Topology in non-Hermitian Chern insulators with skin effect

TL;DR

This paper establishes a bulk-boundary correspondence in non-Hermitian Chern insulators with skin effect, using non-Bloch and biorthogonal approaches to characterize topology and edge states.

Contribution

It introduces a combined framework to analyze non-Hermitian topological phases, including a biorthogonal inverse participation ratio and a non-Bloch Chern number.

Findings

Biorthogonal inverse participation ratio distinguishes bulk and edge states.

Non-Bloch Chern number predicts topological edge bands.

Presence of crossing degeneracies enriches edge state behavior.

Abstract

The non-Hermitian skin effect can arise in materials that have asymmetric hoppings between atoms or resonating units, which makes the bulk eigenspectrum sensitive to boundary conditions. When skin effect emerges, eigenstates in the bulk continuum can become localized on the edges, making the distinction between edge and bulk states challenging. We establish the bulk-boundary correspondence for a Chern insulator model with non-Hermitian skin effect by combining two approaches ("non-Bloch" approach and "biorthogonal" approach). Both approaches can suppress the skin effect but they are based on different mathematical tools. A biorthogonal inverse participation ratio is used as a measure to distinguish between bulk states and edge states, and a non-Bloch Chern number is used to characterize the topology and predict the number of topological edge bands. In addition to tangential degeneracies, crossing degeneracies are found to occur between the bulk and edge bands. Their presence enriches the (de)localization behavior of the edge states but does not affect the Chern number. The phase diagram of the system has interesting features that are not found in Hermitian systems. For example, one topological transition and two non-Hermitian phase transitions can be induced by tuning a single parameter. The gapless phase is topologically protected due to the stable existence of the non-Hermitian band degeneracies guaranteed by nonzero discriminant numbers.