Universal non-Hermitian skin effect in two and higher dimensions

TL;DR

This paper proves the universal presence of the non-Hermitian skin effect in higher dimensions, characterizing its conditions, types, and experimental relevance across various physical systems.

Contribution

It establishes a theorem linking the skin effect to the spectral coverage on the complex plane and introduces new types of skin effects in higher dimensions.

Findings

Skin effect exists iff the spectrum covers a finite area on the complex plane.

Two new skin effect types: corner-skin and geometry-dependent-skin.

Skin effect is linked to the presence of exceptional points/lines in higher dimensions.

Abstract

Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a non-Hermitian hamiltonian are localized at the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodic-boundary spectrum of the hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic non-Hermitian hamiltonian, and, unlike in one dimension, is compatible with all spatial symmetries. We propose two new types of skin effect in two and higher dimensions: the corner-skin effect where all eigenstates are localized at one corner of the system, and the geometry-dependent-skin effect where skin modes disappear for systems of a particular shape, but appear on generic polygons. An immediate corollary of our theorem is that any non-Hermitian system having exceptional points (lines) in two (three) dimensions exhibits skin effect, making this phenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.