Collective modes of a spin-orbit-coupled superfluid Fermi gas in a two-dimensional optical lattice: a comparison between the Gaussian approximation and the Bethe-Salpeter approach

TL;DR

This paper compares the Gaussian approximation and Bethe-Salpeter approach in analyzing collective modes of a spin-orbit-coupled superfluid Fermi gas in a 2D optical lattice, highlighting differences in accuracy and features like the roton-like dispersion.

Contribution

It provides a systematic derivation of collective modes using both approaches and compares their predictions, especially regarding the speed of sound and dispersion features.

Findings

Gaussian approximation overestimates the Goldstone mode speed

Gaussian fails to reproduce roton-like dispersion features

Minimum speed of sound indicates topological phase transition boundary

Abstract

A functional integral technique and a Legendre transform are used to give a systematic derivation of the Schwinger-Dyson equations for the generalized single-particle Green's function and the Bethe-Salpeter equation for the two-particle Green's function and the associated collective modes of a population-imbalanced spin-orbit-coupled atomic Fermi gas loaded in a two-dimensional optical lattice at zero temperature. The collective-mode excitation energy is calculated within the Gaussian approximation, and from the Bethe-Salpeter equation in the generalized random phase approximation assuming the existence of a Sarma superfluid state. It is found that the Gaussian approximation overestimates the speed of sound of the Goldstone mode. More interestingly, the Gaussian approximation fails to reproduced the rotonlike structure of the collective-mode dispersion which appears after the linear part of the dispersion in the Bethe-Salpeter approach. We use the Gaussian approximation and the Bethe-Salpeter approach to investigate the speed of sound of a balanced spin-orbit-coupled atomic Fermi gas near the boundary of the topological phase transition driven by an out-of-plane Zeeman field. It is shown that within both approaches, the minimum of the speed of sound is located at the topological phase transition boundary, and this fact can be used to confirm the existence of a topological phase transition.