Fig. 1. Summary of results.

In this work, we have derived the complete sets of trivial bands [elementary band (co)representations (EBRs), see SN 17] and symmetry-indicated, spinful, stable topological bands in the 1,651 Shubnikov space groups [SSGs]. The EBRs subdivide into the physical EBRs of the 230 Type-II nonmagnetic space groups [SGs] and the magnetic EBRs [MEBRs] of the 1,421 Type-I, III, and IV magnetic SGs [MSGs, see SN 2]^10–12. We have additionally performed the first complete calculation of the small (co)representations [(co)reps] and compatibility relations [see SN 11] for all 1,651 single and double SSGs, which we have made accessible through the tools listed in Table 1. These results comprise the theories of Magnetic Topological Quantum Chemistry (MTQC) and fermionic symmetry-based indicators (double SIs)^7,30–32,51, which apply to all possible 3D magnetic and nonmagnetic crystals with mean-field Hamiltonians. We have also determined the physical bases of all double (spinful) symmetry-based indicators (SIs), and symmetry-indicated topological bulk and anomalous boundary states for all 1,651 double SSGs (SN 26). Lastly, the MEBRs of the Type-III and Type-IV MSGs computed in this work also facilitates the complete enumeration of symmetry-enforced magnetic topological semimetals (SMs) – examples are provided in Fig. 4c and in SN 15. In this figure, we have used red checks to indicate areas of magnetic topological band theory completed in this work, and we have used red stars to indicate areas in which we have solved complete subareas (such as the double SIs of the 1,651 double SSGs), but in which there remain topological features outside of the scope of this work, such as non-symmetry-indicated stable topological bands^25,26,34,55 and bosonic (spinless) topological crystalline insulators (TCIs).