Exact Solutions Disentangle Higher-Order Topology in 2D Non-Hermitian Lattices

TL;DR

This paper provides exact analytical solutions for higher-order topological states in 2D non-Hermitian lattices, revealing how edge and corner states behave under non-Hermitian skin effects and generalized boundary conditions.

Contribution

It introduces explicit solutions and winding number criteria that predict the non-Hermitian skin effect and topological state behavior in 2D non-Hermitian systems, advancing understanding of non-Bloch band topology.

Findings

Exact solutions confirm point-gap topology leads to skin effect.

Winding numbers predict skin effect for edge and bulk states.

Corner states depend on intersection of gapped edge states.

Abstract

We report the exact closed-form solutions for higher-order topological states as well as explicit energy-spectrum relationships in two-dimensional (2D) non-Hermitian multi-orbital lattices with generalized boundary conditions. These analytical solutions unequivocally confirm that topological edge states in a 2D non-Hermitian system which feature point-gap topology must undergo the non-Hermitian skin effect along the edge. Under double open boundary conditions, the occurrence of the non-Hermitian skin effect for either topological edge states or bulk states can be accurately predicted by our proposed winding numbers. We unveil that the zero-energy topological corner state only manifests itself on a corner where two nearby gapped edge states intersect, and thus can either disappear completely or strengthen drastically due to the non-Hermitian skin effect of gapped topological edge states. Our analytical results offer direct insight into the non-Bloch band topology in two or higher dimensions and trigger experimental investigations into related phenomena such as quadrupole topological insulators and topological lasing.