# Dynamic structure factors of a strongly interacting Fermi superfluid near an
orbital Feshbach resonance across the phase transition from BCS to Sarma
superfluid

Peng Zou1<EMAIL_ADDRESS>Huaisong Zhao1, Lianyi He2, Xia-Ji Liu3, and
Hui Hu3 1College of Physics, Qingdao University, Qingdao 266071, China
2Department of Physics, Tsinghua University, Beijing 100084, China  3Centre
for Quantum Technology Theory, Swinburne University of Technology, Melbourne
3122, Australia

###### Abstract

We theoretically investigate dynamic structure factors of a strongly
interacting Fermi superfluid near an orbital Feshbach resonance with random
phase approximation approach at zero temperature, and find their dynamical
characters across the phase transition from a balanced conventional Bardeen-
Cooper-Schrieffer superfluid to a polarized Sarma superfluid by continuously
varying the chemical potential difference of two spin components. In a BEC-
like regime of the Fermi superfluid, dynamic structure factors can help
distinguish the in-phase ground state from the out-of-phase metastable state
by the relative location of molecular excitation and Leggett mode, or the
minimum energy to break a Cooper pair. In the phase transition from BCS to
Sarma superfluid, we find the dynamic structure factor of Sarma superfluid has
its own specific gapless excitation at a small transferred momentum where the
collective phonon excitation acquires a finite width, and also a relatively
strong atomic excitation at a large transferred momentum, because of the
existence of unpaired Fermi atoms. Our results can be used to differentiate
Sarma superfluid from BCS superfluid.

##  I Introduction

Interacting quantum many-body system is always a central research field in
physics, and contains many interesting but also challenging problems. As a
many-body physical quantity, the density-density dynamic structure factor,
i.e., the Fourier transformation of density-density correlation function,
contains rich information about the dynamical excitation of the system
pitaevskiibook . At a low transferred momentum regime, one can observe many
kinds of collective excitations, while molecular and atomic excitation can be
seen at a large transferred momentum. Besides the spectrum of the system,
dynamic structure factors also display the weight distribution of both Fermi
atoms and Cooper-pairs according to the width and strength of excitation peak.
From both the spectrum and weight distribution we can understand rich many-
body properties, like the pair condensation and transition temperature
lingham2014 , Tan’s contact parameter kuhnle2011 ; hoinka2013 , etc,.
Generally the experimental measurement of many-body physical quantities is a
quite difficult task, luckily a two-photon Bragg scattering technique can be
able to measure dynamic structure factors at an almost full regime of
transferred momentum veeravalli2008 ; hoinka2017 ; hoinka2012 ; brunello2001
.

The accumulation of interacting many particles can form lots of quantum matter
states, i.e. Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phases fulde1964 ;
larkin1964 , Sarma superfluid sarma1963 , topological superfluid zhang2008
etc,. The search and distinguishment of new matter states in different
materials or microscopic particles system are interesting and important jobs.
The BCS-type Fermi superfluid is a conventional superfluid state, where two
Fermi atoms with the same amplitude of momentum but unlike direction and spin
can form a Cooper pair with zero momentum of center of mass. It is usually the
ground state of many superconductors or Fermi superfluid. A Sarma superfluid
is a possible candidate ground state for a spin-polarized Fermi superfluid.
Having gapless fermionic excitations, this state can be viewed as a phase
separation state in momentum space: some fermions pair and form a superfluid,
while other unpaired atoms just occupy in certain regions of momentum space
bounded by gapless Fermi surfaces.

In the context of two-component spin-1/2 Fermi atomic gas at the famous BEC-
BCS crossover, Sarma superfluid can only be stable on the BEC side of the
crossover. It may also be stabilized by considering a multiband structure
he2009 , which experimentally can be realized with alkali-earth-metal atoms
(such as Sr) or alkali-earth-metal-like atoms (i.e. Yb), where there are two
electrons outside the atoms. In this case the usual Magnetic Feshbach
resonance fails to tune the effective interacting strength since their zero
magnetic moment difference, but a pioneering work by R. Zhang et al
theoretical proposed an alternative mechanism of orbital Feshbach resonance
(OFR) for 173Yb zhang2015 . In Fermi gases of 173Yb, the long-lived metastable
electric orbital state ${}^{3}P_{0}$ (denoted as $\left|e\sigma\right\rangle$
where $\sigma=\uparrow,\downarrow$ stands for the two internal nuclear spin
states) can be chosen, together with the ground orbital state ${}^{1}S_{0}$
($\left|g\sigma\right\rangle$) . In fact this forms a four-component Fermi
system, where a pair of atoms can be described by using the singlet ($-$) and
triplet ($+$) basis in the absence of external Zeeman field,

$\left|\pm\right\rangle=\frac{1}{2}\left(\left|ge\right\rangle\pm\left|eg\right\rangle\right)\otimes\left(\left|\uparrow\downarrow\right\rangle\mp\left|\downarrow\uparrow\right\rangle\right),$
(1)

or by a usual two-channel basis in the presence of the Zeeman field

$\left|o\right\rangle=\frac{1}{\sqrt{2}}\left(\left|-\right\rangle+\left|+\right\rangle\right)=\frac{1}{\sqrt{2}}\left(\left|g\uparrow,e\downarrow\right\rangle-\left|e\downarrow,g\uparrow\right\rangle\right),$
(2)

$\left|c\right\rangle=\frac{1}{\sqrt{2}}\left(\left|-\right\rangle-\left|+\right\rangle\right)=\frac{1}{\sqrt{2}}\left(\left|g\downarrow,e\uparrow\right\rangle-\left|e\uparrow,g\downarrow\right\rangle\right),$
(3)

where $o$ and $c$ are short for open and close channel, respectively. The
interparticle interaction is described by a $s$-wave scattering length, namely
the singlet scattering length $a_{-}$ and the triplet one $a_{+}$. Because of
a shallow bound state (related to a large $a_{+}\simeq 1900a_{0}$, where
$a_{0}$ is the Bohr radius) caused by the interorbital (nuclear) spin-exchange
interaction, the small difference in the nuclear Landé factor between
different orbital states allows the tunability of scattering length in the
open channel through an external magnetic fieldzhang2015 . The existence of
the predicted OFR has been confirmed soon experimentally pagano2015 ;
hofer2015  . More fascinatingly, OFR is a narrow resonance due to the
significant closed-channel fraction xu2016 , and a two-channel mode he2015  is
really necessary to provide a right description. In the two-channel system,
the exchange interaction between two channels varies significantly the
stability mechanism of the pairing state, and makes Sarma state, which is
always unstable in single-channel system, can be a stable ground state he2009
once a nonzero interband exchange interaction and a large asymmetry between
the two bands are existent. In our recent work, a superfluid phase transition
from a balanced conventional Bardeen-Cooper-Schrieffer superfluid to a
polarized Sarma superfluid in a strongly interacting Fermi superfluid near an
orbital Feshbach resonance has been reported zou2018 .

