# Viscous Flow in a 1D Spin-Polarized Fermi Gas: the Role of Integrability on
Viscosity

Jeff Maki  Department of Physics and HKU-UCAS Joint Institute for Theoretical
and Computational Physics at Hong Kong, The University of Hong Kong, Hong
Kong, China  Shizhong Zhang<EMAIL_ADDRESS>Department of Physics and HKU-
UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong,
The University of Hong Kong, Hong Kong, China

###### Abstract

The transport properties of one-dimensional Fermi gases at low-temperatures
are often described by the Luttinger liquid (LL) model. However, to study
dissipation one needs to examine interactions beyond the LL model. In this
work we provide a simple model which allows for a direct microscopic
calculation of the bulk viscosity, namely the one dimensional spin polarized
p-wave Fermi gas. We calculate the bulk viscosity in both the high- and low-
temperature limits. We find that the bulk viscosity is finite and consistent
with the requirement of scale symmetry, in spite of the inherent integrability
of the microscopic model. We argue how integrability does not forbid a finite
bulk viscosity, and compare our work to previous kinetic theory calculations.

### Introduction.—

Transport in integrable models has attracted much attention recently as they
exhibit behavior that differs fundamentally from their non-integrable
counterparts [1, 2, 3]. In integrable models, a large number of conservation
laws qualitatively changes the long-wavelength dynamics, rendering the
standard hydrodynamic approach inapplicable [4, 5, 6, 7, 8, 9, 10].

However, the long-wavelength dynamics of the system are still governed by a
few transport coefficients, such as the conductivity, thermal conductivity,
and bulk viscosity [11]. Although these quantities were first derived in the
hydrodynamic context, there has been a substantial amount of care in defining
these transport quantities using linear response and Kubo formulae [12, 13,
14, 15]. In this way the definition of the long-wavelength transport
coefficients provide meaningful and continuous results regardless of
integrability. As an example, it is well known that these definitions show
that the conductivity, thermal conductivity, and shear viscosity (for higher-
dimensional models) are infinite when the models are integrable, and become
finite with the inclusion of interactions that break the integrability [16,
17, 18, 19].

Of particular interest for 1D integrable models is the bulk viscosity. Unlike
the other transport coefficients, the bulk viscosity is special as it is
constrained to be zero for scale and Galilean invariant systems [20], due to a
hidden SO(2,1) symmetry [21, 22] that prevents entropy production for a large
class of dynamics [23, 24]. In addition, the microscopic definition of the
bulk viscosity naturally implies that the bulk viscosity spectral function
does not contain a “Drude” peak for a large class of systems [14, 15]. For
these reasons it is important to reconsider the role of the bulk viscosity in
the dynamics of integrable systems.

In this work, we explicitly evaluate the bulk viscosity for one integrable
model, the spin-polarized Fermi gas with zero-range p-wave interactions in 1D
[25]. This system has been the focus of several experimental and theoretical
efforts [26, 27, 28, 29, 30, 31, 32, 33] and provides an ideal platform for
studying the interplay between scale symmetry, integrability and the bulk
viscosity. We find that the bulk viscosity spectral function is indeed finite
for finite interaction strengths at both high- and low-temperatures. Our
results are consistent with scale symmetry, but inconsistent with previous
kinetic theory calculations [34, 35].

### The Bulk Viscosity for the Spin-Polarized Fermi Gas.—

In this letter, we consider the following Hamiltonian for a spin-polarized
Fermi gas :

$\displaystyle H$ $\displaystyle=\int dx\
\psi^{\dagger}(x)\left(-\frac{\partial_{x}^{2}}{2m}-\mu\right)\psi(x)$
$\displaystyle-\frac{g}{4m}\int
dx\psi^{\dagger}(x)\overleftrightarrow{\partial_{x}}\psi^{\dagger}(x)\psi(x)\overleftrightarrow{\partial_{x}}\psi(x)$
(1)

where $\psi(x)$ is the fermionic annihilation operator, $g$ is the odd-wave
coupling constant, $m$ is the atomic mass, and $\mu$ is the chemical
potential.
$\overleftrightarrow{\partial_{x}}=(\overleftarrow{\partial_{x}}-\overrightarrow{\partial_{x}})/2$
is the symmetrized derivative that acts both on the immediate left and right.
In principle, Eq. (1) requires regularization, which is discussed in the
supplementary materials (SM).

The (real part of the) bulk viscosity spectral function, $\zeta(\omega)$, can
be defined in terms of the response of a fluid to a time-dependent strain of
the systems [14]. As shown in the SM, this reduces to:

$\zeta(\omega)=\frac{Im\left[\chi_{\Pi\Pi}(\omega)-\chi_{\Pi\Pi}(0)\right]}{\omega}$
(2)

where $\chi_{\Pi\Pi}(\omega)$ is the retarded correlation function for the
trace of the stress-tensor in frequency space:

$\chi_{\Pi\Pi}(\omega)=\int_{0}^{\infty}e^{i\left(\omega+i\delta\right)t}i\langle[\Pi(t),\Pi(0)]\rangle$
(3)

The specific form of the trace of the stress-tensor, $\Pi(t)$, is defined
below. This definition does not invoke the constitutive relations or any other
hydrodynamic assumptions, only the conservation of momentum [14].

First let us consider the bulk viscosity at low temperatures. In this regime
the dominant low-energy degrees of freedom are those near the two Fermi
points, $\pm k_{F}$, where $k_{F}$ is the Fermi wavenumber which is related to
the chemical potential: $\mu=k_{F}^{2}/2m$. There the dynamics is described by
the Luttinger liquid (LL) model [36, 37, 38]. To obtain the LL model
associated with Eq. (1), we employ the standard procedure of expanding the
fermionic operators around the Fermi surface and obtaining the bosonized
Hamiltonian $H=H_{LL}+V_{-}+V_{+}$, including terms beyond the quadratic order
[38]:

$\displaystyle H_{LL}$ $\displaystyle=\frac{v}{2}\int
dx\left[(\partial_{x}\varphi_{L}(x))^{2}+(\partial_{x}\varphi_{R}(x))^{2}\right]$
(4) $\displaystyle V_{-}$
$\displaystyle=\frac{\sqrt{2\pi}}{6}\tilde{\eta}_{-}\int
dx\left[(\partial_{x}\varphi_{L})^{3}(x)-(\partial_{x}\varphi_{R})^{3}(x)\right]$
(5) $\displaystyle V_{+}$
$\displaystyle=\frac{\sqrt{2\pi}}{6}\tilde{\eta}_{+}\int
dx\left[(\partial_{x}\varphi_{L})^{2}(x)\partial_{x}\varphi_{R}(x)-L\leftrightarrow
R\right]$ (6)

where $H_{LL}$ is the standard LL Hamiltonian.

In Eqs. (4-6), $v$ is the velocity of the chiral bosonic excitations,
$\phi_{r}(x)$ with $r=-\bar{r}=\pm 1$ for right and left movers respectively.
The interactions $V_{-}$ and $V_{+}$ are the leading terms beyond the LL model
allowed by parity: $\varphi_{r}(x)\to\varphi_{\bar{r}}(-x)$. These
interactions couple like chiral modes, $V_{-}$, and opposite chiral modes,
$V_{+}$ [39]. We label such interactions as intra-band and inter-band
scattering, respectively. The derivation of Eqs. (4-6), as well as the exact
expressions for $v$, $\tilde{\eta}_{-}$, $\tilde{\eta}_{+}$, and the LL
parameter, $\mathcal{K}$, are given in the SM. We have also ignored two
further interactions, $V_{2}$ and $V_{2}^{\prime}$, as they are more
irrelevant than $V_{-}$ and $V_{+}$, as shown in the SM.

