text
stringlengths 448
13k
| label
int64 0
1
| doc_idx
stringlengths 8
14
|
|---|---|---|
---
abstract: 'We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) giant quadrupole resonance (GQR) in $^{208}$Pb where the strength function is obtained by the shell model calculation including 1p1h and 2p2h configurations. It is found that at almost all energy scales, fluctuation of the strength function obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. This is contrasted with the results for the GQR in $^{40}$Ca, where at the intermediate energy scale about 1.7 MeV a deviation from the GOE limit was detected. It is found that the physical origin for this different behavior of the local scaling dimension is ascribed to the difference in the properties of the damping process.'
author:
- Hirokazu Aiba
- Masayuki Matsuo
- Shigeru Nishizaki
- Toru Suzuki
title: 'Fluctuation properties of strength function associated with the giant quadrupole resonance in $^{208}$Pb'
---
Introduction {#sec:intro}
============
Giant resonances, excited by various probes, show, at an initial stage of the excitation process, a regular motion with a definite vibrational frequency [@speth; @harakeh]. These regular motions are then damped due to the coupling with a huge number of background states, and finally the so | 0 | non_member_968 |
called compound states are realized.
We now have understood the both ends of these processes: The frequency of the giant resonance, for instance, can be well evaluated by the random phase approximation (RPA). Compound states, on the other hand, are also well described by the random matrix theory with the Gaussian orthogonal ensemble (GOE) [@dyson; @mehta], which characterizes a classical chaotic motion.
It is still not well understood, however, how the dynamics changes from regular to chaotic [@mottelson]. In order to answer this question, it is very useful to study the fluctuation properties of the strength functions: The structure at the large energy scale of the strength function corresponds to the behavior of the initial stage, while the fluctuation properties at small energy scale correspond to the long time behavior.
We proposed and have used a novel fluctuation analysis based on the quantity we call the local scaling dimension to study the fluctuation properties of the strength functions [@aiba]. This method is devised to quantitatively characterize the fluctuation at each energy scale, and is suitable for the investigation of the fine structure of the strength function.
The strength distribution of giant resonances and its fluctuation have also been studied experimentally. | 0 | non_member_968 |
Recently, the fine structure of the strength distribution of the giant quadrupole resonance (GQR) in $^{208}$Pb [@shevchenko; @shevchenko2; @lacroix] or the Gamow-Teller resonance (GTR) in $^{90}$Zr [@kalmykov] were measured and theoretical analysis has also been done.
In the previous paper [@aiba2], we investigated the GQR in $^{40}$Ca, where the strength function was calculated by means of the second Tamm-Dancoff approximation (TDA), namely, the 1p1h and 2p2h model space is included. The results of the local scaling dimension analysis were as follows: At small energy scale, the behavior of the local scaling dimension is almost the same as that of the GOE, which exhibits the complexity of 2p2h background states. On the other hand, a clear deviation from the GOE was found at the intermediate energy scale and it was found that this energy corresponds to the spreading width of 1p1h states. Hence, we can say that the spreading width of 1p1h states is detected as deviation from the GOE limit in $^{40}$Ca.
For $^{40}$Ca the Landau damping is important for the damping process of the giant resonance. Namely, the strength is first fragmented over a wide range of 1p1h states, and this fragmentation characterizes a global profile of the total | 0 | non_member_968 |
strength function.
However, as the mass of nuclei increases, the relative importance of the Landau damping may change. Accordingly, 2p2h states may also contribute to the global profile of the strength function. Therefore, it is very important to investigate how the difference between the damping process of light nuclei and that of heavy nuclei does affect the properties of the fluctuation of the strength function.
In this paper, we study the isoscalar (IS) GQR of $^{208}$Pb, where the strength function is calculated with the second TDA in the same manner as in $^{40}$Ca, and study the fluctuation of the strength function by means of the local scaling dimension. Comparing results with those of $^{40}$Ca we would like to clarify which properties of the damping process are reflected in the fluctuation of the strength function and make clear the physical origin of the difference.
This paper is organized as follows: In Sec. \[sec:lsd\], we briefly explain the local scaling dimension. The strength function for IS GQR in $^{208}$Pb is calculated in Sec. \[sec:numerical\], where the adopted Hamiltonian and the model space are shown. In Sec. \[sec:measures\], we discuss the nearest-neighbor level spacing distribution, $\Delta_3$ statistics as well as a histogram of | 0 | non_member_968 |
the strength distribution. In Sec. \[sec:results\], we apply the local scaling dimension to the IS GQR strength function in $^{208}$Pb. Detail of damping process is studied in Sec. \[sec:damping\], where the physical origin for the difference of the fluctuation property of the strength function between $^{40}$Ca and $^{208}$Pb is also discussed. Finally, Sec. \[sec:conclusion\] is devoted to conclusion.
Local Scaling Dimension {#sec:lsd}
=======================
We briefly explain the local scaling dimension. See Refs. [@aiba; @aiba2] for details.
The strength function is expressed as [@Bohr-Mottelson2] $$S(E)=\sum_i S_i\delta(E-E_i+E_0).
\label{defstr}$$ Here $E_i$ and $S_i$ denote the energy and the strength of exciting the $i$th energy level, respectively. Strengths are normalized as $\sum_i S_i=1$.
To study the fluctuation at each energy scale, we consider binned distribution of the strength by dividing whole energy interval under consideration into $N$ bins with length $\epsilon$. Strength contained in $n$th bin is denoted by $p_n$, $$p_n\equiv\sum_{i\in n{\rm th~ bin}}S_i.
\label{defp}$$ To characterize the distribution of the binned strengths, we introduce the moments of $p_n$, which are called in literature the partition function $\chi_m(\epsilon)$ defined by $$\chi_m(\epsilon)\equiv \sum_{n=1}^N p_n^m \\
=N\langle p_n^m\rangle.
\label{partition}$$ Finally, by extending the idea of the generalized fractal dimensions [@hentschel; @halsey] to non-scaling cases in a | 0 | non_member_968 |
straightforward way, we can define the local scaling dimension as, $$D_m(\epsilon)\equiv \frac{1}{m-1}
\frac{\partial\log\chi_m(\epsilon)}
{\partial\log\epsilon}.
\label{scaledim}$$ Since the local scaling dimension has a definite physical meaning similar to that of the generalized fractal dimension, the value of $D_m(\epsilon)$ can quantitatively characterize the fluctuation of the strength function at each energy scale $\epsilon$.
In the actual calculation of the local scaling dimension, we define it by means of the finite difference under the change of a factor 2, $$D_m(\sqrt{2}\epsilon) = \frac{1}{m-1}\frac{
\log\chi_m(2\epsilon)-
\log\chi_m(\epsilon)}
{\log 2},
\label{approscaledim}$$ rather than the derivative in Eq. (\[scaledim\]).
Numerical Calculation of Strength Function {#sec:numerical}
==========================================
We calculated the strength function of the IS GQR in $^{208}$Pb within the second TDA including the 1p1h and 2p2h excitations. Single-particle wave-functions and energies were obtained for a Woods-Saxon potential including the Coulomb interaction. The effective mass parameter $m^*/m$, which scales the Woods-Saxon single-particle energies $\varepsilon_{\rm WS}$ as $\varepsilon_{\rm HF}=\varepsilon_{\rm WS}/(m^*/m)$ to simulate the bare (Hartree-Fock) single-particle energies $\varepsilon_{\rm HF}$, is set to be 1 in this calculation.
As the residual interaction, the Landau-Migdal-type interaction [@Schwe] including the density-dependence was adopted. The model space was constructed in terms of single-particle states within the four major shells, two below and two above | 0 | non_member_968 |
the Fermi surface, and included all 1p1h states and 2p2h states whose unperturbed energies are less than 15MeV. Resultant number of 1p1h states and 2p2h states are 39 and 8032, respectively. We diagonalized the Hamiltonian within this model space and obtained the strength function for the isoscalar quadrupole operator.
![ Calculated strength function of the IS GQR in$^{208}$Pb. Dotted curve shows the smooth strength function by means of the Strutinsky method with the smoothing width 0.2 MeV. []{data-label="fig_strfun"}](fig1.eps){width="6cm"}
Figure \[fig\_strfun\] shows the calculated strength function. The average of the excitation energy weighted by the strength is about 10.5 MeV, and the standard deviation around the average is about 2.6 MeV, where all levels are considered. The peak position lies at the same value as the average. These values are consistent with the (p,p’) experimental data [@shevchenko]. Moreover, the agreement of the global shape with the experimental data is also good. The dotted curve in Fig. \[fig\_strfun\] represents the smooth strength function by means of the Strutinsky method [@ring-schuck] with the smoothing width 0.2 MeV. The value of the FWHM of this smooth strength function is 0.63 MeV. In order to quantitatively characterize the spreading of the strength function around the | 0 | non_member_968 |
largest peak, the FWHM is more appropriate than the standard deviation [@bertsch2]. Thus, we use the FWHM as a measure of the total width $\Gamma$ of the strength function, which gives $\Gamma=0.63$ MeV.
Hereafter, when we estimate the value of the FWHM, the same procedure as above is adopted, namely, we calculate the FWHM for the smooth strength function by means of the Strutinsky method with the smoothing width 0.2 MeV.
Fluctuation at small scale {#sec:measures}
==========================
![ The nearest-neighbor level spacing distribution for (a) $^{40}$Ca and (b) $^{208}$Pb. For $^{208}$Pb 3321 levels between 9.9 MeV and 13.1 MeV, while for $^{40}$Ca 804 levels between 20 MeV and 30 MeV are considered. Level spacings were unfolded by the Strutinsky method with a smoothing width 0.5 MeV for $^{208}$Pb and 5.0 MeV for $^{40}$Ca, respectively. The solid curve represents the Wigner distribution. []{data-label="fig_nns"}](fig2.eps){width="9.2cm"}
Before going to the detailed discussion of the local scaling dimension, we briefly show the results for other fluctuation measures: the nearest-neighbor level spacing distribution (NND), the strength distribution, and $\Delta_3$ statistics. Here, the NND and the strength distribution are measures characterizing the fluctuation at small energy scale limit. We present the results of $^{40}$Ca as well as | 0 | non_member_968 |
those of $^{208}$Pb for the sake of comparison.
Figure \[fig\_nns\] shows the NND. For both nuclei the NND follows the Wigner distribution well. We present the strength distribution in Fig. \[fig\_strdis\] where a histogram of the square-root of normalized strengths is plotted. We also find that for both $^{208}$Pb and $^{40}$Ca the distribution follows the Porter-Thomas one rather well. These two figures indicate that for both nuclei the fluctuation of the strength as well as that of the energy level spacing is governed by the GOE at least at small energy scale limit as expected.
Figure \[fig\_delta3\] shows the $\Delta_3$ statistics. We again find that at small energy range the $\Delta_3$ follows the GOE line for both $^{208}$Pb and $^{40}$Ca, although at intermediate energy scales, $L_{\rm max}\simeq 20$ or 15 for $^{208}$Pb or $^{40}$Ca, respectively, the $\Delta_3$ starts to deviate from the GOE line to upward.
![ The histogram of the square-root of normalized strengths $\bar{S}_i^{1/2}$ associated with IS GQR in (a) $^{40}$Ca and (b) $^{208}$Pb. The solid curve represents the Porter-Thomas distribution which becomes a Gaussian when plotted as a function of $\bar{S}_i^{1/2}$. See the caption of Fig. \[fig\_nns\] for the number of considered levels and also see Sec. | 0 | non_member_968 |
\[sec:cal\_lsd\] for the normalization of the strengths. []{data-label="fig_strdis"}](fig3.eps){width="9.2cm"}
![ The $\Delta_3$ statistics for (a) $^{40}$Ca and (b) $^{208}$Pb. The horizontal axis $L$ shows the value of the energy interval for the unfolded spectrum. The solid curve represents the $\Delta_3$ for the GOE level fluctuation. See Fig. \[fig\_nns\] for other parameters. []{data-label="fig_delta3"}](fig4.eps){width="9.2cm"}
Results of local scaling dimension {#sec:results}
==================================
Calculation of the local scaling dimension {#sec:cal_lsd}
------------------------------------------
Since we are not interested in the global shape of the strength function, we actually adopt the normalized strength function $\bar{S}(E)$ for the fluctuation analysis as in the case of $^{40}$Ca [@aiba2]. The normalized strength function $\bar{S}(E)$ is given by $${\bar S}(E)=\sum_i{\bar S}_i\delta(E-{\bar E}_i+{\bar E}_0),
\label{eq_norstrfun}$$ where the normalized strength $\bar{S}_i $ of the $i$th level is defined by $$\bar{S}_i \equiv{\cal N}\frac{S_i\tilde{\rho}(E_i)}{
\tilde{S}(E_i)}.
\label{eq_norstr}$$ Here, $\tilde{\rho}(E)$ and $\tilde{S}(E)$ denote the level density and the strength function, respectively, smoothed by the Strutinsky method [@ring-schuck]. ${\cal N}$ is a normalization factor to guarantee $\sum_i\bar{S}_i=1$.
We determine the width parameter $\omega$ of the Strutinsky smoothing function as follows: We note that the smoothed strength function $\tilde{S}(E)$ should represent the global profile of the original strength function $S(E)$ at large energy scale, but at the same time, we | 0 | non_member_968 |
would like to choose $\omega$ as large as possible since we do not want to wash out the fluctuations at smaller energy scales. Figure \[fig\_fwhm\] shows the FWHM of the smoothed strength function $\tilde{S}(E)/\tilde{\rho}(E)$ as the function of the smoothing width $\omega$. The linear increase of the FWHM at large values of $\omega \agt 0.6$ MeV indicates that the value of $\omega$ is too large, while with smaller values $\omega \alt 0.5$ MeV the FWHM stays at an approximately constant value, reflecting the total width. We therefore adopt 0.5 MeV as the value of the smoothing width $\omega$ in order to satisfy the above requirements.
We use the equidistant energy level ${\bar E}_i$ in Eq. (\[eq\_norstrfun\]), namely, ${\bar E}_i=id$, where $d$ denotes the average level spacing. Finally, we adopted the energy range from 9.9 MeV to 13.1 MeV, where 3321 levels are included.
![ FWHM of the smoothed strength function $\tilde{S}(E)/\tilde{\rho}(E)$ of IS GQR in $^{208}$Pb as a function of smoothing width $\omega$ used in the Strutinsky method. The dotted line is fitted to data and gives $\sim 0.98\omega+0.2$. []{data-label="fig_fwhm"}](fig5.eps){width="6cm"}
![ Normalized strength function Eq. (\[eq\_norstrfun\]) of IS GQR in$^{208}$Pb. Smoothing width $\omega=0.5$ MeV was used. []{data-label="fig_norstrfun"}](fig6.eps){width="6cm"}
The normalized strength | 0 | non_member_968 |
function is plotted in Fig. \[fig\_norstrfun\]. The local scaling dimension is derived from this normalized strength function.
Behavior of the local scaling dimension
---------------------------------------
![ Partition function (a) and local scaling dimension (b) for the IS GQR in $^{208}$Pb, and those in $^{40}$Ca are also shown at (c) and (d). Curves in each figure correspond to $m=2$ - 5 from upper to lower. Dotted curves in (b) and (d) represent $D_2(\epsilon)$ for the GOE. []{data-label="fig_pat_ldim"}](fig7.eps){width="9.2cm"}
Figure \[fig\_pat\_ldim\] (a) and (b) represent the partition function and the local scaling dimension, respectively, of IS GQR in $^{208}$Pb. The horizontal axes in both figures represent the bin width $\epsilon$ of energy in unit of $d$, where $d$ represents the average level spacing over the energy range 9.9 - 13.1 MeV ($d=0.96$ keV). The partition function clearly deviates from the linear relation in the log-log plot. This means that for the GQR strength function the self-similar property does not hold. We can also see a more detailed structure in the figure of the local scaling dimension. At the smallest energy scale $\epsilon\simeq d$, the value of the local scaling dimension is small, $D_2\simeq 0.35$, which means that the fluctuation is very large at small | 0 | non_member_968 |
energy scales. As the energy scale or the bin width increases, the values of $D_m(\epsilon)$ monotonically increase. Finally, at about $\epsilon\simeq 100d$ the values of $D_m(\epsilon)$ converges to unity, which indicates that at large energy scales, the strength function appears smooth. The most important feature in Fig. \[fig\_pat\_ldim\] (b) is that the local scaling dimension for $^{208}$Pb almost follows the GOE line at almost all the energy scales.
This should be contrasted with the case of $^{40}$Ca [@aiba2]: The partition function and the local scaling dimension for $^{40}$Ca are shown in Fig. \[fig\_pat\_ldim\] (c) and (d), respectively, for a comparison. When the energy scale is small, the local scaling dimension almost follows the GOE line. As the energy scale increases, however, we can find a dip and a deviation from the GOE line at about 1.7 MeV (Note that $d=12$ keV for $^{40}$Ca). We verified that an occurrence of the dip is not due to a statistical error. Moreover, further studies indicate that the energy where the minimum is located is approximately related to the value of the spreading width of 1p1h states.
Note that if we look only at the small energy scale limit or large energy scale limit, | 0 | non_member_968 |
we can not find the difference between $^{208}$Pb and $^{40}$Ca. Studies of fluctuation at intermediate energy scales lead to the finding of the difference. In the following we shall investigate the mechanism which brings about the difference in fluctuations at intermediate energy scales.
Studies of damping process {#sec:damping}
==========================
Let us now investigate origins of the difference between the cases of $^{40}$Ca and $^{208}$Pb. In our previous study of the GQR in $^{40}$Ca, we have shown that the behavior of the local scaling dimension, shown in Fig. \[fig\_pat\_ldim\] (d), can be interpreted in terms of the doorway damping mechanism. We here employ the same picture in order to clarify the damping mechanism of the GQR in $^{208}$Pb.
The doorway damping mechanism consists of a two-step process which is illustrated in Fig. \[fig\_pic\_Ca\]. The giant resonance is spread over the 1p1h states due to the Landau damping, the width of which is denoted by $\Gamma_{\rm L}$. The average spacing of 1p1h states is denoted by $D_{\rm 1p1h}$. The 1p1h states are considered here as the “doorway" states of the damping process. The 1p1h states then couple to more complicated background states (2p2h states) through the residual two-body interaction. The coupling causes | 0 | non_member_968 |
the spreading width of 1p1h states, which we denote $\gamma_{12}$. We define the GQR TD state as the Tamm-Dancoff (TD) state with the largest quadrupole strength among all TD states, where the TD states mean the states obtained in the TDA, i.e., by the diagonalization within the model space limited to the 1p1h configurations. The GQR TD state also couples to 2p2h states, and hence it should have the spreading width due to the coupling. This is similar to $\gamma_{12}$, but we introduce a separate symbol $\Gamma_2$ since the GQR TD state is a special state consisting of a coherent superposition of many unperturbed 1p1h excitations. $d_{\rm 2p2h}$ is the average spacing of background 2p2h states. The residual interaction also acts among the 2p2h states, and the mixing among the 2p2h states causes a spreading width of the 2p2h states, which we denote $\gamma_{22}$.
