Mean-field theory of strongly disordered superconductors

Qualitative features of the mean-field theory of superconductivity in a strongly disordered system of fermions with short-range attraction are discussed. In this limit the effective theory at low energies is entirely bosonic, and I consider both the artificial infiniterange limit and the more realistic case of ”nearest-neighbor” hopping of bosons between the 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 is finite everywhere at T < TMF . I find that the meanfield transition temperature TMF > 0 always, and argue that the superconductor-insulator transition at T = 0 in models with net attraction between fermions is in the universality class of ”dirty-bosons”.