From symmetry-based indicators to topological states in magnetic space groups

Symmetry-based indicators(SI) is a powerful tool for diagnosing topological materials. The recent development of SI theory in non-magnetic space groups and its application in the high-throughput calculations, which discovered a large amount of new topological materials, has drawn much attention in the community. However, the SI theory for magnetic space groups(MSGs) has only been partly tackled, due to their vast number (1651 MSGs in total) and complicated structure. Here we develop the complete theory of SI in all MSGs, which includes the explicit formulas of SIs, their physical meanings and quantitative mappings to topological invariants, and their correspondence to real-space constructions of magnetic topological states. With the help of the real-space recipe, we enumerate all the gapped topological crystalline states protected by MSG symmetries. Generic Weyl states that can be diagnosed using SIs are also identified as a byproduct. Our results can be readily applied to high-throughput searching for new magnetic topological materials.