In the following we shall evaluate all these quantities in order to clarify the damping mechanism of the GQR in $^{208}$Pb (Sec. \[sec:mechanism\] and Sec. \[sec:spreading\_width\] ). We also study whether there are specific states among 2p2h states which strongly couple with the GQR mode (Sec. \[sec:surfacce\_vib\]) and then discuss the difference of the nature associated | 0 | non_member_968 |
with the fluctuation of strength function between $^{40}$Ca and $^{208}$Pb (Sec \[sec:comparison\]).
![ Schematic drawing of the doorway damping mechanism of the giant resonance, and related quantities. []{data-label="fig_pic_Ca"}](fig8.eps){width="7cm"}
Mechanism producing the total width {#sec:mechanism}
-----------------------------------
### Landau damping
For $^{40}$Ca, the Landau damping is important, so that the strengths are already fragmented in the 1p1h levels. Therefore we first would like to investigate in $^{208}$Pb, how the strength is distributed in the TDA where only the 1p1h states are included.
![ TDA strength function for the IS quadrupole operator in $^{208}$Pb. See Fig. \[fig\_strfun\] for the dotted curve. []{data-label="fig_tdastr"}](fig9.eps){width="6cm"}
Figure \[fig\_tdastr\] shows the TDA strength function, which is obtained by means of the TDA, namely by neglecting 2p2h states, of the IS quadrupole operator. Different from the case of $^{40}$Ca, strengths in the GQR region is considerably concentrated on the single peak located at about 10.7 MeV. Because of this, the TDA strength function is very different from the full strength function in Fig. \[fig\_strfun\]. At the same time, we also see only a small effect of the Landau damping. In fact, the strength concentration on the single peak at $E=10.7$ MeV is 59% of the strengths in the energy | 0 | non_member_968 |
interval 9 - 13 MeV. The Landau damping width $\Gamma_{\rm L}$ may be evaluated in terms of a smoothed profile of the strength function plotted with the dotted curve in Fig. \[fig\_tdastr\]. Its FWHM reads 0.21 MeV. On the other hand, if we closely look at Fig. \[fig\_tdastr\], we find that there is the second largest peak just below the largest one and that these two levels dominate the whole structure. The level spacing between these two levels can be considered as a typical spreading of strength and may be a more direct quantitative measure of the Landau damping width $\Gamma_{\rm L}$: The level spacing 0.18 MeV gives $\Gamma_{\rm L}=0.18$ MeV.
### damping due to 2p2h states
The Landau damping width $\Gamma_{\rm L}=0.18$ MeV is not enough to explain the total width $\Gamma=0.63$ MeV of Sec. \[sec:numerical\]. Then, we would like to study a role of 2p2h states in the damping process, namely, the fragmentation of the GQR TD state located at $E=10.7$ MeV in Fig. \[fig\_tdastr\] over 2p2h states. We shall investigate the damping width $\Gamma_2$ caused by the coupling to 2p2h states. To estimate this width, we perform a calculation where we include only the GQR TD state | 0 | non_member_968 |
and 2p2h states, where the coupling between the GQR TD state and 2p2h states as well as the interaction among 2p2h states are taken into account.
![ Strength function by neglecting all TD states except the GQR TD state. 3342 2p2h states lying in 9 MeV - 13 MeV are considered. See Fig. \[fig\_strfun\] for the dotted curve. []{data-label="fig_GQRTDstr"}](fig10.eps){width="6cm"}
Figure \[fig\_GQRTDstr\] shows the resulting strength function. The estimated FWHM is 0.41 MeV, i.e., $\Gamma_2=0.41$ MeV.
If the Landau damping and the 2p2h damping are independent of each other, and neighboring TD states around the GQR TD states also have the same spreading width as $\Gamma_2$, the following approximate relation holds: $$\Gamma\simeq\Gamma_{\rm L}+
\Gamma_2.
\label{Gamma}$$ The values, $\Gamma_{\rm L}=0.18$ MeV and $\Gamma_2=0.41$ MeV, estimated above indeed satisfy this relation. Consequently, the total width $\Gamma=0.63$ MeV is approximately explained as a sum of the Landau damping width $\Gamma_{\rm L} $ and the 2p2h damping width $\Gamma_2$.
The importance of the 2p2h damping is contrasted with the case of $^{40}$Ca, where the total width can be explained essentially by the Landau damping width, i.e., $\Gamma\simeq \Gamma_\text{L}$.
Spreading width of 1p1h states and 2p2h states {#sec:spreading_width}
----------------------------------------------
For the case of $^{40}$Ca, the strength | 0 | non_member_968 |
is fragmented over many 1p1h states by the Landau damping, and strength in each 1p1h state is further spread due to the coupling with 2p2h states. Let us evaluate the spreading width $\gamma_{12}$ of the 1p1h states due to this coupling. We shall also evaluate the spreading width $\gamma_{22}$ of 2p2h states, which is caused by the residual coupling among 2p2h states.
![ Averaged strength function of (a) TD states and (b) 2p2h states. Average was performed over levels lying in 9 MeV - 13 MeV. The number of levels is 12 and 3342 for TD states and 2p2h states, respectively. []{data-label="fig_avedoorstr"}](fig11.eps){width="9.2cm"}
We evaluate $\gamma_{12}$ by using the strength functions of TD states as in Ref. [@aiba2]. Namely, we calculate the strength function of each TD state. Averaging the strength functions over whole TD states, we obtain Fig. \[fig\_avedoorstr\] (a). The FWHM of this averaged strength function gives an evaluation of the spreading width $\gamma_{12}$. We read $\gamma_{12 }=0.38$ MeV. (Note that we define $\gamma_{12 }$ as the spreading width of TD states instead of that of unperturbed 1p1h states.) The value of spreading width of 2p2h states $\gamma_{22}$ is also evaluated in the same manner. From Fig. \[fig\_avedoorstr\] (b) | 0 | non_member_968 |
we also obtain $\gamma_{22}=0.75$ MeV as the estimate of the spreading width of 2p2h states. These results will be used in Sec. \[sec:comparison\]
For the sake of comparison, let us estimate the spreading width by assuming the Fermi golden rule. The root mean square of matrix elements between 1p1h states and 2p2h states is calculated as $(\overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2})^{1/2}=9.3\times10^{-3}$ MeV. Similarly, we calculate $(\overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2})^{1/2}=1.0\times10^{-2}$ MeV. Since the level spacing of 2p2h states is $d_{\rm 2p2h}=1.2$ keV, the spreading widths $\gamma_{12}$ and $\gamma_{22}$ are approximately estimated in the Fermi golden rule as $\gamma^{\rm FG}_{12}=2\pi \overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2}/d_{\rm 2p2h}=0.46$ MeV and $\gamma^{\rm FG}_{22}=2\pi \overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2}/d_{\rm 2p2h}=0.53$ MeV, respectively, which are in approximate agreement with the direct evaluation within 30%.
Search for strongly coupled states in 2p2h states {#sec:surfacce_vib}
-------------------------------------------------
In the picture of Fig. \[fig\_pic\_Ca\] 2p2h states are assumed to play a role as the chaotic background and provide the GOE fluctuation to the strength function. However, if the GQR TD state couples with not all 2p2h states equally but specific states in 2p2h states strongly, there is a possibility for this hierarchical | 0 | non_member_968 |
structure in 2p2h states to give rise to a deviation from the GOE fluctuation. We, here, would like to investigate whether whole 2p2h states are rather equally coupled with the GQR TD state or whether there are specific states in 2p2h states which strongly couple with that state.
As a candidate of such specific states, we can consider the low-energy surface vibration plus 1p1h states: In Refs. [@bertsch2; @bertsch; @broglia; @bortignon; @lacroix2], the importance of the coupling to the surface vibration in the wide range of damping phenomena including the damping of a single particle motion as well as that of giant resonances was discussed. As for the giant resonance, which is composed of a coherent superposition of 1p1h states, this means that the damping occurs via the coupling with the specific 2p2h states, namely, the surface vibration plus 1p1h (s.v.+1p1h) states.
Since our model does not assume the particle-vibration coupling a priori, it is not trivial whether our model also has a mechanism that enhances the coupling with the low-energy surface vibration. Therefore, we would like to study whether the s.v.+1p1h states are particularly strongly coupled with the GQR TD state within our model. To do so, we calculate | 0 | non_member_968 |
the FWHM of the following approximate strength function: $$S(E)=-\frac{1}{\pi}{\rm Im}\left(
E-E_c-\sum_\alpha \frac{V_{c\alpha}^2}{E-\omega_\alpha+i\gamma_{22}/2}\right)^{-1},
\label{doorwaystr}$$ where, $E_c$ and $\omega_\alpha$ denote the energy of the GQR TD state and the energy of the $\alpha$th s.v.+1p1h state, respectively. $V_{c\alpha}$ represents the coupling matrix element between the GQR TD state and the s.v.+1p1h state $\alpha$.
Only $J^\pi=2^+$, $3^-$ modes are included as surface vibrations: We took only the lowest TD state as $J^\pi=2^+$ surface vibrational mode. On the other hand, we must pay attention to the collectivity of the octupole mode. Figure \[fig\_octstr\] shows the TDA strength function for the IS octupole operator. Compared with the experimental data [@spear], the energy of the lowest state is too high, and strengths are fragmented over several states. Thus, we took into account the lowest nine states for the octupole mode. Note that s.v.+1p1h states thus defined are not orthogonal. In this sense Eq. (\[doorwaystr\]) is an approximation which neglects the non-orthogonality.
![ TDA strength function for the IS octupole operator in $^{208}$Pb. []{data-label="fig_octstr"}](fig12.eps){width="6cm"}
The strength function based on Eq. (\[doorwaystr\]) is presented in Fig. \[fig\_doorwaystr\]. The width $\Gamma_2^{\rm (s.v.)}$ estimated by the FWHM is 0.074 MeV. This value is significantly smaller than the width $\Gamma_2=0.41$ MeV of | 0 | non_member_968 |
the GQR TD state caused by the coupling to the whole 2p2h states.
![ Strength function of the GQR TD state evaluated by considering only surface vibration plus 1p1h (s.v.+1p1h) states based on Eq. (\[doorwaystr\]). $\gamma_{22}=0.75$ MeV is used. []{data-label="fig_doorwaystr"}](fig13.eps){width="6cm"}
From the estimate by the Fermi golden rule, we can give more detailed comparison between the width for the case of s.v.+1p1h states and that for the whole 2p2h states. It is noted in Table \[table1\] that the spreading width $\Gamma_2^{\rm (s.v.)}=0.074$ MeV and $\Gamma_2=0.41$ MeV are well accounted for by the estimate. In the Fermi golden rule the spreading width is governed by two factors; 1) the average value of squared coupling matrix elements $\overline{ V_{c\alpha}^2}$ between the GQR TD state and the states that couple to it, and 2) the level density of the coupling states. From Table \[table1\], we see that the large difference between the two widths simply reflects the difference between the number of s.v.+1p1h states 909 and 2p2h states 3142 whereas the coupling strength of s.v.+1p1h states $\overline{ V_{c\alpha}^2}=0.65\times10^{-4}$ MeV$^2$ is comparable to the coupling strength $\overline{V_{c\alpha}^2}=0.72\times10^{-4}$ MeV$^2$ for the whole 2p2h states.
Table \[table1\] and Fig. \[fig\_doorwaystr\] suggest that our model does not | 0 | non_member_968 |
contain the enhancement of the coupling with the surface vibrations in the damping of the GQR. Therefore we consider in the following the 2p2h states as background states which do not have specific structures.
\# $\overline{V_{c\alpha}^2}$ (MeV$^2$) $\Gamma_2^{\rm FG}$ (MeV) $\Gamma_2$ (MeV)
----------- ------ -------------------------------------- --------------------------- ------------------
s.v.+1p1h 909 $0.65\times 10^{-4}$ 0.092 0.074
2p2h 3342 $0.72\times 10^{-4}$ 0.38 0.41
: Averaged value of squared coupling matrix elements $\overline{V_{c\alpha}^2}$ between the GQR TD state and surface vibration plus 1p1h states or the whole 2p2h states(third column), the associated spreading width $\Gamma_2^{\rm FG}$ of the GQR TD state evaluated by the Fermi golden rule (fourth column), and the spreading width $\Gamma_2$ estimated by the FWHM of the strength function based on Eq. (\[doorwaystr\]) (fifth column). Second column shows the number of states considered. The second row shows the results obtained by including only the s.v.+1p1h states while the third row shows those for the case of the whole 2p2h states. []{data-label="table1"}
Physical origin of the difference between $^{40}$Ca and $^{208}$Pb {#sec:comparison}
------------------------------------------------------------------
In the above subsections, we have evaluated the physical quantities such as the various spreading widths, with which we have discussed the damping process of $^{40}$Ca and $^{208}$Pb, especially the | 0 | non_member_968 |
mechanism of producing the total width of the strength function. Here, using these quantities we would like to discuss the physical origin of the difference between the fluctuation of the strength fluctuation of $^{40}$Ca and that of $^{208}$Pb. Table \[table2\] summarizes the values of the above physical quantities related to the initial stage of the damping process for both $^{40}$Ca and $^{208}$Pb.
We have shown in our previous study [@aiba] that the damping process through the doorway states causes large fluctuations which have characteristic energy scales, and that the fluctuations emerge in the local scaling dimension. For instance, the energy scale of the spreading width $\gamma_{12}$ of the doorway states is the quantity which shows up prior to the other quantities. It is noted, however, the size of the fluctuations depends on the mutual relations among the quantities mentioned above, and indeed we have examined in [@aiba] the relations which are needed to detect the effect of the spreading width $\gamma_{12}$.
$\Gamma$ $\Gamma_{\rm L}$ $\Gamma_2$ $\gamma_{12}$ $D_{\rm 1p1h}$ $\gamma_{22}$ $d_{\rm 2p2h}$
------------ ---------- ------------------ ------------ --------------- ---------------- --------------- ----------------
$^{40}$Ca 4000 4000 1500 1500 500 5200 11
$^{208}$Pb 630 180 410 380 230 750 1.2
: Values of physical quantities | 0 | non_member_968 |
---
abstract: 'We investigate the phase transitions of a nonlinear, parallel version of the Ising model, characterized by an antiferromagnetic linear coupling and ferromagnetic nonlinear one. This model arises in problems of opinion formation. The mean-field approximation shows chaotic oscillations, by changing the couplings or the connectivity. The spatial model shows bifurcations in the average magnetization, similar to what seen in the mean-field approximation, induced by the change of the topology, after rewiring short-range to long-range connection, as predicted by the small-world effect. These coherent periodic and chaotic oscillations of the magnetization reflect a certain degree of synchronization of the spins, induced by long-range couplings. Similar bifurcations may be induced in the randomly connected model by changing the couplings or the connectivity and also the dilution (degree of asynchronism) of the updating. We also examined the effects of inhomogeneity, mixing ferromagnetic and antiferromagnetic coupling, which induces an unexpected bifurcation diagram with a “bubbling” behavior, as also happens for dilution.'
author:
- Franco Bagnoli
- Raúl Rechtman
title: Stochastic Bifurcations in the Nonlinear Parallel Ising Model
---
Introduction
============
There are quite a large number of studies about opinion formation in uniform societies [@Hegselmann; @Deffuant; @review; @Stauffer; @GalamReview; @Galam1; @guazzini; @BagnoliGuazziniLio; | 0 | non_member_969 |
@GrottoGuazziniBagnoli; @GuazziniCiniBagnoliRamasco; @ViloneCarlettibagnoliGuazzini]. Many such models adopt an approach similar to that of the Ising model. In such cases one has two opinions, say A and B or -1 and 1, and one is interested in the establishment of a majority (magnetic phase transitions) or in the effects of borders, or in the influence of some leader (social impact theory) [@latane]. This opinion space can be seen as the first ingredient of these models.
The second ingredient is how to model the response to an external influence. It is common to classify the attitude of people (agents) as either conformist or contrarian (also known as nonconformist). A conformist tends to agree with his neighbors and a contrarian to disagree.
It is also easy to map this attitude onto Ising terms: conformist agents correspond to ferromagnetic coupling and contrarians to antiferromagnetic ones [@review]. The effects induced by the presence of contrarian agents in a society have been studied in models related to the voter model [@masuda2013; @crokidakis2014; @Independence; @schneider2004; @delalama2005; @corcos02; @galam04; @Biswas; @Galam-Gemrev; @Galam-chaotic; @sudoyi2013; @bagnoli2013; @bagnoli2015].
In general, agents that are under a strong social pressure tend to agree with the great majority even when they are certain that | 0 | non_member_969 |
the majority’s opinion is wrong, as shown by Asch [@Asch]. Under a strong social pressure a contrarian may agree with a large majority, an phenomenon that may be modelled using non-linear interactions.
The strategy of following an overwhelming majority may be ecological, since it is probable that this coherent behavior is due to some unknown piece of information, and in any case the competitive loss is minimal since it equally affects all other agents.
A binary opinion model where an agent tends to align with the largest neighboring cluster, similar to an Ising model with plaquette interactions, was studied in Ref. [@biswas2009]. In this model, a single dissenting agent immersed in a cluster of different opinions cannot overcome the social pressure, and therefore the model exhibits absorbing homogeneous phases. The possibility of dissenting, distributed as a quenched disorder, was introduced in Ref [@biswas2011a].
The third ingredient is the connectivity, i.e., how the neighborhood of a given agent is composed. Traditionally, magnetic systems have been studied either on regular lattices, trees or with random connections, whose behavior is similar to that of the mean-field approach. In recent years, much attention has been devoted to other network topologies, like the small-world [@WattsStrogatz] | 0 | non_member_969 |
to scale-free [@Barabasi] etc.
The Ising model has been studied in different topologies [@Klemm; @WuZhou; @HolmeNewman; @barre], in particular, the topological details may affect the critical dynamics [@goswami2011] and the zero-temperature quench [@biswas2011].
In contrast with the Ising model, in the study of opinion formation there is no compulsory obligation to have symmetric interactions, each agent is influenced by those in his neighborhood, which are not necessarily influenced by the first agent.
