s,L^{\prime}),$ where $L^{\prime}$ is the line through $(\psi((1-\alpha (x))x),(1-\alpha (x))x)$ with slope $\psi^{\prime}(x),$ so $$\begin{gathered}
d_{D}(s) \geq \frac{\psi (x) - \psi ((1-\alpha(x))x)}{\sqrt{1+\psi^{\prime}(x)^{2}}} \\
\geq \frac{1}{2} {\alpha (x) \times \psi^{\prime}((1-\alpha (x))x)} \geq \frac{1}{4} \alpha (x) \times \psi^{\prime}(x).\end{gathered}$$ Thus, $\frac{\alpha (x)}{4}x\psi^{\prime}(x)\leq d_{D}(s)\leq \alpha (x)x\psi^{\prime}(x).$ Analogous estimates hold for $r,$ which gives (\[B\]).
The analytic disc $\zeta \mapsto (\psi (x),x\zeta)$ provides immediate upper estimates in (\[C\]) and (\[D\]).
To get lower estimate for $k_{D}(s,q),$ we map $D$ to a domain in $\mathbb{C}$ by the complex affine projection $\pi_{s}$ to $\{z_{1}=\psi (x)\},$ parallel to the complex tangent space to $\partial D$ at the point $(\psi(x),x).$ Then $\pi_{s}(D)=\{\psi (x)\}\times D_{s},$ where $D_{s}$ is a convex domain in $\mathbb{C},$ containing the disc $\{|z_{2}|<x\},$ and its tangent line at the point $x$ is the real line $\{\Re z_{2}=x\}.$ The projection is given by the explicit formula $$\pi_{s}(z_{1},z_{2})=\Big{(}\psi (x),z_{2}+\frac{\psi (x)-z_{1}}{\psi^{\prime}(x)}\Big{)}.$$ We renormalize by setting $f_{+}(z)=1-\frac{1}{x}[\pi_{s}(z)]_{2}.$ Therefore $f_{+}(D) \subset H=\{z\in\mathbb{C}:\Re z>0\},$ so $$\label{E}
k_{D}(s,q)\geq
k_{H}(f_{+}(s),f_{+}(q)) = k_H (\alpha(x),0)
\geq -\frac{1}{2}\log \alpha (x) +C_{2},$$ where $C_{2}>0$ does not depend on $x.$
The estimate for $k_{D}(r,q)$ proceeds along the same lines, but we use the projection $\pi_{r}$ to $\{z_{1}=\psi (x)\}$ along the complex tangent space to $\partial D$ at $(\psi (x),x),$ given by $$\pi_{r}=\Big{(}\psi (x),z_{2}-\frac{\psi (x)-z_{1}}{\psi^{\prime}(x)}\Big{)}.$$ Note that choosing $f_{-}(z)=1+\frac{1}{x}[\pi_{r}(z)]_{2},$ we have $f_{-}(D)\subset\{\Re z>0\}.$
Now we tackle the lower estimates for $k_{D}(r,s).$ Let $\gamma$ be any piecewise $\mathcal{C}^{1}$ curve such that $\gamma (0)=s,\,\gamma (1)=r.$ Let $c_{0}<\frac{1}{2}.$ We claim that there exists $t_{0}\in (0,1)$ such that if we set $u=\gamma (t_{0}),$ then $|f_{+}(u)|,|f_{-}(u)|\geq c_{0}.$
For this write $\gamma=(\gamma_{1},\gamma_{2}).$ Set $\zeta_{1}=1-\frac{\psi (x)\,-\,\gamma_1(t_0)}{x\psi^{\prime}(x)}.$ By the explicit form of $\pi_{s},$ the condition $|f_{+}(u)|\geq c_{0}$ reads $|\zeta_{1}-\frac{\gamma_{2}(t_0)}{x}|\geq c_{0},$ and the condition $|f_{-}(u)|\geq c_{0}$ reads $|\zeta_{1}+\frac{\gamma_{2}(t_0)}{x}|\geq c_{0}.$ We claim that the discs $\overline{\mathbb{D}}(\zeta_{1},c_{0})$ and $\overline{\mathbb{D}}(-\zeta_{1},c_{0})$ are disjoint for any $t.$ Indeed, they would intersect if and only if $0\in\overline{\mathbb{D}}(\zeta_{1},c_{0}),$ which implies $$\Re\Big{(}\frac{\gamma_{1}(t_0)}{x \psi^{\prime}(x)}\Big{)}
\leq -1+c_{0}+\frac{\psi (x)}{x\psi^{\prime}(x)}\leq -\frac{1}{3}$$ for any $x$ such that $\frac{\psi (x)}{x\psi^{\prime}(x)}\leq \frac{1}{6}$, which we may assume by Lemma \[psipsi\]. In particular $\Re\gamma_{1}(t_0)<0,$ which is excluded for any $\gamma (t)\in D.$ Now let $t_{1}=\max\{t:\frac{\gamma_{2}(t)}{x}\in\overline{\mathbb{D}}(\zeta_{1},c_{0})\}.$ Then $\frac{\gamma_{2}(t_{1})}{x}\notin\overline{\mathbb{D}}(-\zeta_{1},c_{0}),$ and by continuity there is $\eta >0$ such that $\frac{\gamma_{2}(t_{1}+\eta)}{x}\notin\overline{\mathbb{D}}(-\zeta_{1},c_{0}),$ and of course $\frac{\gamma_{2}(t_{1}+\eta)}{x}\notin\overline{\mathbb{D}}(\zeta_{1},c_{0})$ by maximality of $t_{1},$ so $t_{0}=t_{1}+\eta$ will provide a point satisfying the claim.
Consequently, taking a curve $\gamma$ such that $$k_D(r,s)+1 > \int_{0}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt,$$ $$\begin{gathered}
\int_{0}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt
\geq\int_{0}^{t_{0}}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt
+\int_{t_{0}}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt \\
\geq k_{D}(r,u)+k_{D}(u,s).\end{gathered}$$ We end the proof by estimating $k_{D}(r,u)$ in the same way as we did $k_D(r,q)$ above, and $k_{D}(u,s)$ as as we did $k_D(s,q)$ above, using the maps $f_+,f_-$ and estimates about the Kobayashi distance in a half plane.
[*Proof of Proposition \[ball\].*]{} Set $G'=G\setminus\overline{D}.$
Assume first that $G'$ is not a domain. Let $G''$ be a bounded connected component of $G'.$ Consider a farthest point $a\in\partial G''$ from the origin. Then $a$ is a concave boundary point of $D$ which a contradiction.
Choose now a smooth domain $E$ such that $D\Subset E\Subset G.$ By smoothness and compactness, there is a constant $C>0$ such that any two points in $G'\cap\overline{E}$ may be jointed by a path in $G'\cap\overline{E}$ of length (at most) $C.$ By Propositions A3 (after integration) and A4, we may find a constant $c>0$ such that $$k_{G'}(z,w)\le c||z-w||^{1/4} \le c^2,\ z,w\in G'\cap\overline{E},$$ $$k_{G'}\le c k_G,\ z,w\in G\setminus E.$$ So there are constants $c_1, c_2$ such that $$k_G\le k_{G'}\le c_1 k_G + c_2,\ z,w\in G\setminus E\mbox{ or }z,w\in G'\cap\overline{E}.$$
Finally, let $z\in G\setminus E$ and $w\in G'\cap\overline{E}.$ Since the Kobayashi distance is intrinsic and $\partial E$ is compact, we may find a point $u\in\partial E$ such that $$k_G(z,w)=k_G(z,u)+k_G(u,w).$$ It follows that $$k_G(z,w)\ge ck_{G'}(z,u)+ck_{G'}(u,w)\ge ck_{G'}(z,w).$$
[*Proof of Proposition \[compact\].*]{} It clear that $G'=G\setminus K$ is a domain. Following the previous proof, let $E$ be a domain such that $K\subset E\Subset G.$ All the arguments in the previous proof work except the estimate on $G'\cap\overline{E}.$ We need to find a constant $c>0$ such that $$k_{G'}(z,w)\le c,\ z,w\in G'\cap\overline{E}.$$
Take a complex line $L$ through $z$ which is disjoint from $K.$ Then the disc in $L$ with center of $z$ and radius $d_{G\cap L}(z)$ lies in $G'.$ Choose a common point $z'$ of this disc and $\partial E$ such that $||z-z'||=d_{E\cap L}(z).$ Then $$\tanh l_{G'}(z,z')\le\frac{||z-z'||}{d_{G\cap L}(z)}\le
1-\frac{r}{d_{G\cap L}(z)}\le 1-\frac{2r}{s},$$ where $r=\mbox{dist}(E,\partial G)$ and $s=\mbox{diam }G.$
Choosing $w'$ for $w$ in the same way, it follows that $$k_{G'}(z,w)\le k_{G'}(z,z')+k_{G'}(z',w')+k_{G'}(w',w)\le 2c'+c''$$ where $c'=\tanh^{-1}(1-2r/s)$ and $c''=\max k_{G'}|_{\partial E\times\partial E}.$
Appendix
========
\(a) Let $D$ be proper convex domain in $\Bbb C^n.$ Then $$c_{D}(z,w)\geq\frac{1}{2}\left|\log\frac{d_{D}(z)}{d_{D}(w)}\right|,\quad z,w\in D.$$
\(b) Let $D$ be proper $\Bbb C$-convex domain in $\Bbb C^n.$ Then $$c_{D}(z,w)\geq\frac{1}{4}\left|\log\frac{d_{D}(z)}{d_{D}(w)}\right|,\quad z,w\in D.$$
Let $b$ be a $\mathcal{C}^{1,1}$-smooth boundary point of a domain $D$ is $\Bbb C^n$ and let $K\Subset D.$ Then there exist a neighborhood $U$ of $b$ and a constant $C>0$ such that $$2k_D(z,w)\le-\log d_D(z)+C,\quad z\in D\cap U,\ w\in K.$$
Let $b$ is a $\mathcal{C}^2$-smooth non-pseudoconvex boundary point of a domain $D$ in $\Bbb C^2.$ Then there exist a neighborhood $U$ of $b$ and a constant $c>0$ such that $$c\kappa_D(z;X)\le\frac{|\langle\nabla d_D(z),X\rangle|}{(d_D(z))^{3/4}}+|X|,\quad z\in D\cap U,\ X\in\Bbb C^n.$$
Let $D$ be a bounded domain in $\Bbb C^n.$ Let $U$ and $V$ be neighborhoods of $\partial D$ with $V\Subset U.$ Then there exists a constant $c>0$ such that for any connected component $D'$ of $D\cap U$ one has that $$ck_{D'}(z,w)\le k_D(z,w),\quad z,w\in D'\cap V.$$
[*Proof.*]{} Let $\varepsilon>0.$ Take a smooth curve $\gamma:[0,1]\to D$ such that $\gamma(0)=z,$ $\gamma(1)=w$ and $$k_D(z,w,\varepsilon):=k_D(z,w)+\varepsilon>\int_0^1\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt.$$ Let $s=\sup\{t\in(0,1):\gamma(0,t)\subset D'\cap V\}$ and $r=\inf\{t\ge s:\gamma([t,1])\subset D'\cap V\}.$ Set $z'=\gamma(s)$ and $w'=\gamma(r).$ The localization property of the Kobayashi metric (cf. ) provides a constant $c'>0$ such that $$c'\kappa_{D'}(u;X)\le\kappa_D(u;X),\quad z\in {D'}\cap V,\ X\in\Bbb C^n.$$ It follows that $$\begin{gathered}
k_D(z,w,\varepsilon)>c'k_{D'}(z,z')+k_D(z',w')+c'k_{D'}(w',w)\\
\ge c'k_{D'}(z,w)+k_D(z',w')-c'k_{D'}(z',w').\end{gathered}$$
If $z'\neq w',$ then $z',w'\in {D'}\cap\partial V\Subset {D'}.$ Then there exists a constant $c_1>0$ such that $$k_{D'}(u,v)\le c_1||u-v||,\quad u,v\in {D'}\cap\partial V.$$ On the other hand, since $D$ is bounded, we may find a constant $c_2>0$ such that $$k_D(u,v)\ge c_2||u-v||,\quad u,v\in {D'}\cap\partial V.$$
Then $$k_D(z,w,\varepsilon)>c'k_{D'}(z,w)+(c_2-c'c_1)||z'-w'||.$$ Since $$k_D(z,w,\varepsilon)>k_D(z',w')\ge c_2||z'-w'||,$$ we get that $$k_D(z,w,\varepsilon)>c'k_{D'}(z,w)-(c'c_1/c_2-1)^+k_D(z,w,\varepsilon).$$
The last inequality also holds if $z'=w'.$ Letting $\varepsilon\to 0,$ we obtain that $$k_D(z,w)\ge\min\{c',c_2/c_1\}k_{D'}(z,w).$$
[M]{} Abouhajar, A., White, M., Young, N.: A Schwarz lemma for a domain related to $\mu$-synthesis, J. Geom. Anal. **17** (2007), 717–750. Andersson, M., Passare, M., Sigurdsson, R., Complex convexity and analytic functionals, Birkhäuser, Basel-Boston-Berlin, 2004. Agler, J., Young, N.J.: A commutant lifting theorem for a domain in $\Bbb C^2$ and spectral interpolation, J. Funct. Anal. **161** (1999), 452–477. Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. **75** (2000), 504–533. Buckley, S.: Gromov hyperbolicity of invariant metrics, preprint (2008); www.uma.es/investigadores/grupos/cfunspot/research/0806pBuckley.pdf. Costara, C.: On the spectral Nevanlinna-Pick problem, Studia Math. **170** (2005), 23–55. Dieu, N.Q., Nikolov N., Thomas P.J.: Estimates for invariant metrics near non-semipositive boundary points , J. Geom. Anal. **23** (2013), 598–610. Forstneric, F., Rosay, J-P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. **279** (1987), 239–252. Gaussier, H., Seshadri, H.: On the Gromov hyperbolicity of convex domains in $\mathbb{C}^{n}$, arXiv:1312.0368. Gromov, M.: Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Ins. Publ., 8, Springer, New York, 1987. Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis, de Gruyter Exp. Math. 9, de Gruyter, Berlin-New York, 1993. Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France **109** (1981), 427–474. Nikolov, N.: Comparison of invariant functions on strongly pseudoconvex domains, J. Math. Anal. Appl. **421** (2015), 180–185. Nikolov, N., Pflug, P., Zwonek, W.: An example of a bounded $\Bbb C$-convex domain which is not biholomorphic to a convex domain, Math. Scan. **102** (2008), 149–155. Nikolov, N., Pflug, P., Zwonek, W.: Estimates for invariant metrics on $\mathbb{C}$-convex domains, Trans. Amer. Math. Soc. **363** (2011), 6245–6256. Nikolov, N., Trybuła, M.: The Kobayashi balls of ($\Bbb C$-)convex domains, Monatsh. Math. DOI 10.1007/s00605-015-0746-3. Rodriguez J.M., Touris, E.: Gromov hyperbolicity through decomposition of metrics spaces, Acta Math. Hung. **103** (2004), 107–138. Väisälä, J.: Gromov hyperbolic spaces, Expo. Math. **23** (2005), 187–231. Zwonek, W.: Geometric properties of the tetrablock, Arch. Math. **100** (2013), 159–165.
