# Magnetic Susceptibility of Andreev Bound States in Superfluid 3He-B

J. W. Scott<EMAIL_ADDRESS>M. D. Nguyen  D. Park  W. P.
Halperin<EMAIL_ADDRESS>Northwestern University, Evanston, IL
60208, USA

###### Abstract

Nuclear magnetic resonance measurements of the magnetic susceptibility of
superfluid 3He imbibed in anisotropic aerogel reveal anomalous behavior at low
temperatures. Although the frequency shift clearly identifies a low-
temperature phase as the B phase, the magnetic susceptibility does not display
the expected decrease associated with the formation of the opposite-spin
Cooper pairs. This susceptibility anomaly appears to be the predicted high-
field behavior corresponding to the Ising-like magnetic character of surface
Andreev bound states within the planar aerogel structures.

††preprint: ASA16-SUS

Unconventional superconductors break symmetries beyond gauge symmetry, and are
classified by the symmetries of the order parameter. The paradigm of
unconventional superconductors is 3He, a spin-triplet $p$-wave BCS superfluid
[1, 2] with two zero field phases $A$ and $B$, breaking and preserving time-
reversal symmetry respectively. One identifying characteristic of
unconventionality is the strong suppression of the transition temperature, and
concomitantly, the amplitude of the order parameter induced by nonmagnetic
impurities [3]. This is in stark contrast to conventional superconductivity
[4]. In the case of 3He, dilute non-magnetic impurities of silica aerogel
particles that are much smaller than the coherence length, reduce both the
amplitude of the order parameter and the transition temperature [5, 6], just
as expected for an unconventional superconductor [7, 8]. The 3He quasiparticle
scattering from these impurities produces surface bound states, Andreev bound
states, [9] irrespective of whether the surface scattering of 3He
quasiparticles is specular or diffuse. Specularity can be achieved by coating
the surfaces with at least two atomic layers of 4He [10]. Under those
conditions the bound states in 3He-$B$ are expected to be Majorana fermions
[11].

The evidence for the existence of these Andreev bound states, both theoretical
and experimental, and their relation to the topological character of the B
phase, have been reviewed by Mizushima et al. 2016 [11]. Acoustic impedance
measurements have resolved a density of states consistent with the existence
of a Majorana cone as specularity is increased [12]. The bound states are also
responsible for dissipation from vibrating wire devices in the low-temperature
limit of the B phase [13]. Theoretical interpretation of ion mobility
measurements at the specular free surface are also consistent with their
Majorana character [14, 15]. Particularly relevant for the present work are
theoretical predictions for the enhancement of the magnetic susceptibility
[16, 17, 18] of the bound states. Here, we report the detection of an
anomalous enhancement to the magnetic susceptibility in the B phase within a
$98\%$ porosity silica aerogel with specular boundary conditions which we
attribute to the magnetic susceptibility of 3He Andreev bound states.

The application of aerogels for investigation of 3He superfluid phases is now
widespread [19]. It is significant that silica aerogels with global anisotropy
[20], couple directly to the 3He orbital angular momentum, producing well-
defined, uniform order parameter textures [21, 22]. The aerogel structure also
determines the stability of different superfluid phases. Silica aerogel
samples can be either grown, or mechanically strained, to produce uniaxial
anisotropy [23], stabilizing the A phase when stretched (positive strain) [24]
and the B phase under compression (negative strain) [25]. In both phases there
is a second transition on cooling at $T_{x}<T_{c}$ from a uniform texture with
angular momentum $\bm{\ell}$ parallel to the anisotropy axis, to a uniform
texture with $\bm{\ell}$ perpendicular to the anisotropy axis [20]. We refer
to this as the orbital flop transition [22], clearly evident in Fig. 1(a) at
$0.77\,T_{c}$, where $T_{c}$ is the superfluid transition temperature. A
different class of nematic alumina aerogels, Nafen, are extremely anisotropic;
they stabilize the polar phase [26] which is a new $p$-wave superfluid phase
that hosts half-quantum vortices [27]. In the present work we find that
positive strain and specular boundary conditions for 3He quasiparticle
scattering are requirements for the enhanced magnetic susceptibility that we
observe in the superfluid $B$ phase.

We determine the magnetic susceptibility $\chi$ relative to the normal Fermi
liquid magnetic susceptibility $\chi_{N}$ from the nuclear magnetic resonance
(NMR) spectra of superfluid 3He imbibed in an anisotropic $16\%$ stretched
aerogel (see the Appendix for details). Computational studies with a biased
diffusion-limited cluster aggregation algorithm show that the microscopic
structure of stretched aerogel has an anisotropic mean free path and planar
voids [20]. Unlike in our prior experiments on similarly stretched aerogel, we
have added $\approx 3.5$ layers of 4He to the surface, changing the
quasiparticle scattering potential [28, 10] and eliminating the contribution
of surface solid 3He to the spectral weight of the NMR signal [29]. The
addition of surface 4He has also been previously shown to alter the stability
of superfluid phases imbibed in aerogel [30, 31, 32].

Figure 1:  Temperature-dependence and tip angle dependence of NMR absorption
spectra of 3He imbibed in $16\%$ stretched aerogel compared with pure 3He. In
panels (a-d) the absorption spectrum is shown at each temperature and tip
angle as a function of frequency by a color scale representing the spectral
amplitude normalized to the maximum absorption amplitude in each panel. (a)
Temperature dependence of spectra at tip angle, $\beta=8^{\circ}$, $P=26.6$
bar, $H_{0}=74.5$ mT. The configurations of the order parameter in order on
warming; below the orbital flop transition $T_{x}=0.77\,T_{c}$,
$\bm{\ell}\perp\bm{H_{0}}$ [22]; above $T_{x}=0.77\,T_{c}$,
$\bm{\ell}\parallel\bm{H_{0}}$; above $T_{ab}=0.93\,T_{c}$ the tip angle
dependence is characteristic of the A phase with
$\bm{\ell}\parallel\bm{H_{0}}$. (b-d) Tip angle dependence of spectra in (a).
(e-g) Theoretical tip angle dependence of spectra for pure 3He corresponding
to (b-d) [33].

