  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.
