# Symmetry and Topology in Non-Hermitian Physics. (arXiv:1812.09133v3 [cond-mat.mes-hall] UPDATED)

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.