We identify a B phase from the NMR frequency shift, Fig. 1. In the superfluid,
macroscopic coherence of the nuclear dipole-dipole interaction causes a
frequency shift $\Delta\omega$ away from the Larmor frequency
$\omega_{L}=\gamma H_{0}$ [1]. The frequency shift $\Delta\omega$ is given by,

$2\omega_{L}\Delta\omega=\Omega(P,T)^{2}F(\beta).$ (1)

The magnitude of the frequency shift is set by the square of the longitudinal
resonance frequency $\Omega^{2}\propto\Delta(P,T)^{2}/\chi$, where
$\Delta(P,T)$ is the pressure- and temperature-dependent order parameter
amplitude. The frequency shift dependence $F(\beta)$ on the NMR tip angle
$\beta$ is set by the structure and orientation of the spin and orbital
degrees of freedom of the order parameter [33]; further discussion of the NMR
experiment, including the tip angle, is given in the Appendix. The theoretical
behavior of $F(\beta)$ for pure 3He is displayed in Fig. 1 (e-g), and can be
compared with our data in panels (b-d) in this figure. The orbital degrees of
freedom are defined by the orbital angular momentum axis, $\bm{\ell}$, which
is oriented by the aerogel anisotropy axis. In the present work the static
magnetic field $\bm{H_{0}}$ is aligned with the axis of anisotropy of the
aerogel.

At temperatures between $T_{c}$ and $T_{ab}$, as seen in Fig. 1 (a), the
nuclear magnetic resonance frequency decreases with decreasing temperature.
This negative frequency shift is characteristic of the A phase with
$\bm{\ell}\parallel\bm{H_{0}}$. Its tip angle dependence is shown in Fig. 1
(d), consistent with theory Fig. 1 (g) [33]. This configuration results from
anisotropic quasiparticle scattering induced by the planar aerogel structure
[20] in a manner analogous to confinement in a slab [34] and consistent with
results from other anisotropic aerogels [21, 35]. For comparison, in the pure
bulk superfluid the energetically favored configuration is
$\bm{\ell}\perp\bm{H_{0}}$ for which the frequency shifts are positive.

At lower temperatures, the superfluid enters the B phase which we identify
from its characteristic NMR frequency shifts, shown in Fig. 1 (a). The shifts
above and below $T_{x}=0.77\,T_{c}$ are characteristic of the B phase with
$\bm{\ell}\parallel\bm{H_{0}}$, Fig. 1 (c,f) and $\bm{\ell}\bot\bm{H_{0}}$,
Fig. 1 (b,e) respectively, where $T_{x}$ is the orbital flop transition [22],
referred to earlier. The magnetic susceptibility in the B phase is temperature
independent within our measured temperature range, Fig. 2, in striking
contrast to the temperature dependent magnetic susceptibility in the pure
superfluid.

In pure 3He $B$, the reduction of the magnetic susceptibility with decreasing
temperature corresponds to the formation of opposite-spin Cooper pairs [36,
37]. This is observed in both pure and impure superfluids [38, 30]. The extent
of the reduction is less in a large magnetic field [39, 40], or in
confinement, [41] which tends to suppress the formation of opposite-spin
pairs. The effect corresponds to a polar distortion of the B phase order
parameter that can be measured independently. We follow the method of Rand et
al. [40], finding that the distortion varies smoothly from $\sim 0.34$ to
$0.09$ in the region below $0.77\,T_{c}$, as shown in Fig. 3 (see Appendix).
The polar distortion $\leq 0.09$ increases the susceptibility in the pure B
phase by $\leq 0.15\,\chi/\chi_{N}$ [39, 40]. Polar distortion cannot account
for the excess susceptibility. Similarly, the effect of quasiparticle
scattering from impurities is also too small as shown in Fig.2.

Figure 2:  Magnetic susceptibility $\chi/\chi_{N}$ (green circle), over a
range of fields at P = 26.6 bar. In contrast to the behavior expected, the
magnetic susceptibility is temperature independent in the B phase (blue
background) for all fields, different pressures, temperatures below $T_{ab}$,
and different orbital configurations. For comparison, $\chi/\chi_{N}$ of the
pure superfluid at $P=27.0$ bar is shown in the lowest panel (purple circle),
taken from Ref. [38]. The blue solid (dashed) lines are the Born (unitary),
Ginzburg-Landau results from the homogeneous isotropic scattering model for
3He in aerogel [9]. Figure 3:  Measured order parameter distortion. The order
parameter amplitude distortion in the direction of the magnetic field,
$1-\Delta_{\parallel}/\Delta_{\bot}$ at $H_{0}=74.5$ mT for the temperature
range between $T_{ab}$ and $T_{x}\,=0.77\,T_{c}$. The degree of order
parameter distortion $\approx 0.34$ close to $T_{ab}$ is comparable to that
seen in nanoconfined planar slabs [41]. At lower temperatures the distortion
we measure $\approx 0.09$ approaches the value for bulk, pure 3He $\approx
0.005$ in a $112$ mT magnetic field [40].

The earliest magnetic susceptibility experiments on the impure B phase in a
nominally isotropic aerogel, were performed by Sprague et al. [30] with 4He
covering the surface of the aerogel, similar to the experiments we report
here. They found a small increase in the susceptibility compared to the pure
superfluid state, subsequently accounted for by microscopic theories for
isotropic impurities including polarization of Andreev bound states [9, 42].
In the treatment by Sharma and Sauls [9], the aerogel was modeled as
uncorrelated impurities with a single mean-free path, from which both
susceptibility and suppression of the critical temperature were determined.
This theory has also been used to account for the temperature dependent
susceptibility observed in both isotropic [43] and compressed aerogel samples
[25] without 4He preplating, which do not exhibit the anomalous behavior we
report here. The susceptibility in the Ginzburg-Landau limit of the
quasiclassical theory is shown in Fig. 2 as a blue solid (dashed) line for the
Born (unitary) scattering limits.

