Symmetry and Topology in Non-Hermitian Physics

By K. Kawabata, Ken Shiozaki, Masahito Ueda, Masatoshi Sato. Physical Review X, 2018

1,199 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/03bd43d7d8c8a3773295e88e076dce0258ed0ef8

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Roadmap on topological photonics (2022, neutral): “In recent years, there has been a rapid development in the theoretical understanding of non-Hermitian topological band structures [164, 165].”
Perspective on topological states of non-Hermitian lattices (2019, background): “Indeed, symmetries such as particle hole symmetry fork into two since complex conjugation and transposition become distinct [52], while others get unified, as antiunitary symmetries which are distinct in the Hermitian case and now can be mapped onto each other [113].”
A review on non-Hermitian skin effect (2022, methodology): “Some typical signatures include the localizations for an extensive number of bulk states, the breakdown of the conventional BBC and the defectiveness of the edge modes.”
Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos (2019, methodology): “[25] K.”
Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect (2022, background): “This reciprocal skin effect is ensured by the Z2 topological invariant unique to non-Hermitian systems [104].”
Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class (2020, background): “[35], this symmetry is called TRS† since it is a Hermitian-conjugate counterpart of time-reversal symmetry (TRS), and non-Hermitian Hamiltonians with Eqs.”
Non-Hermitian systems and topology: A transfer-matrix perspective (2018, background): “[95] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato, “Symmetry and topology in nonHermitian physics,” arXiv:1812.”
Highlighting photonics: looking into the next decade (2021, background): “For examples, is there a general theory to characterize a complex system with underlying physics driven simultaneously by nonlinearity, topology and non-Hermiticity, and how would nonlinearity affect overall topological features and the global classifications of symmetry and topology [91, 168, 169]?”
Higher-order non-Hermitian skin effect (2020, background): “, H (k) is point-gapped in terms of a reference point E [45, 61]].”
Exceptional topology of non-Hermitian systems (2019, methodology): “In certain regimes, such open systems in contact with an environment can accurately be described by approaches such as Lindblad quantum master equations (Lindblad, 1976), FeynmanVernon theory (Feynman and Vernon, 1963), and the Keldysh formalism (Keldysh, 1964).”