# Shear Viscosity in the Strong Interaction Regime of a $p$-wave Superfluid
Fermi Gas

Seyed Mostafa Moniri  Faculty of Science, Golpayegan University of Technology,
P.O. Box 87717-67498, Golpayegan, Iran<EMAIL_ADDRESS>Heshmatollah
Yavari  Department of Physics, University of Isfahan, Isfahan 81746, Iran
Elnaz Darsheshdar  Departamento de Física, Universidade Federal de São Carlos,
P.O. Box 676, 13565-905, São Carlos, São Paulo, Brazil

###### Abstract

The $p$-wave superfluid state is a promising spin-triplet and non $s$-wave
pairing state in an ultracold Fermi gas. In this work we study the low-
temperature shear viscosity of a one-component $p$-wave superfluid Fermi gas,
by means of Kubo formalism. Our study is done in the strong-coupling limit
where Fermi superfluid reduces into a system of composite bosons. Taking into
account ${{p}_{x}}$-wave Cooper channel in the self-energy, the viscous
relaxation rates are determined. The relaxation rates related to these
interactions are calculated as a function of temperature. Their temperature
dependence is different from the s-wave superfluid Fermi gas, and this is due
to the anisotropic pairing interaction in the $p$-wave superfluid. Our results
contribute to understand how this anisotropy affects transport properties of
this unconventional superfluid Fermi gas in low temperature limit.

###### keywords:

$P$-wave superfluid Fermi gas , T-matrix approximation , Shear viscosity

††journal: Physics Letters A

##  1 Introduction

As a spin-triplet pairing superfluid, $p$-wave superfluid Fermi gas attracts
more attention during last years and was explored in both experimental [1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11] and theoretical side [12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26]. In the Feshbach resonance of these systems,
the strength of a $p$-wave interaction can be tuned from a weak-coupling to a
strong-coupling regimes by controlling an external magnetic field. This was
studied experimentally in 40K [1, 3] and 6Li [4, 5] Fermi gases. The weak-
coupling limit is characterized by $(p_{F}^{3}v_{i})^{-1}\lesssim 0$ and the
strong-coupling by $(p_{F}^{3}v_{i})^{-1}\gtrsim 0$, where $p_{F}$ is the
Fermi momentum and $v_{i}(i=x,y,z)$ is the $p_{i}$-wave scattering volume
[27]. Considering the fact that the experimental realizations have been done
on $p$-wave Feshbach molecules [2, 4, 7, 8, 9, 10], in order to achieve
molecules of $p$-wave interaction, it is needed to overcome a bunch of
problems such as the dipolar relaxation [4, 7, 28], and the three-body loss
[29, 30, 19].
On the other hand, applying an external magnetic field in x direction leads to
the splitting of the $p$-wave Feshbach resonance into a $p_{x}$-wave channel
and degenerate $p_{y}$-wave and $p_{z}$-wave channels by a magnetic dipole-
dipole interaction. This uniaxial anisotropy is studied in a 40K Fermi gas
[3]. It has been shown that such an anisotropic $p$-wave pairing interaction
may lead to two superfluid phases with different $p$-wave pairing symmetries
[12, 20]. The observed resonance field of a $p_{x}$-wave Feshbach resonance is
higher than that of the other channels [3]. It leads stronger $p_{x}$-wave
pairing interaction ($U_{x}$) comparing to $p_{y}$-wave ($U_{y}$) and
$p_{z}$-wave ($U_{z}$) ones, consequently, the highest superfluid phase
transition temperature $T_{C}^{p_{x}}$ for $p_{x}$-wave superfluid phase.
Specific condition for this uniaxial anisotropy is discussed in [12, 20]
noticing that the system has another phase transition from the $p_{x}$-wave
state to the $p_{x}+ip_{y}$-wave state at
$T_{C}^{p_{x}+ip_{y}}(<T_{C}^{p_{x}})$. This indicates that plural superfluid
phases can be studied in realization of a $p$-wave superfluid Fermi gas, in
all coupling ranges from the weak to the strong ones. In addition to plural
superfluid phases, an unconventional Fermi superfluid with a nodal superfluid
order parameter, in its $p_{x}$-wave state, has an interesting pseudogap
phenomenon near $T_{C}^{p_{x}+ip_{y}}$. In this state, the system undergos a
phase transition at $T_{C}^{p_{x}}$, but single-particle excitations are
gapless in the nodal direction ($\perp p_{x}$) thus $p_{y}$-wave and
$p_{z}$-wave pairing fluctuations exist. They will be the strongest at
$T_{C}^{p_{x}+ip_{y}}$, consequently, far below $T_{C}^{p_{x}}$, an
anisotropic pseudogap phenomenon is anticipated in the nodal direction of the
$p_{x}$-wave superfluid order parameter near $T_{C}^{p_{x}+ip_{y}}$. This
fluctuation is expected only for non-zero $p_{y}$-wave and a $p_{z}$-wave
interaction.
The above explained novel phenomena of an unconventional Fermi superfluid with
a nodal superfluid order parameter in the $p_{x}$-wave state, are encouraging
to investigate the shear viscosity in this system. Transport properties in the
condensed matter systems have been attracted attentions continuously [31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], and was studied using
Boltzmann equation approach in many works [31, 32, 34, 33], while the linear
response theory and Kubo formula approach has attracted much more attentions
[35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. The superiority of the Kubo formula
is due to its direct relation to diagrammatic Green’s function, thus a good
control and using proper limits can be achieved. For example using fast
calculation techniques for scattering in the T-matrix method [45], one can do
the calculations, or in the low temperature limits, proper approximations can
be applied [40]. In the high temperature and weak interaction limits, the
obtained results from Kubo formula will converge to the Boltzmann equation
ones. In Boltzmann equation, it is assumed that quasiparticles having a well-
defined energy-momentum relation. On the other hand, in the case of a
broadened spectral functions and not well-defined quasiparticles, the effect
of this broadening and the related kinetic equation should be considered in
the calculation of the transport properties. Due to the absence of a small
parameter in the strong-coupling limit, this is too far ambitious. So,
applying Kubo formula will be required.

