# Superfluidity and collective modes in Rashba spin-orbit coupled Fermi gases

###### Abstract

We present a theoretical study of the superfluidity and the corresponding collective modes in two-component atomic Fermi gases with -wave attraction and synthetic Rashba spin-orbit coupling. The general effective action for the collective modes is derived from the functional path integral formalism. By tuning the spin-orbit coupling from weak to strong, the system undergoes a crossover from an ordinary BCS/BEC superfluid to a Bose-Einstein condensate of rashbons. We show that the properties of the superfluid density and the Anderson-Bogoliubov mode manifest this crossover. At large spin-orbit coupling, the superfluid density and the sound velocity become independent of the strength of the -wave attraction. The two-body interaction among the rashbons is also determined. When a Zeeman field is turned on, the system undergoes quantum phase transitions to some exotic superfluid phases which are topologically nontrivial. For the two-dimensional system, the nonanalyticities of the thermodynamic functions and the sound velocity across the phase transition are related to the bulk gapless fermionic excitation which causes infrared singularities. The superfluid density and the sound velocity behave nonmonotonically: they are suppressed by the Zeeman field in the normal superfluid phase, but get enhanced in the topological superfluid phase. The three-dimensional system is also studied.

###### keywords:

Fermi superfluidity, BCS-BEC crossover, Rashba spin-orbit coupling^{†}

^{†}journal: Annals of Physics (N. Y.)

## 1 Introduction

It is generally believed that, by tuning the strength of the attractive interaction in a many-fermion system, we can realize a smooth crossover from the Bardeen–Cooper–Schrieffer (BCS) superfluidity at weak attraction to Bose–Einstein condensation (BEC) of difermion molecules at strong attraction Eagles ; Leggett ; BCSBEC1 ; BCSBEC2 ; BCSBEC3 ; BCSBEC4 ; BCSBEC5 ; BCSBEC6 ; BCSBEC7 ; BCSBEC8 . One typical example is the dilute Fermi gas in three dimensions with short-range attractive interaction, where the effective range of the interaction is much smaller than the inter-particle distance characterized by where is the Fermi momentum in the absence of interaction. The attraction strength is characterized by a dimensionless parameter where is the -wave scattering length of the short-range interaction. The BCS-BEC crossover has been confirmed in the experiments of ultracold fermionic atoms BCSBECexp1 ; BCSBECexp2 ; BCSBECexp3 , where the -wave scattering length and hence the parameter was tuned by means of the Feshbach resonance.

On the other hand, the effect of a nonzero Zeeman field has been a longstanding problem of fermionic superconductivity/superfluidity for several decades LOFFreview . It is generally believed that the superfluidity is completely destroyed when the Zeeman field becomes large enough. The well-known theoretical result for -wave weak-coupling superconductors is that, at a critical Zeeman field (called Chandrasekhar-Clogston limit) where is the zero temperature gap at , a first-order phase transition from the BCS state to the normal state occurs CClimit ; Sarma . Further theoretical studies showed that the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov state FFLO may survive in a narrow window between and . The Zeeman field effects in the BCS-BEC crossover have been experimentally studied by using cold fermionic atoms Imexp ; Imth . The atom numbers of the two lowest hyperfine states of Li are adjusted to create a population imbalance which simulates effectively the Zeeman field . The experimental results show that the fermionic superfluidity in the BCS-BEC crossover regime is also completely destroyed when the Zeeman field is large enough.

The recent experimental breakthroughs in generating synthetic non-Abelian gauge field and synthetic spin-orbit coupling SOC ; SOCF01 ; SOCF02 ; SOCrmp ; SOC01 ; SOC02 ; SOC03 ; 3DSOC have opened up the way to study the spin-orbit coupling effects as well as the combined spin-orbit coupling and Zeeman field effects on the BCS-BEC crossover SOC-BCSBEC ; SOC-Hu ; SOC-Yu ; SOC-Gong ; SOC-Iskin ; SOC-He ; SOC-Han ; SOC-Yi ; SOC-other . For solid state systems, it was shown that the topologically nontrivial superconducting phase appears in spin-orbit coupled systems if the Zeeman field is large enough TSC01 ; TSC02 ; TSC03 ; TSC04 ; TSC05 ; TSC06 ; TSC07 ; TSC08 ; TSC09 . For neutral atoms, the spin-orbit coupling can be generated through a synthetic non-Abelian gauge potential SOCrmp . The well-known Rashba spin-orbit coupling for spin-1/2 fermions can be generated via a 2D synthetic vector potential SOC02 ; SOC03

(1) |

where for any vector . The single-particle Hamiltonian for a fermion moving in the synthetic gauge field is given by where is the momentum operator. In this paper we use the natural units for convenience.

