Microscopic free energy functional of superconductive amplitude and phase: Superfluid density in disordered superconductors

Recent experiments on disordered superconductors find that the superfluid density $n_s(T)$ decreases dramatically and characteristically with disorder, differently from what is expected for a mean order parameter field or BCS, amplitude only, picture for the superconductor. We describe here a new microscopic free energy functional which explicitly describes its dependence on both the amplitude and phase of superconducting order, in a gauge invariant manner. We use this here in an approximation of noninteracting phase fluctuations (Gaussian or harmonic approximation) to obtain $n_s(T)$ in the presence of (static) disorder. We compare our results successfully with experiment.