Electromagnetic response of superconductors in the presence of multiple collective modes

Abstract

We revisit the importance of collective-mode fluctuations and gauge invariance in the electromagnetic response of superconducting systems. In particular, we show that order-parameter fluctuations, gapless or not, have no contribution to the Meissner effect in both $s$- and $p$-wave superconductors. More generally, we extend this result to uniform and nonuniform superfluids with no external wavevector scale. To facilitate this analysis, we formulate a path-integral-based matrix methodology for computing the electromagnetic response of fermionic fluids in the presence of concomitantly fluctuating collective modes. Closed-form expressions for the electromagnetic response in different scenarios are provided, including the case of fluctuations of electronic density and the phase and amplitude of the order parameter. All microscopic symmetries and invariances are manifestly satisfied in our formalism, and it can be straightforwardly extended to other scenarios.