We determine the relative stability of different ground-state phases of spin-imbalanced populations of attractive fermions in square lattices. The ground state is determined within Hartree-Fock-Bogoliubov theory, with care taken to remove finite-size effects. The phases are systematically characterized by the symmetry of the order parameter and their real- and momentum-space structures. For quarter- to half-filled lattices, where the Fermi surfaces are most distorted from their spherical counterpart in the continuum, we predominantly find unidirectional Larkin-Ovchinikov-type phases. We discuss the effect of commensuration between the ordering wave vector and the density imbalance and describe the mechanism of Fermi surface reconstruction and pairing for various orders. A robust supersolid phase exists when the ordering wave vector is diagonally directed. Charge and pairing order coexist, rather than compete, and are responsible for the opening of the gap on different portions of the Fermi surface. A variational determination of the correct pair momentum of the Larkin-Ovchinikov phases shows that phase separation does not occur in the considered regime of density and magnetization.
We study a two-species fermion mixture with different populations on a square lattice modeled by a Hubbard Hamiltonian with onsite interspecies repulsive interaction. Such a model can be realized in a cold-atom system with fermionic atoms in two different hyperfine states loaded on an optical lattice and with tunable interspecies interaction strength via external fields. For a two-dimensional square lattice, when at least one of the fermion species is close to half-filling, the system is highly affected by lattice effects. With the majority species near half-filling and varying densities for the minority species, we find that several correlated phases emerge as the ground state, including a spin density wave state, a charge density wave state with stripe structure, and various
p-wave BCS pairing states for both species. We study this system using a functional renormalization group method, determine its phase diagram at weak coupling, discuss the origin and characteristics of each phase, and provide estimates for the critical temperatures.