In order to calculate the bulk viscosity we note that the stress-tensor can be
defined through the current conservation law:
$\partial_{t}J(x,t)+\partial_{x}\Pi(x,t)=0$, where $J(x,t)$ is the particle
current, and $\Pi(x,t)$ is the local stress-tensor. The trace of the stress-
tensor, $\Pi(t)$, for our LL model can be obtained analytically:

$\displaystyle\Pi(t)$
$\displaystyle=\sqrt{\frac{\mathcal{K}}{2\pi}}\frac{4\sqrt{2\pi}}{6}(3\tilde{\eta}_{-}+\tilde{\eta}_{+})H+2v\sqrt{\frac{\mathcal{K}}{2\pi}}\frac{\sqrt{2\pi}\tilde{\eta}_{+}}{6}$
$\displaystyle\times\int
dx\sum_{r}\left[\partial_{x}\phi_{r}(x,t)\partial_{x}\phi_{\bar{r}}(x,t)-\left(\partial_{x}\phi_{r}(x,t)\right)^{2}\right]$
(7)

From Eq. (7), one can see that the intra- and inter-band couplings enter
$\Pi(t)$ differently. The intra-band coupling only modifies the first term
proportional to the total Hamiltonian [40]. Hence this term will not produce a
finite bulk viscosity. This can be also seen by examining the density-density
correlation function, as shown in Ref. [39]. There one can evaluate the
density-density correlation function perturbatively with respect to $V_{-}$.
The result exactly reproduces the non-interacting result but with a
renormalized Fermi velocity, $v$, and a renormalized mass,
$1/\tilde{\eta}_{-}$. Since the non-interacting gas has a vanishing bulk
viscosity, the bulk viscosity for a Luttinger liquid with only band curvature
corrections is zero. This result is intuitive since in the non-interacting
limit $\tilde{\eta}_{-}=1/m$, while $\tilde{\eta}_{+}=-3k_{F}\ell/(\pi m)$
[40].

The true non-trivial interaction effects are contained in $V_{+}$, the inter-
band scattering. This interaction introduces a new term to $\Pi(t)$. When
evaluating Eq. (2), only this term contributes to the bulk viscosity as the
commutator between $H$ and any operator vanishes in thermal equilibrium. One
can evaluate the bulk viscosity from Eq. (7). The details are shown in the SM,
here we state the result:

$\displaystyle\zeta(\omega)$
$\displaystyle=\frac{\mathcal{K}\tilde{\eta}_{+}^{2}}{72v}\frac{\omega}{\tanh(\beta\omega/4)}$
(8)

where $\beta=1/T$ is the inverse temperature. This is one of the main results
of the Letter.

Eq. (8) is valid to leading order in perturbation theory with respect to
$V_{+}$. In terms of the scattering volume $\ell$, the zero-frequency limit of
the bulk viscosity at finite temperatures is: $\zeta(\omega\to
0)=1/(2\pi)^{2}mk_{F}\ell^{2}T$. On the other hand, at strictly zero
temperature, the bulk viscosity depends linearly on the frequency: $\lim_{T\to
0}\zeta(\omega)=(mk_{F}/8\pi^{2})\ell^{2}\omega$, which is consistent with the
vanishing of the density of states near the Fermi surface,
$\rho(\omega)\propto\omega^{\frac{1}{2}(\mathcal{K}+1/\mathcal{K})-1}$ [38],
i.e. there is a lack of flow.

Moving towards the strong coupling limit, $V_{-}$ and $V_{+}$ are still the
only allowed leading irrelevant interactions consistent with parity
constraint. We also expect that both the LL parameter and the speed of sound
are well behaved around resonance, namely they can be expanded in powers of
$(n\ell)^{-1}$, where $n$ is the density, i.e.
$\mathcal{K}=\bar{\mathcal{K}}\left(n^{2}/mT\right)$, $v=n/m\
\bar{v}\left(n^{2}/mT\right)$ and
$\tilde{\eta}_{+}=(n\ell)^{-1}\bar{\eta}_{+}\left(n^{2}/mT\right)$. We can
combine these basic scaling relations to state that bulk viscosity must be of
the form:

$\zeta(\omega\to
0)=\frac{1}{n\ell^{2}}\bar{\zeta}\left(\frac{n^{2}}{mT}\right)$ (9)

The behaviour of these scaling functions can in principle be obtained from
Bethe Ansatz calculations [41].

At high-temperatures, $T\gg k_{F}^{2}/2m$, one can also obtain explicit
expressions for the bulk viscosity using the virial expansion in both weak and
strong coupling limits. In both limits, Eq. (2) reduces to the evaluation of
the contact-contact correlation function just as in the 3D case [42, 43, 44,
45]. In the weak coupling limit, one finds

$\zeta(\omega\to
0)\approx\frac{2}{\pi^{5/2}}\left(k_{F}\ell\right)^{2}T^{1/2},$ (10)

Combining Eq. (10) with the result from low-temperature LL calculation for
weak coupling, Eq. (8), we see that the zero frequency bulk viscosity is
proportional to $\ell^{2}$ and vanishes in the limit $\ell\to 0$, consistent
with requirement of scale symmetry. In addition, $\zeta(\omega\to 0)$ exhibits
a smooth temperature dependence since there is no finite temperature phase
transition in 1D.

Near resonance when $k_{F}\ell\gg 1$, the high temperature bulk viscosity is
given by

$\zeta(\omega\to
0)\propto\frac{k_{F}^{2}}{\ell^{2}T^{3/2}}\log\left(\frac{4}{\ell^{2}\omega
e^{-\gamma_{E}}}\right).$ (11)

Eq. (11) is consistent with the general scaling form presented in Eq. (9), up
to a logarithmic factor. In this limit $\bar{\zeta}(x)\propto x^{3/2}$.

Figure 1: Bulk viscosity for a weakly interacting spin-polarized 1D Fermi gas
with odd-wave interactions as a function of density, $n$.
$\lambda_{th}=\sqrt{2\pi/mT}$ is the thermal de Broglie wavelength. The bulk
viscosity at low-temperatures is evaluated via the LL model, Eq. (8), while
the high-temperature result follows the virial expansion, Eq. (10). In both
limits the bulk viscosity is proportional to $\ell^{2}$ as required by
conformal symmetry. Our results in the high- and low-temperature limits
smoothly connect with another, consistent with the lack of a superfluid
transition in 1D.

### Comparison to Traditional Kinetic Theory.—

Eqs. (8-11) were calculated using the microscopic definition in Eq. (2). As
one can see these results are consistent with conformal symmetry and provide
sensible results at finite and zero temperature. However, our results disagree
with previous calculations using kinetic theory [34, 35].

In the kinetic theory approach, the bulk viscosity is calculated by examining
the effects of three-body interactions in a 1D spin polarized Fermi gas
subjected to a slowly varying velocity gradient. According to standard
arguments, the two-body scattering only leads to forward and backward
scattering and can’t relax the momentum, hence three-body processes are needed
[46, 47]. This argument leads to a bulk viscosity $\zeta\propto
k_{F}^{5}\ell^{-2}T^{-3}$ at low-temperatures. The kinetic theory then
predicts a divergent bulk viscosity in the weakly interacting limit at low-
temperatures which is inconsistent with the requirement of conformal symmetry
[20]. A similar analysis at high-temperatures for zero-range p-wave
interactions gives a vanishing bulk viscosity for arbitrary scattering volume
because the real part of the quasiparticle self-energy leads only to the
renormalization of the chemical potential and the effective mass.

It is then important to ask why the previous kinetic theory calculations are
inadequate for evaluating the bulk viscosity. Recently studies have shown that
the kinetic approach is deficient in calculating the bulk viscosity, while it
is accurate at describing the other transport quantities [15]. The kinetic
theory approach misses an important piece of the bulk-viscosity due to the
correlation of pairs, as indicated explicitly in Eq. (2). Such correlation
effects do not occur at the single-particle level, and is absent in the
standard kinetic theory calculation [48].

### A Modified Kinetic Theory Approach.—

From our response function calculations, we know the bulk viscosity arises
solely from the two-body interactions, $V_{+}$, which has been previously
ignored [34]. In principle, one may ask whether a kinetic theory calculation
including $V_{+}$ can produce qualitatively correct results. Consider the
following Hamiltonian of fermionic quasiparticles from the LL model:

$\displaystyle H_{0}$ $\displaystyle=\sum_{r=\pm 1}\int
dx:\psi^{\dagger}_{r}(x)\left(-irv\partial_{x}\right)\psi_{r}(x)$
$\displaystyle+\frac{\tilde{\eta}_{-}}{2}\partial_{x}\psi^{\dagger}(x)\partial_{x}\psi(x)\colon$
(12) $\displaystyle V_{+}$ $\displaystyle=\tilde{\eta}_{+}\int
dx\sum_{r}\colon
ir\psi^{\dagger}_{r}(x)\overleftrightarrow{\partial_{x}}\psi_{r}(x)\psi_{\bar{r}}^{\dagger}(x)\psi_{\bar{r}}(x)\colon$
(13)

where $\colon\ \colon$ represents normal ordering. In defining Eq. (12) we
note the first line comes from the LL Hamiltonian and defines the renormalized
velocity, $v$. The second term is due to $V_{-}$ and describes the quadratic
dispersion of the quasiparticles with an effective mass $1/\tilde{\eta_{-}}$.
We have again neglected any term that is more irrelevant than $V_{+}$.

We then follow the calculation in Ref. [34] and subject the fermionic quasi-
particles to a slowly varying velocity field $u(x)$. The presence of the
external velocity gradient will induce non-equilibrium dynamics and
dissipation. For small velocity gradients, the power dissipated can be written
as: $W=\zeta L\left(\partial_{x}u\right)^{2}$. Since the power dissipated is
related to entropy production, $W=T\dot{S}$, one can relate the bulk viscosity
in terms of the quasiparticle distribution via:

$\zeta(\partial_{x}u)^{2}=\frac{T}{L}\sum_{p}\frac{(\dot{n}_{p}^{0})^{2}}{n_{p}^{0}(1-n_{p}^{0})}\tau$
(14)

where $n_{p}^{0}(t)$ is the equilibrium quasiparticle distribution function
$n_{p}^{0}=\left[\exp(\beta(\epsilon_{p}-\mu))+1\right]^{-1}$, $\epsilon_{p}$
is the fermionic quasiparticle dispersion, $\Delta\mu=\mu(T)-\mu(0)$ is the
residual chemical potential at finite $T$, and $\dot{n}_{p}^{0}(t)$ is
formally of order $\partial_{x}u$. In defining Eq. (14) we have used the
relaxation time approximation, and $\tau$ is the scattering rate of the
quasiparticle due to $V_{+}$.

Now, the crucial question is to determine the value of $\tau$. Traditionally,
one associates $\tau$ with three-body interactions [34]. This leads to results
inconsistent with our microscopic calculation. To understand how kinetic
theory could produce a finite bulk viscosity, we instead associate $\tau$ with
the two-particle scattering time produced by $V_{+}$. As shown in the SM, the
scattering rate is:

$\frac{1}{\tau}\approx A\frac{\tilde{\eta}_{+}^{2}T^{3}}{v^{4}}$ (15)

where $A$ is some constant and we have ignored the $p$ dependence of the
inter-band scattering rate.

Given $\tau$, we turn next to the time derivative of the quasiparticle
distribution function, $\dot{n}_{p}^{0}(t)$. This can be determined by noting
that the quasiparticle scattering conserves number and energy. The result is:

$\dot{n}_{p}^{0}=n_{p}^{0}(1-n_{p}^{0})\frac{3v^{2}p^{2}-\pi^{2}T^{2}}{6\tilde{\eta}_{-}^{-1}v^{2}T}\left(n\partial_{n}\ln(\nu)\right)\partial_{x}u$
(16)

and $\nu=\pi n\tilde{\eta}_{-}/v$. Substituting Eqs. (15-16) into Eq. (14) and
expanding to leading order in $k_{F}\ell$, we obtain the final expression for
the bulk viscosity in the weak coupling limit:

$\zeta\approx Cmk_{F}\ell^{2}T+O(T^{2})$ (17)

for some constant $C$ constant. Eq. (17) is consistent with the LL result, Eq.
(8).

This result is encouraging because it is consistent with scale symmetry.
However, we have assumed that the two-body scattering can lead to a relaxation
of the quasiparticle distribution function. It is still unknown how these
results could arise from a proper kinetic theory calculation, especially since
the bulk viscosity is determined by multi-particle correlations. On the other
hand, it is also important to understand within the response function
formalism, how three-body interactions can affect the bulk viscosity.

### Discussions.—

We have shown in this Letter that there is indeed a bulk viscosity for a
Galilean invariant integrable model, the spin polarized Fermi gas in 1D. This
is interesting as naively one associates integrability with divergent
transport physics. A natural question is to ask whether the dual of this
model, the Lieb Liniger model of interacting bosons [49, 25, 50, 51], also has
a finite bulk viscosity.

From the virial expansion we can establish explicit duality relations between
the fermionic bulk viscosity $\zeta_{F}(\ell)$ and the bosonic counterpart
$\zeta_{B}(a)$ at high-temperatures (see SM):

$\zeta_{F}(\ell)=\zeta_{B}\left(\frac{1}{a}\right).$ (18)

For these two models, the duality is of the strong-to-weak variety, mapping
the strongly interacting bulk viscosity of one gas to the weakly interacting
limit of the other. We do not expect the temperature to affect the duality
relation. As a result, we can deduce that the bulk viscosity of the Lieb-
Liniger model near resonance and at low-temperatures is given by Eq. (8) with
the replacement of $\ell$ with $a^{-1}$ via Eq. (18). Similarly, the bulk
viscosity for the spin-polarized Fermi gas near resonance and at low-
temperatures can be determined by evaluating the bulk viscosity for a weakly
interacting Bose gas in 1D. This gives a bulk viscosity consistent with Eq.
(9) where $\bar{\zeta}(x)=x^{4}$, i.e. the bulk viscosity near resonance
diverges as $T^{-4}$ [52].

### Acknowledgements

– The authors would like to thank Xiwen Guan, Ian Affleck, Wade DeGottardi,
Manuel Valiente, and Tilman Enss for useful discussions. This work is
supported by HK GRF 17304820, 17304719 and CRF C6009-20G and C7012-21G.

### Note Added:

During the preparation of this manuscript, we learned of another work in
preparation on this subject [52]. There the authors calculate the bulk
viscosity spectral function for the spin-polarized Fermi gas and interacting
Bose gas. Their results are consistent with ours.

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##  I Supplementary Materials for Viscous Flow in a 1D Spin-Polarized Fermi
Gas: the Role of Integrability on Viscosity

The goal of this supplementary materials is to provide a detailed analysis on
the bulk viscosity of a zero-ranged p-wave Fermi gas confined to one spatial
dimension (1D). We begin by examining the two-body scattering properties of
this system.

##  II Two-Body Scattering of the Spin-Polarized Fermi Gas

The Hamiltonian for this system is given by:

$\displaystyle H$ $\displaystyle=\int dx\
\psi^{\dagger}(x)\left(-\frac{\partial_{x}^{2}}{2m}-\mu\right)\psi(x)$
$\displaystyle-\int dx\
\frac{g}{4m}\psi^{\dagger}(x)\overleftrightarrow{\partial_{x}}\psi^{\dagger}(x)\psi(x)\overleftrightarrow{\partial_{x}}\psi(x)$
(19)

where $\psi^{(\dagger)}(x)$ is the fermionic annihilation (creation) operator,
$m$ is the atomic mass, $\mu$ is the chemical potential,
$\overleftrightarrow{\partial_{x}}=(\overleftarrow{\partial_{x}}-\overrightarrow{\partial_{x}})/2$
is the symmetrized derivative that acts both on the immediate left and right,
and $g$ is the bare p-wave coupling constant that depends on the ultraviolet
cut-off for the theory, $\Lambda$. In this note we have set $\hbar$ to unity.

In order to regularize the theory, we solve for the two-body T-matrix.
Consider two fermions with momenta $Q/2\pm k$ and total energy
$E=Q^{2}/4m+k^{2}/m$ scattering into a state of two fermions with momenta
$Q/2\pm k^{\prime}$. The corresponding matrix element for the two-body
T-matrix is:

$\displaystyle\langle\frac{Q}{2}\pm k|$ $\displaystyle T|\frac{Q}{2}\pm
k^{\prime}\rangle=kk^{\prime}\tilde{T}(E)$
$\displaystyle\frac{1}{\tilde{T}(E)}$
$\displaystyle=\frac{1}{g}-\frac{1}{2}\int_{-\Lambda}^{\Lambda}\frac{dl}{2\pi}\frac{l^{2}}{E-\frac{Q^{2}}{4m}-\frac{l^{2}}{m}+i\delta}$
$\displaystyle=\frac{1}{g}+\frac{m\Lambda}{2\pi}-\frac{m}{4}\sqrt{-E+Q^{2}/4-i\delta}$
(20)

Given Eq. (20) one can show that the scattered relative wavefunction is given
by:

$\displaystyle\psi(x)$ $\displaystyle=e^{ikx}+f_{k}\frac{x}{|x|}e^{ik|x|}$
$\displaystyle f_{k}$ $\displaystyle=-ik\frac{m\tilde{T}(k^{2}/m)}{4}$ (21)

where $f_{k}$ is the scattering amplitude. According to the effective range
expansion, the scattering amplitude can be written as:

$f_{k}=ik\left[-\frac{1}{\ell}-ik+O(k^{2})\right]^{-1}$ (22)

where $\ell$ is the one-dimensional p-wave scattering volume. Comparing Eq.
(20) to Eq. (22), one can identify:

$\frac{m}{4\ell}=\frac{1}{g}+\frac{m\Lambda}{2\pi}.$ (23)

For most of this work we will focus on the weakly interacting gas. In this
limit, $\ell k\ll 1$, the T-matrix can be approximated by the bare coupling
constant $g$

$g\approx\frac{4\ell}{m}.$ (24)

##  III Definition of the Bulk Viscosity

There are numerous studies discussing the definition of the bulk viscosity
spectral function $\zeta(\omega)$ using linear response. Although several
studies obtain microscopic expressions for the bulk viscosity by invoking
hydrodynamics, these expressions are actually quite general and are
independent of the applicability of hydrodynamics. Here we report an
equivalent definition of the bulk viscosity [14, 15]:

$\displaystyle\zeta(\omega)$
$\displaystyle=\frac{\chi_{\Pi,\Pi}(\omega)}{i\omega}$
$\displaystyle-\frac{1}{i\omega}\left(\left\langle\left.\frac{d\Pi_{b}}{db}\right|_{b=0}\right\rangle-\left.\frac{d}{db}\langle\Pi\rangle_{b}\right|_{b=0}\right)$
(25)

In Eq. (25), the first term is the retarded stress-tensor correlation
function:

$\chi_{\Pi,\Pi}(\omega)=\int_{-\infty}^{\infty}dte^{i\left(\omega+i\delta\right)t}i\theta(t)\langle\left[\Pi(t),\Pi(0)\right]\rangle$
(26)

where $\Pi$ is the stress-tensor defined below in Eq. (28). The second term is
the so called ”Drude peak” or contact term. For this term we define the
dilated stress-tensor, $\Pi_{b}$ as:

$\Pi_{b}=e^{iDb}\Pi e^{-iDb}$ (27)

where $D$ being the generator of scale transformations. The Drude peak in Eq.
(25) has a natural interpretation as the difference between how the stress-
tensor operator behaves under a scale transformation and how the expectation
value of the stress-tensor changes under a scale transformation. To make a
clear connection with previous formulations of the problem, we note that
$\langle\Pi\rangle=PL$, where $P$ is the pressure and $L$ is the volume.

In order to examine the Drude peak, it is best to invoke the original
microscopic model for the spin-polarized Fermi gas Eq. (19). The stress-tensor
for this model is defined as:

$\Pi=2H+\frac{1}{\ell}C_{\ell}$ (28)

where we also define the thermodynamic contact [33]:

$\displaystyle C_{\ell}$ $\displaystyle=-\frac{\partial H}{\partial\ell^{-1}}$
$\displaystyle=-\int
dx\frac{g^{2}}{4}\psi^{\dagger}(x)\overleftrightarrow{\partial_{x}}\psi^{\dagger}(x)\psi(x)\overleftrightarrow{\partial_{x}}\psi(x)$
(29)

Under a scale transformation the field operator transforms as:

$\psi_{b}(x)=e^{-b}\psi\left(xe^{-b}\right).$ (30)

From the transformation, the stress-tensor changes according to:

$\Pi_{b}=2e^{-2b}H(\ell e^{-b})+\frac{e^{-b}}{\ell}C_{\ell}(\ell e^{-b})$ (31)

Note that we have explicitly included the dependence of $\ell$ in the
Hamiltonian and the contact to emphasize how these operators change under a
scale transformation. It readily follows that:

$\displaystyle\left\langle\left.\frac{d\Pi_{b}}{db}\right|_{b=0}\right\rangle=-2PL-\frac{1}{\ell}\langle
C_{\ell}\rangle$ (32)

The second term in the Drude peak can be evaluated by noting:
$\langle\Pi\rangle=PL$. From dimensional analysis [15]:

$PL=\frac{1}{L^{2}}\bar{p}\left(\frac{L}{\ell},S,N\right)$ (33)

for some dimensionless function $\bar{p}$. Under a scale transformation the
length of the system changes as $L\to Le^{b}$, which corresponds to:

$\left.\frac{d}{db}\langle\Pi_{b}\rangle\right|_{b=0}=-2PL-\frac{1}{\ell}\langle
C_{\ell}\rangle-\frac{1}{\ell^{2}}\frac{\partial}{\partial\ell^{-1}}\langle
C_{\ell}\rangle$ (34)

The combination of Eqs. (32) and $(\ref{eq:p_scale})$ with Eq. (25) show that
the Drude peak is related to the change in the contact with respect to the
scattering volume. This quantity is naturally related to the stress-tensor
correlation function and linear response Theory. This is evident when one
considers how the contact changes under an infinitesimal time independent
change in the scattering volume, $\ell\to\ell+\delta\ell$:

$-\frac{\partial\langle C_{\ell}\rangle}{\partial\ell}\approx\chi_{\Pi\Pi}(0)$
(35)

This leads us to the final result that the bulk viscosity spectral function is
given by:

$\zeta(\omega)=\frac{\chi_{\Pi\Pi}(\omega)-\chi_{\Pi\Pi}(0)}{i\omega}$ (36)

##  IV Linearization and Bosonization of the Spin-Polarized Fermi Gas

In order to describe the low-temperature and low-energy properties of the
system, we follow the standard procedure of linearizing the fermionic
operators around the Fermi surface and bosonizing the result [38]. First we
write the fermionic operators as:

$\displaystyle\psi(x)$ $\displaystyle=\sum_{r=\pm 1}e^{irk_{F}x}\psi_{r}(x)$
$\displaystyle\psi_{r}(x)$
$\displaystyle=\frac{1}{\sqrt{L}}\sum_{k}\psi(rk_{F}+k)e^{ikx}$ (37)

where $k_{F}$ is the Fermi wavenumber and $\mu=k_{F}^{2}/2m$. In terms of
these chiral modes the Hamiltonian in Eq. (19) is written as:

$\displaystyle H_{0}$ $\displaystyle=\sum_{r}\int dx\
\psi_{r}^{\dagger}(x)\left[-irv_{F}\partial_{x}-\frac{\partial_{x}^{2}}{2m}\right]\psi_{r}(x)$
(38) $\displaystyle V_{0}$ $\displaystyle=\int dxgk_{F}^{2}n_{R}(x)n_{L}(x)$
(39) $\displaystyle V_{1}$ $\displaystyle=\sum_{r}\int dx\
irgk_{F}\left[\psi^{\dagger}_{r}(x)\overleftrightarrow{\partial_{x}}\psi_{r}(x)n_{\bar{r}}(x)\right]$
(40) $\displaystyle V_{2}$ $\displaystyle=\int dx\
g\psi_{R}^{\dagger}(x)\overleftrightarrow{\partial_{x}}\psi_{L}^{\dagger}(x)\psi_{L}(x)\overleftrightarrow{\partial_{x}}\psi_{R}(x)$
(41) $\displaystyle V_{2}^{\prime}$ $\displaystyle=\sum_{r}\int dx\
g\psi_{r}^{\dagger}(x)\overleftrightarrow{\partial_{x}}\psi_{r}^{\dagger}(x)\psi_{r}(x)\overleftrightarrow{\partial_{x}}\psi_{r}(x)$
(42)

In Eqs. (38-42) $n_{r}(x)=\psi_{r}^{\dagger}(x)\psi_{r}(x)$ is the density of
$r$-moving fermions. We have also neglected writing the implicit normal
ordering operator for clarity.

Secondly we write the chiral fermionic operators, $\psi_{r}(x)$ as:

$\displaystyle\psi_{r}(x)$
$\displaystyle=\frac{1}{\sqrt{2\pi\alpha}}e^{\sqrt{2\pi}i\phi_{r}(x)}$
$\displaystyle=\frac{1}{\sqrt{L}}e^{-\sqrt{2\pi}i\phi_{r}^{+}(x)}e^{-\sqrt{2\pi}i\phi_{r}^{-}(x)}$
(43)

where $\phi_{r}(x)$ is the chiral bosonic operator, $\alpha$ is a short
distance cut-off that we take to zero at the end of the calculation, and $L$
is the length of the system. We have also neglected the Klein factors for
simplicity. In the second equality we write the bosonic field in terms of
their annihilation, $\phi^{-}_{r}(x)$, and creation, $\phi^{+}_{r}(x)$ parts.
The commutation relation between the creation and annihilation operators is:

$\left[\phi_{r}^{-}(x),\phi_{r^{\prime}}^{+}(y)\right]=-\frac{1}{2\pi}\ln\left[\frac{-2\pi
ir}{L}(x-y+i\alpha)\right]\delta_{r,r^{\prime}}.$ (44)

From Eqs. (43-44), one can then write Eqs. (38-42) in terms of the bosonic
field $\phi_{r}(x)$ by the method of point-splitting. The result is:

$\displaystyle H$ $\displaystyle\approx H_{LL}+H_{bc}+V_{1}$ $\displaystyle
H_{LL}$ $\displaystyle=\int
dx\left\\{\frac{v_{F}}{2}\left[\left(\partial_{x}\phi_{R}\right)^{2}+\left(\partial_{x}\phi_{L}\right)^{2}\right]\right.$
$\displaystyle\left.-\frac{gk_{F}^{2}}{2\pi}\left(\partial_{x}\phi_{R}\right)\left(\partial_{x}\phi_{L}\right)\right\\}$
(45) $\displaystyle H_{bc}$ $\displaystyle=\frac{\sqrt{2\pi}}{6m}\int
dx\left[\left(\partial_{x}\phi_{L}\right)^{3}-\left(\partial_{x}\phi_{R}\right)^{3}\right]$
(46) $\displaystyle V_{1}$ $\displaystyle=\frac{gk_{F}}{2\sqrt{2\pi}}\int
dx\left[\left(\partial_{x}\phi_{R}\right)^{2}\left(\partial_{x}\phi_{L}\right)-L\leftrightarrow
R\right]$ (47)

As we will discuss below Eq. (45) gives the standard Luttinger Liquid model
that describes sound waves with linear dispersions. The contributions from
Eqs. (46-47) give the leading irrelevant interactions.

