Electromagnetic response of disordered superconductors

We use the exact eigenstate formalism to construct a gauge-invariant theory for correlation functions for disordered superconductors. The density response and the longitudinal and transverse current response at zero temperature are calculated explicitly for Bardeen-Cooper-Schrieffer (BCS) superconductors with arbitrary amounts of disorder. At small frequencies and wave numbers this can be done analytically, and we discuss the coherence length, the penetration depth, and the Anderson-Bogolubov mode at various degrees of disorder. We then calculate the longitudinal dielectric function numerically for larger frequencies and wave numbers. We find that the disorder does not change the absence of low-lying collective modes in charged superconductors. Implications for the quasiparticle inelastic lifetime are discussed.