[^1]: Then D is biholomorphic to a bounded domain (cf. [@Jarnicki Theorem 7.1.8]).
[^2]: One can show that these distances are not bilipschitz equivalent.
[^3]: One can show that these distances are not bilipschitz equivalent if, for example, $K$ is a closed polydisc.
| ArXiv |
---
abstract: 'We present constraints on the dark energy equation-of-state parameter, $w=P/(\rho c^2)$, using [60]{} Type Ia supernovae ([SNe Ia]{}) from the ESSENCE supernova survey. We derive a set of constraints on the nature of the dark energy assuming a flat Universe. By including constraints on ([${\Omega}_{\rm M}$]{}, $w$) from baryon acoustic oscillations, we obtain a value for a static equation-of-state parameter $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} and [${\Omega}_{\rm M}$]{}=[$0.274^{+0.033}_{-0.020}~{\rm (stat}~1\sigma{)}$]{} with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}. These results are consistent with those reported by the SuperNova Legacy Survey in a similar program measuring supernova distances and redshifts. We evaluate sources of systematic error that afflict supernova observations and present Monte Carlo simulations that explore these effects. Currently, the largest systematic currently with the potential to affect our measurements is the treatment of extinction due to dust in the supernova host galaxies. Combining our set of ESSENCE [SNe Ia]{} with the SuperNova Legacy Survey [SNe Ia]{}, we obtain a joint constraint of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, [${\Omega}_{\rm M}$]{}=[$0.267^{+0.028}_{-0.018}~{\rm (stat}~1\sigma{)}$]{}with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.91$]{}. The current [SNe Ia]{} data are fully consistent with a cosmological constant.'
author:
- '[W. M. Wood-Vasey]{}, [G. Miknaitis]{}, [C. W. Stubbs]{}, [S. Jha]{}, [A. G. Riess]{}, [P. M. Garnavich]{}, [R. P. Kirshner]{}, [C. Aguilera]{}, [A. C. Becker]{}, [J. W. Blackman]{}, [S. Blondin]{}, [P. Challis]{}, [A. Clocchiatti]{}, [A. Conley]{}, [R. Covarrubias]{}, [T. M. Davis]{}, [A. V. Filippenko]{}, [R. J. Foley]{}, [A. Garg]{}, [M. Hicken]{}, [K. Krisciunas]{}, [B. Leibundgut]{}, [W. Li]{}, [T. Matheson]{}, [A. Miceli]{}, [G. Narayan]{}, [G. Pignata]{}, [J. L. Prieto]{}, [A. Rest]{}, [M. E. Salvo]{}, [B. P. Schmidt]{}, [R. C. Smith]{}, [J. Sollerman]{}, [J. Spyromilio]{}, [J. L. Tonry]{}, [N. B. Suntzeff]{}, and [A. Zenteno]{}'
bibliography:
- 'apj-jour.bib'
- 'cos\_paper.bib'
title: 'Observational Constraints on the Nature of Dark Energy: First Cosmological Results from the ESSENCE Supernova Survey'
---
Introduction: Supernovae and Cosmology {#sec:introduction}
======================================
We report the analysis of [60]{} Type Ia supernovae ([SNe Ia]{}) discovered in the course of the ESSENCE program (Equation of State: Supernovae trace Cosmic Expansion—an NOAO Survey Program) from 2002 to 2005. The aim of ESSENCE is to measure the history of cosmic expansion over the past 5 billion years with sufficient precision to distinguish whether the dark energy is different from a cosmological constant at the $\sigma_w=\pm0.1$ level. Here we present our first results and show that we are well on our way towards that goal. Our present data are fully consistent with a $w=-1$, flat Universe, and our uncertainty in $w$, the parameter that describes the cosmic equation of state, analyzed in the way we outline here, will shrink below $0.1$ for models of constant $w$ as the ESSENCE program is completed. Other approaches to using the luminosity distances have been suggested to constrain possible cosmological models. We here provide the ESSENCE observations in a convenient form suitable for a testing a variety of models.[^1]
As reported in a companion paper [@miknaitis07], ESSENCE is based on a supernova search carried out with the 4-m Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) with the prime-focus MOSAIC II 64 Megapixel CCD camera. Our search produces densely sampled $R$-band and $I$-band light curves for supernovae in our fields. As described in that paper, we optimized the search to provide the best constraints on $w$, given fixed observing time and the properties of the MOSAICII camera and CTIO 4-m telescope. Spectra from a variety of large telescopes, including Keck, VLT, Gemini, and Magellan, allow us to determine supernova types and redshifts. We have paid particular attention to the central problems of calibration and systematic errors that, when the survey is complete in 2008, will be more important to the final precision of our cosmological inferences than statistical sampling errors for about 200 objects.
This first cosmological report from the ESSENCE survey derives some properties of dark energy from the sample presently in hand, which is still small enough that the statistics of the sample size make a noticeable contribution to the uncertainty in dark-energy properties. But our goal is to set out the systematic uncertainties in a clear way so that these are exposed to view and so that we can concentrate our efforts where they will have the most significant effect. To infer luminosity distances to the ESSENCE supernovae over the redshift interval $0.15$–$0.70$, we employ the relations developed for [SNe Ia]{} at low redshift [@jha06c] among their light-curve shapes, colors, and intrinsic luminosities. The expansion history from $z\approx0.7$ to the present provides leverage to constrain the equation-of-state parameter for the dark energy as described below. In §\[sec:introduction\] we sketch the context of the ESSENCE program. In §\[sec:distances\] we show from a set of simulated light curves that this particular implementation of light-curve analysis is consistent, with the same cosmology emerging from the analysis as was used to construct the samples, and that the statistical uncertainty we ascribe to the inference of the dark-energy properties is also correctly measured. This modeling of our analysis chain gives us confidence that the analysis of the actual data set is reliable and its uncertainty is correctly estimated. Section \[sec:systematics\] delineates the systematic errors we confront, estimates their present size, and indicates some areas where improvement can be achieved. Section \[sec:cosmology\] describes the sample and provides the estimates of dark energy properties using the ESSENCE sample. The conclusions of this work are given in §\[sec:conclusions\].
Context
-------
Supernovae have been central to cosmological measurements from the very beginning of observational cosmology. @shapley19 employed supernovae against the “island universe” hypothesis arguing that objects such as SN 1885A in Andromeda would have $M=-16$ which was “out of the question.” Edwin Hubble [@hubble29b] noted “a mysterious class of exceptional novae which attain luminosities that are respectable fractions of the total luminosities of the systems in which they appear.” These extra-bright novae were dubbed “supernovae” by @baade34 and divided into two classes, based on their spectra, by @minkowski41. Type I supernovae (SNe I) have no hydrogen lines while Type II supernovae (SNe II) show H$\alpha$ and other hydrogen lines.
The high luminosity and observed homogeneity of the first handful of SN I light curves prompted @wilson39 to suggest that they be employed for fundamental cosmological measurements, starting with time dilation of their characteristic rise and fall to distinguish true cosmic expansion from “tired light.” After the [SN Ib]{} subclass was separated from the [SNe Ia]{} [see @filippenko97 for a review] this line of investigation has grown more fruitful as techniques of photometry have improved and as the redshift range over which supernovae have been well observed and confirmed to have standard light-curve shapes and luminosities has increased [@rust74; @leibundgut96; @riess97; @goldhaber01; @riess04b; @foley05; @hook05; @conley06; @blondin06]. Within the uncertainties, the results agree with the predictions of cosmic expansion and provide a fundamental test that the underlying assumption of an expanding universe is correct.
Evidence for the homogeneity of [SNe Ia]{} comes from their small scatter in the Hubble diagram. @kowal68 compiled data for the first well-populated Hubble diagram of SNe I. The 1$\sigma$ scatter about the Hubble line was $0.6$ mag, but Kowal presciently speculated that supernova distances to individual objects might eventually be known to 5-10% and suggested that “\[i\]t may even be possible to determine the second-order term in the redshift-magnitude relation when light curves become available for very distant supernovae.” Precise distances to [SNe Ia]{} enable tests for the linearity of the Hubble law and provide evidence for local deviations from the local Hubble flow, attributed to density inhomogeneities in the local universe [@riess95; @riess97; @zehavi98; @bonacic00; @radburn-smith04; @jha06c]. While [SN Ia]{} cosmology is not dependent on the value of $H_0$, it is sensitive to deviations from a homogeneous Hubble flow and these regional velocity fields may limit our ability to estimate properties of dark energy, as emphasized by @hui06 and by @cooray06. Whether the best strategy is to map the velocity inhomogeneities thoroughly or to skip over them by using a more distant low-redshift sample remains to be demonstrated. We have used a lower limit of redshift $z>0.015$ in constructing our sample of [SNe Ia]{}.
