A gauge-invariant formulation of optical responses in superconductors

Superconductors are often discussed in the mean-field approximation that breaks $U(1)$ symmetry. Since the $U(1)$ symmetry underlies the charge conservation, naive application of response theory sometimes gives results that are not gauge-invariant. We study the effect of vertex corrections on the electromagnetic responses of superconductors by employing the gauge-invariant formulation, called consistent fluctuations of order parameters, that manifestly satisfy the Ward identity. For inversion broken two-band superconductors, we find that the linear and second-order optical responses are significantly affected by vertex corrections and the conductivities near the gap energy almost completely disappear. We also investigate the linear optical responses in $d$-wave superconductors and obtained qualitatively different results compared to the result using the solution to the Bethe-Salpeter equation. The full photon vertex used in our study, manifestly satisfies the Ward identity. Our results demonstrate the fundamental importance of vertex corrections in exploring optical responses in superconductors.