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On the Symmetries of Anisotropic Spin Interaction Models

  • Arist Zhenyuan Yang
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Abstract

We show that anisotropic spin interactions do not merely break spin-space group (SSG) symmetries, but instead twist them through cohomology invariants, yielding symmetry classes beyond subgroups of $ O(3)\times \operatorname{Isom}(\mathbb{R}^3) $. This requires redefining the spin-only group $ S_0 $ in terms of proper spin rotations. Based on this unitary $ S_0 $, we formulate a twisted SSG (tSSG) theory that captures the complete set of spin-space symmetries. We then study a spin-1 model with tSSG symmetry using linear flavor wave theory and find ℤ2 topological quadrupolar excitations defined on a spin Brillouin Klein bottle. Specifically, the quadrupolar excitations possess a momentum-space glide-reflection symmetry and the edge states exhibit a nonlocal momentum twist. These results establish the symmetry language required for interacting spin systems, whether realized in magnetic materials or on programmable quantum simulators, and open a route to unconventional magnetism.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026symmetries,
  title = {{On the Symmetries of Anisotropic Spin Interaction Models}},
  author = {Arist Zhenyuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2605.14969},
  eprint = {2605.14969},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.14969},
}
RIS
TY  - UNPB
TI  - On the Symmetries of Anisotropic Spin Interaction Models
AU  - Arist Zhenyuan Yang
PY  - 2026
DA  - 2026/07/22
PB  - arXiv
DO  - 10.48550/arxiv.2605.14969
UR  - https://arxiv.org/abs/2605.14969
AB  - We show that anisotropic spin interactions do not merely break spin-space group (SSG) symmetries, but instead twist them through cohomology invariants, yielding symmetry classes beyond subgroups of $ O(3)\times \operatorname{Isom}(\mathbb{R}^3) $. This requires redefining the spin-only group $ S_0 $ in terms of proper spin rotations. Based on this unitary $ S_0 $, we formulate a twisted SSG (tSSG) theory that captures the complete set of spin-space symmetries. We then study a spin-1 model with tSSG symmetry using linear flavor wave theory and find ℤ_2 topological quadrupolar excitations defined on a spin Brillouin Klein bottle. Specifically, the quadrupolar excitations possess a momentum-space glide-reflection symmetry and the edge states exhibit a nonlocal momentum twist. These results establish the symmetry language required for interacting spin systems, whether realized in magnetic materials or on programmable quantum simulators, and open a route to unconventional magnetism.
ER  -