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We study s-wave superconductivity in the two dimensional attractive Hubbardmodel in an applied magnetic field, assume the extreme Pauli limit, and examinethe role of spatial fluctuations in the coupling regime corresponding toBCS-BEC crossover. We use a decomposition of the interaction in terms of anauxiliary pairing field, retain the static mode, and sample the pairing fieldvia Monte Carlo. The method requires iterative solution of theBogoliubov-de-Gennes (BdG) equations for amplitude and phase fluctuatingconfigurations of the pairing field. We establish the full thermal phasediagram of this strong coupling problem, revealing $T_c$ scales an order ofmagnitude below the mean field estimate, highlight the spontaneousinhomogeneity in the field induced magnetization, and discover a strong nonmonotonicity in the temperature dependence of the low energy density of states.We compare our results to the experimental phase diagram of the imbalancedFermi gas at unitarity. This paper focuses on the magnetized but homogeneous(breached pair) superconducting state, a companion paper deals with theFulde-Ferrell-Larkin-Ovchinnikov (FFLO) regime.

Population imbalanced lattice fermions near the BCS-BEC crossover: I.The breached pair and metastable FFLO phases