Topological classification and diagnosis in magnetically ordered electronic materials

We show that lattice symmetry groups without time reversal, also known as magnetic space group symmetries, protect topological invariants as well as surface states that are distinct from those of nonmagnetic topological states. We obtain, by explicit and exhaustive construction of topological crystals using lower-dimensional building blocks, the full topological classification of electronic band insulators that are magnetically ordered for each one of the 1421 magnetic space groups (MSGs) in three dimensions with significant spin-orbit coupling. Compared with previous works on classification which include the results for some of the MSGs, our unified framework systematically gives the complete classification for all MSGs. We have also computed the symmetry-based indicators (SIs) for each nontrivial class and, by doing so, establish the complete mapping from symmetry representations to topological invariants. Our work goes beyond Elcoro et al. [L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Nat. Commun. 12, 5965 (2021)] by constructing all gapped magnetic topological crystalline states, among which many cannot be diagnosed by SIs. By doing so, we have also identified all SIs that do not correspond to any gapped states but gapless Weyl semimetals.