Optical response of Higgs mode in superconductors at clean limit

TL;DR

This paper clarifies the optical response of the Higgs mode in clean-limit superconductors, resolving previous theoretical controversies by correcting mathematical flaws and extending the analysis to include electromagnetic effects, confirming finite Higgs-mode generation and charge conservation.

Contribution

The study corrects previous theoretical derivations of the second-order optical response, demonstrating finite Higgs-mode generation and charge conservation in superconductors at the clean limit.

Findings

Finite Higgs-mode generation via vector potential drive effect.

Charge-density fluctuation vanishes at clean limit.

Phase mode excitation cancels unphysical charge fluctuations.

Abstract

The phenomenological Ginzburg-Landau theory and the charge conservation directly lead to the finite Higgs-mode generation and vanishing charge-density fluctuation in the second-order optical response of superconductors at clean limit. Nevertheless, recent microscopic theoretical studies of the second-order optical response, apart from the one through the gauge-invariant kinetic equation [Yang and Wu, Phys. Rev. B {\bf 100}, 104513 (2019)], have derived a vanishing Higgs-mode generation but finite charge-density fluctuation at clean limit. We resolve this controversy by re-examining the previous derivations with the vector potential alone within the path-integral and Eilenberger-equation approaches, and show that both previous derivations contain mathematical flaws. After fixing these flaws, a finite Higgs-mode generation through the drive effect of vector potential is derived at clean limit, exactly recovering the previous result from the gauge-invariant kinetic equation as well as Ginzburg-Landau theory. By further extending the path-integral approach to include electromagnetic effects from the scalar potential and phase mode, in the second-order response, a finite contribution from the drive effect of scalar potential to the Higgs-mode generation at clean limit as well as the vanishing charge-density fluctuation are derived, also recovering the results from the gauge-invariant kinetic equation. Particularly, we show that the phase mode is excited in the second-order response, and exactly cancels the previously reported unphysical excitation of the charge-density fluctuation, guaranteeing the charge conservation.