Abstract
In topological mechanics, the identification of a mechanical system's rigidity matrix with an electronic tight-binding model allows to infer topological properties of the mechanical system, such as the occurrence of `floppy' boundary modes, from the associated electronic band structure. Here we introduce an approach to systematically construct topological mechanical systems by an exact supersymmetry (SUSY) that relates the bosonic (mechanical) and fermionic (e.g. electronic) degrees of freedom. As examples we discuss mechanical analogues of the Kitaev honeycomb model and of a second-order topological insulator with floppy corner modes. Our SUSY construction naturally defines hitherto unexplored topological invariants for bosonic (mechanical) systems, such as bosonic Wilson loop operators that are formulated in terms of a SUSY-related fermionic Berry curvature.
Highlights
In topological mechanics, the identification of a mechanical system’s rigidity matrix with an electronic tightbinding model allows one to infer topological properties of the mechanical system, such as the occurrence of “floppy” boundary modes, from the associated electronic band structure
Our SUSY construction naturally defines a set of topological invariants for bosonic systems, such as bosonic Wilson loop operators that are formulated in terms of a SUSY-related fermionic Berry curvature
Supersymmetry (SUSY) explicitly relates bosonic and fermionic degrees of freedom—a fundamental concept that has first been introduced [1,2,3] in highenergy physics and widely been adopted in the formulation of extensions of the standard model [4]
Summary
We report our discovery that we can use SUSY to map fermionic systems to bosonic ones while preserving locality in both and give a recipe for how to fabricate these bosonic systems with metamaterials As examples of this construction, we discuss mechanical incarnations of the Z2 spin-liquid phase in the Kitaev honeycomb model and of a second-order topological insulator. This SUSY construction allows us to identify topological properties of the bosonic (mechanical) system by explicitly associating it with a fermionic Berry curvature. That connects the annihilation operator of a (complex) boson b with the creation operator of a (complex) fermion c via
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