In defining $H$ we have neglected the contributions due to $V_{2}$ and
$V_{2}^{\prime}$. These terms are more irrelevant than $H_{bc}$ and $V_{1}$.
To facilitate this discussion consider the bosonized form of $V_{2}$:

$\displaystyle V_{2}$ $\displaystyle=-\frac{g}{24\pi^{2}}\int
dx\left[\partial_{x}^{3}\phi_{R}\partial_{x}\phi_{L}+\phi_{x}\phi_{R}\partial_{x}^{3}\phi_{L}\right]$
$\displaystyle+\frac{g}{4\pi}\int
dx\left(\partial_{x}^{2}\phi_{R}\right)\left(\partial_{x}^{2}\phi_{L}\right)$
$\displaystyle+\frac{g}{4\pi}\int
dx\left(\partial_{x}\phi_{R}\right)^{2}\left(\partial_{x}\phi_{L}\right)^{2}$
$\displaystyle-\frac{g}{6\pi}\int
dx\left[\left(\partial_{x}\phi_{R}\right)^{3}\partial_{x}\phi_{L}+\partial_{x}\phi_{R}\left(\partial_{x}\phi_{L}\right)^{3}\right]$
(48)

$V_{2}$ has two distinct effects. First, $V_{2}$ modifies the dispersion of
the sound waves. Equivalently one can say that the LL parameter and the
renormalized sound velocity, defined below, become momentum dependent. As we
will discuss later, this momentum dependence is not important to the
discussion of the bulk viscosity and can be neglected. Secondly $V_{2}$
generates interactions that are quartic in the bosonic fields. These
interactions are more irrelevant than the cubic interactions and can also be
ignored.

To bring Eqs. (45-47) to the standard LL form define:

$\displaystyle\phi_{L}$
$\displaystyle=\frac{\tilde{\theta}+\tilde{\phi}}{\sqrt{2\pi}}$
$\displaystyle\phi_{R}$
$\displaystyle=\frac{\tilde{\theta}-\tilde{\phi}}{\sqrt{2\pi}}$ (49)

In terms of these new fields one obtains:

$\displaystyle H_{LL}$ $\displaystyle=\int dx\
\frac{v}{2\pi}\left[\mathcal{K}\left(\partial_{x}\tilde{\theta}\right)^{2}+\frac{1}{\mathcal{K}}\left(\partial_{x}\tilde{\phi}\right)^{2}\right]$
$\displaystyle H_{bc}$ $\displaystyle=\int dx\ \frac{1}{6\pi
m}\left[3\left(\partial_{x}\tilde{\theta}\right)^{2}\left(\partial_{x}\tilde{\phi}\right)+\left(\partial_{x}\tilde{\phi}\right)^{3}\right]$
$\displaystyle V_{1}$ $\displaystyle=\int dx\
\frac{gk_{F}}{(2\pi)^{2}}\left[-\left(\partial_{x}\tilde{\theta}\right)^{2}\left(\partial_{x}\tilde{\phi}\right)+\left(\partial_{x}\tilde{\phi}\right)^{3}\right]$
(50)

where the renormalized $v$ and LL paramete, $\mathcal{K}$ are defined as:

$\displaystyle v$
$\displaystyle=v_{F}\sqrt{1-\left(\frac{mgk_{F}}{2\pi}\right)^{2}}$
$\displaystyle\mathcal{K}$
$\displaystyle=\sqrt{\frac{1-mgk_{F}/2\pi}{1+mgk_{F}/2\pi}}$ (51)

For our purposes it is more convenient to define renormalized chiral modes via
[39]:

$\displaystyle\varphi_{L}(x)$
$\displaystyle=\sqrt{\frac{1}{2\pi}}\left(\mathcal{K}\tilde{\theta}(x)+\frac{1}{\mathcal{K}}\tilde{\phi}(x)\right)$
(52) $\displaystyle\varphi_{R}(x)$
$\displaystyle=\sqrt{\frac{1}{2\pi}}\left(\mathcal{K}\tilde{\theta}(x)-\frac{1}{\mathcal{K}}\tilde{\phi}(x)\right)$
(53)

In terms of the dressed chiral modes, Eq. (52-53), the Hamiltonian becomes:

$\displaystyle H_{LL}$ $\displaystyle=\int
dx\frac{v}{2}\left[\left(\partial_{x}\varphi_{R}\right)^{2}+\left(\partial_{x}\varphi_{L}\right)^{2}\right]$
(54) $\displaystyle V_{-}$ $\displaystyle=\eta_{-}\int
dx\left((\partial_{x}\varphi_{L})^{3}-(\partial_{x}\varphi_{R})^{3}\right)$
(55) $\displaystyle V_{+}$ $\displaystyle=\eta_{+}\int
dx\left((\partial_{x}\varphi_{L})^{2}\partial_{x}\varphi_{R}-(\partial_{x}\varphi_{R})^{2}\partial_{x}\varphi_{L}\right)$
(56) $\displaystyle\eta_{-}$
$\displaystyle=\frac{\sqrt{2\pi}\tilde{\eta}_{-}}{6}=\sqrt{\frac{2\pi}{\mathcal{K}}}\frac{1}{24m}\left[3+\mathcal{K}^{2}+\frac{3mgk_{F}}{2\pi}\left(\mathcal{K}^{2}-1\right)\right]$
(57) $\displaystyle\eta_{+}$
$\displaystyle=\frac{\sqrt{2\pi}\tilde{\eta}_{+}}{6}=\sqrt{\frac{2\pi}{\mathcal{K}}}\frac{1}{8m}\left[(1-\mathcal{K}^{2})-\frac{mgk_{F}}{2\pi}\left(1+3\mathcal{K}^{2}\right)\right]$
(58)

In this representation the density fluctuations are given by:

$n(x)=-\sqrt{\frac{\mathcal{K}}{2\pi}}\sum_{r=\pm
1}r\partial_{x}\varphi_{r}(x)$ (59)

Similarly, we define the current and stress-tensor according to:

$\displaystyle 0$ $\displaystyle=\partial_{t}n(x,t)+\partial_{x}J(x,t)$
$\displaystyle 0$ $\displaystyle=\partial_{t}J(x,t)+\partial_{x}\Pi(x,t)$ (60)

From these definitions, and the Heisenberg equations of motion, the current
$J(x)$ and stress-tensor $\Pi(x)$ operators are found to be:

$\displaystyle J(x)$
$\displaystyle=\sum_{r}-\sqrt{\frac{\mathcal{K}}{2\pi}}\left[v\partial_{x}\varphi_{r}(x)\right.$
$\displaystyle\left.-\frac{\sqrt{2\pi}r}{6}\left(3\tilde{\eta}_{-}+\tilde{\eta}_{+}\right)\left(\partial_{x}\varphi_{r}(x)\right)^{2}\right]$
(61) $\displaystyle\Pi(x)$
$\displaystyle=\sum_{r}-\sqrt{\frac{\mathcal{K}}{2\pi}}\left[v^{2}r\partial_{x}\varphi_{r}(x)\right.$
$\displaystyle\left.-2v\frac{\sqrt{2\pi}\tilde{\eta}_{+}}{6}\left(\partial_{x}\varphi_{r}(x)\partial_{x}\varphi_{\bar{r}}(x)-\left(\partial_{x}\varphi_{r}(x)\right)^{2}\right)\right.$
$\displaystyle\left.-\frac{4\sqrt{2\pi}}{6}(3\tilde{\eta}_{-}+\tilde{\eta}_{+})\mathcal{H}(x)\right]$
(62)

The current form of $\Pi(x)$ can not be transparently connected to Eq. (28).
That said, when one performs the trace in the non-interacting limit the
stress-tensor satisfies:

$\lim_{\ell\to 0}\Pi=\lim_{\ell\to 0}\int dx\Pi(x)=\frac{2H}{m}.$ (63)

This is the relation expected by conformal symmetry, up to a factor of $m$.

##  V Calculation of the Bulk Viscosity via Stress-Tensor

Next we calculate the bulk viscosity Eq. (36) in both the low- and high-
temperature limits. At low-temperatures, the physics is described by the LL
model presented in Eqs. (54-58). In order to calculate the bulk viscosity, we
consider the trace of the stress-tensor:

$\displaystyle\Pi(t)=\sqrt{\frac{\mathcal{K}}{2\pi}}$
$\displaystyle\left[\frac{4\sqrt{2\pi}}{6}\left(3\tilde{\eta}_{-}+\tilde{\eta}_{+}\right)H\right.$
$\displaystyle\left.-\frac{2v\sqrt{2\pi}\tilde{\eta}_{+}}{6}\sum_{r}\int
dx\left(\partial_{x}\varphi_{r}(x,t)\partial_{x}\varphi_{\bar{r}}(x,t)\right.\right.$
$\displaystyle\left.\left.-\left(\partial_{x}\varphi_{r}(x,t)\right)^{2}\right)\right]$
(64)

Since the commutator between the Hamiltonian and any operator vanishes in
thermal equilibrium, one only needs to worry about the second line of Eq. (64)
in calculating Eq. (36). To explicitly calculate Eq. (36) we use finite
temperature field theory. In doing so we replace the retarded correlation
function in Eq. (26) with the imaginary time-ordered Green’s function and
evaluate it to leading order in perturbation theory. The result is:

$\displaystyle\chi_{\Pi,\Pi}(q,i\omega_{n})$
$\displaystyle=\mathcal{K}\frac{v^{2}\tilde{\eta}_{+}^{2}}{9}\sum_{r}\frac{1}{\beta}\sum_{i\eta_{n}}\int_{-\infty}^{\infty}\frac{dp}{2\pi}$
$\displaystyle\left[D_{r}(p,i\eta_{n})D_{\bar{r}}(-p,-i\eta_{n}+i\omega_{n})\right.$
$\displaystyle\left.+3D_{r}(p,i\eta_{n})D_{r}(-p,-i\eta_{n}+i\omega_{n})\right]$
(65)

where the bare bosonic propagator, $D_{r}(q,i\omega_{n})$ is given by:

$D_{r}(q,i\omega_{n})=\frac{-rq}{i\omega_{n}-rvq}$ (66)

After performing the frequency summation in Eq. (65) only one term survives:

$\displaystyle\chi_{\Pi,\Pi}(q,i\omega_{n})$
$\displaystyle=-\mathcal{K}\frac{v^{2}\tilde{\eta}_{+}^{2}}{9}\sum_{r}$
$\displaystyle\int_{-\infty}^{\infty}\frac{dp}{2\pi}\frac{p^{2}}{i\omega_{n}-2rvp}\left(1+2n_{B}(rvp)\right)$
(67)

where $n_{b}(x)=\left[\exp(\beta x)-1\right]^{-1}$ is the Bose-Einstein
distribution.

The bulk viscosity can now be evaluated by substituting the result of Eq. (67)
into Eq. (36) and then performing the analytic continuation,
$i\omega_{n}\to\omega+i\delta$. This procedure gives the following expression
for the bulk viscosity:

$\zeta(\omega)\approx\mathcal{K}\frac{\tilde{\eta}_{+}^{2}}{72v}\frac{\omega}{\tanh\left(\frac{\beta\omega}{4}\right)}$
(68)

In the weakly interacting limit this gives:

$\zeta(\omega)\approx\frac{m\ell^{2}k_{F}}{8\pi^{2}}\frac{\omega}{\tanh\left(\frac{\omega}{4T}\right)}$
(69)

where we note that the definition of $\Pi$ used in this work, Eq. (62) differs
from the traditional definition by a factor of $m$. Thus we have added an
additional factor of $m^{2}$, which is correct to leading order in
$k_{F}\ell$.

##  VI Calculation of the Bulk Viscosity Using Microscopic Theory

In this section we present a derivation of the bulk viscosity in the high
temperature limit by using the full microscopic theory. We begin with Eq.
(36). Although the bulk viscosity is defined in terms of the stress-tensor
correlation function, we note that the commutator of the Hamiltonian with any
operator vanishes in thermal Equilibrium. Following Eq. (28), the bulk
viscosity can also be defined in terms of the the contact correlation function
[42, 43, 44, 45]:

$\zeta(\omega)=\frac{Im\left[\chi_{C,C}(\omega)-\chi_{C,C}(0)\right]}{\ell^{2}\omega}$
(70)

where:

$\chi_{C,C}(\omega)=\int_{-\infty}^{\infty}dte^{i(\omega+i\delta)t}i\theta(t)\langle\left[C_{\ell}(t),C_{\ell}(0)\right]\rangle$
(71)

and the contact operator $C_{\ell}$ is given by Eq. (29).

The evaluation of Eq. (71) can be calculated using finite temperature field
theory. According to the virial expansion, the leading contribution to Eq.
(70) comes from the pair-pair propagator:

$\displaystyle\zeta(\omega)\approx\int_{-\infty}^{\infty}$
$\displaystyle\frac{dQ}{2\pi}\int_{-\infty}^{\infty}\frac{dx}{\pi}\frac{n_{B}(x)-n_{B}(x+\omega)}{\ell^{2}\omega}$
$\displaystyle
Im\left[\frac{T(Q,x-i\delta)}{4}\right]Im\left[\frac{T(Q,x+\omega-i\delta)}{4}\right]$
(72)

with $n_{B}(x)$ is the Bose-Einstein distribution at temperature $T$, and
$T(Q,z)$ is the T-matrix, Eq. (20).

In the high temperature limit the chemical potential is large and negative,
and an expansion in terms of the fugacity, $z=e^{\beta\mu}$, is permissible.
In the high-temperature limit one finds:

$\displaystyle\zeta(\omega)$ $\displaystyle\approx
z^{2}\frac{1-e^{-\beta\omega}}{\ell^{2}\omega}\int_{-\infty}^{\infty}\frac{dQ}{2\pi}\int_{-\infty}^{\infty}\frac{dx}{\pi}e^{-\beta\frac{Q^{2}}{4}-\beta
x}$ $\displaystyle
Im\left[\frac{T(x-i\delta)}{4}\right]Im\left[\frac{T(x+\omega-i\delta)}{4}\right]$
(73)

and

$\displaystyle\left(\frac{T(x-i\delta)}{4}\right)^{-1}=\frac{1}{\ell}+\sqrt{-x+i\delta}$
(74)

In the weakly interacting limit one finds:

$\zeta(\omega\to
0)\approx\frac{2}{\pi^{5/2}}\left(k_{F}\ell\right)^{2}T^{1/2}$ (75)

To obtain Eq. (75) we have used the equation of state to relate the fugacity
to the density, $n$, and the Fermi momentum, $k_{F}$:

$\displaystyle z$ $\displaystyle=n\sqrt{2\pi\beta}$ $\displaystyle n$
$\displaystyle=\frac{k_{F}}{\pi}$ (76)

Conversely, in the resonantly interacting limit, $\ell^{-1}=0$, one obtains:

$\displaystyle\lim_{\ell^{-1}\to 0}\lim_{\omega\to 0}\zeta(\omega)$
$\displaystyle\approx
2\frac{k_{F}^{2}}{\pi^{5/2}}\frac{1}{\ell^{2}}\frac{1}{T^{3/2}}\ln\left(m\ell^{2}T\right)$
(77) $\displaystyle\lim_{\omega\to 0}\lim_{\ell^{-1}\to 0}\zeta(\omega)$
$\displaystyle\approx
2\frac{k_{F}^{2}}{\pi^{5/2}}\frac{1}{\ell^{2}}\frac{1}{T^{3/2}}\ln\left(\frac{4T}{\omega}\right)$
(78)

We briefly note that Eq. (72) also describes the low-temperature bulk
viscosity to leading order in perturbation theory. A direct evaluation of Eq.
(72) using the microscopic model at low-temperatures gives results consistent
with Eq. (69) in the zero-frequency limit. However, one needs to use the two-
body T-matrix in the presence of the many-body background. We won’t provide
the details here, but provide a similar analysis for the Lieb-Liniger model
below. This further confirms that the LL model provides an accurate evaluation
of the bulk viscosity.

