AC Conductivity Crossover in Localized Superconductors

An important experimental signature of localization is the low-frequency AC conductivity, which typically vanishes as $\omega^{\phi}$. The exponent $\phi = 2$ for Anderson insulators, whereas for many body localized insulators $\phi$ is a continuously varying exponent $1 \le \phi \le 2$. In this work, we study the low-frequency AC conductivity of localized superconductors, in which disorder is strong enough to localize all quasiparticles, while remaining weak enough to leave superconductivity intact. We find that while the ac conductivity still follows the general form $\sigma(\omega) \sim \omega^{\phi}$, the exponent $\phi$ can be markedly different from the characteristic value for localized insulators. In particular, in certain symmetry classes at zero temperature, we obtain $\phi > 2$. We further identify an interesting temperature dependent crossover in the scaling form of the AC conductivity, which could be useful for the experimental characterization of localized superconductors.