Non-Bloch band theory and bulk–edge correspondence in non-Hermitian systems

Kazuki Yokomizo,Shuichi Murakami

doi:10.1093/ptep/ptaa140

Kazuki Yokomizo, Shuichi Murakami

Open Access

https://doi.org/10.1093/ptep/ptaa140

Copy DOI

Abstract

Abstract In this paper, we review our non-Bloch band theory in 1D non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su–Schrieffer–Heeger model, the bulk–edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge states.

Highlights

In recent years, interest in studies of non-Hermitian systems has been rapidly growing both in theories and in experiments
In terms of the non-Bloch band theory, we investigate the non-Hermitian SSH model, which has been studied in some previous works [16, 19,20,21, 28]
We explain how to construct the generalized Brillouin zone (GBZ), which is given by the trajectories of βM and βM+1 satisfying the condition |βM | = |βM+1| for continuum energy bands, and show that the Bloch wave number becomes complex in an infinite open chain in general

Summary

Introduction

Interest in studies of non-Hermitian systems has been rapidly growing both in theories and in experiments. We study the constructions of the GBZ and of the continuum energy bands in a simple non-Hermitian tight-binding model. In the limit of L → ∞, the continuum energy bands and the GBZ are independent of boundary conditions in an open chain. This Hamiltonian can be non-Hermitian, meaning that ti,μν is not necessarily equal to t∗−i,νμ In this situation, the real-space eigen-equation is written as H|ψ = E|ψ , where the eigenvector is given by |ψ = (· · · , ψ1,1, · · · , ψ1,q, ψ2,1, · · · , ψ2,q, · · · )T. The non-Bloch band theory explained here says that the eigenenergies for the continuum energy bands are determined by βM and βM+1, and that the GBZ Cβ and a set of the eigenenergies are independent of boundary conditions in an open chain.

Non-Hermitian SSH model

Bulk-edge correspondence

Topological semimetal phase with exceptional points

Summary