##  VII Calculation of the Scattering rate in Kinetic Theory

In this section we consider using a pseudo kinetic theory to evaluate the bulk
viscosity. In this approach we examine the response of Fermionic
quasiparticles to a two-body dephasing process beyond the LL model. Although
these processes do not physically relax the momentum, we find results
consistent with the more microscopic definition of the bulk viscosity. This
approach is in contrast to previous studies where the bulk viscosity is
explicitly related to three-body processes which do relax the momentum of the
system [34, 35], but provide answers inconsistent with conformal symmetry
[20].

Consider the following Hamiltonian for fermionic quasiparticles:

$\displaystyle H$ $\displaystyle=H_{0}+V_{+}$ $\displaystyle H_{0}$
$\displaystyle=\sum_{r}\int
dx\colon\psi^{\dagger}_{r}(x)\left(-irv\partial_{x}\right)\psi_{r}(x)+\frac{1}{\tilde{m}}\partial_{x}\psi^{\dagger}(x)\partial_{x}\psi(x)\colon$
$\displaystyle V_{+}$ $\displaystyle=\int dx\sum_{r}\colon
ir\eta_{+}\psi^{\dagger}_{r}(x)\frac{\overleftarrow{\partial_{x}}-\overrightarrow{\partial_{x}}}{2}\psi_{r}(x)\psi_{\bar{r}}^{\dagger}(x)\psi_{\bar{r}}(x)\colon$
(79)

In Eq. (79) we have already included the LL renormalization effects and the
non-perturbative summation of the $V_{-}$ interactions in order to define
quasiparticles with velocity, $v$, Eq. (51), and effective mass
$\tilde{m}^{-1}=\tilde{\eta}_{-}$, Eq. (57). The interactions beyond the LL
model couple right and left moving fermions together with an interaction
strength $\eta_{+}$ from Eq. (58).

To examine the bulk viscosity we subject the system to a slowly changing
velocity field $u$. Because of the interactions, there will necessarily be
power dissipated. In the limit of small velocity gradients, one expects the
power dissipated to be of the form:

$W=\zeta L(\partial_{x}u)^{2}$ (80)

where $\zeta$ is the zero-frequency limit of the bulk viscosity spectral
function.

From standard thermodynamic arguments the power dissipated is $W=T\dot{S}$,
where $S$ is the entropy. The entropy can be explicitly calculated using the
fermionic statistics of the quasiparticles:

$S=-\sum_{r,p}\left[n_{r,p}\ln(n_{r,p})+(1-n_{r,p})\ln(1-n_{r,p})\right]$ (81)

where $n_{r,p}$ is the time-dependent distribution function of a $r$-moving
quasiparticle with momentum $p$. Assuming the velocity gradient is weak
enough, the momentum distribution can be expanded as
$n_{r,p}=n_{r,p}^{0}+\delta n_{r,p}$, with $\delta
n_{r,p}\propto\partial_{x}u$. The equilibrium distribution function can be
written as:

$n_{r,p}^{0}=\left(\exp\left[\beta\left(rvp+\frac{p^{2}}{2\tilde{m}}-\Delta\mu\right)\right]+1\right)^{-1}.$
(82)

In defining Eq. (82), we note that $\Delta\mu$ is the shift in the chemical
potential away from its zero-temperature value: $\Delta\mu=\mu(T)-\mu(T=0)$,
as the zero-temperature chemical potential is used in writing down the
linearized Hamiltonian. From the Sommerfeld expansion one finds:

$\Delta\mu=\frac{\pi^{2}T^{2}}{12\tilde{m}v^{2}}.$ (83)

Then expanding $W$ to linear order in $\delta n_{r,p}$ gives:

$W=-T\sum_{r,p}\frac{\dot{n}_{r,p}^{0}\delta
n_{r,p}}{n_{r,p}^{0}(1-n_{r,p}^{0})}.$ (84)

where $\dot{n}_{r,p}^{0}$ represents the total time derivative of
$n_{r,p}^{0}$.

To proceed further we need to evaluate the time-dependence of the equilibrium
distribution function: $\dot{n}_{r,p}^{0}$. Following Refs. [34, 35] we note
that the velocity gradient can be imagined as a box with a size $L(t)$
expanding such that: $\partial_{x}u=\dot{L}(t)/L(t)$. In this picture the time
dependence of the density and momentum $p$ are trivial:

$\displaystyle\frac{\dot{n}}{n}$ $\displaystyle=-\partial_{x}u$
$\displaystyle\frac{\dot{p}}{p}$ $\displaystyle=-\partial_{x}u$ (85)

In order to obtain the time-dependence of the temperature and the chemical
potential, we note that the quasi-particle collisions preserve the quasi-
particle number and energy:

$\displaystyle\sum_{p}\dot{n}_{r,p}(t)$ $\displaystyle=0$
$\displaystyle\sum_{p}\epsilon_{r}(p)\dot{n}_{r,p}(t)$ $\displaystyle=0$ (86)

where $\epsilon_{r}(p)\approx rvp$. Requiring the equilibrium distribution
function satisfy Eq. (86), one finds the following time-dependences for
$\beta=1/T$ and $\Delta\mu$:

$\displaystyle\frac{\dot{\beta}}{\beta}$
$\displaystyle=\left[n\partial_{n}\ln(nv)\right]\partial_{x}u$
$\displaystyle\frac{\dot{\Delta\mu}}{\Delta\mu}$
$\displaystyle=\left[n\partial_{n}\ln\left(\frac{\tilde{m}}{n^{2}}\right)\right]\partial_{x}u$
(87)

which is valid to first order in $1/(\beta\tilde{m}v^{2})$. Finally we can
obtain the time-derivative of the distribution function as:

$\dot{n}_{r,p}^{0}=n_{r,p}^{0}(1-n_{p}^{0})\frac{6v^{2}p^{2}-\pi^{2}T^{2}}{12\tilde{m}v^{2}T}\left(n\partial_{n}\ln(\nu)\right)\partial_{x}u$
(88)

where $\nu=\frac{\pi n}{\tilde{m}v}$.

Next we need to calculate $\delta n_{r,p}$. This is done using the relaxation
time-approximation: $\delta n_{r,p}=-\tau_{r,p}\dot{n}^{0}_{r,p}$, where
$\tau_{r,p}$ is the scattering time associated with the two-body interactions
beyond the LL model, $V_{+}$. Using the relaxation time-approximation the bulk
viscosity is found to be:

$\zeta=\frac{T}{L}\sum_{p}\frac{(\dot{n}_{p}^{0})^{2}}{n_{p}^{0}(1-n_{p}^{0})}\tau.$
(89)

The definition of $\tau$ follows from Fermi’s golden rule. Consider the
scattering time for a right moving fermion with momentum $p$.

$\frac{1}{\tau}=\sum_{k_{1},k_{2},k_{1}^{\prime},k_{2}^{\prime}}\sum_{r^{\prime}}\delta_{p,k_{1}}W_{1,2}^{1^{\prime},2^{\prime}}n_{L}(k_{2})(1-n_{r^{\prime}}(k_{1}^{\prime}))(1-n_{\bar{r}^{\prime}}(k_{2}^{\prime}))$
(90)

where $\bar{r}=-r$ and

$\displaystyle W_{1,2}^{1^{\prime},2^{\prime}}=2\pi$
$\displaystyle\delta\left(\epsilon_{r}(k_{1})+\epsilon_{\bar{r}}(k_{2})-\epsilon_{r^{\prime}}(k_{1}^{\prime})-\epsilon_{\bar{r}^{\prime}}(k_{2}^{\prime})\right)$
$\displaystyle|\langle k_{1}^{\prime}\ r_{1}^{\prime},k_{2}^{\prime}\
r_{2}^{\prime}|V_{+}|k_{1}R,k_{2}L\rangle|^{2}.$ (91)

The matrix element due to $V_{+}$ is given by:

$\displaystyle\langle k_{1}^{\prime}\ r_{1}^{\prime},k_{2}^{\prime}\
r_{2}^{\prime}|$ $\displaystyle V_{+}|k_{1}R,k_{2}L\rangle=$
$\displaystyle\frac{\eta_{+}}{L}\left[(q+q^{\prime})\delta_{R,r_{1}^{\prime}}+(q-q^{\prime})\delta_{R,\bar{r}_{1}^{\prime}}\right].$
(92)

A straightforward calculation provides:

$\displaystyle\frac{1}{\tau}=\frac{4\eta_{+}^{2}}{v}\int\frac{dq}{2\pi}q^{2}n_{L}^{0}(p-2q)(1-n_{L}^{0}(p-2q))(1-n_{R}^{0}(p)),$
(93)

