On the Symmetries of Anisotropic Spin Interaction Models

TL;DR

This paper introduces a twisted spin-space group theory for anisotropic spin interactions, revealing new symmetry classes and topological excitations with nontrivial boundary states.

Contribution

It formulates a twisted SSG theory capturing all spin-space symmetries and uncovers topological quadrupolar excitations on a Klein-bottle with M"obius boundary states.

Findings

Topological quadrupolar excitations on a Klein-bottle

Ribbon spectrum with M"obius boundary states

Classification of excitations by Z2 symmetry

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 topological quadrupolar excitations defined on a spin Brillouin Klein-bottle rather than the conventional torus. Specifically, the bosonic BdG Hamiltonian satisfies a glide reflection sewing relation, the ribbon spectrum exhibits M\"obius boundary states. These topological excitations are classified by $ \mathbb{Z}_2 $, enforced by the nonorientability of the Klein-bottle.