It is of great interest to explore the many-body physical eigenstates of OFR,
and investigate their rich dynamics. The existence of two order parameters in
OFR has opened the possibility of observing the long-sought massive Leggett
mode resulted from the fluctuation of the relative phase of the two order
parameters, which can be observed in both BCS and Sarma superfluid leggett1966
; blumberg2007 ; lin2012 ; bittner2015 ; klimin2019 . So it is interesting to
study all dynamics of OFR , and find their specific characters in different
matter states, from which to research possibility to distinguish different
many-body eigenstates, or different ground states of OFR by an experimental
observable many-body physical quantity. Following this idea, we would like to
investigate the dynamic structure factor across the phase transition process
from BCS superfluid to Sarma superfluid in OFR, and find their own specific
dynamical characters which may service as fingerprints of different matter
states.

This paper is organized as follows. In the next section, we will use the
language of Green’s function to introduce the microscopic model of a three-
dimensional Fermi superfluid near OFR with both spin-balanced and spin-
polarized population , outline the mean-field approximation and how to
calculate response function with random phase approximation, and give the
results of dynamic structure factor of BCS superfluid in Sec. III, and the
results of Sarma superfluid in Sec. IV. In Sec. V and VI, we will give our
conclusions and acknowledgement, respectively.

##  II Mean-field theory

###  II.1 Model Hamiltonian

The spin-population imbalanced Fermi gases near orbital Feshbach resonance can
be described by a two-channel Hamiltonian

$\mathcal{H}=\int
d{\rm\textit{{r}}}\left[\underset{ni}{\sum}\psi_{ni}^{\dagger}\mathcal{H}_{ni}^{0}\psi_{ni}+\underset{nm}{\sum}V_{nm}\varphi_{n}^{\dagger}\varphi_{m}\right]$
(4)

where $\mathcal{H}_{ni}^{0}=-\hbar^{2}\nabla^{2}/2M-\mu_{ni}$ is the single
particle Hamiltonian of spin $i=1,2$ Fermi atoms with mass $M$ and chemical
potential $\mu_{ni}$, the subscript $n=o,c$ denotes the open or close channel,
and $\psi_{ni}\left(\textit{{r}}\right)$ and
$\psi_{ni}^{\dagger}\left(\textit{{r}}\right)$ are the corresponding
annihilation and creation field operator of Fermi atoms, respectively.
$\varphi_{n}\equiv\psi_{n2}\psi_{n1}$ is an anomalous density operator. In
each channel we assume spin-up chemical potential
$\mu_{n1}=\mu_{n}+\delta\mu_{n}$ and spin-down one
$\mu_{n2}=\mu_{n}-\delta\mu_{n}$, where $\delta\mu$ is the chemical potential
difference of different spin around an average value $\mu_{n}$. In the
presence of a Zeeman field, a pair of atoms in the open and close channel has
different Zeeman energy with a difference
$\delta\left(B\right)=\left(g_{g}m_{2}+g_{e}m_{1}\right)\mu_{B}B-\left(g_{g}m_{1}+g_{e}m_{2}\right)\mu_{B}B$,
arising from their difference in magnetic momentum. So we may define the
effective chemical potentials of the open and close channel as $\mu_{o}=\mu$
and $\mu_{c}=\mu-\delta\left(B\right)/2$. The interaction potentials between
atomic pairs can be well approximated by $s$-wave contact pseudopotentials

$V_{\pm}\left(r\right)\simeq\frac{4\pi\hbar^{2}a_{\pm}}{M}\delta\left(r\right).$
(5)

When using the external Zeeman field as a control knob, it will be convenient
to use the two-channel language to describe contact interaction potentials,
which become

$V_{oo}\left(r\right)=V_{cc}\left(r\right)=\frac{V_{-}+V_{+}}{2}=\frac{4\pi\hbar^{2}a_{0}}{M}\delta\left(r\right),$
(6)

$V_{oc}\left(r\right)=V_{co}\left(r\right)=\frac{V_{-}-V_{+}}{2}=\frac{4\pi\hbar^{2}a_{1}}{M}\delta\left(r\right).$
(7)

The two scattering lengths $a_{0}$ and $a_{1}$ are given by
$a_{0}=\left(a_{-}+a_{+}\right)/2$ and $a_{1}=\left(a_{-}-a_{+}\right)/2$, and
the bare interaction strengths $V_{nm}$ ($n,m=o,c$) need to be regularized
with the two scattering lengths $a_{0}$ and $a_{1}$ by the standard
renormalization procedure

$\left[\begin{array}[]{cc}V_{oo}&V_{oc}\\\
V_{co}&V_{cc}\end{array}\right]^{-1}=\frac{M}{4\pi\hbar^{2}}\left[\begin{array}[]{cc}a_{0}&a_{1}\\\
a_{1}&a_{0}\end{array}\right]^{-1}-\sum_{k}\frac{M}{\hbar^{2}k^{2}}.$ (8)

Following the standard mean-field theoretical treatment, we may introduce two
order parameters in open and close channel

$\left[\begin{array}[]{c}\triangle_{o}\left(r\right)\\\
\triangle_{c}\left(r\right)\end{array}\right]\equiv\left[\begin{array}[]{cc}V_{oo}&V_{oc}\\\
V_{co}&V_{cc}\end{array}\right]\left[\begin{array}[]{c}\left\langle\varphi_{o}\left(r\right)\right\rangle\\\
\left\langle\varphi_{c}\left(r\right)\right\rangle\end{array}\right].$ (9)

After a Fourier transformation to field operator
$\psi_{ni}=\sum_{k}c_{ni,k}\phi_{k}$ ($\phi_{k}$ is the plane wave function),
we obtain the mean-field Hamiltonian in the momentum representation

$\begin{array}[]{ll}\mathcal{H}_{{\rm
mf}}&=\underset{nk}{\sum}\left[\left(\xi_{nk}-\delta\mu\right)c_{n1,k}^{\dagger}c_{n1,k}+\left(\xi_{nk}+\delta\mu\right)c_{n2,k}^{\dagger}c_{n2,k}\right]\\\
&+\underset{nk}{\sum}\left(\Delta_{n}c_{n1,k}^{\dagger}c_{n2,-k}^{\dagger}+\Delta_{n}^{*}c_{n2,-k}c_{n1,k}\right)\end{array}$
(10)

where $\xi_{nk}=\hbar^{2}k^{2}/2M-\mu_{n}$, and order parameter satisfies
$\Delta_{n}=\sum_{m}V_{nm}\sum_{k}\left\langle
c_{m2,-k}c_{m1,k}\right\rangle$. Since we just discuss the eigenstate, two
order parameters should be both real number and satisfy
$\Delta_{n}\equiv\Delta_{n}^{*}$.

