Duality of Topological Edge States in a Mechanical Kitaev Chain

TL;DR

This paper explores topological edge states in a mechanical analog of the Kitaev chain, revealing a duality that links trivial and nontrivial topologies and demonstrating conditions for Majorana-like modes.

Contribution

It introduces a mechanical system exhibiting dual topological edge states and uncovers a duality linking trivial and nontrivial topologies in finite chains.

Findings

Topological edge states exist in both trivial and nontrivial bulk systems.

A duality maps free boundary trivial systems to fixed boundary nontrivial systems.

Conditions for degenerate in-gap modes similar to Majorana zero modes are identified.

Abstract

We theoretically investigate and experimentally demonstrate the existence of topological edge states in a mechanical analog of the Kitaev chain with a non-zero chemical potential. Our system is a one-dimensional monomer system involving two coupled degrees of freedom, i.e., transverse displacement and rotation of elastic elements. Due to the particle-hole symmetry, a topologically nontrivial bulk leads to the emergence of edge states in a finite chain with fixed boundaries. In contrast, a topologically trivial bulk also leads to the emergence of edge states in a finite chain, but with free boundaries. We unravel a duality in our system that predicts the existence of the latter edge states. This duality involves the iso-spectrality of a subspace for finite chains, and as a consequence, a free chain with topologically trivial bulk maps to a fixed chain with a nontrivial bulk. Lastly, we provide the conditions under which the system can exhibit perfectly degenerate in-gap modes, akin to Majorana zero modes. These findings suggest that mechanical systems with fine-tuned degrees of freedom can be fertile testbeds for exploring the intricacies of Majorana physics.