# Superfluid transition temperature and fluctuation theory of
spin-orbit and Rabi coupled fermions with tunable interactions

Philip D. Powell  Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, California 94550, USA  Department of Physics, University of
Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801,
USA  Gordon Baym  Department of Physics, University of Illinois at Urbana-
Champaign, 1110 W. Green Street, Urbana, Illinois 61801, USA  C. A. R. Sá de
Melo  School of Physics, Georgia Institute of Technology, 837 State Street,
Atlanta, Georgia 30332, USA

###### Abstract

We obtain the superfluid transition temperature of equal Rashba-Dresselhaus
spin-orbit- and Rabi-coupled Fermi superfluids, from the Bardeen-Cooper-
Schrieffer (BCS) to Bose-Einstein condensate (BEC) regimes in three dimensions
for tunable $s$-wave interactions. In the presence of Rabi coupling, we find
that spin-orbit coupling enhances (reduces) the critical temperature in the
BEC (BCS) limit. For fixed interactions, we show that spin-orbit coupling can
convert a first-order (discontinuous) phase transition into a second-order
(continuous) phase transition, as a function of Rabi coupling. We derive the
Ginzburg-Landau free energy to sixth power in the superfluid order parameter
to describe both continuous and discontinuous phase transitions as a function
of spin-orbit and Rabi couplings. Lastly, we develop a time-dependent
Ginzburg-Landau fluctuation theory for an arbitrary mixture of Rashba and
Dresselhaus spin-orbit couplings at any interaction strength.

##  I Introduction

The ability to simulate magnetic and other external fields spielman-2009a ;
spielman-2009b ; ketterle-2015 ; spielman-2011 ; zwierlein-2012 ; wang-2012 ;
spielman-2013a ; galitski-2013 ; zhang-2014 ; fallani-2015 ; huang-2016 ;
wu-2016  in cold atomic gases has created the opportunity to explore a wide
variety of new interactions and complex phase structures otherwise
inaccessible in the laboratory. Moreover, the capacity to generate these
synthetic fields in both bosonic and fermionic systems, and to continuously
tune two-body interactions by means of a Feshbach resonance, has opened up a
wonderland of tunable systems, previously restricted to theorists’ dreams. For
example, the possibility of simulating quantum chromodynamics (QCD) on an
optical lattice cirac-2012 ; wiese-2013 ; zohar-2015 ; dalmonte-2016  is a
tantalizing prospect for researchers whose current theoretical tools remain
limited by QCD’s non-perturbative character and the restriction of lattice
techniques to near-zero chemical potential.

Previous theoretical analyses of three-dimensional spin-orbit-coupled Fermi
gases (e.g., 6Li, 40K) have focused mainly on the zero-temperature limit, in
which several exotic phases characterized by unconventional pairing are
expected to emerge zhang-2011 ; zhai-2011 ; pu-2011 ; han-2012a ; seo-2012a ;
seo-2012b . However, the Raman laser platforms currently employed to produce
synthetic spin-orbit fields also induce heating that prevents the realization
of temperatures sufficiently low to observe the superfluid transition in
either the weakly coupled Bardeen-Cooper-Schrieffer (BCS) or the strongly
coupled Bose-Einstein condensate (BEC) regimes zwierlein-2012 ; spielman-2013a
. Thus, while two-body bound states (Feshbach molecules) have been observed in
the BEC limit of 40K spielman-2013a ; doga-2016 , the observation of
superfluid states remain elusive. Future experiments, however, may break this
impasse by employing a new platform currently under development—the radio-
frequency atom chip—which avoids heating of the atom cloud entirely
spielman-2010 . While rf atom chips are somewhat more restricted than the
Raman scheme in the maximum obtainable spin-orbit coupling, its potential to
reach superfluid temperatures is leading to its adoption in the next
generation of experiments probing the topological superfluid phases of spin-
orbit-coupled fermions spielman-private .

One class of systems of particular interest in the context of quantum
simulation is that of Rashba-Dresselhaus spin-orbit-coupled gases zhang-2011 ;
zhai-2011 ; pu-2011 ; han-2012a ; seo-2012a ; seo-2012b ; Juzeliunas-2010 ;
campbell-2011 . These systems are intriguing both because they reflect physics
studied extensively in the context of semiconductors dresselhaus-1955 ;
rashba-1960 , and because they provide a platform for realizing tunable non-
Abelian fields in the laboratory. Thus, while the holy grail of a full optical
simulation of QCD remains years in the future, there do exist notable
analogies between quark matter and cold atomic systems (e.g., non-Abelian
fields, evolution between strongly and weakly coupled limits) within near-term
experimental reach sademelo-2008 ; ozawa-2010 ; powell-2013 ; doga-2018 .
Investigations of spin-orbit-coupled ultracold gases have also included
optical lattices yamamoto-2017 ; ye-2018 ; ye-2018a ; rey-2019 ; yi-2019 ;
yau-2019 ; liu-2019 ; marsiglio-2020 ; hofstetter-2020 ; hofstetter-2021 ,
thus enlarging the number of possible physical systems that can be accessible
experimentally.

To date, most experimental realizations of these systems have adopted equal
Rashba-Dresselhaus couplings spielman-2011 ; zwierlein-2012 ; wang-2012 ;
spielman-2012 , but systems exhibiting Rashba-only couplings have also been
created huang-2016 ; zhang-2016a ; spielman-2021 . Other experiments have
generated spin-orbit coupling dynamically lev-2019  or even created three-
dimensional spin-orbit coupling pan-2021 . Due to the versatility of Rashba-
Dresselhaus coupled systems, the ability to realize these systems in the
laboratory, and the myriad technical challenges inherent in reaching
arbitrarily low temperatures, it is increasingly important to provide a
theoretical framework for guiding and testing these simulators against
experimental probes at realistic (nonzero) temperatures.

This problem bears a close relation to spin-orbit coupling in solids, where
the role of the Rabi frequency is played by an external Zeeman magnetic field.
While a mean-field treatment describes well the evolution from the BCS to the
BEC regime at zero temperature leggett-1980 ; sademelo-1997 , this order of
approximation fails to describe the correct critical temperature of the system
in the BEC regime because the physics of two-body bound states, i.e., Feshbach
molecules, is not captured when the pairing order parameter goes to zero
sademelo-1993 . To remedy this problem, we include the effects of order-
parameter fluctuations in the thermodynamic potential.

In this paper, we investigate the impact of a specific class of spin-orbit
coupling, namely, an equal mixture of Rasha and Dresselhaus terms, on the
superfluid transition temperature of a three-dimensional Rabi-coupled Fermi
gas, but also give general results for an arbitrary mixture of Rashba and
Dresselhaus components. This paper is the longer version of our preliminary
work powell-2017 . We stress that the present results are applicable to both
neutral cold atomic and charged condensed-matter systems. We show that spin-
orbit coupling, in the presence of a Rabi field (or Zeeman field, in solids),
enhances the critical temperature of the superfluid in the BEC regime and
converts a discontinuous first-order phase transition into a continuous
second-order transition, as a function of the Rabi frequency for given two-
body interactions. We analyze the nature of the phase transition in terms of
the Ginzburg-Landau free energy, calculating it to the sixth power of the
superfluid order parameter, as required to describe both discontinuous
transitions as a function of the spin-orbit coupling, Rabi frequency, and two-
body interactions.

This paper is organized as follows. In Sec. II, we describe the Hamiltonian
and action for three-dimensional Fermi gases in the presence of a general
Rashba-Dresselhaus spin-orbit coupling, Rabi field, and tunable $s$-wave
interactions. We also obtain the inverse Green operator that is used in the
calculation of the thermodynamic potential and Ginzburg-Landau theory of
subsequent sections. In Sec. III, we analyze the thermodynamic potential
across the entire BCS-to-BEC evolution, including contributions from both the
mean-field and Gaussian fluctuations, and obtain the order parameter and
number equations. In Sec. IV, we study the combined effects of Rabi fields and
spin-orbit coupling on the superfluid critical temperature, constructing the
finite-temperature phase diagram versus Rabi fields and scattering parameter.
In Sec. V, we present the Ginzburg-Landau (GL) theory for the superfluid order
parameter and investigate further corrections to the critical temperature in
the BEC limit by including interactions between bosonic bound states. The GL
action is obtained to sixth order in the order parameter to allow for the
existence of discontinuous (first-order) phase transitions. In Sec. VI, we
compare our work on the experimentally relevant equal Rashba-Dresselhaus spin-
orbit coupling with earlier work that has considered different forms of
theoretically motivated spin-orbit couplings. In Sec. VII, we conclude and
look toward the future of experimental work in this field.

In the interest of readability, we relegate a number of detailed calculations
to appendices. In Appendix A, we discuss the Hamiltonian and effective
Lagrangian for a general Rashba-Dresselhaus spin-orbit coupling. In Appendix
B, we analyze the saddle-point approximation for general Rashba-Dresselhaus
spin-orbit coupling. In Appendix C, we derive the modified number equation,
including the contribution arising from Gaussian fluctuations, which
renormalizes the chemical potential obtained at the saddle-point level. In
Appendix D, using a general Rashba-Dresselhaus spin-orbit coupling, we obtain
expressions for the coefficients of the Ginzburg-Landau theory up to sixth
order in order parameter.

##  II Hamiltonian and Action

Throughout this paper, we adopt units in which $\hbar=k_{B}=1$. The
Hamiltonian density of a three-dimensional Fermi gas in the presence of
Rashba-Dresselhaus spin-orbit coupling and Rabi field is

${\cal H}({\bf r})={\cal H}_{k}({\bf r})+{\cal H}_{so}({\bf r})+{\cal
H}_{I}({\bf r})-\mu n({\bf r}).$ (1)

The first term in Eq. (1) is the kinetic energy,

$\displaystyle{\cal{H}}_{k}({\bf r})=\sum_{s}\psi^{\dagger}_{s}({\bf
r})\frac{{\hat{\bf k}}^{2}}{2m}\psi_{s}({\bf r}),$ (2)

where ${\hat{\bf k}}=-i{\bm{\nabla}}$ is the momentum operator, $\psi_{s}({\bf
r})$ is the fermion field at position ${\bf r}$ with (real or pseudo-) spin
$s$ and mass $m$. The second term is the spin-orbit interaction,

${\cal H}_{so}({\bf r})=\sum_{ss^{\prime}}\psi_{s}^{\dagger}({\bf
r})\left[{\bf H}_{so}({\hat{\bf
k}})\right]_{ss^{\prime}}\psi_{s^{\prime}}({\bf r}),$ (3)

with the spin-orbit coupling matrix in momentum $({\bf k})$ space being

${\bf H}_{so}(\hat{\bf
k})=\frac{\kappa}{m}({\hat{k}}_{x}{\bm{\sigma}}_{x}+\eta{\hat{k}}_{y}{\bm{\sigma}}_{y})-\frac{\Omega_{R}}{2}\hskip
1.42262pt{\bm{\sigma}}_{z},$ (4)

where $({\bm{\sigma}}_{x},{\bm{\sigma}}_{y},{\bm{\sigma}}_{z})$ are the Pauli
matrices in spin space, $\kappa$ is the momentum transfer to the atoms in a
two-photon Raman process spielman-2013a  or on a radio frequency atom chip
spielman-2010 , $\eta$ is the anisotropy of the Rashba-Dresselhaus field, and
$\Omega_{R}$ is the Rabi frequency. The third term is the two-body $s$-wave
contact interaction,

${\cal H}_{I}({\bf r})=-g\psi^{\dagger}_{\uparrow}({\bf
r})\psi^{\dagger}_{\downarrow}({\bf r})\psi_{\downarrow}({\bf
r})\psi_{\uparrow}({\bf r}),$ (5)

where $g>0$ corresponds to a constant attraction between opposite spins.
Finally, $\mu$ is the chemical potential and $n({\bf
r})=\sum_{s}\psi^{\dagger}_{s}({\bf r})\psi_{s}({\bf r})$ is the local
density. While the general Rashba-Dresselhaus spin-orbit coupling is discussed
in Appendix A, in what follows we focus on the more experimentally relevant
situation of equal Rashba and Dresselhaus couplings ($\eta=0$).

Standard manipulations (see Appendix A) lead to the Lagrangian density,

$\displaystyle{\cal L}({\bf r,\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hskip 1.42262pt\Psi^{\dagger}({\bf r},\tau){\bf
G}^{-1}(\hat{\bf k},\tau)\Psi({\bf r},\tau)+\frac{1}{g}|\Delta({\bf
r},\tau)|^{2}$ (6) $\displaystyle\hskip 14.22636pt+K(\hat{\bf k})\delta({\bf
r}-{\bf r}^{\prime}),$

where $\tau=it$ is the imaginary time, $\Psi=(\psi_{\uparrow}\hskip
2.84526pt\psi_{\downarrow}\hskip 2.84526pt\psi^{\dagger}_{\uparrow}\hskip
2.84526pt\psi^{\dagger}_{\downarrow})^{T}$ is the Nambu spinor, $K(\hat{\bf
k})={\hat{\bf k}}^{2}/2m-\mu$ is the kinetic energy operator with respect to
the chemical potential, and $\Delta({\bf
r},\tau)=-g\langle\psi_{\downarrow}({\bf r},\tau)\psi_{\uparrow}({\bf
r},\tau)\rangle$ is the pairing field describing the formation of pairs of two
fermions with opposite spins. Note that $\mu$ includes the overall positive
shift $\kappa^{2}/2m$ in the single-particle kinetic energies due to spin-
orbit coupling. The inverse Green’s operator appearing in Eq. (6) is

$\displaystyle{\bf G}^{-1}(\hat{\bf
k},\tau)=\begin{pmatrix}\partial_{\tau}-K_{\uparrow}&-i\kappa{\hat{k}}_{x}/m&0&-\Delta\\\
i\kappa{\hat{k}}_{x}/m&\partial_{\tau}-K_{\downarrow}&\Delta&0\\\
0&\Delta^{*}&\partial_{\tau}+K_{\uparrow}&-i\kappa{\hat{k}}_{x}/m\\\
-\Delta^{*}&0&i\kappa{\hat{k}}_{x}/m&\partial_{\tau}+K_{\downarrow}\end{pmatrix},$

where $K_{{\uparrow,\downarrow}}=K(\hat{\bf k})\mp\Omega_{R}/2,$ are the
kinetic energy terms shifted by the Rabi coupling.

As noted above, a mean-field treatment of this Lagrangian fails to correctly
describe the superfluid critical temperature in the BEC regime. However, the
inclusion of Gaussian fluctuations of $\Delta$ captures the effects of two-
body bound states and leads to a physical superfluid transition temperature.
It is to this task that we now turn.

##  III Thermodynamic Potential

The system’s partition function may be expressed in terms of the functional
integral,

${\cal{Z}}=\int{\cal{D}}\Delta{\cal{D}}\Delta^{*}{\cal{D}}\Psi{\cal{D}}\Psi^{\dagger}\hskip
1.42262pte^{-{\cal{S}}},$ (8)

where the Euclidean action is

${\cal S}=\int_{0}^{\beta}d\tau\int d^{3}{\bf r}\hskip 1.42262pt{\cal L}({\bf
r},\tau),$ (9)

$\beta=1/T$ is the inverse temperature, and the Lagrangian density is given by
Eq. (6). Integrating over the fermion fields yields the thermodynamic
potential,

$\Omega=-T\ln{\cal{Z}}=\Omega_{0}+\Omega_{F},$ (10)

where $\Omega_{0}=-T\ln{\cal Z}_{0}=TS_{0}$ is the mean-field (saddle-point)
contribution, for which $\Delta({\bf r},\tau)=\Delta_{0}$, and
$\Omega_{F}=-T\ln{\cal Z}_{F}$ is the contribution arising from order-
parameter fluctuations. Detailed derivations of the thermodynamic potential
for a general Rashba-Dresselhaus spin-orbit coupling, as well as the
associated order parameter and number equations, are given in Appendices B and
C. The contributions to the thermodynamic potential for the experimentally
relevant situation of equal Rashba-Dresselhaus spin-orbit coupling are
discussed below in Sec. III.1 at the mean-field and in Sec. III.2 at the
Gaussian fluctuation level.

###  III.1 Mean-Field Approximation

The mean-field, or saddle-point, term in the thermodynamic potential is

$\Omega_{0}=V\frac{|\Delta_{0}|^{2}}{g}-\frac{T}{2}\sum_{{\bf
k},j}\ln\left[1+e^{-\beta E_{j}({\bf k})}\right]+\sum_{\bf k}\xi_{\bf k},$
(11)

where $\xi_{\bf k}=\varepsilon_{\bf k}-\mu$, $\varepsilon_{\bf k}={\bf
k}^{2}/2m$, and the $E_{j}({\bf k})$, with $j=\\{1,2,3,4\\}$, are the
eigenvalues of the momentum space Nambu Hamiltonian matrix,

${\bf H}_{0}({\bf k})={\bm{\partial}}_{\tau}-{\bf G}^{-1}({\bf
k},\tau)|_{\Delta=\Delta_{0}},$ (12)

where the operator ${\bm{\partial}}_{\tau}={\bf I}\partial_{\tau}$, and ${\bf
I}$ is the identity matrix. The first set of eigenvalues,

$E_{1,2}({\bf k})=\left[\zeta^{2}_{\bf k}\pm 2\sqrt{E_{0,{\bf k}}^{2}h_{\bf
k}^{2}-\left(\frac{\kappa k_{x}}{m}\right)^{2}|\Delta_{0}|^{2}}\hskip
1.42262pt\right]^{1/2}$ (13)

describe quasiparticle excitations, with the plus $(+)$ associated with
$E_{1}$ and the minus $(-)$ with $E_{2}$. The second set of eigenvalues,
$E_{3,4}({\bf k})=-E_{2,1}({\bf k}),$ corresponds to quasiholes. Further,
$\zeta^{2}_{\bf k}=E^{2}_{0,{\bf k}}+h^{2}_{\bf k}$, where $E_{0,{\bf
k}}=\sqrt{\xi^{2}_{\bf k}+|\Delta_{0}|^{2}},$ and $h_{\bf k}=\sqrt{(\kappa
k_{x}/m)^{2}+\Omega^{2}_{R}/4}$ is the magnitude of the combined spin-orbit
and Rabi couplings.

