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.
Highlights
For over 100 years, the group-theoretic characterization of crystalline solids has provided the foundational language for diverse problems in physics and chemistry
Using the smallreps and MEBRs, we construct a complete position-space theory of mean-field band topology in the 1,651 single and double SSGs – Magnetic Topological Quantum Chemistry (MTQC) – that subsumes the earlier theory of TQC5,6 [see Fig. 2]
We find that many of the symmetry-indicated spinful magnetic topological phases consist of familiar Weyl SMs with surface Fermi arcs59–61, 3D quantum anomalous Hall (QAH) phases
Summary
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. The symmetry and group theory of crystalline solids have been used to characterize phase transitions[3], identify biological structures like the DNA double helix[4], and, most recently, to elucidate the position-space origin of topological bands through the theories of Topological Quantum Chemistry (TQC)[5,6] and equivalent works[7,8,9]. Using the small (co)reps and MEBRs, we construct a complete position-space theory of mean-field band topology in the 1,651 single (spinless) and double (spinful) SSGs – Magnetic Topological Quantum Chemistry (MTQC) – that subsumes the earlier theory of TQC5,6 [see Fig. 2]. We find that many of the symmetry-indicated spinful magnetic topological phases consist of familiar Weyl SMs with surface Fermi arcs59–61, 3D quantum anomalous Hall (QAH) phases
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