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A functional integral technique and a Legendre transform are used to give asystematic derivation of the Schwinger-Dyson equations for the generalizedsingle-particle Green's function and the Bethe-Salpeter equation for thetwo-particle Green's function and the associated collective modes of apopulation-imbalanced spin-orbit-coupled atomic Fermi gas loaded in atwo-dimensional optical lattice at zero temperature. The collective-modeexcitation energy is calculated within the Gaussian approximation, and from theBethe-Salpeter equation in the generalized random phase approximation assumingthe existence of a Sarma superfluid state. It is found that the Gaussianapproximation overestimates the speed of sound of the Goldstone mode. Moreinterestingly, the Gaussian approximation fails to reproduced the rotonlikestructure of the collective-mode dispersion which appears after the linear partof the dispersion in the Bethe-Salpeter approach.  We use the Gaussian approximation and the Bethe-Salpeter approach toinvestigate the speed of sound of a balanced spin-orbit-coupled atomic Fermigas near the boundary of the topological phase transition driven by anout-of-plane Zeeman field. It is shown that within both approaches, the minimumof 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 phasetransition.

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