To the best of our knowledge the shear viscosity in the strong interaction
regime of a ${{p}_{x}}$-wave superfluid Fermi gas using the self-energy and
the T-matrix approximation in the Kubo formula method has not been studied. In
this paper following the method of Ref. [40], we study the low temperature
shear viscosity of one component ${{p}}$-wave superfluid Fermi gas with a
uniaxially anisotropic ${{p}_{x}}$-wave pairing interaction
$(U_{x}>U_{y}=U_{z})$ in the strong-coupling limit. We use many-body methods
to study the self-energy of the system in the T-matrix approximation (TMA). To
calculate the self-energy of the system, the ${{p}_{x}}$-wave interactions are
taken into account in ${{p}_{x}}$-wave Cooper channel in superfluid phases.

##  2  Formulation

The Shear viscosity using the Kubo formula is expressed by [46],

$\displaystyle\eta=-\underset{\omega\to
0}{\mathop{\lim}}\left[\frac{\operatorname{Im}{{\Xi}_{ret}}\left(\omega\right)}{\omega}\right]$
(1)

and the quantity of $\Xi_{ret}\left({{\omega}_{n}}\right)$ named current-
current correlation function and can be obtained from [46],

$\displaystyle\Xi\left(i{{\omega}_{n}}\right)=-\frac{1}{V}\int_{0}^{\beta}{d\tau\,{{e}^{i{{\omega}_{n}}\tau}}}\left\langle{{T}_{\tau}}S\left(\beta\right){{e}^{{{H}_{0}}\tau}}{{\Pi}_{xy}}{{e}^{-{{H}_{0}}\tau}}{{\Pi}_{xy}}\right\rangle$
(2)

where $S\left(\beta\right)$ is the S-matrix of the perturbed part of the
Hamiltonian [46]. $\Xi_{ret}\left({{\omega}}\right)$ is obtained by letting
$i{{\omega}_{n}}\rightarrow\omega+i{{\delta}}$ in
$\Xi\left(i{{\omega}_{n}}\right)$ ($\delta$ is an infinitesimal parameter),
and we have

$\displaystyle\eta=\frac{\beta
mn{{\nu}^{2}}}{16}\int{d{{\varepsilon}_{p}}}\frac{d\varepsilon}{2\pi}\,{{\operatorname{sech}}^{2}}\left(\frac{\beta\varepsilon}{2}\right)A{{\left(\mathbf{p},\varepsilon\right)}^{2}}$
(3)

where the spectral function can be approximated as [46]

$\displaystyle
A{{\left(\mathbf{p},\varepsilon\right)}^{2}}\approx\frac{2\pi\delta\left(\varepsilon-{{E}_{p}}\right)}{\operatorname{Im}\Sigma\left(\mathbf{p},\varepsilon\right)}$
(4)

and viscous relaxation rate can be then defined as

$\displaystyle\tau_{\eta}^{-1}=\left|2\operatorname{Im}\Sigma\left(\mathbf{p},\tau\right)\right|$
(5)

In the strong-coupling region a Feshbach resonance leads to tightly bound
dimer molecules. Accordingly, in this paper we analyze the strong-coupling
side of the crossover where fermions get rearranged into a system of composite
bosons (dimers). The ${{p}_{x}}$-wave superfluid state, is a spin-triplet
pairing state, thus even in one component superfluid Fermi gas, fermions will
behave as dimers in the strong-coupling limit. We consider specifically the
limit of low temperature, whereby most of dimers are condensed. The
Hamiltonian of one component superfluid Fermi gas with a uniaxially
anisotropic ${{p}_{x}}$-wave interaction can be written as [13],

