Abstract
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, point-like and line-like vacant regions. On the basis of these concepts and K-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.
Highlights
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 illustrated in detail with concrete examples
This work provides the following fundamental insights into symmetry and topology in non-Hermitian physics: (i) Symmetry ramification.—We discover that nonHermiticity ramifies symmetry due to the distinction between complex conjugation and transposition, which are equivalent for Hermitian Hamiltonians, as described in Sec
Whereas symmetries are unified in nonHermitian physics [126], they can ramify due to the distinction between complex conjugation and transposition for non-Hermitian Hamiltonians
Summary
While Hermiticity is a common assumption that underlies physics of isolated systems, non-Hermitian Hamiltonians [1] have recently attracted growing attention. Certain topological phases survive even in the presence of non-Hermiticity [100,102], including non-Hermitian extensions of the Su-Schrieffer-Heeger model (i.e., onedimensional system with chiral or sublattice symmetry [158]) [29,102,105,108,114,116,118,119,121,148,150,154], the Chern insulator (i.e., two-dimensional system without any symmetry [159,160,161]) [10,11,117,119,121,124,151,155], and the quantum spin Hall insulator (i.e., two-dimensional system with time-reversal symmetry [162]) [126]. In view of the rapid theoretical and experimental advances in non-Hermitian physics, there has been a great interest and an urgent need for comprehensive topological classification that provides a reference point for experiments and predicts novel non-Hermitian topological phases
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