Theoretical studies of the magnetic susceptibility from bound states in the B
phase near a solid plane boundary have found two effects, the emergence of a
spontaneous polarization normal to the surface of the plane [16] and the
enhancement of the magnetic susceptibility in a confined slab [17]. In the
latter case, it was predicted that confining the B phase to a slab causes an
increase in the susceptibility, with the highest levels of confinement
recovering the susceptibility of the normal liquid [17, 44]. A detailed study
of the field dependence of Andreev bound states found that the susceptibility
increase is associated with a gap in the Andreev bound state dispersion
spectrum induced by magnetic field [45].

The NMR frequency shift of the superfluid below $T_{ab}$ precludes the
identification of the superfluid state as any of the experimentally observed
or theoretically well-established equal spin pairing states, such as the polar
or planar phases. The magnetic susceptibility also does not display the
decrease characteristic of the isotropic impure superfluid models, and is
inconsistent with the Ginzburg-Landau limit of these theories. Simulations of
the growth of anisotropic aerogels [20], together with small angle x-ray
scattering [23] provide strong evidence for the existence of planar structures
in stretched aerogels. As a consequence, we propose that planar structures in
the aerogel are responsible for the enhanced susceptibility in a manner
analogous to the predictions for Andreev bound states in confined slabs.

Figure 4:  Magnetic phase diagram at $P=26.6$ bar. The green trace (green
triangle) denotes the experimentally observed $T_{ab}$ in the aerogel. A
linear fit shows the extrapolated $T_{ab}$ at zero field to be
$T_{ab}=0.93\,T_{c}$. The purple trace (purple triangle) denotes the $A-B$
transition in pure bulk superfluid at $P=27.0$ bar [46, 38] for comparison.
(Inset) The pressure-temperature phase diagram at $H_{0}=195$ mT, with
critical temperature $T_{c}$ (green open circle), A-B transition temperature
$T_{ab}$ (green open triangle). The black curve is a fit for the observed
$T_{c}$ based on an impurity model [7, 47, 48]. The purple line denotes the
pure liquid $T_{c}$, the dashed black line is a guide to the eye.

According to theory, a slab of thickness of $\approx 5\,\xi_{0}$ is required
to recover the full normal state susceptibility from bound states [17]. In the
pressure range of the current work this corresponds to $130$ nm taking the
zero temperature cohrence length to be $\xi_{0}=(\hbar v_{F})/(2\pi
k_{B}T_{c})$; $v_{F}$ is the Fermi velocity. Numerical simulations of the
stretched aerogel structure indicate an approximately planar mass distribution
with plane separations of $\sim 50-100$ nm. This is similar to the calculated
quasiparticle mean free path of $90-120$ nm [20]. Although aerogel and slab
confinements are very different, the slab thickness for the expected enhanced
magnetic susceptibility is roughly the same as that of the planar structure of
the stretched aerogel.

The $A$ to $B$ transition temperature, $T_{ab}$, decreases with a
quadratically increasing field with a slope comparable to that of the pure
superfluid and similar for all isotropic aerogels and anisotropic silica
aerogels with 4He preplating including the present work, and shown in Fig. 4
[49, 38, 50, 51, 52, 32]. This slope taken at comparable pressures in aerogel
samples with and without preplating is included in Supplementary Material as
Table 1[43, 25, 32]. Since this transition is first-order its field dependence
is given by a Clausius-Clapeyron relation between the differences in
susceptibility $\chi$ and entropy $S$ for the two superfluid phases,

$\frac{dT_{ab}}{dH^{2}}=-\frac{1}{2}\frac{\chi_{A}-\chi_{B}}{S_{A}-S_{B}}.$
(2)

Our measurements of the magnetic susceptibility $\chi_{B}$ are in stark
contrast with the Clausius-Clapeyron relation.

In summary, we have discovered an anomalous contribution to the susceptibility
of superfluid 3He-B imbibed in a stretched aerogel. This susceptibility
appears to be a consequence of planar regions in the aerogel structure and is
surprisingly large, essentially identical to that of the normal state, similar
to predictions for the susceptibility of 3He-B confined to a planar slab with
specular boundary conditions [17]. How this can be reconciled with the field
dependence of the $AB$ transition remains an open question.

This work was supported by NSF Division of Materials Research grant
DMR-2210112. We are grateful to J. A. Sauls and A. M. Zimmerman for useful
discussion and to V.P. Mineev for communication. We also thank the
Northwestern University Instrument Shop for use of its facilities.

##  I Appendix on Experimental Methods

We grow the $16$% stretched aerogel with the rapid supercritical extraction
method presented in Pollanen et al. [23]. The aerogel is grown in a $4.20$ mm
inner diameter glass NMR tube, however, during the growth process the aerogel
shrinks radially inwards from the walls of the NMR tube. The NMR sample is
approximately $3.53$ mm in diameter and it measures $5.5$ mm long. Density
measurements show the sample to be approximately $97.5\%$ porous. Optical
birefringence measurements reveal an axis of structural anisotropy aligned
with the cylindrical axis of the aerogel. The cylindrical axis is oriented
along the direction of the static NMR field $H_{0}$, and connected by a fill
line to the silver heat exchanger of the cryostat.

The NMR experiments require a radio frequency pulse of length
$\tau_{p}\propto\beta$, the proportion calibrated in the normal Fermi liquid
by observing the dependence of the intensity of the NMR signal on the length
of the pulse $\tau_{p}$; maximal intensity corresponds to $\beta=90^{\circ}$.
After cooling by nuclear demagnetization, intermittent NMR pulses of several
fixed lengths are delivered as the sample warms, producing the color map in
Fig. 1 (a). During this process we measure temperature via a capacitive
pressure transducer 3He melting curve thermometer [46] and a 195Pt
susceptibility thermometer. The cryostat can also be held at a roughly fixed
temperature while spectra are captured for a variety of pulse lengths,
producing the color maps in Fig. 1 (b-d). Each NMR spectrum in the temperature
sweep data is accumulated over a small region of temperature, typically $\sim
8$ $\mu$K.

Measurements were performed between pressures of 19.0 bar and 26.6 bar and
magnetic fields from 74.5 to 195 mT. The pressure-dependent critical
temperature suppression of the $16\%$ stretched aerogel is consistent with a
quasiparticle mean free path $\lambda=131$ nm and aerogel correlation length
$\xi_{a}=23$ nm [47].

