  A low-energy theory of the Nambu-Goldstone excitation spectrum and the
corresponding speed of sound of an interacting Fermi mixture of Lithium-6 and
Potassium-40 atoms in a two-dimensional optical lattice at finite temperatures
with the Fulde-Ferrell order parameter has been formulated. It is assumed that
the two-species interacting Fermi gas is described by the one-band Hubbard
Hamiltonian with an attractive on-site interaction. The discussion is
restricted to the BCS side of the Feshbach resonance where the Fermi atoms
exhibit superfluidity. The quartic on-site interaction is decoupled via a
Hubbard-Stratonovich transformation by introducing a four-component boson field
which mediates the Hubbard interaction. 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. The numerical solution of the Bethe-Salpeter
equation in the generalized random phase approximation shows that there exist
two distinct sound velocities in the long-wavelength limit. In addition to the
long-wavelength mode (Goldstone mode), the two-species Fermi gas has a
superfluid phase revealed by two rotonlike minima in the asymmetric
collective-mode energy.
