Remarks on the mean-field theory of strongly disordered superconductors

Qualitative features of the mean-field BCS-like theory of superconductivity in a strongly disordered system of fermions with short-range attraction are discussed. Specifically, I consider two versions of the effective bosonic low-energy theory for the electron pairs: with the infinite range, and with the nearest neighbor hoping between localized states. In the first case the superconducting gap is uniform in space, and there is a smooth BCS-BEC crossover with decrease in density, at weak disorder; at moderate densities, or larger disorder, the mean-field ground state is the ”localized superconductor” of Ma and Lee. In the latter case, the gap becomes highly non-uniform in space, but surprisingly stays everywhere finite at T < TMF . I find that the mean-field 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”.