We determined the polar distortion of the amplitude of the order parameter,
which is the suppression of $\Delta_{\parallel}$ aligned with the magnetic
field relative to the perpendicular component $\Delta_{\bot}$, following the
method of Rand et al. [40]. The frequency shift was measured at two tip angles
above and below the critical tipping angle (nominally $\arccos(-1/4)\approx
104^{\circ}$), separating the regions where $\Delta\omega$ is small and large
(e.g. $10^{\circ}$ vs. $\approx 135^{\circ}$ in Fig. 1 (c)). The frequency
shifts for two different pulse lengths taken on warming determine the
distortion of the order parameter as a function of temperature. This gap
distortion is given by,

$\frac{\Delta_{\parallel}}{\Delta_{\bot}}=\frac{1+\frac{\Delta\omega_{1}}{\Delta\omega_{2}}+2\frac{\Delta\omega_{1}}{\Delta\omega_{2}}\cos{\beta_{2}}}{1-2\frac{\Delta\omega_{1}}{\Delta\omega_{2}}\cos{\beta_{2}}}$
(3)

with $\Delta\omega_{1}$ and $\Delta\omega_{2}$ the frequency shifts of the
small and large pulses, respectively, and $\beta_{2}$ the tipping angle of the
large pulse[53]. This ratio is independent of the magnetic susceptibility of
the superfluid and the B phase longitudinal resonance frequency $\Omega_{B}$.

The susceptibility is proportional to the integral of the absorption spectrum
of the NMR signal. At temperatures above those shown in Fig. 2 there is an
additional contribution to the susceptibility from the 3He in the fill lines
and from that in the near vicinity of the aerogel, which we refer to as the
3He bath. At temperatures substantially higher than $T_{c}$ in the aerogel,
the bath is in the superfluid A phase. As a consequence, the contribution of
the bath to the signal, $\sim 20\,\%$ of the total spectral weight, is shifted
in frequency away from the signal of the normal liquid in the aerogel, as
shown in Supplemental Material Fig. 1. At and below $T_{ab}$ in the pure
superfluid, the B phase bath signal abruptly becomes very broad. At lower
temperatures, below $T_{c}$ in the aerogel as shown Fig. 2, this contribution
cannot be distinguished from the background noise. The influence of the
superfluid bath on the susceptibility in both stretched and compressed aerogel
can be seen in Supplemental Material Fig. 1[22, 54].

