Nuts and bolts of supersymmetry

A topological mechanism is a zero-elastic-energy deformation of a mechanical structure that is robust against smooth changes in system parameters. Here, we map the nonlinear elasticity of a paradigmatic class of topological mechanisms onto a supersymmetric field theory introduced by Witten and Olive. Heuristically, this approach entails taking the square root of a nonlinear Hamiltonian. It generalizes the standard procedure of obtaining two copies of the Dirac equation by taking the square root of the linear Klein-Gordon equation. Our real-space formalism goes beyond topological band theory by incorporating nonlinearities and spatial inhomogeneities, such as domain walls (i.e., kinks), where topological states are typically localized. We interpret the two components of the real fermionic field as site and bond displacements, respectively. The constraint of zero elastic energy insures that kinks in the mechanical system saturate the Bogomolny-Prasad-Sommerfield bound, while forbidding antikinks. This mechanism can be viewed as a manifestation of the underlying supersymmetry being half broken.