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Abstract For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry. However, the group theory of crystals with commensurate magnetic order has remained incomplete for the past 70 years, due to the complicated symmetries of magnetic crystals. In this work, we complete the 100-year-old problem of crystalline group theory by deriving the small corepresentations, momentum stars, compatibility relations, and magnetic elementary band corepresentations of the 1,421 magnetic space groups (MSGs), which we have made freely accessible through tools on the Bilbao Crystallographic Server. We extend Topological Quantum Chemistry to the MSGs to form a complete, real-space theory of band topology in magnetic and nonmagnetic crystalline solids – Magnetic Topological Quantum Chemistry (MTQC). Using MTQC, we derive the complete set of symmetry-based indicators of electronic band topology, for which we identify symmetry-respecting bulk and anomalous surface and hinge states. 

The existence of topological hinge states is a key signature for a newly proposed class of topological matter, the second-order topological insulators. In the present paper, a universal mechanism to generate chiral hinge states in the ferromagnetic axion insulator phase is introduced, which leads to an exotic transport phenomenon, the quantum anomalous Hall effect (QAHE) on some particular surfaces determined by both the crystalline symmetry and the magnetization direction. A realistic material system, Sm-doped Bi2Se3, is then proposed to realize such exotic hinge states by combining first-principles calculations and Green’s function techniques. A physically accessible way to manipulate the surface QAHE is also proposed, which makes it very different from the QAHE in ordinary 2D systems.