$\displaystyle
H=\sum\limits_{\mathbf{p}}{{}}{{\xi}_{\mathbf{p}}}c_{\mathbf{p}}^{\dagger}{{c}_{\mathbf{p}}}-\frac{1}{2}\sum\limits_{\mathbf{p},\mathbf{{p}^{\prime}},\mathbf{q},i}{{}}{{F}_{n}}\left(\mathbf{p}\right){{p}_{i}}{{U}_{i}}{p}^{\prime}_{i}{{F}_{n}}{{\left({\mathbf{{p}^{\prime}}}\right)}}$
(6) $\displaystyle
c_{\mathbf{p}+\mathbf{q}/2}^{\dagger}c_{-\mathbf{p}+\mathbf{q}/2}^{\dagger}{{c}_{-\mathbf{{p}^{\prime}}+\mathbf{q}/2}}{{c}_{\mathbf{{p}^{\prime}}+\mathbf{q}/2}}$

where, $c_{\mathbf{p}}^{\dagger}$ is the creation operator of a Fermi atom and
the kinetic energy
${{\xi}_{\mathbf{p}}}={{\varepsilon}_{\mathbf{p}}}-\mu={{p}^{2}}/2m-\mu$, is
conventionally measured relative to Fermi chemical potential $\mu$ where $m$
is the atomic mass. The pairing interaction in the ${{p}_{i}}$-wave Cooper
channel $\left(i=x,y,z\right)$ is ${{U}_{i}}$ and we consider uniaxial
anisotropy $\left({{U}_{x}}>{{U}_{y}}={{U}_{z}}\right)$ in order to study
${{p}_{x}}$-wave superfluid with the highest superfluid phase transition
temperature. Here, we do not consider $p$-wave Feshbach resonance, but we
treat $U_{x}$, $U_{y}$ and $U_{z}$ as tunable parameters [3].
A cutoff function is deffined in (6) in order to solve ultraviolet divergency
as ${{F}_{n}}(p)=\text{ }1/\left[1\text{ }+{{(p/{{p}_{c}})}^{2n}}\right]$ with
$n=3$, where ${{p}_{c}}$ is a cutoff momentum. The $p$-wave interaction
strength is conveniently measured in terms of the scattering volume
${{v}_{i}}$ and the effective range ${{k}_{0}}$, respectively are given by,
[26, 47, 27, 48],

$\displaystyle\frac{4\pi{{v}_{i}}}{m}=-\frac{{{U}_{i}}}{3}\frac{1}{1-\frac{{{U}_{i}}}{3}\sum\limits_{p}{\frac{{{p}^{2}}}{2{{\varepsilon}_{p}}}F_{n}^{2}\left(p\right)}}$
(7)
$\displaystyle{{k}_{0}}=-\frac{4\pi}{{{m}^{2}}}\sum\limits_{p}{\frac{{{p}^{2}}}{2\varepsilon_{p}^{2}}F_{n}^{2}\left(p\right)}$
(8)

In the strong-coupling side in the superfluid phase, it is convenient to
rewrite the Hamiltonian of Eq. (6) into the sum of the mean-field BCS part and
the fluctuation corrections term. It is useful to introduce the Nambu
representation for the field operators [50],

$\psi_{\mathbf{p}}=\left(\begin{matrix}&c_{\mathbf{p}}\\\
&c_{\mathbf{p}}^{\dagger}\end{matrix}\right)$ (9)

In Nambu formalism [50], we can rewrite Hamiltonian in Eq. (6) as below,

$\displaystyle
H=\frac{1}{2}\sum\limits_{\mathbf{p}}{{}}\psi_{\mathbf{p}}^{\dagger}\left({{\xi}_{\mathbf{p}}}{{\tau}_{3}}-\hat{\Delta}\left(\mathbf{p}\right)\right){{\psi}_{\mathbf{p}}}-\frac{1}{2}\sum\limits_{\mathbf{q},i}{{}}{{U}_{i}}\rho_{i}^{+}\left(\mathbf{q}\right)\rho_{i}^{-}\left(-\mathbf{q}\right)$
(10)

where ${{\tau}_{i}}\left(i=x,y,z\right)$ are Pauli matrices. The first term in
Eq. (10) is just the mean-field BCS Hamiltonian, where

$\displaystyle\hat{\Delta}\left(\mathbf{p}\right)=\left(\begin{matrix}0&\Delta\left(\mathbf{p}\right)\\\
{{\Delta}^{*}}\left(\mathbf{p}\right)&{}\\\ \end{matrix}\right)$ (11)

is a $p$-wave superfluid order parameter and,

$\displaystyle\Delta\left(\mathbf{p}\right)=\mathbf{b}.\mathbf{p}{{F}_{n}}\left(\mathbf{p}\right)$
(12)

where $\mathbf{b}=\left({{b}_{x}},{{b}_{y}},{{b}_{z}}\right)$ has the form of

$\displaystyle{{b}_{i}}={{U}_{i}}\sum\limits{{{p}_{i}}{{F}_{n}}\left(\mathbf{p}\right)\left\langle{{c}_{-\mathbf{p}}}{{c}_{\mathbf{p}}}\right\rangle}$
(13)

The $\rho_{i}^{\pm}\left(\mathbf{q}\right)$ is a generalized density operator
for a pair of fermions that physically describes ${{p}_{i}}$-wave superfluid
fluctuations and can be written as

