A Replica-BCS theory for dirty superconductors

In this paper, we revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply the RS-BCS theory to dirty superconductor grains of size $L^d$, where $L$ is the localization length and $d$ is the dimension of the system, assuming that the macroscopic system is composed of weakly coupled grains. Besides disordered potential, We also consider the case where regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Within the RS-BCS mean field theory, the system phase diagram, single-particle tunneling density of states and the superfluid density are computed within the RS-BCS theory for different strengths of disorder, We show that our result agrees qualitatively with previous numerical studies in the case with pure attractive interaction, and that a Cooper-pair-glass state may exist when regions with opposite sign of interaction exists and the net interaction between electrons is close to zero. The plausible relevance of our result to the anomalous metal state is discussed.