Classification of Lifshitz invariant in multiband superconductors: an application to Leggett modes in the linear response regime in Kagome lattice models

TL;DR

This paper classifies Lifshitz invariants in multiband superconductors and demonstrates their role in exciting Leggett modes via linear optical response, with applications to Kagome lattice models.

Contribution

It identifies the Lifshitz invariant in Ginzburg-Landau theory, classifies its symmetry conditions, and applies this to Kagome lattice models to explain Leggett mode excitations.

Findings

Lifshitz invariant can exist even with inversion symmetry.

Leggett modes appear as resonance peaks in optical conductivity.

Theoretical predictions suggest experimental detection via linear optical response.

Abstract

Multiband superconductors are sources of rich physics arising from multiple order parameters, which show unique collective dynamics including Leggett mode as relative phase oscillations. Previously, it has been pointed out that the Leggett mode can be optically excited in the linear response regime, as demonstrated in a one-dimensional model for multiband superconductors[T. Kamatani, et al., Phys. Rev. B 105, 094520 (2022)]. Here we identify the linear coupling term in the Ginzburg-Landau free energy to be the so-called Lifshitz invariant, which takes a form of $\boldsymbol{d}\cdot\left(\Psi^{*}_{i}\nabla\Psi_{j} - \Psi_{j}\nabla\Psi^{*}_{i}\right)$, where $\boldsymbol{d}$ is a constant vector and $\Psi_{i}$ and $\Psi_{j}$ $(i\neq j)$ represent superconducting order parameters. We have classified all pairs of irreducible representations of order parameters in the crystallographic point groups that allow for the existence of the Lifshitz invariant. We emphasize that the Lifshitz invariant can appear even in systems with inversion symmetry. The results are applied to a model of $s$-wave superconductors on a Kagome lattice with various bond orders, for which in some cases we confirm that the Leggett mode appears as a resonance peak in a linear optical conductivity spectrum based on microscopic calculations. We discuss a possible experimental observation of the Leggett mode by a linear optical response in multiband superconductors.