Evolution of Collective (Higgs and phase) Modes in Disordered s-wave Superconductors.*

The destruction of superconductivity by strong disorder and the resultant superconductor-insulator phase transition (SIT) is an active area of research in condensed matter physics. We use functional integral techniques to study the attractive Hubbard model on a square lattice, in the presence of random on-site potential. Expanding the theory in quantum fluctuations around the spatially inhomogeneous saddle point given by BdG mean field theory, we study the evolution of the amplitude (Higgs) and phase fluctuations with disorder. We find a phase transition where the Higgs gap vanishes and low energy phase fluctuations develop large spectral weight. Near the transition, the system breaks up into patches of superconducting islands that host localized low energy Higgs modes, separated from regions with low energy incoherent phase fluctuations. We discuss the implications of our results for optical conductivity and Josephson spectroscopy disordered superconductors.