The utility of [SNe Ia]{} as distance indicators results from the demonstration that the intrinsic brightness of each [SN Ia]{} is closely connected to the shape of its light curve. As the sample of well-observed [SNe Ia]{} grew, some distinctly bright and faint objects were found. For example, SN 1991T [@filippenko92a; @phillips92] and SN 1991bg [@filippenko92b; @leibundgut93] were of different luminosity, and their light curves were not the same, either. The possible correlation of the shapes of supernova light curves with their luminosities had been explored by @pskovskii77b. More homogeneous photometry from CCD detectors, more extreme examples from larger samples, and more reliable distance estimators enabled @phillips93 to establish the empirical relation between light-curve shapes and supernova luminosities. The Calán-Tololo sample [@hamuy96] and the CfA sample [@riess99; @jha06a], have been used to improve the methods for using supernova light curves to measure supernova distances. Many variations on Phillips’ idea have been developed, including [${\Delta{m_{15}}}$]{} [@phillips99], MLCS [@riess96; @jha06c], DM15 [@prieto06], stretch [@goldhaber01], CMAGIC [@wang03], and SALT [@guy05].
These methods are capable of achieving the 10% precision for supernova distances that [@kowal68] foresaw 40 years ago. In the ESSENCE analysis, we have used a version of the @jha06c method called MLCS2k2. We have compared it with the results of the SALT [@guy05] light-curve fitter used by the SuperNova Legacy Survey [SNLS; @astier06]. This comparison provides a test: if the two approaches do not agree when applied to the same data they cannot both be correct. As shown in §\[sec:distances\], SALT and this version of MLCS2K2, with our preferred extinction prior, are in excellent accord when applied to the same data. While gratifying, this agreement does not prove they are both correct. Moreover, as described in §\[sec:cosmology\], the cosmological results depend somewhat on the assumptions about SN host-galaxy extinction that are employed. This has been an ongoing problem in supernova cosmology. The work of @lira95 demonstrated the empirical fact that although SN Ia have a range of colors at maximum light, they appear to reach the same intrinsic color about 30–90 days past maximum light, independent of light curve shape.
@riess96 used de-reddened [SN Ia]{} data to show that near maximum light intrinsic color differences existed with fainter [SNe Ia]{}appearing redder than brighter objects and then used this information to construct an absorption-free Hubble diagram. Given a good set of observations in several bands, the reddening for individual supernovae can then be determined and the general relations between supernova luminosity and the light curve shapes in many bands can be established [@hamuy96; @riess99; @phillips99]. The initial detections of cosmic acceleration employed either these individual absorption corrections [@riess98] or a full-sample statistical absorption correction [@perlmutter99]. Finding the best approach to this problem, whether by shifting observations to the infrared, limiting the sample to low-extinction cases, or making other restrictive cuts on the data, is an important area for future work. Some ways to explore this issue are sketched in §\[sec:cosmology\].
@kowal68 recognized that second-order terms in cosmic expansion might be measured with supernovae once the precision and redshift range grew sufficiently large. More direct approaches with the [*Hubble Space Telescope (HST)*]{} were imagined by @colgate79 and with special clarity by @tammann79. Tammann anticipated that [*HST*]{} photometry of [SNe Ia]{} at $z\approx0.5$ would lead to a direct determination of cosmic deceleration and that the time dilation of [SN Ia]{} light curves would be a fundamental test of the expansion hypothesis. While [*HST*]{} languished on the ground after the Challenger disaster, this line of research was attempted from the ground at the European Southern Observatory (ESO) by a Danish group in 1986–1988. Their cyclic CCD imaging of the search fields used image registration, convolution and subtraction, and real-time data analysis [@hansen87]. Alas, the rate of [SNe Ia]{} in their fields was lower than they had anticipated, and only one [SN Ia]{}, SN 1988U was discovered and monitored in two years of effort [@hansen87; @norgaard-nielsen89]. More effective searches by the Lawrence Berkeley Lab (LBL) group exploiting larger CCD detectors and sophisticated detection software showed that this approach could be made practical and used to find significant numbers of high-redshift [SNe Ia]{} [@perlmutter95].
By 1995, two groups, the LBL-based Supernova Cosmology Project (SCP) and the High-Z Supernova Search Team [HZT; @schmidt98]) were working in this field. The first [SN Ia]{} cosmology results using 7 high-redshift [SNe Ia]{}[@perlmutter97] found a Universe consistent with [${\Omega}_{\rm M}$]{}$=1$ but subsequent work by the SCP [@perlmutter98] and by the HZT [@garnavich98] revised this initial finding to favor a lower value of [${\Omega}_{\rm M}$]{}. At the January 1998 meeting of the American Astronomical Society both teams reported that the [SN Ia]{} results favored a universe that would expand without limit, but at that time neither team claimed the Universe was accelerating. The subsequent publication of stronger results based on larger samples by the HZT [@riess98] and by the SCP [@perlmutter99] provided a big surprise. The supernova data showed that [SNe Ia]{} at $z\approx0.5$ were about $0.2$ mag dimmer than expected in an open universe and pointed firmly at an accelerating universe [for first-hand accounts, see @overbye99; @riess00b; @filippenko01b; @kirshner02; @perlmutter03]; reviews are given by @leibundgut01, @filippenko05a and others.
The supernova route to cosmological understanding continues to improve. One source of uncertainty has been the small sample of very well observed low-redshift supernovae [@hamuy96; @riess99]. The most recent contribution is the summary of CfA data obtained in 1997–2001 [@jha06a], but significantly enhanced samples from the CfA [@hicken06] together with new data from the Katzman Automatic Imaging Telescope [KAIT; @li00; @filippenko01a; @filippenko05b], from the Carnegie SN Program [@hamuy06], from the Supernova Factory [@wood-vasey04; @copin06], and from the Sloan Digital Sky Survey II Supernova Survey [SDSS II; @frieman04; @dilday05] are in prospect. As the low-$z$ sample approaches 200 objects, the size of the sample will cease to be a source of statistical uncertainty for the determination of cosmological parameters. As described in §\[sec:systematics\], systematic errors of calibration and K-correction will ultimately impose the limits to understanding dark energy’s properties, and we are actively working to improve these areas [@stubbs06].
Some of the potential sources of systematic error in the high-$z$ sample have been examined. The fundamental assumption is that distant [SNe Ia]{} can be analyzed using the methods developed for the low-$z$ sample. Since nearby samples show that the [SNe Ia]{} in elliptical galaxies have a different distribution in luminosity than the [SNe Ia]{} in spirals [@hamuy00; @gallagher05; @neill06; @sullivan06b], morphological classification of the distant sample may provide some useful clues to help improve the cosmological inferences [@williams03]. For example, @sullivan03 showed that restricting the SCP sample to [SNe Ia]{} in elliptical galaxies gave identical cosmological results to the complete sample, which is principally in spiral galaxies. The possibility of grey dust raised by @aguirre99a [@aguirre99b] was examined by @riess00a and and by @nobili05 through infrared observations of high-$z$ supernovae and was put to rest by the very high-redshift observations of @riess04b. Improved methods for handling the vexing problems of absorption by dust have been developed by @knop03 and by @jha06c. These questions are described in more detail in §\[sec:extinction\].
The question of whether distant supernovae have spectra that are the same as nearby supernovae has been investigated by @coil00, @lidman05, @matheson05, @hook05, @howell05, and @blondin06. The more telling question of whether these spectra evolve in the same way as those of nearby objects was approached by @foley05. In all cases, the evidence points toward nearby supernovae behaving in the same way as distant ones, bolstering confidence in the initial results. This observed consistency does not mean that the samples are identical, only that the variations between the nearby and distant samples are successfully accounted for by the methods currently in hand. We do not know whether this will continue to be the case as future investigations press for more stringent limits on cosmological parameters [@albrecht06].
The highest redshift [SN Ia]{} data [@riess04b] show the qualitative signature expected from a mixed dark-energy/dark-matter cosmology. That is, they show cosmic deceleration due to dark matter preceded the current era of cosmic acceleration due to dark energy. The sign of the observed effect on supernova apparent magnitudes reverses—[SNe Ia]{} at $z\sim 0.5$ appear $0.2$ mag dimmer than expected in a coasting cosmology but the very distant supernovae whose light comes from $z>1$ appear brighter than they would in that cosmology. By itself, this turnover is a very encouraging sign that supernova cosmology does not founder on grey dust or even on a simple evolution of supernova properties with cosmic epoch. As part of this analysis, @riess04b constructed the “gold” sample of high-$z$ and low-$z$ supernovae whose observations met reasonable criteria for inclusion in an analysis of all of the published light curves and spectra using a uniform method of deriving distances from the light curves.
The analysis of the gold sample provided an estimate of the time derivative of the equation-of-state parameter, $w$, for dark energy. These observations are very important conceptually because the simplest fact about the cosmological constant as a candidate for dark energy is that it should be constant (i.e., $w' = dw/dz = 0$). The observations are consistent with a constant dark energy over the redshift range out to $z\approx 1.6$. Other forms of dark energy could satisfy the observed constraints, but this observational test is one that the cosmological constant could have failed. In the analysis of the ESSENCE data presented in §\[sec:cosmology\], we use the supernova data to constrain the properties of $w$, as first carried out by @white98 and by @garnavich98. This parameterization of dark energy by $w$ is not the only possible approach. A more detailed approach is to compare the observational data to a specific model and, for example, try to reconstruct the dark energy scalar-field potential [see, for example, @li06]. A more agnostic view is that we are simply measuring the expansion history of the universe, and a kinematic description of that history in terms of expansion rate, acceleration, and jerk [@riess04b; @rapetti06] covers the facts without assuming the nature of dark energy.
The ESSENCE project was conceived to tighten the constraints on dark energy at $z\approx0.5$ to reveal any discrepancy between the observations and the leading candidate for dark energy, the cosmological constant. A simple way to express this is that we aim for a 10% uncertainty in the value of $w$. This program is similar to the approach of the SuperNova Legacy Survey (SNLS) being carried out at the Canada-France-Hawaii Telescope, and we compare our methods and results to theirs [@guy05; @astier06] at several points in the analysis below.
The SNLS has taken the admirable step of publishing their light curves online and making the code of their light-curve fitting program, SALT, available for public inspection and use[^2]. Making the light curves public, as was done for the results of the HZT and its successors @riess98 [@tonry03; @barris04; @krisciunas05; @clocchiatti06], by @knop03, by @riess04b for the very high redshift [*HST*]{} supernova program, and for the low-$z$ data of @hamuy96, @riess99, and @jha06a, provides the opportunity for others to perform their own analysis of the results. In addition to exploring a variety of approaches to analyzing our own [SN Ia]{} observations, we show the first joint constraints from ESSENCE and SNLS, and some joint constraints derived from combining these with the @riess04b gold sample in §\[sec:cosmology\].
Luminosity Distance Determination {#sec:distances}
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The physical quantities of interest in our cosmological measurements are the redshifts and distances to a set of space-time points in the Universe. The redshifts come from spectra and the luminosity distances, $D_L$, come from the observed flux of the supernova combined with our understanding of [SN Ia]{} light curves from nearby objects.
Extracting a luminosity distance to a supernova from observations of its light curve necessitates a number of assumptions. We use the observations of nearby supernovae to establish the relations between color, light-curve shape in multiple bands, and peak luminosity. These nearby observations attain high signal-to-noise ratios, and the nearby objects can be observed in more passbands (including infrared) than faint distant objects. We assume that the resulting method of converting light curves to luminosity distances applies at all redshifts. The observed spectral uniformity of supernovae over a range of redshift [@lidman05; @hook05; @blondin06] supports this approach. We assume that $R_V$, the ratio of selective to absolute extinction, is independent of redshift. Below in §\[sec:extinction\], we test the potential systematic effect of departures from this assumption. We adopt an astrophysically sensible prior distribution of host-galaxy extinction properties, with a redshift dependence that is derived from the simulations we present below.
Our approach is to conduct comprehensive simulations of the ESSENCE data and analysis. As described by @miknaitis07, we use this same approach to explore our photometric performance. For the aspects of our analysis that are “downstream” of the light-curve generation, we generate sets of synthetic light curves and subject them to our analysis pipeline. In this way we can test the performance of our distance-fitting tools, and by exaggerating various systematic errors (zeropoint offsets, etc.) we can assess the impact of these effects on our determination of $w$.
We must recognize and emphasize that in the era of precision [SN Ia]{}cosmology (constraining dark energy properties, rather than just detecting its existence), careful attention to systematic errors is of paramount importance: shifts of a few hundredths of a magnitude can lead to constraints on $w$ that change by $0.1$. Different, yet defensible, choices in the analysis chain may show such effects.
