Conductivity due to classical phase fluctuations in a model for high-$T_c$ superconductors

We consider the real part of the conductivity, ${\sigma }_{1;\alpha \alpha }\left(\omega \right),$ arising from classical phase fluctuations in high-$T_c$ superconductors. We show that ${\int }_0^{\infty }{\sigma }_{1;\alpha \alpha }d\omega \ne 0$ below the superconducting transition temperature $T_c,$ provided there is some quenched disorder in the system. Furthermore, for a fixed amount of quenched disorder, this integral at low temperatures is proportional to the zero-temperature superfluid density, in agreement with experiment. We calculate ${\sigma }_{1;\alpha \alpha }\left(\omega \right)$ explicitly for a model of overdamped phase fluctuations.