Topological band theory of a generalized eigenvalue problem with Hermitian matrices: Symmetry-protected exceptional rings with emergent symmetry

TL;DR

This paper develops a topological band theory for generalized eigenvalue problems with Hermitian matrices, revealing symmetry-protected exceptional rings and explaining hyperbolic metamaterials' dispersion.

Contribution

It introduces a novel topological framework for GEVPs with Hermitian matrices, uncovering emergent symmetry-protected exceptional rings.

Findings

Symmetry-protected exceptional rings (SPERs) emerge in Hermitian GEVPs.

SPERs are protected by an emergent symmetry unique to GEVPs.

The theory explains the dispersion characteristics of hyperbolic metamaterials.

Abstract

So far, topological band theory is discussed mainly for systems described by eigenvalue problems. Here, we develop a topological band theory described by a generalized eigenvalue problem (GEVP). Our analysis elucidates that non-Hermitian topological band structures may emerge for systems described by a GEVP with Hermitian matrices. The above result is verified by analyzing a two-dimensional toy model where symmetry-protected exceptional rings (SPERs) emerge although the matrices involved are Hermitian. Remarkably, these SPERs are protected by emergent symmetry, which is unique to the systems described by the GEVP. Furthermore, these SPERs elucidate the origin of the characteristic dispersion of hyperbolic metamaterials which is observed in experiments.