Extracting Luminosity Distances from Light Curves: Distance Fitters
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We use the MLCS2k2 method of @jha06c as the primary tool to derive relative luminosity distances to our [SNe Ia]{}. For comparison, we also provide the results obtained using the Spectral Adaptive Lightcurve Template (SALT) fitter of @guy05 on the ESSENCE light curves. SALT was used in the recent cosmological results paper from the SNLS [@astier06, hereafter A06]. We provide a consistent and comprehensive set of distances obtained to nearby, ESSENCE, and SNLS supernovae for each luminosity-distance fitting technique. The ESSENCE light curves used in this analysis were presented by @miknaitis07 and we provide them online, together with our set of previously published light curves for nearby [SNe Ia]{}, for the convenience of those interested.[^3] Additional [SN Ia]{} light-curve fitting methods will be further explored in future ESSENCE analyses. Understanding the behavior of our distance determination method is critical to our goal of quantifying the uncertainties of our analysis chain.
MLCS2k2 and SALT, as well as the light-curve “stretch” approach used by @perlmutter97 [@perlmutter99], @goldhaber01 and @knop03, exploit the fact that the rate of decline, the color, and the intrinsic luminosity of [SNe Ia]{}are correlated. At present we treat [SNe Ia]{} as a single-parameter family, and the distance fitting techniques use multi-color light curves to deduce a luminosity distance and host-galaxy reddening for each supernova. Previous papers have shown that the different techniques produce relative luminosity distances that scatter by $\sim 0.10$ mag for an individual [SN Ia]{} [e.g., @tonry03], but this scatter is uncorrelated with redshift. As a consequence, the cosmology results are insensitive to the distance fitting technique. However, as described by @miknaitis07, the measurement of the equation-of-state parameter hinges on subtle distortions in the Hubble diagram, so we have undertaken a comprehensive set of simulations to understand potential biases introduced by MLCS2k2.
The MLCS2k2 approach [@riess96; @riess98; @jha06c] to determining luminosity distances uses well-observed nearby [SNe Ia]{} to establish a set of light-curve templates in multiple passbands. The parameters $\Delta$ (roughly equivalent to the variation in peak visual luminosity, this parameter characterizes intrinsic color, rate of decline, and peak brightness), $A_V$ (the $V$-band extinction of the supernova light in its host galaxy), and $\mu$ (the distance modulus) are then determined by fitting each multi-band set of distant supernova light curves to redshifted versions of these templates. Jha et al. (2006c) present results from MLCS2k2 based on nearby SN Ia. Here we have modified MLCS2k2 for application to both high and low-redshift [SNe Ia]{}. We begin with a rest-frame model of the [SN Ia]{} in its host galaxy, and then propagate the model light curves through the host-galaxy extinction, K-correction, Milky Way extinction to the detector, incorporating the measured passband response (including the atmosphere for ground-based observations). We then fit this model directly to the natural-system observations. This forward-modeling approach has particular advantages in application to the more sparsely sampled (in color and time) data typical of high-redshift SN searches.
The SALT method of @guy05, which was used for the SNLS first-results analysis of A06, constructs a fiducial [SN Ia]{} template using combined spectral and photometric information, then transforms this template into the rest frame of the [SN Ia]{}, and finally calculates a flux, stretch, and generalized color. The color parameter in SALT is notable in that it includes both the intrinsic variation in [SN Ia]{} color and the extinction from dust in the host galaxy within a single parameter (in contrast, MLCS2k2 attempts to separate these components of the observed colors for each supernova). While the reddening vector (attenuation vs. color excess) is similar to the [SN Ia]{} color vs. absolute magnitude relation, the two sources of correlated color and luminosity variation are not identical.
The stretch and color parameters of SALT were used by A06 to estimate luminosity distances by fitting for the stretch-luminosity and color-luminosity relationships in the nearby sample and applying those to the full [SNe Ia]{} sample. Given that the SALT color parameter conflates the two physically distinct phenomena of host-galaxy extinction and [SN Ia]{} color variation, it is remarkable and perhaps a source of deep insight that this treatment works as well as it does. Because of both survey selection effects and possible demographic shifts in the host environments of [SNe Ia]{}we would not expect that the proportion of reddening from dust and from intrinsic variation would remain constant with redshift as this approach assumes. However, the SALT/A06 method does seem to work quite well in practice.
Sensitivity to Assumptions about the Host-Galaxy Extinction Distribution: Extinction Priors {#sec:extinction_prior}
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The best way to treat host-galaxy extinction is a serious question for this work and for the field of supernova cosmology. The Bayesian approach we use is detailed in §\[sec:priors\]. Here we describe simulations that are designed to evaluate the effects of those methods.
There have been four basic approaches to combining reddening measurements with astrophysical knowledge to determine the host galaxy extinction along the line-of sight: (1) assume that linear $A_V$ is the natural space for extinction and assume a flat prior [@perlmutter99; @knop03]; (2) use models of the dust distribution in galaxies [@hatano98; @commins04; @riello05] to model line-of-sight extinction values [@riess98; @tonry03; @riess04b]; (3) assume that the distribution of host-galaxy $A_V$ follows an exponential form [@jha06c], based on observed distributions of $A_V$ in nearby [SNe Ia]{}; and (4) self-calibrate within a set of low-$z$ [SNe Ia]{} to obtain a consistent color+$A_V$ relationship and assume that relation for the full set [@astier06].
Approach (1) assumes the least prior knowledge about the distribution of $A_V$ and produces a Gaussian probability distribution for the fitted luminosity distance. However, this approach weakens the ability to separate intrinsic [SN Ia]{}color from $A_V$ and results in a fit parameter $A_V$ that is a mixture of the two. An $A_V$ that is truly related to the dust extinction should never be negative. The probability prior with $-\infty < A_V < +\infty$ is not the natural range over which to assume a flat distribution. The physically reasonable prior on $A_V$ should be strictly positive. One approach is to base the prior for absorption on the distribution of dust in galaxies. Theoretical modeling of dust distributions in galaxies, such as that of @hatano98, @commins04, and @riello05, provides a physically motivated dust distribution. This method represents approach (2) above and is the method we adopt here. In contrast, @jha06c empirically derived an exponential $A_V$ distribution from MLCS2k2 fits to nearby [SNe Ia]{} by assuming a particular color distribution of [SNe Ia]{}. This distribution was derived using the empirical fact that [SNe Ia]{} reach a common color about 40 days past maximum light [@lira95]. They found an exponential distribution of $A_V$, $$p(A_V) \propto \exp \left(\frac{-A_V}{\tau}\right),
\label{eq:default}$$ where $\tau=0.46$ mag. Unfortunately, the highest-extinction objects drive the tail of this exponential and significantly affect the fit, resulting in a prior sensitive to sample selection, which differs significantly in high-redshift searches compared to the nearby objects studied by @jha06c.
A06 analyzed the results of the SALT [SN Ia]{} light-curve fitter with approach (4) and have systematic sensitivities that are similar to those of approach (1).
We use MLCS2k2 as our main analysis tool. We designate approach (1) the “flatnegav” prior and approach (3) the “default” prior and discuss both of these further in §\[sec:priors\]. Approach (2) is based on a galactic line-of-sight or “glos” prior on $A_V$: $$\hat{p}(A_V) \propto \frac{A}{\tau} \exp\left({\frac{-A_V}{\tau}}\right) +
\frac{2B}{\sqrt{2\pi}\sigma} \exp\left(\frac{-A_V^2}{2\sigma^2}\right),
\label{eq:glos}$$ where $A=1$, $B=0.5$, $\tau=0.4$, $\sigma=0.1$, and $\hat{p}(A_V) \equiv 0$ for $A_V < 0$. This exponential plus one-sided narrow Gaussian “glos” prior is based on the host-galaxy dust models of @hatano98, @commins04, and @riello05. As described below, we have modeled our selection effects with redshift to adapt the “glos” prior into the “glosz” prior that is the basis for our analysis. We feel this approach leverages our best understanding of the effects of extinction and selection.
Figs. \[fig:mlcs\_fits\_prior\_glosz\] and \[fig:salt\_fits\] show the distribution of the fit parameters and overlay the prior distribution assumed for each of these approaches. Fig. \[fig:mlcs\_vs\_salt\] compares the fit distances and extinction/color parameters of the MLCS2k2 “glosz” and SALT fit results for the ESSENCE, SNLS, and nearby samples. The distribution of recovered $\Delta$ and $A_V$ match their imposed priors for MLCS2k2 “glosz” while the stretch and color fit parameters from SALT show a consistent distribution for the three different sets of [SNe Ia]{}.
ESSENCE Selection Effects and the Motivation for a Redshift-Dependent Extinction Prior {#sec:selection}
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We examined the effect of the survey selection function on the expected demographics of the ESSENCE [SNe Ia]{} and explored the interplay between extinction, Malmquist bias, and our observed light curves. To determine the impact of the selection bias, we developed a Monte Carlo simulation of the ESSENCE search. We created a range of supernova light curves that match the properties of the nearby sample, added noise based on statistics from actual ESSENCE photometry, and then fit the resulting light curves in the same way the real events are analyzed. In this way we estimated the impact of subtle biases, although this simulation cannot test for errors in our light-curve model or population drift with redshift.
Based on its low-redshift training set, MLCS2K2 is able to output a finely sampled light curve given a redshift ($z$), distance modulus ($\mu$), light-curve shape parameter ($\Delta$), host extinction ($A_V$), host extinction law ($R_V$), date of rest-frame $B$-band maximum light ($t_0$), Milky Way reddening ($E(B-V)_\mathrm{MW}$), and the bandpasses of the observations. At a given redshift we calculated a distance modulus, $\mu_\mathrm{true}$, from the luminosity distance for the standard cosmology ($\Omega_m=0.3$, $\Omega_\Lambda =0.7$) and that distance modulus plus an assumed $M_B=-19.5$ for [SNe Ia]{} set the brightness for our simulated supernovae. Varying the assumed cosmology does not significantly impact the simulation results since we are comparing the input distance modulus with the recovered distance modulus, $\mu_\mathrm{obs}$, which is independent of the cosmology.
At each of a series of fixed redshifts, we created $\sim1000$ simulated light curves with parameters chosen from random distributions. The light-curve width, $\Delta$, was selected from the @jha06c distribution measured from the low-$z$ sample. The $\Delta$ distribution is approximately a Gaussian peaking at $\Delta =-0.15$ with an extended tail out to $\Delta =1.5$. The host extinction for each simulated event, $A_V$, was selected from either the @jha06c distribution (“default”) estimated from the local sample or from a “galaxy line-of-sight” estimation (“glos”). The “default” distribution was an exponential decay with index $0.46$ mag and set to zero for $A_V < 0.0$ mag. The “glos” distribution is also set to zero for $A_V < 0.0$ mag and combines a narrow Gaussian with a exponential tail for $A_V > 0.0$ mag (see Eq. \[eq:glos\]). The extinction law is assumed to be $R_V = 3.1$. The Milky Way reddening \[$E(B-V)_\mathrm{MW}$\] distribution was constructed from the @schlegel98 (hereafter SFD) reddening maps that cover the ESSENCE fields. The $E(B-V)_\mathrm{MW}$ was measured for 10,000 random locations in each ESSENCE field and the reddening was selected from the sum of the histograms (see Figure 2). The dates of observation for a simulated [SN Ia]{} were based on the actual dates of ESSENCE 4-m observations. An ESSENCE field was chosen at random from the list of monitored fields and a date of maximum, $t_0$, selected to fall randomly between the Modified Julian Date (MJD) of the first and last observation of an observing season. The simulated light curve was then interpolated for only those dates that ESSENCE took images. With each ESSENCE field observation, we estimated the magnitude in $R$ and $I$ that provided a 10$\sigma$ photometric detection based on the seeing and sky brightness. The signal-to-noise ratio (SNR) for each simulated light-curve point was then scaled from the 10$\sigma$ detection magnitude, assuming the noise was dominated by the sky background.
For each date of ESSENCE observation, we have a simulated noiseless magnitude and an estimate of the SNR of the observation. To each simulated observation we added an appropriate random value in flux space selected from a normal distribution with a width corresponding to the predicted SNR.