For the 2D synthetic vector potential given in (1), the single-particle Hamiltonian can be reduced to

(2) |

where an irrelevant constant has been omitted. The spin-dependent term can be mapped to the standard Rashba spin-orbit coupling by a spin rotation and . The gauge field strength characterizes the strength of the spin-orbit coupling, which can be tuned from weak to strong in cold atom experiments. Since the final physical results depend only on , we set in this paper without loss of generality. For many-fermion systems, the spin-orbit coupling strength can be characterized by the dimensionless ratio . While for solid state systems this ratio is very small, it can reach the order in cold atom systems SOCF01 ; SOCF02 . Therefore, cold fermionic atoms provide the way to study the fermionic superfluidity in the presence of a strong spin-orbit coupling.

Motivated by the experimental progress of realizing spin-orbit coupled atomic Fermi gases SOCF01 ; SOCF02 , the fermionic superfluidity with spin-orbit coupling has been extensively studied SOC-BCSBEC ; SOC-Hu ; SOC-Yu ; SOC-Gong ; SOC-Iskin ; SOC-He ; SOC-Han ; SOC-Yi ; SOC-Zhou ; SOC-other . It was shown that, in the presence of the Rashba spin-orbit coupling, the two-body bound state exists even for where the bound state does not exist for 3Dbound . With increased , the binding energy is generally enhanced. The bound state at possesses a nontrivial effective mass which is generally larger than twice of the fermion mass SOC-Hu ; SOC-Yu ; SOC-He . Such a novel bound state is referred to as rashbon in the studies rashbon . For many-fermion systems, it has been proposed that a spin-orbit coupled Fermi gas can undergo a smooth crossover from the ordinary BCS/BEC superfluidity to the Bose-Einstein condensation of rashbons if is tuned from small to large values SOC-BCSBEC ; SOC-Hu ; SOC-Yu ; SOC-Gong ; SOC-Iskin ; SOC-He ; SOC-Han . On the other hand, if a Zeeman field is turned on, some topologically nontrivial superfluid phases emerge SOC-Gong ; SOC-Han ; SOC-Yi .

In this paper, we study the bulk superfluid properties and the collective modes in Rashba spin-orbit coupled Fermi superfluids. We mainly consider two aspects: (1) the bulk superfluid properties and the collective modes from weak to strong spin-orbit coupling at zero Zeeman field, which manifest the crossover from ordinary Fermi superfluidity to the Bose-Einstein condensation of rashbons, and (2) the quantum phase transitions from the normal superfluid phase to topologically nontrivial superfluid phases in the presence of nonzero Zeeman field and their effects on the bulk superfluid properties and the collective modes.

The paper is organized as follows. In Sec. 2, we derive the general effective action for the superfluid ground state and the collective modes with arbitrary spin-orbit coupling and Zeeman field by using the functional path integral method. In Sec. 3 and Sec. 4, we study the systems with zero Zeeman field in three and two spatial dimensions, respectively. The systems with nonzero Zeeman fields are studied in Sec. 5. We summarize in Sec. 6.

## 2 General formalism

We consider a homogeneous spin-1/2 Fermi gas with a short-range -wave attractive interaction in the spin-singlet channel. For cold atom experiments, the attractive strength can be tuned from weak to strong BCSBEC6 . In the dilute limit where the effective range is much smaller than the characteristic length scales of the system, that is, , the attractive interaction can be modeled by a contact one Cui . The many-body Hamiltonian of the system can be written as

(3) |

where

(4) |

Here represents the two-component fermion fields, is the chemical potential, and is the Zeeman magnetic field. We set in this paper without loss of generality. The contact coupling denotes the attractive -wave interaction between unlike spins.

In the functional path integral formalism, the partition function of the system at finite temperature is

(5) |

where

(6) |

Here is obtained by replacing the field operators and with the Grassmann variables and , respectively. To decouple the interaction term we introduce the auxiliary complex pairing field and apply the Hubbard-Stratonovich transformation. Using the Nambu-Gor’kov representation

(7) |

we express the partition function as

(8) |

where

(9) |

The inverse single-particle Green’s function is given by

(10) |

where

(11) |

Integrating out the fermion fields, we obtain

(12) |

where the effective action reads

(13) |

The effective action cannot be evaluated precisely. In this work, we consider mainly the zero temperature case. Therefore, we follow the conventional approach to the BCS-BEC crossover problem, that is, we first consider the superfluid ground state which corresponds to the saddle point of the effective action, and then study the Gaussian fluctuations around the saddle point. The Gaussian fluctuations correspond to the collective modes, including the gapless Goldstone mode and the massive Higgs mode. In ordinary fermionic superfluids, only the Goldstone mode or the so-called Anderson-Bogoliubov mode remains at low energy whereas the Higgs mode is pushed up to the two-particle continuum. Therefore, the Higgs mode usually appears as a broad resonance at the large characteristic energy scale of the system.

In the superfluid ground state, the pairing field acquires a nonzero expectation value , which serves as the order parameter of the superfluidity. Due to the U symmetry, we can set to be real without loss of generality. Then we decompose the pairing field as , where is the fluctuation around the mean field. The effective action can be expanded in powers of the fluctuation , that is,

(14) |

where is the saddle-point or mean-field effective action with determined by the saddle point condition . Note that under the saddle point condition the linear terms in and in Eq. (14) vanish automatically.