$\displaystyle\rho_{i}^{\pm}\left(\mathbf{q}\right)=\sum\limits{{{p}_{i}}{{F}_{n}}\left(\mathbf{p}\right)\psi_{\mathbf{p}+\mathbf{q}/2}^{\dagger}{{\tau}_{\pm}}{{\psi}_{\mathbf{p}-\mathbf{q}/2}}}$
(14)

and ${{\tau}_{\pm}}\text{ }=\text{
}\left({{\tau}_{x}}\pm{{\tau}_{y}}\right)/2$. The second term in Eq. (10)
gives fluctuation corrections to the mean-field Hamiltonian. The BCS mean-
field Green’s function is,

$\displaystyle{{\hat{G}}_{0}}\left(\mathbf{p},i{{\omega}_{m}}\right)=\frac{1}{i{{\omega}_{m}}-{{\xi}_{\mathbf{p}}}{{\tau}_{3}}+\hat{\Delta}\left(\mathbf{p}\right)}$
(15)

where ${{\omega}_{m}}$ is the fermionic Matsubara frequency. Fluctuation
corrections to the single-particle mean-field Hamiltonian can be conveniently
considered by the self-energy $\Sigma\left(\mathbf{p},i{{\omega}_{m}}\right)$
in the single-particle thermal Green’s function,

$\displaystyle\hat{G}\left(\mathbf{p},i{{\omega}_{m}}\right)=\frac{1}{\hat{G}_{0}^{-1}\left(\mathbf{p},i{{\omega}_{m}}\right)-\hat{\Sigma}\left(\mathbf{p},i{{\omega}_{m}}\right)}$
(16)

Figure 1: Feynman diagrams describing the self-energy
$\hat{\Sigma}\left(\mathbf{p},i{{\omega}_{m}}\right)$ and particle-particle
scattering matrix
$\Gamma_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$ that is
shown by wavy line in the TMA. The solid line and the dashed line describe the
mean-field single-particle Green’s function
$\hat{G}_{0}\left(\mathbf{p},i{{\omega}_{m}}\right)$ and the p-wave
interaction, respectively.

The self-energy $\Sigma\left(\mathbf{p},i{{\omega}_{m}}\right)$ within the TMA
can be written as,

$\displaystyle\hat{\Sigma}\left(\mathbf{p},i{{\omega}_{n}}\right)=\frac{2}{\beta}\sum\limits_{i,j}{{}}\sum\limits_{\mathbf{q},i{{\nu}_{n}}}{{}}F_{n}^{2}\left(\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left[{{p}_{i}}-\frac{{{q}_{i}}}{2}\right]\left[{{p}_{j}}-\frac{{{q}_{j}}}{2}\right]$
(17)
$\displaystyle\left(\begin{matrix}G_{0}^{22}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-i{{\nu}_{n}}\right)\Gamma_{i,j}^{-+}\left(\mathbf{q},i{{\nu}_{n}}\right)&G_{0}^{21}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-i{{\nu}_{n}}\right)\Gamma_{i,j}^{--}\left(\mathbf{q},i{{\nu}_{n}}\right)\\\
G_{0}^{12}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-i{{\nu}_{n}}\right)\Gamma_{i,j}^{++}\left(\mathbf{q},i{{\nu}_{n}}\right)&G_{0}^{11}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-i{{\nu}_{n}}\right)\Gamma_{i,j}^{+-}\left(\mathbf{q},i{{\nu}_{n}}\right)\end{matrix}\right)$

The self-energy $\Sigma\left(\mathbf{p},i{{\omega}_{m}}\right)$ is
diagrammatically given in Fig. (1). The TMA particle-particle scattering
matrix $\Gamma_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$
gives the equation,

$\displaystyle\Gamma_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)=-{{U}_{i}}{{\delta}_{i,j}}{{\delta}_{s,-{s}^{\prime}}}-{{U}_{i}}\sum\limits_{{{s}^{\prime}}^{\prime}=\pm}{{}}\sum\limits_{k}{{}}\Pi_{i,k}^{s,{{s}^{\prime}}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)\Gamma_{k,j}^{-{{s}^{\prime}}^{\prime},{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$
(18)

and,

$\displaystyle\Pi_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)=\frac{1}{\beta}\sum\limits_{\mathbf{p},i{{\omega}_{m}}}{{}}{{p}_{i}}{{p}_{j}}F_{n}^{2}\left(p\right)\text{Tr}$
$\displaystyle\left[{{\tau}_{s}}{{G}_{0}}\left(\mathbf{p}+\mathbf{q}/2,i{{\omega}_{n}}\right){{\tau}_{{{s}^{\prime}}}}{{G}_{0}}\left(\mathbf{p}-\mathbf{q}/2,i{{\omega}_{n}}-i{{\nu}_{n}}\right)\right]$

is the pair correlation function. In particular,
$\Pi_{i,i}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$ describes
fluctuations in the ${{p}_{i}}$-wave Cooper channel, and
$\Pi_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)\left(i\neq
j\right)$ describes coupling between ${{p}_{i}}$-wave and ${{p}_{j}}$-wave
pairing fluctuations. We performed the Matsubara summation, for the self-
energy $\hat{\Sigma}\left(\mathbf{p},i{{\omega}_{n}}\right)$ in Eq. (17), it
gives,