## References

  * Leggett [1973] A. J. Leggett, Microscopic theory of NMR in an anisotropic superfluid (${}^{3}\mathrm{He}{A}$), Phys. Rev. Lett. 31, 352 (1973).
  * Wheatley [1975] J. C. Wheatley, Experimental properties of superfluid ${}^{3}\mathrm{He}$, Rev. Mod. Phys. 47, 415 (1975).
  * Larkin [1965] A. I. Larkin, Vector pairing in superconductors of small dimensions, JETP Lett. 2, 130 (1965).
  * Anderson [1959] P. W. Anderson, Theory of dirty superconductors, Journal of Physics and Chemistry of Solids 11, 26 (1959).
  * Porto and Parpia [1995] J. V. Porto and J. M. Parpia, Superfluid ${}^{3}\mathrm{He}$ in aerogel, Phys. Rev. Lett. 74, 4667 (1995).
  * Sprague  _et al._ [1995] D. T. Sprague, T. M. Haard, J. B. Kycia, M. R. Rand, Y. Lee, P. J. Hamot, and W. P. Halperin, Homogeneous equal-spin pairing superfluid state of ${}^{3}\mathrm{He}$ in aerogel, Phys. Rev. Lett. 75, 661 (1995).
  * Thuneberg  _et al._ [1996] E. V. Thuneberg, M. Fogelström, S. K. Yip, and J. A. Sauls, Localized vs. delocalized scattering in superfluid ${}^{3}\mathrm{He}$-aerogel, Czechoslovak Journal of Physics 46, 113 (1996).
  * Thuneberg  _et al._ [1998] E. V. Thuneberg, S. K. Yip, M. Fogelström, and J. A. Sauls, Models for superfluid ${}^{3}\mathrm{He}$ in aerogel, Phys. Rev. Lett. 80, 2861 (1998).
  * Sharma and Sauls [2001] P. Sharma and J. A. Sauls, Magnetic susceptibility of the Balian–Werthamer phase of ${}^{3}\mathrm{He}$ in aerogel, Journal of Low Temperature Physics 125, 115 (2001).
  * Heikkinen  _et al._ [2021] P. J. Heikkinen, A. Casey, L. V. Levitin, X. Rojas, A. Vorontsov, P. Sharma, N. Zhelev, J. M. Parpia, and J. Saunders, Fragility of surface states in topological superfluid ${}^{3}\mathrm{He}$, Nature Communications 12, 1 (2021).
  * Mizushima  _et al._ [2016] T. Mizushima, Y. Tsutsumi, T. Kawakami, M. Sato, M. Ichioka, and K. Machida, Symmetry-protected topological superfluids and superconductors —from the basics to ${}^{3}\mathrm{He}$—, Journal of the Physical Society of Japan 85, 022001 (2016), https://doi.org/10.7566/JPSJ.85.022001 .
  * Murakawa  _et al._ [2009] S. Murakawa, Y. Tamura, Y. Wada, M. Wasai, M. Saitoh, Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, S. Higashitani, and K. Nagai, New anomaly in the transverse acoustic impedance of superfluid ${}^{3}\mathrm{He}\mathrm{\text{$-$}}\mathbf{B}$ with a wall coated by several layers of ${}^{4}\mathrm{He}$, Phys. Rev. Lett. 103, 155301 (2009).
  * Autti  _et al._ [2020] S. Autti, S. L. Ahlstrom, R. P. Haley, A. Jennings, G. R. Pickett, M. Poole, R. Schanen, A. A. Soldatov, V. Tsepelin, J. Vonka, T. Wilcox, A. J. Woods, and D. E. Zmeev, Fundamental dissipation due to bound fermions in the zero-temperature limit, Nature Communications 11, 1 (2020).
  * Tsutsumi [2017] Y. Tsutsumi, Scattering theory on surface Majorana fermions by an impurity in ${}^{3}\mathrm{He}\text{$-$}\mathrm{B}$, Phys. Rev. Lett. 118, 145301 (2017).
  * Ikegami and Kono [2019] H. Ikegami and K. Kono, Observation of Majorana bound states at a free surface of ${}^{3}\mathrm{He}\text{-}{B}$, Journal of Low Temperature Physics 195, 343 (2019).
  * Chung and Zhang [2009] S. B. Chung and S.-C. Zhang, Detecting the Majorana fermion surface state of ${}^{3}\mathrm{He}\mathrm{\text{$-$}}{B}$ through spin relaxation, Phys. Rev. Lett. 103, 235301 (2009).
  * Nagato  _et al._ [2009] Y. Nagato, S. Higashitani, and K. Nagai, Strong anisotropy in spin susceptibility of superfluid ${}^{3}\mathrm{He}\text{-}{B}$ film caused by surface bound states, Journal of the Physical Society of Japan 78, 123603 (2009), https://doi.org/10.1143/JPSJ.78.123603 .
  * Nagato  _et al._ [2018] Y. Nagato, S. Higashitani, and K. Nagai, Subgap in the surface bound states spectrum of superfluid ${}^{3}\mathrm{He}\text{-}{B}$ with rough surface, Journal of Low Temperature Physics 190, 277 (2018).
  * Halperin [2019] W. Halperin, Superfluid 3He in aerogel, Annual Review of Condensed Matter Physics 10, 155 (2019).
  * Nguyen  _et al._ [2023] M. D. Nguyen, J. S. Simon, J. W. Scott, Y. C. C. Tsai, A. M. Zimmerman, and W. P. Halperin, Planar and nematic aerogels: DLCA and superfluid ${}^{3}\mathrm{He}$ (2023), arXiv:2301.11261 [cond-mat.mtrl-sci] .
  * Li  _et al._ [2014a] J. I. A. Li, A. M. Zimmerman, J. Pollanen, C. A. Collett, W. J. Gannon, and W. P. Halperin, Orientation of the angular momentum in superfluid ${}^{3}\mathrm{He}\text{-}{A}$ in a stretched aerogel, Journal of Low Temperature Physics 175, 31 (2014a).
  * Zimmerman  _et al._ [2018] A. M. Zimmerman, J. I. A. Li, M. D. Nguyen, and W. P. Halperin, Orbital-flop transition of angular momentum in a topological superfluid, Phys. Rev. Lett. 121, 255303 (2018).
  * Pollanen  _et al._ [2008] J. Pollanen, K. R. Shirer, S. Blinstein, J. P. Davis, H. Choi, T. M. Lippman, W. P. Halperin, and L. B. Lurio, Globally anisotropic high porosity silica aerogels, Journal of Non-Crystalline Solids 354, https://doi.org/10.1016/j.jnoncrysol.2008.05.047 (2008).
  * Pollanen  _et al._ [2012] J. Pollanen, J. I. A. Li, C. A. Collett, W. J. Gannon, W. P. Halperin, and J. A. Sauls, New chiral phases of superfluid ${}^{3}\mathrm{He}$ stabilized by anisotropic silica aerogel, Nature Physics 8, 317 (2012).
  * Li  _et al._ [2014b] J. I. A. Li, A. M. Zimmerman, J. Pollanen, C. A. Collett, W. J. Gannon, and W. P. Halperin, Stability of superfluid ${}^{3}\mathrm{He}\text{$-$}{B}$ in compressed aerogel, Phys. Rev. Lett. 112, 115303 (2014b).
  * Dmitriev  _et al._ [2015] V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and A. N. Yudin, Polar phase of superfluid ${}^{3}\mathrm{He}$ in anisotropic aerogel, Phys. Rev. Lett. 115, 165304 (2015).
  * Autti  _et al._ [2016] S. Autti, V. V. Dmitriev, J. T. Mäkinen, A. A. Soldatov, G. E. Volovik, A. N. Yudin, V. V. Zavjalov, and V. B. Eltsov, Observation of half-quantum vortices in topological superfluid ${}^{3}\mathrm{He}$, Phys. Rev. Lett. 117, 255301 (2016).
  * Mineev [2018] V. P. Mineev, Influence of exchange scattering on superfluid ${}^{3}\mathrm{He}$ states in nematic aerogel, Phys. Rev. B 98, 014501 (2018).
  * Collin  _et al._ [2009] E. Collin, S. Triqueneaux, Y. M. Bunkov, and H. Godfrin, Fast-exchange model visualized with ${}^{3}\text{H}\text{e}$ confined in aerogel: A fermi liquid in contact with a ferromagnetic solid, Phys. Rev. B 80, 094422 (2009).
  * Sprague  _et al._ [1996] D. T. Sprague, T. M. Haard, J. B. Kycia, M. R. Rand, Y. Lee, P. J. Hamot, and W. P. Halperin, Effect of magnetic scattering on the ${}^{3}\mathrm{He}$ superfluid state in aerogel, Phys. Rev. Lett. 77, 4568 (1996).
  * Dmitriev  _et al._ [2018] V. V. Dmitriev, A. A. Soldatov, and A. N. Yudin, Effect of magnetic boundary conditions on superfluid ${}^{3}\mathrm{He}$ in nematic aerogel, Phys. Rev. Lett. 120, 075301 (2018).
  * Zimmerman  _et al._ [2020] A. M. Zimmerman, M. D. Nguyen, J. W. Scott, and W. P. Halperin, Effect of magnetic impurities on superfluid ${}^{3}\mathrm{He}$, Phys. Rev. Lett. 124, 025302 (2020).
  * Zimmerman  _et al._ [2019] A. M. Zimmerman, M. D. Nguyen, and W. P. Halperin, NMR frequency shifts and phase identification in superfluid ${}^{3}\mathrm{He}$, Journal of Low Temperature Physics 195, 358 (2019).
  * Ahonen  _et al._ [1975] A. Ahonen, T. Haavasoja, M. Haikala, M. Krusius, and M. Paalanen, Orbital textures and negative NMR shift in ${}^{3}\mathrm{He}$ A, Physics Letters A 55, 157 (1975).
  * Dmitriev  _et al._ [2020] V. V. Dmitriev, M. S. Kutuzov, A. Y. Mikheev, V. N. Morozov, A. A. Soldatov, and A. N. Yudin, Superfluid ${}^{3}\mathrm{He}$ in planar aerogel, Phys. Rev. B 102, 144507 (2020).
  * Balian and Werthamer [1963] R. Balian and N. R. Werthamer, Superconductivity with pairs in a relative $p$ wave, Phys. Rev. 131, 1553 (1963).
  * Leggett [1965] A. J. Leggett, Spin susceptibility of a superfluid fermi liquid, Phys. Rev. Lett. 14, 536 (1965).
  * Scholz [1981] H. N. Scholz,  _Magnetic properties of superfluid and normal Helium-3, and a search for superconductivity in gold_ , Ph.D. thesis, The Ohio State University (1981).
  * Fishman and Sauls [1986] R. S. Fishman and J. A. Sauls, Response functions and collective modes of superfluid ${}^{3}\mathrm{He}\text{$-$}{B}$ in strong magnetic fields, Phys. Rev. B 33, 6068 (1986).
  * Rand  _et al._ [1996] M. R. Rand, D. T. Sprague, T. M. Haard, J. B. Kycia, H. H. Hensley, Y. Lee, P. J. Hamot, D. M. Marks, W. P. Halperin, T. Mizusaki, and T. Ohmi, Nonlinear spin dynamics and magnetic field distortion of the superfluid ${}^{3}\mathrm{He}\text{-}{B}$ order parameter, Phys. Rev. Lett. 77, 1314 (1996).
  * Levitin  _et al._ [2013] L. V. Levitin, R. G. Bennett, E. V. Surovtsev, J. M. Parpia, B. Cowan, A. J. Casey, and J. Saunders, Surface-induced order parameter distortion in superfluid ${}^{3}\mathrm{He}\mathrm{\text{$-$}}{B}$ measured by nonlinear NMR, Phys. Rev. Lett. 111, 235304 (2013).
  * Mineev and Krotkov [2001] V. P. Mineev and P. L. Krotkov, Spin susceptibility of the superfluid ${}^{3}\mathrm{He}-\mathrm{B}$ in aerogel, Phys. Rev. B 65, 024501 (2001).
  * Pollanen  _et al._ [2011] J. Pollanen, J. I. A. Li, C. A. Collett, W. J. Gannon, and W. P. Halperin, Identification of superfluid phases of ${}^{3}\mathrm{He}$ in uniformly isotropic $98.2\%$ aerogel, Phys. Rev. Lett. 107, 195301 (2011).
  * Mizushima [2012] T. Mizushima, Superfluid ${}^{3}\mathrm{He}$ in a restricted geometry with a perpendicular magnetic field, Phys. Rev. B 86, 094518 (2012).
  * Mizushima  _et al._ [2012] T. Mizushima, M. Sato, and K. Machida, Symmetry protected topological order and spin susceptibility in superfluid ${}^{3}\mathrm{He}\text{$-$}{B}$, Phys. Rev. Lett. 109, 165301 (2012).
  * Greywall [1986] D. S. Greywall, ${}^{3}\mathrm{He}$ specific heat and thermometry at millikelvin temperatures, Phys. Rev. B 33, 7520 (1986).
  * Sauls and Sharma [2003] J. A. Sauls and P. Sharma, Impurity effects on the ${A}_{1}-{A}_{2}$ splitting of superfluid ${}^{3}\mathrm{He}$ in aerogel, Phys. Rev. B 68, 224502 (2003).
  * Aoyama and Ikeda [2006] K. Aoyama and R. Ikeda, Pairing states of superfluid ${}^{3}\mathrm{He}$ in uniaxially anisotropic aerogel, Phys. Rev. B 73, 060504(R) (2006).
  * Tang  _et al._ [1991] Y. H. Tang, I. Hahn, H. M. Bozler, and C. M. Gould, Magnetic suppression of the ${B}$ phase of superfluid ${}^{3}\mathrm{He}$, Phys. Rev. Lett. 67, 1775 (1991).
  * Gervais  _et al._ [2002] G. Gervais, K. Yawata, N. Mulders, and W. P. Halperin, Phase diagram of the superfluid phases of ${}^{3}\mathrm{He}$ in 98% aerogel, Phys. Rev. B 66, 054528 (2002).
  * Bhupathi  _et al._ [2010] P. Bhupathi, B. H. Moon, M. Gonzalez, and Y. Lee, Transverse acoustic spectroscopy of superfluid ${}^{3}\mathrm{He}$ in compressed aerogel, Journal of Low Temperature Physics 158, 176 (2010).
  * Li  _et al._ [2015] J. I. A. Li, A. M. Zimmerman, J. Pollanen, C. A. Collett, and W. P. Halperin, Anisotropic phases of superfluid ${}^{3}\mathrm{He}$ in compressed aerogel, Phys. Rev. Lett. 114, 105302 (2015).
  * Hasegawa [1983] Y. Hasegawa, Effect of the Gap Distortion on the Motion of Magnetization of 3He-B, Progress of Theoretical Physics 70, 1141 (1983).
  * Li  _et al._ [2013] J. I. A. Li, J. Pollanen, A. M. Zimmerman, C. A. Collett, W. J. Gannon, and W. P. Halperin, The superfluid glass phase of ${}^{3}\mathrm{He}\text{$-$}{A}$, Nature Physics 9, 775 (2013).


