New topological invariants and real-space constructions of topological states protected by magnetic space group symmetries

Using real-space recipes, we construct all gapped topological states protected by magnetic space groups (MSGs) for free fermion systems with spin-orbit coupling, and enumerate all topological invariants for MSG symmetries and their corresponding surface states, many of which are discovered for the first time. Among these real-space topological states, a vast number of them are beyond the scope of layer constructions. We also derive explicit formulas of symmetry-based indicators (SIs) for all MSGs and make quantitative mappings between SIs and topological invariants. A byproduct of our work is that we also find all Weyl semimetal states that can be determined by SIs in MSGs. Their existence are characterized by either quantized Berry phase of a certain loop or the difference between Chern numbers of two high-symmetry planes in Brillouin zone. We hope our work will facilitate the discovery of more magnetic topological materials in the future.