Topology shared between classical metamaterials and interacting superconductors

Supersymmetry has been studied at a linear level between normal modes of metamaterials described by rigidity matrices and non-interacting quantum Hamiltonians. Recently, insight into the behavior of nonlinear mechanical systems was found by defining topological indices via the Poincaré-Hopf index. It turns out, because of the mathematical similarity, this topological index shows a way to approach supersymmetric quantum theory from classical mechanics. Using this mathematical similarity, we establish a topological connection between isostatic mechanical metamaterials and supersymmetric quantum systems, such as certain electron-phonon problems in metals and superconductors. Specifically, we show that a topological invariant derived from the Poincaré-Hopf index of a classical metamaterial is exactly the same as the Witten index of the supersymmetric electron-phonon problem. Our result shows that (1) classical metamaterials can be used to study the topology of interacting quantum systems with aid of supersymmetry, and (2) with fine-tuning between anharmonicity of phonons and couplings among Majorana fermions and phonons, it is possible to realize such a supersymmetric quantum system that shares the same topology as classical mechanical systems.