  We study s-wave superconductivity in the two dimensional attractive Hubbard
model in an applied magnetic field, assume the extreme Pauli limit, and examine
the role of spatial fluctuations in the coupling regime corresponding to
BCS-BEC crossover. We use a decomposition of the interaction in terms of an
auxiliary pairing field, retain the static mode, and sample the pairing field
via Monte Carlo. The method requires iterative solution of the
Bogoliubov-de-Gennes (BdG) equations for amplitude and phase fluctuating
configurations of the pairing field. We establish the full thermal phase
diagram of this strong coupling problem, revealing $T_c$ scales an order of
magnitude below the mean field estimate, highlight the spontaneous
inhomogeneity in the field induced magnetization, and discover a strong non
monotonicity in the temperature dependence of the low energy density of states.
We compare our results to the experimental phase diagram of the imbalanced
Fermi gas at unitarity. This paper focuses on the magnetized but homogeneous
(breached pair) superconducting state, a companion paper deals with the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) regime.

Quantum Gases;Strongly Correlated Electrons;Superconductivity