Pairing in a population imbalanced Fermi system in a two-dimensional optical lattice is studied using Determinant Quantum Monte Carlo (DQMC) simulations and mean-field calculations. The approximation-free numerical results show a wide range of stability of the Fulde-Ferrell-Larkin-Ovshinnikov (FFLO) phase. Contrary to claims of fragility with increased dimensionality we find that this phase is stable across wide range of values for the polarization, temperature and interaction strength. Both homogeneous and harmonically trapped systems display pairing with finite center of mass momentum, with clear signatures either in momentum space or real space, which could be observed in cold atomic gases loaded in an optical lattice. We also use the harmonic level basis in the confined system and find that pairs can form between particles occupying different levels which can be seen as the analog of the finite center of mass momentum pairing in the translationally invariant case. Finally, we perform mean field calculations for the uniform and confined systems and show the results to be in good agreement with QMC. This leads to a simple picture of the different pairing mechanisms, depending on the filling and confining potential.
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