The classification and diagnosis of magnetic topological states

Topological band theory, essentially single-particle, can be extended to magnetic materials, as long as we assume that the electrons that contribute to topology are separate from those that contribute to local moments, which are static, and that the electrons interact with the local moments only via exchange coupling (effective Zeeman). When these assumptions hold, the topological classification of magnetic insulators reduces to that of non-interacting fermions protected by magnetic space groups. In this talk, I first review a general method called the "real-space recipes" for exhaustively constructing all gapped topological states protected by spatial symmetries. Then I apply this method to the case of magnetic space groups, obtaining a full classification of magnetic topological insulators in 3D. Finally, I briefly discuss a new type of symmetry, the spin-space groups, and their representations. The study of spin-space groups has been recently revived due to the experimental advances in altermagnetism.