# Nucleon, Delta and Omega excited state spectra at three pion mass values

John Bulava
NIC, DESY, Platanenallee 6, D-15738, Zeuthen, Germany
Email<EMAIL_ADDRESS>Robert G. Edwards, Bálint Joó, David G. Richards
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Eric Engelson
In-Depth Engineering Corp., 11350 Random Hills Road, Fairfax, VA 22030, USA
Email<EMAIL_ADDRESS>Huey-Wen Lin
Department of Physics, University of Washington, Seattle, WA 98195, USA
Email<EMAIL_ADDRESS>Colin Morningstar
Department of Physics, Carnegie Mellon University,Pittsburgh, PA 15213, USA
Email<EMAIL_ADDRESS>
Department of Physics, University of Maryland, College Park, MD 20742, USA
E-mail

###### Abstract:

The energies of the excited states of the Nucleon, $\Delta$ and $\Omega$ are
computed in lattice QCD, using two light quarks and one strange quark on
anisotropic lattices. The calculations are performed at three values of the
pion mass: $m_{\pi}$ = 392(4), 438(3) and 521(3) MeV. We employ the
variational method with a basis of about ten interpolating operators enabling
six energies to be distinguished clearly in each irreducible representation of
the octahedral group. We compare our calculations of nucleon excited states
with the low-lying experimental spectrum. There is reasonable agreement for
the pattern of states.