MLCS2k2 was then used to fit the simulated light curves and provide estimates of $\mu$, $\Delta$, $A_V$, and $t_0$, assuming a fixed $R_V = 3.1$. MLCS2k2 required an initial guess of the date of maximum, an estimate achieved by selecting from a normal distribution about the true date with a $1\sigma$ width of 2 days. The SFD Milky Way reddening was also required in MLCS2k2 and was provided from the true reddening after adding an uncertainty of 10%. Finally, in the real ESSENCE data we discarded supernovae when the MLCS2k2 reduced $\chi^2$ indicated a very poor fit. For the simulated light curves, we dropped events from the sample if the reduced $\chi^2$ exceeded 2.
### Deriving an Extinction Prior from the Simulation Results
Simulated ESSENCE samples were created at a range of redshifts out to $z=0.70$ and the light curves that passed the detection criteria from the actual ESSENCE search were fit with MLCS2k2. The fitting was done with the “default” prior and the “glos” prior (with corresponding $A_V$ distributions). The difference between the “true” (input) distance modulus and recovered (fit) distance modulus, $\Delta\mu$, was calculated for each event and the mean, median, and dispersion for the ensemble were calculated at each redshift. The median $\Delta\mu$ of the simulations was within $0.03$ mag for $z < 0.45$, but at higher redshift the simulated supernovae were estimated to be brighter than the input supernovae by more than $0.2$ mag. This bias results from the loss of faint events (large $A_V$ and large $\Delta$) from the sample as the distance increases. In a sense, this is a classic Malmquist bias, but here it is caused by an uninformed prior. These results are shown in Fig. \[fig:simulatepriors\].
The decreasing ability to observe large $A_V$ events as the redshift increases (see Fig. \[fig:cut\]) makes it clear that a using single $A_V$ prior for all redshifts is not correct. Because events with large $A_V$ and large $\Delta$ are lost at large redshift due to the magnitude limits of the search, we should adjust the prior as a function of $z$ to account for these predictable losses. Applying redshift-dependent window functions to the basic “glos” prior provides a much better prior as a function of redshift.
We fit the recovered $A_V$ distributions derived from the simulations, which start with a uniform $A_V$, to a window function based on the error function (integral of a Gaussian), and two parameters describe where that function drops to half its peak value ($A_{1/2}$) and the width of the transition ($\sigma_A$). The window function $W$ has the form $$W(A_V,A_{1/2},\sigma_A) = 1-
{1\over{\sqrt{\pi}}}\int_{-\infty}^{(A_V-A_{1/2})\sigma_A}
e^{-x^2}\;dx
\label{eq:gloszwindow}$$ where $A_{1/2}$ and $\sigma_A$ are functions of $z$ and estimated from the simulations. A similar process was applied to the $\Delta$ distribution and a table providing the parameters is given in Table \[tab:gloszwindow\]. We embody this prescription in the “glosz” prior we use for our main MLCS2k2 light-curve fitting. The “glosz” prior is the “glos” prior modified by the window functions in $A_V$ and $\Delta$. The simulations using the “glosz” prior provide a median $\Delta\mu$ within 0.03 mag for $z < 0.7$, which we judge to be satisfactory performance.
Comparison of MLCS2k2 and SALT Luminosity Distance Fitters
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The release of the source code to the SALT fitter [@guy05] makes a modern [SN Ia]{} light-curve fitter fully accessible and available to the community. This public release of SALT allows us to compare the results of our MLCS2k2 distance fitter with the SALT fitter used in the SNLS first results paper [@astier06]. We present the results of SALT fits to our nearby and ESSENCE samples in Table \[tab:salt\_fits\]. To compute the distance moduli we quote in that table, we assume the $\alpha=1.52$, $\beta=1.57$ values from A06. To calibrate the additional dispersion to add to the distance moduli of MLCS2k2 and SALT, we fit a [$\Lambda{\rm CDM}$]{} model to the nearby sample alone and derived the additional [$\sigma_{\rm add}$]{} to added in quadrature to recover [$\chi^2/{\rm DoF}$]{}$=1$ for the nearby sample. This [$\sigma_{\rm add}$]{} is related to the intrinsic dispersion of the absolute luminosity of [SNe Ia]{}, but is not precisely the same both because the light-curve fitters include varying degrees of model uncertainty and because the light curves of the [SNe Ia]{} are subject to photometric uncertainty. We find [$\sigma_{\rm add}$]{}$=0.10$ for MLCS2k2 with the “glosz” prior and [$\sigma_{\rm add}$]{}$=0.13$ for SALT. These values should be added to the $\sigma_\mu$ uncertainties given Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\] Fig. \[fig:mlcs\_vs\_salt\] visually demonstrates that the relative luminosity distances using the SALT light-curve fitter agree, within uncertainties, with the MLCS2k2 distances when the latter are fit using the “glosz” $A_V$ prior.
Testing the Recovery of Cosmological Models Using Simulations of the ESSENCE Dataset {#sec:simulation}
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In order to assess the reliability with which we recover cosmological parameters, we have simulated 100 sets of 100 light curves representing both the nearby and the ESSENCE light curves. Table \[tab:mlcs\_cuts\] presents the quality cuts for MLCS2k2 we derived from these simulated light-curve sets. Our light-curve goodness-of-fit cuts, when applied to these simulated light curves (see Table \[tab:mlcs\_cuts\]) and combined with the same external constraints of baryon acoustic oscillations [BAO; @eisenstein05] and flatness, allow us to recover our input cosmology of $(\Omega_M=0.3, \Omega_\Lambda=0.7, w=-1)$ to within $\pm0.11$ in $w$. This $\pm0.11$ uncertainty on an individual measurement of $w$ is matched by the $\sigma=0.11$ distribution of recovered $w$ values from the 100 sets of simulated light curves. This confirms our statistical error estimate on $w$; the estimated uncertainty matches the distribution, and within the self-consistent realm of synthetic and analyzed light curves based on MLCS2k2 our estimates of luminosity distance are not biased.
Potential Sources of Systematic Error {#sec:systematics}
=====================================
Here we identify and assess sources of systematic error that could afflict our measurements. These can be divided into two groups. Certain sources of systematic error may introduce perturbations either to individual photometric data points or to the distances or redshifts estimated to the [SNe Ia]{}. Others affect the data in a more or less random fashion and produce excess [*scatter*]{} in the Hubble diagram. Errors that are uncorrelated with either distance or redshift will not bias the cosmological result. These sources of photometric error are detailed by @miknaitis07; we summarize those results here in Table \[tab:photscatter\]. We add these effects in quadrature to the statistical uncertainties given by the luminosity distance fitting codes for each [SN Ia]{} distance measurement: $\sigma_\mu^{\rm phot scatter}=0.026$ mag.
In §\[sec:distances\] we discussed our testing of the MLCS2k2 fitter on simulated data sets that replicate the data quality of the ESSENCE and nearby [SNe Ia]{}. We explore the issue of host-galaxy extinction further in §\[sec:extinction\] & \[sec:priors\]. The interaction of Malmquist bias and selection effects with the extinction and color distribution of [SNe Ia]{}is discussed in §\[sec:malmquist\].
Any non-cosmological difference in measurements of nearby and distant [SNe Ia]{} has the potential to perturb our measurement of $w$. Table \[tab:systematics\] lists potential systematic effects of this sort. We present both our estimate of the sensitivity $dw/dx$ of the equation-of-state parameter to each potential systematic effect and our best estimate of the potential size of the perturbation, ${\Delta}x$. The upper bound on the bias introduced in $w$ is then ${\Delta}w = dw/dx\times{\Delta}x$. @miknaitis07 discusses the systematic uncertainties on $\mu$, which we convert here to systematic uncertainties on $w$, due to photometric errors from astrometric uncertainty on faint objects (${\Delta}w = 0.005$), potential biases from the difference imaging (${\Delta}w = 0.001$), and linearity of the MOSAIC II CCD (${\Delta}w = 0.005$). None of these contributed noticeably to the systematic uncertainty in our measurement of $w$. The rest of this section describes how we appraised our additional potential sources of systematic uncertainty.
The conclusion of this section is that our current overall estimate for the 1$\sigma$ equivalent systematic uncertainty in a static equation-of-state parameter is ${\Delta}w =$ [$0.13$]{} for our “glosz” analysis.
Photometric Zeropoints {#sec:zeropoint}
----------------------
Supernova cosmology fundamentally depends on the ability to accurately measure fluxes of objects over a range in redshift. Errors in photometric calibration translate to errors in cosmology in two basic ways.
Nearby objects at redshifts $<0.1$ play a crucial role in establishing a comparison reference for cosmological measurements. ESSENCE is inefficient at finding and observing low-redshift objects with the same telescope and detector system, so we use photometry of low-redshift [SNe Ia]{} in the literature from our own work and that of others [for the full list see @jha06c]. Using these external [SNe Ia]{} requires understanding the photometric calibration of our high-redshift sample relative to this low-redshift sample. Every supernova cosmology result to date has made use of more or less the same low-redshift photometry, so any inaccuracies in the nearby sample are a source of common systematic error for all [SN Ia]{} cosmology experiments. Calibration of photometry at the $\sim1\%$ level required to make precise inferences about the nature of dark energy is notoriously difficult [@stubbs06].
Photometric miscalibration can result in a second, more insidious systematic error if there is an error in the relative flux scaling between the broad-band passbands. This offset would distort the observed colors for the entire sample. Since these colors are used to infer the extinction, even small color errors result in significant biases in the measured distances. After all, the inferred host galaxy extinction, $A_{V}$, is related to the measured color excess, $E(B-V)$, by $A_{V}\approx3.1 E(B-V)$ (for Milky Way-like dust). A color error in rest-frame $B-V$ (observer-frame $R$, $I$ for ESSENCE) of $0.01$ mag can result in $0.03$ mag error in extinction, an inaccuracy that would lead directly to a $3\%$ error in the distance modulus, or a $1.5\%$ error in the distance. We currently estimate our color zeropoint uncertainty at $0.02$ mag and our absolute zeropoint (relative to the nearby [SNe Ia]{}) uncertainty to be $0.02$ mag. These respectively translate to $0.04$ and $0.02$ shifts in $w$ (see Table \[tab:systematics\]).
@miknaitis07 describe the calibration program we undertook to measure the transmission of the CTIO 4-m MOSAIC II system with the $R$ and $I$ filters of the ESSENCE survey. The calibration of the ESSENCE survey fields will be further improved by an intensive calibration program we are undertaking on the CTIO 4-m in 2006. Together with the improved calibration of the SDSS Southern Stripe by the SDSSII project, which overlaps 25% of our ESSENCE fields, we aim to achieve 1% photometric calibration of our CTIO 4-m MOSAIC II BVRI natural system.
We here use MLCS2k2 v004 with the @bohlin04 values for the magnitudes of Vega: i.e., `alpha_lyr_stis_002.fits` with $R_{\rm
Vega}=0.033$ mag. This value for $R_{\rm Vega}$ comes from @bessell98 but has been shifted down by $0.004$ mag as @bohlin04 suggest (from their $V_{\rm Vega}=0.026$ mag compared to @bessell98 $V_{\rm Vega}=0.030$ mag).
K-Corrections and Bandpass Uncertainty {#sec:kcorrection}
--------------------------------------
Uncertainty in the transmission function, typically called the bandpass, of the optical path of the telescope+detector is an important and potentially systematic effect. In this context, bandpass refers to the wavelength-dependent throughput of the entire optical path, including atmospheric transmission, mirror reflectivity, filter function, and CCD response. Since an error in the assumed bandpasses translates into a redshift-dependent error in the supernova flux, it is important to account for possible errors in the bandpass estimates.
The *relative* error due to bandpass miscalibration is small for objects with similar spectra, such as [SNe Ia]{}. Bandpass shape errors are largely accounted for by the filter zeropoint calibration, with residual errors corresponding to the difference between the spectral energy distribution of the objects of interest and those of the calibration sources. In the case of [SN Ia]{} observations, any residual zeropoint error is absorbed when we marginalize over the “nuisance parameter,” [$\cal{M}$]{}$=M_B - 5\log_{10} (H_0) + 25$ [@kim04]. This relative comparison results in a very small systematic error in the cosmological parameters from a global calibration error across bandpasses. Moreover, variations in atmospheric transmission are expected to contribute only random uncertainty.
