A gauge-invariant formulation of optical responses in superconductors

TL;DR

This paper develops a gauge-invariant approach to analyze optical responses in superconductors, emphasizing the importance of vertex corrections for accurate and consistent electromagnetic response calculations.

Contribution

It introduces a gauge-invariant formulation that satisfies the Ward identity, providing new insights into optical responses affected by vertex corrections in superconductors.

Findings

Vertex corrections significantly alter optical responses near the gap energy.

Inversion broken two-band superconductors show almost complete disappearance of certain conductivities.

Qualitative differences are observed in $d$-wave superconductors compared to Bethe-Salpeter solutions.

Abstract

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.