In order to evaluate Eq. (93) we need to perform the normal ordering, which
essentially subtracts off the $T=0$ contribution. However, the $T=0$
contribution is identically zero. Therefore the relaxation time is finally
given by:

$\frac{1}{\tau}=\frac{\eta_{+}^{2}T^{3}}{3v^{4}}$ (94)

Substituting Eqs. (88) and Eq. (94) into Eq. (89) gives the final result for
the bulk viscosity:

$\zeta\approx\frac{1}{24\pi}\frac{T}{v\tilde{m}^{2}\eta_{+}^{2}}\left[n\partial_{n}\ln(\nu)\right]^{2}.$
(95)

From the definition of $\nu=\pi n/\tilde{m}v$, one can show that:

$\displaystyle
n\partial_{n}\ln(\nu)=-\frac{5}{4}\left(\frac{2k_{F}\ell}{\pi}\right)^{2}.$
(96)

This gives us the final expression for the bulk viscosity:

$\displaystyle\zeta$ $\displaystyle\approx\frac{25}{6\pi^{4}}mk_{F}\ell^{2}T$
(97)

Eq. (97) has the same functional form as the LL calculation using the
microscopic definition of the bulk viscosity.

The main point to note is that one has to go beyond the standard kinetic
theory calculation to produce results consistent with the microscopic
evaluation of the bulk viscosity.

##  VIII Bulk Viscosity for the Lieb-Liniger Model

Let us now consider the Lieb-Liniger model of interacting bosons:

$\displaystyle H$ $\displaystyle=\int dx\
\frac{1}{2m}\partial_{x}\phi^{\dagger}(x)\partial_{x}\phi(x)$
$\displaystyle+\int
dx\frac{g}{4}\phi^{\dagger}(x)\phi^{\dagger}(x)\phi(x)\phi(x)$ (98)

where $\phi(x)$ is the bosonic operator, $m$ is the atomic mass and $g=-4/a$
relates the interaction strength to the scattering length, $a$.

In the high temeprature limit, the two-body T-matrix has the form:

$\displaystyle\left(\frac{T(x-i\delta)}{4}\right)^{-1}=-a+\frac{1}{\sqrt{-x+i\delta}}$
(99)

where $\xi_{k}=\frac{k^{2}}{2m}-\mu$, and $n_{B}(x)=\left(e^{\beta
x}-1\right)^{-1}$ is the Bose-Einstein distribution.

We can again define the stress-tensor according to:

$\Pi=2H+aC_{a}$ (100)

where the contact operator is:

$C_{a}=\frac{\partial H}{\partial a}=\int dx\
\frac{g}{16}\phi^{\dagger}(x)\phi^{\dagger}(x)\phi(x)\phi(x).$ (101)

From Eq. (25) the bulk viscosity for the Lieb-Liniger model is defined as:

$\zeta(\omega)=\frac{\chi_{\Pi\Pi}(\omega)-\chi_{\Pi\Pi}(0)}{i\omega}+\frac{2a\langle
C_{a}\rangle}{i\omega}$ (102)

The extra term in Ref. (102) arises from the behaviour of the contact operator
under a scale transformation. Since $g=-4/a$, the contact operator does not
simply transform according to its scaling dimension of $-3$ but obeys the
modified relation:

$\left.\frac{dC_{a}}{db}\right|_{b=0}=-C_{a}.$ (103)

This leads to the following behavior of $\Pi$ under a scale transformation:

$\left.\frac{d\Pi_{b}}{db}\right|_{b=0}=-2\Pi-aC_{a}.$ (104)

This ought to be compared to how the expectation value $\langle\Pi\rangle$
changes under a scale transformation:

$\left.\frac{d}{db}\langle\Pi\rangle_{b}\right|_{b=0}=-2\langle\Pi\rangle-3a\langle
C_{a}\rangle-a^{2}\frac{\partial\langle C_{a}\rangle}{\partial a}$ (105)

The last term can again be identified as $\chi_{\Pi\Pi}(0)$ via linear
response theory, similar to the Fermionic case. This substitution brings the
original definition of the bulk viscosity, Eq. (25), into the form of Eq.
(102). The presence of the Drude peak is still consistent with scale symmetry,
but suggests a divergent bulk viscosity spectral function at zero-frequency
for fixed arbitrary interaction strengths. If one is considered with the real
part of the bulk viscosity spectral function, the Drude peak only appears at
zero-frequency. In this work, we always assume that the frequency is finite
but vanishingly small, so as to ignore the Drude peak.

At leading order the bulk viscosity can be written as:

$\displaystyle\zeta(\omega)\approx\frac{a^{2}}{\pi\beta}$
$\displaystyle\int_{-\infty}^{\infty}\frac{dx}{\pi}\frac{n_{B}(x)-n_{B}(x+\omega)}{\omega}$
$\displaystyle
Im\left[\frac{T(x-i\delta)}{4}\right]Im\left[\frac{T(x+\omega-i\delta)}{4}\right]$
(106)

where $T(x-i\delta)$ is the two-body T-matrix defined in the vacuum. Eq. (106)
is valid in the high-temperature limit for arbitrary interaction strengths.
Near resonance $na\ll 1$ the bulk viscosity can be analytically evaluated:

$\zeta(\omega\to 0)\approx\frac{2}{\pi^{5/2}}\left(k_{F}a\right)^{2}T^{1/2}$
(107)

where we have used $n=k_{F}/\pi$, and similarly for the weakly interacting
limit $na\gg 1$:

$\lim_{\omega\to 0}\lim_{a^{-1}\to 0}\zeta(\omega)\approx
2\frac{k_{F}^{2}}{\pi^{5/2}}\frac{1}{a^{2}}\frac{1}{T^{3/2}}\ln\left(\frac{4T}{\omega}\right)$
(108)

Upon inspection of Eqs. (107-108), one finds that they are equivalent to Eqs.
(75,78) in the opposing limit. This allows one to establish the duality
relations:

$\zeta_{F}(\ell)=\zeta_{B}\left(\frac{1}{a}\right).$ (109)

Hence the bulk viscosity for a strongly (weakly) interacting Fermi gas is
equal to the bulk viscosity of a weakly (strongly) interacting Bose gas.

Figure 2: Bulk viscosity of a spin-polarized Fermi gas near resonance. At high
temperatures, Eq. (78), the bulk viscosity vanishes like
$\ell^{-2}n^{2}T^{-3/2}$, while for low temperatures the bulk viscosity
diverges as: $\ell^{-2}n^{7}T^{-4}$. From the duality relations, this plot
also describes the bulk viscosity for a weakly interacting 1D bose gas.

In the low-temperature limit the bulk viscosity can be evaluated at low-
temperatures. The only required modification to Eq. (106) is to use the
T-matrix calculated within the many-body background. To leading order in
perturbation theory one can write the many-body T-matrix as:

$\displaystyle Im\left[\frac{T(Q,x-i\delta)}{4}\right]$ $\displaystyle\approx$
$\displaystyle\frac{1}{a^{2}}\int\frac{dk}{\pi}$
$\displaystyle\left[1+n_{B}\left(\xi_{\frac{Q}{2}+k}\right)+n_{B}\left(\xi_{\frac{Q}{2}-k}\right)\right]$
$\displaystyle\times\pi\delta(x-\xi_{\frac{Q}{2}+k}-\xi_{\frac{Q}{2}-k}-i\delta)$
(110)

From this expression, one can show that the integrand is dominated near
$\epsilon\approx 0$ and $Q\approx 0$. Expanding near this point and evaluating
the resulting integrals give:

$\zeta_{B}(\omega\to
0)\approx\frac{16}{\pi^{2}}\frac{n^{7}}{a^{2}T^{4}}\log\left(16\frac{T^{2}}{n^{2}\omega}\right)$
(111)

In order to obtain Eq. (111) we have used the non-interacting equation of
state:

$n=\frac{1}{\sqrt{2\pi\beta}}Li_{1/2}\left(e^{\beta\mu}\right)$ (112)

where $Li_{s}(x)$ is the polylogarithm function. As one approaches the zero-
temperature limit for fixed density, one can show
$\sqrt{2\beta}n=(-\beta\mu)^{-1/2}$, i.e. $\mu\to 0^{-}$ as $T\to 0$.

From Eqs. (78) and (111) one can construct the bulk viscosity for arbitrary
temperatures near resonance, as shown in Fig. (2). From the duality relations
Figs. (1) and (2) describe the bulk viscosity for a 1D Bose gas near resonance
and for weak interactions respectively.