Topological phases of non-Hermitian systems

TL;DR

This paper develops a comprehensive framework for understanding topological phases in non-Hermitian systems, revealing unique phases, classification schemes, and robustness properties that extend the topological insulator paradigm.

Contribution

It introduces a systematic classification of non-Hermitian topological phases across all symmetry classes and dimensions, unifying them with Hermitian topological theory.

Findings

Non-Hermitian systems exhibit unique topological phases characterized by an integer winding number.

Robustness of non-Hermitian topological phases against disorder demonstrated in the Hatano-Nelson model.

A periodic table of non-Hermitian topological phases is constructed using K-theory, predicting the absence of phases in 2D.

Abstract

Recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi-)edge modes. We then apply the K-theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a Z2 topological index for arbitrary quantum channels. Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.