Mean-field theory of strongly disordered superconductors

Qualitative features of the mean-field theory of superconductivity in a strongly disordered systems of fermions with short-range attraction are discussed. In this limit the effective theory is entirely bosonic, and I consider both the artificial infinite-range limit, and the more realistic case of "nearest neighbor" hopping of bosons between localized states. In the infinite range case the mean-field theory is exact, and the superconducting gap is uniform in space. There is a smooth BCS-BEC crossover with decrease in density, at weak enough disorder; at moderate densities, or larger disorder, the mean-field ground state is the BCS-like localized superconductor. In the latter case, the gap is highly non-uniform in space, but surprisingly stays everywhere finite below the mean-field transition temperature.