Classification of magnetic frustration from topology

Studies of frustrated magnets have mostly concentrated on analyzing their response to perturbations leaving open the fundamental question of why they are frustrated. In a recent work we demonstrated that topological mechanics can be used as an efficient tool to answer this question when frustration is encoded in a “rigidity matrix”, a non-Hermitian matrix found in all frustrated magnets whose ground states are determined by Moessner-Chalker-Maxwell counting. Here we classify all topological invariants associated with these rigidity matrices by generalizing methods used in the construction of the 10-fold way from Hermitian to non-Hermitian matrices resulting in a 3-fold way classification for each counting index ν= D - K where D are the number of degrees of freedom and K the number of constraints. We illustrate the classification by demonstrating the existence of a new vortex-like invariant for real rigidity matrices using random matrices and models of kagome Heisenberg antiferromagnets. Surprisingly in the latter we discover topological properties of kagome coplanar states. So by classifiying all rigidity matrices, we answer the question of the origin of frustration in the form of accidental degeneracy in a wide class of frustrated magnets by linking it to topological invariants.