Topology shared between classical metamaterials and interacting superconductors

TL;DR

This paper reveals a topological connection between classical mechanical metamaterials and interacting quantum systems like superconductors, using supersymmetry and topological indices to bridge classical and quantum physics.

Contribution

It establishes a novel topological link between isostatic mechanical metamaterials and supersymmetric quantum systems, enabling classical systems to model quantum topological properties.

Findings

Defined a topological index $Q_{net}$ for mechanical systems.

Linked $Q_{net}$ to the Witten index in supersymmetric quantum systems.

Showed conditions under which classical and quantum topologies coincide.

Abstract

Supersymmetry has been studied at a linear level between normal modes of metamaterials described by rigidity matrices and non-interacting quantum Hamiltonians. The connection between classical and quantum was made through the matrices involved in each problem. Recently, insight into the behavior of nonlinear mechanical systems was found by defining topological indices via the Poincar\'e-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 electrons coupled to phonons in metals and superconductors. Firstly, we define $Q_{net}$ for an isostatic mechanical system that counts the minimum number of zero-energy configurations. Secondly, we write a supersymmetric Hamiltonian that describes a metal or a superconductor interacting with anharmonic phonons. This Hamiltonian has a Witten index, a topological invariant that captures the balance of bosonic and fermionic zero-energy states. We are able to connect these two systems by showing that $Q_{net}=W$ under very general conditions. 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.