Figure 1:  The free energy of both spin-population balanced BCS state (black
line) and spin-polarized Sarma state (red line) at scattering length
$1/k_{F}a_{0}=-0.5$, $1/k_{F}a_{1}=-0.05$ and Zeeman detuning
$\delta(B)=0.4\varepsilon_{F}$. $F_{{\rm IG}}$ is the free energy of ideal
Fermi gases with the same particle number.

With the language of Green’s function, we start with equations of motion of
any operators $A$ and $B$ in the momentum-energy representation
$\omega\left\langle\left\langle
A|B\right\rangle\right\rangle_{\omega}=\left\langle\left[A,B\right]_{+}\right\rangle+\left\langle\left\langle\left[A,\mathcal{H}_{{\rm
mf}}\right]|B\right\rangle\right\rangle_{\omega}$, and obtain three
independent Green’s function

$\begin{array}[]{cll}G_{1}^{\left(n\right)}\left(K\right)&\equiv\left\langle\left\langle
c_{n1,k}|c_{n1,k}^{\dagger}\right\rangle\right\rangle_{\omega}&=\underset{l}{\sum}\frac{\frac{1}{2}\left(1+l\frac{\xi_{nk}}{E_{nk}}\right)}{i\omega-
E_{nk}^{l}},\\\
G_{2}^{\left(n\right)}\left(K\right)&\equiv\left\langle\left\langle
c_{n2,k}|c_{n2,k}^{\dagger}\right\rangle\right\rangle_{\omega}&=\underset{l}{\sum}\frac{\frac{1}{2}\left(1-l\frac{\xi_{nk}}{E_{nk}}\right)}{i\omega+E_{nk}^{l}},\\\
\Gamma^{\left(n\right)}\left(K\right)&\equiv\left\langle\left\langle
c_{n1,k}|c_{n2,-k}\right\rangle\right\rangle_{\omega}&=\underset{l}{\sum}\frac{l\frac{\Delta_{n}}{2E_{nk}}}{i\omega-
lE_{nk}^{l}},\end{array}$ (11)

where $K=\left(k,i\omega\right)$ is a four-dimensional momentum, and
$\begin{array}[]{cc}E_{nk}^{l}=-\delta\mu+lE_{nk},&E_{nk}=\sqrt{\xi_{nk}^{2}+\Delta_{n}^{2}}\end{array},$
$l=\pm 1$ stands for the particle or hole solution. These Green’s functions
should be self-consistently solved with density equations

$\rho_{o}=\sum_{k}\left\\{1-\frac{\xi_{ok}}{E_{ok}}\left[1-f\left(E_{ok}^{+1}\right)-f\left(-E_{ok}^{-1}\right)\right]\right\\},$
(12)

$\rho_{c}=\sum_{k}\left\\{1-\frac{\xi_{ck}}{E_{ck}}\left[1-f\left(E_{ck}^{+1}\right)-f\left(-E_{ck}^{-1}\right)\right]\right\\},$
(13)

and order parameters equations

$\frac{\lambda_{1}\Delta_{o}}{\Delta_{c}}+\lambda_{0}=\sum_{k}\left[\frac{M}{\hbar^{2}k^{2}}-\frac{1-f\left(E_{ck}^{+1}\right)-f\left(-E_{ck}^{-1}\right)}{2E_{ck}}\right]$
(14)

$\frac{\lambda_{1}\Delta_{c}}{\Delta_{o}}+\lambda_{0}=\sum_{k}\left[\frac{M}{\hbar^{2}k^{2}}-\frac{1-f\left(E_{ok}^{+1}\right)-f\left(-E_{ok}^{-1}\right)}{2E_{ok}}\right]$
(15)

where

$\begin{array}[]{cc}\lambda_{0}=+\frac{Ma_{0}}{4\pi\hbar^{2}\left(a_{0}^{2}-a_{1}^{2}\right)},&\lambda_{1}=-\frac{Ma_{1}}{4\pi\hbar^{2}\left(a_{0}^{2}-a_{1}^{2}\right)}\end{array},$
(16)

and $f\left(x\right)=1/\left(e^{x/k_{B}T}+1\right)$ is the famous Fermi-Dirac
distribution function. The free energy of the system in mean-field
approximation reads

$\begin{array}[]{cl}F_{{\rm
mf}}=&-\Delta^{\dagger}\left[\begin{array}[]{cc}\lambda_{0}&\lambda_{1}\\\
\lambda_{1}&\lambda_{0}\end{array}\right]\Delta+\underset{nk}{\sum}\left(\xi_{nk}-E_{nk}+\frac{\Delta_{n}^{2}M}{\hbar^{2}k^{2}}\right)\\\
&-k_{B}T\underset{nkl}{\sum}{\rm ln}\left(1+e^{-E_{nk}^{l}/k_{B}T}\right)+\mu
N\end{array}$ (17)

where
$\Delta\equiv\left[\begin{array}[]{cc}\Delta_{o},&\Delta_{c}\end{array}\right]^{T}$.
As shown in Fig.1, a phase transition from a spin-balanced BCS superfluid to a
spin-polarized Sarma superfluid is observed by continuously increasing
chemical potential difference $\delta\mu$ over a critical value
$\delta\mu_{{\rm cr}}=0.486\varepsilon_{F}$.

###  II.2 Random phase approximation and response function

The fluctuation term of interaction Hamiltonian plays a great role in the
calculation of dynamics of the system, random phase approximation has been
proved to be an effective way to collect the contribution from the fluctuation
term. To introduce this story about how to obtain an accurate response
function with random phase approximation approach, let us begin with the
linear response theory.

