Bulk-boundary correspondence in non-Hermitian systems: stability analysis for generalized boundary conditions

Rebekka Koch,Jan Carl Budich

doi:10.1140/epjd/e2020-100641-y

Abstract

The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk topological invariants defined for infinite periodic systems and the occurrence of protected zero-energy surface states in finite samples, is a ubiquitous and widely observed phenomenon in topological matter. In non-Hermitian generalizations of topological systems, however, this fundamental correspondence has recently been found to be qualitatively altered, largely owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary conditions. In this work, we report on two contributions towards comprehensively explaining this remarkable behavior unique to non-Hermitian systems with theory. First, we analytically solve paradigmatic non-Hermitian topological models for their zero-energy modes in the presence of generalized boundary conditions interpolating between open and periodic boundary conditions, thus explicitly following the breakdown of the conventional BBC. Second, addressing the aforementioned spectral fragility of non-Hermitian matrices, we investigate as to what extent the modified non-Hermitian BBC represents a robust and generically observable phenomenon.Graphical abstract

Highlights

In a broad variety of physical situations ranging from classical settings to open quantum systems, non-Hermitian Hamiltonians have proven to be a powerful and conceptually simple tool for effectively describing dissipation
We address several remaining issues regarding the bulk-boundary correspondence (BBC) in non-Hermitian systems, focusing on the biorthogonal basis approach reported in reference [56]
We focus on one-dimensional tight-binding models with N unit cells and generally asymmetric hopping amplitudes between the sites that render the Hamiltonian nonHermitian. Compared to their Hermitian counterparts, some non-Hermitian tight-binding models show qualitatively different eigenspectra depending on the imposed boundary conditions [51,52,56], i.e. periodic boundary conditions (PBC) or open boundary conditions (OBC)

Summary

Introduction

In a broad variety of physical situations ranging from classical settings to open quantum systems, non-Hermitian Hamiltonians have proven to be a powerful and conceptually simple tool for effectively describing dissipation. Considering one-dimensional (1D) non-Hermitian systems, we derive analytical expressions for the occurrence of exceptional point (EP) transitions and the formation of zeroenergy edge modes as a function of a generalized boundary condition parameter Γ continuously interpolating between periodic (Γ = 1) and open (Γ = 0) boundaries (see Fig. 1 for an illustration). In this context, the discrepancy between the topological phase diagrams of systems with different boundary conditions is intuitively explained by the occurrence of topological phase transitions, in which the boundary condition parameter Γ plays the role of a control parameter.

Non-Hermitian tight-binding models with generalized boundary conditions

Biorthogonal bulk boundary correspondence approach

Analytical solution with generalized BC

Stability of edge states

Discussion