Classically Frustrated Magnets, Symmetries and ${\mathbb{Z}}_2$-Equivariant Topology

A novel result in ${\mathbb{Z}}_2$-equivariant homotopy theory is stated, proven, and applied to the topological classification of classically frustrated magnets in the presence of canonical time-reversal symmetry. This result generalizes a lemma that had been key to the homotopical derivation of the renowned Bott-Kitaev periodic table for topological insulators and superconductors. The methods used in the classification of topological insulators and superconductors are here generalized and their generalizations applied to systems that are not quantum mechanical. We distinguish between three symmetry classes $\mathrm{A}\mathrm{I}\mathrm{I}\mathrm{I}$, $\mathrm{B}\mathrm{D}\mathrm{I}$, and $\mathrm{C}\mathrm{I}\mathrm{I}$ depending on the existence and type of canonical time-reversal symmetry. For each of these classes, the relevant objects to classify are ${\mathbb{Z}}_2$-equivariant maps into a Stiefel manifold. The topological classification is illustrated through examples of classically frustrated spin models and is compared to that of Roychowdhury and Lawler (RL).