##  1 Introduction

The goal of determining the spectrum of hadron masses from lattice QCD is
addressed in this work by calculations of the excited state spectrum of the
nucleon, $\Delta$ and $\Omega$ using anisotropic lattices.[1] Earlier excited-
baryon analyses were based on quenched QCD [2] and two-light-flavor
($N_{f}=2$) QCD [3]. In this work we use ensembles of gauge configurations
developed in Ref. [4] for $N_{f}=2+1$ QCD with two dynamical light quarks and
one strange quark. The lattices are $16^{3}\times 128$ with spatial and
temporal lattice spacings $a_{s}$= 0.122 fm and $a_{t}$ = 0.035 fm.

For families of particles with given isospin and strangeness, spectra are
calculated in the six double-valued irreducible representations (irreps) of
the octahedral group. There are three irreps for even-parity that are labeled
with a $g$ subscript (gerade) and three for odd-parity that are labeled with a
$u$ subscript (ungerade). They are: $G_{1g},H_{g},G_{2g},G_{1u},H_{u}$ and
$G_{2u}$.

Sets of seven to eleven three-quark operators are used in each irrep and the
variational method [5, 6] is used to extract energies of six states. Most
operators incorporate gauge-covariant displacements of the quarks relative to
one another in order to obtain nontrivial shapes. [7] The recently developed
“distillation” method [8] is used for quark smearing. We start with a large
set of operators in each irrep and then “prune” them to sets of about 10 that
have the lowest condition numbers. That yields sets of approximately linearly-
independent operators that are suitable for calculations based on
diagonalizing a matrix of correlation functions.

##  2 Results

A detailed presentation of all of our results is given in Ref. [1]. Here we
present selected results for the nucleon and $\Delta$ excited states.

Plots of the nucleon effective energies, calculated as

$E_{\rm
eff}(t)=\frac{1}{2}ln\left(\frac{\widetilde{\lambda}(t-1)}{\widetilde{\lambda}(t+1)}\right),$
(1)

where $\widetilde{\lambda}(t)$ is an eigenvalue of the generalized eigenvalue
problem, are shown in Figure 1 for the $G_{1g}$ and $G_{1u}$ irreps. These
plots show the values of $E_{\rm eff}$ obtained from Eq. (1) as vertical bars
and $E_{\rm eff}$ calculated using the fit function,

$\lambda_{fit}(t)=(1-A)e^{-E(t-t_{0})}+Ae^{-E^{\prime}(t-t_{0})},$ (2)

in place of $\widetilde{\lambda}(t)$ in Eq. (1) as curved dashed lines.
Comparison of the curved dashed lines with the bars from the lattice ensembles
shows the usefulness of two-exponential fits. The term
$Ae^{-E^{\prime}(t-t_{0})}$ models the contributions of higher energy states
at early times allowing the exponential term $(1-A)e^{-E(t-t_{0})}$ to be
determined over a larger fit window $(t_{i},t_{f})$ than would be possible
using a single exponential. Fit energy $E$ and uncertainty of the fit energy,
$\sigma$, are shown by dashed horizontal lines at $E+\sigma$ and $E-\sigma$
extending over the fit window. Note that the statistics allow credible
determinations of six energy levels in each irrep. This provides evidence that
quark smearing based on “distillation” is effective with regard to suppressing
high-frequency fluctuations in the gauge ensembles.

Figure 1:  Nucleon $G_{1g}$ effective energies are shown for the lowest states
in the upper six graphs. The effective energy increases from left to right
along the first row and continues to increase from left to right along the
second row. The lower six graphs show nucleon $G_{1u}$ effective energies
increasing in the same pattern. Calculations are for $m_{\pi}=392(4)$ MeV.
Vertical bars show the effective energy and the curved dashed line shows the
effective energy calculated from the fit function. Horizontal dashed lines
show the fit results for $E\pm\sigma$ and their extent shows the fitting
interval $(t_{i},t_{f})$.

The energies obtained from the $G_{1g}$ and $G_{1u}$ effective mass plots of
Fig. 1 are shown as boxes extending from $E-\sigma$ to $E+\sigma$ in Fig. 2.
We show nucleon energies that are obtained in the same manner as shown in Fig.
1 for all irreps of the octahedral group and three pion masses. Experimental
spectra are shown to the left of lattice energies for the spins and parities
that have subductions to the lattice irreps. Lattice and experimental spectra
are shown in Fig. 3 for the $\Delta$ family. See Ref. [1] for the $\Omega$
spectra.

In the nucleon spectra, there is good evidence for a spin $\frac{5}{2}^{-}$
state. We find nearly degenerate $H_{u}$ and $G_{2u}$ partner states, which is
the signature of spin $\frac{5}{2}^{-}$. However, other spins are difficult to
identify because there are many nearly-degenerate states, within
uncertainties. It is a near-term goal within the collaboration to address spin
identification by using operators that are subduced from continuum spins.

Our lattice spectra show scant evidence for multiparticle states even though
many energies lie above the relevant thresholds. This is probably because
single-hadron operators are used. There is an inference for a multiparticle
contribution in that we find four low-lying states in $H_{u}$ while there are
three low-lying experimental states that have subductions to $H_{u}$. There is
a threshold for a multiparticle state in the same energy range. Our lattice
results agree with the experimental pattern if one of the four low-lying
$H_{u}$ states is multiparticle. However, we cannot identify the multiparticle
states in the spectrum. It is a near-term goal within the collaboration to
incorporate multiparticle operators that couple directly to such states.