Each individual may be a conformist or a contrarian and this character does not change in time. In these terms, the simple ferromagnetic Ising model represents a uniform society of conformists with local symmetric interactions.
The fourth ingredient is the update scheduling, that may be completely asynchronous, like in standard Monte Carlo simulations, or completely parallel, like in Cellular Automata, or something in between [@Derrida; @NewmannDerrida; @Cirillo]. It is not clear which scheme is the most representative of reality. Real human interactions are indeed continuous, but also clocked by days, elections, etc.
An effect that is favoured by parallelism is synchronization in the presence of complex dynamics. As happens in physics, a macroscopic irregular behavior (macroscopic chaos) implies a coherent, although irregular motion of many elements (the microscopic | 0 | non_member_969 |
constituents).
One of most intriguing effects is the hipster’s one, in which a society of contrarians tends to behave in a uniform way [@hipster]. Clearly, “conformist hipsters” always change their behavior, when they realize to be still in the mainstream, but since they do so all together, they remain synchronized: the parallelism is a crucial element of such a behavior.
Finally, the fifth ingredient is homogeneity. There are many possibilities of introducing mixtures of agents or spins with different coupling. We investigate what happens when one mixes ferromagnetic and antiferromagnetic interactions, and we shall show that this mixture promotes a “bubbling” behavior in the bifurcation, meaning that the bifurcation appears first for intermediate values of the parameters, similar to what happens with asynchronism.
In previous studies we presented “reasonable contrarian” agents whose response to the average opinion of their neighbors is nonlinear and discusssed the collective behavior of societies composed of reasonable contrarians only and by mixtures of these agents and nonlinear conformists [@bagnoli2013; @bagnoli2015]. The rationale was that in some cases, and in particular in the presence of frustrated situations like in minority games [@minority; @minority1; @minority2; @minority3], it is not convenient to always follow the majority, since in | 0 | non_member_969 |
this case one is always on the “losing side” of the market. This is one of the main reasons for the emergence of a contrarian attitude. On the other hand, if all or almost all agents in a market take the same decision, it is often wise to follow such a trend. We can denote such a situation with the word “social norm”.
A society composed by a strong majority of reasonable contrarians exhibits interesting behaviors when changing the topology of the connections. On a one-dimensional regular lattice, there is no long-range order, the evolution is disordered and the average opinion is always halfway between the extreme values [@bagnoli2005]. However, adding long-range connections or rewiring existing ones, we observe the Watts-Strogatz “small-world” effect, with a transition towards a mean-field behavior. But since in this case the mean-field equation is, for a suitable choice of parameters, chaotic, we observe the emergence of coherent oscillations, with a bifurcation cascade eventually leading to a chaotic-like behavior of the average opinion.
The small-world transition is essentially a synchronization effect. Similar effects with a bifurcation diagram resembling that of the logistic map have been observed in a different model of “adapt if novel - drop | 0 | non_member_969 |
if ubiquitous” behavior, upon changing the connectivity [@Dodds; @Harris].
The main goal of the present study is that of reformulating the opinion formation models mentioned above [@bagnoli2013; @bagnoli2015], in terms of a parallel, nonlinear Ising model both on a regular lattice, where the spin at any site is influenced by its nearest neighbors, and on small-world networks.
In the first case the mean-field behavior of the magnetization is described by a nonlinear equation for which chaos can be evaluated by the Lyapunov exponent [@ott02], which is a measure of the stability of trajectories.
The Lyapunov exponent is the time average of the growth rate of an initial infinitesimal perturbation of a trajectory. Clearly, this quantity cannot be simply defined for stochastic systems, since in this case one would essentially measure the effects of the noise. However, in many cases and in particular the present one, we would like to compare the dynamical properties of a stochastic microscopic model and its mean-field approximation. We show here that the Boltzmann’s entropy of an aggregate variable like the magnetization is a quantity that can be defined for both deterministic and stochastic systems. In the first case, Boltzmann’s entropy can be used as a | 0 | non_member_969 |
measure of chaos [@Boltzmann].
The scheme of the paper is the following. We discuss the “nonlinear” parallel Ising model in Section \[sec:parallelising\]. We can therefore introduce the mean-field approximation of the model in Section \[sec:meanfield\], showing the bifurcation phase diagrams as a function of the parameters. The definition of the entropy and the results of microscopic simulations $\eta$ are reported in Section \[sec:simulations\]. Finally, conclusions are drawn in the last Section. In this Section we discuss also the differences between the present and the original model of Refs. [@bagnoli2013; @bagnoli2015].
![\[plaquette\] (Color online.) A $K=3$ spin neighborhood, with the interaction terms corresponding to the external field $\tilde H$, the two-spin $\tilde J$, three-spin $\tilde Z$, and four-spin $\tilde W$ interaction constants.](plaquette){width="\columnwidth"}
Parallel nonlinear Ising model {#sec:parallelising}
==============================
We consider a system with $N$ sites, each one in a state $s_i\in
\{-1,1\}$, $i=1,\dots,N$. The state of the system is $@s=(s_1,\dots,s_{N})$. The topology of the system is defined by the adjacency matrix $a$ with $a_{ij}=1$ if site $j$ belongs to site $i$’s neighborhood and is zero otherwise. The connectivity $k_i$, the local field $\tilde{h}_i$, and the rescaled local field $h_i$ at site $i$ are $$k_i=\sum_j a_{ij},\qquad
\tilde{h}_i=\sum_j a_{ij}s_j, \qquad
h_i=\dfrac{\tilde{h}_i}{k_i},$$ with $h_i\in | 0 | non_member_969 |
[-1,1]$. In this paper we shall use a uniform connectivity $k_i=K\,\,\forall\, i$. The magnetization $m$ is defined as $$m=\dfrac{1}{N}\sum_is_i,$$ with $m\in [-1,1]$.
In the following we consider multi-spin (plaquette) interactions. We moreover consider only completely asymmetric interactions [@Suzuki; @Sherrington], arranged to give a preferred direction that corresponds to time in the standard cellular automata language [@bagnoli2013; @SS-bif].
Considering up to 4-spin interactions, the Hamiltonian is $$\begin{aligned}
\label{Hamiltonian}
\mathcal{H} (@s) =& -\sum_i s_i'\Bigl(\tilde H+ \tilde J\sum_j a_{ij} s_j +%
\tilde Z \sum_{jk} a_{ij} a_{ik}s_j s_k + \nonumber\\
&\tilde W \sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l\Bigr),\end{aligned}$$ where $\tilde H$ is the external field, and $\tilde J$, $\tilde Z$, $\tilde W$ the two-spin, three-spin and four-spin interaction constants respectively as shown in Fig. \[plaquette\].
It is possible to recast the interaction constants in terms of the local field $h_i$. The terms containing $s'_i$ at “time” $t+1$ are, $$\begin{split}
&\text{2-spin: }\; s_i'\sum_j a_{ij} s_j = s_i' \tilde h_i,\\
&\text{3-spin: }\; s_i'\sum_{jk} a_{ij} a_{jk}s_j s_k=s_i'Q^{(2)}_i,\\
&\text{4-spin: }\; s_i'\sum_{jkl} a_{ij} a_{ik} a_{il} s_j s_k s_l = s_i'Q^{(3)}_i.\\
\end{split}$$ These expressions define $ Q^{(2)}$ and $ Q^{(3)}$. Since $$\begin{split}
&\tilde{h_i}^2 = K + 2Q_i^{(2)},\\
&\tilde{h_i}^3 = (3K-2)\tilde h_i + 6 Q_i^{(3)},\\
\end{split}$$ the Hamiltonian can be | 0 | non_member_969 |
written as $$\mathcal{H} (@s) = -\sum_i s'_i
(H + J h_i + Zh_i^2 + W h_i^3),$$ where the correspondences among coupling constants are $$\begin{split}
H &= \tilde H - \frac{1}{2} K\tilde Z,\\
J &= K\left(\tilde J - \frac{3K-2}{6}\tilde W\right),\\
Z &= \frac{1}{2} K^2\tilde Z, \\
W &= \frac{1}{6}K^3\tilde W.\\
\end{split}$$ In the following, we shall consider pair ($J$) and four-spin ($W$) terms, *i.e.*, $H=Z=0$, in agreement with previous investigations [@bagnoli2013].
The coupling term $J$ modulates the “linear” effects of neighbors, so $J>0$ gives a conformist (ferromagnetic) behavior and $J<0$ a contrarian (antiferromagnetic) one. The term $W$ modulates the nonlinear effects of the crowd. In this way one can model the Asch effect by inserting $J<0$ (contrarian attitude) and $W>0$ (social norms).
The time evolution of the spins is given by the parallel application of the transition probabilities $\tau(s_i'|h_i)$ that gives the probability that the spin at site $i$ and time $t+1$ takes value $s_i'$ given the local field $h_i$ at time $t$, see Fig. \[plaquette\]. The local transition probability is defined by a heat bath probability $$\begin{aligned}
\label{eq:ntau}
\tau(s'_i| h_i) &= \dfrac{1}{1+\exp(-2 s'_i (J h_i + W h_i^3))}\nonumber\\
&=\frac{1}{2}\left[1+s'_i\tanh (J h_i + W h_i^3)\right].\end{aligned}$$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(a) (b)
![\[fig:tau\] (Color | 0 | non_member_969 |
online). (a) The transition probability $\tau=\tau(1|h)$, Eq. as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](PI00-o1-k20-J-10p00-W15p0 "fig:"){width="0.45\columnwidth"} ![\[fig:tau\] (Color online). (a) The transition probability $\tau=\tau(1|h)$, Eq. as a function of the local field $h$ for three values of the coupling constant $J$. (b) Graph of the magnetization $m'$ at time $t+1$ as a function of the magnetization $m$ at time $t$, Eq. , with some iterates for $J=-7.5$. ](m-m-I00-o1-k20-J-7p50 "fig:"){width="0.45\columnwidth"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The parallel version of the linear ($W=0$) Ising model does not show many differences with respect to the standard serial one [@Derrida]. The observables that depend only on single-site properties take the same values in parallel or sequential dynamics [@NewmannDerrida], although differences arise for two-site correlations [@SS-meta]. In general the resulting dynamics is no more reversible with respect to the Gibbs measure induced by any Hamiltonian [@Cirillo].
In the following, unless otherwise specified, we always use $W=15$ and $K=20$.
Mean-field approximation {#sec:meanfield}
========================
The mean-field approximation for the magnetization $m$ of the fully parallel | 0 | non_member_969 |
case with fixed connectivity $K$, follows from the Markov equation assuming no spatial correlations. Then $$\begin{aligned}
\label{eq:mf}
m'=&f(m) = \frac{1}{2^{K}}\sum_{k=0}^K\binom{K}{k}(1+m)^k(1-m)^{K-k}\nonumber\\
&\times\tanh\left[J\left(\frac{2k}{K}-1\right)+W\left(\frac{2k}{K}-1\right)^3\right],\end{aligned}$$ with $m=m(t)$ and $m'=m(t+1)$. We show in Fig. \[fig:tau\] (b) the graph of $m'$ together with some iterates of the map.
The mean-field magnetization exhibits chaos that can be characterized by the Lyapunov exponent $\lambda$ [@ott02]. However, on spatially extended networks $m$ changes stochastically and cannot be characterized in the same way.
In order to compare microscopic and mean-field models within the same framework we use the Boltzmann’s entropy $\eta$ [@Boltzmann; @bagnoli2013] of the magnetization $m$. The interval $[-1,1]$ is partitioned in $L$ disjoint intervals $I_i$ of equal size and the probability $q_i$ of $I_i$ is the fraction of visits to $I_i$ after $T$ time steps with $T\gg 1$.
Once these probabilities are known, $\eta$ is defined by $$\label{eq:s}
{\eta}=- \dfrac{1}{\log L}\sum_{i=1}^Lq_i\log q_i,$$ so that $0\leq \eta\leq 1$, the lower bound corresponding to a fixed point, the upper one to the uniform distribution $q_i=1/L$.
For a periodic orbit of period $2^p$ and $L=2^b$, $\eta=p/b$. For low-dimensional dynamical systems, like the mean-field equation, the mid-value threshold $\eta=0.5$ effectively separates the contracting dynamics (cycles) from the chaotic ones. For spatially-extended systems, | 0 | non_member_969 |
there is always a stochastic noise that increases the value of the entropy in the “fixed-point” part of the parameter space. This base-level value is related to the size of the sample, and slowly vanishes for large samples.
In order to use finite-size samples, we set the onset of the phase in the stochastic systems corresponding to the chaotic phase in the deterministic ones to the mid-value of the range of $\eta$. Taking the limits $T\to\infty$, $L\to\infty$ leads to the Kolmogorov-Sinai entropy [@kolmogorov58; @kolmogorov59; @sinai59; @ott02].
Before presenting the different scenarios, let us illustrate the type of bifurcations that are present. In Figs. \[fig:mf-bif-c0p60\] (a) and (b) we show parts of the bifurcation diagram of the map of Eq. as a function of $J$ staring with different values of the initial magnetization $m_0$.
Referring to the values in the Figure, at $J=J_0$ there is a pitchfork bifurcation, i.e., a separation of basins, that reunite at $J=J_1$, which is another pitchfork bifurcation, in the reverse direction. Intermixed, there are period-doubling bifurcations. There are other pitchfork bifurcations for different intervals of $J$.
In Fig. \[fig:mf-bif-c0p60\] (c) we show the return map of the mean-field map for $J=-5.2$. We can see that there | 0 | non_member_969 |
are four basins of attraction. For small values of the initial magnetization $m_0$, the orbit is attracted to $m=-1$ and for large $m_0$ to $m=1$. The regions where this occurs are marked by the vertical dotted lines in the Figure. For other values of $m_0$, $m$ ends in one of two period-two orbits. This figure shows, in the lower part, the two basins of attraction that are symmetric in the sense that if $m_0$ belongs to one basin of attraction, $-m_0$ belongs to the other one.
In what follows we present the bifurcation diagrams of the mean-field map Eq. by varying the coupling constants $J$ and $W$. Unless otherwise noticed, we computed the Lyapunov exponent $\lambda$ by averaging over 10,000 time steps after a transient of another $10,000$ steps. The entropy $\eta$ was computed using 256 boxes and $25,000$ time steps.
![\[fig:mf-J\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the linear coupling $J$. In (b) the corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $J$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond | 0 | non_member_969 |
to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $J$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-m-lambda-eta-PI00-o2-k20-W15p00){width="0.9\columnwidth"}
![\[fig:mf-W\] (Color online.) (a) Bifurcation diagram of the mean-field map of the magnetization $m$, Eq. , as a function of the coupling constant $W$ with $J=-10$. In (b) The corresponding Lyapunov exponent $\lambda$ and in (c) the entropy $\eta$. The vertical dotted lines are drawn at the estimated values of $W$ for which $\lambda=0$. The horizontal dotted lines in (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$, orbits. For every value of $W$, two initial values of the magnetization were used, $m_0=-0.3$ and $m_0=0.3$.](mf-lambda-eta-PI00-o13--14-k20-J-10){width="0.9\columnwidth"}
In Fig. \[fig:mf-J\] (a) we show the bifurcation diagram of the magnetization $m$, Eq. , as a function of $J$ with $W$ and $K$ fixed. The diagram exhibits a period doubling cascade towards chaos with periodic windows and pitchfork bifurcations.
The bifurcation diagram of $m$ as a function of $W$ with $J$ and $K$ fixed is shown in Fig. \[fig:mf-W\] (a). In this case, there is an inverse period doubling cascade to chaos with pitchfork bifurcations.
The next row of figures, Figs. \[fig:mf-J\] (b) and Fig. \[fig:mf-W\] (b), show | 0 | non_member_969 |
the corresponding Lyapunov exponent $\lambda$, and the last one, Figs. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c), the entropy $\eta$. The dotted vertical lines are drawn at some of the values of $J$ or $W$ where $\lambda$ passes from a negative to a positive value or vice versa. These values coincide to jumps of $\eta$ from values smaller to $1/2$ to larger ones or vice versa and mark the appearance of chaos or periodic windows in the bifurcation diagrams.
Therefore, $\eta$ can be used as a measure of chaos. To stress this, we show in Figs. \[fig:mfpdWJ\] the mean-field phase diagrams of $\lambda$ (top) and $\eta$ (bottom). These diagrams are similar. The horizontal lines at $W=15$ correspond to Figs. \[fig:mf-J\] (b) and (c) and the vertical ones at $J=-10$ to Figs. \[fig:mf-W\] (b) and (c) respectively. We find a similar behavior of the mean-field map as $K$ varies with fixed $J$ and $W$. The three quantities $J$, $W$ and $K$ are related by scaling relations, as shown in the Appendix.
![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of | 0 | non_member_969 |
$(-J,W)$ where $\eta>1/2$. ](lambda-J-W-PI00-o15-k20 "fig:"){width="0.9\columnwidth"}\
![\[fig:mfpdWJ\] (Color online) (a) Mean-field phase diagram of the Lyapunov exponent $\lambda$ showing the values of $(-J,W)$ where $\lambda>0$. (b) The mean-field phase diagram of the entropy showing the values of $(-J,W)$ where $\eta>1/2$. ](eta-J-W-PI00-o15-m8-k20 "fig:"){width="0.9\columnwidth"}
The dotted horizontal lines in Fig. \[fig:mf-J\] (c) and Fig. \[fig:mf-W\] (c) correspond to period 2, $\eta=1/8$, and period 4, $\eta=1/4$ orbits. Looking at the bifurcation diagram in Fig. \[fig:mf-J\] (a), for small $-J$, the map has a fixed point and as $-J$ grows there is a first bifurcation to a period 2 orbit and another one to what looks like a period 4 orbit, but $\eta=1/8$ instead of $\eta=1/4$ for period 4 orbits. What appears like a bifurcation to period four orbits is actually a pitchfork bifurcation to two period-two orbits that depend on the initial magnetization $m_0$ as mentioned before. There are other pitchfork bifurcations for other values of $J$ with $W$ fixed and also for values of $W$ with $J$ fixed.
Small-World stochastic bifurcations {#sec:simulations}
===================================
In the Watts-Strogatz small-world model [@WattsStrogatz], starting with a network where every site has $K$ nearest neighbors, at any site $i$, with probability $p$, known as the long-range connection probability, | 0 | non_member_969 |
each one of its $K$ neighbors is replaced by a random one. Then the spin at each site is updated according to Eq. . As $p$ grows, coherent oscillations of a majority of spins begin to appear so that the magnetization $m$ shows noisy periodic or irregular oscillations. The noise is the manifestation of the stochasticity of the updating rule. Similar patterns can be seen in Ref. [@bagnoli2005], where the effect of the size of neighborhood is studied.