In any superfluid state of OFR, there are usually four different densities in
both open and close channels. Besides the normal spin-up and down densities
($\rho_{i}^{n}\equiv\left\langle\psi_{ni}^{\dagger}\psi_{ni}\right\rangle$,
$i=1,2$), the other two are anomalous density
$\rho_{3}^{n}\equiv\left\langle\psi_{n2}\psi_{n1}\right\rangle$ and its
complex conjugate
$\rho_{4}^{n}\equiv\left\langle\psi_{n1}^{\dagger}\psi_{n2}^{\dagger}\right\rangle$.
When a weak external perturbation potential $V_{{\rm ext}}$ is imposed to all
densities in two channels with form $V_{{\rm
ext}}=\left[\begin{array}[]{cc}V^{o},&V^{c}\end{array}\right]^{T}$ and
$V^{n}=\left[\begin{array}[]{cccc}V_{1}^{n},&V_{2}^{n},&V_{3}^{n},&V_{4}^{n}\end{array}\right]^{T}$,
the corresponding perturbation Hamiltonian reads

$H_{p}=\sum_{nkqj}\varPsi_{nk+q}^{\dagger}V_{j}^{n}\sigma_{j}\varPsi_{nk}$
(18)

where $j=1,2,3,4$ stands for four different densities index in $n$ channel,
and
$\varPsi_{nk}=\left[\begin{array}[]{cc}c_{n1,k},&c_{n2,-k}^{\dagger}\end{array}\right]^{T}$
is the field operator in Nambu spinor representation. Also with Pauli matrices
$\sigma_{x,y,z}$ and unit matrix $I$ we define
$\sigma_{1}\equiv\left(I+\sigma_{z}\right)/2$,
$\sigma_{2}\equiv\left(I-\sigma_{z}\right)/2$,
$\sigma_{3}\equiv\left(\sigma_{x}-i\sigma_{y}\right)/2$ and
$\sigma_{4}\equiv\left(\sigma_{x}+i\sigma_{y}\right)/2$. This perturbation
Hamiltonian $H_{p}$ introduces a fluctuation of density matrix

$\rho_{q}=\chi V_{{\rm ext}},$ (19)

which connects to the external potential $V_{{\rm ext}}$ by the response
function $\chi$ of the system.

$\rho_{q}^{n}=\sum_{k}\left[\begin{array}[]{c}\varPsi_{nk}^{\dagger}\sigma_{1}\varPsi_{nk+q}\\\
\varPsi_{nk}^{\dagger}\sigma_{2}\varPsi_{nk+q}\\\
\varPsi_{nk}^{\dagger}\sigma_{3}\varPsi_{nk+q}\\\
\varPsi_{nk}^{\dagger}\sigma_{4}\varPsi_{nk+q}\end{array}\right]$ (20)

is the expression of density fluctuation in $n$ channel. Usually the exact
calculation of response function $\chi$ is quite hard and difficult. However,
random phase approximation suggests that we can treat the fluctuation term of
interaction Hamiltonian $H_{{\rm sf}}=\sum_{nq}\rho_{q}^{n\dagger}\cdot
A_{q}^{n}$ as a self-consistent dynamical potential $V_{{\rm
sf}}^{n}=M_{I}A_{q}^{n},$ in which
$A_{q}^{n}=\left[\begin{array}[]{cccc}0,&0,&\Delta_{n,q},&\Delta_{n,-q}^{*}\end{array}\right]^{T}$is
the strength of fluctuation potential with

$\left[\begin{array}[]{c}\Delta_{o,q}\\\ \Delta_{o,-q}^{*}\\\ \Delta_{c,q}\\\
\Delta_{c,-q}^{*}\end{array}\right]=\sum_{k}\left[\begin{array}[]{cccc}V_{oo}&0&V_{oc}&0\\\
0&V_{oo}&0&V_{oc}\\\ V_{co}&0&V_{cc}&0\\\
0&V_{co}&0&V_{cc}\end{array}\right]\left[\begin{array}[]{c}\left\langle\varPsi_{ok}^{\dagger}\sigma_{3}\varPsi_{ok+q}\right\rangle\\\
\left\langle\varPsi_{ok}^{\dagger}\sigma_{4}\varPsi_{ok+q}\right\rangle\\\
\left\langle\varPsi_{ck}^{\dagger}\sigma_{3}\varPsi_{ck+q}\right\rangle\\\
\left\langle\varPsi_{ck}^{\dagger}\sigma_{4}\varPsi_{ck+q}\right\rangle\end{array}\right],$
(21)

and $M_{I}$ is just a constant matrix

$M_{I}=\left[\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}\right].$ (22)

We should notice that here we neglect the fluctuation from the normal spin-up
and down density. This is reasonable in three dimensional space where the
utilization of $s$-wave contact interaction induces two anomalous densities to
be divergent, and play a much more important role than normal densities. Now
the system can be treated as a mean-field quasiparticle gases experienced by
an effective potential

$V_{{\rm eff}}\equiv V_{{\rm ext}}+V_{{\rm sf}},$ (23)

which induces the density fluctuations $\rho_{q}$ of the system following

$\rho_{q}=\Pi\cdot V_{{\rm eff}},$ (24)

in which $\Pi$ is the response function matrix in the mean-field level, whose
calculation is feasible and relatively quite easy. Finally, with Eqs.19, 23
and 24, we find the desired response function $\chi$ can be calculated by its
connection with mean-field response function $\Pi$ by

$\chi=\frac{\Pi}{1-\Pi M_{G}},$ (25)

where $M_{G}$ is another constant matrix

$M_{G}=\left[\begin{array}[]{cc}M_{I}V_{oo},&M_{I}V_{oc}\\\
M_{I}V_{co},&M_{I}V_{cc}\end{array}\right].$ (26)

The mean-field response function $\Pi$ is a $8\times 8$ matrix, its explicit
form is given by

