The Crystallographic Spin Point Groups and their Representations

Abstract

The spin point groups are finite groups whose elements act on both real space and spin space. Among these groups are the magnetic point groups in the case where the real and spin space operations are locked to one another. The magnetic point groups are central to magnetic crystallography for strong spin-orbit coupled systems and the spin point groups generalize these to the intermediate and weak spin-orbit coupled cases. The spin point groups were introduced in the 1960's in the context of condensed matter physics and enumerated shortly thereafter. In this paper, we complete the theory ofcrystallographic spin point groups by presenting an account of these groups and their representation theory. Our main findings are that the so-called nontrivial spin point groups (numbering $598$ groups) have co-irreps corresponding exactly to the (co-)-irreps of regular or black and white groups and we tabulate this correspondence for each nontrivial group. However a total spin group, comprising the product of a nontrivial group and a spin-only group, has new co-irreps in cases where there is continuous rotational freedom. We provide explicit co-irrep tables for all these instances. We also discuss new forms of spin-only group extending the Litvin-Opechowski classes. To exhibit the usefulness of these groups to physically relevant problems we discuss a number of examples from electronic band structures of altermagnets to magnons.

99 pages, 73 figures (mostly tables). v2: Citations added, and on grey groups. v3: Funding source acknowledgmed for JR and HS. v5: Specified groups are crystallographic, added details about rutile & MnTe altermagnetism in section 4, updated figures 2 and 3. Slightly restructured to differentiate new results from known group/rep theory (now in appendix)

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• Altermagnetic Instabilities from Quantum Geometry
• Constructions and Applications of Irreducible Representations of Spin-Space Groups
• Persistent spin textures, altermagnetism and charge-to-spin conversion in metallic chiral crystals TM${}_3$X${}_6$
• Anisotropic Piezomagnetism in Noncollinear Antiferromagnets
• Zeeman-type spin splittings in strained d-wave altermagnets
• Quantum spin models of commensurate $p$-wave magnets
• Finite-momentum mixed singlet-triplet pairing in chiral antiferromagnets induced by even-parity spin texture
• High-throughput search for topological magnon materials
• Tailoring Corner States and Exceptional Points in Altermagnets
• Nonrelativistic Functional Properties in Collinear Antiferromagnets Based on Multipole Representation Theory
• Optical phonons as a testing ground for spin group symmetries
• Nonlinear opto-magnetic signature of d-wave altermagnets
• Competitive Orders in Altermagnetic Chiral Magnons