Electromagnetic response of superconductors in the presence of multiple collective modes

Collective-mode fluctuations play an essential role in ensuring the electromagnetic response of superfluids is gauge invariant. The contribution of these fluctuations, however, is known to drop out from the Meissner response of uniform superfluids. The same phenomenon is not so established in the context of nonuniform superfluids. To clarify this issue, we revisit how collective modes appear in the Meissner effect. We find that their contribution vanishes both in uniform and nonuniform systems, unless an external length scale is present---as in Fulde-Ferrell or finite-sized superfluids. As examples, we consider $s$-wave and chiral $p+ip$ superconductors. To facilitate this analysis, we formulate a path-integral matrix methodology for computing the response of fermionic fluids in the presence of multiple collective modes. Closed-form expressions are provided, incorporating effects from phase and amplitude of the superconducting order parameter and electronic density fluctuations. All microscopic symmetries and invariances are manifestly satisfied in this approach, and it can be straightforwardly extended to other scenarios.