Periodic table for topological bands with non-Hermitian symmetries

Classifications of symmetry-protected topological (SPT) phases provide a framework to systematically understand the physical properties and potential applications of topological systems. While such classifications have been widely explored in the context of Hermitian systems, a complete understanding of the roles of more general non-Hermitian symmetries and their associated classification is still lacking. Here, we derive a periodic table for noninteracting SPTs with general non-Hermitian symmetries. Our analysis reveals additional non-Hermitian topological classes, while also naturally incorporating the entire classification of Hermitian systems as a special case of our scheme. Building on top of these results, we derive two independent generalizations of Kramers theorem to the non-Hermitian setting, which constrain the spectra of the system and lead to different topological invariants. To elucidate the physics behind the periodic table, we provide explicit constructions of non-Hermitian topological invariants, focusing on the symmetry classes in zero, one, and two dimensions with non-Hermitian topological classifications beyond those previously discussed (e.g., Z in zero dimensions, Z2 in one and two dimensions). These results thus provide a framework for the design and engineering of non-Hermitian symmetry-protected topological systems.