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Supersymmetry "protected" topological phases of isostatic lattices and kagome antiferromagnets

I generalize the theory of phonon topological band structures of isostatic lattices to frustrated antiferromagnets. I achieve this with a discovery of a many-body supersymmetry (SUSY) in the phonon problem of balls and springs and its connection to local constraints satisfied by ground states. The Witten index of the SUSY model demands the Maxwell-Calladine index of mechanical structures. "Spontaneous supersymmetry breaking" is identified as the need to gap all modes in the bulk to create the topological isostatic lattice state. Since ground states of magnetic systems also satisfy local constraint conditions (such as the vanishing of the total spin on a triangle) I identify a similar SUSY structure for many common models of antiferromagnets including the square, triangluar, kagome, pyrochlore nearest neighbor antiferromagnets, and the $J_2 = J_1/2$ square lattice antiferromagnet. Remarkably, the kagome family of antiferromagnets is the analog of topological isostatic lattices among this collections of models. Thus, a solid state realization of the theory of phonon topological band structure may be found in frustrated magnetic materials.