However, the bandpass uncertainty becomes important when we compare [SNe Ia]{} at different redshifts for which the bandpass samples different spectral regions. In order to compare [SNe Ia]{} at multiple redshifts, we need to perform a K-correction [@leibundgut90; @hamuy93a; @kim96; @nugent02]. That is, we assume a spectral distribution for the supernova and convert the observed magnitude to what it would have been had the supernova been at another redshift. This process involves performing synthetic photometry of the assumed spectral distribution over the assumed bandpass. We address the issue of systematics arising from errors in the assumed spectral distribution in the supernova evolution section, §\[sec:snevolution\]. Here we address systematics arising from errors in our determination of the CTIO 4-m MOSAIC II $R$ and $I$ bandpass functions. Systematic effects on supernova cosmology that result from bandpass uncertainties are discussed more thoroughly by @davis06.
\[sec:bandpass\]
Calculating the effect of bandpass uncertainty is fairly difficult because of the arbitrary nature of the shape changes that might affect the bandpass. However, we can make several general calculations. As a first step, we take standard bandpasses and add white noise to represent a miscalibrated filter. White noise contributes power on all scales, so this approach adds small-scale discrepancies as well as large-scale warps or shifts in the filter. By averaging over many such miscalibrated filters, we can estimate the effect of filter miscalibration. Fig. 16 of @davis06 shows photometric error as a function of noise amplitude. A noise amplitude of $0.02$ produces a typical deviation of $2$% from the nominal filter shape at any wavelength. Calibrating the bandpass to better than $3\%$ allows us to keep the K-correction error introduced from a mismeasurement of our effective bandpass to less than 0.005 mag (0.5% in flux) and a systematic uncertainty of ${\Delta}w = 0.005$.
Extinction {#sec:extinction}
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The most significant cause of variation in luminosity of [SNe Ia]{} is the extinction experienced by the light from the [SN Ia]{}due to scattering and absorption from dust in the host galaxy.
Dust introduces a wavelength-depended diminution of a supernova’s light. In the case of Milky Way dust, we correct for its effects by using tabulated values as a function of Galactic longitude and latitude measured by other means [SFD @schlegel98], being sure in our MLCS2k2 fits to properly account for its uncertainty and correlation across all observations. For dust in the supernova’s host galaxy, we infer the extinction from the reddening of each supernova’s light curve.
However, the slope of differential reddening, characterized in the @cardelli89 extinction model by the parameter $R_V$, may vary. The nominal value of $R_V$ for the Milky Way is $3.1$, but different lines of sight within our galaxy have values of $R_V$ that vary from $2.1$ to $5.1$. Studies of $R_V$ in other galaxies have been more limited because we lack sources of known color and luminosity with which to probe the dust.
Because we use the supernova rest-frame $B-V$ color to determine the reddening of each [SN Ia]{}, and the distance modulus to a supernova is corrected by a value approximately three times the inferred reddening, extinction correction magnifies any source of systematic error in a supernova’s observed effective color. Systematic color errors can result from photometry errors, redshift-dependent K-correction errors, and evolution in the colors of supernovae.
Using the IR-emission maps of the Galaxy from the all-sky COBE/DIRBE and IRAS/ISSA maps, SFD have estimated the dust column density around the sky, which can then be translated to a color excess. This analysis has largely superseded the work of @burstein78, who used radio HI measurements and a relationship between gas and extinction to estimate the color excess across the sky. @burstein03 has reanalyzed the IR and HI measurements and finds that Milky Way extinctions are more precisely derived using the IR method. However, @burstein03 still finds a discrepant value for extinction at the poles, with SFD providing extinctions that are $E(B-V)=0.02$ mag higher than what the HI measurements indicate. @burstein03 suggests as a possible explanation for the discrepancy that SFD may predict too large an extinction in areas with high gas-to-dust ratios. @finkbeiner99 precisely estimated their sensitivities to these systematics and concluded they had controlled them to $0.01$ mag. The ESSENCE program targets fields at high Galactic latitude to minimize Galactic extinction. Although nearby and distant [SNe Ia]{}are both affected by the assumed Milky Way extinction, the nearby objects are observed in $B-V$, whereas the $z\approx0.5$ objects are observed in $R-I$. An $E(B-V)=0.02$ difference in extinction at the pole leads to approximately a $0.02$ mag difference in the relative distances between $z=0$ and $z=0.5$ objects, assuming a Galactic reddening law, host-galaxy corrections based on rest-frame $B-V$ color, and distances based on $V$. For this analysis, we use the SFD extinction map values with an uncertainty of 16% for each individual [SN Ia]{} but assume an additional $0.01$ mag of systematic uncertainty in our distance moduli to account for the known source of uncertainty of extinction at the pole.
In most supernova work we assume the Galactic reddening law [@cardelli89] applies to external galaxies ($R_V=3.1$), but studies of individual [SNe Ia]{} have found a range of values extending to much smaller values of $R_V$ [@riess96; @tripp98; @phillips99; @krisciunas00; @wang03; @altavilla04; @reindl05; @elias-rosa06]. These measurements are dominated by objects with large extinction values, where a significant measurement can be made can be made of the extinction law (lessening the effects of intrinsic color scatter and systematic color variations with luminosity), and it is possible that $R_V$ is correlated with total extinction [@jha06c]. In principle, with photometry in three or more passbands, it is possible to fit for $R_V$, but in practice, at $z>0.2$, there are only a few [SNe Ia]{} in the literature with the requisite high-precision photometry extending from the rest-frame UV to the near-IR. The systematic error on our measurement of $D_L$ caused by assuming a particular value of $R_V$ depends on the average extinction as a function redshift, assuming $R_V$ is constant with $z$, except for a small correction caused by the rest-frame effective bandpass of our filters drifting away from the low-$z$ values, depending on the precise redshift of each object. To quantify this effect, we fit our complete distance set with three different values of $R_V$: $2.1$, $3.1$, and $4.1$.
Color and Extinction Distributions and Priors {#sec:priors}
---------------------------------------------
To evaluate the systematic effects produced by various prior assumptions about extinction, we have fit the entire data set with a variety of plausible priors: the “exponential” prior of @jha06c, a flat prior from $-\infty$ to $+\infty$ (the “flatnegav” prior), and an exponential prior with an added Gaussian around zero that is based on models of the dust distribution in galaxies (“glos” and the redshift-dependent “glosz”). These results are presented in §\[sec:cosmology\] and form the basis for Table \[tab:w\_rv\_sys\].
To separate the effects of color and extinction, @jha06c noted that the distribution of color excess in their nearby sample was consistent with a Gaussian distribution of $\sigma=0.2$ convolved with a one-sided exponential, $\exp{(-A_V/\tau)}$, where $\tau=0.46$ mag. As discussed in §\[sec:extinction\_prior\], the “glosz” prior we adopt here is derived from models of line-of-sight dust distributions in galaxies. It has more parameters than the simple exponential model of @jha06c, but we believe these additional parameters are well motivated.
The power of MLCS2k2 to distinguish between color and extinction lies in the ability to treat the two phenomena independently. A06 uses SALT and makes the assumption that the color$+$extinction distribution is the same in the nearby and in the high-redshift samples; the separation of the $A_V$ component in the MLCS2k2 model allows us to model our expected distribution of $A_V$ based on both models of dust in galaxies and selection effects of the ESSENCE survey. This separation allows us to take the nominal “glos” model and create the “glosz” prior that combines the distribution of dust in galaxies with the redshift-dependent selection effects.
The difference in the mean estimated parameter for a constant $w$ is given in Table \[tab:w\_rv\_sys\] for the different MLCS2k2 $A_V$ priors discussed above. For the main MLCS2k2 “glosz” analysis we present here, we find a slope of ${\Delta}w / {\Delta}R_V=0.02$ in the dependence of $w$ on the assumed $R_V$. The effect on $w$ of varying $R_V$ is substantially greater for the less restrictive $A_V$ priors because the covariance between $A_V$ and $\mu$ is substantially greater for these priors. A reasonable variation of $0.5$ in the value for $R_V$ contributes a systematic uncertainty of ${\Delta}w=0.01$
Differences in the inferred value of $w$ for various assumed absorption priors shows that this is a significant systematic effect. The maximum difference between two priors, “exponential” and “glosz,” for the nominal $R_V=3.1$ case is ${\Delta}w=0.165$. While we have conducted careful simulations to determine the most appropriate prior for our sample (see §\[sec:selection\]) and it is clear that the “exponential” is not appropriate for this analysis, we nonetheless take half of the difference between the two as representative of our systematic uncertainty, $\Delta_w^{\rm prior}=0.08$, due to the choice of prior. The residual $0.02$ mag shift of the simulations with the “glosz” prior shown in Fig. \[fig:cut\] for $z\approx0.65$ results in a very small shift in ${\Delta}w$ of only $0.001$. Since we use an $A_V$, that obviously interacts strongly with our understanding of the intrinsic color distribution of [SNe Ia]{}. We estimate this contribution to our systematic error budget at ${\Delta}w=0.06$ We have not undertaken a similar analysis with the SALT fitter, but the underlying assumption that the color, extinction, luminosity relationship for [SNe Ia]{} is constant with redshift is subject to uncertainties analogous to those considered here in the context of the MLCS2k2 $A_V$ prior. The issue of color and extinction distributions clearly needs to be addressed for substantial further progress to be made in the field of supernova cosmology.
Malmquist Bias and Other Selection Effects {#sec:malmquist}
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As with all magnitude-limited surveys, at the faint limits of the survey we are more likely to observe objects drawn from the bright end of the [SN Ia]{} luminosity distribution. This Malmquist bias is particularly dangerous for inferences about cosmology based on supernova observations. However, it is not necessarily troubling that we may observe more luminous, broad events at high redshift, as long as the known empirical luminosity-width relation is valid at those redshifts. Rather, the concern for cosmological measurements is that at high redshift, we may preferentially find [SNe Ia]{} which are bright *for their light curve shape*. A second and more subtle concern is that at higher redshifts we are also less likely to detect [SNe Ia]{} whose light suffers significant absorption due to dust in their host galaxies.
We have modeled both of these effects (see §\[sec:selection\] & \[sec:extinction\]) and have controlled for their impact. Our current limits on systematics due to uncontrolled selection effects is $\Delta_w^{\rm selection}=0.02$. A thorough study of the efficiency of the ESSENCE survey will be presented by @pignata07. We aim for this future work to allow us to reduce this contribution to our systematic error to no more than 1%.
Type Ia Supernova Evolution {#sec:snevolution}
---------------------------
A persistent concern for any standard-candle cosmology is the possibility that the distant candles may differ slightly from their low-redshift counterparts. In a recent paper [@blondin06] we compare the spectra of the high-redshift [SNe Ia]{} in this sample with low-redshift [SNe Ia]{} and demonstrate that there is no evidence for any systematic difference in their properties. This conclusion is based on line-profile morphology and measurements of the phase-evolution of the velocity location of maximum absorption and peak emission.
These results confirm a number of other studies of distant [SNe Ia]{}[e.g., @coil00; @sullivan03; @lidman04] that all confirm that, to the accuracy of current observations, the high and low redshift supernova populations are indistinguishable. Recently @hook05 used spectral dating, spectral time sequences, and measurements of expansion velocities to compare distant and nearby [SNe Ia]{}; they also find no evidence for evolution in [SN Ia]{} properties up to $z\approx0.8$.
Although we are confident that the subtypes of distant [SNe Ia]{} are well represented by the subtypes seen nearby, we cannot rule out a subtle shift in the population demographics that may yet bias the estimates of cosmological parameters. This potential bias is of particular concern for future experiments that plan to measure the equation-of-state parameter, $w$, with an accuracy of a few percent. There is now evidence that [SN Ia]{} properties are correlated with host-galaxy morphology. @hamuy96 and @riess99 show that the brightest [SNe Ia]{} occur only in galaxies with on-going star formation. However, they observe no residual correlation after light-curve shape correction. Because the galactic demographics over the redshift range of interest change less than current variations in stellar population of [SN Ia]{}host galaxies, we remain confident that our one-parameter correction for supernova luminosity adequately corrects any shift in the average luminosity of [SNe Ia]{} to the same precision as in the nearby Universe, $\sigma_\mu < 0.02$ mag. We thus estimate a systematic uncertainty from possible [SN Ia]{}evolution on our measurement of $w$ of ${\Delta}w=0.02$.
