Superfluid density and phase relaxation in superconductors with strong disorder

As a prototype of a disordered superconductor we consider the attractive Hubbard model with on-site disorder. We solve the Bogoljubov-de-Gennes equations on two-dimensional finite clusters at zero temperature and evaluate the electromagnetic response to a vector potential. We find that the standard decoupling between transverse and longitudinal response does not apply in the presence of disorder. Moreover the superfluid density is strongly reduced by the relaxation of the phase of the order parameter already at mean-field level when disorder is large. We also find that the anharmonicity of the phase fluctuations is strongly enhanced by disorder. Beyond mean-field, this provides an enhancement of quantum fluctuations inducing a zero-temperature transition to a non-superconducting phase of disordered preformed pairs. Finally, the connection of our findings with the glassy physics for extreme dirty superconductors is discussed.