Probing non-Hermitian skin effect and non-Bloch phase transitions

Stefano Longhi

doi:10.1103/physrevresearch.1.023013

Stefano Longhi

Open Access

https://doi.org/10.1103/physrevresearch.1.023013

Copy DOI

Abstract

In non-Hermitian crystals showing the non-Hermitian skin effect, ordinary Bloch band theory and Bloch topological invariants fail to correctly predict energy spectra, topological boundary states, and symmetry breaking phase transitions in systems with open boundaries. Recently, it has been shown that a correct description requires to extend Bloch band theory into complex plane. A still open question is whether non-Hermitian skin effect and non-Bloch symmetry-breaking phase transitions can be probed by real-space wave dynamics far from edges, which is entirely governed by ordinary Bloch bands. Here it is shown that the Lyapunov exponent in the long-time behavior of bulk wave dynamics can reveal rather generally non-Bloch symmetry breaking phase transitions and the existence of the non-Hermitian skin effect.

Highlights

Bloch band theory describes energy spectra and single electronic bulk states in crystals with either periodic boundary conditions (PBCs) or open boundary conditions (OBCs)
Among the most relevant features observed in non-Hermitian systems, one should mention the strong sensitivity of the energy spectra on boundary conditions [7,15,16,17,18,19,20,21], the nonHermitian skin effect (NHSE) [9,17,18,20,21,22,23,24], i.e., the exponential localization of continuum-spectrum eigenstates to the edges, and the failure of the bulk-boundary correspondence based on Bloch band topological invariants [4,17,18,24,25,26,27,28,29,30,31,32,33,34,35,36,37]
In this work we have shown that, even though the bulk dynamics in non-Hermitian systems is entirely described by Bloch band theory, the Lyapunov exponent in the long-time dynamics is determined by the turning points of non-Bloch bands, which can reveal both non-Bloch symmetry-breaking phase transitions and the existence of the non-Hermitian skin effect

Summary

INTRODUCTION

Bloch band theory describes energy spectra and single electronic bulk states in crystals with either periodic boundary conditions (PBCs) or open boundary conditions (OBCs). Bloch bulk invariants can be introduced to classify topological bands and to predict the appearance of topological edge states in crystals with OBCs (bulk-boundary correspondence) [1,2,3] Such major results are challenged when trying to apply Bloch band theory to non-Hermitian systems. Bloch band theory in non-Hermitian systems is demonstrated by restoration of the (non-Bloch) bulk-boundary correspondence [18,25,26,27,28,29] and in the study of non-Hermitian wave scattering and domain walls [35,38] Another major consequence of the NSHE is that distinct bulk symmetry-breaking phase transitions are observed when considering Bloch and non-Bloch bands, i.e., systems with PBCs and OBCs. For example, for certain non-Hermitian extensions of the SuSchrieffer-Heeger (SSH) model [39], the bulk eigenenergies in the case of OBCs are entirely real over a wide range of system parameters as a consequence of pseudo-Hermiticity, while they are complex for PBCs [4,18,20,28].

TWO-BAND NON-HERMITIAN MODELS

Some specific models

NON-HERMITIAN SKIN EFFECT AND SADDLE-POINT CRITERION

REAL-SPACE WAVE-PACKET DYNAMICS AND LYAPUNOV EXPONENT

PROBING NON-BLOCH SYMMETRY-BREAKING PHASE TRANSITIONS

CONCLUSION AND OUTLOOK