Some lattice states appear to be “squeezed” by the small lattice volume used.
They show up at higher energies than would be the case in a larger volume. The
$G_{2}$ states require partner states in other irreps, such as $H$, in order
to realize all the magnetic substates for a given spin. The partners should be
close to the same energy. However, in the $\Delta$ spectra of Fig. 3 we find
$G_{2}$ states at high energies without suitable partners being evident.
Possibly they have been “squeezed”. It is also a goal to perform calculations
of spectra at larger volumes.

Although we do not attempt to extrapolate energies to $m_{\pi}$ = 140 MeV, it
is evident from Figs. 2 and 3 that the lowest-energy states on the lattice
tend toward the energies of the physical resonances as the pion mass
decreases. Decreasing the pion mass is an obvious goal but we recognize that
it entails a more complex analysis for excited states that can decay.

Figure 2: Spectra for isospin $\frac{1}{2}$ (nucleon family) at three values
of $m_{\pi}$ in each irrep of the cubic group are compared with experimental
spectra. Columns labeled by $m_{\pi}$ = 392, 438 and 521 MeV show lattice
spectra. The $G_{1g}$ and $G_{1u}$ spectra in the $m_{\pi}$ = 392 MeV column
are obtained from the plots of Fig. 1. Boxes extend from $E-\sigma$ to
$E+\sigma$. Two, three and four-star experimental resonances are shown to the
left of lattice spectra in columns labeled by their $J^{P}$ values. Each
$J^{P}$ value listed has a subduction to the lattice irrep shown. Each box for
an experimental resonance has height equal to the full decay width and an
inner box (color aqua) showing the uncertainty in the Breit-Wigner energy.
Open boxes show the thresholds for multiparticle states.  Figure 3: Spectra
for isospin $\frac{3}{2}$ ($\Delta$ family) at three values of $m_{\pi}$ in
each irrep of the cubic group are compared with experimental spectra. Columns
labeled by $m_{\pi}$ = 392, 438 and 521 MeV show lattice spectra. Boxes extend
from $E-\sigma$ to $E+\sigma$. Two, three and four-star experimental
resonances are shown to the left of lattice spectra in columns labeled by
their $J^{P}$ values. Each $J^{P}$ value listed has a subduction to the
lattice irrep shown. Each box for an experimental resonance has height equal
to the full decay width and an inner box (color aqua) showing the uncertainty
in the Breit-Wigner energy.

##  3 Summary

This work represents a milestone in our long-term research program aimed at
determining the spectra of baryons in QCD. It provides the first spectrum for
N, $\Delta$ and $\Omega$ baryons based on $N_{f}=2+1$ QCD with high
statistics. A large number of baryon operators is used to calculate matrices
of correlation functions. They are analyzed using the variational method with
fixed eigenvectors. The analysis provides spectra at three pion masses:
$m_{\pi}$ = 392(4) MeV, 438(3) MeV and 521(3) MeV.

The lattice volume and pion masses used give considerably higher energies than
the experimental resonance energies. However, there is reasonable agreement of
the overall pattern of lattice and experimental states. One exception is that
almost all $\Delta$ states in the $G_{2}$ irrep are too high. That may be
caused by a volume that is too small for highly excited states.

###### Acknowledgments.

This work was done using the Chroma software suite [9] on clusters at
Jefferson Laboratory and the Fermi National Accelerator Laboratory using time
awarded under the USQCD Initiative. JB and CM acknowledge support from U.S.
National Science Foundation Award PHY-0653315. EE and SW acknowledge support
from U.S. Department of Energy contract DE-FG02-93ER-40762. HL acknowledges
support from U.S. Department of Energy contract DE-FG03-97ER4014. BJ, RE and
DR acknowledge support from U.S. Department of Energy contract DE-
AC05-060R23177, under which Jefferson Science Associates, LLC, manages and
operates Jefferson Laboratory. BJ and RE acknowledge support under U.S. Dept.
of Energy SciDAC contracts DE-FC02-06ER41440 and DE-FC02-06ER41449. The U.S.
Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce this manuscript for U.S. Government purposes.

## References

  * [1] J. M. Bulava et al., _Nucleon, $\Delta$ and $\Omega$ excited state spectra in $N_{f}$ = 2+1 lattice QCD_, _Phys. Rev. D_ 82, 014507 (2010) [arXiv:1004.5072].
  * [2] S. Basak et al., _Lattice QCD determination of patterns of excited baryon states_ , _Phys. Rev. D_ 76, 074504 (2007) [arXiv:0709.0008].
  * [3] J. M. Bulava et al. , _Excited State Nucleon Spectrum with Two Flavors of Dynamical Fermions_ , _Phys. Rev. D_ 79 034505 (2009), [arXiv:0901.0027].
  * [4] H-W. Lin et al., _First results from 2+1 dynamical quark flavors on an anisotropic lattice: light-hadron spectroscopy and setting the strange-quark mass_ , _Phys. Rev. D_ 79, 034502 (2009) [arXiv:0810.3588].
  * [5] C. Michael, _Adjoint sources in lattice gauge theory_ , _Nucl. Phys. B_ 259, 58 (1985).
  * [6] M. Lüscher and U. Wolff, _How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation_ , _Nucl. Phys. B_ 339, 222 (1990).
  * [7] S. Basak et al., _Group-theoretical construction of extended baryon operators in lattice QCD_ , _Phys. Rev. D_ 72 094506 (2005) [arXiv:hep-lat/0506029].
  * [8] M. Peardon et al., _Novel quark-field creation operator construction for hadronic physics in lattice QCD_ , _Phys. Rev. D_ 80, 054506 (2009) [arXiv:0905.2160].
  * [9] R. G. Edwards and B. Joó , _The Chroma Software System for Lattice QCD_ , _Nucl. Phys. B. Proc. Suppl._ 140, 832 (2005) [hep-lat/0409003].