We express the two-body interaction parameter $g$ in terms of the renormalized
$s$-wave scattering length $a_{s}$ via the relation sademelo-1993

$\frac{1}{g}=-\frac{m}{4\pi a_{s}}+\frac{1}{V}\sum_{\bf
k}\frac{1}{2\varepsilon_{\bf k}}.$ (14)

Note that $a_{s}$ is the $s$-wave scattering length in the absence of spin-
orbit and Rabi fields. It is, of course, possible to express $g$, and all
subsequent relations, in terms of a scattering length which is renormalized by
the presence of the spin-orbit and Rabi fields goldbart-2011 ; ozawa-2012 ,
but for both simplicity and the sake of referring to the more experimentally
accessible quantity, we do not do so here.

The order-parameter equation is obtained from the saddle-point condition
$\delta\Omega_{0}/\delta\Delta_{0}^{*}|_{T,V,\mu}=0$, leading to

$\frac{m}{4\pi a_{s}}=\frac{1}{2V}\sum_{\bf
k}\bigg{[}\frac{1}{\varepsilon_{\bf k}}-A_{+}({\bf
k})-\frac{\Omega^{2}_{R}}{4\xi_{\bf k}h_{\bf k}}\hskip 1.42262ptA_{-}({\bf
k})\bigg{]},$ (15)

where we introduced the notation

$A_{\pm}({\bf k})=\frac{1-2n_{1}({\bf k})}{2E_{1}({\bf
k})}\pm\frac{1-2n_{2}({\bf k})}{2E_{2}({\bf k})},$ (16)

with $n_{j}({\bf k})=1/\left[e^{\beta E_{j}({\bf k})}+1\right]$ being the
Fermi function. In addition, the particle number at the saddle point
$N_{0}=-\partial\Omega_{0}/\partial\mu|_{T,V},$ is given by

$N_{0}=\sum_{\bf k}\bigg{\\{}1-\xi_{\bf k}\bigg{[}A_{+}({\bf k})+\frac{(\kappa
k_{x}/m)^{2}}{\xi_{\bf k}h_{\bf k}}\hskip 1.42262ptA_{-}({\bf
k})\bigg{]}\bigg{\\}}.$ (17)

The mean-field temperature $T_{0}$ is determined by solving Eq. (15) for the
given $\mu$. The corresponding number of particles is given by Eq. (17). This
mean-field treatment leads to a transition temperature $\sim
e^{1/k_{F}a_{s}}$, where $k_{F}$ is the Fermi momentum. This result gives the
correct transition temperature on the BCS limit; however, it is unphysical on
the BEC regime for $k_{F}a_{s}\to 0$. In order to find a physical result, we
need to include order-parameter fluctuations, which we now do.

###  III.2 Gaussian Fluctuations

In discussing Gaussian fluctuations, we concentrate on equal Rasha-Dresselhaus
couplings, leaving details for general Rashba-Dresselhaus coupling to Appendix
C.

To obtain the correct superfluid transition temperature in the BEC limit we
must include the physics of two-body bound states near the transition, as
described by the two-particle $T$-matrix nozieres-1985 ; baym-2006 .
Accounting for all two-particle channels, the T-matrix calculation leads to a
two-particle scattering amplitude $\Gamma$, where

$\Gamma^{-1}({\bf q},z)=\frac{m}{4\pi a_{s}}-\frac{1}{2V}\sum_{\bf
k}\bigg{[}\frac{1}{~{}\varepsilon_{\bf
k}}+\sum^{2}_{i,j=1}\alpha_{ij}W_{ij}\bigg{]};$ (18)

$z$ is the complex frequency and

$W_{ij}=\frac{1-n_{i}({\bf k})-n_{j}({\bf k+q})}{z-E_{i}({\bf k})-E_{j}({\bf
k+q})}.$ (19)

At the superfluid phase boundary $\Delta_{0}\to 0$, the eigenvalues appearing
in Eq. (19) reduce to $E_{1,2}({\bf k})=\left||\xi_{\bf k}|\pm h_{\bf
k}\right|$, but it is straightforward to show that ignoring the absolute
values does not result in any change in either the mean-field order parameter
or number equation. Meanwhile, the coefficients

$\displaystyle\alpha_{11}$ $\displaystyle=$ $\displaystyle\alpha_{22}=|u_{\bf
k}u_{{\bf k}+{\bf q}}-v_{\bf k}v^{*}_{{\bf k}+{\bf q}}|^{2},$ (20)
$\displaystyle\alpha_{12}$ $\displaystyle=$ $\displaystyle\alpha_{21}=|u_{\bf
k}v_{{\bf k}+{\bf q}}+u_{{\bf k}+{\bf q}}v_{\bf k}|^{2},$ (21)

are the coherence factors associated with the quasi-particle amplitudes for
$\Delta_{0}=0$:

$u_{\bf k}=\sqrt{\frac{1}{2}\left(1+\frac{\Omega_{R}}{2h_{\bf
k}}\right)},\hskip 14.22636ptv_{\bf
k}=i\sqrt{\frac{1}{2}\left(1-\frac{\Omega_{R}}{2h_{\bf k}}\right)}.$ (22)

The Gaussian fluctuation correction to the thermodynamic potential is

$\Omega_{F}=-T\sum_{{\bf q},iq_{n}}\ln\left[\beta\Gamma({\bf
q},iq_{n})/V\right].$ (23)

over the entire BCS-to-BEC evolution. The fluctuation contribution to the
particle number is therefore $N_{F}=-\partial\Omega_{F}/\partial\mu|_{T,V}$,
where

$N_{F}=\sum_{\bf
q}\int^{\infty}_{-\infty}\frac{d\omega}{\pi}n_{B}(\omega)\left[\frac{\partial\delta({\bf
q},\omega)}{\partial\mu}-\frac{\partial\delta({\bf
q},0)}{\partial\mu}\right]_{T,V},$ (24)

with the phase shift $\delta({\bf q},\omega)$ defined via the relation

$\Gamma({\bf q},\omega\pm i\epsilon)=|\Gamma({\bf q},\omega)|e^{\pm
i\delta({\bf q},\omega)}.$ (25)

When two-body states are present, the fluctuation contribution can be written
as $N_{F}=N_{sc}+N_{b}$, where

$N_{sc}=\sum_{\bf q}\int^{\infty}_{\omega_{tp}({\bf
q})}\frac{d\omega}{\pi}n_{B}(\omega)\left[\frac{\partial\delta({\bf
q},\omega)}{\partial\mu}-\frac{\partial\delta({\bf
q},0)}{\partial\mu}\right]_{T,V}$ (26)

is the number of particles in scattering states, and $\omega_{tp}({\bf q})$ is
the two-particle continuum threshold corresponding to the branch point of
$\Gamma^{-1}({\bf q},z)$ nozieres-1985 ; pethick-2011 ,

$N_{b}=2\sum_{\bf q}n_{B}(E_{bs}({\bf q})-2\mu),$ (27)

is the number of fermions in bound states, where
$n_{B}(\omega)=1/(e^{\beta\omega}-1)$ is the Bose distribution function, and
$E_{bs}({\bf q})$ is the energy of the bound states obtained from
$\Gamma^{-1}({\bf q},z=E-2\mu)=0$, corresponding to a pole in the scattering
amplitude $\Gamma({\bf q},z)$. In the limit of large and negative fermion
chemical potential, the system becomes non-degenerate and $\Gamma^{-1}({\bf
q},z)=0$ becomes the exact eigenvalue equation for the two-body bound state in
the presence of spin-orbit and Rabi coupling doga-2016 . The total number of
fermions, as a function of $\mu$, thus becomes

$N=N_{0}+N_{F},$ (28)

where $N_{0}$ is given in Eq. (17) and $N_{F}$ is the sum of $N_{sc}$ and
$N_{b}$, as discussed above nozieres-1985 ; sademelo-1993 .

##  IV Critical Temperature

We calculate numerically the transition temperature $T_{c}$ between the normal
and uniform superfluid states, as a function of the scattering parameter
$1/k_{F}a_{s}$, by simultaneously solving the order parameter and number
equations (15) and (28). The solutions correspond to the minima of the free
energy, ${\cal F}=\Omega+\mu N$. We do not discuss the cases of Fulde-Ferrell
fulde-1964  or Larkin-Ovchinnikov larkin-1965  nonuniform superfluid phases
since they only exist over a very narrow region of the phase diagram deep in
the BCS regime fulde-1964 ; larkin-1965 , which is not experimentally
accessible for ultracold fermions.

Figure 1:  (Color online) (a) The superfluid transition temperature
$T_{c}/T_{F}$, where $T_{F}$ is the Fermi temperature, vs the scattering
parameter $1/k_{F}a_{s}$ for equal Rashba-Dresselhaus spin-orbit coupling and
two different Rabi coupling strengths, $\Omega_{R}=0$ and $\varepsilon_{F}$.
For $\Omega_{R}=0$ [solid (black) curve], $T_{c}$ is the same as for zero
spin-orbit coupling since the equal spin-orbit field can be gauged away. The
dashed (blue) line shows $T_{c}$ for zero spin-orbit coupling, with
$\Omega_{R}=\varepsilon_{F}$, while the dotted (green) line shows $T_{c}$ for
$\Omega_{R}=\varepsilon_{F}$ and $\tilde{\kappa}=\kappa/k_{F}=0.5$. (b)
$T_{c}$ is drawn at unitarity, $1/k_{F}a_{s}=0$, and in the inset at
$1/k_{F}a_{s}=-2.0$, as a function of
$\widetilde{\Omega}_{R}=\Omega_{R}/\varepsilon_{F}$. The solid (red) curves
represent $\tilde{\kappa}=0$ and the dotted (blue) curves represent
$\tilde{\kappa}=0.5$. Across the dotted (red) curves, the phase transition is
first order.

Figure 1, in which we scale temperatures by the Fermi temperature
$T_{F}=k_{F}^{2}/2m$, shows the effects of spin-orbit and Rabi couplings on
$T_{c}$. The solid (black) line in Fig. 1(a) shows $T_{c}$ versus
$1/k_{F}a_{s}$ for zero Rabi coupling $(\Omega_{R}=0)$ and zero spin-orbit
coupling $\kappa$. If $\Omega_{R}=0$, the spin-orbit coupling can be removed
by a simple gauge transformation, and thus plays no role. In this situation,
the pairing is purely $s$-wave. The dashed (blue) line shows $T_{c}$ for
$\Omega_{R}\neq 0$, with vanishing equal Rashba-Dresselhaus spin-orbit
coupling. We see that for fixed interaction strength, the pair-breaking effect
of the Rabi coupling suppresses superfluidity, compared with $\Omega_{R}=0$;
the Rabi field here plays the pair-breaking role of the Zeeman field in an
superconductor.

With both spin-orbit and Rabi couplings present, the two-particle pairing is
no longer purely singlet $s$-wave, but obtains a triplet $p$-wave component;
the admixture stabilizes the superfluid phase, as shown by the dotted (green)
line. The latter curve shows that in the BEC regime with large positive
$1/k_{F}a_{s}$, the superfluid transition temperature is enhanced by the
presence of spin-orbit and Rabi couplings, a consequence of the reduction of
the bosonic effective mass in the $x$ direction below $2m$. However, for
sufficiently large $\Omega_{R}$, the geometric mean bosonic mass $M_{B}$
increases above $2m$ and $T_{c}$ decreases. This renormalization of the mass
of the bosons can be traced back to a change in the energy dispersion of the
fermions when both spin-orbit coupling and Rabi fields are present.

Figure 1(b) shows $T_{c}$ versus $\Omega_{R}$ for fixed $1/k_{F}a_{s}$, both
with and without equal Rashba-Dresselhaus spin-orbit coupling at
$\kappa=0.5k_{F}$. When both $\kappa$ and $T$ are zero, superfluidity is
destroyed at a critical value of $\Omega_{R}$ corresponding to the Clogston
limit clogston-1962 . At low temperature, the phase transition to the normal
state is first order because the Rabi coupling is sufficiently large to break
singlet Cooper pairs. However, at higher temperatures the singlet $s$-wave
superfluid starts to become polarized by thermally excited quasiparticles that
produce a paramagnetic response. Thus, above the characteristic temperature
indicated by the large (red) dots, the transition becomes second order, as
pointed out by Sarma sarma-1963 . The change in the transition order occurs
not only for $\kappa=0$, but also for nonzero values of $\kappa$ both in the
BCS regime and near unitarity, depending on the choice of parameters, as
illustrated in Fig. 1(b).

The critical temperature for $\kappa\neq 0$ vanishes only asymptotically in
the limit of large $\Omega_{R}$. We note that for $\Omega_{R}=E_{F}$ and
$\kappa=0$, the transition from the superfluid to the normal state is
continuous at unitarity, but very close to a discontinuous transition. In the
range $1.05\lesssim\Omega_{R}/E_{F}\lesssim 1.10$, numerical uncertainties as
$\kappa\to 0$ prevent us from predicting exactly whether the transition at
unitarity is continuous or discontinuous.

Figure 2:  (Color online) Chemical potential at the superfluid critical
temperature ($T_{c}$) for $\tilde{\kappa}=\kappa/k_{F}=0.5$ and various Rabi
fields, ${\widetilde{\Omega}}_{R}=\Omega_{R}/\varepsilon_{F}$.

Figure 2 shows $\mu(T_{c})$ for fixed spin-orbit coupling and several Rabi
couplings. The solid (black) curve, which represents the situation in which no
Rabi field is present, is equivalent to the situation in which spin-orbit
coupling is also absent, as noted in the discussion of Fig. 1. It is evident
that while the Rabi field reduces the chemical potential in the BCS limit, it
also shifts the onset of the system’s evolution to the BEC limit to larger
inverse scattering lengths, and produces a non-monotonic behavior of
$\mu(T_{c})$ near unitarity.

Figure 3 shows $T_{c}$ for equal Rashba-Dresselhaus coupling
$\kappa=0.5k_{F}$, as a function of Rabi field and scattering parameter. We
also superpose the zero-temperature phase diagram to illustrate the different
superfluid ground states of this system. According to the zeros of the lowest
quasiparticle energy $E_{2}({\bf k})$, the uniform superfluid phases that
emerge are seo-2012a  direct gapped with zero rings (line nodes), indirectly
gapped with zero rings, gapless with one ring, and gapless with two rings.

Figure 3:  (Color online) Phase diagram of critical temperature $T_{c}/T_{F}$
vs $1/k_{F}a_{s}$ and $\Omega_{R}/\varepsilon_{F}$ for equal Rashba-
Dresselhaus coupling $\kappa/k_{F}=0.5$. The finite-temperature uniform
superfluid phases reflect those at $T=0$ shown in the background. These phases
are distinguished by the number of rings (line nodes) in the quasiparticle
excitation spectrum [i.e., where $E_{2}({\bf k})=0$] and type of gap: (1)
direct gapped superfluid with zero rings (magenta diamonds), (2) indirect
gapped superfluid with zero rings (red circles), (3) gapless superfluid with
two rings (blue square), and (4) gapless one-ring superfluid (green stars).

Figure 4 shows the fractional number $N_{b}/N$ of bound fermions at $T_{c}$ as
a function of $1/k_{F}a_{s}$ for two sets of external fields. In the BCS (BEC)
regime, the relative contribution to $N$ is dominated by unbound (bound)
fermions. The main effect of spin-orbit and Rabi fields on $N_{b}/N$ is to
shift the location where the two-body bound states emerge. For fixed spin-
orbit coupling (Rabi field) and increasing Rabi field (spin-orbit coupling),
two-body bound states emerge at larger (smaller) scattering parameters. These
shifts are in agreement with the calculated shifts in binding energies of
Feshbach molecules in the presence of equal Rashba-Dresselhaus spin-orbit
coupling and Rabi fields doga-2016 .

Figure 4:  (Color online) Fractional number $N_{b}/N$ of bound fermions as a
function of the interaction parameter $1/k_{F}a_{s}$, for equal Rashba-
Dresselhaus coupling $\kappa/k_{F}=0.5$ and Rabi frequencies
${\widetilde{\Omega}}_{R}=\Omega_{R}/\varepsilon_{F}=0$ (black solid line) and
${\widetilde{\Omega}}_{R}=\Omega_{R}/\varepsilon_{F}=2$ (red dot-dashed line).

##  V Ginzburg-Landau Theory

To further elucidate the effects of fluctuations on the order of the
superfluid transition, as well as to assess the impact of spin-orbit and Rabi
couplings near the critical temperature, we now derive the Ginzburg-Landau
description of the free energy near the transition. In the limit of small
order parameter, the fluctuation action ${\cal S}_{F}$ can be expanded in
powers of the order parameter $\Delta({q})$ beyond Gaussian order. The
expansion of ${\cal S}_{F}$ to quartic order is sufficient to describe the
continuous (second-order) transition in $T_{c}$ versus $1/k_{F}a_{s}$ in the
absence of a Rabi field sademelo-1993 . However, to correctly describe the
first-order transition clogston-1962 ; sarma-1963  at low temperature (Fig.
1), it is necessary to expand the free energy to sixth order in $\Delta$.

