Classifying Order-Two Spatial Symmetries in Non-Hermitian Hamiltonians: Point-gapped AZ and AZ$^†$ Classes

Abstract: Crystalline topological insulators and superconductors have been a prominent topic in the field of condensed matter physics. These systems obey certain crystalline (spatial) symmetries that depend on the geometry of the lattice. The presence of spatial symmetries can lead to shift in the classification of ten fold Altland Zirnbauer class, given rise to new symmetry protected topological phases. If the constraint of Hermiticity is broken, the classification expand into 38 fold. In this paper, following procedures in Hermitian systems, we classify all possible types of order-two spatial symmetries for point gapped non Hermitian systems within 16 out of 38 non Hermitian topological classes. These 16 classes are denoted by AZ and AZ^$\dag$ classes. We show that, similar to the Hermitian case, spatial symmetries will also lead to a shift in the classification of AZ and AZ^$\dag$ classes. There also exist novel symmetry protected topological phases exclusive to point gapped non Hermitian Hamiltonians. Toy models are also given based on our classifications.