One way to verify this confidence is to search for additional parameters that allow tighter luminosity groupings of the low-redshift population. In a first, reassuring step, Hubble diagrams for subsets of [SNe Ia]{} based on host-galaxy type separately confirm the accelerated expansion of the Universe [@sullivan03].
Hubble Bubble and Local Large-Scale Structure {#sec:bubble}
---------------------------------------------
The local large-scale structure and associated correlated flows of the Universe should not yet present a significant contribution to the systematic error budget of the current survey [@hui06; @cooray06]. However, at the lowest multipoles we are sensitive to local correlated flows, and, at the most extreme, our cosmological results would be sensitive to a local velocity monopole or “Hubble bubble.” @jha06c see such an effect in their analysis of nearby [SNe Ia]{}. We use only the subset of [SNe Ia]{} from @jha06c with $z>0.015$ and find that this effect could contribute as much as $0.065$ to our systematic error budget in $w$. We will rely on future sets of nearby [SNe Ia]{} ($0.01<z<0.05$) that are now being acquired at the CfA, by the Carnegie Supernova program, by the Lick Observatory Supernova Search, and by the SNfactory to reduce this uncertainty below 2% to help achieve the desired systematic uncertainty required for the final ESSENCE analysis.
Gravitational Lensing {#sec:lensing}
---------------------
Gravitational lensing can increase or decrease the observed flux from a distant object. The expected distribution is asymmetric about the average flux multiplier of unity. @holz05 calculate the effect for [SN Ia]{}surveys and determine that any systematic effect from neglecting the asymmetry of the probability distribution function for magnification (as we do here) decreases quickly with the number of [SNe Ia]{} per effective bin. Roughly speaking, at a $z\approx0.5$, in a redshift bin width of ${\Delta}z \sim 0.1$, ten [SNe Ia]{} per bin are sufficient to reduce any systematic effect in luminosity distance to less than $0.3\%$, which makes no noticeable contribution to our systematic error budget. For the redshifts of interest in the ESSENCE survey, lensing has a more significant effect in the scatter it adds to the observed brightness of [SNe Ia]{}. @holz05 calculate a 3% increase in the dispersion in distance modulus at $z\approx0.5$. We include the effect of lensing in our analysis by adding a statistical dispersion of $\sigma^{\rm lensing}_{\mu}=0.03$ to our luminosity distance uncertainty for the ESSENCE and SNLS [SNe Ia]{}.
Grey Dust {#sec:greydust}
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When the first cosmological results with [SNe Ia]{} were announced, that distant [SNe Ia]{} were dimmer than they would be in a decelerating Universe, @aguirre99a [@aguirre99b] suggested various models for intergalactic grey dust that could explain this dimming without producing observable reddening. To explain [SNe Ia]{}becoming consistently dimmer with distance, this dust would need to be distributed throughout intergalactic space beginning at least at $z=2$ [@goobar02]. The most naive model of such dust distribution and creation would predict that [SNe Ia]{} should continue to get dimmer relative to a flat, [${\Omega}_{\rm M}$]{}$=1$, cosmology all the way up to at least a redshift of $2$. The high-redshift [SN Ia]{} work of @riess04b demonstrated that this continued dimming is not what is observed: the apparent magnitudes of [SNe Ia]{} become first a little dimmer and then a little brighter with redshift than they would in an empty Universe. This is exactly what we expect from an early phase of deceleration followed by a recent phase of acceleration in a mixed, dark-matter/dark-energy cosmology.
A more complicated model of dust was contrived by @goobar02. It involves the creation of intergalactic dust at just the right rate to match the decrease in opacity due to expansion of the Universe. This carefully constructed model mimics the signal of an accelerating universe and is difficult to distinguish from a universe that is presently dominated by dark energy. This model does not have a strong underpinning in the behavior of known dust and represents a form of fine-tuning. In the larger context of converging cosmological evidence, this particular scheme for matching the data seems less plausible than a universe with dark energy.
Recent observational constraints from non-[SN Ia]{} sources have independently placed significant constraints on the amount of intergalactic dust [@petric06; @ostman06]. In particular, the observations of @petric06 limit intergalactic dust to contributing no more than one percent to potential dimming of light out to a redshift of $0.5$, based on upper limits to X-ray scattering by dust along the line of sight to a quasar at $z=4.3$.
Cosmological Results from the ESSENCE Four-Year Data {#sec:cosmology}
====================================================
The ESSENCE [SNe Ia]{} allow us test the hypothesis of a [$\Lambda{\rm CDM}$]{} concordance model and constrain flat, constant-$w$ models of the Universe. We use our MLCS2k2 light-curve fitting technique to measure luminosity distances to nearby and ESSENCE [SNe Ia]{} (Table \[tab:mlcs\_fits\_prior\_glosz\]). When then fit cosmological models to constrain the properties of the dark energy. We compare the results we obtain using MLCS2k2 with those obtained using the SALT light-curve fitter [@guy05]. The SALT fitter was used to fit the nearby light curves, our ESSENCE light curves, and the SNLS light curves.[^4] To verify that we were making appropriate use of the fitter, we fit the nearby and SNLS light curves with SALT, taking the same $\alpha=1.52$ and $\beta=1.57$ width and color parameters used in A06. We recovered the $w$ result of A06 to within $0.01$ in best-fit constant $w$ in a model with a flat Universe using the cosmology fitter that we employ here[^5]. We have compiled our light curves of nearby [SNe Ia]{} from the literature independently of the SNLS analysis and used slightly different quality cuts, so it is quite encouraging that we can replicate these results. Table \[tab:salt\_fits\] gives the SALT fit parameters for the nearby, ESSENCE, and SNLS [SNe Ia]{} discussed here.
ESSENCE [SN Ia]{} Sample {#sec:essence}
------------------------
For the ESSENCE project we find that using photometric selection criteria based on the color and rise time of the candidate object, similar to those used by the SNLS [@howell05; @sullivan06a], and in good weather and seeing conditions, 80% of the candidates we take spectra of are [SNe Ia]{}. We use a deterministic analysis [@blondin07], as described in @miknaitis07, to determine final types and redshifts for our SNe and to cull objects that are not [SNe Ia]{} from our sample. All of the ESSENCE supernovae used in this analysis were spectroscopically confirmed as [SNe Ia]{}.
From 2002–2005 the ESSENCE project discovered and spectroscopically confirmed 113 [SNe Ia]{}. As discussed by @miknaitis07, which gives full details of these [SNe Ia]{} including their RA and Dec, only 4 of the 15 [SNe Ia]{} from 2002 have been fully analyzed so that leaves us with 102 [SNe Ia]{}. Although we kept 91T-like [SNe Ia]{} such as d083, d085, and d093, we rejected the peculiar [SN Ia]{} d100 [@matheson05]. Three [SNe Ia]{} were rejected from the nearby+ESSENCE only fits because they were at redshifts greater than $0.67$ (see below). After we applied the cuts in Tables \[tab:mlcs\_cuts\] and Tables \[tab:salt\_cuts\], we were left with 57 and 60 [SNe Ia]{} for MLCS2k2 and SALT respectively.
With the MLCS2k2 fitter, the largest cut was the 32 [SNe Ia]{} rejected because they had fewer than 8 data points with an SNR $> 5$, no such points after +9 days, or no such points before +4 days. Two of the 102 [SNe Ia]{} were located near edges of the detector field-of-view that we later determined were photometrically less reliable. Due to high [$\chi^2/{\rm DoF}$]{} or related poor light-curve goodness-of-fit values, we eliminated an additional 6 [SNe Ia]{}. This left us with a total of 57 [SNe Ia]{} for our main MLCS2k2 nearby+ESSENCE analysis. The SALT fitter successfully fit three more [SNe Ia]{} than MLCS2k2, but, in general, our SALT quality cuts accepted the same [SNe Ia]{} as our MLCS2k2 quality cuts. The requirements we imposed here on the light curves were stringent cuts to ensure reliable fit parameters. We are currently engaged in an active program to improve the sensitivity of [SN Ia]{} light-curve fitters and we anticipate recovering 50% of the [SNe Ia]{} rejected here in the final ESSENCE analysis.
Nearby [SN Ia]{} Sample {#sec:nearby}
-----------------------
The [SN Ia]{} cosmological measurement is fundamentally a comparison of the luminosity distance vs. redshift relation at low redshift and high redshift. The ESSENCE [SNe Ia]{} alone provide a homogeneous set of luminosity distance vs. redshift measurements covering the redshift range $0.15<z<0.7$. We selected our nearby [SNe Ia]{} from the set compiled by @jha06c. We applied the light-curve criteria from Tables \[tab:mlcs\_cuts\] and \[tab:salt\_cuts\] and also rejected known peculiar [SNe Ia]{} such as SN 2000cx [@li01] and SN 2002cx [@li03; @jha06b]. Our list of nearby [SNe Ia]{} has 41 [SNe Ia]{} in common with the set used by A06. To minimize complications from loosely constrained local peculiar and coordinated flows, we only considered [SNe Ia]{} with CMB-frame redshifts of $z>0.015$. Our final sample consists of [45]{} nearby [SNe Ia]{} as listed in the fit parameter tables (Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\]). We used the re-derived Landolt/Vega calibration of A06 to determine the passbands for this set of nearby [SNe Ia]{}. The light curves we used for these [SNe Ia]{} are also included with the ESSENCE light curves available on our website.[^6]
External Constraints
--------------------
To provide complementary cosmological constraints on our [SN Ia]{} observations, we include external information from baryon acoustic oscillations [BAO; @eisenstein05]. The BAO constraints on ([${\Omega}_{\rm M}$]{}, $w$) from @eisenstein05 are the most complementary measurement in the ([${\Omega}_{\rm M}$]{}, $w$) plane to our [SN Ia]{} measurements, relying only on the observed redshift and angular size of the first doppler peak in the CMB and not on $H_0$. In addition, because the BAO constraints on [${\Omega}_{\rm M}$]{} are similar in precision (and value) to those derived from large scale structure [@percival01; @percival02], WMAP directly [@spergel06], and from the study of X-ray clusters [for a review see @voit05], we choose to combine our results only with the BAO results.
We compare the specific model of a flat Universe with either $w=-1$ or constant $w$ of any value to our data. [SNe Ia]{} have very little leverage on the global flatness of the Universe because they effectively measure the difference between [${\Omega}_{\rm M}$]{} and [${\Omega}_{\Lambda}$]{}, and flatness depends on the sum. @eisenstein05 have constrained curvature to be within [${\Omega}_{\rm K}$]{}$=\pm0.01$ of flat. The results presented here (from the [SNe Ia]{}) on $w$ are not significantly affected by variation of [${\Omega}_{\rm K}$]{} by this amount, because the effects of curvature are not noticeable until looking back to much higher redshift. However, non-flat models will significant alter the BAO results on ([${\Omega}_{\rm M}$]{}, $w$) and therefore our joint constraints.
For our analysis of the ESSENCE and nearby [SNe Ia]{}, we have chosen to additionally limit our redshift range to $z<0.670$ to avoid using the rest-frame $U$ band. Since this remove just three ESSENCE [SNe Ia]{} from our sample, the tradeoff is worthwhile to minimize this source of uncertainty (see §\[sec:distances\]). When we add in the SNLS or Riess gold samples, we relax this constraint to incorporate those higher-redshift [SNe Ia]{}.
In Figs. \[fig:joint\_mlcs\_prior\_glosz\_hubble\_diagram\] and \[fig:joint\_salt\_hubble\_diagram\] we show Hubble diagrams of the nearby, ESSENCE, and SNLS samples for the two different fitters we consider in this paper. We overplot an empty Universe ([${\Omega}_{\rm M}$]{},[${\Omega}_{\Lambda}$]{},$w$) = $(0,0,-1)$, a matter-only open Universe $(0.3,0,-1)$, and a [$\Lambda{\rm CDM}$]{} concordance cosmology $(0.27,0.73,-1)$. The residuals in luminosity distance are then shown with respect to the [$\Lambda{\rm CDM}$]{} model. MLCS2k2 appears to be more suited for the ESSENCE data sample than SALT, although the latter benefits from its flux-based fitting by being able to extract useful luminosity distances from a few more [SNe Ia]{}. One [SN Ia]{}, “d083,” is a particular outlier in both fitters at $\sim0.5$ mag brighter than expected in the best-fit or [$\Lambda{\rm CDM}$]{} cosmologies. @matheson05 found the spectrum of this object to be like that of SN 1991T, which is the archetype of over-luminous [SNe Ia]{}. This [SN Ia]{} is likely an interesting object worthy of further study and is potentially similar to a similarly super-luminous object, SN 2003fg, found in the SNLS survey [@howell06]. However, given that our sample comprises [60]{} objects, we certainly allow for the reasonable statistical possibility of a 3$\sigma$ outlier such as “d083” and thus retain it in our sample.