As shown in the following, by changing several parameters, we can obtain stochastic bifurcation diagrams similar to the mean-field ones. The following microscopic simulations were carried using lattices of $N=10,000$ sites, with a transient of $10,000$ time steps. The entropy $\eta$ was computed with 256 boxes and $25,600$ time steps.
In Fig. \[bifp\] we show the bifurcation map and the entropy $\eta$ as functions of $p$. There is always some disorder, even for small values of $p$ where $m\sim 0$, and as $p$ grows we find bifurcations and more disorder. Indeed, the entropy $\eta$ is a good measure of this behavior, small values of $\eta$ corresponding to noisy “periodic orbits” while larger ones to disorder (“chaos”).
In the mean-field case, we found | 0 | non_member_969 |
$\eta=1/2$ to be a good threshold to separate order from chaos. For the stochastic dynamics on small-world networks we choose as the threshold the approximate value of the entropy at the first bifurcation as shown in the figures. For values smaller than this threshold there are noisy periodic orbits. The bifurcation diagram of the figure is reminiscent of the mean-field one, Fig. \[fig:mf-J\].
Notice that pitchfork bifurcations (dependence on the initial magnetization) are present also in the microscopic simulations, as shown in Fig. \[bifp\]
In Fig. \[fig:sw-J\] we show the bifurcation diagrams and entropy of the magnetization $m$ for the small-world networks obtained for different values of $p$. As before, the entropy is a good indicator of disorder.
![\[bifp\] (Color online) Small-world stochastic bifurcation diagram of the magnetization $m$, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the long-range connection probability $p$ with $J=-10$. The “jump” of $m$ for $p\simeq 0.45$ corresponds to a pitchfork bifurcation (dependence on the initial magnetization).](sw-bif-eta-PI-1-o2-N10000-k20-J-10p00-W15p00-c0p50){width="\columnwidth"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$p=0.3$ $p=0.5$
![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling | 0 | non_member_969 |
constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p30-c0p50 "fig:"){width="0.45\columnwidth"} ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p50-c0p50 "fig:"){width="0.45\columnwidth"}
$p=0.8$ $p=1.0$
![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p0p80-c0p50 "fig:"){width="0.45\columnwidth"} ![\[fig:sw-J\] (Color online) Small-world bifurcation diagram of the magnetization $m$, dots (in magenta) and the corresponding entropy $\eta$, continuous line (in green), as functions of of the linear coupling constant $J$ and different values of the long-range connection probability $p$.](sw-bif-PI-1-o5-N10000-W15p00-k20-p1p00-c0p50 "fig:"){width="0.45\columnwidth"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Clearly, by setting the rewiring probability $p$ large enough, one can also recover the mean-field bifurcation diagrams as function of $J$, $K$ and $W$, with a good correspondence of the critical values of parameters.
Partial asynchronism (dilution)
-------------------------------
The dilution $d$ is the fraction of sites chosen at random that are not updated at every time step. We define the diluted | 0 | non_member_969 |
rule as $$\label{p}
s_{i}(t+1) = \begin{cases}
1 & \text{with probability $ (1-d) \tau(1|h_i)$,}\\
-1 & \text{with probability $(1-d) \left[1-\tau(1|h_i)\right]$,}\\
s_{i}(t) & \text{otherwise, i.e., with probability $d$,}
\end{cases}$$ so that for $d=0$ one has the standard parallel updating rule. One time step is defined when on the average every site of the lattice is updated once. For a system with $N$ sites, the smallest value of the dilution is $d=1/N$ and then $t_d=1/d$ updates are needed to complete one time step. If $d=1/2$, $t_d=2$, etc.
The mean-field equation corresponding to dilution is $$m(t+1) = (1-d)m(t) + d f(m(t));$$ where $f$ is the the map of Eq. . The mean-field phase diagram is reported in Fig. \[fig:mfpddJ\]. Notice that the border at $d=0$ corresponds to the horizontal line in Fig. \[fig:mfpdWJ\].
![\[fig:mfpddJ\] (Color online) Mean-field bifurcation diagram of the magnetization $m$ (dots in magenta, two initial conditions), and the entropy $\eta$ (continuous curve in green), as functions of the dilution probability $d$ with $J=-10$. ](m-f-dil-k20-J-10p00-W15p00){width="\columnwidth"}
The bifurcation diagrams and the entropy $\eta$ of the magnetization as functions of of the dilution $d$ are shown in Fig. \[bifd\] for different value of the long-range connection probability $p$. It is interesting to note the | 0 | non_member_969 |
“bubbling” transition: the oscillations are favored, for intermediate values of the rewiring $p$, by a non-complete parallelism.
As shown in the figure, for values of $p$ larger than $0.1$, the dilution is able to trigger bifurcations also in the spatial model. In contrast with the linear Ising model [@Cirillo], where even a small amount of asynchronism is able to destroy the “effective” antiferromagnetic coupling, here the behavior is smooth with respect to dilution. See Ref. [@SS-meta] for a study about metastable effects in the linear model.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$p=0.1$ $p=0.2$
![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p10 "fig:"){width="0.45\columnwidth"} ![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p20 "fig:"){width="0.45\columnwidth"}
$p=0.5$ $p=1.0$
![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of | 0 | non_member_969 |
the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p0p50 "fig:"){width="0.45\columnwidth"} ![\[bifd\] (Color online). Bifurcation diagrams of the magnetization $m$ on small-world networks, dots (in magenta), and the entropy $\eta$, continuous curve (in green), as functions of the dilution $d$ with $J=-10$, and different values of the long-range connection probability $p$.](PI5-o4-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Heterogeneity
-------------
In order to measure the effects of heterogeneity, we let a fraction $\xi$ of spins interact ferromagnetically ($J>0$) with their $K$ neighbors and a fraction $1-\xi$ interact antiferromagnetically ($J<0$). We show in Fig. \[fig:mix\] the bifurcation diagrams of the magnetization $m$ together with the entropy $\eta$ as functions of $\xi$ for different values of $p$. Again, the entropy is a good measure of disorder. One can see a “bubbling” effect very similar to what observed by changing the dilution. In other words, oscillations, which are a product of antiferromagnetism and parallelism, are actually favoured by a small percentage of asynchronism and/or of ferromagnetic nodes, for a partial long-range rewiring of links.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$p=0.05$ $p=0.2$
![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization | 0 | non_member_969 |
$m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p05 "fig:"){width="0.45\columnwidth"} ![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p20 "fig:"){width="0.45\columnwidth"}
$p=0.8$ $p=1.0$
![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p0p80 "fig:"){width="0.45\columnwidth"} ![\[fig:mix\] (Color online.) Small-world ferro-anti ferro bifurcation diagram (left axis, dots in magenta) and entropy $\eta$ (right axis, continuous curve in green) of the magnetization $m$ as a function of $\xi$ | 0 | non_member_969 |
for different values of the long range probability $p$ and $N=10,000$, $|J|=10$,. for $P=0.05$ there is no threshold for $\eta$, for $p=0.2$ it is $\eta=0.7$, for $p=0.8$ and $p=1.0$ it is $\eta=0.6$.](sw-bif-eta-m-xi-PI5-o8-N10000-k20-J-10p00-W15p00-p1p00 "fig:"){width="0.45\columnwidth"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Conclusions {#sec:conclusions}
===========
We investigated the phase transitions of a nonlinear, parallel version of the Ising model, characterized by a linear coupling $J<0$ and a nonlinear one $W>0$. The mean-field approximation shows chaotic oscillations, by changing the couplings $J$ and $W$ or the connectivity $K$. We showed in the Appendix that there is a scaling relation among these parameters.
The nonlinear Ising model was studied on small-world networks, where $p$ is the probability of long-range rewiring of links. Here, entropy of the magnetization becomes a measure of disorder which is adequate once a threshold between the presence and absence of noisy periodic orbits is established. The noisy periodic and disordered behavior of $m$ imply a certain degree of synchronization of the spins, induced by long-range couplings.
We have shown also that similar bifurcations may be induced in the randomly connected model by changing the parameters $J$, the dilution factor $d$ and the heterogeneity $\xi$, by mixing ferromagnetic and antiferromagnetic interactions.
In particular, we observed that | 0 | non_member_969 |
---
abstract: 'We study statistics of resonances in a one-dimensional disordered chain coupled to an outer world simulated by a perfect lead. We consider a limiting case for weak disorder and derive some results which are new in these studies. The main focus of the present study is to describe statistics of the scattered complex energies. We derive compact analytic statistical results for long chains. A comparison of these results has been found to be in good agreement with numerical simulations.'
address:
- '$^1$Instituto de Ciencias F'' isicas, Universidad Nacional Aut'' onoma de M'' exico, Cuernavaca, M'' exico'
- '$^2$Department of Physics, Technion-Israel, Institute of Technology, Haifa 32000, Israel'
author:
- 'Vinayak$^{1,2}$'
title: 'Statistics of resonances in a one-dimensional chain: a weak disorder limit'
---
Introduction
============
Resonant phenomena have received much attention in atomic and nuclear physics and more recently in chaotic and disordered systems [@disordered; @GG:00; @Casati:99; @KS:06; @KS:08; @JF:09; @chaos]. Complex energies, $\tilde{{E}}_{\alpha}=E_{\alpha}-\frac{i}{2}\,\Gamma_{\alpha}$, which correspond to poles of the scattering matrix on the unphysical sheet, characterize resonances [@LandauL]. Resonances correspond to the long-lived quasi-stationary states which eventually decay to continuum while distribution of resonance widths, $P(\Gamma)$, determines decay of the corresponding survival probability with time.
In recent | 0 | non_member_970 |
years, $P(\Gamma)$ has been a subject of investigations [@disordered; @GG:00] for a, simple but much studied, discrete tight-binding one dimensional random chain which is coupled to a perfect lead at one side. A numerical study [@GG:00] shows that in a broad range of $\Gamma$, $P(\Gamma)\sim \Gamma^{-\gamma}$, where the exponent $\gamma$ is very close to $1$. Intuitively the $1/\Gamma$ behaviour can be deduced by assuming a uniform distribution for the localization centers of exponentially localized states [@Casati:99]. However, from analytic point of view one usually considers an infinitely long chain in which case the average density of resonances (DOR) has a well defined limit. For a finite size system, the difference between the DOR and $P(\Gamma)$ is the normalization by the system size [@KS:06; @KS:08]. Recently, Kunz and Shapiro have derived analytic expression of the DOR for a semi-infinite disordered chain [@KS:08]. They have obtained an exact integral representation of the DOR which is valid for arbitrary lead-chain coupling strength. This has been further simplified for small lead-chain coupling strength where a universal scaling formula is found. In this limit they have proved the $1/\Gamma$-behavior of the DOR [@KS:06; @KS:08]. Besides, for the continuous limit of this model an integral representation | 0 | non_member_970 |
of DOR has been obtained [@JF:09].
Kunz and Shapiro’s work has established a universal $1/\Gamma$ law for arbitrary strength of disorder in a semi-infinite chain. Numerically one can verify $1/\Gamma$ law of the DOR, similar to what has been done by Terraneo and Guarnery [@GG:00] in finite samples for $P(\Gamma)$. Such verifications require the localization lengths to be much smaller than the size of the sample. In case of weak disorder an analytic result for the localization lengths is particularly useful. It comes from a second order perturbation theory. It states that the localization length is maximum near the middle of the energy band and is proportional to $W^{-2}$ where $W$ is the width of the disorder [@thou; @krammac; @KWegner:1981]. On the other hand, this result also leads to an interesting limiting situation where the localization lengths are much longer than the sample size. This is what we refer to as a weak disorder limit in this paper. This limit has scarcely been studied hitherto although it is relevant in the study of localization through resonances. Besides, there has been a believe for some sort of universality in the weak disorder limit. In this paper we address to this limit | 0 | non_member_970 |
and derive analytic results which describe the statistics of resonances. Our work probes a fresh area and studies a weak disorder limit which has never been addressed before.
For open systems, instead of studying the scattering matrix in a complex plane we follow an alternative approach where one solves the Schrödinger equation by describing a particle ejected from the system or equivalently with a boundary condition of outgoing waves (Siegert boundary condition [@Siegert]). In this approach one naturally turns up to a problem of solving a non-Hermitian effective Hamiltonian which admits complex eigenvalues $\tilde{{E}}_{\alpha}$ [@GG:00; @KS:06; @KS:08; @HKF:09]. For details of such non-Hermitian effective Hamiltonians, we refer to a recent study [@JF:10] and references therein.
We derive the statistics which describe scattered complex energies of disordered chain around those regular ones which correspond to an open chain without any disorder (clean chain). For instance, we derive average of square of the absolute values of the shifts in complex energies from the regular ones over all realizations of the set of random site energies. Similarly we obtain results for the statistics of real and imaginary parts of those shifts. These results lead to compact expressions for long chains. To show the | 0 | non_member_970 |
generality of our approach we also derive these results for the so-called parametric resonances which have been particularly useful in numerical studies [@GG:00]. Finally, we give numerical verifications of our analytic results.
The paper is organized as follows. Although the system and its effective Hamiltonian have been nicely explained earlier in [@GG:00; @KS:06; @KS:08], for the sake of completeness of this paper we will describe these briefly in section II. In the same section we will also describe the exact and the parametric resonances. In Sec. III we will derive result for resonances in an open-clean chain of finite length, in terms of a polynomial equation. For long chains, we will solve this polynomial equation in the leading order of the inverse of the length. In Sec. IV we will use the perturbation theory to obtain the first and the second order corrections in the complex energies for a weak disorder. In Sec. V we will calculate statistics of the scattered complex energies. In the same section we will simplify our results for long chains and obtain compact expressions. In Sec. VI we will briefly discuss about the numerical methods to calculate complex energies of non-Hermitian effective Hamiltonians and numerically | 0 | non_member_970 |
verify our analytical results. This will be followed by the conclusion in Sec. VII.
![A one-dimensional disordered chain with $N$ sites, represented in the figure by black dots, is coupled to a lead. Open circles represent sites of the lead. The outgoing plane wave is shown by the arrow where $0<\Re\{\tilde{k}\}<\pi$ and $\Im \{\tilde{k}\}<0$, so that it propagates left in the lead and its amplitude grows in the lead.[]{data-label="System"}](system.eps){width="50.00000%"}
Model and Its Effective Hamiltonian
===================================
A discrete tight-binding one dimensional chain of length $N$ (shown by positive integers, $n=1,\,2,...,\,N$, used for indexing the sites of the chain in Fig. \[System\]) is connected to an outer world (represented by a perfect lead which sites are shown by a zero and negative integers, $n=0,\,-1,\,-2,...$). Each site of the chain has the site energy $\epsilon_{n}$ where $\epsilon_{n}$ are statistically independent random variables chosen from some symmetric distribution. Each nearest neighbor site of the chain as well as of the lead is coupled by a hopping amplitude $t$. The hopping amplitude for the pair $n=0$ and $n=1$ is $t'$ which takes values from $t'=0$ (closed chain) to $t'=t$ (fully coupled chain). With this hopping, a particle, which is initially located somewhere in the chain, | 0 | non_member_970 |
eventually escapes to the outer world.
Now we write down the Schrödinger equation for the entire system, $$\begin{aligned}
\label{Sch1}
-t\psi_{n+1}-t\psi_{n-1}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $n<0$},
\\
\label{Sch2}
-t\psi_{-1}-t'\psi_{1}&=&\tilde{\mathcal{E}}\psi_{0}, ~~~~~~~~\text{for $n=0$},
\\
\label{Sch3}
-t'\psi_{0}-t\psi_{2}+\epsilon_{1}\psi_{1}&=&\tilde{\mathcal{E}}\psi_{1},~~~~~~~~\text{for $n=1$},
\\
\label{Sch4}
-t\psi_{n-1}-t\psi_{n+1}+\epsilon_{n}\psi_{n}&=&\tilde{\mathcal{E}}\psi_{n},~~~~~~~~\text{for $2\leq n\leq N$.}\end{aligned}$$ In order to avoid cluttering of notations we always represent quantities corresponding to disordered system by [*script letters*]{} while quantities for the clean system are represented in usual math notations. Tilde is used to discriminate the open system case from the closed one. Equation (\[Sch1\]) is for the lead where $\epsilon_{n}=0$. Equations (\[Sch2\], \[Sch3\]) describe the lead-chain coupling and Eq. (\[Sch4\]) is for the chain. As in, [@KS:08] we solve Eqs. (\[Sch1\]-\[Sch4\]) with a boundary condition of an outgoing plane wave in the lead, i.e., $\psi_{n_{\leq 0}}\propto \exp(-i\tilde{k}n)$ where $0<\Re\{\tilde{k}\}<\pi$ and $\Im \{\tilde{k}\}<0$. The condition on $\Re\{\tilde{k}\}$ ensures that the outgoing wave propagates to left in the lead. The condition on $\Im \{\tilde{k}\}$ is considered so that the amplitude of the resonance wave function grows in the lead. It comes from Eq. (\[Sch1\]) that the complex energy $\tilde{\mathcal{E}}$ is related to the complex wave vector $\tilde{k}$ via the dispersion relation $\tilde{\mathcal{E}}=-2 t \cos(\tilde{k})$. Now we eliminate all $\psi_{n}$ for $n<1$ from Eqs. (\[Sch1\]-\[Sch4\]) and | 0 | non_member_970 |
obtain $$\label{reduced}
-t\psi_{n+1}-t\psi_{n-1}+\tilde{\epsilon}_{n}\psi_{n}=\tilde{\mathcal{E}}\psi_{n},$$ where $$\label{energy}
\tilde{\epsilon}_{n}=\epsilon_{n}-t\eta\, \exp(i\tilde{k})\delta_{n1},$$ for $n=1,\,2,...,\,N$. The parameter $\eta=(t'/t)^2$ measures the coupling strength to the outside world.
An effective Hamiltonian defined by the Eq. (\[reduced\]) is non-Hermitian. For instance, if $\mathcal{H}$ is the $N\times N$ tridiagonal Hermitian matrix which represents the Hamiltonian of the closed-disordered chain then one may write the effective Hamiltonian, $\tilde{\mathcal{H}}$, as $$\label{Hamil}
\tilde{\mathcal{H}}=\mathcal{H}-t\,\eta\,\lambda(\tilde{k})\, P.$$ Here $P=|1\rangle\langle1|$ is the projection for site $n=1$ and $\lambda=\exp(i\tilde{k})$. The above non-Hermitian effective Hamiltonian has been first obtained by Terraneo and Guarnery [@GG:00]. The underlying result here is that the same relation (\[Hamil\]) is valid for any Hermitian $\mathcal{H}$ representing a (closed) quantum system [@JF:10] which has $N$-dimensional state space. Resonances are characterized by the complex eigenvalues, $\tilde{\mathcal{E}}_{\alpha}$, of $\tilde{\mathcal{H}}$.
