Non-Hermitian Topology and Exceptional-Point Geometries

TL;DR

This paper reviews non-Hermitian topological phenomena, focusing on exceptional points and recent advances in non-Hermitian band topology, highlighting their unique mathematical structures and physical implications.

Contribution

It provides a comprehensive overview of non-Hermitian topology, emphasizing the role of exceptional points and recent developments in the field.

Findings

Exceptional points exhibit non-trivial topological properties.

Non-Hermitian systems display unique phenomena like the skin effect.

New classifications of non-Hermitian topological phases have been proposed.

Abstract

Non-Hermitian theory is a theoretical framework that excels at describing open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom (DOFs) of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this paper, we comprehensively review non-Hermitian topology by establishing its relationship with the behaviors of complex eigenvalues and biorthogonal eigenvectors. Special attentions are given to exceptional points - branch-point singularities on the complex eigenvalue manifolds that exhibit non-trivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications