Winding numbers and generalized mobility edges in non-Hermitian systems

Qi-Bo Zeng,Yong Xu

doi:10.1103/physrevresearch.2.033052

Qi-Bo Zeng, Yong Xu

Open Access

https://doi.org/10.1103/physrevresearch.2.033052

Copy DOI

Abstract

The Aubry-Andr\'e-Harper (AAH) model with a self-dual symmetry plays an important role in studying the Anderson localization. Here we find a self-dual symmetry determining the quantum phase transition between extended and localized states in a non-Hermitian AAH model and show that the eigenenergies of these states are characterized by two types of winding numbers. By constructing and studying a non-Hermitian generalized AAH model, we further generalize the notion of the mobility edge, which separates the localized and extended states in the energy spectrum of disordered systems, to the non-Hermitian case and find that the generalized mobility edge is of a topological nature even in the open boundary geometry in the sense that the energies of localized and extended states exhibit distinct topological structures in the complex energy plane. Finally, we propose an experimental scheme to realize these models with electric circuits.

Highlights

Anderson localization [1] is a ubiquitous phenomenon in disordered physical systems
We study the self-dual symmetry, the winding numbers, and the mobility edges in non-Hermitian AAH models. (i) We show that there are two types of winding numbers: Wh arising from asymmetric hopping and Wo arising from the complex on-site potential. (ii) We find a self-dual
By generalizing the mobility edge in a Hermitian system to a non-Hermitian one, we find that the generalized mobility edges under both periodic boundary conditions (PBCs) and open boundary conditions (OBCs) are topological in the sense that the energy spectra of the localized and extended states exhibit nonzero and zero winding numbers

Summary

INTRODUCTION

Anderson localization [1] is a ubiquitous phenomenon in disordered physical systems. Due to the destructive interference of scattered waves, the states in the system can become localized [2,3]. We study the self-dual symmetry, the winding numbers, and the mobility edges in non-Hermitian AAH models. Symmetry in a non-Hermitian AAH model with asymmetric hopping determining the quantum phase transition between extended and localized states. The energies of both localized and extended states form loop structures in the complex energy plane that are characterized by the winding number Wo and Wh, respectively, under periodic boundary conditions (PBCs), and by Wo under open boundary conditions (OBCs). With weak asymmetric hopping breaking the PT symmetry, we find that the energy spectra of both localized and extended states obtained under PBCs form loop structures characterized by the winding number Wo and Wh, respectively.

SELF-DUAL SYMMETRY

TOPOLOGICAL MOBILITY EDGE

SUMMARY