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The speed of the Goldstone sound mode of a spin-orbit-coupled atomic Fermi gas loaded in a square optical lattice with a non-Abelian gauge field in the presence of a Zeeman field is calculated within the Gaussian approximation and from the Bethe-Salpeter equation in the generalized random phase approximation. It is found that (i) there is no sharp change of the slope of the Goldstone sound mode across the topological quantum phase transition point and (ii) the Gaussian approximation significantly overestimates the speed of sound of the Goldstone mode compared to the value provided by the BS formalism.

It is well known that topological superfluids are new states of matter that can be observed in two-dimensional atomic Fermi gases with strong Rashba spin-orbit coupling (SOC), conventional s-wave pairing, and out-of-plane magnetic field (Zeeman field) which breaks time-reversal symmetry. In what follows, we study the speed of sound of a two-component pseudospin-

Recent numerical calculations of the sound speed of atomic Fermi gases with s-wave attraction, synthetic Rashba SOC, and out-of-plane Zeeman field in two- and three-dimensional

It is widely accepted among the cold-atom community that the system of a two-component Fermi gas with s-wave attraction in square optical lattice with an external non-Abelian gauge field (or a synthetic Rashba SOC) and out-of-plane Zeeman field can be described by a negative-U Hubbard model [

Hamiltonian (

In the case of a square lattice the vectors

The opening or closing of the gap can be achieved by varying the tunable parameter

The single-particle and collective excitations of the system under consideration manifest themselves as poles of the single-particle and two-particle Green’s functions, but the corresponding expressions for the Green’s functions cannot be evaluated exactly because the interaction part of the Hubbard Hamiltonian is quartic in the fermion fields. The simplest way to solve this problem is to apply the so-called mean field decoupling of the quartic interaction. According to the standard mean field theory on the Hubbard Hamiltonian, the on-site interaction is decoupled (up to Hartree-Fock correction terms) as

Instead of applying the mean field decoupling, we shall transform the quartic terms to a quadratic form by introducing a four-component boson field which mediates the interaction of fermions in the same manner as in quantum electrodynamics, where the photons mediate the interaction of electric charges. Green’s functions are thermodynamic averages of the

The functional-integral formulation of the Hubbard model requires the representation of the Hubbard interaction

As it is well known, Green’s functions in the functional-integral approach are defined by means of the so-called generating functional with sources for the boson and fermion fields, but the corresponding functional integrals cannot be evaluated exactly because the interaction part of the Hubbard Hamiltonian is quartic in the Grassmann fermion fields. However, we can transform the quartic terms to a quadratic form by introducing a model system which consists of a four-component boson field

The field operators (

The action of the above-mentioned model system is assumed to be of the following form

The Fourier transform of noninteracting Green’s function is

The Fourier transform of zero-temperature single-particle Green’s function (

The matrix elements

For a fixed filing factor

As we mentioned before, in order to keep the polarization fixed, one has to adjust the corresponding Zeeman field. This means that

Another interesting feature is that because of the SOC term in Hamiltonian (

With the help of the pairing amplitudes, one can calculate the total condensate fraction

The spectrum of the collective modes will be obtained by solving the BS equation in the GRPA. As we have already mentioned, the kernel of the BS equation is a sum of the direct

At a given vector

In contrast to our functional-integral formalism, one can use the Hubbard-Stratonovich transformation to introduce the energy gap as an order parameter field. In this approach, one can integrate out the fermion fields and to arrive at an effective action. The next steps are to consider the state which corresponds to the saddle point of the effective action and to write the effective action as a series in powers of the fluctuations and their derivatives. The exact result can be obtained by explicitly calculating the terms up to second order in the fluctuations and their derivatives. This approximation is called the Gaussian approximation. Within this approximation, the collective excitation spectrum

There are various mean field quantities of physical interest, such as the chemical potential, the pairing gap, the singlet and triplet pairing amplitudes, and the singlet and triplet condensate fractions. We focus on the zero-temperature case assuming two different filling factors of

In Figure

Chemical potential

The singlet

Next, we have calculated the speed of sound as a function of the SOC parameter

It is clear that the single-particle dispersion, given by (

In Figure

The slope of the Goldstone sound mode of a Fermi gas in a square optical lattice subject to a non-Abelian gauge field

The boundaries for the TQPT in [

We have fixed the non-Abelian parameter

Chemical potential

In this paper, we are concerned with the question which naturally arises here as to whether there is a possibility of detecting TQPTs by monitoring the behavior of slope of the sound mode if a non-Abelian gauge field is used instead of a synthetic Rashba SOC. The answer is not trivial because the non-Abelian gauge field is taken into account via the Peierls substitution. As a result, the tight-binding energy

In summary, we have derived the BS equation in the GRPA for the collective excitation energy of a Fermi gas loaded on a square optical lattice with a non-Abelian gauge field in the presence of a Zeeman field. To the best of our knowledge, there is no other calculation of the speed of sound in a lattice geometry as a function of the non-Abelian parameter with which we can make comparisons. According to our numerical calculations, there is no sharp change of the slope of the Goldstone sound mode across the phase transition point. It is found that the Gaussian approximation significantly overestimates the speed of sound of the Goldstone mode compared to the value obtained within the BS formalism.

The two-particle propagator in BS equation (

The elements

The authors declare that they have no conflicts of interest.

Israel Chávez Villalpando gratefully acknowledges funding from CONACyT Grant no. 291001 and UNAM-DGAPA-PAPIIT IN102417.

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