The quadratic (Gaussian-order) term in the action is

$\displaystyle{\cal S}_{G}$ $\displaystyle=$ $\displaystyle\beta
V\sum_{q}\frac{|\Delta_{q}|^{2}}{\Gamma({\bf q},z)}.$ (29)

For an order parameter varying slowly in space and time, we may expand
$\Gamma^{-1}$ as

$\Gamma^{-1}({\bf
q},z)=a+\sum_{\ell}c_{\ell}\frac{q^{2}_{\ell}}{2m}-d_{0}z+\cdots,$ (30)

with the sum over $\ell=\\{x,y,z\\}$. The full result, as a functional of
$\Delta({\bf r},\tau)$, has the form

$\displaystyle{\cal S}_{F}$ $\displaystyle=$
$\displaystyle\int_{0}^{\beta}d\tau\int d^{3}{\bf
r}\Big{(}d_{0}\Delta^{*}\frac{\partial}{\partial\tau}\Delta+a|\Delta|^{2}$
(31)
$\displaystyle+\sum_{\ell}c_{\ell}\frac{|\nabla_{\ell}\Delta|^{2}}{2m}+\frac{b}{2}|\Delta|^{4}+\frac{f}{3}|\Delta|^{6}\Big{)}.$

The full time-dependent Ginzburg-Landau action describes systems in and near
equilibrium (e.g., with collective modes). The imaginary part of $d_{0}$
measures the non-conservation of $|\Delta|^{2}$ in time (i.e., the Cooper pair
lifetime). Details of the derivation of ${\cal S}_{F}$ are found in Appendix
D.

We are interested in systems at thermodynamic equilibrium, where the order
parameter is independent of time, that is, $\Delta({\bf r},\tau)=\Delta({\bf
r})$. In this situation, minimizing the free energy $T{\cal S}_{F}$ with
respect to $\Delta^{*}$ yields the Ginzburg-Landau equation,

$\left(-\sum_{\ell}c_{\ell}\frac{\nabla^{2}_{\ell}}{2m}+b|\Delta({\bf
r})|^{2}+f|\Delta({\bf r})|^{4}+a\right)\Delta({\bf r})=0.$ (32)

For $b>0$, the system undergoes a continuous phase transition when $a$ changes
sign. However, when $b<0$, the system is unstable in the absence of $f$. For
$b<0$ and $a>0$, a first-order phase transition occurs when $3b^{2}=16af$.
Positive $f$ stabilizes the system even when $b<0$.

In the BEC regime, where $d_{0}$ is purely real, we define an effective
bosonic wave function $\Psi({\bf r})=\sqrt{d_{0}}\Delta({\bf r})$ to recast
Eq. (32) in the form of the Gross-Pitaevskii equation for a dilute Bose gas,

$\left(-\sum_{\ell}\frac{\nabla^{2}_{\ell}}{2M_{\ell}}+U_{2}|\Psi({\bf
r})|^{2}+U_{3}|\Psi({\bf r})|^{4}-\mu_{B}\right)\Psi({\bf r})=0.$ (33)

Here, $\mu_{B}=-a/d_{0}$ is the bosonic chemical potential,
$M_{\ell}=m(d_{0}/c_{\ell})$ are the anisotropic bosonic masses, and
$U_{2}=b/d_{0}^{2}$ and $U_{3}=f/d_{0}^{3}$ represent contact interactions of
two and three bosons. In the BEC regime, these terms are always positive,
leading to a dilute gas of stable bosons. The boson chemical potential
$\mu_{B}$ is $\approx 2\mu+E_{b}<0$, where $E_{b}=-E_{bs}({\bf q}={\bf 0})$ is
the two-body binding energy in the presence of spin-orbit coupling and Rabi
frequency, obtained from the condition $\Gamma^{-1}({\bf q},E-2\mu)=0$,
discussed earlier.

The anisotropy of the effective bosonic masses, $M_{x}\neq M_{y}=M_{z}\equiv
M_{\perp}$, stems from the anisotropy of the equal Rashba-Dresselhaus spin-
orbit coupling, which together with the Rabi coupling modifies the dispersion
of the constituent fermions along the $x$ direction. In the limit
$k_{F}a_{s}\ll 1$, the many-body effective masses reduce to those obtained by
expanding the two-body binding energy, $E_{bs}({\bf q})\approx-
E_{b}+\sum_{\ell}q^{2}_{\ell}/2M_{\ell},$ and agree with known results
doga-2016 . However, for $1/k_{F}a_{s}\lesssim 2$, many-body and thermal
effects produce deviations from the two-body result.

In the absence of two- and three-body boson-boson interactions, $U_{2}$ and
$U_{3}$, we directly obtain an analytic expression for $T_{c}$ in the Bose
limit from Eq. (27),

$T_{c}=\frac{2\pi}{M_{B}}\left(\frac{n_{B}}{\zeta(3/2)}\right)^{2/3},$ (34)

with $M_{B}=(M_{x}M_{\perp}^{2})^{1/3}$, by noting that $\mu_{B}=0$ or
$E_{bs}({\bf q}={\bf 0})-2\mu=0$, and using the condition that $n_{B}\simeq
n/2$ [with corrections exponentially small in $(1/k_{F}a_{s})^{2}$], where
$n_{B}$ is the density of bosons. In the BEC regime, the results shown in Fig.
1 include the effects of the mass anisotropy, but do not include the effects
of boson-boson interactions.

To account for boson-boson interactions, we adopt the Hamiltonian of Eq. (33)
with $U_{2}\neq 0$, but with $U_{3}=0$, and apply the method developed in Ref.
baym-1999  to show that these interactions further increase $T_{BEC}$ to

$T_{c}(a_{B})=(1+\gamma)T_{BEC},$ (35)

where $\gamma=\lambda n_{B}^{1/3}a_{B}$. Here, $a_{B}$ is the $s$-wave boson-
boson scattering length, $\lambda$ is a dimensionless constant $\sim 1$, and
we use the relation $U_{2}=4\pi a_{B}/M_{B}$. Since $n_{B}=k_{F}^{3}/6\pi^{2}$
and the boson-boson scattering length is $a_{B}=U_{2}M_{B}/4\pi$, we have
$\gamma={\tilde{\lambda}}{\widetilde{M}}_{B}{\widetilde{U}}_{2},$ where
${\widetilde{M}}_{B}=M_{B}/2m,$
${\widetilde{U}}_{2}=U_{2}k_{F}^{3}/\varepsilon_{F},$ and
${\tilde{\lambda}}=\lambda/4(6\pi^{5})^{1/3}\approx\lambda/50.$ For fixed
$1/k_{F}a_{s}$, $T_{c}$ is enhanced by the spin-orbit field, a
$\Omega_{R}$-dependent decrease in the effective boson mass $M_{B}$
($\sim$10-15%), as well as a stabilizing boson-boson repulsion $U_{2}$
($\sim$2-3%), for the parameters used in Fig. 1.

In closing our discussion of the strongly bound BEC limit, we note that in the
absence of spin-orbit coupling, a Gaussian-order calculation of the two-boson
scattering length yields the erroneous Born approximation result
$a_{B}=2a_{s}$. However, an analysis of the $T$-matrix beyond Gaussian order,
which includes the effects of two-body bound states, obtains the correct
result $a_{B}=0.6a_{s}$ at very low densities iskin-2008  and agrees with
four-body calculations petrov-2005 . The same method can be used to estimate
$U_{2}$ or $a_{B}$ beyond the Born approximation discussed above.
Nevertheless, while the precise quantitative relation between $a_{B}$ and
$a_{s}$ in the presence of spin-orbit coupling is yet unknown, the trend of
increasing $T_{c}$ due to spin-orbit coupling has been clearly shown.

##  VI Comparison to earlier work

In this section, we briefly compare our results with earlier investigations of
different types of theoretically motivated spin-orbit couplings, worked in
different dimensions, or at zero temperature. Our results focus mainly on an
analysis of the critical superfluid temperature and the effects thereon of
order-parameter fluctuations for a three-dimensional Fermi gas in the presence
of equal Rashba-Dresselhaus spin-orbit coupling and Rabi fields. The
appendices consider the more general situation of arbitrary Rashba and
Dresselhaus components.

Several works have analyzed the effects of spin-orbit-coupled fermions in
three dimensions at zero temperature zhang-2011 ; zhai-2011 ; pu-2011 ;
han-2012a ; seo-2012a ; seo-2012b ; shenoy-2011 ; feng-2020 ; dellanna-2011 ;
dellanna-2012 . While some authors have described the situation of Rashba-only
couplings shenoy-2011 ; zhang-2011 ; zhai-2011 ; pu-2011 , others have
assessed the case of equal Rashba and Dresselhaus components seo-2012a ;
seo-2012b  or a general mixture of the two han-2012a . It has been
demonstrated that in the absence of a Rabi field, the zero-temperature
evolution from BCS to BEC superfluidity is a crossover for $s$-wave systems,
not only for Rashba-only couplings shenoy-2011 ; zhang-2011 ; zhai-2011 ;
pu-2011 ; han-2012a , but also for arbitrary Rashba and Dresselhaus components
han-2012a . This result directly follows from the fact that the quasiparticle
excitation spectrum remains fully gapped throughout the evolution.

In contrast, the addition of a Rabi field gives rise to topological phase
transitions for Rashba-only couplings zhang-2011  and equal Rashba and
Dresselhaus components seo-2012a ; seo-2012b , a situation which certainly
persists for general Rashba-Dresselhaus couplings. The simultaneous presence
of a general Rashba-Dresselhaus spin-orbit coupling and Rabi fields leads to a
qualitative change in the quasiparticle excitation spectrum and to the
emergence of topological superfluid phases zhang-2011 ; seo-2012a ; seo-2012b
. Two-dimensional systems have also been investigated at zero temperature,
where topological phase transitions have been identified for Rashba-only
tewari-2011  and equal Rashba-Dresselhaus han-2012b  couplings, in the
presence of a Rabi field.

While early papers in this field focused mainly on the zero-temperature limit,
progress toward finite-temperature theories was made first in two dimensions
devreese-2014 ; devreese-2015  and later in three dimensions shenoy-2015 ;
ohashi-2016 ; levin-2015 . The effects of a general Rashba-Dresselhaus spin-
orbit coupling and Rabi field on the Berezenskii-Kosterlitz-Thouless
transition were thoroughly investigated for two-dimensional Fermi gases at
finite temperatures devreese-2014 ; devreese-2015 , including both Rashba-only
and equal Rashba-Dresselhaus spin-orbit couplings as examples.

The superfluid critical temperature in three dimensions was investigated using
a spherical (3D) spin-orbit coupling $\lambda{\bf k}\cdot{\bm{\sigma}}$ in the
absence of a Rabi field shenoy-2015 ; ohashi-2016 , and also for Rashba-only
(2D) couplings in the presence of a Rabi field levin-2015 . In a recent review
article dellanna-2021 , the critical temperature throughout the BCS-BEC
evolution was discussed both in the absence sademelo-1993  and presence
powell-2017  of Rashba-Dresselhaus spin-orbit coupling. In Secs. 5 and 6 of
this review, the authors describe the same method and expressions we obtained
in our earlier preliminary work powell-2017  for the analytical relations
required to obtain the critical temperature at the Gaussian order; they
include, however, only the contribution of bound states discussed earlier in
the literature for Rashba-only spin-orbit coupling without Rabi fields
zhai-2011 . In contrast, here we develop a complete Gaussian theory to compute
the superfluid critical temperature of a three-dimensional Fermi gas in the
presence of both a general Rashba-Dresselhaus (2D) spin-orbit coupling and
Rabi fields. We focus our numerical calculations on the specific situation of
equal Rashba-Dresselhaus components, which is easier to achieve experimentally
in the context of ultracold atoms. Our key results, already announced in our
earlier work powell-2017 , include the contributions of bound and scattering
states at the Gaussian level. As seen in Fig. 4 of this present paper, there
is a wide region of interaction parameters for which the contribution of
scattering states cannot be neglected. Furthermore, unlike previous work
shenoy-2015 ; ohashi-2016 ; levin-2015 ; dellanna-2021 , we provide a
comprehensive analysis of the Ginzburg-Landau fluctuation theory and include
the effects of boson-boson interactions on the superfluid critical temperature
in the BEC regime.

##  VII Conclusion

We have evaluated the superfluid critical temperature throughout the BCS-to-
BEC evolution of three-dimensional Fermi gases in the presence of equal
Rashba-Dresselhaus spin-orbit couplings, Rabi fields, and tunable $s$-wave
interactions. Furthermore, we have developed the Ginzburg-Landau theory up to
sixth power in the order parameter to elucidate the origin of first-order
phase transitions when the spin-orbit field is absent and the Rabi field is
sufficiently large. Lastly, in the appendices, we have presented the finite-
temperature theory of $s$-wave interacting fermions in the presence of a
generic Rashba-Dresselhaus coupling and external Rabi fields, as well as the
corresponding time-dependent Ginzburd-Landau theory near the superfluid
critical temperature.

###### Acknowledgements.

We thank I. B. Spielman for discussions. The research of P.D.P. was supported
in part by NSF Grant No. PHY1305891 and that of G.B. by NSF Grants No.
PHY1305891 and No. PHY1714042. Both G.B. and C.A.R. SdM. thank the Aspen
Center for Physics, supported by NSF Grants No. PHY1066292 and No. PHY1607611,
where part of this work was done. This work was performed under the auspices
of the U.S. Department of Energy by Lawrence Livermore National Laboratory
under Contract No. DE-AC52-07NA27344.

##  Appendix A Hamiltonian and effective Lagrangian for general Rashba-
Dresselhaus spin-orbit coupling

In this appendix, we consider a larger class of spin-coupled fermions in three
dimensions with a general Rashba-Dresselhaus (GRD) coupling. The Hamiltoninan
density for equal Rashba-Dresselhaus (ERD) discussed in Sec. II is a
particular case of the general Rashba-Dresselhaus Hamiltonian density,

${\cal H}({\bf r})={\cal H}_{0}({\bf r})+{\cal H}_{so}({\bf r})+{\cal
H}_{I}({\bf r}).$ (36)

Adopting units in which $\hbar=k_{B}=1$, the independent-particle Hamiltonian
density without spin-orbit coupling is

${\cal H}_{0}({\bf r})=\sum_{\alpha}\left(\frac{|\nabla\psi_{\alpha}({\bf
r})|^{2}}{2m_{\alpha}}-\mu_{\alpha}\psi^{\dagger}_{\alpha}({\bf
r})\psi_{\alpha}({\bf r})\right),$ (37)

where $\psi_{\alpha}$, $m_{\alpha}$, and $\mu_{\alpha}$ are the fermion field
operator, mass, and chemical potentials for internal state $\alpha$,
respectively. The spin-orbit Hamiltonian can be written as

${\cal H}_{so}({\bf r})=-\sum_{i\alpha\beta}\psi^{\dagger}_{\alpha}({\bf
r})\sigma_{i,\alpha\beta}h_{i}({\bf r})\psi_{\beta}({\bf r}),$ (38)

where the ${\bm{\sigma}}_{i}$ are the Pauli matrices in isospin (internal
state) space and ${\bf h}=(h_{x},h_{y},h_{z})$ includes both the spin-orbit
coupling and Zeeman fields. Finally, we consider a two-body $s$-wave contact
interaction,

${\cal H}_{I}({\bf r})=-g\psi^{\dagger}_{\uparrow}({\bf
r})\psi^{\dagger}_{\downarrow}({\bf r})\psi_{\downarrow}({\bf
r})\psi_{\uparrow}({\bf r}),$ (39)

where $g>0$ corresponds to an attractive interaction.

By introducing the pairing field $\Delta({\bf
r},\tau)=-g\langle\psi_{\downarrow}({\bf r},\tau)\psi_{\uparrow}({\bf
r},\tau)\rangle,$ we remove the quartic interaction and obtain the Lagrangian
density,

$\displaystyle{\cal L}({\bf r,\tau})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hskip 1.42262pt\Psi^{\dagger}({\bf r},\tau){\bf
G}^{-1}(\hat{\bf k},\tau)\Psi({\bf r},\tau)+\frac{|\Delta({\bf
r},\tau)|^{2}}{g}$ (40) $\displaystyle\hskip
14.22636pt+\widetilde{K}_{+}(\hat{\bf k})\delta({\bf r}-{\bf r}^{\prime}),$

where we introduced the momentum operator $\hat{\bf k}=-i{\bf\nabla}$, the
Nambu spinor $\Psi=(\psi_{\uparrow}\hskip 2.84526pt\psi_{\downarrow}\hskip
2.84526pt\psi^{\dagger}_{\uparrow}\hskip
2.84526pt\psi^{\dagger}_{\downarrow})^{T}$, and defined
$\widetilde{K}_{\pm}=(\widetilde{K}_{\uparrow}\pm\widetilde{K}_{\downarrow})/2.$
Here, $\widetilde{K}_{\uparrow}=K_{\uparrow}-h_{z},$ and
$\widetilde{K}_{\downarrow}=K_{\downarrow}+h_{z},$ with $K_{\alpha}(\hat{\bf
k})={\hat{\bf k}}^{2}/(2m_{\alpha})-\mu_{\alpha}$ being the kinetic energy
operator of internal state $\alpha$ with respect to its chemical potential.
Lastly, the inverse Green’s operator appearing in Eq. (40) is

$\displaystyle{\bf G}^{-1}(\hat{\bf
k},\tau)=\begin{pmatrix}\partial_{\tau}-\widetilde{K}_{\uparrow}&h^{*}_{\perp}&0&-\Delta\\\
h_{\perp}&\partial_{\tau}-\widetilde{K}_{\downarrow}&\Delta&0\\\
0&\Delta^{*}&\partial_{\tau}+\widetilde{K}_{\uparrow}&-h_{\perp}\\\
-\Delta^{*}&0&-h^{*}_{\perp}&\partial_{\tau}+\widetilde{K}_{\downarrow}\end{pmatrix},$

where $h_{\perp}(\hat{\bf k})=h_{x}(\hat{\bf k})+ih_{y}(\hat{\bf k})$ plays
the role of the spin-orbit coupling, and $h_{z}$ is the Zeeman field along the
$z$ direction.