$\displaystyle{{\Sigma}_{11}}\left(\mathbf{p},i{{\omega}_{m}}\right)=2\sum\limits_{i,j}{{}}\sum\limits_{\mathbf{q}}{{}}F_{n}^{2}\left(\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left[{{p}_{i}}-\frac{{{q}_{i}}}{2}\right]\left[{{p}_{j}}-\frac{{{q}_{j}}}{2}\right]$
(20)
$\displaystyle\bigg{(}\left[\frac{1}{2}-\frac{{{\xi}_{\mathbf{p}-\mathbf{q}}}}{2{{E}_{\mathbf{p}-\mathbf{q}}}}\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}}}\right)\right]-$
$\displaystyle\int{{}}\frac{d\varepsilon}{2\pi}{{n}_{B}}\left(\varepsilon\right)B_{i,j}^{-+}\left(\mathbf{p}-\mathbf{q},\varepsilon\right)G_{22}^{0}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-\varepsilon\right)\bigg{)}$

and,

$\displaystyle{{\Sigma}_{21}}\left(\mathbf{p},i{{\omega}_{m}}\right)=2\sum\limits_{i,j}{{}}\sum\limits_{\mathbf{q}}{{}}F_{n}^{2}\left(\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left[{{p}_{i}}-\frac{{{q}_{i}}}{2}\right]\left[{{p}_{j}}-\frac{{{q}_{j}}}{2}\right]$
(21)
$\displaystyle\Delta\left(\mathbf{p}-\mathbf{q}\right)\left(\frac{1}{2{{E}_{\mathbf{p}-\mathbf{q}}}}-\int{{}}\frac{d\varepsilon}{2\pi}{{n}_{B}}\left(\varepsilon\right)B_{i,j}^{++}\left(\mathbf{q},\varepsilon\right)G_{12}^{0}\left(\mathbf{p}-\mathbf{q},i{{\omega}_{m}}-\varepsilon\right)\right)$

where ${{n}_{B}}\left(\varepsilon\right)$ and
${{n}_{F}}\left(\varepsilon\right)$ are the bosonic and fermionic distribution
functions respectively,
${{E}_{\mathbf{p}}}=\sqrt{\xi_{\mathbf{p}}^{2}+{{\Delta}^{2}}\left(\mathbf{p}\right)}$
is the Bogoliubov single-particle excitation spectrum and
$B_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},\varepsilon\right)=-2\operatorname{Im}\Gamma_{i,j}^{s,{s}^{\prime},\text{ret}}\left(\mathbf{q},\varepsilon\right)$.
The order parameter $\Delta\left(\mathbf{p}\right)$ is related to the
${{p}_{x}}$-wave superfluid phase [27] and it differs from the ${s}$-wave one
by its momentum dependency. The other components of self-energy can be
obtained by using
$\Pi_{i,j}^{--}\left(\mathbf{q},i{{\nu}_{n}}\right)=\Pi_{i,j}^{++}\left(\mathbf{q},i{{\nu}_{n}}\right)$
and
$\Pi_{i,j}^{+-}\left(\mathbf{q},i{{\nu}_{n}}\right)=\Pi_{i,j}^{-+}\left(\mathbf{q},-i{{\nu}_{n}}\right)$.
In the TMA for the particle-particle scattering matrix
$\Gamma_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$ we
consider the first two terms as,

$\displaystyle\Gamma_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)=-{{U}_{i}}{{\delta}_{i,j}}{{\delta}_{s,-{s}^{\prime}}}+{{U}_{i}}{{U}_{j}}\Pi_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$
(22)

The momentum dependent nature of the interaction in $p$-wave superfluids,
allows us to use this approximation in low momentum limits, in order to remove
higher power of the momentum. The pair correlation function,
$\Pi_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},i{{\nu}_{n}}\right)$, can be
obtained after doing the Matsubara summation in Eq. (2) as below,

$\displaystyle\Pi_{i,j}^{++}\left(\mathbf{q},i{{\nu}_{n}}\right)=\frac{1}{4}\sum\limits_{\mathbf{p}}{{}}{{p}_{i}}{{p}_{j}}F_{n}^{2}\left(p\right)\frac{\Delta_{\mathbf{p}+\mathbf{q}/2}^{*}\Delta_{\mathbf{p}-\mathbf{q}/2}^{*}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}$
(23)
$\displaystyle\Bigg{(}\frac{{{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}}{\left({{\nu}^{2}}_{n}+{{\left({{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}^{2}}\right)}\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)+\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle-\frac{{{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}}{\left({{\nu}^{2}}_{n}+{{\left({{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}^{2}}\right)}\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)-\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]\Bigg{)}$