$\left[\begin{array}[]{cccccccc}\Pi_{11}^{o}&\Pi_{12}^{o}&\Pi_{13}^{o}&\Pi_{14}^{o}&0&0&0&0\\\
\Pi_{12}^{o}&\Pi_{22}^{o}&\Pi_{23}^{o}&\Pi_{24}^{o}&0&0&0&0\\\
\Pi_{14}^{o}&\Pi_{24}^{o}&-\Pi_{12}^{o}&\Pi_{34}^{o}&0&0&0&0\\\
\Pi_{13}^{o}&\Pi_{23}^{o}&\Pi_{43}^{o}&-\Pi_{12}^{o}&0&0&0&0\\\
0&0&0&0&\Pi_{11}^{c}&\Pi_{12}^{c}&\Pi_{13}^{c}&\Pi_{14}^{c}\\\
0&0&0&0&\Pi_{12}^{c}&\Pi_{22}^{c}&\Pi_{23}^{c}&\Pi_{24}^{c}\\\
0&0&0&0&\Pi_{14}^{c}&\Pi_{24}^{c}&-\Pi_{12}^{c}&\Pi_{34}^{c}\\\
0&0&0&0&\Pi_{13}^{c}&\Pi_{23}^{c}&\Pi_{43}^{c}&-\Pi_{12}^{c}\end{array}\right].$
(27)

Different from spin-population balanced case, there are $9$ independent matrix
elements in mean-field response function of each channel, or $16$ matrix
elements in two channels, which make the calculation of response function
become much heavy. In the final appendix of this paper, we give all
expressions of these independent matrix elements. Although there are no cross
terms in matrix of Eq.27 between open and close channel, the coupling effect
between these two channels is indeed existent. First, it is shown in the
definition of two order parameters (Eq.9). Also the random phase approximation
brings a correction to mean-field response function, which is shown in the
denominator of Eq.25. This correction brings not only the coupling effect of
these two channels, but also the right correction to investigate all
collective excitations.

Finally, we obtain the intra and inter response functions between these two
channels

$\begin{array}[]{c}\chi^{\left(oo\right)}=\chi_{11}+\chi_{12}+\chi_{21}+\chi_{22},\\\
\chi^{\left(oc\right)}=\chi_{15}+\chi_{16}+\chi_{25}+\chi_{26},\\\
\chi^{\left(co\right)}=\chi_{51}+\chi_{52}+\chi_{61}+\chi_{62},\\\
\chi^{\left(cc\right)}=\chi_{55}+\chi_{56}+\chi_{65}+\chi_{66}.\end{array}$
(28)

###  II.3 Dynamic structure factors

We now turn to discuss dynamic structure factors (DSF for short). Similarly we
can define the intra and inter dynamic structure factor between two channels,
which here is named the total and relative density dynamic structure factor,
respectively zhang2016 ,

$S_{{\rm tot}}\left(q,\omega\right)=-\frac{{\rm
Im\left[\chi^{\left(oo\right)}+\chi^{\left(oc\right)}+\chi^{\left(co\right)}+\chi^{\left(cc\right)}\right]}}{\pi\left(1-e^{-\hbar\omega/k_{B}T}\right)},$
(29)

$S_{{\rm rel}}\left(q,\omega\right)=-\frac{{\rm
Im\left[\chi^{\left(oo\right)}-\chi^{\left(oc\right)}-\chi^{\left(co\right)}+\chi^{\left(cc\right)}\right]}}{\pi\left(1-e^{-\hbar\omega/k_{B}T}\right)},$
(30)

where the subscript ’rel’ denotes a relative density oscillation between the
open and close channel. The total density dynamic structure factor satisfies
the famous $f$-sum rule

$\int d\omega S_{{\rm tot}}\left(q,\omega\right)\omega=N\frac{q^{2}}{2M},$
(31)

while at limit $q\rightarrow 0$ the relative density dynamic structure factor
satisfies

$\int d\omega S_{{\rm
rel}}\left(q,\omega\right)\omega=N\frac{q^{2}}{2M}-8\left\langle
V_{oc}\right\rangle,$ (32)

where $\langle V_{oc}\rangle$ is the coupling energy between open and closed
channels zhang2016 . In our following discussion, we will focus on zero
temperature $T=0$, and set Plank constant $\hbar=1$ for simple.

##  III DSF in balanced BCS superfluid

Let us first overview a spin-population balanced case with chemical potential
difference $\delta\mu=0$. The ground state of the system is a BCS-type
superfluid in both open and close channel, namely $\left[{\rm
BCS}\right]_{o}\left[{\rm BCS}\right]_{c}$. There are two stable eigenstates,
one is in-phase state with order parameters satisfy $\Delta_{o}\Delta_{c}>0$,
and the other one is an out-of-phase state with $\Delta_{o}\Delta_{c}<0$. Both
of them are possible candidates of ground state, depending on specific
interaction parameters. The collective excitation at transferred momentum
$q\ll k_{F}$ or $q\sim k_{F}$ had been discussed in Ref. zhang2016 . We will
first focus our discussion on a large transferred momentum $q\gg k_{F}$, and
then discuss the situation related to phase transition.

Figure 2:  The total density dynamic structure factor (blue line) and relative
one(red line) of BCS superfluid in in-phase ground state (a) and out-of-phase
metastable state (b). Two inset figures are their own enlarge of the pair-
breaking excitation, the left and right arrow mark the initial position of
pair-breaking excitation in the open and closed channel, respectively.

