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We study fermion pairing in a population-imbalanced mixture of $^{6}$Liatomic gas loaded in a three-dimensional lattice at very low temperatures.Using the number equation for each population, the gap equation and theequation for the Helmholtz free energy, we determine the gap, chemicalpotentials and pair-momentum as functions of polarization. These parametersdefine the stability regions for: a Fulde-Ferrell-Larkin-Ovchinnikov phase; aphase separation region where BCS and normal phases coexist; a Sarma phase whenthe pair-momentum vanishes, and the transition to the normal phase when the gapdisappears. The collective-mode energies are then calculated using aBethe-Salpeter approach in the generalized random phase approximation assumingthat the system is well described by the single-band Hubbard model. A novelresult is that this fermionic gas has a superfluid phase revealed by rotonlikeminima in the asymmetric collective-mode energy spectrum.

Superfluidity of a spin-imbalanced Fermi gas in a three-dimensional optical lattice