Note here dependency of $\tilde{\mathcal{H}}$ on the complex wave vector $\tilde{k}$ which is related to the complex energies via the dispersion relation mentioned above - this is not a standard eigenvalue problem. To standardize this problem “parametric resonances” are often used as an alternative. In this approach the dependence of $\lambda$ on $\tilde{k}$ is typically neglected, reducing thereby the problem of finding the eigenvalues of the effective Hamiltonian at chosen value of $\tilde{k}$. As expected, parametric resonances yield | 0 | non_member_970 |
approximate statistical results which are close to those for the exact resonances in strongly localized regime [@GG:00]. Parametric resonances depend on a chosen parameter, for instance let $\tilde{k}=k_{0}$ and we fix it in the middle of the energy band, $k_{0}=\pi/2$. Writing explicitly $$\lambda(\tilde{k})=
\begin{cases}
\exp(i \tilde{k}), & \text {for exact resonances,}
\\
i, & \text {for parametric resonances.}
\end{cases}$$
From now on we set the energy scale by taking $t=1$, denoting the complex variable $\tilde{\mathcal{E}}/t$ by $\tilde{\mathcal{Z}}$. We denote the Hamiltonian matrix representing the closed-clean chain by $H$. It differs from $\mathcal{H}$ only at the diagonal as, for the clean chain case, all the site energies are zero. Calculation of the eigenvalues of $H$ is a standard exercise where one derives $z_{\alpha}=-2 \cos[\alpha\pi/(N+1)]$ for $\alpha=1,...,N$.
Before going into a detail treatment to the problem, we should first sketch the outline of our approach. We are interested in a weak disorder regime. Since our approach rely on perturbation theory, we need complex energies of open-clean chain, i.e., the $\tilde{z}_{\alpha}$s. So we will begin with calculating the resonances for open-clean chain of finite length. Then we will do the perturbation series expansion up to the second order of strength of the disorder. | 0 | non_member_970 |
This will be followed by the derivation of the statistical results. Finally, we will consider the large-$N$ limit of these results.
Open-clean Chain
================
We begin with defining the resolvent $\tilde{\mathcal{G}}(z)=(z-\tilde{\mathcal{H}})^{-1}$. Using Eq. (\[Hamil\]) we may also write $$\label{resolvent}
\tilde{\mathcal{G}}(z)=(z-\mathcal{H}+\eta\,\lambda\,P)^{-1}.$$ For the open-clean chain we define the resolvent $$\begin{aligned}
\tilde{G}(z)&=&(z-H+\eta\,\lambda\,P)^{-1}
\nonumber
\\
&=&
(1+\eta\,\lambda\,GP)^{-1}\,G,\end{aligned}$$ where we have introduced $G(z)=(z-H)^{-1}$ as the resolvent for the “unperturbed" closed-clean chain. Resonances correspond to the singularities of the matrix $\tilde{G}_{mn}(z)$, or to the roots of the secular equation $$\label{charcpol}
F(z)=0
=1+\eta\lambda G_{11}(z),$$ where $G_{11}$ is the $\{1,\,1\}$ element of the matrix $G$ in site representation. ($G_{nm}(z)=\langle n|(z-H)^{-1}|m\rangle$.)
![Comparison of the result (\[resz\]) (pluses) with the numerical solution of the polynomial equation (\[fincharpol\]) (circles, squares and diamonds) for the exact resonances where $\eta=0.5,\,0.81$ and $0.99$. We have considered $N=100$.[]{data-label="ansatz-exact"}](ansatz-exact.eps){width="75.00000%"}
To obtain $G_{11}$ for finite $N$, we use the ordinary difference equation (ODE), $$\label{ODE}
\psi_{n+1}+\psi_{n-1}+z\,\psi_{n}=0,$$ with the boundary conditions $\psi_{0}=\psi_{N+1}=0$. This equation is obtained from Eq. (\[reduced\]) by setting all $\epsilon_{n}=0$. Next we consider $u_{n}(z)$ and $v_{n}(z)$ to be the two linearly independent functions which satisfy the ODE $$\begin{aligned}
\label{un}
u_{n+1}+u_{n-1}+z\,u_{n}&=&0,
\\
\label{vn}
v_{n+1}+v_{n-1}+z\,v_{n}&=&0,\end{aligned}$$ where $u_{0}=v_{N+1}=0$. Since norm of $u_{n},\,v_{n}$ is arbitrary, we fix $u_{1}=v_{N}=1$. Further | 0 | non_member_970 |
we claim that the resolvent is given by $$\label{G0uv}
G_{nm}=-\dfrac{u_{n}\,v_{m}\Theta(m-n)+u_{m}\,v_{n}\Theta(n-m)}{W_{n}}.$$ Here $\Theta(n)$ is the unit-step function and $W_{n}=u_{n}v_{n-1}-u_{n-1}v_{n}$ is the Wronskian. Using Eqs. (\[un\], \[vn\]) it is straight forward to see that the Wronskian is independent of $n$. One can also check that $$G_{n+1m}+G_{n-1m}+z\,G_{nm}=\delta_{nm}.$$ We now set $u_{n}=v_{N+1-n}$ to match the initial value problem (\[un\], \[vn\]) to the boundary value problem (\[ODE\]). We find $$\label{Gun}
G_{11}=-\dfrac{u_{N}}{u_{N+1}}.$$ The ODE (\[un\]) is satisfied by the Chebyshev polynomial of the second kind, $U_{m}(-z/2)$, defined as $$U_{m}(x)=\dfrac{\sin[(m+1)\cos^{-1}(x)]}{\sin[\cos^{-1}(x)]},$$ for $U_{0}(x)=1$ and $U_{1}(x)=2x$. Since we have fixed $u_{1}=1$, therefore $u_{n}=U_{n-1}$, thus we can write Eq. (\[Gun\]) as $$\label{GF}
G_{11}=-\dfrac{U_{N-1}(-z/2)}{U_{N}(-z/2)}=-\dfrac{\sin[N\,k]}{\sin[(N+1)k]}.$$ Here the last equality follows from the energy dispersion relation. Using Eq. (\[GF\]) in Eq. (\[charcpol\]), we end up with an algebraic equation $$\label{fincharpol}
F(z)
=0=
1-\eta\lambda\dfrac{U_{N-1}(-z/2)}{U_{N}(-z/2)}.$$
![Repeated on the same pattern of Fig. \[ansatz-exact\] but for parametric resonances.[]{data-label="ansatz-parametric"}](ansatz-parametric.eps){width="75.00000%"}
Zeros of $F(z)$ are the roots of a polynomial of order $N$. For exact resonances Eq. (\[fincharpol\]) can be easily transformed into $$\label{Az}
[\mathsf{a}(z)]^{(2N+1)}=
\dfrac{\mathsf{a}(z)^{-1}-\eta\,\mathsf{a}(z)}{1-\eta},$$ where $$\label{Aexp}
\mathsf{a}(z)=-\exp[ik(z)].$$ In order to solve Eq. (\[Az\]), we propose an ansatz assuming that opening of the system at one end causes $\mathcal{O}(N^{-1})$ complex corrections to the $k_{\alpha}$’s. Let $$\label{ansatz}
\tilde{k}_{\alpha}=k_{\alpha}+\dfrac{\Phi_{\alpha}}{N},$$ where | 0 | non_member_970 |
$\Phi_{\alpha}$ is a complex quantity and $k_{\alpha}=\alpha\pi/(N+1)$. Inserting this ansatz into Eqs. (\[Az\], \[Aexp\]) we obtain $$\label{res}
\tilde{k}_{\alpha}= k_{\alpha}-
\dfrac{i}{2N}\,
\text{ln}\left[\Omega(k_{\alpha};\eta)\right]+\mathcal{O}\left(\dfrac{1}{N^2}\right),$$ where $$\label{Omega}
\Omega(k;\eta)=
\dfrac{1-\eta \,e^{2i\,k_{\alpha}}}{1-\eta}.$$ Now, up to $\mathcal{O}(N^{-1})$, $\tilde{z}_{\alpha}$ may be written as $$\label{resz}
\tilde{z}_{\alpha}=-2\cos(k_{\alpha})-
\dfrac{i\sin(k_{\alpha})}{N}\,
\text{ln}(\Omega).$$ The same result can be obtained for the parametric resonances, after repeating the similar steps, but with different $\Omega$: $$\label{Omgpara}
\Omega(k_{\alpha};\eta)=
\dfrac{1-i\eta \,e^{i\,k_{\alpha}}}
{1-i\eta \,e^{-i\,k_{\alpha}}}.$$
![Scatter plot for exact resonances where $N=100$, $\eta=0.81$. Dense points in the graph represent exact resonances in the disordered chain for $2500$ realizations where $W=0.015$. These are scattered around dots which represent exact resonances in the clean chain.[]{data-label="scatter_exact"}](scatter-exact.eps){width="75.00000%"}
One should bear in mind that there is no resonance for $\eta=1$, as the system is fully coupled to the lead. However, for parametric resonances, one artificially gets resonances even when $\eta=1$. Note that the result (\[resz\]) is symmetric about the imaginary axis for both cases. In Fig. \[ansatz-exact\] and Fig. \[ansatz-parametric\] we compare the numerical solutions of the polynomial equation (\[fincharpol\]) with our results (\[resz\], \[Omega\], \[Omgpara\]), for $N=100$, $\eta=0.50,\,0.81$ and $0.99$ and $N=100$, respectively for exact and parametric resonances. Eq. (\[fincharpol\]) has been solved by using the Newton’s method with the initial guess $\tilde{k}_{\alpha}=k_{\alpha}$. These figures | 0 | non_member_970 |
show that our result (\[resz\]) is close to the numerical solution. The agreement gets better as $\eta\rightarrow1$ (not shown here separately). However, the ansatz (\[ansatz\]) is not valid near the band edges. Moreover, the agreement fails for parametric resonances near the middle of the band as $\eta\rightarrow 1$; see Fig. \[ansatz-parametric\] for $\eta=0.99$.
The Weak disorder limit
=======================
![Scatter plot for parametric resonances where $N=100$, $\eta=0.81$ and $W=0.015$, for $5000$ realizations. As in Fig.\[scatter\_exact\], here also dots represent the clean chain and points represent the disordered chain.[]{data-label="scatter_parametric"}](scatter-parametric.eps){width="75.00000%"}
In the next stage of the problem we switch on a very weak disorder in the chain. From a second order perturbation theory we know that for a disordered infinitely long chain the localization length, $\xi(E)$, is maximum at the middle of the band. For small $W$ it is given by [@krammac; @thou] $$\begin{aligned}
\label{xi}
\xi(E)=\dfrac{24(4t^2-E^2)}{W^2},\end{aligned}$$ implying thereby, $\xi(0)=\dfrac{96 t^2}{W^2}$. However, the exact result shows a small deviation at the band center due to the breakdown of the second-order perturbation theory [@KWegner:1981]. We consider a limiting situation when $\xi(0)/N >> 1$. For instance, in Fig. \[scatter\_exact\] and in Fig. \[scatter\_parametric\], we show the scatter plot ($\Re\{\tilde{\mathcal{Z}}_{\alpha}\}$ vs $\Im\{\tilde{\mathcal{Z}}_{\alpha}\}$) for exact and parametric resonances | 0 | non_member_970 |
respectively. In both cases we have considered $N=100$, $\eta=0.81$ and $W=0.015$ so that $\xi(0)>>N$. As seen in these figures, complex energies of the disordered chain are scattered around the $\tilde{z}_{\alpha}$s.
We now calculate the corrections to $\tilde{z}_{\alpha}$ for such weak disorder case. It is suggestive here to deal with the self-energy. Let $\mathcal{S}_{1}(\epsilon_{2},...,\epsilon_{N};z)$ be the self-energy for the first site, defined via $$\begin{aligned}
\label{GS}
\mathcal{G}_{11}(z)=\dfrac{1}{z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)}.\end{aligned}$$ Here $\{\epsilon\}$ denotes the set $\epsilon_{2},...,\epsilon_{N}$ and $\mathcal{G}_{11}$ is the $\{1,\,1\}$ element of the resolvent $\mathcal{G}(z)=(z-\mathcal{H})^{-1}$, defined for the Hermitian matrix $\mathcal{H}$. For the later convenience we write $$\begin{aligned}
\label{Hborn}
\mathcal{H}=H+\mathcal{W}, \end{aligned}$$ where $\mathcal{W}=\sum_{\ell=1}^{N}\epsilon_{\ell}P_{\ell}$ and $P_{\ell}=|\,\ell\,\rangle\langle\,\ell\,|$ is the projection for the $\ell$’th site.
In the rest of the paper we will work out results only for the exact resonances. For the parametric resonance theses results can be carried out following similar steps, so we skip all the intermediate steps merely by stating the result at the end.
As before in Eq. (\[charcpol\]), for disordered chain, resonances correspond to the roots of the secular equation $$\label{secular}
\mathcal{F}(z)=0=z-\epsilon_{1}-\mathcal{S}_{1}(\{\epsilon\};z)+\lambda\eta.$$ Preserving $\tilde{z}_{\alpha}$ as the roots of Eq. (\[charcpol\]), we define $\tilde{\mathcal{Z}}_{\alpha}$ as the roots of Eq. (\[secular\]). Now we expand the roots $\tilde{\mathcal{Z}}_{\alpha}=\tilde{z}_{\alpha}+(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})+(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$, assuming that $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$ are | 0 | non_member_970 |
linear while $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})$ are quadratic in the $\epsilon_{j}$ , for $j=1,...,N$. Then for $\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})$, up to $\mathcal{O}(\{\epsilon\}^{2})$, we get $$\begin{aligned}
\label{Selfenergy}
\mathcal{S}_{1}(\{\epsilon\};\tilde{\mathcal{Z}}_{\alpha})
&=&
S_{1}(\{0\};\tilde{z}_{\alpha})+
\sum_{n=2}^{N}\epsilon_{n}
\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
(\delta_{1} \tilde{\mathcal{Z}}_{\alpha}+\delta_{2} \tilde{\mathcal{Z}}_{\alpha})\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}\sum_{n,m=2}^{N}\epsilon_{n}\epsilon_{m}
\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial\epsilon_{n}\partial\epsilon_{m}}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber
\\
&+&
\dfrac{1}{2}(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}\left(\dfrac{\partial^{2} \mathcal{S}_{1}(\{\epsilon\};z)}{\partial^{2} z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}.\end{aligned}$$ We will use this expansion in Eq. (\[secular\]). Before that we evaluate $$\begin{aligned}
\label{SGdz}
1-\left(\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial z}\right)_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
=
\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}},\end{aligned}$$ and, $$\begin{aligned}
\label{dGdz}
\dfrac{\partial \mathcal{S}_{1}(\{\epsilon\};z)}{\partial \epsilon_{n}}\bigg |_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=&
\dfrac{1}{\left(G_{11}\right)^2}\,
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}},
\nonumber
\\\end{aligned}$$ for $n\geq2$. These equalities come from Eq. (\[GS\]). Finally, we calculate derivatives of $\mathcal{G}_{11}$, at $\{\epsilon\}=0$ and $z=\tilde{z}_{\alpha}$ with respect to $\{\epsilon\}$ by using Eq. (\[Hborn\]) for the [*Born-series*]{} expansion of $\mathcal{G}(z)$. We find $$\begin{aligned}
\label{dGdeps}
\dfrac{\partial \mathcal{G}_{11}}{\partial \epsilon_{n}}\bigg|_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
&=&
G_{1n}G_{n1}\bigg|_{z=\tilde{z}_{\alpha}}.\end{aligned}$$ Grouping all these, for the first order corrections, we obtain $$\begin{aligned}
\label{FPT1}
&&(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
=
\dfrac{\epsilon_{1}+\sum_{n=2}^{N}\epsilon_{n}
\dfrac{G_{1n}G_{n1}}
{\left[G_{11}\right]^{2}}\Bigg|_{z=\tilde{z}_{\alpha}}}
{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha}]}
{2\sin(\tilde{k}_{\alpha})}
}. \end{aligned}$$ Similarly for the second order corrections we get $$\begin{aligned}
\label{dz2}
(\delta_{2} \tilde{\mathcal{Z}}_{\alpha})
&=&
\Bigg[\sum_{n,m=2}^{N}\epsilon_{n} \epsilon_{m}
\left\lbrace
\dfrac{G_{1n}G_{nm}G_{m1}}{[G_{11}]^{2}}
-
\dfrac{[G_{1n}G_{1m}]^{2}}
{[G_{11}]^{3}}
\right\rbrace
\nonumber
\\
&-&
\dfrac{(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})^{2}}{2}
\left\lbrace\left(\dfrac{\partial^{2}}{\partial^{2} z}\dfrac{1}{G_{11}}\right)+\eta\left(\dfrac{d^{2}\exp(ik(z)}{d^{2}z}\right)\right\rbrace
\Bigg]_{\{\epsilon\}=0,z=\tilde{z}_{\alpha}}
\nonumber\\
&\times &
\Bigg[{\dfrac{\partial}{\partial z} \dfrac{1}{G_{11}(z)}\bigg|_{z=\tilde{z}_{\alpha}}
+
\dfrac{i\eta\exp[i\tilde{k}_{\alpha})}
{2\sin(\tilde{k}_{\alpha})}
}\Bigg]^{-1}.
\nonumber\\\end{aligned}$$
Note that $(\delta_{1} \tilde{\mathcal{Z}}_{\alpha})$ and $(\delta_{2} \tilde{\mathcal{Z}}_{\alpha}) $ have been | 0 | non_member_970 |
obtained in terms of the resolvent of the closed-clean chain which we already know in terms of Chebyshev polynomials; see Eq. (\[G0uv\]) and the relation between $u_{n}$ and $v_{n}$ with Chebyshev polynomials.
