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Low-energy theory of the Nambu-Goldstone modes of an ultracold $^6Li-$ $^{40}K$ mixture in an optical lattice

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.