We take interaction parameters $1/k_{F}a_{0}=12/7$ and $1/k_{F}a_{1}=-12$ (or
$1/k_{F}a_{-}=2.0$ and $1/k_{F}a_{+}=1.5$), at which the system is in the BEC
regime of the famous BCS-BEC crossover. Then in-phase state is the ground
state, while the out-of-state is an energetical metastable state. We set
Zeeman detuning $\delta\left(B\right)=0.5\varepsilon_{F}$. The results of
total and relative dynamic structure factors of in-phase and out-of-phase
state at transferred momentum $q=5k_{F}$ are shown in Fig.2. Panel (a) is from
the in-phase state with chemical potential $\mu=-3.83\varepsilon_{F}$, open
channel order parameter $\Delta_{o}=1.38\varepsilon_{F}$ and close channel one
$\Delta_{c}=1.23\varepsilon_{F}$, while panel (b) from out-of-phase state with
$\mu=-2.05\varepsilon_{F}$, $\Delta_{o}=1.03\varepsilon_{F}$ and
$\Delta_{c}=-1.23\varepsilon_{F}$. Something in common in these two panels is
that, all dynamic structure factors consist of four different excitations,
which are molecular excitation, collective Leggett mode and Cooper-pair
breaking excitation in both open and close channel. The molecular excitation
of both panels always locates at $q^{2}/4M\simeq 12.5\varepsilon_{F}$, which
are only dependent on the value of transferred momentum $q$, no matter what
state it is. Since the chemical potential in two states both satisfy $\mu<0$,
the minimum energy required to break a Cooper-pair is
$\omega_{th}^{\left(o\right)}=2\sqrt{(q^{2}/8M-\mu)^{2}+\Delta_{o}^{2}}$ in
open channel and
$\omega_{th}^{\left(c\right)}=2\sqrt{(q^{2}/8M-\mu+\delta\left(B\right)/2)^{2}+\Delta_{c}^{2}}$
in close channel, respectively. These two initial excitations generate two
twist points in two inset figures, located by two arrows. Panel (a) is the
ground state because the in-phase state requires more energy to break a
Cooper-pair than the case of out-of-phase state shown in Panel (b). This is
also proved by a relatively longer energy distance between molecular
excitation and ${\rm
min}\left[\omega_{th}^{\left(o\right)},\omega_{th}^{\left(c\right)}\right]$,
whose value is almost equal to the bigger bound energy
$\varepsilon_{b}^{-}=-\hbar^{2}/Ma_{-}^{2}=8.0\varepsilon_{F}$, while the
energy distance between Leggett mode and ${\rm
min}\left[\omega_{th}^{\left(o\right)},\omega_{th}^{\left(c\right)}\right]$
take the other smaller bound energy
$\varepsilon_{b}^{+}=-\hbar^{2}/Ma_{+}^{2}$ (these two energy differences will
be exactly equal to
$\varepsilon_{b}^{\pm}=-\hbar^{2}/Ma_{\pm}^{2}=4.5\varepsilon_{F}$ when
$\delta\left(B\right)=0$). This is also the reason why the Leggett mode
locates on the right hand side of molecular excitation in Panel (a), or on the
left hand side in Panel (b). Here dynamic structure factors display rich
information about large transferred momentum dynamical excitations, which do a
great help to distinguish ground in-phase state from metastable out-of-phase
state.

Figure 3:  The total (upper panel) and relative (lower panel) dynamic
structure factor in BCS superfluid side of phase transition, with scattering
length $1/k_{F}a_{0}=-0.5$, $1/k_{F}a_{1}=-0.05$, Zeeman detuning
$\delta(B)=0.4\varepsilon_{F}$. The result is independent on the value of
chemical potential difference when $\delta\mu<\delta\mu_{{\rm cr}}$. (a)
phonon mode, (b) Leggett mode, (c) $2\Delta_{c}$, (d) $2\Delta_{o}$, (e) pair-
breaking excitations at a large $q$ in two channels.

Next we discuss the phase transition from BCS to Sarma superfluid with the
same parameters in Fig.1. When chemical potential difference $\delta\mu$ is
smaller than the critical value $\delta\mu_{{\rm cr}}=0.486\varepsilon_{F}$, a
BCS superfluid state becomes the ground state of the system. These parameters
are like the results of single-channel Fermi superfluid close to unitary
regime of BCS-BEC crossover, with a positive chemical
$\mu=0.43\varepsilon_{F}$. As shown in Fig.3, at a small transferred momentum
$q$, we observe two collective excitations (red curves in two panels), the
phonon (a) and Leggett mode (b). Relatively, the gapless phonon mode is easier
to investigate in total dynamic structure factor than in the relative dynamic
structure factor, while the gapped Leggett mode just in inverse. Two short
horizontal lines at
$\omega^{\left(c\right)}=2|\Delta_{c}|=0.72\varepsilon_{F}$ (c) and
$\omega^{\left(o\right)}=2|\Delta_{o}|=1.10\varepsilon_{F}$ (d) represent two
minimum values of energy to break a Cooper pair in open and close channel,
respectively, while the same physics evolves into a two-peak signal at a large
$q$ (e). Here in fact all results are independent the value of chemical
potential difference $\delta\mu$, when $\delta\mu<\delta\mu_{{\rm cr}}$. The
system is always a spin-population balanced BCS superfluid state.

##  IV DSF in Sarma superfluid

Figure 4:  The total (upper panel) and relative (lower panel) dynamic
structure factor in Sarma superfluid side of phase transition, with scattering
length $1/k_{F}a_{0}=-0.5$, $1/k_{F}a_{1}=-0.05$, Zeeman detuning
$\delta(B)=0.4\varepsilon_{F}$, and chemical potential difference
$\delta\mu=0.5\varepsilon_{F}$. (${\rm a^{\prime}}$) phonon mode mixed with
single-particle gapless excitation in close channel, (b) Leggett mode, (d)
$2\Delta_{o}$, (e) pair-breaking excitations at a large $q$ in two channels.

For the same scattering length $1/k_{F}a_{0}=-0.5$, $1/k_{F}a_{1}=-0.05$ and
Zeeman detuning $\delta(B)=0.4\varepsilon_{F}$, but varying the chemical
potential difference over the critical value $\delta\mu_{{\rm cr}}$ to
$\delta\mu=0.5\varepsilon_{F}$, the system comes into a spin-population
polarized Sarma superfluid state $\left[{\rm BCS}\right]_{o}\left[{\rm
Sarma}\right]_{c}$. The chemical potential difference $\delta\mu$ is larger
than the order parameter in close channel $\Delta_{c}=-0.08\varepsilon_{F}$,
but smaller than one in open channel $\Delta_{o}=0.54\varepsilon_{F}$. Then
the close channel becomes a gapless Sarma superfluid, while the open channel
is a BCS superfluid. Our main results of the total and relative dynamic
structure factor in this Sarma superfluid are shown respectively in two panels
of Fig.4, in the form of two-dimensional contour plot in the $q$-$\omega$
plane. The dynamic structure factors as a function of $\omega$ at small and
large transferred momenta are reported in Fig.5 and Fig.6, respectively.

Figure 5:  The total (upper panel) and relative (lower panel) dynamic
structure factors of Sarma superfluid at small transferred momentums,
$q=0.3,0.5,0.7,0.9k_{F}$. A green arrow in lower panel locates the Leggett
mode.