Statistics of the Scattered Complex Energies
============================================
We are interested in the statistics of the scattered complex energies. For instance, using the first order result (\[FPT1\]) of the perturbation theory, we calculate average of square of absolute shift in complex energies defined as, $\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle\equiv\langle|(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})|^2\rangle$. The angular brackets are used here to represent the averaging over many realizations of set of all random site energies $\{\epsilon_{n}\}$. This quantity gives a statistical account for the scattered complex energies. We also calculate $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$, viz, average of square of the real and the imaginary part of the shift $(\tilde{\mathcal{Z}}_{\alpha}-\tilde{z}_{\alpha})$, respectively. To obtain the latter quantities we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, since $$\begin{aligned}
(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}=\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}+2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4},
\nonumber
\\
\\
(\Im\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}=-\dfrac{(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}+[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}-2(|\Delta \tilde{\mathcal{Z}}_{\alpha})\,|^{2})}{4}.
\nonumber
\\\end{aligned}$$ Here we have used $\{^{*}\}$ to represent the complex conjugate (c.c.).
For all these three statistics we simplify $(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})$, given in Eq. (\[FPT1\]), in terms of Chebyshev polynomials as $$\begin{aligned}
\label{FPT2}
(\delta_{1}\tilde{\mathcal{Z}}_{\alpha})
&=&
\Bigg[\dfrac{\tilde{z}_{\alpha}^2-4}{2}
\sum_{n=1}^{N}\epsilon_{n}
\left(U_{N-n}(\tilde{z}_{\alpha}/2)\right)^{2}
| 0 | non_member_970 |
\Bigg]
\nonumber\\
&\times&
\Bigg[U_{N-1}(\tilde{z}_{\alpha}/2)T_{N+1}(\tilde{z}_{\alpha}/2)-N
\nonumber\\
&-&
i\eta\exp[i\tilde{k}_{\alpha}]\sin(\tilde{k}_{\alpha})[U_{N-1}(\tilde{z}_{\alpha}/2)]^2
\Bigg]^{-1},\end{aligned}$$ where $T_{m}(z)=\cos[m\cos^{-1}(z)]$ is the Chebyshev polynomial of the first kind. Further simplifications occur when these polynomials are expressed in their trigonometric forms. For instance, let’s calculate $|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2$, with $ \tilde{z}_{\alpha}/2=\cos( \tilde{\theta}_{\alpha})$ where $\tilde{\theta}_{\alpha}=\pi- \tilde{k}_{\alpha}$. We obtain $$\begin{aligned}
\label{abdz2}
\dfrac{|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2}{4}
=
\dfrac{\sum_{n,m=1}^{N}\epsilon_{n'}\epsilon_{m'}\sin^{2}(n' \tilde{\theta}_{\alpha})\sin^{2}(m'\tilde{\theta}^{*}_{\alpha})}
{|D(\tilde{z_{\alpha}})|^{2}}.\end{aligned}$$ Here $n'$ and $m'$ are respectively $N+1-n$ and $N+1-m$, and $D(\tilde{z_{\alpha}})$ is simply the quantity in the second bracket of Eq. (\[FPT2\]). Averaging releases one of the summation as the $\epsilon_{j}'$s are statistically independent-identically-distributed (i.i.d.) random variables. We simply have $$\begin{aligned}
\label{abdz3}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
=
\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{2}(n\tilde{\theta}_{\alpha})\sin^{2}(n\tilde{\theta}^{*}_{\alpha})}
{|D|^{2}},\end{aligned}$$ where $\sigma^{2}$ is variance of the $\epsilon_{j}'$s.
Summation in the above equality can be performed by using trigonometric identities. For instance, we first write $$\begin{aligned}
\label{abdz4}
4\sin^{2}(n\theta)\sin^{2}(n\theta^{*})
&=&
1-\cos(2n\theta)-\cos(2n\theta^{*})
\nonumber\\
&+&
\dfrac{\cos(4\,n\,\Re\{\theta\})+\cos(4\,i\,n\,\Im\{\theta\})}{2},\end{aligned}$$ and we use the summation formula $$\begin{aligned}
\label{abdz5}
\sum_{n=1}^{N}\cos(n\theta)
&=&
\dfrac{1}{2}
\left[
\dfrac{\sin[(N+1/2)\theta]}{\sin(\theta/2)}-1
\right].\end{aligned}$$ It turns out after some trigonometry that one can write the summation in a closed form. We find $$\begin{aligned}
\sum_{n=1}^{N}4\sin^{2}(n\theta)\sin^{2}(n\theta^{*})
&=&N+\dfrac{1}{2}-\dfrac{U_{2N}+U_{2N}^{*}}{2}+
\dfrac{T^{*}_{2N+2}T_{2N}-T_{2N+2}T^{*}_{2N}}{2[T^{*}_{2}-T_{2}]}.
\nonumber
\\\end{aligned}$$ Here the argument of the polynomials is $\tilde{z}_{\alpha}/2$ and for their complex conjugate it is $\tilde{z}^{*}_{\alpha}/2$. Finally, we write down finite-$N$ result for average of the absolute square of the | 0 | non_member_970 |
shift, $$\begin{aligned}
\label{abdz6}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
=
\sigma^{2}\dfrac{N+\dfrac{1}{2}-\dfrac{U_{2N}+U_{2N}^{*}}{2}
+
\dfrac{T^{*}_{2N+2}T_{2N}-T_{2N+2}T^{*}_{2N}}{2[T^{*}_{2}-T_{2}]}}
{|D|^{2}}.\end{aligned}$$
We now turn our attention to large-$N$ behavior of the result (\[abdz6\]). For this purpose we use the ansatz (\[ansatz\]) and result the (\[res\]) for $\tilde{k}_{\alpha}$. Large-N behavior for the Chebyshev polynomials, with argument $\tilde{z}_{\alpha}$, may be calculated as $$\begin{aligned}
T_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[2iN\tilde{\theta}_{\alpha}]+\exp[-2iN\tilde{\theta}_{\alpha}]}{2}
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-2i{k}_{\alpha})+\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(2i{k}_{\alpha})}{2},$$ $$\begin{aligned}
T_{2N+2}(\tilde{z}_{\alpha}/2)
=
\dfrac{\Omega(k_{\alpha};\eta)+\left[\Omega(k_{\alpha};\eta)\right]^{-1}}{2}+\mathcal{O}(N^{-1}),\end{aligned}$$ $$\begin{aligned}
U_{2N}(\tilde{z}_{\alpha}/2)&=&\dfrac{\exp[i(2N+1)\tilde{\theta}_{\alpha}]-\exp[-i(2N+1)\tilde{\theta}_{\alpha}]}
{\exp(i\tilde{\theta}_{\alpha})-\exp(-i\tilde{\theta}_{\alpha})}
\nonumber
\\
&\approx&
\dfrac{\Omega({k}_{\alpha};\eta)\exp(-i{k}_{\alpha})-\left[\Omega({k}_{\alpha};\eta)\right]^{-1}\exp(i{k}_{\alpha})}{\exp(-i{k}_{\alpha})-\exp(i{k}_{\alpha})}.\end{aligned}$$ Finally, $$\begin{aligned}
T_{2}(\tilde{z}^{*}_{\alpha}/2)- T_{2}(\tilde{z}_{\alpha}/2)
&=& -\dfrac{2i\,\Im\{\Phi_{\alpha}\}}{N}\,
{z}_{\alpha}+\mathcal{O}(N^{-2})
\nonumber
\\
&\approx&
\dfrac{4i}{N}\, \cos(k_{\alpha})\,\Im\{\Phi_{\alpha}\},\end{aligned}$$ where we have used the ansatz (\[ansatz\]) in the second order polynomial $T_{2}(z)=2z^{2}-1$ and $\Im\{\Phi_{\alpha}\}=-\sin({k}_{\alpha})\,\text{ln}(|\Omega|)$, as obtained from Eqs. (\[ansatz\], \[res\]).
We can now plug in these results in Eq. (\[abdz6\]). These asymptotic results gives the numerator as ($N+a1+a2/(a3/N)$) where $a1,\,a2/a3$ are $\mathcal{O}(N^{0})$. Similarly we obtain denominator as ($\,N^{2}+b1\,N+b2$) where $b1$ and $b2$ are $\mathcal{O}(N^{0})$; see Appendix A for details. Thus in the leading order, we obtain $$\begin{aligned}
\label{abdz7}
\langle|(\Delta \tilde{\mathcal{Z}}_{\alpha})|^2\rangle
&=&
\dfrac{\sigma^{2}}{N}\,
\left(
1+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
\right).\end{aligned}$$
![Asymptotic results for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$, shown by filled circles, squares and diamonds, vs energy index $\alpha$. We have compared here the finite-$N$ results, shown by open circles, for the exact resonances where $N=100$ and $\eta=0.81$. In the set we | 0 | non_member_970 |
show these results for 14 energy indices near the middle of the energy band but on a different scale.[]{data-label="ex_finitevsLarge"}](ex-finitevsLarge.eps){width="75.00000%"}
What follows next is the calculation of large-$N$ results for $\langle\,( \Re\{\Delta\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. Since we need first to calculate $\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})\,^{2}\rangle$ and $\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle$, from Eq. (\[FPT2\]) we obtain after averaging $$\begin{aligned}
\label{dzsq}
\langle\,(\Delta \tilde{\mathcal{Z}}_{\alpha})^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}_{\alpha})}
{D^{2}},\end{aligned}$$ and $$\begin{aligned}
\label{cdzsq}
\langle\,[(\Delta \tilde{\mathcal{Z}}_{\alpha})^{*}]^{2}\,\rangle
=\sigma^{2}\dfrac{\sum_{n=1}^{N}4\sin^{4}(n\tilde{\theta}^{*}_{\alpha})}
{(D^{*})^{2}}.\end{aligned}$$
![Shown on the same pattern of Fig. \[ex\_finitevsLarge\] but for the parametric resonances where $N=500$ and $\eta=0.81$. In this figure $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ are shown respectively by pluses, crosses and stars. The inset is shown for the indices near the middle of the band.[]{data-label="pa_finitevsLarge"}](pa-finitevsLarge.eps){width="75.00000%"}
For summation we use the formula [@GR] $$\begin{aligned}
\label{sumsin4}
\sum_{n=1}^{N} \sin^{4}(n\,\theta)
&=&
\dfrac{1}{8}
\Bigg[
3N-\dfrac{\sin(N\theta)}{\sin(\theta)}
\big(4\cos[(N+1)\theta]
\nonumber
\\
&-&
\dfrac{\cos[2(N+1)\theta]\,\cos(N\theta)}{\cos(\theta)}
\big)\,
\Bigg],
\nonumber
\\
\sum_{n=1}^{N} \sin^{4}(n\,\tilde{\theta}_{\alpha})&=&
\dfrac{1}{8}
\left[
3N-4U_{N-1}T_{N+1}+\dfrac{T_{2N+2}U_{2N-1}}{\tilde{z}_{\alpha}}
\right],\end{aligned}$$ where in the second equality the polynomials have argument $\tilde{z}_{\alpha}/2$ with $2\cos(\tilde{\theta}_{\alpha})=\tilde{z}_{\alpha}$. Similarly for the summation in Eq. (\[cdzsq\]) one gets the polynomials with argument $\tilde{z}^{*}_{\alpha}/2$. One can now use the equality (\[sumsin4\]) in Eqs. (\[cdzsq\], \[cdzsq\]) in order to derive finite-$N$ result for $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. For large-$N$ we make use of the ansatz (\[ansatz\]) and calculate | 0 | non_member_970 |
the leading order contribution as $$\begin{aligned}
\label{LargeNdx}
\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=&
\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{5}{2}+\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},
\\
\nonumber
\\
\label{LargeNdy}
\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle
&=&
-\dfrac{\sigma^{2}}{2N}
\Bigg\{\dfrac{1}{2}-\dfrac{1}{8}\,
\dfrac{
\left(
|\Omega|^{2}-|\Omega|^{-2}
\right)}
{\text{ln}(|\Omega|)}
+
g(k_{\alpha})\Bigg\},\end{aligned}$$ where $$\begin{aligned}
g(k_{\alpha})&=&
\dfrac{1}{8}
\left(
\dfrac{e^{-2ik_{\alpha}}\Omega^{2}-e^{2ik_{\alpha}}\Omega^{-2}}
{4i \sin(k_{\alpha})\left(2N\cos(k_{\alpha})+i\sin(k_{\alpha})\text{ln}(\Omega)\right)}
\right)
+
(\text{c.c.}).\end{aligned}$$
Equations (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) are our main analytical results and they are also valid for parametric resonances with the $\Omega$ given in Eq. (\[Omgpara\]). In Fig. \[ex\_finitevsLarge\] we verify the asymptotic results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) against their finite-$N$ counterparts, for exact resonances with $N=100$ and $\eta=0.81$. Fig. \[pa\_finitevsLarge\] is repeated on the same pattern but for parametric resonances where $N=500$ and $\eta=0.81$. They confirm that the asymptotic results give a good account for the finite-$N$ results. However, there are some exception near the edges (not visible on the scale of the plot) where the ansatz (\[ansatz\]) is not valid.
It turns out that in order to calculate the DOR we need the second order corrections $(\delta_{2}\tilde{\mathcal{Z}}_{\alpha})$, derived in Eq. (\[dz2\]). We have followed the method used earlier [@FZ:99] for Hatano-Nelson Model [@HN:97]. However, we have not been able to obtain a closed expression of the DOR. This is discussed in Appendix B where we | 0 | non_member_970 |
leave the calculations with a formal expression for the DOR.
Numerical Methods and Verification of The Eqs. (\[abdz7\], \[LargeNdx\], \[LargeNdy\])
======================================================================================
![Comparison of asymptotic results with numerics, for exact resonances where $N=100$, $W=0.015$ and $\eta=0.81$. In this figure, filled circles, squares and diamonds are the numerical results respectively for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]). In the inset we show a comparison for 14 indices near the middle of the energy band on a different scale of the plot.[]{data-label="ex_numericsvsLarge"}](ex-numericsvsLarge.eps){width="75.00000%"}
Numerical simulations for parametric resonance are always cost efficient. The reason being that there one deals with standard eigenvalue problem for which many fast subroutine packages are available, for instance LAPACK. On the other hand to verify the results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) for exact resonances, where one needs to obtain numerical solutions of a characteristic polynomial equation of order $N$ in a complex plane, there is no as good algorithm. In this paper we show results for the exact resonances by calculating roots of the characteristic polynomial where we have used a cost efficient numerical subroutine [*ezero*]{}. The subroutine is available on the CPC program library. There is one major advantage of | 0 | non_member_970 |
using this subroutine over other methods, for instance the Newton’s method. This subroutine does not require initial guesses for the roots but only the contour which encloses all the roots of the polynomial. Besides, it also avoids calculating the derivatives which may result into numerical overflow.
In alternative to [*ezero*]{} we have used a different approach for calculating the roots. We survey the complex $\tilde{k}$-plane for the zeros of the $\text{Det}[M(\tilde{k})M(\tilde{k})^{\dagger}]$ where $M_{rs}=-2\cos(\tilde{k})\delta_{rs}-\tilde{\mathcal{H}}_{rs}$ for $r,s=1,...,N$ [@Neuberger]. (In our system $-\pi<\Re\{\tilde{k}\}<\pi$ and $\Im\{\tilde{k}\}<0$.) These zeros give the eigenvalues of $\tilde{\mathcal{H}}$. However, in the latter approach it is advisable to disintegrate the complex plane into small cells at first and then at every iteration into smaller one - only for $N$ cells which contain minima of the lowest eigenvalue and throwing the rest out. In this way one makes the algorithm faster and obtain the zeros in a reasonable precision. For a tridiagonal matrix this algorithm consumes a time which roughly grows with $N^3$. However, while comparing the two methods on a simple machine we find that the method used in [*ezero*]{} is much faster than the method described here. We refer to [@ezero] for further details of this subroutine.
In Fig. | 0 | non_member_970 |
\[ex\_numericsvsLarge\], we compare asymptotic results with simulation done for the total number of realizations $L=2500$, for exact resonances. In Fig. \[pa\_numericsvsLarge\] we compare numerical results obtained for parametric resonances, where $N=500$, $\eta=0.81$ and $L=5000$, with our theory for large-$N$. Though we have considered only the flat disorder yet our results are valid for the Gaussian or other symmetric distribution functions. These figures show that our asymptotic results are in fair agreement with the numerical results for almost all $\alpha$. For instance, near the middle of the energy band it describes reasonably well a dip and a peak, respectively in the $\langle\,(\Im\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$ and $\langle\,(\Re\{\Delta \tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle$. These two opposite effects, however, cancel out in $\langle\,(|\{\Delta \tilde{\mathcal{Z}}_{\alpha}\}|)^{2}\,\rangle$.
![ Shown on the same pattern of Fig. \[ex\_numericsvsLarge\] but for the parametric resonances where $N=500$, $W=0.015$ and $\eta=0.81$. These numerical results are obtained from the diagonalization of $N$-dimensional matrices for $5000$ realizations. In this figure $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ are shown respectively by pluses, crosses and stars while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]).[]{data-label="pa_numericsvsLarge"}](pa-numericsvsLarge.eps){width="75.00000%"}
Conclusion
==========
In conclusion, we have studied resonances in a one dimensional discrete tight-binding open chain in a weak disorder limit. In this study we have | 0 | non_member_970 |
calculated complex energies in an open-clean chain of finite length. The result we obtain is a polynomial equation which we have been able to solve for long chains using an ansatz for the solution. To the best of our knowledge, this result has never been derived before. We have used a perturbation theory up to the second order where we have derived the first and the second order corrections to the complex energies in terms of Chebyshev polynomials. The first order corrections have been useful to obtain closed form of the statistical results for the scattered complex energies. These results have been further simplified for long chains where we obtain compact results. The asymptotic results have been verified against numerics. Evidently, in the weak disorder limit the perturbation theory predicts nice statistical results. Our results are new in these studies and they could be useful in the further studies of such systems.
It would be interesting to study statistics of resonances in the weak disorder limit for higher dimensional models as well as for the cases when the site energies are not independent random variables but they are correlated with each other [@Izrailev:99]. Besides, there has been growing interest for | 0 | non_member_970 |
the case when $M$ sites are connected to the outer world where $1\le M\le N$ [@Borgonovi:2012]. We believe that our methods could be useful for the study of such models. Finally, we mention the case where $\xi(0)\sim\mathcal{O}(N)$. It requires a separate investigation as our perturbative analysis fails in this limit.
The author is thankful to Boris Shapiro for suggesting the problem to him. The author would also like to give credit to Joshua Feinberg for the derivation of some of the equations in Secs. III and IV and also for the help in Appendix B. Discussions with both of them are gratefully acknowledged. The author also acknowledges Marko Žnidarič and Thomas H. Seligman for reading the manuscript.
