Abstract
Non-Hermiticity enriches topological phases beyond the existing Hermitian framework. Whereas their unusual features with no Hermitian counterparts were extensively explored, a full understanding about the role of symmetry in non-Hermitian physics has still been elusive, and there remains an urgent need to establish their topological classification in view of rapid theoretical and experimental progress. Here, we develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 instead of 10, which describe intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Furthermore, due to the complex nature of energy spectra, non-Hermitian systems feature two different types of complex-energy gaps, pointlike and linelike vacant regions. On the basis of these concepts and -theory, we complete classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. Remarkably, non-Hermitian topology depends on the type of complex-energy gaps, and multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Moreover, the bulk-boundary correspondence in non-Hermitian systems is elucidated within our framework, and symmetries preventing the non-Hermitian skin effect are identified. Our classification not only categorizes recently observed lasing and transport topological phenomena, but also predicts a new type of symmetry-protected topological lasers with lasing helical edge states and dissipative topological superconductors with nonorthogonal Majorana edge states. Furthermore, our theory provides topological classification of Hermitian and non-Hermitian free bosons. Our work establishes a theoretical framework for the fundamental and comprehensive understanding of non-Hermitian topological phases and paves the way toward uncovering unique phenomena and functionalities that emerge from the interplay of non-Hermiticity and topology.
- Received 22 December 2018
- Revised 15 May 2019
DOI:https://doi.org/10.1103/PhysRevX.9.041015
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Topology describes geometrically invariant properties under bending and stretching and plays a pivotal role in physics for characterizing phases of matter. In thermal equilibrium, such topological phases are fundamentally understood on the basis of the tenfold internal symmetry of time reversal and charge conjugation. In recent years, researchers have shown that topological phases are enriched by non-Hermiticity, which effectively describes the external environment and is realized, for instance, in nonequilibrium open systems with gain and loss of energy and particles. However, a thorough understanding of symmetry in non-Hermitian physics has remained elusive, and a general theoretical framework for non-Hermitian topology has yet to be established. Such a framework, which we develop here, provides a benchmark for experiments and may lead to novel phenomena and functionalities.
We demonstrate that non-Hermiticity ramifies and unifies symmetry, leading to 38-fold symmetry beyond the tenfold symmetry in conventional physics. We also show that two distinct types of complex-energy gaps can be introduced because of the complex nature of the spectrum (e.g., of light, sound, electrons, cold atoms). Finally, we complete the topological classification of non-Hermitian systems for all the symmetry classes and the complex-energy gaps. Our findings establish a theoretical framework for a fundamental and comprehensive understanding of non-Hermitian topology, which justifies the lasing and transport topological phenomena observed in recent experiments.
We expect that this general theory may lead to new applications in non-Hermitian physics. For example, it predicts symmetry-protected topological lasers that have unique lasing and transport properties. It also describes dissipative topological superconductors, which may be useful for quantum computation.
