Symmetry and Topology in Non-Hermitian Physics

We develop a complete theory of symmetry and topology in non-Hermitian\nphysics. We demonstrate that non-Hermiticity ramifies the celebrated\nAltland-Zirnbauer symmetry classification for insulators and superconductors.\nIn particular, charge conjugation is defined in terms of transposition rather\nthan complex conjugation due to the lack of Hermiticity, and hence chiral\nsymmetry becomes distinct from sublattice symmetry. It is also shown that\nnon-Hermiticity enables a Hermitian-conjugate counterpart of the\nAltland-Zirnbauer symmetry. Taking into account sublattice symmetry or\npseudo-Hermiticity as an additional symmetry, the total number of symmetry\nclasses is 38 instead of 10, which describe intrinsic non-Hermitian topological\nphases as well as non-Hermitian random matrices. Furthermore, due to the\ncomplex nature of energy spectra, non-Hermitian systems feature two different\ntypes of complex-energy gaps, point-like and line-like vacant regions. On the\nbasis of these concepts and K-theory, we complete classification of\nnon-Hermitian topological phases in arbitrary dimensions and symmetry classes.\nRemarkably, non-Hermitian topology depends on the type of complex-energy gaps\nand multiple topological structures appear for each symmetry class and each\nspatial dimension, which are also illustrated in detail with concrete examples.\nMoreover, the bulk-boundary correspondence in non-Hermitian systems is\nelucidated within our framework, and symmetries preventing the non-Hermitian\nskin effect are identified. Our classification not only categorizes recently\nobserved lasing and transport topological phenomena, but also predicts a new\ntype of symmetry-protected topological lasers with lasing helical edge states\nand dissipative topological superconductors with nonorthogonal Majorana edge\nstates. Furthermore, our theory provides topological classification of\nHermitian and non-Hermitian free bosons.\n