To make progress, we expand the order parameter about its saddle-point (mean-
field) value $\Delta_{0}$ by writing $\Delta({\bf
r},\tau)=\Delta_{0}+\eta({\bf r},\tau).$ Next, we integrate over the fermionic
fields and use the decomposition ${\bf G}^{-1}(\hat{\bf k},\tau)={\bf
G}_{0}^{-1}(\hat{\bf k},\tau)+{\bf G}_{F}^{-1}(\hat{\bf k},\tau),$ where ${\bf
G}_{0}^{-1}(\hat{\bf k},\tau)$ is the mean-field Green’s operator, given by
Eq. (A) with $\Delta({\bf r},\tau)=\Delta_{0}$, and ${\bf G}_{F}^{-1}(\hat{\bf
k},\tau)$ is the contribution to the inverse Green’s operator arising from
fluctuations. These steps yield the saddle-point Lagrangian density,

${\cal L}_{0}({\bf r},\tau)=-\frac{T}{2V}\hskip 1.42262pt{\rm Tr}\ln(\beta{\bf
G}^{-1}_{0})+\frac{|\Delta_{0}|^{2}}{g}+\widetilde{K}_{+}(\hat{\bf
k})\delta({\bf r}-{\bf r}^{\prime}),$ (42)

and the fluctuation contribution,

${\cal L}_{F}({\bf r},\tau)=-\frac{T}{2V}\hskip 1.42262pt{\rm Tr}\ln({\bf
I}+{\bf G}_{0}{\bf G}^{-1}_{F})+\Lambda({\bf r},\tau)+\frac{|\eta({\bf
r},\tau)|^{2}}{g},$ (43)

resulting in the effective Lagrangian density ${\cal L}_{\rm eff}({\bf
r},\tau)={\cal L}_{0}({\bf r},\tau)+{\cal L}_{F}({\bf r},\tau).$ In the
expressions above, we work in a volume $V$ and take traces over both discrete
and continuous indices. Notice that the term $\Lambda({\bf
r},\tau)=\left[\Delta_{0}\eta^{*}({\bf r},\tau)+\Delta_{0}^{*}\eta({\bf
r},\tau)\right]/g$ in the fluctuation Lagrangian cancels out the linear terms
in $\eta$ and $\eta^{*}$ when the logarithm is expanded, due to the saddle
point condition

$\frac{\delta S_{0}}{\delta\Delta_{0}^{*}}=0,$ (44)

where $S_{0}=\int_{0}^{\beta}{d\tau}d^{3}{\bf r}{\cal L}_{0}({\bf r},\tau)$ is
the saddle-point action.

##  Appendix B Saddle Point Approximation for general Rashba-Dresselhaus spin-
orbit coupling

We first analyze the saddle-point contribution. The saddle-point thermodynamic
potential $\Omega_{0}=-T\ln{\cal Z}_{0}$ can be obtained for the saddle-point
partition function ${\cal Z}=e^{-S_{0}}$ as $\Omega_{0}=TS_{0}$. Transforming
the saddle-point Lagrangian ${\cal L}_{0}$ from Eq. (42) into momentum space
and integrating over spatial coordinates and imaginary time leads to the
saddle-point thermodynamic potential,

$\Omega_{0}=V\frac{|\Delta_{0}|^{2}}{g}-\frac{T}{2}\sum_{{\bf
k},j}\ln(1+e^{-\beta E_{{\bf k},j}})+\sum_{\bf k}\widetilde{K}_{+}({\bf k}),$
(45)

where $K_{\alpha}({\bf k})={\bf k}^{2}/2m_{\alpha}-\mu_{\alpha}$ and the
eigenvalues $E_{{\bf k},j}$ are the poles of ${\bf G}_{0}({\bf k},z)$, with
$j=\\{1,2,3,4\\}$.

Next, we restrict our analysis to mass balanced systems
($m_{\uparrow}=m_{\downarrow}$) in diffusive equilibrium
($\mu_{\uparrow}=\mu_{\downarrow}$). We also consider the general Rashba-
Dresselhaus (GRD) spin-orbit field $h_{\perp}({\bf k})=\kappa(k_{x}+i\eta
k_{y})/m,$ where $\kappa$ and $\eta$ are the magnitude and anisotropy of the
spin-orbit coupling, respectively. Note that this form is equivalent to
another common form of the Rashba-Dresselhaus coupling found in the literature
seo-2012a ; seo-2012b : ${\bf h}_{so}={\bf h}_{R}+{\bf h}_{D}$ where ${\bf
h}_{R}=v_{R}(k_{x}\hat{{\bf y}}-k_{y}\hat{{\bf x}})$ and ${\bf
h}_{D}=v_{D}(k_{x}\hat{{\bf y}}+k_{y}\hat{{\bf x}})$. The two forms are
related via a momentum-space rotation and the correspondences
$\kappa=m(v_{R}+v_{D})$ and $\eta=(v_{R}-v_{D})/(v_{R}+v_{D})$. The equal
Rashba-Dresselhaus limit (ERD) corresponds to $v_{R}=v_{D}=v$, leading to
$\eta=0$ and $\kappa=2mv$. The specific case of equal Rashba-Dresselhaus spin-
orbit coupling discussed in the main part of the paper corresponds to the case
where $\eta=0$, that is, $h_{\perp}({\bf k})=\kappa k_{x}/m.$

For the general Rashba-Dresselhaus case, the four eigenvalues are

$\displaystyle E_{1,2}({\bf k})$ $\displaystyle=$
$\displaystyle\left[\zeta_{\bf k}^{2}\pm 2\sqrt{E_{0,{\bf k}}^{2}h_{\bf
k}^{2}-|\Delta_{0}|^{2}|h_{\perp}({\bf k})|^{2}}\right]^{1/2},$ (46)
$\displaystyle E_{3,4}({\bf k})$ $\displaystyle=$ $\displaystyle-E_{2,1}({\bf
k}),$ (47)

where the $+$ $(-)$ sign within the outermost square root corresponds to
$E_{1}$ $(E_{2})$, and the functions inside the square roots are $\zeta_{\bf
k}^{2}=E_{0,{\bf k}}^{2}+h_{\bf k}^{2}$, with contributions

$\displaystyle E_{0,{\bf k}}=\sqrt{\xi^{2}_{\bf k}+|\Delta_{0}|^{2}},$ (48)
$\displaystyle h_{\bf k}=\sqrt{|h_{\perp}({\bf k})|^{2}+h^{2}_{z}},$ (49)

where $\xi_{\bf k}=\varepsilon_{\bf k}-\mu,$ and $\varepsilon_{\bf k}={\bf
k}^{2}/2m.$ The order-parameter equation is found from the saddle point
condition $\delta\Omega_{0}/\delta\Delta_{0}^{*}|_{T,V,\mu}=0$. At the phase
boundary between the superfluid and normal phases, $\Delta_{0}\to 0$, and the
order-parameter equation becomes

$\displaystyle\frac{m}{4\pi a_{s}}$ $\displaystyle=$
$\displaystyle\frac{1}{2V}\sum_{\bf k}\bigg{[}\frac{1}{\varepsilon_{\bf
k}}-\frac{\tanh(\beta E_{1}/2)}{2E_{1}}-\frac{\tanh(\beta E_{2}/2)}{2E_{2}}$
$\displaystyle-$ $\displaystyle\frac{h^{2}_{z}}{\xi_{\bf k}h_{\bf k}}\hskip
1.42262pt\left(\frac{\tanh(\beta E_{1}/2)}{2E_{1}}-\frac{\tanh(\beta
E_{2}/2)}{2E_{2}}\right)\bigg{]},$

after expressing the interaction parameter $g$ in terms of the $s$-wave
scattering length via the relation

$\frac{1}{g}=-\frac{m}{4\pi a_{s}}+\frac{1}{V}\sum_{\bf
k}\frac{1}{2\varepsilon_{\bf k}}.$ (51)

We note that $a_{s}$ is the $s$-wave scattering length in the absence of spin-
orbit and Zeeman fields. It is, of course, possible to express all relations
obtained in terms of a scattering length which is renormalized by the presence
of the spin-orbit and Rabi fields goldbart-2011 ; ozawa-2012 . However, in
addition to complicating our already cumbersome expressions, it would make
reference to a quantity that is more difficult to measure experimentally and
that would hide the explicit dependence of the properties that we analyze in
terms of the spin-orbit and Rabi fields, so we do not consider such
complications here. Note that since $\Delta_{0}=0$ at the phase boundary, the
eigenvalues in Eq. (46) reduce to $E_{1}({\bf k})=\Big{|}|\xi_{\bf k}|+h_{\bf
k}\Big{|}$, $E_{2}({\bf k})=\Big{|}|\xi_{\bf k}|-h_{\bf k}\Big{|}$, which is
the absolute value of the normal-state energy dispersions. However, it is
straightforward to show that ignoring the absolute values does not result in
any change in either the mean-field order parameter given by Eq. (B) or number
equation shown in Eq. (B), when $\Delta_{0}\to 0$.

The saddle-point critical temperature $T_{0}$ is determined by solving Eq. (B)
subject to the thermodynamic constraint
$N_{0}=-\partial\Omega_{0}/\partial\mu|_{T,V},$ which yields

$\displaystyle N_{0}$ $\displaystyle=$ $\displaystyle\sum_{\bf
k}\bigg{\\{}1-\xi_{\bf k}\bigg{[}\frac{1}{\varepsilon_{\bf
k}}+\frac{\tanh(\beta E_{1}/2)}{2E_{1}}+\frac{\tanh(\beta E_{2}/2)}{2E_{2}}$
$\displaystyle+$ $\displaystyle\frac{|h_{\perp}({\bf k})|^{2}}{\xi_{\bf
k}h_{\bf k}}\hskip 1.42262pt\left(\frac{\tanh(\beta
E_{1}/2)}{2E_{1}}-\frac{\tanh(\beta
E_{2}/2)}{2E_{2}}\right)\bigg{]}\bigg{\\}}.$

A mean-field description of the system, which involves a simultaneous solution
of Eqs. (B) and (B), yields the asymptotically correct description of the
system in the BCS limit; however, such a description fails miserably in the
BEC regime where it does not account for the formation of two-body bound
states. The general Rashba-Dresselhaus spin-orbit saddle-point equations (B)
and (B) reduce to the equal Rashba-Dresselhaus equations (15) and (17) of the
main part of the paper with the explicit use of $h_{z}=\Omega_{R}/2$ and
$h_{\perp}({\bf k})=\kappa k_{x}/m$, where $\Omega_{R}$ is the Rabi coupling.

##  Appendix C Derivation of the modified number equation with Gaussian
fluctuations

We begin by deriving the modified number equation arising from Gaussian
fluctuations of the order parameter near the superfluid phase boundary. The
fluctuation thermodynamic potential $\Omega_{F}$ results from the Gaussian
integration of the fields $\eta({\bf r},\tau)$ and $\eta^{*}({\bf r},\tau)$ in
the fluctuation partition function ${\cal Z}_{F}=\int d\eta^{*}d\eta
e^{-S_{F}}$, where the action $S_{F}=\int d\tau_{0}^{\beta}\int d^{3}{\bf
r}{\cal L}_{F}({\bf r},\tau)$ is calculated to quadratic order. The
contribution to the thermodynamic potential due to Gaussian fluctuations is

$\Omega_{F}=-T\sum_{iq_{n},{\bf q}}\ln\left[\beta\Gamma({\bf
q},iq_{n})/V\right]$ (53)

where $q_{n}=2\pi nT$ are the bosonic Matsubara frequencies and $\Gamma({\bf
q},iq_{n})$ is directly related to the pair fluctuation propagator
$\chi_{pair}({\bf q},iq_{n})=V\Gamma^{-1}({\bf q},iq_{n}).$

The Matsubara sum can be evaluated via contour integration,

$\Omega_{F}=-T\sum_{\bf q}\oint_{\cal{C}}\frac{dz}{2\pi i}\hskip
1.42262ptn_{B}(z)\ln\left[\beta\Gamma({\bf q},z)/V\right],$ (54)

where $n_{B}(z)=1/(e^{z}-1)$ is the Bose function and the countour ${\cal{C}}$
encloses all of the Matsubara poles of the Bose function. Next, we deform the
contour around the Matsubara frequencies towards infinity, taking into account
the branch cut and the possibility of poles coming from the logarithmic term
inside the countour integral. We take the branch cut to be along the real
axis, then add and subtract the pole at $iq_{n}=0$ to obtain

$\Omega_{F}=-T\sum_{\bf q}\int^{\infty}_{-\infty}\frac{d\omega}{\pi}\hskip
1.42262ptn_{B}(\omega)\left[\delta({\bf q},\omega)-\delta({\bf q},0)\right],$
(55)

where the phase shift $\delta({\bf q},\omega)$ is defined via $\Gamma({\bf
q},\omega\pm i\epsilon)=|\Gamma({\bf q},\omega)|e^{\pm i\delta({\bf
q},\omega)},$ and arises from the contour segments above and below the real
axis.

The thermodynamic identity $N=-\partial\Omega/\partial\mu|_{T,V}$ then yields
to the fluctuation correction,

$N_{F}=T\sum_{\bf q}\int^{\infty}_{-\infty}\frac{d\omega}{\pi}\hskip
1.42262ptn_{B}(\omega)\left[\frac{\partial\delta({\bf
q},\omega)}{\partial\mu}-\frac{\partial\delta({\bf
q},0)}{\partial\mu}\right],$ (56)

to the the saddle-point number equation, and has a similar analytical
structure as in the case without spin-orbit and Zeeman fields nozieres-1985 ;
sademelo-1993 . Thus, we can write the final number equation at the critical
temperature $T_{c}$ as $N=N_{0}+N_{F}$. Since the phase shift $\delta({\bf
q},z)$ vanishes everywhere that $\Gamma({\bf q},z)$ is analytic, the only
contributions to Eq. (56) arise from a possible isolated pole at
$\omega_{p}({\bf q})$ and a branch cut extending from the two-particle
continuum threshold $\omega_{tp}({\bf q})=\min_{\\{i,j,{\bf
k}\\}}\left[E_{i}({\bf k})+E_{j}({\bf k+q})\right]$ to $z\to\infty$ along the
positive real axis. The explicit form of $\Gamma({\bf q},z)$ can be extracted
from Eq. (74) of Appendix D.

When there is a pole corresponding to the emergence of a two-body bound state,
we can explicitly write $\Gamma({\bf q},z)\sim R({\bf q})/(z-\omega_{p}({\bf
q})),$ from which we obtain $\partial\delta({\bf
q},\omega)/\partial\mu=2\delta(z-\omega_{p}({\bf q})),$ leading to the bound
state density

$N_{b}=2\sum_{\bf q}n_{B}(\omega_{p}({\bf q})),$ (57)

where the energy $\omega_{p}({\bf q})$ must lie below the two-particle
continuum threshold $\omega_{tp}({\bf q})$. The factor of 2, which arises
naturally, is due to the two fermions comprising a bosonic molecule.
Naturally, the presence of this term in the fluctuation-modified number
equation is dependent upon the existence of such a pole, that is, a molecular
bound state. These bound states correspond to the Feshbach molecules in the
presence of spin-orbit coupling and Zeeman fields spielman-2013a ; doga-2016 .

Having extracted the pole contribution to Eq. (56), when it exists, the
remaining integral over the branch cut corresponds to scattering state
fermions,

$N_{sc}=T\sum_{\bf q}\int^{\infty}_{\omega_{tp}({\bf
q})}\frac{d\omega}{\pi}\hskip
1.42262ptn_{B}(\omega)\left[\frac{\partial\delta({\bf
q},\omega)}{\partial\mu}-\frac{\partial\delta({\bf
q},0)}{\partial\mu}\right],$ (58)

whose energy is larger than the minimum energy $\omega_{tp}({\bf q})$ of two
free fermions. Thus, when bound states are present, we arrive at the modified
number equation,

$N=N_{0}+N_{sc}+N_{b}$ (59)

where $N_{0}$ is the number of free fermions obtained from the saddle-point
analysis in Eq. (B), and $N_{b}$ and $N_{sc}$ are the bound state and
scattering contributions given in Eqs. (57) and (58), respectively. These
general results are particularized to the equal Rashba-Dresselhaus case in
Sec. III.2 of this paper.

The number of unbound states $N_{u}$ is then easily seen to be
$N_{u}=N_{0}+N_{sc}$, that is, the sum of the free-fermion $(N_{0})$ and
scattering $(N_{sc})$ contributions. Naturally, the number of unbound states
is also equal to the total number of states, $N$, minus the number of bound
states, $N_{b}$, that is, $N_{u}=N-N_{b}$.

##  Appendix D Derivation of Ginzburg-Landau coefficients for general Rashba-
Dresselhaus spin-orbit coupling

Next, we derive explicit expressions for the coefficients of the time-
dependent Ginzburg-Landau theory valid near the critical temperature of the
superfluid. We start from the fluctuation Lagrangian,

${\cal L}_{F}({\bf r},\tau)=-\frac{T}{2V}\hskip 1.42262pt{\rm Tr}\ln({\bf
I}+{\bf G}_{0}{\bf G}^{-1}_{F})+\Lambda({\bf r},\tau)+\frac{|\eta({\bf
r},\tau)|^{2}}{g},$ (60)

in a volume $V$, and take the traces over both discrete and continuous
indices. Notice that the term $\Lambda({\bf
r},\tau)=\left[\Delta_{0}\eta^{*}({\bf r},\tau)+\Delta_{0}^{*}\eta({\bf
r},\tau)\right]/g$ in the fluctuation Lagrangian cancels out the linear terms
in $\eta$ and $\eta^{*}$ when the logarithm is expanded, due to the saddle-
point condition. Since the expansion is performed near $T_{c}$, we take the
saddle-point order parameter $\Delta_{0}\to 0$ and redefine the fluctuation
field as $\eta({\bf r},\tau)=\Delta({\bf r},\tau)$ to obtain

${\cal L}_{F}({\bf r},\tau)=\frac{|\Delta|^{2}}{g}-\frac{T}{2V}{\rm
Tr}\ln({\bf I}+{\bf G}_{0}[0]{\bf G}^{-1}_{F}[\Delta]).$ (61)

Notice that the arguments in ${\bf G}_{0}[0]$ and ${\bf G}^{-1}_{F}[\Delta]$
represent the values of $\Delta_{0}=0$ and $\eta=\Delta$, respectively.

