Classification of nodal topological band theories with combined twofold antiunitary spatial symmetries

A band theory that describes an electronic system can be invariant under a combined symmetry composed of two distinct operations. For example, a system can carry a magnetic symmetry, which is a combination of time-reversal and a spatial symmetry, such as an antiferromagnetic translation. We consider mirror, twofold rotation, and parity symmetry combined with one of the three non-spatial symmetries (time-reversal, particle-hole and chiral symmetries) as composite symmetries. In the presence of a composite symmetry, topological phases can emerge, while the two symmetries constituting the composite symmetry are not necessarily preserved individually. In this work, we systemically classify topological (semi)metals and nodal superconductors with different composite symmetries in any dimensions. We also discuss model realization, topological-index characterization as well as potential material applications.