Support from ISF-1067 and generous hospitality of Technion Institute are also acknowledged. Additional support by the project 79613 by CONACyT, Mexico, is acknowledged.
Large-$N$ behavior of the denominator in (\[FPT2\])
===================================================
The denominator in Eq. (\[FPT2\]) can be simplified as follows: $$\begin{aligned}
\label{app1}
D(z_{\alpha})
&=&
-N+U_{N-1}(z_{\alpha}/2)T_{N+1}(z_{\alpha}/2)-i\eta\exp[I(z_{\alpha})]
\nonumber
\\
&\times&
\sin[k(z_{\alpha})][U_{N-1}(z_{\alpha}/2)]^{2}
\nonumber
\\
&\approx&
-N+\Big[-\exp(-ik_{\alpha})[\Omega(1-\eta)+1]
\nonumber
\\
&+&
\exp(ik_{\alpha})[\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta]
\Big]
\nonumber
\\
&\times&
\Big[2[\exp(-ik_{\alpha})-\exp(ik_{\alpha})]\Big]^{-1}.\end{aligned}$$ Using now $$\Omega(1-\eta)+1=2-\eta \exp(2ik_{\alpha}),$$ and $$\Omega^{-1}(1+\eta\exp(2ik_{\alpha}))+1-2\eta
=
\dfrac{2-3\eta+\eta^{2}\exp(2ik_{\alpha})}
{1-\eta\exp(2ik_{\alpha})},
\nonumber
\\$$ in (\[app1\]) we get $$\begin{aligned}
D
&\simeq&
-N-\dfrac{1}{1-\eta\exp(2ik_{\alpha})}.\end{aligned}$$ For | 0 | non_member_970 |
---
author:
- Timothy Porter
title: '$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories'
---
The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguña, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, , or any equivalent text if one can be found!
What do the notes set out to do?
“Aims and Objectives!” or should it be “Learning Outcomes”? {#aims-and-objectives-or-should-it-be-learning-outcomes .unnumbered}
===========================================================
- To revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today’s resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions;
- To introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas;
- To introduce Joyal’s quasicategories, (previously called weak Kan complexes but I agree with André that his | 0 | non_member_971 |
nomenclature is better so will adopt it) and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and myself;
- To ask lots of questions of myself and of the reader.
The notes include some material from the ‘Cubo’ article, [@cubo], which was itself based on notes for a course at the *Corso estivo Categorie e Topologia* in 1991, but the overlap has been kept as small as is feasible as the purpose and the audience of the two sets of notes are different and the abstract homotopy theory has ‘moved on’, in part, to try the new methods out on those same ‘old’ problems and to attack new ones as well.
As usual when you try to specify ‘learning outcomes’ you end up asking who has done the learning, the audience? Perhaps. The lecturer, most certainly!
**Acknowledgements**
I would like to thank Heiner Kamps and his colleagues at the Fern Univeristät for the invitation to give the talks of which these notes are a summary and to the Fern Univeristät for the support that made the visit possible, to José Manuel García-Calcines, Josué Remedios and their colleagues and for | 0 | non_member_971 |
the Departamento de Mathematica Fundamental in the Universidad de La Laguna, Tenerife, simlarly and also to Carlos Simpson, Bertrand Toen, André Joyal, Clemens Berger, André Hirschowitz and others at the Nice meeting in May 2003, since that is where bits of ideas that I had gleaned over a longish period of time fitted together so that I think I begin to understand the way that a lot of things interlock in this area better than I did before!
These notes have also benefitted from comments by Jim Stasheff and some of his colleagues on an earlier version.
$\mathcal{S}$-categories
========================
Categories with simplicial ‘hom-sets’
-------------------------------------
We assume we have a category $\mathcal{A}$ whose objects will be denoted by lower case letter, $x$,$y$,$z$, …, at least in the generic case, and for each pair of such objects, $(x,y)$, a simplicial set $\mathcal{A}(x,y)$ is given; for each triple $x, y, z$ of objects of $\mathcal{A}$, we have a simplicial map, called *composition* $$\mathcal{A}(x,y)\times \mathcal{A}(y,z)\longrightarrow \mathcal{A}(x,z);$$ and for each object $x$ a map $$\Delta[0] \to \mathcal{A}(x,x)$$ that ‘names’ or ‘picks out’ the ‘identity arrow’ in the set of 0-simplices of $\mathcal{A}(x,x)$. This data is to satisfy the obvious axioms, associativity and identity, suitably adapted | 0 | non_member_971 |
to this situation. Such a set up will be called a *simplicially enriched category* or more simply *an $\mathcal{S}$-category*. Enriched category theory is a well established branch of category theory. It has many useful tools and not all of them have yet been exploited for the particular case of $\mathcal{S}$-categories and its applications in homotopy theory.
Some authors use the term simplicial category for what we have termed a simplicially enriched category. There is a close link with the notion of simplicial category that is consistent with usage in simplicial theory *per se*, since any simplicially enriched category can be thought of as a simplicial object in the ‘category of categories’, but a simplicially enriched category is not just a simplicial object in the ‘category of categories’ and not all such simplicial objects correspond to such enriched categories. That being said that usage need not cause problems provided the reader is aware of the usage in the paper to which reference is being made.
**Examples**
\(i) $\mathcal{S}$, the category of simplicial sets:\
here $$\mathcal{S}(K,L)_n:= S(\Delta[n] \times K,L);$$ Composition : for $f \in \mathcal{S}(K,L)_n$, $ g \in \mathcal{S}(L,M)_n$, so $f : \Delta[n] \times K \to L$, $g : \Delta[n] \times L | 0 | non_member_971 |
\to M$, $$g\circ f := (\Delta[n] \times K \stackrel{diag \times K}{\longrightarrow} \Delta[n] \times\Delta[n] \times K\stackrel{\Delta[n] \times f}{\longrightarrow }\Delta[n] \times L \stackrel{g}{\to} M);$$ Identity : $id_K : \Delta[0] \times K \stackrel{\cong}{\to} K$,
\(ii) $\mathcal{T}op$, ‘the’ category of spaces (of course, there are numerous variants but you can almost pick whichever one you like as long as the constructions work): $$\mathcal{T}op(X,Y)_n := Top(\Delta^n \times X, Y)$$ Composition and identities are defined analogously to in (i).
\(iii) For each $X$, $Y \in Cat$, the category of small categories, then we similarly get $\mathcal{C}at(X,Y)$, $$\mathcal{C}at(X,Y)_n = {Cat}([n] \times X, Y).$$ We leave the other structure up to the reader.
\(iv) $\mathcal{C}rs$, the category of crossed complexes: see for background and other references, and Tonks, [@andythesis] for a more detailed treatment of the simplicially enriched category structure; $$\mathcal{C}rs(A,B) := Crs(\pi(n)\otimes C, D)$$ Composition has to be defined using an approximation to the identity, again see [@andythesis].
\(v) $\mathcal{C}h^+_K$, the category of positive chain complexes of modules over a commutative ring $K$. (Details are left to the reader, or follow from the Dold-Kan theorem and example (vi) below.)
\(vi) $\mathcal{S}(Mod_K)$, the category of simplicial $K$-modules. The structure uses tensor product with the free simplicial $K$-module on | 0 | non_member_971 |
$\Delta[n]$ to define the ‘hom’ and the composition, so is very much like (i).
In general any category of simplicial objects in a ‘nice enough’ category has a simplicial enrichment, although the general argument that gives the construction does not always make the structure as transparent as it might be.
There is an evident notion of $\mathcal{S}$-enriched functor, so we get a category of ‘small’ $\mathcal{S}$-categories, denoted $\mathcal{S}\!-\!Cat$. Of course, none of the above examples are ‘small’. (With regard to ‘smallness’, although sometimes a smallness condition is essential, one can often ignore questions of smallness and, for instance, consider simplicial ‘sets’ where actually the collections of simplices are not truly ‘sets’ (depending on your choice of methods for handling such foundational questions).)
From simplicial resolutions to $\mathcal{S}$-cats.
--------------------------------------------------
The forgetful functor $U : Cat \to DGrph_0$ has a left adjoint, $F$. Here $DGrph_0$ denotes the category of directed graphs with ‘identity loops’, so $U$ forgets just the composition within each small category but remembers that certain loops are special ‘identity loops’. The free category functor here takes, between any two objects, all strings of composable *non-identity* arrows that start at the first object and end at the second. One can | 0 | non_member_971 |
think of $F$ identifying the old identity arrow at an object $x$ with the empty string at $x$.
This adjoint pair gives a comonad on $Cat$ in the usual way, and hence a functorial simplicial resolution, which we will denote $S(\mathbb{A})\to \mathbb{A}$. In more detail, we write $T = FU$ for the functor part of the comonad, the unit of the adjunction $\eta : Id_{DGrph_0} \to UF$ gives the comultiplication $F\eta U: T \to T^2$ and the counit of the adjuction gives $\varepsilon : FU \to Id_{Cat}$, that is, $\varepsilon : T \to Id$. Now for $\mathbb{A}$ a small category, set $S(\mathbb{A})_n = T^{n+1}(\mathbb{A})$ with face maps $d_i : T^{n+1}(\mathbb{A}) \to T^n(\mathbb{A})$ given by $d_i = T^ {n-i}\varepsilon T^i$, and similarly for the degeneracies which use the comultiplication in an analogous formula. (Note that there are two conventions possible here. The other will use $d_i = T^i\varepsilon T^{n-i}$. The only effect of such a change is to reverse the direction of certain ‘arrows’ in diagrams later on. The two simplicial structures are ‘dual’ to each other.)
This $S(\mathbb{A})$ is a simplicial object in $Cat$, $S(\mathbb{A}) : \mathbf{\Delta}^{op} \to Cat$, so does not immediately gives us a simplicially enriched category, however | 0 | non_member_971 |
its simplicial set of objects is constant because $U$ and $F$ took note of the identity loops.
In more detail, let $ob : Cat \to Sets$ be the functor that picks out the set of objects of a small category, then $ob(S(\mathbb{A})) : \mathbf{\Delta}^{op} \to Sets$ is a constant functor with value the set $ob(\mathbb{A})$ of objects of $\mathbb{A}$. More exactly it is a discrete simplicial set, since all its face and degeneracy maps are bijections. Using those bijections to identify the possible different sets of objects, yields a constant simplicial set where all the face and degeneracy maps are identity maps, i.e. we do have a *constant* simplicial set.
\
Let $\mathcal{B} : \mathbf{\Delta}^{op} \to Cat$ be a simplicial object in $Cat$ such that $ob(\mathcal{B})$ is a constant simplicial set with value $B_0$, say. For each pair $(x,y)\in B_0$, let $$\mathcal{B}(x,y)_{n} = \{\sigma \in \mathcal{B}_{n} | ~ {\rm dom}(\sigma) = x, {\rm codom}(\sigma) = y\},$$ where, of course, ${\rm dom}$ refers to the domain function in $\mathcal{B}_n$ since otherwise ${\rm dom}(\sigma)$ would have no meaning, similarly for $\rm codom$.
\(i) The collection $\{\mathcal{B}(x,y)_n |~ n \in \mathbb{N}\}$ has the structure of a simplicial set $\mathcal{B}(x,y)$ with face and degeneracies | 0 | non_member_971 |
induced from those of $\mathcal{B}$.
\(ii) The composition in each level of $\mathcal{B}$ induces $$\mathcal{B}(x,y) \times \mathcal{B}(y,z)\to \mathcal{B}(x,z).$$ Similarly the identity map in $\mathcal{B}(x,x)$ is defined as $id_x$, the identity at $x$ in the category $\mathcal{B}_0$.
\(iii) The resulting structure is an $\mathcal{S}$-enriched category, that will also be denoted $\mathcal{B}$.$\blacksquare$
The proof is easy. In particular, this shows that $S(\mathbb{A})$ is a simplicially enriched category. The description of the simplices in each dimension of $S(\mathbb{A})$ that start at $a$ and end at $b$ is intuitively quite simple. The arrows in the category, $T(\mathbb{A})$ correspond to strings of symbols representing non-identity arrows in $\mathbb{A}$ itself, those strings being ‘composable’ in as much as the domain of the $i^{th}$ arrow must be the codomain of the $(i-1)^{th}$ one and so on. Because of this we have:\
$S(\mathbb{A})_0$ consists exactly of such composable chains of maps in $\mathbb{A}$, none of which is the identity;\
$S(\mathbb{A})_1$ consists of such composable chains of maps in $\mathbb{A}$, none of which is the identity, together with a choice of bracketting;\
$S(\mathbb{A})_2$ consists of such composable chains of maps in $\mathbb{A}$, none of which is the identity, together with a choice of two levels of bracketting;\
and so | 0 | non_member_971 |
on. Face and degeneracy maps remove or insert brackets, but care must be taken when removing innermost brackets as the compositions that can then take place can result in chains with identities which then need removing, see [@cordier82], that is why the comandic description is so much simpler, as it manages all that itself.
To understand $S(\mathbb{A})$ in general it pays to examine the simplest few cases. The key cases are when $\mathbb{A} = [n]$, the ordinal $\{0< \ldots <n\}$ considered as a category in the usual way. The cases $n=0$ and $n=1$ give no surprises. $S[0]$ has one object 0 and $S[0](0,0$ is isomorphic to $\Delta[0]$, as the only simplices are degenerate copies of the identity. $S[1]$ likewise has a trivial simplicial structure, being just the category $[1]$ considered as an $\mathcal{S}$-category. Things do get more interesting at $n = 2$. The key here is the identification of $S[2](0,2)$. There are two non-degenerate strings or paths that lead from 0 to 2, so $S[2](0,2)$ will have two vertices. The bracketted string $((01)(12))$ on removing inner brackets gives $(02)$ and outer brackets, $(01)(12)$ so represents a 1-simplex $$\xymatrix@+15pt{(01)(12)\ar[r]^{\quad (01)(12))}&(02)}$$ Other simplicial homs are all $\Delta[0]$ or empty. It thus is | 0 | non_member_971 |
possible to visualise $S[2]$ as a copy of $[2]$ with a 2-cell going towards the bottom: $$\xymatrix{ &1\ar[dr]&\\
0\ar[rr]\ar[ur]_{\hspace{.5cm}\Downarrow}&&2}$$ The next case $n = 3$ is even more interesting. $S[3](i,j)$ will be (i) empty if $j<i$, (ii) isomorphic to $\Delta[0] $ if $i = j$ or $i = j-1$, (iii) isomorphic to $\Delta[1]$ by the same reasoning as we just saw for $j = i + 2$ and that leaves $S[3](0,3)$. This is a square, $\Delta[1]^2$, as follows: $$\xymatrix@+25pt{(02)(23)\ar[r]^{((02)(23))}&(03)\\
(01)(12)(23)\ar[u]^{((01)(12))((23))}_{\hspace{.5cm} a}\ar[ur]^{diag}\ar[r]_{((01))((12)(23))}&(01)(13)\ar[u]_{((01)(13))}^{b\hspace{6mm}}
}$$ where the diagonal $diag = ((01)(12)(23))$, $a = (((01)(12))((23)))$ and $b = (((01))((12)(23)))$. (It is instructive to check that this is correct, firstly because I may have slipped up (!) as well as seeing the mechanism in action. Removing the innermost brackets is $d_0$, and so on.)
The case of $S[4]$ is worth doing. I will not draw the diagrams here although aspects of it have implications later, but suggest this as an exercise. As might be expected $S[4](0,4)$ is a cube.
**Remark**
The history of this construction is interesting. A variant of it, but with topologically enriched categories as the end result, is in the work of Boardman and Vogt, [@boardmanvogt] and also in Vogt’s paper, [@vogt73]. | 0 | non_member_971 |
Segal’s student Leitch used a similar construction to describe a homotopy commutative cube (actually a *homotopy coherent cube*), cf. [@leitch], and this was used by Segal in his famous paper, [@segal], under the name of the ‘explosion’ of $\mathbb{A}$. All this was still in the topological framework and the link with the comonad resolution was still not in evidence. Although it seems likely that Kan knew of this link between homotopy coherence and the comonadic resolutions by at least 1980, (cf. ), the construction does not seem to appear in his work with Dwyer as being linked with coherence until much later. Cordier made the link explicit in [@cordier82] and showed how Leitch and Segal’s work fitted in to the pattern. His motivation was for the description of homotopy coherent diagrams of topological spaces. Other variants were also apparent in the early work of May on operads, and linked in with Stasheff’s work on higher associativity and commutativity ‘up to homotopy’.
Cordier and Porter, [@cordierporter86], used an analysis of a locally Kan simplicially enriched category involving this construction to prove a generalisation of Vogt’s theorem on categories of homotopy coherent diagrams of a given type. (We will return to this | 0 | non_member_971 |
aspect a bit later in these notes, but an elementary introduction to this theory can be found in .) Finally Bill Dwyer, Dan Kan and Justin Smith, [@DKS], introduced a similar construction for an $\mathbb{A}$ which is an $\mathcal{S}$-category to start with, and motivated it by saying that $\mathcal{S}$-functors with domain this $\mathcal{S}$-category corresponded to *$\infty$-homotopy commutative $\mathbb{A}$-diagrams*, yet they do not seem to be aware of the history of the construction, and do not really justify the claim that it does what they say. Their viewpoint is however important as, basically, within the setting of Quillen model category structures, this provides a cofibrant replacement construction. Of course, any other cofibrant replacement could be substituted for it and so would still allow for a description of homotopy coherent diagrams in that context. This important viewpoint can also be traced to Grothendieck’s *Pursuing Stacks*, [@stacks].
The DKS extension of the construction, [@DKS], although simple to do, is often useful and so will be outlined next. If $\mathbb{A}$ is already a $\mathcal{S}$-category, think of it as a simplicial category, then applying the $\mathcal{S}$-construction to each $\mathbb{A}_n$ will give a bisimplicial category, i.e. a functor $S(\mathbb{A}) : \mathbf{\Delta}^{op} \times \mathbf{\Delta}^{op} \to Cat$. Of | 0 | non_member_971 |
this we take the diagonal so the collection of $n$-simplices is $S(\mathbb{A})_{n,n}$, and by noticing that the result has a constant simplicial set of objects, then apply the lemma.
The Dwyer-Kan ‘simplicial groupoid’ functor.