In Fig. \[fig:mlcs\_prior\_glosz\_salt\_om\_w\] we show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ probability contours for our measurement of $w$ vs. [${\Omega}_{\rm M}$]{} for ESSENCE+nearby alone, the BAO constraints from @eisenstein05, and the combination of the [SN Ia]{} and BAO constraints.
Table \[tab:results\] shows the cosmological parameters $w$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology and the 1-D marginalized values. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the MLCS2k2-analyzed ESSENCE+nearby sample with a [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}and a residual standard deviation of [$0.20$]{} mag. Thus, while the estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, a flat, $w=-1$ model of the Universe is consistent with our data.
Our results from these [60]{} [SNe Ia]{} from the ESSENCE survey are consistent with the results of A06. It is reassuring that two independent teams using different telescopes and studying different regions of the sky find that [SNe Ia]{} at high redshift exhibit the same luminosity distance vs. redshift relationship. These samples strengthen and extend the evidence from [SNe Ia]{} for dark energy and, together with complementary constraints on [${\Omega}_{\rm M}$]{}, point toward simple [$\Lambda{\rm CDM}$]{} models for our Universe.
Joint ESSENCE+SNLS Cosmological Constraints
-------------------------------------------
A new opportunity presents itself with the release of the SNLS light curves from A06 and the light curves presented in this paper. For the first time it is possible to do a proper, self-consistent joint fit of two large, independent sets of distant [SNe Ia]{}.
When fitting the SNLS [SNe Ia]{} with MLCS2k2 and the “glosz” prior we shift the assumed $A_V$ and $\Delta$ prior selection window functions by ${\Delta}z=+0.20$ to represent the greater depth of the SNLS survey. The proper way to derive this prior for SNLS would be to model the SNLS survey efficiency and and fit simulated [SNe Ia]{} with MLCS2k2 as we presented in §\[sec:selection\] for the ESSENCE survey. Similar concerns apply for possible selection effects in the heterogeneously nearby sample. Nevertheless, we believe our use of the “glosz” prior is appropriate for the low-redshift sample (where it is just the “glos” prior) and the simple extension in redshift to be a reasonable approach for the SNLS sample. The additional systematic errors introduced by this joint comparison center on the photometric calibration of the distant sample relative to the nearby [SNe Ia]{}. We estimate that uncertainty to be the same as the calibration uncertainty to the nominal Vega system used by each project: $\Delta{\rm zpt}=0.02$ mag. We have not modeled different offsets between the two data sets, but merely express the uncertainty as an additional uncertainty in our inferred cosmological parameters. This relative zeropoint uncertainty adds an additional ${\Delta}w=0.02$ to our overall systematic uncertainty on $w$.
With our combined analysis, we start with the traditional [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{}contour plot that was the first clear evidence for dark energy.
Table \[tab:joint\_results\] shows the cosmological parameters $w_0$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the SNLS+ESSENCE+nearby sample analyzed using MLCS2k2 “glosz” with a [$\chi^2/{\rm DoF}$]{} of [$0.90$]{} from [162]{} [SNe Ia]{}and a residual standard deviation of [$0.23$]{} mag. A joint analysis of the luminosity distances from the SALT fitter results in a [$\chi^2/{\rm DoF}$]{} of [$2.76$]{} from [178]{} [SNe Ia]{}and a residual standard deviation of [$0.28$]{} mag.. Fig. \[fig:joint\_mlcs\_prior\_glosz\_salt\_om\_w\] show the joint MLCS2k2 and SALT results for this joint sample. The estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, and a flat, $w=-1$ model of the Universe remains consistent with the current generation of [SN Ia]{} data.
Joint ESSENCE+SNLS+Riess Gold Sample Cosmological Constraints
-------------------------------------------------------------
In order to explore models with varying $w$, we now include the gold sample from @riess04b to extend our reach out to $z\approx1.5$. The high-quality intermediate-redshift samples of the ESSENCE and SNLS surveys provide an excellent complement to the high-redshift [SNe Ia]{}in this set. The heterogeneous nature of the collection of [SNe Ia]{} in the gold sample makes it beyond the scope of this paper to produce definite estimates of the systematic errors that result from including this additional set, but it is tempting to add these [SNe Ia]{} and examine the new constraints on cosmological parameters.
We used the 39 nearby [SNe Ia]{} in common between the nearby [SN Ia]{} sample we discuss here and the gold sample to normalize the luminosity distances between the two sets. To avoid double-counting of [SNe Ia]{} in this joint analysis, we then drop the nearby [SNe Ia]{} from the gold sample and use only the nearby [SNe Ia]{} fit in this paper.
We first compute the [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{} contours to update the case for dark energy from [SNe Ia]{}. Fig. \[fig:OMOL\_OMw\_joint\_riess04\] represents the most stringent demonstration to date of the existence of dark energy. The [SNe Ia]{} data alone rule out an empty Universe at $4.5$ $\sigma$, an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(0.3, 0)$ Universe at $10$ $\sigma$, and an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(1, 0)$ $\sigma$ Universe at $>20$ $\sigma$. The joint constraints on constant-$w$ models from this full set are $w=-1.09^{+0.09}_{-0.10}$. The highest-redshift data do not noticeably improve constraints for these models over the set of intermediate-redshift [SNe Ia]{} from ESSENCE+SNLS. It is for models with variable $w$ that the high-redshift data summarized by @riess04b provide the most utility. We here provide the global constraints on models characterized by $w=w_0+w_a(1-a)$ [@linder03; @albrecht06]. Using the BAO constraints on variable $w$ models would require integration from $z=0.35$ to $z\sim1089$ and the corresponding assumption that $w=w_0+w_a(1-a)$ is the proper parameterization over this stretch. If one is willing to make this assumption, then BAO+CMB already places significant constraints on the allowed $(w_0,w_a)$ parameter space. However, given that our multi-variable parameterizations of $w$ are arbitrary models with no clear theoretical motivation, we instead choose to regard $w=w_0+w_a(1-a)$ as a local expansion valid out to a redshift of $\sim2$ but not necessarily to $z\sim1089$. We then explicitly assume [${\Omega}_{\rm M}$]{}$=0.27\pm0.03$. Fig. \[fig:w0wa\_joint\_riess04\] shows the $(w_0, w_a)$ contours for this combined analysis. These constraints represent the advances of our understanding of dark energy. It is clear that work remains to constrain models of variable $w$.
Conclusions {#sec:conclusions}
===========
The ESSENCE survey has successfully discovered, confirmed, and followed 119 [SNe Ia]{} in our first four years of operation. We presented results from an analysis of [60]{} of those [SNe Ia]{} here, chosen so as to maximize insight while minimizing susceptibility to systematic errors. We have expended considerable effort to make quantitative estimates of various sources of systematic uncertainty that may afflict the ESSENCE results; of these, host-galaxy extinction and a potential local velocity monopole are currently the predominant concerns. We are working to devise ways to better estimate extinction, using both spectroscopic and photometric observations. Ideally, we would use all available information to arrive at an extinction prior customized for each supernova (e.g., different priors for elliptical and spiral host galaxies), rather than applying a single prior to the collection of all light curves.
The ESSENCE photometric calibration uncertainties will be reduced by an intensive calibration campaign this fall on the CTIO 4-m telescope in conjunction with the improved calibration of the SDSS southern stripe from the SDSS II project [@frieman04; @dilday05]. We hope to reduce our overall systematic uncertainty to the 5% level through this improved photometric calibration and an improved external nearby [SN Ia]{} sample from KAIT, the Nearby Supernova Factory, CfA, SDSS II, and the Carnegie SN Program to reduce our systematic sensitivity to a potential velocity monopole in the local [SN Ia]{} sample.
Combining our [SN Ia]{} observations with the BAO results of @eisenstein05 we find that a fit to a constant-$w$, flat-Universe model to our data finds an estimated parameter value of $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} with a [$\chi^2/{\rm DoF}$]{}$=$[$0.96$]{} using our full set analyzed with the MLCS2k2 fitter of @jha06c. A $w=-1$, flat-Universe model is consistent with our data. A combined analysis of ESSENCE+SNLS+nearby results in a estimated mean parameter of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}. We have no reliable estimate of the systematic effects involving the SALT fitter but take our general systematic uncertainty of [$0.13$]{} as representative of the issues currently confronting supernova cosmology.
The statistical increase from the [SNe Ia]{} from the entire 6-year ESSENCE data set plus improved photometric calibration of our detector and photometric measurements will reduce our statistical uncertainty to 7% and, together with an improvement in our systematic uncertainties to the level 5%, allow us to surpass our goal of a 10% measurement of a constant $w$ in a flat Universe.
Establishing the nature of dark energy is among the most pressing issues in the physical sciences today. The emerging impression that the equation-of-state parameter is close to $w=-1$ makes it difficult to determine whether the underlying physics arises in the particle physics sector or from the classical cosmological constant of general relativity. A value of $w=-1$ is perhaps the least informative possible outcome. In our view, this state of affairs motivates a vigorous effort to push the observational constraints to improve our sensitivity to the value and derivative of $w$ and strongly encourages searching for other indications of new physics, as we well may need another piece to solve the puzzle handed us by Nature.
Acknowledgments
===============
Based in part on observations obtained at the Cerro Tololo Inter-American Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation (NSF); the European Southern Observatory, Chile (ESO Programmes 170.A-0519 and 176.A-0319); the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the NSF (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONICET (Argentina) (Programs GN-2002B-Q-14, GS-2003B-Q-11, GN-2003B-Q-14, GS-2004B-Q-4, GN-2004B-Q-6, GS-2005B-Q-31, GN-2005B-Q-35); the Magellan Telescopes at Las Campanas Observatory; the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona; and the F. L. Whipple Observatory, which is operated by the Smithsonian Astrophysical Observatory. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration; the Observatory was made possible by the generous financial support of the W. M. Keck Foundation.
The ESSENCE survey team is very grateful to the scientific and technical staff at the observatories we have been privileged to use.
[*Facilities:*]{} , , , , , , , , .
The survey is supported by the US National Science Foundation through grants AST-0443378, AST-057475, and AST-0607485. The Dark Cosmology Centre is funded by the Danish National Research Foundation. SJ thanks the Stanford Linear Accelerator Center for support via a Panofsky Fellowship. AR thanks the NOAO Goldberg fellowship program for its support. PMG is supported in part by NASA Long-Term Astrophysics Grant NAG5-9364 and NASA/HST Grant GO-09860. RPK enjoy support from AST06-06772 and PHY99-07949 to the Kavli Institute for Theoretical Physics. AC acknoledges the support of CONICYT, Chile, under grants FONDECYT 1051061 and FONDAP Center for Astrophysics 15010003.
Our project was made possible by the survey program administered by NOAO, and builds upon the data reduction pipeline developed by the SuperMacho collaboration. IRAF is distributed by the National Optical Astronomy Observatory, which is operated by AURA under cooperative agreement with the NSF. The data analysis in this paper has made extensive use of the Hydra computer cluster run by the Computation Facility at the Harvard-Smithsonian Center for Astrophysics. We also acknowledge the support of Harvard University.
This paper is dedicated to the memory of our friend and colleague Bob Schommer.
[^1]: <http://www.ctio.noao.edu/essence/>
[^2]: <http://snls.in2p3.fr/conf/release/>
[^3]: <http://www.ctio.noao.edu/essence/>
[^4]: <http://snls.in2p3.fr/conf/release/>
[^5]: <http://qold.astro.utoronto.ca/conley/simple_cosfitter/>
[^6]: <http://www.ctio.noao.edu/essence/>
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