Projective representation theory of spin space groups
Abstract
Spin space groups provide the natural symmetry framework for magnetic crystals with weak spin-orbit coupling, but extracting their physical consequences requires a general theory of irreducible representations. The central difficulty is that spin space groups are represented projectively: their factor systems can render lattice translations noncommuting, induce nonsymmorphic actions in momentum space, and modify the projective structure of little cogroups. These effects lie beyond conventional space-group and double-group representation theory. Here, using Mackey's theory of group extensions, we develop a unified constructive framework for all collinear, coplanar, and noncoplanar spin space groups, including antiunitary symmetries and general factor systems. A central technical result is a canonical decomposition of the relevant factor system ν into a translational factor σ, a mixed factor γ coupling translations to point-group operations, and a point-group factor α. This decomposition makes transparent how ordinary representation theory is modified: σ determines the projective translation algebra and the appropriate Brillouin zone, γ controls the momentum-space group action and can make it nonsymmorphic, and α contributes to the factor systems of little cogroups. On this basis, we construct all projective irreducible corepresentations by induction over momentum-space orbits. The framework identifies which nonsymmorphic momentum-space symmetries can be realized by spin space groups and reveals Brillouin spaces that are compact flat manifolds rather than tori, symmetry-enforced Zak phases, reconstructed high-symmetry momenta and band degeneracies, and new types of quasiparticles. Our results establish the representation-theoretic foundation for systematic studies of weak-spin-orbit-coupled magnetic materials.
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BibTeX
@misc{zhang2026projective,
title = {{Projective representation theory of spin space groups}},
author = {Zheng Zhang and Peiyuan Wang and Y. X. Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21289},
eprint = {2609.21289},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21289},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Projective representation theory of spin space groups
AU - Zheng Zhang
AU - Peiyuan Wang
AU - Y. X. Zhao
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21289
UR - https://arxiv.org/abs/2609.21289
AB - Spin space groups provide the natural symmetry framework for magnetic crystals with weak spin-orbit coupling, but extracting their physical consequences requires a general theory of irreducible representations. The central difficulty is that spin space groups are represented projectively: their factor systems can render lattice translations noncommuting, induce nonsymmorphic actions in momentum space, and modify the projective structure of little cogroups. These effects lie beyond conventional space-group and double-group representation theory. Here, using Mackey's theory of group extensions, we develop a unified constructive framework for all collinear, coplanar, and noncoplanar spin space groups, including antiunitary symmetries and general factor systems. A central technical result is a canonical decomposition of the relevant factor system ν into a translational factor σ, a mixed factor γ coupling translations to point-group operations, and a point-group factor α. This decomposition makes transparent how ordinary representation theory is modified: σ determines the projective translation algebra and the appropriate Brillouin zone, γ controls the momentum-space group action and can make it nonsymmorphic, and α contributes to the factor systems of little cogroups. On this basis, we construct all projective irreducible corepresentations by induction over momentum-space orbits. The framework identifies which nonsymmorphic momentum-space symmetries can be realized by spin space groups and reveals Brillouin spaces that are compact flat manifolds rather than tori, symmetry-enforced Zak phases, reconstructed high-symmetry momenta and band degeneracies, and new types of quasiparticles. Our results establish the representation-theoretic foundation for systematic studies of weak-spin-orbit-coupled magnetic materials.
ER -