Symmetry and Topology in Non-Hermitian Physics

Non-Hermiticity enriches topological phases beyond the existing framework for Hermitian topological phases. Here, we develop a general theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies and unifies the celebrated Altland-Zirnbauer symmetry for insulators and superconductors, leading to 38-fold symmetry instead of the 10-fold one. Moreover, we reveal that two types of energy gaps are relevant for non-Hermitian systems because of the complex-valued nature of energy spectra, both of which constitute non-Hermitian topology. Based on these fundamental insights in non-Hermitian physics, we completely classify topological phases of non-Hermitian insulators and superconductors, as well as semimetals that support exceptional points. Our work paves the way toward unique phenomena and functionalities due to the interplay of non-Hermiticity and topology, such as symmetry-protected topological lasers and dissipative topological quantum computation.

K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019).