--------------------------------------------
Let $K$ be a simplicial set. Near the start of simplicial homotopy theory, Kan showed how, if $K$ was reduced (that is, if $K_0$ was a singleton), then the free group functor applied to $K$ in a subtle way, gave a simplicial group whose homotopy groups were those of $K$, with a shift of dimension. With Dwyer in , he gave the necessary variant of that construction to enable it to apply to the non-reduced case. This gives a ‘simplicial groupoid’ $G(K)$ as follows:
The object set of all the groupoids $G(K)_n$ will be in bijective correspondence with the set of vertices $K_0$ of $K$. Explicitly this object set will be written $\{\overline{x} ~|~ x \in K_0\}$.
The groupoid $G(K)_n$ is generated by edges $$\overline{y} : \overline{d_1d_2 \ldots d_{n+1} y} \to \overline{d_0d_2\ldots d_{n+1}y} \quad \mbox{ for } y \in K_{n+1}$$with relations $\overline{s_0x} = id_{\overline{d_1d_2 \ldots d_n x}}$. Note since these just ‘kill’ some of the generating edges, the resulting groupoid $G(K)_n$ is still a free | 0 | non_member_971 |
groupoid.
Define $\sigma_i\overline{x} = \overline{s_{i+1}x} \quad \mbox{ for } i\geq 0$, and, for $i > 0$, $\delta_i\overline{x} = \overline{d_{i +1}x}$, but for $i = 0$, $\delta_0\overline{x} = (\overline{d_1x})(\overline{d_0x})^{-1}$.
These definitions yield a simplicial groupoid as is easily checked and, as is clear, its simplicial set of objects is constant, so it also can be considered as a simplicially enriched groupoid, $G(K)$.
(NB. Beware there are several ‘typos’ in the original paper relating to these formulae for the construction and in some of the related material.)
As before it is instructive to compute some examples and we will look at $G(\Delta[2])$ and $G(\Delta[3]$. ) simplicially enriched groupoid are free groupoids in each simplicial dimension, their structure can be clearly seen from the generating graphs. For instance, $G(\Delta[2])_0$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{12}}&\\
\overline{0}\ar[rr]_{\overline{02}}\ar[ur]^{\overline{01}}&&\overline{2}}$$ whilst $G(\Delta[2])_1$ is the free groupoid on the graph $$\xymatrix{&\overline{1}\ar[dr]^{\overline{122}}&\\
\overline{0}\ar[rr]_{\overline{022}}\ar[ur]<.5ex>^{\overline{011}}\ar[ur]<-.5ex>_{\overline{012}}&&\overline{2}}$$ Here it is worth noting that $\delta_0(\overline{012}) = (\overline{02}).(\overline{12})^{ -1}$. Higher dimensions do not have any non-degenrate generators.
Again with $G(\Delta[3])$, in dimension 0 we have the free groupoid on the directed graph give by the 1-skelton of $\Delta[3]$. In dimension 2, the generating directed graph is $$\xymatrix@+20pt{&\overline{1}\ar[dr]^{\overline{133}}\ar[dd]<.5ex>\ar[dd]<-.5ex>&\\
\overline{0}\ar[ur]<.7ex>\ar[ur]\ar[ur]<-.7ex>\ar'[r][rr]\ar[dr]<.5ex>^{\overline{023}}\ar[dr]<-.5ex>_{\overline{022}}&&\overline{3}\\
&\overline{2} \ar[ur]_{\overline{233}} & }$$ Here | 0 | non_member_971 |
only a few of the arrow labels have been given. Others are easy to provide (but moderately horrible to typeset in a sensible way!). Those from $\overline{0}$ to $\overline{1}$ are $\overline{012}$, $\overline{011}$ and $\overline{013}$; those from $\overline{1}$ to $\overline{2}$ are $\overline{122}$ and $\overline{123}$, and finally from $\overline{0}$ to $\overline{3}$, we have $\overline{033}$.
The next dimension is only a little more complicated. It has extra degenerate arrows such as $\overline{0112}$ and $\overline{0122}$ from $\overline{0}$ to $\overline{1}$ but also between these two vertices has $\overline{0123}$, coming from the non-degenerate three simplex of $\Delta[3]$. The full diagram is easy to draw (and again a bit tricky to typeset in a neat way), and is therefore left ‘as an exercise’.
**Remarks**
\(i) The functor $G$ has a right adjoint $\overline{W}$ and the unit $K\to\overline{W}G(K)$ is a weak equivalence of simplicial sets. This is part of the result that shows that simplicially enriched groupoids model all homotopy types, for which see the original paper, or the book by Goerss and Jardine, [@GoerssJardine].
\(ii) It is tantalising that the definition of $S(\mathbb{A})$ for a category $\mathbb{A}$ and of $G(K)$ for a simplicial set $K$ are very similar yet very different. Why does the twist occur in | 0 | non_member_971 |
the $d_0$ of $G(K)$? Why do the source and target maps at each level in $G(K)$ end up at the zeroth and first vertex rather than the seemingly more natural zeroth and $n^{th}$ that occur in $S(\mathbb{A})$? In fact is there a variant of the $G(K)$ construction that is nearer to the $S(\mathbb{A})$ construction without being merely artificially so?
Structure
=========
The ‘homotopy’ category
-----------------------
If $\mathcal{C}$ is an $\mathcal{S}$-category, we can form a category $\pi_0\mathcal{C}$ with the same objects and having $$(\pi_0\mathcal{C})(X,Y) =
\pi_0({\mathcal{C}}(X,Y)).$$ For instance, if $\mathcal{C} =
\mathcal{CW}$, the category of CW-complexes, then $\pi_0\mathcal{CW} = {Ho\mathcal(CW)}$, the corresponding homotopy category. Similarly we could obtain a groupoid enriched category using the fundamental groupoid (cf. Gabriel and Zisman, [@gabrielzisman]).
One can ‘do’ some elementary homotopy theory in any $\mathcal{S}$-category, $\mathcal{C}$, by saying that two maps $f_0, f_1 : X \to Y$ in $\mathcal{C}$ are homotopic if there is an $H \in {\mathcal{C}}(X,Y)_1$ with $d_0H =
f_1$, $d_1H = f_0$.
This theory will not be very rich unless at least some low dimensional Kan conditions are satisfied. The $\mathcal{S}$-category, $\mathcal{C}$, is called *locally Kan* if each $\mathcal{C}(X,Y)$ is a Kan complex, *locally weakly Kan* if …, etc. (If you have | 0 | non_member_971 |
not met ‘weak Kan complexes’ before, you will soon meet them in earnest!)
The theory is ‘geometrically’ nicer to work with if $\mathcal{C}$ is *tensored* or *cotensored*.
Tensoring and Cotensoring
-------------------------
**Tensored**
If for all $K \in \mathcal{S}$, $X, Y, \in \mathcal{C}$, there is an object $K\bar{\otimes} X$ in $\mathcal{C}$ such that $$\mathcal{C}(K\bar{\otimes}X,Y) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$naturally in $K$, $X$ and $Y$, then $\mathcal{C}$ is said to be *tensored* over $\mathcal{S}$.
**Cotensored**
Dually, if we require objects $\bar{\mathcal{C}}(K,Y)$ such that $$\mathcal{C}(X,\bar{\mathcal{C}}(K,Y)) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$then we say $\mathcal{C}$ is *cotensored* over $\mathcal{S}$.
To gain some intuitive feeling for these, think of $K\bar{\otimes} X$ as being $K$ product with $X$, and $\bar{\mathcal{C}}(K,Y)$ as the object of functions from $K$ to $Y$. These words do not, as such, make sense in general, but do tell one the sort of tasks these constructions will be set to do. They will not be much used explicitly here however.
(cf. Kamps and Porter, )\
If $\mathcal{C}$ is a locally Kan $\mathcal{S}$-category tensored over $\mathcal{S}$ then taking $I\times X = \Delta[1]\bar{\otimes}X$, we get a good cylinder functor such that for the cofibrations relative to $I$ and weak equivalences taken to be homotopy equivalences, the category $\mathcal{C}$ has a cofibration category | 0 | non_member_971 |
structure.$\blacksquare$
A cofibration category structure is just one of many variants of the abstract homotopy theory structure introduced to be able to push through homotopy type arguments in particular settings. There are variants of this result, due to Kamps, see references in , where $\mathcal{C}$ is both tensored and cotensored over $\mathcal{S}$ and the conclusion is that $\mathcal{C}$ has a Quillen model category structure. The examples of locally Kan $\mathcal{S}$-categories include $\mathcal{T}op$, $\mathcal{K}an$, $\mathcal{G}rpd$ and $\mathcal{C}rs$, but not $\mathcal{C}at$ or $\mathcal{S}$ itself.
Nerves and Homotopy Coherent Nerves.
====================================
Kan and weak Kan conditions
---------------------------
Before we get going on this section, it will be a good idea to bring to the fore the definitions of *Kan complex* and *weak Kan complex* (or *quasi-category*).
As usual we set $\Delta[n] = {\mathbf \Delta}( - , [n]) \in \mathcal{S}$, then for each $i$, $0 \leq i \leq n$, we can form a subsimplicial set, $\Lambda^i[n]$, of $\Delta[n] $ by discarding the top dimensional $n$-simplex (given by the identity map on $[n]$) and its $i^{th}$ face. We must also discard all the degeneracies of these simplices. This informal definition does not give a ‘picture’ of what we have, so we will list the various | 0 | non_member_971 |
cases for $n = 2$. $$\Lambda^0[2] = \xymatrix{
&1\ar@{ ..}[dr]^{\leftarrow 0^{th} \mbox{~\scriptsize{face missing}}}&\\
0\ar[ur]\ar[rr]&&2}$$ $$\Lambda^1[2] = \xymatrix{
&1\ar[dr]^{\hspace{2.3cm}}&\\
0\ar[ur]\ar@{ ..}[rr]&&2}$$ $$\Lambda^2[2] = \xymatrix{
&1\ar[dr]^{\hspace{2.3cm}}&\\
0\ar@{ ..}[ur]\ar[rr]&&2}$$ A map $p: E \rightarrow B$ is a *Kan fibration* if given any $n$, $i$ as above and any $(n,i)$-horn in $E$, i.e. any map $f_1 : \Lambda^i[n]
\rightarrow E$, and $n$-simplex, $f_0 : \Delta[n] \rightarrow B$, such that $$\xymatrix{\Lambda^i[n]\ar[r]^{f_1}\ar[d]_{inc}& E\ar[d]^p\\
\Delta[n]\ar[r]_{f_0}&B}$$ commutes, then there is an $f : \Delta[n] \rightarrow E$ such that $pf = f_0$ and $f.inc = f_1$, i.e. $f$ lifts $f_0$ and extends $f_1$.
A simplicial set, $K$, is a *Kan complex* if the unique map $K
\rightarrow \Delta[0]$ is a Kan fibration. This is equivalent to saying that every horn in $K$ has a filler, i.e. any $f_1 : \Lambda^i[n]\rightarrow Y$ extends to an $f : \Delta[n] \rightarrow Y$. This condition looks to be purely of a geometric nature but in fact has an important algebraic flavour; for instance if $f_1 : \Lambda^1[2]\rightarrow Y$ is a horn, it consists of a diagram $$\xymatrix{ & \ar[dr]^b&\\\ar[ur]^a&&}$$ of ‘composable’ arrows in $K$. If $f$ is a filler, it looks like $$\xymatrix{ & \ar[dr]^b&\\ \ar[rr]_c\ar[ur]^a_{\quad f}&&}$$ and one can think of the | 0 | non_member_971 |
third face $c$ as a composite of $a$ and $b$. This ‘composite’ $c$ is not usually uniquely defined by $a$ and $b$, but is ‘up to homotopy’. If we write $c = ab$ as a shorthand then if $g_1 :\Lambda^0[2]
\rightarrow K$ is a horn, we think of $g_1$ as being $$\xymatrix{ & &\\ \ar[ur]^d\ar[rr]_e&&}$$and to find a filler is to find a diagram $$\xymatrix{ & \ar@{--}[dr]^x&\\ \ar[ur]^d\ar[rr]_e&&}$$ and thus to ‘solve’ the equation $dx = e$ for $x$ in terms of $d$ and $e$. It thus requires in general some approximate inverse for $d$, in fact, taking $e$ to be a degenerate 1-simplex puts one in exactly such a position.
In many useful cases, we do not always have inverses and so want to discard any requirement that would imply they always exist. This leads to the weaker form of the Kan condition in which in each dimension no requirement is made for the existence of fillers on horns that miss out the zeroth or last faces. More exactly:
**Definition**
A simplicial set ${\bf K}$ is *a weak Kan complex* or *quasi-category* if for any $n$ and $0< k < n$, any $(n,k)$-horn in $K$ has a filler.
**Remarks**
| 0 | non_member_971 |
\(i) Joyal, [@joyal], uses the term *inner horn* for any $(n,k)$-horn in $K$ with $0< k < n$. The two remaining cases are then conveniently called *outer horns*.
\(ii) For any space $X$, its singular complex, $Sing(X)$ is given by $Sing(X)_n = Top(\Delta^n,X)$ with the well known face and degeneracy maps. This simplicial set is always a Kan complex as is the related $\mathcal{T}op(X,Y)$ as mentioned above.
Nerves
------
The categorical analogue of the singular complex is, of course, the nerve: if $\mathbb{C}$ is a category, its *nerve*, $Ner(\mathbb{C})$, is the simplicial set with $Ner(\mathbb{C})_n = Cat([n],\mathbb{C})$, where $[n]$ is the category associated to the finite ordinal $[n] = \{ 0 < 1< \ldots < n\}$. The face and degeneracy maps are the obvious ones using the composition and identities in $\mathbb{C}$.
The following is well known and easy to prove.
\
(i) $Ner(\mathbb{C})$ is always weakly Kan.\
(ii) $Ner(\mathbb{C})$ is Kan if and only if $\mathbb{C}$ is a groupoid.$\blacksquare$
Of course more is true. Not only does any inner horn in $Ner(\mathbb{C})$ have a filler, it has exactly one filler. To express this with maximum force the following idea, often attributed to Graeme Segal or to Grothendieck, is very | 0 | non_member_971 |
useful.
Let $p>0$, and consider the increasing maps $e_i : [1] \to [p]$ given by $e_i(0) = i$ and $e_i(1) = i+1$. For any simplicial set $A$ considered as a functor $A : \mathbf{\Delta}^{op} \to Sets$, we can evaluate $A$ on these $e_i$ and, noting that $e_i(1) = e_{i+1}(0)$, we get a family of functions $A_p \to A_1$, which yield a cone diagram, for instance, for $p =3$: $$\xymatrix{A_p \ar[drrr]^{A(e_1)}\ar[ddrr]^{A(e_2)}\ar[dddr]_{A(e_3)}&&&\\
&&&A_1\ar[d]^{d_0}\\
&&A_1\ar[r]^{d_1}\ar[d]^{d_0}&A_0\\
&A_1\ar[r]_{d_1}&A_0&&}$$ and in general, thus yield a map $$\delta[p]: A_p \to A_1\times_{A_0}A_1 \times_{A_0}\ldots \times_{A_0}A_1.$$ The maps, $\delta[p]$, have been called the *Segal maps* and will recur throughout the rest of these notes.
\
If $A = Ner(\mathbb{C})$ for some small category $\mathbb{C}$, then for $A$, the Segal maps are bijections.
**Proof**
A simplex $\sigma \in Ner(\mathbb{C})_p$ corresponds uniquely to a composable $p$-chain of arrows in $\mathbb{C}$, and hence exactly to its image under the relevant Segal map. $\blacksquare$
Better than this is true:
\[GrotSegal\] \
If $A$ is a simplicial set such that the Segal maps are bijections then there is a category structure on the directed graph $$\xymatrix{A_1 \ar[r]<1ex> \ar[r]&A_0\ar[l]<1ex>}.$$making it a category whose nerve is isomorphic to the given $A$.
**Proof**
To get composition you use | 0 | non_member_971 |
$$A_1\times_{A_0}A_1 \stackrel{\cong}{\to} A_2\stackrel{d_1}{\to}A_1.$$ Associativity is given by $A_3$. The other laws are easy, and illuminating, to check. $\blacksquare$
The condition ‘Segal maps are a bijection’ is closely related to notions of ‘thinness’ as used by Brown and Higgins in the study of crossed complexes and their relationship to $\omega$-groupoids, see, for instance, , and also to Duskin’s ‘hypergroupoid’ condition, [@duskin].
Another result that is sometimes useful, is a refinement of ‘groupoids give Kan complexes’:. The proof is ‘the same’:
\
Let $A = Ner(\mathbb{C})$, the nerve of a category $\mathbb{C}$.
\(i) Any $(n,0)$-horn $$f : \Lambda^0[n]\rightarrow A$$ for which $f(01)$ is an isomorphism has a filler. Similarly any $(n,n)$-horn $g : \Lambda^n[n]\rightarrow A$ for which $g(n-1~n)$ is an isomorphism, has a filler.
\(ii) Suppose $f$ is a morphism in $\mathbb{C}$ with the property that for any $n$, any $(n,0)$-horn $\phi : \Lambda^0[n]\rightarrow A$ having $f$ in the $(0,1)$ position, has a filler, then $f$ is an isomorphism. (Similarly with $(n,0)$ replaced by $(n,n)$ with the obvious changes.) $\blacksquare$
**Remark**
Joyal in [@joyal] suggested that the name ‘weak Kan complex’, as introduced by Boardman and Vogt, [@boardmanvogt], could be changed to that of ‘quasi-category’ to stress the analogy with categories *per | 0 | non_member_971 |
se* as ‘*Most concepts and results of category theory can be extended to quasi-categories*’, [@joyal].
It would have been nice to have explored Joyal’s work on quasi-categories more fully, e.g. [@joyal], but time did not allow it. The following few sections just skate the surface of the theory.
Quasi-categories
----------------
Categories yield quasi-categories via the nerve construction. Quasi-categories yield categories by a ‘fundamental category’ construction that is left adjoint to nerve. This can be constructed using the free category generated by the 1-skeleton of $A$, and then factoring out by a congruence generated by the basic relations : $gf \equiv h$, one for each commuting 1-sphere $(g,h,f)$ in $A$. By a *1-sphere* is meant a map $a : \partial \Delta[2] \to A$, thus giving three faces, $(a_0,a_1,a_2)$ linked in the obvious way. The 1-sphere is said to be *commuting* if there is a 2-simplex, $b\in A_2$, such that $a_i = d_ib$ for $i = 0,1,2$.
This ‘fundamental category’ functor also has a very neat description due to Boardman and Vogt. (The treatment here is adapted from [@joyal].)
We assume given a quasi-category $A$. Write $gf \sim h$ if $(g,h,f)$ is a commuting 1-sphere. Let $x,y \in A_0$ and let $A_1(x,y) | 0 | non_member_971 |
Subsets and Splits