Gauge-invariant electromagnetic responses in superconductors

TL;DR

This paper develops a comprehensive, gauge-invariant theoretical framework for calculating electromagnetic responses in both conventional and unconventional superconductors, covering linear and nonlinear effects.

Contribution

It generalizes the CFOP approach to include full photon vertices and provides a diagrammatic Feynman diagram representation for response kernels.

Findings

Provides a unified gauge-invariant response theory for all orders.

Applicable to both conventional and unconventional superconductors.

Enables accurate predictions of electromagnetic responses beyond linear order.

Abstract

Gauge invariance is essential for making physically meaningful predictions. In superconductors, mean-field Hamiltonians that explicitly break $U(1)$ symmetry often yield gauge-dependent results. While this issue has been resolved for linear responses in conventional superconductors, a unified framework that also covers unconventional superconductors and nonlinear responses has yet to be established. In this study, we present a comprehensive theoretical framework that enables gauge-invariant calculations of electromagnetic responses at arbitrary orders in external fields, applicable to both conventional and unconventional superconductors. Our construction generalizes the consistent-fluctuation-of-the-order-parameter (CFOP) approach to full photon vertices and admits a diagrammatic representation of the response kernel in terms of Feynman diagrams.