Interplay of disorder and superconductivity at quantum critical points*

We study the superconducting instability of a two-dimensional disordered Fermi liquid weakly coupled to the soft fluctuations associated with proximity to an Ising-ferromagnetic quantum critical point. We derive interaction-induced corrections to the Usadel equation governing the superconducting gap function, and show that diffusion and localization effects drastically modify the interplay between fermionic incoherence and strong pairing interactions. In particular, we obtain the phase diagram, and demonstrate that (i) there is an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at stronger disorder; and (ii) diffusive particle-particle modes (so-called "Cooperons") acquire anomalous dynamical scaling z=4, indicating strong non-Fermi liquid behavior.