We expand the logarithm to sixth order in $\Delta$ to obtain

$\displaystyle{\cal L}_{F}({\bf r},\tau)$ $\displaystyle=$
$\displaystyle\frac{|\Delta|^{2}}{g}+\frac{T}{2V}{\rm
Tr}\bigg{[}\frac{1}{2}({\bf G}_{0}{\bf G}^{-1}_{F})^{2}+\frac{1}{4}({\bf
G}_{0}{\bf G}^{-1}_{F})^{4}$ (62) $\displaystyle+\frac{1}{6}({\bf G}_{0}{\bf
G}^{-1}_{F})^{6}+...\bigg{]},$

where the higher-order odd (cubic and quintic) terms in the order-parameter
amplitudes expansion can be shown to vanish due to conservation laws and
energy or momentum considerations.

The traces can be evaluated explicitly by using the momentum-space inverse
single-particle Green’s function

${\bf G}^{-1}_{0}(k,k^{\prime})=\begin{pmatrix}{\bf A}^{-1}(k)&{\bf 0}\\\ {\bf
0}&-\left[{\bf A}^{-1}(-k)\right]^{T}\end{pmatrix}\delta_{kk^{\prime}},$ (63)

derived from Eq. (A). Here, we use the shorthand notation
$k\equiv(i\omega,{\bf k})$, where $\omega_{n}=2\pi nT$ are bosonic Matsubara
frequencies and define the $2\times 2$ matrix,

${\bf A}^{-1}(k)=\begin{pmatrix}i\omega_{n}-\widetilde{K}_{\uparrow}({\bf
k})&h^{*}_{\perp}({\bf k})\\\ h_{\perp}({\bf
k})&i\omega_{n}-\widetilde{K}_{\downarrow}({\bf k})\end{pmatrix},$ (64)

where $\widetilde{K}_{\uparrow}=\xi_{\mathbf{k}}-h_{z}$,
$\widetilde{K}_{\downarrow}=\xi_{\mathbf{k}}+h_{z}$, with
$\xi_{\mathbf{k}}=\mathbf{k}^{2}/2m-\mu$ the kinetic energy relative to the
chemical potential, $h_{z}$ the external Zeeman field, and $h_{\perp}({\bf
k})=h_{x}({\bf k})+ih_{y}({\bf k})$ the spin-orbit field. We also define the
fluctuation contribution to the inverse Green’s function,

${\bf G}^{-1}_{F}(k,k^{\prime})=\begin{pmatrix}{\bf
0}&-i\bm{\sigma}_{y}\Delta_{k-k^{\prime}}\\\
i\bm{\sigma}_{y}\Delta^{\dagger}_{k^{\prime}-k}&{\bf 0}\end{pmatrix},$ (65)

where $\bm{\sigma}_{y}$ is the second Pauli matrix in isospin (internal state)
space and

$\Delta_{k}=\frac{\beta}{V}\int_{0}^{\beta}d\tau\int d^{3}{\bf r}e^{i({\bf
k}\cdot{\bf r}-\omega\tau)}\Delta(r)$ (66)

is the Fourier transform of $\Delta(r)$, with $r\equiv({\bf r},\tau)$, and
also has dimensions of energy. Recall that we set $\hbar=k_{B}=1$, such that
energy, frequency and temperature have the same units.

Inversion of Eq. (63) yields

${\bf G}_{0}(k,k^{\prime})=\begin{pmatrix}{\bf A}(k)&{\bf 0}\\\ {\bf
0}&-\left[{\bf A}(-k)\right]^{T}\end{pmatrix}\delta_{kk^{\prime}},$ (67)

where the matrix ${\bf A}(k)$ is

${\bf A}(k)=\frac{1}{\det[{\bf
A}^{-1}(k)]}\begin{pmatrix}i\omega_{n}-\widetilde{K}_{\downarrow}({\bf
k})&-h^{*}_{\perp}({\bf k})\\\ -h_{\perp}({\bf
k})&i\omega_{n}-\widetilde{K}_{\uparrow}{\bf k})\end{pmatrix}.$ (68)

with $\det[{\bf A}^{-1}(k)]=\prod^{2}_{j=1}\left[i\omega_{n}-E_{j}({\bf
k})\right]$ and where the independent-particle eigenvalues $E_{j}({\bf k})$
are two of the poles of ${\bf G}_{0}(k,k)$. These poles are exactly the
general eigenvalues described in Eqs. (46) in the limit of $\Delta_{0}\to 0$.
Note that setting $\Delta_{0}=0$ in the general eigenvalue expressions yields
$E_{1,2}({\bf k})=\left||\xi_{\bf k}|\pm h_{\bf k}\right|$. The other set of
poles of ${\bf G}_{0}(k,k)$ corresponds to the eigenvalues $E_{3,4}({\bf
k})=-E_{2,1}({\bf k})$ found from $\det\left[{\bf A}^{-1}(-k)\right]^{T}=0$.

Using Eq. (62) to write the fluctuation action as ${\cal
S}_{F}=\int_{0}^{\beta}d\tau\int d^{3}{\bf r}{\cal L}_{F}({\bf r},\tau),$
results in

$\displaystyle{\cal S}_{F}$ $\displaystyle=$ $\displaystyle\beta
V\sum_{q}\frac{|\Delta_{q}|^{2}}{\Gamma(q)}+\frac{\beta
V}{2}\sum_{q_{1},q_{2},q_{3}}b_{1,2,3}\Delta_{1}\Delta^{*}_{2}\Delta_{3}\Delta^{*}_{1-2+3}\hskip
14.22636pt$ (69) $\displaystyle+\frac{\beta V}{3}\sum_{q_{1}\cdots
q_{5}}f_{1\cdots
5}\Delta_{1}\Delta^{*}_{2}\Delta_{3}\Delta^{*}_{4}\Delta_{5}\Delta^{*}_{1-2+3-4+5},$

where summation over $q\equiv(iq_{n},{\bf q})$ indicates sums over both the
bosonic Matsubara frequencies $q_{n}=2\pi nT$ and momentum ${\bf q}$. Here, we
used the shorthand notation $j\equiv q_{j}$ to represent the labels of
$\Delta_{q_{j}}$ or $\Delta_{q_{j}}^{*}$.

The quadratic order appearing in Eq. (69) arises from the terms $|\Delta({\bf
r},\tau)|^{2}/g$ and $(T/2V){\rm Tr}({\bf G}_{0}{\bf G}^{-1}_{F})^{2}/2$ in
Eq. (62), and is directly related to the pair propagator
$\chi_{pair}(q)=V\Gamma^{-1}(q)$, with

$\Gamma^{-1}(q)=\frac{1}{g}-\frac{T}{2V}\sum_{k}\frac{{\rm Tr}\left[{\bf
A}(k){\bf A}^{-1}(q-k)\right]}{\det[{\bf A}^{-1}(q-k)]},$ (70)

where we use the identity $\bm{\sigma}_{y}{\bf A}\bm{\sigma}_{y}={\rm
det}({\bf A})({\bf A}^{T})^{-1}.$ The fourth-order contribution arises from
$\frac{1}{4}({\bf G}_{0}{\bf G}^{-1}_{F})^{4}$ and leads to

$b(q_{1},q_{2},q_{3})=\frac{T}{2V}\sum_{k}\frac{{\rm Tr}\left[{\bf A}(k){\bf
A}^{-1}(q_{1}-k){\bf A}(k-q_{1}+q_{2}){\bf
A}^{-1}(q_{1}-q_{2}+q_{3}-k)\right]}{{\rm det}\left[{\bf
A}^{-1}(q_{1}-k)\right]{\rm det}\left[{\bf
A}^{-1}(q_{1}-q_{2}+q_{3}-k)\right]},$ (71)

while the sixth order contribution emergences from $\frac{1}{6}({\bf
G}_{0}{\bf G}^{-1}_{F})^{6}$, giving

$\displaystyle f(q_{1},\cdots,q_{5})$ $\displaystyle=$
$\displaystyle\frac{T}{2V}\sum_{k}{\rm det}\left[{\bf A}(q_{1}-k)\right]{\rm
det}\left[{\bf A}(q_{1}-q_{2}+q_{3}-k)\right]{\rm det}\left[{\bf
A}(q_{1}-q_{2}+q_{3}-q_{4}+q_{5}-k)\right]$ (72) $\displaystyle\hskip
42.67912pt\times{\rm Tr}\bigg{[}{\bf A}(k){\bf A}^{-1}(q_{1}-k){\bf
A}(k-q_{1}+q_{2}){\bf A}^{-1}(q_{1}-q_{2}+q_{3}-k)$ $\displaystyle\hskip
71.13188pt\times{\bf A}(k-q_{1}+q_{2}-q_{3}+q_{4}){\bf
A}^{-1}(q_{1}-q_{2}+q_{3}-q_{4}+q_{5}-k)\bigg{]}.$

Evaluating the expressions given in Eqs. (70) through (72) requires us to
perform summations over Matsubara frequencies of the type

$T\sum_{i\omega_{n}}\frac{1}{i\omega_{n}\pm E({\bf k})}=\begin{cases}\hskip
8.53581ptn({\bf k})\hskip 17.07164pt\mbox{if ``$+$''}\\\ 1-n({\bf k})\hskip
8.53581pt\mbox{if ``$-$''}\\\ \end{cases},$ (73)

where $n({\bf k})=1/\left[e^{\beta E({\bf k})}+1\right]$ is the Fermi
function. For the quadratic term, we obtain the result

$\displaystyle\Gamma^{-1}({\bf q},iq_{n})$ $\displaystyle=$
$\displaystyle-\frac{m}{4\pi a_{s}}+\frac{1}{2V}\sum_{\bf
k}\bigg{[}\frac{1}{\varepsilon_{\bf k}}$ (74) $\displaystyle+$
$\displaystyle\sum_{i,j=1}^{2}\alpha_{ij}({\bf k},{\bf q})W_{ij}({\bf k},{\bf
q},iq_{n})\bigg{]},$

where the functions in the last term are

$W_{ij}({\bf k},{\bf q},iq_{n})=\frac{1-n_{i}({{\bf k}})-n_{j}({\bf k}+{\bf
q})}{iq_{n}-E_{i}({\bf k})-E_{j}({\bf k}+{\bf q})},$ (75)

corresponding to the contribution of bubble diagrams to the pair
susceptibility. The coherence factors are

$\displaystyle\alpha_{11}({\bf k},{\bf q})$ $\displaystyle=$
$\displaystyle|u_{\bf k}u_{{\bf k}+{\bf q}}-v_{\bf k}v^{*}_{{\bf k}+{\bf
q}}|^{2},$ (76) $\displaystyle\alpha_{12}({\bf k},{\bf q})$ $\displaystyle=$
$\displaystyle|u_{\bf k}v_{{\bf k}+{\bf q}}+u_{{\bf k}+{\bf q}}v_{\bf
k}|^{2},$ (77)

with $\alpha_{11}({\bf k},{\bf q})=\alpha_{22}({\bf k},{\bf q})$ and
$\alpha_{12}({\bf k},{\bf q})=\alpha_{21}({\bf k},{\bf q}),$ where the
quasiparticle amplitudes are

$\displaystyle u_{\bf k}=\sqrt{\frac{1}{2}\left(1+\frac{h_{z}}{h_{\bf
k}}\right)},$ (78) $\displaystyle v_{\bf k}=e^{i\theta_{\bf
k}}\sqrt{\frac{1}{2}\left(1-\frac{h_{z}}{h_{\bf k}}\right)}.$ (79)

The angle $\theta_{\bf k}$ is the phase associated with the spin-orbit field
$h_{\perp}({\bf k})=|h_{\perp}({\bf k})|e^{i\theta_{\bf k}},$ and we replaced
the interaction parameter $g$ by the $s$-wave scattering length $a_{s}$ via
Eq. (51), recalling that $\varepsilon_{\bf k}={\bf k}^{2}/2m$. The phase and
modulus of $h_{\perp}({\bf k})$ are

$\displaystyle\theta_{\bf k}=\arctan\left(\frac{\eta k_{y}}{k_{x}}\right),$
(80) $\displaystyle|h_{\perp}({\bf k})|=\frac{|\kappa|}{m}\sqrt{k_{x}^{2}+\eta
k_{y}^{2}},$ (81)

and the total effective field is

$h_{\bf k}=\sqrt{h_{z}^{2}+|h_{\perp}({\bf k})|^{2}}.$ (82)

Since we are interested only in the long-wavelength and low-frequency regime,
we perform an analytic continuation to real frequencies
$iq_{n}=\omega+i\delta$ after calculating the Matsubara sums for all
coefficients appearing in Eq. (69) and perform a small momentum ${\bf q}$ and
low-frequency $\omega$ expansion resulting in the Ginzburg-Landau action,

$\displaystyle{\cal S}_{F}={\cal S}_{GL}$ $\displaystyle=$ $\displaystyle\beta
V\sum_{q}\left(a+\sum_{\ell}c_{\ell}\frac{q^{2}_{\ell}}{2m}-d_{0}\omega\right)|\Delta_{q}|^{2}+\frac{\beta
V}{2}\sum_{q_{1},q_{2},q_{3}}b(q_{1},q_{2},q_{3})\Delta_{q_{1}}\Delta^{*}_{q_{2}}\Delta_{q_{3}}\Delta^{*}_{q_{1}-q_{2}+q_{3}}\hskip
14.22636pt$ (83) $\displaystyle+\frac{\beta V}{3}\sum_{q_{1}\cdots
q_{5}}f(q_{1},q_{2},q_{3},q_{4},q_{5})\Delta_{q_{1}}\Delta^{*}_{q_{2}}\Delta_{q_{3}}\Delta^{*}_{q_{4}}\Delta_{q_{5}}\Delta^{*}_{q_{1}-q_{2}+q_{3}-q_{4}+q_{5}}.$

Here, the label $\ell$ appearing explicitly in the term
$\sum_{\ell}c_{\ell}q_{\ell}^{2}/(2m)$ represents the spatial directions
$\\{x,y,z\\}$, while the $q_{j}$’s in the sums correspond to $({\bf
q}_{j},\omega_{j})$ and the summations $\sum_{q_{j}}$ represent integrals
$\beta V\int d\omega_{j}\int d^{3}{\bf q}_{j}$, where $j$ labels a fermion
pair and can take values in the set $\\{1,2,3,4,5\\}$. In the expression
above, we used the result

$\Gamma^{-1}({\bf
q},\omega)=a+\sum_{\ell}c_{\ell}\frac{q_{\ell}^{2}}{2m}-d_{0}\omega+\cdots$
(84)

for the analytically continued expression of $\Gamma^{-1}({\bf q},iq_{n})$
appearing in Eq. (74). To write the coefficients above in a more compact
notation, we define

$\displaystyle X_{i}=X_{i}({\bf k})=\tanh\left[\beta E_{i}({\bf k})/2\right],$
(85) $\displaystyle Y_{i}=Y_{i}({\bf k})={\rm sech}^{2}\left[\beta E_{i}({\bf
k})/2\right].$ (86)

The frequency- and momentum-independent coefficient is

$\displaystyle a=-\frac{m}{4\pi a_{s}}+\frac{1}{V}\sum_{\bf
k}\bigg{[}\frac{1}{2\varepsilon_{\bf
k}}-\left(\frac{X_{1}}{4E_{1}}+\frac{X_{2}}{4E_{2}}\right)$
$\displaystyle-\frac{h^{2}_{z}}{\xi_{\bf k}h_{\bf
k}}\left(\frac{X_{1}}{4E_{1}}-\frac{X_{2}}{4E_{2}}\right)\bigg{]},$ (87)

where $E_{1}=E_{1}({\bf k})$ and $E_{2}=E_{2}({\bf k})$. The coefficient
$d_{0}=d_{R}+id_{I}$ multiplying the linear term in frequency has a real
component given by

$d_{R}=\frac{1}{2V}{\cal{P}}\sum_{\mathbf{k}}\sum^{2}_{i,j=1}\alpha_{ij}(\mathbf{k},\mathbf{0})\frac{1-n_{i}({\bf
k})-n_{j}({\bf k})}{\left[E_{i}({\bf k})+E_{j}({\bf k})\right]^{2}}.$ (88)

Using the explicit forms of the coherence factors $u_{\bf k}$ and $v_{\bf k}$
that define $\alpha_{ij}({\bf k},{\bf q}={\bf 0})$, the above expression can
be rewritten as

$\displaystyle d_{R}=\frac{1}{2V}\hskip 1.42262pt{\cal{P}}\sum_{\bf
k}\biggr{[}\left(1+\frac{h^{2}_{z}}{\xi^{2}_{\bf
k}}\right)\left(\frac{X_{1}}{4E^{2}_{1}}+\frac{X_{2}}{4E^{2}_{2}}\right)$
$\displaystyle+\frac{2h^{2}_{z}}{\xi_{\bf k}h_{\bf
k}}\left(\frac{X_{1}}{4E^{2}_{1}}-\frac{X_{2}}{4E^{2}_{2}}\right)\biggr{]},$
(89)

which defines the time scale for temporal oscillations of the order parameter.
Here, the symbol ${\cal{P}}$ denotes the principal value, and the coefficient
$d_{R}$ is obtained from

${\rm Re}\left[\Gamma^{-1}({\bf q}={\bf
0},\omega+i\delta)\right]=-\frac{m}{4\pi a_{s}}+\frac{1}{2V}\sum_{\bf
k}\left[\frac{1}{\varepsilon_{\bf k}}+{\cal P}\sum^{2}_{i,j=1}\alpha_{ij}({\bf
k},{\bf q}={\bf 0})\frac{1-n_{i}({\bf k})-n_{j}({\bf k})}{\omega-E_{i}({\bf
k})-E_{j}({\bf k})}\right].$ (90)

The imaginary component of the coefficient $d$ has the form

$d_{I}=\frac{\pi}{2V}\sum_{\mathbf{k}}\sum^{2}_{i,j=1}\alpha_{ij}(\mathbf{k},\mathbf{0})\left[1-n_{i}({\bf
k})-n_{j}({\bf k})\right]\delta^{\prime}\left(E_{i}({\bf k})+E_{j}({\bf
k})\right),$ (91)

where the derivative of the delta function is
$\delta^{\prime}(\lambda)=\partial\delta(x+\lambda)/\partial x|_{x=0}.$ Using
again the expressions of the coherence factors $u_{\bf k}$ and $v_{\bf k}$
leads to

$d_{I}=\frac{\pi}{2V}\sum_{\bf
k}\left\\{(X_{1}+X_{2})\delta^{\prime}(2\xi_{\bf
k})+\frac{|h_{\perp}|^{2}}{h^{2}_{\bf
k}}\bigg{[}X_{1}\delta^{\prime}(2E_{1})+X_{2}\delta^{\prime}(2E_{2})-(X_{1}+X_{2})\delta^{\prime}(2\xi_{\bf
k})\bigg{]}\right\\},$ (92)

which determines the lifetime of fermion pairs. This result originates from

${\rm Im}\left[\Gamma^{-1}({\bf q}={\bf
0},\omega+i\delta)\right]=-\frac{\pi}{2V}\sum_{\bf
k}\sum_{i,j=1}^{2}\alpha_{ij}({\bf k},{\bf q}={\bf 0})\left[1-n_{i}({\bf
k})-n_{j}({\bf k})\right]\delta\left(\omega-E_{i}({\bf k})-E_{j}({\bf
k})\right),$ (93)

which immediately reveals that below the two-particle threshold
$\omega_{tp}({\bf q}={\bf 0})={\rm min}_{\\{i,j,{\bf k}\\}}\left[E_{i}({\bf
k})+E_{j}({\bf k})\right]$ at center-of-mass momentum ${\bf q}={\bf 0}$, the
lifetime of the pairs is infinitely long due to the emergence of stable two-
body bound states. Note that collisions between bound states are not yet
included.