and,

$\displaystyle\Pi_{i,j}^{-+}\left(\mathbf{q},i{{\nu}_{n}}\right)=\frac{1}{4}\sum\limits_{\mathbf{p}}{{}}{{p}_{i}}{{p}_{j}}F_{n}^{2}\left(p\right)$
(24)
$\displaystyle\Bigg{(}\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)+\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle\Bigg{[}\left(1+\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\frac{1}{i{{\nu}_{n}}-\left({{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}$
$\displaystyle+\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1-\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\frac{i{{\nu}_{n}}}{\nu_{n}^{2}+{{\left({{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}^{2}}}\Bigg{]}$
$\displaystyle+\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)-\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle\Bigg{[}\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\frac{1}{i{{\nu}_{n}}-\left({{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}$
$\displaystyle+\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1+\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\frac{i{{\nu}_{n}}}{\nu_{n}^{2}+{{\left({{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)}^{2}}}\Bigg{]}\Bigg{)}$

By substituting Eqs.(23) and (24) in Eq. (22) and using analytic continuation
we can obtain spectral function of
$B_{i,j}^{s,{s}^{\prime}}\left(\mathbf{q},\varepsilon\right)=-2\operatorname{Im}\Gamma_{i,j}^{s,{s}^{\prime},\text{ret}}\left(\mathbf{q},\varepsilon\right)$
as below,

$\displaystyle
B_{i,j}^{++}\left(\mathbf{q},\varepsilon\right)=\frac{\pi}{4}{{U}_{i}}{{U}_{j}}\sum\limits_{\mathbf{p}}{{}}{{p}_{i}}{{p}_{j}}F_{n}^{2}\left(p\right)\frac{\Delta_{\mathbf{p}+\mathbf{q}/2}^{*}\Delta_{\mathbf{p}-\mathbf{q}/2}^{*}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}$
(25)
$\displaystyle\Bigg{(}\left[\delta\left(\varepsilon-{{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)-\delta\left(\varepsilon+{{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)+\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle+\left[\delta\left(\varepsilon+{{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)-\delta\left(\varepsilon-{{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)-\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]\Bigg{)}$

and,

$\displaystyle
B_{i,j}^{-+}\left(\mathbf{q},\varepsilon\right)=\frac{\pi}{2}{{U}_{i}}{{U}_{j}}\sum\limits_{\mathbf{p}}{{}}{{p}_{i}}{{p}_{j}}F_{n}^{2}\left(p\right)$
(26)
$\displaystyle\Bigg{(}\Bigg{[}\frac{1}{2}\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1-\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\delta\left(\varepsilon+{{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)$
$\displaystyle+\left(1+\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}+\frac{1}{2}\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1-\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\right)$
$\displaystyle\delta\left(\varepsilon-{{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\Bigg{]}\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)+\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]$
$\displaystyle+\Bigg{[}\frac{1}{2}\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1+\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\delta\left(\varepsilon+{{E}_{\mathbf{p}+\mathbf{q}/2}}-{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)$
$\displaystyle+\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}{{E}_{\mathbf{p}-\mathbf{q}/2}}}+\frac{1}{2}\left(1-\frac{{{\xi}_{\mathbf{p}+\mathbf{q}/2}}}{{{E}_{\mathbf{p}+\mathbf{q}/2}}}\right)\left(1+\frac{{{\xi}_{\mathbf{p}-\mathbf{q}/2}}}{{{E}_{\mathbf{p}-\mathbf{q}/2}}}\right)\right)$
$\displaystyle\delta\left(\varepsilon-{{E}_{\mathbf{p}+\mathbf{q}/2}}+{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\Bigg{]}\left[\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}+\mathbf{q}/2}}\right)-\tanh\left(\frac{\beta}{2}{{E}_{\mathbf{p}-\mathbf{q}/2}}\right)\right]\Bigg{)}$

As we have the anisotropy of $U_{x}>U_{y}=U_{z}$, the superfluid phase is
dominated by the $p_{x}$-wave pairing interaction in BEC limit. Then we limit
our calculation to $p_{x}$-wave cooper channel. We approximate
$B_{xx}^{s,{s}^{\prime}}\left(\mathbf{q},\varepsilon\right)$ at low momentum
and temperature. Then after integrating, we have,

$\displaystyle
B_{xx}^{++}\left(\mathbf{q},\varepsilon\right)=\frac{U_{x}^{2}}{24\pi}\sqrt{{{\Delta}^{2}}\left({{\mathbf{p}}_{0}}-\mathbf{q}\right)+{{\mu}^{2}}}{{p}_{0}}^{3}F_{n}^{2}\left({{p}_{0}}\right)\frac{{{\Delta}^{2}}\left({{\mathbf{p}}_{0}}-\mathbf{q}\right)}{{{E}_{{{\mathbf{p}}_{0}}+\mathbf{q}/2}}{{E}_{{{\mathbf{p}}_{0}}-\mathbf{q}/2}}}$
(27)
$\displaystyle\left(1-{{\text{e}}^{-2\beta{{E}_{{{\mathbf{p}}_{{}_{0}}}+\mathbf{q}/2}}}}-{{\text{e}}^{-2\beta{{E}_{{{\mathbf{p}}_{{}_{0}}}-\mathbf{q}/2}}}}\right)$