### 2.1 Saddle point: mean-field approximation

The mean-field effective action or the thermodynamic potential can be expressed as

(15) |

where the inverse fermion Green’s function reads

(16) |

and is given by

(17) |

The dispersion is defined as with . In this paper denotes the energy and the momentum of fermions with ( integer) being the fermion Matsubara frequency. We use the notation with for the 3D system.

The determinant of the inverse fermion propagator, , can be evaluated as

(18) | |||||

where . The quasiparticle excitation spectra are given by

(19) |

The quantities and are defined as and . Completing the Matsubara frequency sum we obtain the explicit form of the mean-field effective action

(20) |

where . Here the term is added to recover the correct limit for . The integral over the fermion momentum is divergent and the contact coupling needs to be regularized. For a short-range interaction potential with its -wave scattering length , it is natural to regularize by the two-body T-matrix in the absence of SOC. We have

(21) |

The superfluid order parameter should satisfy the saddle-point condition , or the so-called gap equation

(22) |

where is the Fermi-Dirac distribution. Meanwhile, if the total fermion density is imposed, the chemical potential should be determined by the number equation , that is,

(23) |

In general, and are obtained by solving the gap and number equations simultaneously. As a convention, we define the Fermi momentum through the noninteracting form , and the Fermi energy is given by .

In the Nambu-Gor’kov space, the fermion propagator takes the form

(24) |

The matrix elements can be evaluated as

(25) |

where are given by

(26) |

To evaluate the collective mode propagator, we also express the fermion propagator in an alternative form by using the following projectors

(27) |

which possess the following properties

(28) |

With the help of these projectors, the fermion propagator can be expressed as

(29) |

We note that the anomalous Green’s function is not diagonal in the spin space for . Therefore, the spin-orbit coupling generates spin-triplet pairing even though the order parameter has the -wave symmetry. According to the Green’s function relation, the spin-singlet and spin-triplet pairing amplitudes can be read from the diagonal and off-diagonal components of . We obtain

(30) |

for the spin-singlet pairing amplitudes and

(31) |

for the spin-triplet pairing amplitudes. In the mean-field theory, the condensation density is half of the summation of all pairing amplitudes squared FC ; FCFM , that is,

(32) |

It is also useful to reexpress the mean-field theory in the helicity representation TSC08 . The helicity basis is related to basis by a -dependent SU transformation. In the helicity basis, is diagonal, that is

(33) |

Therefore, the spin-orbit coupled Fermi gas can be viewed as a two-band system. The Zeeman field provides a band gap at . In the presence of attraction, the mean-field approximation for reads

(34) |

The new momentum-dependent pair potentials read

(35) |

where the interband and intraband pair potentials are given by

(36) |

Using these new pair potentials, the quasiparticle spectra can be expressed as

(37) |

The above expressions in the helicity basis will help us understand some results in Sec. 5.

### 2.2 Gaussian fluctuation: collective excitations

Then we consider the fluctuations around the mean field. The linear terms which are of order vanish precisely once the saddle-point condition is imposed. The quadratic terms, corresponding to the Gaussian fluctuations, can be evaluated as

(38) |

where is defined as

(39) |

In this paper denotes the energy and momentum of bosons with being the boson Matsubara frequency.

After taking the trace in the Nambu-Gor’kov space, we find that can be written in a bilinear form

(40) |

where the inverse boson propagator is a matrix,

(41) |

The matrix elements of can be expressed in terms of the fermion propagator . We have

(42) |

The explicit forms of these functions are evaluated in Appendix A. It is straightforward to show that . However, we have if the spin-orbit coupling and the Zeeman field are both nonzero. Taking the analytical continuation , the dispersion of the collective mode is determined by the equation

(43) |

We can decompose as , where and are even and odd functions of , respectively. Their explicit forms read

(44) | |||||

and

(45) | |||||

where for convenience. On the other hand and are even functions of and can be expressed as

(46) |

Here and read

(47) | |||||

and

(48) | |||||

The explicit expressions of the functions and () are presented in Appendix A. We note that and are odd functions of , that is, they are proportional to . However, the determinant of the matrix is an even function of , as we expect.

To make the results more physically transparent, we decompose the complex fluctuation field into its amplitude part and phase part , . Converting to the variables and , we obtain

(49) |

where the matrix reads

(50) |

From the expressions of and , we have and . Therefore the amplitude and phase modes decouple completely at and . Furthermore, at the saddle point we have precisely

(51) |

Therefore the phase mode at is gapless, that is, the Goldstone mode. For neutral fermionic superfluids, this mode is also called the Anderson-Bogoliubov mode. Another collective mode or the so-called Higgs mode is massive. It is likely heavily damped since its mass gap is generally larger than the two-particle continuum at .

We are interested in the low energy behaviors of these collective modes. For this purpose, we make a small and expansion of