Type II Lifshitz invariant and optically active Higgs mode in time-reversal symmetry broken superconductors

TL;DR

This paper introduces a new type of Lifshitz invariant in superconductors that break time-reversal symmetry, enabling the Higgs mode to be observed in optical conductivity, supported by theoretical classification and numerical calculations.

Contribution

The paper identifies and classifies a novel 'type II' Lifshitz invariant specific to time-reversal symmetry broken superconductors, revealing an optically active Higgs mode.

Findings

Type II Lifshitz invariant appears only in time-reversal symmetry broken superconductors.

The Higgs mode becomes visible in optical conductivity due to the type II invariant.

Numerical models confirm the group-theoretical classification results.

Abstract

Lifshitz invariant is a symmetry-allowed term in the Ginzburg-Landau free energy of an ordered phase, involving the order parameters and a single spatial derivative, which serves as a source of unusual optical responses. Here we introduce a ``type II" Lifshitz invariant for superconductors, which changes its sign under the particle-hole transformation and can be distinguished from the ordinary particle-hole even ``type I" Lifshitz invariant. We show that the type II Lifshitz invariant appears only in superconductors that break time-reversal symmetry and allows the Higgs mode to be visible in the optical conductivity spectrum. We provide a classification of all pairs of irreducible corepresentations of order parameters in the magnetic point groups that admit a type II Lifshitz invariant. We also numerically calculate the optical conductivity for various models of time-reversal symmetry broken multiband superconductors, finding agreement with the group-theoretical analysis. Our results establish a universal class of time-reversal symmetry broken superconductors hosting an optically active Higgs mode.