Different from the results of BCS superfluid whose phonon excitation has an
obvious delta-like structure, the phonon excitation here has a finite
expansion width because of the scattering with the gapless single-particle
excitation in the close channel, which is denoted as (${\rm a^{\prime}}$) in
Fig.4. In greater detail, all collective excitations are focused in a small
transferred momentum $q$ regime $\left(q<k_{F}\right)$. A single-particle
gapless excitation of Sarma superfluid can be observed in both total and
relative dynamic structure factors, and it is mixed with another gapless
excitation, the collective phonon excitation. This monotonically increasing
linear behavior of dynamic structure factor is also displayed in Fig.6 at
$q=2k_{F}$, where the transferred energy starts from a very small value. The
results in this regime are quite similar to behavior of ideal Fermi gases
mazzanti1996 , and it can be understood from the fact that some fermionic
atoms are not paired, and behave like free fermions in close channel. This
gapless dynamical behavior is the most specific character of Sarma superfluid.

Also we can observe an horizontal gapped Leggett excitation, whose position
almost does not vary with the transferred momentum $q$. This is also shown in
the peak around $\omega\simeq 0.6\varepsilon_{F}$ of relative dynamic
structure factor in Fig.5, located by a green arrow. Going on increasing
transferred momentum $q$, a merge procedure between Leggett and phonon mode
can be clearly investigated in the relative dynamic structure factor (Panel
(b) of Fig.4, and lower panel of Fig.5), the critical value of transferred
momentum is around $q\approx 0.7k_{F}$, after which the phonon and Leggett
excitations are mixed together and show only a wide peak, as shown by gray dot
line in Fig.5.

Another physics we are interested in is the minimum value to break Cooper
pairs in two channels. Different from the BCS superfluid, only one short
horizontal line at $\omega^{(o)}=2\Delta_{o}=1.08\varepsilon_{F}$, who is the
minimum energy to break a Cooper-pair in open channel, is observed, while the
other one in close channel is completely immersed in the gapless excitation.

Figure 6:  The total (upper panel) and relative (lower panel) dynamic
structure factors of Sarma superfluid at big transferred momentums,
$q=2,3,4k_{F}$. Figure 7:  The comparison of total dynamic structure factor
between BCS superfluid state (blue line) and Sarma superfluid state (grey dot
line) at $q=1k_{F}$ (a) and $q=5k_{F}$ (b).

At a large $q$, the molecular excitation and atomic excitation are both
observed in Fig.6 and Panel (b) of Fig.7. Different from the BCS superfluid,
the molecular excitation does not have a clear two-peak signal related to
pair-breaking excitations in two channels, which is because that the Cooper
pair breaking in close channel is quite weak, compared with one in open
channel. We can only observe a very small peak in total dynamic structure
factors, or a very small dip in relative dynamic structure factors, whose
location is after the first big Cooper-pair breaking peak in Fig.6. The atomic
excitation, whose location is around $q^{2}/2M$, becomes more and more obvious
when increasing transferred momentum $q$. Also Sarma superfluid has a relative
stronger atomic excitation than one in BCS superfluid since it has more
surplus of unpaired Fermi atoms.

Finally we give Fig.7 by which to present the main difference between BCS
superfluid (gray line) and Sarma superfluid (blue line). Compared with BCS
superfluid, Sarma superfluid has a linear gapless excitation at $q=k_{F}$,
mixed with the collective phonon excitation (Panel (a)), and a stronger atomic
excitation (Panel (b)) at $q=5k_{F}$.

##  V Conclusions

In summary, we investigate dynamic structure factors of a two-channel Fermi
superfluid with orbital Feshbach resonance. In spin-population balanced BCS
superfluid state, dynamic structure factors can do help to distinguish the
ground in-phase from metastable out-of-phase state by the location of minimum
energy to break a Cooper-pair, or the relative location between collective
Leggett mode and molecular excitation in a BEC-like parameters regime. For
another different interaction parameters where Sarma superfluid may replace
the BCS superfluid as a possible ground state of the system, we numerically
calculate dynamic structure factors and find their dynamical characters in
both spin-population balanced BCS superfluid and spin-polarized Sarma
superfluid. An obvious gapless excitation at a small transferred energy is
observed only in Sarma superfluid, which also has a stronger atomic excitation
at a large transferred momentum. These discoveries can serve to differentiate
all possible eigenstates of the system.

##  VI Acknowledgement

Our research was supported by the National Natural Science Foundation of
China, Grants No. 11804177 (P.Z.), No. 11547034 (H.Z.) and No. 11775123
(L.H.), by the Shandong Provincial Natural Science Foundation, China, Grant
No. ZR2018BA032 (P.Z.), and by Australian Research Council’s (ARC) Discovery
Projects No. DP180102018 (X.-J.L), and No. DP170104008 (H.H.). L.H.
acknowledges the support of the Recruitment Program for Young Professionals in
China (i.e., the Thousand Young Talent Program).

###  VI.1 Appendix

In this appendix, we will give the derivation of mean-field response functions
$\Pi$ in both open and close channels. Actually the response functions in
these two channels are the same as each other, but with their corresponding
values of chemical potential $\mu_{n}$ and order parameter $\Delta_{n}$ ,
where $n=o,c$ is the channel index. For example, in $n$ channel, there are $9$
independent matrix elements, which are expressed as

$\Pi_{11}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{1}^{l}\left(k\right)P_{1}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{12}^{n}\left(q,\omega\right)=-\sum_{kll^{\prime}}P_{3}^{l}\left(k\right)P_{3}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{13}^{n}\left(q,\omega\right)=-\sum_{kll^{\prime}}P_{1}^{l}\left(k\right)P_{3}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{14}^{n}\left(q,\omega\right)=-\sum_{kll^{\prime}}P_{3}^{l}\left(k\right)P_{1}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{22}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{2}^{l}\left(k\right)P_{2}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{23}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{3}^{l}\left(k\right)P_{2}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{24}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{2}^{l}\left(k\right)P_{3}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{34}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{2}^{l}\left(k\right)P_{1}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

$\Pi_{43}^{n}\left(q,\omega\right)=+\sum_{kll^{\prime}}P_{1}^{l}\left(k\right)P_{2}^{l^{\prime}}\left(k+q\right)F_{k}^{ll^{\prime}}\left(q,\omega\right),$

where

$P_{1}^{l}\left(k\right)=\frac{1}{2}\left(1+l\frac{\xi_{nk}}{E_{nk}}\right),$

$P_{2}^{l}\left(k\right)=\frac{1}{2}\left(1-l\frac{\xi_{nk}}{E_{nk}}\right),$

$P_{3}^{l}\left(k\right)=l\frac{\Delta_{n}}{2E_{nk}},$

$F_{k}^{ll^{\prime}}\left(q,\omega\right)=\frac{f\left(E_{nk}^{l}\right)-f\left(E_{nk+q}^{l^{\prime}}\right)}{\omega+i\eta+E_{nk}^{l}-E_{nk+q}^{l^{\prime}}}.$

$q$ and $\omega$ are respectively the transferred momentum and energy, $l=\pm
1$ stands for the particle or hole solution, $\eta$ is a small positive
number. Specifically one should notice that expressions of $\Pi_{34}^{n}$ and
$\Pi_{43}^{n}$ are both divergent. The random phase approximation, shown in
the denominator of Eq.25, do help to cure the divergence of them.