and,

$\displaystyle
B_{xx}^{-+}\left(\mathbf{q},\varepsilon\right)=\frac{1}{12\pi}U_{x}^{2}\,\sqrt{{{\Delta}^{2}}\left({{\mathbf{p}}_{0}}-\mathbf{q}\right)+{{\mu}^{2}}}{{p}_{0}}^{3}F_{n}^{2}\left({{p}_{0}}\right)$
(28)
$\displaystyle\left(1-{{\text{e}}^{-2\beta{{E}_{{{\mathbf{p}}_{{}_{0}}}+\mathbf{q}/2}}}}-{{\text{e}}^{-2\beta{{E}_{{{\mathbf{p}}_{{}_{0}}}-\mathbf{q}/2}}}}\right)$
$\displaystyle\left(1+\frac{{{\xi}_{{{\mathbf{p}}_{0}}+\mathbf{q}/2}}{{\xi}_{{{\mathbf{p}}_{0}}-\mathbf{q}/2}}}{{{E}_{{{\mathbf{p}}_{0}}+\mathbf{q}/2}}{{E}_{{{\mathbf{p}}_{0}}-\mathbf{q}/2}}}+\frac{1}{2}\left(1-\frac{{{\xi}_{{{\mathbf{p}}_{0}}+\mathbf{q}/2}}}{{{E}_{{{\mathbf{p}}_{0}}+\mathbf{q}/2}}}\right)\left(1-\frac{{{\xi}_{{{\mathbf{p}}_{0}}-\mathbf{q}/2}}}{{{E}_{{{\mathbf{p}}_{0}}-\mathbf{q}/2}}}\right)\right)$

where
${{p}_{0}}=\sqrt{\left(2{{\Delta}^{2}}(\textbf{q})+2{{\mu}^{2}}-\varepsilon\sqrt{{{\Delta}^{2}}(\textbf{q})+{{\mu}^{2}}}\right)/\mu}$.
Substituting above equations to Eq. (20), (21) and performing analytic
continuation, the imaginary parts of the diagonal and the off-diagonal parts
of the self-energy matrix are,

$\displaystyle\operatorname{Im}\left[{{\Sigma}_{11}}\left(\mathbf{p},{{\omega}_{m}}\right)\right]=\frac{1}{2}\sum\limits_{\mathbf{q}}{{}}F_{n}^{2}\left(\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left[{{p}_{x}}-\frac{{{q}_{x}}}{2}\right]\left[{{p}_{x}}-\frac{{{q}_{x}}}{2}\right]$
(29)
$\displaystyle\Bigg{(}\left(1-\frac{{{\xi}_{\mathbf{p}-\mathbf{q}}}}{{{E}_{\mathbf{p}-\mathbf{q}}}}\right){{n}_{B}}\left({{\omega}_{m}}-{{E}_{\mathbf{p}-\mathbf{q}}}\right)B_{xx}^{-+}\left(\mathbf{p}-\mathbf{q},{{\omega}_{m}}-{{E}_{\mathbf{p}-\mathbf{q}}}\right)$
$\displaystyle+\left(1+\frac{{{\xi}_{\mathbf{p}-\mathbf{q}}}}{{{E}_{\mathbf{p}-\mathbf{q}}}}\right){{n}_{B}}\left({{\omega}_{m}}+{{E}_{\mathbf{p}-\mathbf{q}}}\right)B_{xx}^{-+}\left(\mathbf{p}-\mathbf{q},{{\omega}_{m}}+{{E}_{\mathbf{p}-\mathbf{q}}}\right)\Bigg{)}$

and,

$\displaystyle\operatorname{Im}\left[{{\Sigma}_{21}}\left(\mathbf{p},{{\omega}_{m}}\right)\right]=\sum\limits_{\mathbf{q}}{{}}F_{n}^{2}\left(\mathbf{p}-\frac{\mathbf{q}}{2}\right)\left[{{p}_{x}}-\frac{{{q}_{x}}}{2}\right]\left[{{p}_{x}}-\frac{{{q}_{x}}}{2}\right]$
(30)
$\displaystyle\frac{{{\Delta}^{2}}\left(\mathbf{p}-\mathbf{q}\right)}{2{{E}_{\mathbf{p}-\mathbf{q}}}}\big{(}{{n}_{B}}\left({{\omega}_{m}}+{{E}_{\mathbf{p}-\mathbf{q}}}\right)B_{xx}^{++}\left(\mathbf{q},{{\omega}_{m}}+{{E}_{\mathbf{p}-\mathbf{q}}}\right)$
$\displaystyle-{{n}_{B}}\left({{\omega}_{m}}-{{E}_{\mathbf{p}-\mathbf{q}}}\right)B_{xx}^{++}\left(\mathbf{q},{{\omega}_{m}}-{{E}_{\mathbf{p}-\mathbf{q}}}\right)\big{)}$