The expressions for the $c_{\ell}$ coefficients appearing in Eq. (84) are
quite long and complex. Since these coefficients are responsible for the mass
renormalization and anisotropy within the Ginzburg-Landau theory, we outline
below their derivation in detail. These coefficients can be obtained from the
last term in Eq. (74), which we define as

$F({\bf q})=\frac{1}{2V}\sum_{\bf k}\sum^{2}_{i,j=1}\alpha_{ij}({\bf k},{\bf
q})W_{ij}({\bf k},{\bf q},iq_{n}=0).$ (94)

The relation between $c_{\ell}$ and the function $F({\bf q})$ defined above is

$c_{\ell}=m\left[\frac{\partial^{2}F({\bf q})}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}.$ (95)

A more explicit form of $c_{\ell}$ is obtained by analyzing the symmetry
properties of $F({\bf q})$ under inversion and reflection symmetries. To make
these properties clear, we rewrite the summand in Eq. (94) by making use of
the transformation ${\bf k}\to{\bf k}-{\bf q}/2$. This procedure leads to the
symmetric form,

$F({\bf q})=\frac{1}{2V}\sum_{\bf
k}\sum_{i,j=1}^{2}{\widetilde{\alpha}}_{ij}({\bf k}_{-},{\bf
k}_{+}){\widetilde{W}}_{ij}\left[E_{i}({\bf k}_{-}),E_{j}({\bf
k}_{+})\right].$ (96)

Here, ${\bf k}_{+}={\bf k}+{\bf q}/2$ and ${\bf k}_{-}={\bf k}-{\bf q}/2$ are
new momentum labels, and

$\displaystyle{\widetilde{\alpha}}_{11}({\bf k}_{-},{\bf k}_{+})$
$\displaystyle=$ $\displaystyle|u_{{\bf k}_{-}}u_{{\bf k}_{+}}-v_{{\bf
k}_{-}}v^{*}_{{\bf k}_{+}}|^{2},$ (97)
$\displaystyle{\widetilde{\alpha}}_{12}({\bf k}_{-},{\bf k}_{+})$
$\displaystyle=$ $\displaystyle|u_{{\bf k}_{-}}v_{{\bf k}_{+}}-v_{{\bf
k}_{-}}u_{{\bf k}_{+}}|^{2}$ (98)

are coherence factors, with ${\widetilde{\alpha}}_{11}({\bf k}_{-},{\bf
k}_{+})={\widetilde{\alpha}}_{22}({\bf k}_{-},{\bf k}_{+})$ and
${\widetilde{\alpha}}_{12}({\bf k}_{-},{\bf
k}_{+})={\widetilde{\alpha}}_{21}({\bf k}_{-},{\bf k}_{+})$ The functions
$u_{{\bf k}_{\pm}}$ and $v_{{\bf k}_{\pm}}$ are defined in Eqs. (78) and (79).
It is now very easy to show that ${\widetilde{\alpha}}_{ij}({\bf k}_{-},{\bf
k}_{+})={\widetilde{\alpha}}_{ij}({\bf k}_{+},{\bf k}_{-})$, that is,
${\widetilde{\alpha}}_{ij}({\bf k}_{-},{\bf k}_{+})$ is an even function of
${\bf q}$, since taking ${\bf q}\to-{\bf q}$ leads to ${\bf k}_{-}\to{\bf
k}_{+}$ and ${\bf k}_{+}\to{\bf k}_{-}$ leaving ${\widetilde{\alpha}}_{ij}$
invariant. It is also clear, from its definition, that
${\widetilde{\alpha}}_{ij}$ is symmetric in the band indices $\\{i,j\\}$.
Furthermore, the function

${\widetilde{W}}_{ij}\left[E_{i}({\bf k}_{-}),E_{j}({\bf
k}_{+})\right]=\frac{{\cal N}_{ij}}{{\cal D}_{ij}},$ (99)

defined above, is the ratio between the numerator,

${\cal N}_{ij}=\tanh\left[\beta E_{i}({\bf k}_{-})/2\right]+\tanh\left[\beta
E_{j}({\bf k}_{+})/2\right],$ (100)

representing the Fermi occupations and the denominator,

${\cal D}_{ij}=2\left[E_{i}({\bf k}_{-})+E_{j}({\bf k}_{+})\right],$ (101)

representing the sum of the quasi-particle excitation energies. To elliminate
the Fermi distributions $n_{i}({\bf k})$ in the numerator, we used the
relation $1-2n_{i}({\bf k})=\tanh\left[\beta E_{i}({\bf k}_{-})/2\right]$.
Notice that ${\widetilde{W}}_{ij}\left[E_{i}({\bf k}_{-}),E_{j}({\bf
k}_{+})\right]$ is not generally symmetric under inversion ${\bf q}\to-{\bf
q}$, that is, under the transformation ${\bf k}_{-}\to{\bf k}_{+}$ and ${\bf
k}_{+}\to{\bf k}_{-}$. This means that ${\widetilde{W}}_{ij}\left[E_{i}({\bf
k}_{-}),E_{j}({\bf k}_{+})\right]\neq{\widetilde{W}}_{ij}\left[E_{i}({\bf
k}_{+}),E_{j}({\bf k}_{-})\right]$, unless when $i=j$, where it is trivially
an even function of ${\bf q}$. However, ${\widetilde{W}}_{ij}\left[E_{i}({\bf
k}_{-}),E_{j}({\bf k}_{+})\right]$ is always symmetric under simultaneous
momentum inversion $({\bf q}\to-{\bf q})$ and band index exchange, that is,

${\widetilde{W}}_{ij}\left[E_{i}({\bf k}_{-}),E_{j}({\bf
k}_{+})\right]={\widetilde{W}}_{ji}\left[E_{j}({\bf k}_{+}),E_{i}({\bf
k}_{-})\right]$ (102)

for any $\\{i,j\\}$. This property will be used later to write a final
expression for $c_{\ell}$. Next, we write

$\left[\frac{\partial^{2}F({\bf q})}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}=\frac{1}{2V}\sum_{\bf k}\sum_{i,j=1}^{2}{\cal F}_{ij},$ (103)

where the function inside the summation is

${\cal F}_{ij}=\left[\frac{\partial^{2}{\widetilde{\alpha}}_{ij}}{\partial
q_{\ell}^{2}}{\widetilde{W}}_{ij}+\alpha_{ij}\frac{\partial^{2}{\widetilde{W}}_{ij}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}.$ (104)

Notice the absence of terms containing the product of the first-order
derivatives of ${\widetilde{\alpha}}_{ij}$ and ${\widetilde{W}}_{ij}$. These
terms vanish due to parity since ${\widetilde{\alpha}}_{ij}$ is an even
function of ${\bf q}$, leading to
$\left[\partial{\widetilde{\alpha}}_{ij}/\partial q_{\ell}\right]_{{\bf
q}={\bf 0}}=0$. The last expression can be further developed upon summation
over the band indices, leading to

$\left[\frac{\partial^{2}F({\bf q})}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}={\cal A}+{\cal B}.$ (105)

The first contribution is given by

${\cal A}=\frac{1}{2V}\sum_{{\bf
k}}\left[\frac{\partial^{2}{\widetilde{\alpha}}_{11}}{\partial
q_{\ell}^{2}}{\widetilde{W}}_{\rm
di}+\frac{\partial^{2}{\widetilde{\alpha}}_{12}}{\partial
q_{\ell}^{2}}{\widetilde{W}}_{\rm od}\right]_{{\bf q}={\bf 0}},$ (106)

and contains the second derivatives of ${\widetilde{\alpha}}_{ij}$ and the
symmetric terms

$\displaystyle{\widetilde{W}}_{\rm
di}=\left({\widetilde{W}}_{11}+{\widetilde{W}}_{22}\right),$ (107)
$\displaystyle{\widetilde{W}}_{\rm
od}=\left({\widetilde{W}}_{12}+{\widetilde{W}}_{21}\right),$ (108)

The second contribution is given by

${\cal B}=\frac{1}{2V}\sum_{{\bf
k}}\left[{\widetilde{\alpha}}_{11}\frac{\partial^{2}{\widetilde{W}}_{\rm
di}}{\partial
q_{\ell}^{2}}+{\widetilde{\alpha}}_{12}\frac{\partial^{2}{\widetilde{W}}_{\rm
od}}{\partial q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}.$ (109)

Next, we explicitly write ${\widetilde{\alpha}}_{ij}$, ${\widetilde{W}}_{ij}$
and their second derivatives with respect to $q_{\ell}$ at ${\bf q}={\bf 0}$.
We start with

$\left[{\widetilde{W}}_{ij}\right]_{{\bf q}={\bf
0}}=\frac{X_{i}+X_{j}}{2\left[E_{i}+E_{j}\right]}$ (110)

and for the second derivative, we write

$\left[\frac{\partial^{2}{\widetilde{W}}_{ij}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}=\left[\frac{1}{{\cal
D}_{ij}}\frac{\partial^{2}{\cal N}_{ij}}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}-\left[\frac{2}{{\cal D}_{ij}^{2}}\frac{\partial{\cal
D}_{ij}}{\partial q_{\ell}}\frac{\partial{\cal N}_{ij}}{\partial
q_{\ell}}\right]_{{\bf q}={\bf 0}}+\left[\frac{2{\cal N}_{ij}}{{\cal
D}_{ij}^{3}}\left(\frac{\partial{\cal D}_{ij}}{\partial
q_{\ell}}\right)^{2}\right]_{{\bf q}={\bf 0}}-\left[\frac{{\cal N}_{ij}}{{\cal
D}_{ij}^{2}}\frac{\partial^{2}{\cal D}_{ij}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}.$

Each one of the four terms in the above expression is evaluated at ${\bf
q}={\bf 0}$ and can be written in terms of specific expressions that are given
below. The numerator is

$\left[{\cal N}_{ij}\right]_{{\bf q}={\bf 0}}=X_{i}+X_{j},$ (111)

the first derivative of ${\cal N}_{ij}$ is

$\left[\frac{\partial{\cal N}_{ij}}{\partial q_{\ell}}\right]_{{\bf q}={\bf
0}}=\frac{Y_{j}^{2}}{4T}\frac{\partial E_{j}}{\partial
k_{\ell}}-\frac{Y_{i}^{2}}{4T}\frac{\partial E_{i}}{\partial k_{\ell}},$ (112)

and the second derivative of ${\cal N}_{ij}$ is

$\displaystyle\left[\frac{\partial^{2}{\cal N}_{ij}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf
0}}=-\frac{X_{j}Y_{j}^{2}}{8T^{2}}\left(\frac{\partial E_{j}}{\partial
k_{\ell}}\right)^{2}+\frac{Y_{i}}{8T}\frac{\partial^{2}E_{i}}{\partial
k_{\ell}^{2}}$
$\displaystyle-\frac{X_{i}Y_{i}^{2}}{8T^{2}}\left(\frac{\partial
E_{i}}{\partial
k_{\ell}}\right)^{2}+\frac{Y_{i}^{2}}{8T}\frac{\partial^{2}E_{i}}{\partial
k_{\ell}^{2}}.$ (113)

The denominator ${\cal D}_{ij}$ and its first derivative are

$\displaystyle\left[{\cal D}_{ij}\right]_{{\bf q}={\bf 0}}=2(E_{i}+E_{j}),$
(114) $\displaystyle\left[\frac{\partial{\cal D}_{ij}}{\partial
q_{\ell}}\right]_{{\bf q}={\bf 0}}=\frac{\partial E_{j}}{\partial
k_{\ell}}-\frac{\partial E_{i}}{\partial k_{\ell}},$ (115)

while the second derivative of ${\cal D}_{ij}$ is

$\left[\frac{\partial^{2}{\cal D}_{ij}}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}=\frac{1}{2}\left[\frac{\partial^{2}E_{i}}{\partial
k_{\ell}^{2}}+\frac{\partial^{2}E_{j}}{\partial k_{\ell}^{2}}\right].$ (116)

When the order parameter is zero, that is, $|\Delta_{0}|=0$, the energies
$E_{1}({\bf k})$ and $E_{2}({\bf k})$ become

$\displaystyle E_{1}({\bf k})=\Big{|}|\xi_{\bf k}|+h_{\bf k}\Big{|}$ (117)
$\displaystyle E_{2}({\bf k})=\Big{|}|\xi_{\bf k}|-h_{\bf k}\Big{|}.$ (118)

The first derivatives of these energies are

$\displaystyle\frac{\partial E_{1}({\bf k})}{\partial k_{\ell}}=S_{1}({\bf
k})\frac{k_{\ell}}{m}+\frac{\partial h_{\bf k}}{\partial k_{\ell}},$ (119)
$\displaystyle\frac{\partial E_{2}({\bf k})}{\partial k_{\ell}}=S_{2}({\bf
k})\frac{k_{\ell}}{m}-\frac{\partial h_{\bf k}}{\partial k_{\ell}},$ (120)

with the functions $S_{1}({\bf k})={\rm sgn}\left[|\xi_{\bf k}|+h_{\bf
k}\right]{\rm sgn}\left[\xi_{\bf k}\right]$ and $S_{2}({\bf k})={\rm
sgn}\left[|\xi_{\bf k}|-h_{\bf k}\right]{\rm sgn}\left[\xi_{\bf k}\right].$
The derivative of the effective Zeeman field is

$\frac{\partial h_{\bf k}}{\partial k_{\ell}}=\frac{1}{h_{\bf
k}}\frac{\kappa^{2}}{m^{2}}\left(k_{x}\delta_{\ell x}+\eta k_{y}\delta_{\ell
y}\right).$ (121)

The second derivatives of the energies are

$\displaystyle\frac{\partial^{2}E_{1}({\bf k})}{\partial
k_{\ell}^{2}}=\frac{S_{1}({\bf k})}{m}+\frac{\partial^{2}h_{\bf k}}{\partial
k_{\ell}^{2}}$ (122) $\displaystyle\frac{\partial^{2}E_{2}({\bf k})}{\partial
k_{\ell}^{2}}=\frac{S_{2}({\bf k})}{m}-\frac{\partial^{2}h_{\bf k}}{\partial
k_{\ell}^{2}},$ (123)

where the second derivative of the effective field is

$\frac{\partial^{2}h_{\bf k}}{\partial k_{\ell}^{2}}=\frac{1}{h_{\bf
k}}\frac{\kappa^{2}}{m^{2}}\left[\left(\delta_{\ell x}+\eta\delta_{\ell
y}\right)-\frac{1}{h_{\bf
k}^{2}}\frac{\kappa^{2}}{m^{2}}\left(k_{x}^{2}\delta_{\ell
x}+\eta^{2}k_{y}^{2}\delta_{\ell y}\right)\right].$ (124)

Since the diagonal elements ${\widetilde{W}}_{ii}$ are even functions of ${\bf
q}$ and so are ${\cal N}_{ii}$ and ${\cal D}_{ii}$, their expressions are
simpler than in the general case discussed above, because the first order
derivatives of ${\cal N}_{ii}$ and ${\cal D}_{ii}$ vanish. The surviving terms
involve only the second derivatives of ${\cal N}_{ii}$ and ${\cal D}_{ii}$
leading to the expression

$\left[\frac{\partial^{2}{\widetilde{W}}_{ii}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}=\left[\frac{1}{{\cal
D}_{ii}}\frac{\partial^{2}{\cal N}_{ii}}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}-\left[\frac{{\cal N}_{ii}}{{\cal
D}_{ii}^{2}}\frac{\partial^{2}{\cal D}_{ii}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}.$ (125)

