Fig. 2. Magnetic Topological Quantum Chemistry in the scheme of topological band theory.

The complete scheme of topological band theory for 3D crystals, following the framework and notation established in refs. ^5,6,31,66,67. Through crystal symmetry eigenvalues [small (co)reps] in momentum space (SN 13), the compatibility relations (SN 16) indicate whether a set of bands is allowed by symmetry to be energetically isolated from other bands in the energy spectrum. If the bands are energetically isolated, then there exists a wide range of methods for diagnosing whether the bands exhibit the stable topology of topological insulators (TIs) and TCIs^{17–20,23–34,54}, fragile topology^{52,54,55,64–67}, or the polarization-nontrivial topology of obstructed atomic limits^5,54,84. For example, as detailed in refs. ^{7,30–32,51,66,67}, the small (co)reps of a set of isolated bands comprise momentum-space symmetry data that can be mapped to position-space topology and boundary states through stable and fragile SIs and real-space invariants. If the bands are instead required by symmetry to cross, then the bands characterize a topological SM, which may exhibit surface³⁸ or hinge^34,54 states. In this figure, the pink boxes indicate areas of topological band theory completed in this work.