Doing the integration over ${{d}^{3}}q$ in Eq. (29) and (30) in low
temperature limit, one can obtain temperature behaviour of the imaginary part
of the self-energy as,

$\displaystyle\operatorname{Im}\left[{{\Sigma}_{11}}\left(\mathbf{p},{{\omega}_{m}}\right)\right]\propto
e^{-E_{p}/k_{B}T}\left(T^{4}+T^{3}\right)$ (31)

and,

$\displaystyle\operatorname{Im}\left[{{\Sigma}_{21}}\left(\mathbf{p},{{\omega}_{m}}\right)\right]\propto
T^{3}\left(e^{-E_{p}/k_{B}T}-e^{-3E_{p}/k_{B}T}\right)$ (32)

Thus from Eqs. (5), (31) and (32), the variation of the viscous relaxation
rate with temperature can be described as

$\displaystyle\tau_{11}^{-1}\propto e^{-E_{p}/k_{B}T}\left(T^{4}+T^{3}\right)$
(33) $\displaystyle\tau_{21}^{-1}\propto
T^{3}\left(e^{-E_{p}/k_{B}T}-e^{-3E_{p}/k_{B}T}\right)$ (34)

By using Eqs. (4) and (5) into Eq. (3) one can write,

$\displaystyle\eta=\frac{\beta
nm{{v}^{2}}{{\tau}_{\eta}}}{8}\int{d{{\varepsilon}_{p}}}\operatorname{sech}{{\left(\frac{\beta{{E}_{p}}}{2}\right)}^{2}}$
(35)

Integrating of Eq. (35) in the long wavelength limit $(p\to 0)$, we can write
the temperature dependence of shear viscosity as,

$\displaystyle{{\eta}_{11}}\propto
e^{E_{p}/k_{B}T}\left(T^{4}+T^{3}\right)^{-1}$ (36)
$\displaystyle{{\eta}_{21}}\propto
T^{-3}\left(e^{-E_{p}/k_{B}T}-e^{-3E_{p}/k_{B}T}\right)^{-1}$ (37)

Low temperature behaviour of the relaxation rates is shown in Fig. (2) for
${{(p_{F}^{3}{{v}_{x}})}^{-3}}\approx 0.0$ in the strong coupling limit. In
this plot we used low temperature values of the order parameter obtained in
[27]. The difference in $\tau_{11}^{-1}$ and $\tau_{21}^{-1}$ plots is due to
the different integrals in Eqs. (20) and (21) leading to differences in (33)
and (34). The obtained temperature dependences of the relaxation rates and the
shear viscosities have not been reported in the non-interacting $s$-wave
superfluid Fermi gas where we used the same method [40], this novelty is due
to the uniaxially anisotropic ${{p}_{x}}$-wave pairing interaction
$(U_{x}>U_{y}=U_{z})$ and the momentum dependent order parameter. Although
equation form of the temperature dependence is new, the general trend of the
temperature behavior in the shear viscosities is similar to other superfluid
Fermi gases like in Fig. (1) of ref. [34], and in a superfluid Fermi gas in
the unitarity limit [52], as they all diverge at infinite $\beta$.

Figure 2:  The viscous relaxation rates in the unit of
$\frac{{{\varepsilon}_{F}}}{\hbar}$ (${{\varepsilon}_{F}}$ is the Fermi energy
and $T_{F}$ is the Fermi temperature).

##  3 Conclusion

Using the linear response theory and Green’s function method, shear viscosity
of one component $p$-wave superfluid Fermi gas with a uniaxially anisotropic
pairing interaction has been studied. The system was studied in the strong-
coupling limit where fermions get rearranged into a system of composite
bosons, and in the limit of low temperature, where most of composite bosons
are condensed. The viscous relaxation rates
(${{\tau}^{-1}}=\left|2\operatorname{Im}\Sigma\right|$) related to collisions
in the diagonal and off-diagonal parts of the self-energy, have been
calculated, taking into account the ${{p}_{x}}$-wave Cooper channel
interaction. It has been shown that the $\tau_{11}^{-1}$ viscous relaxation
rate has $e^{-E_{p}/k_{B}T}\left(T^{4}+T^{3}\right)$ temperature behavior,
while the $\tau_{21}^{-1}$ varies as
$T^{3}\left(e^{-E_{p}/k_{B}T}-e^{-3E_{p}/k_{B}T}\right)$ with the temperature.
These new forms of temperature dependences in viscous relaxation rates have
never been reported in a non-interacting $s$-wave superfluid Fermi gas where
we used the same method [40]. The general trend of low temperature shear
viscosity stays similar to other superfluid Fermi gases as reported in ref.
[34], and in a superfluid Fermi gas in the unitarity limit [52], i.e. they all
have divergence behavior at infinite $\beta$. This is the main result of our
paper, which is important for ongoing experiential and theoretical works on
${{p}_{x}}$-wave unconventional superfluid Fermi gas in low temperature limit.

## Acknowledgments

E.D. is supported by FAPESP (Brazil) through Grant No. 2018/10813-2.

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