Exceptional points and non-Hermitian skin effects under nonlinearity of eigenvalues

TL;DR

This paper explores how nonlinear eigenvalue problems in metamaterials can lead to unique non-Hermitian topological phenomena like exceptional points and skin effects, extending topological band theory into nonlinear regimes.

Contribution

It introduces topological invariants for nonlinear systems and demonstrates the emergence of exceptional points even in single-component systems, expanding understanding of non-Hermitian topology.

Findings

Nonlinear systems can exhibit exceptional points and skin effects.

Topological invariants are extended to nonlinear eigenvalue problems.

Exceptional points can occur in single-component systems without internal degrees of freedom.

Abstract

Band structures of metamaterials described by a nonlinear eigenvalue problem are beyond the existing topological band theory. In this paper, we analyze non-Hermitian topology under the nonlinearity of eigenvalues. Specifically, we elucidate that such nonlinear systems may exhibit exceptional points and non-Hermitian skin effects which are unique non-Hermitian topological phenomena. The robustness of these non-Hermitian phenomena is clarified by introducing the topological invariants under nonlinearity which reproduce the existing ones in linear systems. Furthermore, our analysis elucidates that exceptional points may emerge even for systems without an internal degree of freedom where the equation is single component. These nonlinearity-induced exceptional points are observed in mechanical metamaterials, e.g., the Kapitza pendulum.