## References

  * (1) L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, (Oxford: Oxford University Press, 2003).
  * (2) M. G. Lingham, K. Fenech, S. Hoinka, and C. J. Vale, Local Observation of Pair Condensation in a Fermi Gas at Unitarity, Phys. Rev. Lett. 112, 100404 (2014).
  * (3) E. D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford, and C. J. Vale, Temperature Dependence of the Universal Contact Parameter in a Unitary Fermi Gas, Phys. Rev. Lett. 106, 170402 (2011).
  * (4) S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut, and S. Gandolfi, Precise Determination of the Structure Factor and Contact in a Unitary Fermi Gas, Phys. Rev. Lett. 110, 055305 (2013).
  * (5) G. Veeravalli, E. Kuhnle, P. Dyke, and C. J. Vale, Bragg Spectroscopy of a Strongly Interacting Fermi Gas, Phys. Rev. Lett. 101, 250403 (2008).
  * (6) S. Hoinka, M. Lingham, M. Delehaye, and C. J. Vale, Dynamic Spin Response of a Strongly Interacting Fermi Gas, Phys. Rev. Lett. 109, 050403 (2012).
  * (7) Sascha Hoinka, Paul Dyke, Marcus G. Lingham, Jami J. Kinnunen, Georg M. Bruun and Chris J. Vale, Goldstone mode and pair-breaking excitations in atomic Fermi superfluids, Nat. Phys. 13, 943 (2017).
  * (8) A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari, and F. Zambelli, Momentum transferred to a trapped Bose-Einstein condensate by stimulated light scattering, Phys. Rev. A 64, 063614 (2001).
  * (9) P. Fulde and R. A. Ferrell, Superconductivity in a Strong Spin-Exchange Field, Phys. Rev. 135, A550 (1964).
  * (10) A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)].
  * (11) G. Sarma, On the influence of a uniform exchange field acting on the spins of the conduction electrons in a superconductor, J. Phys. Chem. Solids 24, 1029 (1963).
  * (12) C. Zhang, S. Tewari, R. M. Lutchyn, and S. D. Sarma, $p_{x}+ip_{y}$ Superfluid from $s$-Wave Interactions of Fermionic Cold Atoms, Phys. Rev. Lett. 101, 160401 (2008).
  * (13) L. He and P. Zhuang, Stable Sarma state in two-band Fermi systems, Phys. Rev. B 79, 024511 (2009).
  * (14) R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, Orbital Feshbach resonance in Alkali-Earth Atoms, Phys. Rev. Lett. 115, 135301 (2015).
  * (15) P. Zou, L. He, X. -J. Liu,and Hui Hu, Strongly interacting Sarma superfluid near orbital Feshbach resonances, Phys. Rev. A 97, 043618 (2018).
  * (16) G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Strongly Interacting Gas of Two-Electron Fermions at an Orbital Feshbach Resonance, Phys. Rev. Lett. 115, 265301 (2015).
  * (17) M. Höfer, L. Riegger, F. Scazza, C. Hofrichter, D. R. Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S. Fölling, Observation of an Orbital Interaction-Induced Feshbach Resonance in 173Yb, Phys. Rev. Lett. 115, 265302 (2015).
  * (18) A. J. Leggett, Number-phase fluctuations in two-band superconductors, Prog. Theor. Phys. 36, 901 (1966).
  * (19) G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhigadlo, and J. Karpinski, Observation of Leggett’s Collective Mode in a Multiband $\rm{MgB}_{2}$ Superconductor, Phys. Rev. Lett. 99, 227002 (2007).
  * (20) S.-Z. Lin and X. Hu, Massless Leggett Mode in Three-Band Superconductors with Time-Reversal-Symmetry Breaking, Phys. Rev. Lett. 108, 177005 (2012).
  * (21) N. Bittner, D. Einzel, L. Klam, and D. Manske, Leggett Modes and the Anderson-Higgs Mechanism in Superconductors without Inversion Symmetry. Phys. Rev. Lett. 115, 227002 (2015).
  * (22) S. N. Klimin, H. Kurkjian and J. Tempere, Leggett collective excitations in a two-band Fermi superfluid at finite temperatures, New J. Phys. 21, 113043 (2019).
  * (23) J. Xu, R. Zhang, Y. Cheng. P. Zhang, R. Qi, and H. Zhai, Reaching a Fermi-superfluid state near an orbital Feshbach resonance, Phys. Rev. A 94, 033609 (2016).
  * (24) L. He, H. Hu, and X.-J. Liu, Two-band description of resonant superfluidity in atomic Fermi gases, Phys. Rev. A 91, 023622 (2015).
  * (25) L. He, J. Wang, S.-G. Peng, X.-J. Liu, and H. Hu, Strongly correlated Fermi superfluid near an orbital Feshbach resonance: Stability, equation of state, and Leggett mode, Phys. Rev. A. 94, 043624 (2016).
  * (26) H. Guo, C.-C. Chien, and Y. He, Theories of Linear Response in BCS Superfluids and How They Meet Fundamental Constraints, J. Low Temp. Phys. 172, 5 (2013).
  * (27) L. He, Dynamic density and spin responses of a superfluid Fermi gas in the BCS-BEC crossover: Path integral formulation and pair fluctuation theory, Ann. Phys. 373, 470 (2016).
  * (28) Y.-C. Zhang, S. Ding, and S. Zhang, Collective modes in a two-band superfluid of ultracold alkaline-earth-metal atoms close to an orbital Feshbach resonance, Phys. Rev. A 95, 041603(R) (2017).
  * (29) F. Mazzanti, A. Polls, J. Boronat, Coherent and incoherent dynamic structure functions of the free Fermi gas, Phys. Lett. A, 220 (1996), p. 251.