Here, the numerator and denominator functions are

$\left[{\cal N}_{ii}\right]_{{\bf q}={\bf 0}}=2X_{i}\quad{\rm
and}\quad\left[{\cal D}_{ii}\right]_{{\bf q}={\bf 0}}=4E_{i},$ (126)

while their second derivatives are

$\displaystyle\left[\frac{\partial^{2}{\cal N}_{ii}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}$ $\displaystyle=$
$\displaystyle-\frac{X_{i}Y_{i}^{2}}{4T^{2}}\left(\frac{\partial
E_{i}}{\partial
k_{\ell}}\right)^{2}+\frac{Y_{i}^{2}}{4T}\frac{\partial^{2}E_{i}}{\partial
k_{\ell}^{2}},$ (127) $\displaystyle\left[\frac{\partial^{2}{\cal
D}_{ii}}{\partial q_{\ell}^{2}}\right]_{{\bf q}={\bf 0}}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}E_{i}}{\partial k_{\ell}^{2}}.$ (128)

The next step in obtaining the $c_{\ell}$ coefficients is to analyze the
functions ${\widetilde{\alpha}}_{ij}$ and their second derivatives. We begin
by writing ${\widetilde{\alpha}}_{11}$ at ${\bf q}={\bf 0}$ :

$\left[{\widetilde{\alpha}}_{11}\right]_{{\bf q}={\bf 0}}=\big{|}u_{\bf
k}^{2}-|v_{\bf k}|^{2}\big{|}^{2}=\frac{h_{z}^{2}}{h_{\bf k}^{2}}.$ (129)

To investigate the second derivative of ${\widetilde{\alpha}}_{11}$, we write

${\widetilde{\alpha}}_{11}=\gamma_{11}\gamma_{11}^{*},$ (130)

where the complex function is given by

$\gamma_{11}=u_{{\bf k}_{-}}u_{{\bf k}_{+}}-v_{{\bf k}_{-}}v_{{\bf k}_{+}}.$
(131)

In this case, we write the first derivative of ${\widetilde{\alpha}}_{11}$ as

$\frac{\partial{\widetilde{\alpha}}_{11}}{\partial
q_{\ell}}=\frac{\partial\gamma_{11}}{\partial
q_{\ell}}\gamma_{11}^{*}+\gamma_{11}\frac{\partial\gamma_{11}^{*}}{\partial
q_{\ell}}$ (132)

and the second derivative as

$\frac{\partial^{2}{\widetilde{\alpha}}_{11}}{\partial
q_{\ell}^{2}}=\frac{\partial^{2}\gamma_{11}}{\partial
q_{\ell}^{2}}\gamma_{11}^{*}+2\frac{\partial\gamma_{11}}{\partial
q_{\ell}}\frac{\partial\gamma_{11}^{*}}{\partial
q_{\ell}}+\gamma_{11}\frac{\partial^{2}\gamma_{11}^{*}}{\partial
q_{\ell}^{2}}.$ (133)

To explore the symmetry with respect to ${\bf q}$, we express $\gamma_{11}$ in
terms of its odd and even components via the relation
$\gamma_{11}=\gamma_{11,e}+\gamma_{11,o}$, where the even component
$\gamma_{11,e}=\left[\gamma_{11}({\bf q})+\gamma_{11}(-{\bf q})\right]/2$ is

$\gamma_{11,e}=u_{{\bf k}_{-}}u_{{\bf k}_{+}}-|v_{{\bf k}_{-}}||v_{{\bf
k}_{+}}|\cos\left(\theta_{{\bf k}_{-}}-\theta_{{\bf k}_{+}}\right)$ (134)

and the odd component $\gamma_{11,o}=\left[\gamma_{11}({\bf
q})-\gamma_{11}(-{\bf q})\right]/2$ is

$\gamma_{11,o}=i|v_{{\bf k}_{+}}||v_{{\bf k}_{-}}|\sin\left(\theta_{{\bf
k}_{+}}-\theta_{{\bf k}_{-}}\right).$ (135)

Expressed via the even $\gamma_{11,e}$ and odd $\gamma_{11,o}$ components, the
second derivative in Eq. (133) is

$\frac{\partial^{2}{\widetilde{\alpha}}_{11}}{\partial
q_{\ell}^{2}}=\frac{\partial^{2}\gamma_{11,e}}{\partial
q_{\ell}^{2}}\gamma_{11.e}^{*}+2\frac{\partial\gamma_{11,o}}{\partial
q_{\ell}}\frac{\partial\gamma_{11,o}^{*}}{\partial
q_{\ell}}+\gamma_{11,e}\frac{\partial^{2}\gamma_{11,e}^{*}}{\partial
q_{\ell}^{2}}.$ (136)

Notice that the even component is purely real, that is,
$\gamma_{11,e}^{*}=\gamma_{11,e}$, and that the odd component is purely
imaginary, $\gamma_{11,o}^{*}=-\gamma_{11,o}$. Use of this property leads to

$\frac{\partial^{2}{\widetilde{\alpha}}_{11}}{\partial
q_{\ell}^{2}}=2\gamma_{11,e}\frac{\partial^{2}\gamma_{11,e}}{\partial
q_{\ell}^{2}}-2\left(\frac{\partial\gamma_{11,o}}{\partial
q_{\ell}}\right)^{2}.$ (137)

The contribution from the even term $\gamma_{11,e}$ is

$\left[\gamma_{11,e}\right]_{{\bf q}={\bf 0}}=u_{\bf k}^{2}-|v_{\bf
k}|^{2}=\frac{h_{z}}{h_{\bf k}},$ (138)

and from its second derivative is

$\left[\frac{\partial^{2}\gamma_{11,e}}{\partial q_{\ell}^{2}}\right]_{{\bf
q}={\bf 0}}=\frac{1}{2}\left(\frac{\partial|v_{\bf k}|}{\partial
k_{\ell}}\right)^{2}-\frac{1}{2}|v_{\bf k}|\frac{\partial^{2}|v_{\bf
k}|}{\partial k_{\ell}^{2}}+|v_{\bf k}|^{2}\left(\frac{\partial\theta_{\bf
k}}{\partial k_{\ell}}\right)^{2},$ (139)

while the contribution from the odd term $\gamma_{11,o}$ is

$\left[\frac{\partial\gamma_{11,o}}{\partial q_{\ell}}\right]_{{\bf q}={\bf
0}}=i|v_{\bf k}|^{2}\frac{\partial\theta_{\bf k}}{\partial k_{\ell}}.$ (140)

Now, we turn our attention to ${\widetilde{\alpha}}_{12}$ and its second
derivative. From Eq. (98), we notice that $\gamma_{12}$ is explicitly odd in
${\bf q}$ because $\gamma_{12}({\bf q})=-\gamma_{12}(-{\bf q})$, since the
operation ${\bf q}\to-{\bf q}$ takes ${\bf k}_{-}\to{\bf k}_{+}$ and vice
versa, leading to

$\left[{\widetilde{\alpha}}_{12}\right]_{{\bf q}={\bf 0}}=0.$ (141)

To calculate the second derivative of ${\widetilde{\alpha}}_{12}$, we write

${\widetilde{\alpha}}_{12}=\gamma_{12}\gamma_{12}^{*},$ (142)

where the complex function

$\gamma_{12}=u_{{\bf k}_{-}}v_{{\bf k}_{+}}-v_{{\bf k}_{-}}u_{{\bf k}_{+}}.$
(143)

We relate $\partial^{2}{\widetilde{\alpha}}_{12}/{\partial q_{\ell}^{2}}$ to
$\gamma_{12}$ and its first and second derivatives via

$\frac{\partial^{2}{\widetilde{\alpha}}_{12}}{\partial
q_{\ell}^{2}}=\frac{\partial^{2}\gamma_{12}}{\partial
q_{\ell}^{2}}\gamma_{12}^{*}+2\frac{\partial\gamma_{12}}{\partial
q_{\ell}}\frac{\partial\gamma_{12}^{*}}{\partial
q_{\ell}}+\gamma_{12}\frac{\partial^{2}\gamma_{12}^{*}}{\partial
q_{\ell}^{2}}.$ (144)

Given that $\left[\gamma_{12}\right]_{{\bf q}={\bf 0}}=0$ and
$\left[\gamma_{12}^{*}\right]_{{\bf q}={\bf 0}}=0$, the expression above
simplifies to

$\left[\frac{\partial^{2}{\widetilde{\alpha}}_{12}}{\partial
q_{\ell}^{2}}\right]_{{\bf q}={\bf
0}}=2\left[\frac{\partial\gamma_{12}}{\partial
q_{\ell}}\frac{\partial\gamma_{12}^{*}}{\partial q_{\ell}}\right]_{{\bf
q}={\bf 0}}=\left[\Lambda_{\ell}({\bf q})\right]^{2},$ (145)

where we used the expressions

$\left[\frac{\partial\gamma_{12}}{\partial q_{\ell}}\right]_{{\bf q}={\bf
0}}=e^{i\theta_{\bf k}}\Lambda_{\ell}({\bf k})$ (146)

for the derivatives of $\gamma_{12}$ at ${\bf q}={\bf 0}$ with the function

$\Lambda_{\ell}({\bf k})=u_{\bf k}\frac{\partial|v_{\bf k}|}{\partial
k_{\ell}}-|v_{{\bf k}}|\frac{\partial u_{\bf k}}{\partial k_{\ell}}+u_{\bf
k}|v_{\bf k}|\frac{\partial\theta_{\bf k}}{\partial k_{\ell}}.$ (147)

The last information needed is the derivatives of $u_{\bf k}$, $|v_{\bf k}|$,
and $\theta_{\bf k}$, which are given by

$\displaystyle\frac{\partial u_{\bf k}}{\partial k_{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{h_{z}}{h_{\bf
k}^{3}}\frac{\kappa^{2}}{m^{2}}\frac{(k_{x}\delta_{\ell x}+\eta
k_{y}\delta_{\ell y})}{(1+h_{z}/h_{\bf k})^{1/2}},$ (148)
$\displaystyle\frac{\partial|v_{\bf k}|}{\partial k_{\ell}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{h_{z}}{h_{\bf
k}^{3}}\frac{\kappa^{2}}{m^{2}}\frac{(k_{x}\delta_{\ell x}+\eta
k_{y}\delta_{\ell y})}{(1-h_{z}/h_{\bf k})^{1/2}},$ (149)
$\displaystyle\frac{\partial\theta_{\bf k}}{\partial k_{\ell}}$
$\displaystyle=$ $\displaystyle\eta\frac{(k_{x}\delta_{\ell
y}-k_{y}\delta_{\ell x})}{k_{x}^{2}+\eta^{2}k_{y}^{2}}.$ (150)

The long steps discussed above complete the derivation of all the functions
needed to compute the $c_{\ell}$ coefficients for an arbitrary spin-orbit
coupling, expressed as a general linear combination of Rashba and Dresselhaus
terms.

As announced earlier, the calculation of $c_{\ell}$, defined in Eq. (95), is
indeed very long and requires the use of all the expressions given from Eq.
(96) to Eq. (150). Despite this complexity, that are a few important comments
about the symmetries of the $c_{\ell}$ coefficients that are worth mentioning.
Given that $c_{\ell}$ determines the mass anisotropies in the Ginzburg-Landau
(GL) theory, we discuss next the anisotropies of $c_{\ell}$ as a function of
the spin-orbit coupling parameters $\kappa$ and $\eta$. First, in the limit of
zero spin-orbit coupling, where $\kappa$ and $\eta$ are equal to zero, all the
$c_{\ell}$ coefficients are identical reflecting the isotropy of the system,
that is, $c_{x}=c_{y}=c_{z}$ and reduce to previously known results
sademelo-1993 . In this case, the GL effective masses
$m_{\ell}=md_{R}/c_{\ell}$ are isotropic: $m_{x}=m_{y}=m_{z}$. Second, in the
limit of $\kappa\neq 0$ and $\eta=\pm 1$, the spin-orbit coupling has the same
strength along the $x$ and $y$ directions, and thus for the Rashba $(\eta=1)$
or Dresselhaus $(\eta=-1)$ cases, the coefficients obey the relation
$c_{x}=c_{y}\neq c_{z}$. This leads to effective masses $m_{x}=m_{y}\neq
m_{z}$. Third, in the limit $\kappa\neq 0$, but $\eta=0$, corresponding to the
ERD case, the coefficients have the symmetry $c_{x}\neq c_{y}=c_{z}$. Now the
effective masses obey the relation $m_{x}\neq m_{y}=m_{z}$. Finally, in the
case where $\kappa\neq 0$, and $0\neq|\eta|<1$, all the $c_{\ell}$
coefficients are different, that is, $c_{x}\neq c_{y}\neq c_{z}$. Therefore,
the effective masses are also different in all three directions: $m_{x}\neq
m_{y}\neq m_{z}$.

Following an analogous procedure, we analyze the coefficients
$b(q_{1},q_{2},q_{3})$, and $e(q_{1},q_{2},q_{3},q_{4},q_{5})$ with all
$q_{i}=(0,0)$, and define

$Z_{ij}=X_{i}+\beta E_{i}Y_{j}/2.$ (151)

Using the notation $b(0,0,0)=b(0)$, we obtain

$b(0)=\frac{1}{8V}\sum_{\bf k}\left[\left(1+\frac{h^{4}_{z}}{\xi^{2}_{\bf
k}h^{2}_{\bf
k}}\right)\left(\frac{Z_{11}}{E^{3}_{1}}+\frac{Z_{22}}{E^{3}_{2}}\right)+\frac{2h^{2}_{z}}{\xi_{\bf
k}h_{\bf
k}}\left(\frac{Z_{11}}{E^{3}_{1}}-\frac{Z_{22}}{E^{3}_{2}}\right)+\frac{h^{4}_{z}}{\xi^{3}_{\bf
k}h^{3}_{\bf k}}\left(\frac{X_{1}}{E_{1}}-\frac{X_{2}}{E_{2}}\right)\right],$
(152)

which is a measure of the local interaction between two pairing fields. Using
the notation $f(0,0,0,0,0)=f(0)$, we obtain

$\displaystyle f(0)$ $\displaystyle=$ $\displaystyle\frac{3}{32V}\sum_{\bf
k}\bigg{[}-\left(1+\frac{3h^{4}_{z}}{\xi^{2}_{\bf k}h^{2}_{\bf
k}}\right)\left(\frac{Z_{11}}{E^{5}_{1}}+\frac{Z_{22}}{E^{5}_{2}}\right)-\frac{h^{2}_{z}}{\xi_{\bf
k}h_{\bf k}}\left(3+\frac{h^{4}_{z}}{\xi^{2}_{\bf k}h^{2}_{\bf
k}}\right)\left(\frac{Z_{11}}{E^{5}_{1}}-\frac{Z_{22}}{E^{5}_{2}}\right)$
(153) $\displaystyle\hskip 56.9055pt-\frac{h^{6}_{z}}{\xi^{4}_{\bf
k}h^{4}_{\bf
k}}\left(\frac{Z_{11}}{E^{3}_{1}}+\frac{Z_{22}}{E^{3}_{2}}\right)-\frac{h^{4}_{z}}{\xi^{3}_{\bf
k}h^{3}_{\bf
k}}\left(\frac{Z_{11}}{E^{3}_{1}}-\frac{Z_{22}}{E^{3}_{2}}\right)$
$\displaystyle\hskip
56.9055pt+\frac{\beta^{2}}{6}\left(\frac{X_{1}Y_{1}}{E^{3}_{1}}+\frac{X_{2}Y_{2}}{E^{3}_{2}}\right)+\frac{\beta^{2}h^{2}_{z}}{6\xi_{\bf
k}h_{\bf
k}}\left(\frac{X_{1}Y_{1}}{E^{3}_{1}}-\frac{X_{2}Y_{2}}{E^{3}_{2}}\right)-\frac{h^{6}_{z}}{\xi^{5}_{\bf
k}h^{5}_{\bf k}}\left(\frac{X_{1}}{E_{1}}-\frac{X_{2}}{E_{2}}\right)\bigg{]},$

which is a measure of the local interaction between three pairing fields. It
is important to mention that in the absence of spin-orbit and Zeeman fields,
the Ginzburg-Landau coefficients obtained above reduce to those reported in
the literature sademelo-1993 .

As we proceed to explicitly write the Ginzburg-Landau action and Lagrangian
density, we emphasize that in contrast to the standard crossover that one
observes in the absence of an external Zeeman field sademelo-1993 , for fixed
$h_{z}\neq 0$ it is possible for the system to undergo a first-order phase
transition with increasing $1/k_{F}a_{s}$. The same applies for fixed
$1/k_{F}a_{s}$ with increasing $h_{z}$. Thus, while an expansion of ${\cal
S}_{F}$ to quartic order is sufficient when no Zeeman fields are present, when
Zeeman fields are turned on, the fourth-order coefficient $b(0)=b$ may become
negative. Such a situation requires the analysis of the sixth-order
coefficient $f(0)=f$ to describe this first-order transition correctly and to
stabilize the theory since $f>0$.

