Periodic table for topological bands with non-Hermitian symmetries

Classifications of symmetry-protected topological phases provide a framework to understand systematically the physical properties and potential applications of topological systems. Here, the authors derive a comprehensive 38-fold topological classification of non-Hermitian systems with generic non-Hermitian symmetry classes. Two independent generalizations of Kramers' degeneracy to the non-Hermitian setting are presented. The nature of the non-Hermitian topological invariants obtained in this classification is explained through worked-out examples, thus providing a framework for the experimental design and engineering of non-Hermitian symmetry-protected topological systems.