The Ginzburg-Landau action in Euclidean space can be written as ${\cal
S}_{GL}=\int dt\int d^{3}{\bf r}{\cal L}_{GL}(r),$ where $r\equiv({\bf r},t)$.
Here, the Lagrangian density is

$\displaystyle{\cal L}_{GL}(r)$
$\displaystyle=a|\Delta(r)|^{2}+\frac{b}{2}|\Delta(r)|^{4}+\frac{f}{3}|\Delta(r)|^{6}$
(154)
$\displaystyle+\sum_{\ell}c_{\ell}\frac{|\nabla_{\ell}\Delta(r)|^{2}}{2m}-id_{0}\Delta^{*}(r)\hskip
1.42262pt\frac{\partial\Delta(r)}{\partial t},\ $

where $\ell=\\{x,y,z\\}$, $b=b(0)$ and $f=f(0)$. A variation of ${\cal
S}_{GL}$ with respect to $\Delta^{*}(r)$ via $\delta{\cal
S}_{GL}/\delta\Delta^{*}(r)=0$ yields the time-dependent Ginzburg-Landau
(TDGL) equation,

$\left(-id_{0}\frac{\partial}{\partial
t}-\sum_{\ell}c_{\ell}\frac{\nabla^{2}_{\ell}}{2m}+b|\Delta|^{2}+f|\Delta|^{4}+a\right)\Delta(r)=0$
(155)

with cubic and quintic terms, where $\Delta=\Delta(r)$ are dependent on space
and time. This equation describes the spatio-temporal behavior of the order
parameter $\Delta({\bf r},t)$ in the long-wavelength and long-time regime.

In the static homogeneous case with $b>0$, Eq. (155) reduces to either the
trivial (normal-state) solution $\Delta=0$ when $a>0$ or to the nontrivial
(superfluid state) $|\Delta|=\sqrt{|a|/b}$, when $a<0$. The coefficient $d$
provides the timescale of the TDGL equation, and thereby determines the
lifetime associated with the pairing field $\Delta(r)$. This can be seen
directly by again considering the homogeneous case to linear order in
$\Delta(r)$, in which case the TDGL equation has the solution
$\Delta(t)\approx\Delta(0)e^{iat/d_{0}}.$ This last expression can be
rewritten more explicitly as
$\Delta(t)\approx\Delta(0)e^{-i\omega_{0}t}e^{-t/\tau_{0}},$ where
$\omega_{0}=|a|d_{R}/|d_{0}|^{2}$ is the oscillation frequency of the pairing
field, and $\tau_{0}=|d_{0}|^{2}/\left(|a|d_{I}\right)$ is the lifetime of the
pairs, where both $d_{R}$ and $d_{I}$ are positive definite, that is,
$d_{R}>0$ and $d_{I}>0$.

In the BEC regime, where stable two-body bound states exist, the imaginary
part of $d_{0}$ vanishes ($d_{I}=0$), and the lifetime time of the pairs is
infinitely long. In this case, $d_{0}=d_{R}$ and we can define the effective
bosonic wave function $\Psi=\sqrt{d_{R}}\Delta$ to recast Eq. (155) in the
form of the Gross-Pitaevskii equation,

$\left(-i\frac{\partial}{\partial
t}-\sum_{\ell}\frac{\nabla^{2}_{\ell}}{2M_{\ell}}+U_{2}|\Psi|^{2}+U_{3}|\Psi|^{4}-\mu_{B}\right)\Psi(r)=0,$
(156)

with cubic and quintic nonlinearities, where $\Psi=\Psi(r)$, to describe a
dilute Bose gas. Here, $\mu_{B}=-a/d_{R}$ is the bosonic chemical potential,
$M_{\ell}=m(d_{R}/c_{\ell})$ are the anisotropic masses of the bosons, and
$U_{2}=b/d_{R}^{2}$ and $U_{3}=f/d_{R}^{3}$ represent contact interactions of
two and three bosons, respectively. In the Bose regime, the lifetime $\tau$ of
the composite boson is $\tau\propto 1/d_{I}\to\infty$ and the interactions
$U_{2}$ and $U_{3}$ are always repulsive, thus leading to a system consisting
of a dilute gas of stable bosons. In this regime, the chemical potential of
the bosons is $\mu_{B}\approx 2\mu+E_{b}<0$, where $E_{b}$ is the two-body
bound state energy in the presence of spin-orbit coupling and Zeeman fields
obtained from the condition $\Gamma^{-1}({\bf q},E-2\mu)=0$ discussed in the
main text. Notice that when $\mu_{B}\to 0^{-}$, in the absence of boson-boson
interactions, the bosons condense.

## References

  * (1) Y-J. Lin, R. L. Compton, K. Jiminéz-García, J. V. Porto, and I. B. Spielman, Synthetic magnetic fields for ultracold neutral atoms, Nature (London) 462, 628 (2009).
  * (2) Y-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose-Einstein condensate in a uniform light-induced vector potential, Phys. Rev. Lett. 102, 130401 (2009).
  * (3) C. J. Kennedy, W. C. Burton, W. C. Chung and W. Ketterle, Observation of Bose–Einstein condensation in a strong synthetic magnetic field, Nature Physics, 11, 859 (2015).
  * (4) Y.-J. Lin, K. Jiménez-García and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471, 83 (2011).
  * (5) L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spectroscopy of a spin-orbit coupled Fermi gas, Phys. Rev. Lett. 109, 095302 (2012).
  * (6) P. Wang, Z.-Q. Yu, Z, Fu, J, Miao, L, Huang, S. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109, 095301 (2012).
  * (7) R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jiménez-García and I. B. Spielman, Raman-Induced Interactions in a Single-Component Fermi Gas Near an s-Wave Feshbach Resonance, Phys. Rev. Lett. 111, 095301 (2013).
  * (8) V. Galitski and I. B. Spielman, Spin–orbit coupling in quantum gases, Nature 494, 49 (2013).
  * (9) Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang, H. Zhai, P. Zhang, and J. Zhang, Production of Feshbach molecules induced by spin-orbit coupling in Fermi gases, Nat. Phys. 10, 110 (2014).
  * (10) M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015).
  * (11) L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases, Nature Physics 12, 540 (2016).
  * (12) Z. Wu, L.Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates Science 354, 83 (2016).
  * (13) J. I. Cirac and P. Zoller, Goals and opportunities in quantum simulation, Nature Phys. 8, 264 (2012).
  * (14) U.-J. Wiese, Ultracold quantum gases and lattice systems: Quantum simulation of lattice gauge theories, Annalen der Physik 525, 777 (2013).
  * (15) E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices, Rep. Prog. Phys. 79, 1 (2015).
  * (16) M. Dalmonte and S. Montangero, Lattice gauge theory simulations in the quantum information era, Contemp. Phys. 57, 388 (2016).
  * (17) M. Gong, S. Tewari, and C. Zhang, BCS-BEC crossover and topological phase transition in 3D spin-orbit coupled degenerate Fermi gases Phys. Rev. Lett. 107, 195303 (2011).
  * (18) Z.-Q. Yu and H. Zhai, Spin-orbit coupled Fermi gases across a Feshbach resonance, Phys. Rev. Lett. 107, 195305 (2011).
  * (19) H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Probing anisotropic superfluidity in atomic Fermi gases with Rashba spin-orbit coupling, Phys. Rev. Lett. 107, 195304 (2011).
  * (20) L. Han and C. A. R. Sá de Melo, Evolution from BCS to BEC superfluidity in the presence of spin-orbit coupling, Phys. Rev. A 85, 011606(R) (2012).
  * (21) K. Seo, L. Han, and C. A. R. Sá de Melo, Emergence of Majorana and Dirac particles in ultracold fermions via tunable interactions, spin-orbit effects, and Zeeman fields, Phys. Rev. Lett. 109, 105303 (2012).
  * (22) K. Seo, L. Han, and C. A. R. Sá de Melo, Topological phase transitions in ultracold Fermi superfluids: The evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids under artificial spin-orbit fields, Phys. Rev. A 85, 033601 (2012).
  * (23) D. M. Kurkcuoglu and C. A. R. Sá de Melo, Formation of Feshbach molecules in the presence of artificial spin-orbit coupling and Zeeman fields, Phys. Rev. A 93, 023611 (2016).
  * (24) N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, Realistic time-reversal invariant topological Insulators with neutral atoms, Phys. Rev. Lett. 105, 255302 (2010).
  * (25) I. B. Spielman, private communication.
  * (26) G. Juzeliünas, J. Ruseckas, and J. Dalibard, Generalized Rashba-Dresselhaus spin-orbit coupling for cold atoms, Phys. Rev. A 81, 053403 (2010).
  * (27) D. L. Campbell, G. Juzeliünas, and I. Spielman, Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms Phys. Rev. A 84, 025602 (2011).
  * (28) G. Dresselhaus, Spin-orbit coupling effects in zinc blende structures Phys. Rev. 100, 580 (1955).
  * (29) E. I. Rashba, Properties of semiconductors with an extremum loop: I. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop, Sov. Phys. Solid State 2, 1224 (1960)
  * (30) C. A. R. Sá de Melo, When fermions become bosons: Pairing in ultracold gases, Physics Today vol. 61 issue 10, page 45 (2008).
  * (31) T. Ozawa and G. Baym, Population imbalance and pairing in the BCS-BEC crossover of three-component ultracold fermions, Phys. Rev. A 82, 063615 (2010).
  * (32) P. D. Powell, From Quarks to Cold Atoms: The Phases of Strongly-Interacting Systems, Ph.D. Thesis, University of Illinois at Urbana-Champaign (2013).
  * (33) D. M. Kurkcuoglu and C. A. R. Sá de Melo, Color superfluidity of neutral ultracold fermions in the presence of color-flip and color-orbit fields, Phys. Rev. A 97, 023632 (2018)
  * (34) Daisuke Yamamoto, I. B. Spielman, and C. A. R. Sá de Melo, Quantum phases of two-component bosons with spin-orbit coupling in optical lattices, Phys. Rev. A 96, 061603(R) (2017).
  * (35) S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, and J. Ye, Dynamics of interacting fermions under spin–orbit coupling in an optical lattice clock, Nature Physics 14, 399 (2018).
  * (36) Yu Yi-Xiang, Fadi Sun, and Jinwu Ye, Class of topological phase transitions of Rashba spin-orbit coupled fermions on a square lattice, Phys. Rev. B 98, 174506 (2018).
  * (37) M. Mamaev, R. Blatt, J Ye, and A. M. Rey, Cluster state generation with spin-orbit coupled fermionic atoms in optical lattices, Phys. Rev. Lett. 122, 160402 (2019).
  * (38) Baihua Gong, Shuai Li, Xin-Hui Zhang, Bo Liu, and Wei Yi, Bloch bound state of spin-orbit-coupled fermions in an optical lattice, Phys. Rev. A 99, 012703 (2019).
  * (39) Man Hon Yau and C. A. R. Sá de Melo Chern-number spectrum of ultracold fermions in optical lattices tuned independently via artificial magnetic, Zeeman, and spin-orbit fields, Phys. Rev. A 99, 043625 (2019)
  * (40) Wei Jia, Zhi-Hao Huang, Xian Wei, Qing Zhao, and Xiong-Jun Liu, Topological superfluids for spin-orbit coupled ultracold Fermi gases, Phys. Rev. B 99, 094520 (2019).
  * (41) Majid Kheirkhah, Zhongbo Yan, Yuki Nagai, and Frank Marsiglio, First- and second-order topological superconductivity and temperature-driven topological phase transitions in the extended hubbard model with spin-orbit coupling, Phys. Rev. Lett. 125, 017001 (2020).
  * (42) Urs Gebert, Bernhard Irsigler, and Walter Hofstetter, Local Chern marker of smoothly confined Hofstadter fermions, Phys. Rev. A 101, 063606 (2020).
  * (43) Irakli Titvinidze, Julian Legendre, Maarten Grothus, Bernhard Irsigler, Karyn Le Hur, and Walter Hofstetter, Spin-orbit coupling in the kagome lattice with flux and time-reversal symmetry, Phys. Rev. B 103, 195105 (2021).
  * (44) R. A. Williams, L. J. LeBlanc, K. Jiménez-García, M. C. Beeler, A. R. Perry, W. D. Phillips, and I. B. Spielman, Synthetic partial waves in ultracold atomic collisions, Science 335, 314 (2012).
  * (45) Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C. Zhang, P. Wang, and J. Zhang Experimental observation of a topological band gap opening in ultracold Fermi gases with two-dimensional spin-orbit coupling, Phys. Rev. Lett. 117, 235304 (2016).
  * (46) L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental realization of two-dimensional synthetic spin–orbit coupling in ultracold Fermi gases, Nature Physics 12, 540 (2016).
  * (47) A. Valdés-Curiel, D. Trypogeorgos, Q.-Y. Liang, R. P. Anderson, and I. B. Spielman, Topological features without a lattice in Rashba spin-orbit coupled atoms, Nature Communications 12, 593 (2021).
  * (48) Ronen M. Kroeze, Yudan Guo, and Benjamin L. Lev, Dynamical spin-orbit coupling of a quantum gas Phys. Rev. Lett. 123, 160404 (2019).
  * (49) Zong-Yao Wang, Xiang-Can Cheng, Bao-Zong Wang, Jin-Yi Zhang, Yue-Hui Lu, Chang-Rui Yi, Sen Niu, Youjin Deng, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan, Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling, Science 372, 271 (2021).
  * (50) A. J. Leggett in Modern Trends in the Theory of Condensed Matter, pp. 13–27, edited by A. Pekalski and R. Przystawa, Springer-Verlag, Berlin (1980).
  * (51) J. R. Engelbrecht, M. Randeria, and C. A. R. Sá de Melo, BCS to Bose crossover: Broken-symmetry state, Phys. Rev. B 55, 15153 (1997).
  * (52) C. A. R. Sá de Melo, M. Randeria, and J. Engelbrecht, Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory, Phys. Rev. Lett. 71, 3202 (1993).
  * (53) P. D. Powell, G. Baym, C. A. R. Sá de Melo, Superfluid transition temperature of spin-orbit and Rabi coupled fermions with tunable interactions, arXiv:1709.07042v1 (2017).
  * (54) S. Gopalakrishnan, A. Lamacraft, and P. M. Goldbart, Universal phase structure of dilute Bose gases with Rashba spin-orbit coupling, Phys. Rev. A 84, 061604 (2011).
  * (55) T. Ozawa, Topics in Multi-component Ultracold Gases and Gauge Fields, Ph.D. Thesis, University of Illinois at Urbana-Champaign (2012).
  * (56) P. Nozières and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys. 59, 195 (1985).
  * (57) Z. Yu and G. Baym, Spin correlation functions in ultracold paired atomic-fermion systems: Sum rules, self-consistent approximations, and mean fields, Phys. Rev. A 73, 063601 (2006).
  * (58) Z. Yu, G. Baym, and C. J. Pethick, Calculating energy shifts in terms of phase shifts, J. Phys. B: At. Mol. Opt. Phys. 44, 195207 (2011).
  * (59) P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135, A550 (1964).
  * (60) A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Sov. Phys. JETP 20, 762 (1965).
  * (61) A. M. Clogston, Upper limit for the critical field in hard superconductors, Phys. Rev. Lett. 9, 266 (1962).
  * (62) G. Sarma, On the influence of a uniform exchange field acting on the spins of the conduction electrons in a superconductor, J. Phys. Chem. Sol. 24, 1029 (1963).
  * (63) G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë and D. Vautherin, The transition temperature of the dilute interacting Bose gas, Phys. Rev. Lett. 83, 1703 (1999).
  * (64) M. Iskin and C. A. R. Sá de Melo, Fermi-Fermi mixtures in the strong-attraction limit, Phys. Rev. A 77, 013625 (2008).
  * (65) D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Diatomic molecules in ultracold Fermi gases: Novel composite bosons, J. Phys. B 38, S645 (2005).
  * (66) J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS-BEC crossover induced by a synthetic non-Abelian gauge field, Phys, Rev. B 84, 014512 (2011)
  * (67) X.-J. Feng and L. Yin, Phase diagram of a spin-orbit coupled dipolar Fermi gas at $T=0$ K, Chin. Phys. Lett. 37, 020301 (2020).
  * (68) L. Dell’Anna, G. Mazzarella, and L. Salasnich, Condensate fraction of a resonant Fermi gas with spin-orbit coupling in three and two dimensions, Physical Review A 84, 033633 (2011).
  * (69) L. Dell’Anna, G. Mazzarella, and L. Salasnich, Tuning Rashba and Dresselhaus spin-orbit couplings: Effects on singlet and triplet condensation with Fermi atoms, Physical Review A 86, 053632 (2012).
  * (70) S. Tewari, T. D, Stanescu, J. D. Sau and S. Das Sarma, Topologically non-trivial superconductivity in spin–orbit-coupled systems: Bulk phases and quantum phase transitions, New Journal of Physics, 13, 065004 (2011).
  * (71) Li Han and C. A. R. Sá de Melo, Ultra-cold fermions in the flatland: Evolution from BCS to Bose superfluidity in two-dimensions with spin-orbit and Zeeman fields, arXiv:1206.4984v1 (2012)
  * (72) J. P. A. Devreese, J. Tempere, and C. A. R. Sá de Melo Effects of spin-orbit coupling on the Berezinskii-Kosterlitz-Thouless transition and the vortex-antivortex structure in two-dimensional Fermi gases, Phys. Rev. Lett. 113, 165304 (2014).
  * (73) J. P. A. Devreese, J. Tempere, and C. A. R. Sá de Melo Quantum phase transitions and Berezinskii-Kosterlitz-Thouless temperature in a two-dimensional spin-orbit-coupled Fermi gas, Phys. Rev. A 92, 043618 (2015)
  * (74) J. P. Vyasanakere and V. B. Shenoy, Fluctuation theory of Rashba Fermi gases: Gaussian and beyond, Phys. Rev. B. 92, 121111(R) (2015).
  * (75) T. Yamaguchi, D. Inotani, and Y. Ohashi, Rashbon bound states associated with a spherical spin–orbit coupling in an ultracold Fermi gas with an s-wave interaction, J. Low Temp. Phys. 183, 161 (2016).
  * (76) B. M. Anderson, C.-T. Wu, R. Boyack, and K. Levin Topological effects on transition temperatures and response functions in three-dimensional Fermi superfluids, Phys. Rev. B 92, 134523 (2015).
  * (77) L. Dell’Anna, and S. Grava, Critical temperature in the BCS-BEC crossover with spin-orbit coupling, Condens. Matter 6, 16 (2021).