Mechanical Topological Insulators

Jacob Moran with Harsh Mathur

In this project we will study mechanical topological insulators. Yu and Mathur [1] have found that the elastic modes of bilayer graphene are analogous to the electronic states of topological insulators. Using a continuum elasticity model they find that the elastic waves are gapped in the bulk but have gapless edge modes leading them to conjecture that bilayer graphene is a mechanical topological insulator. In this project we will attempt to place this conjecture on a firmer footing by applying a formalism developed by Kane and Lubensky [2] to characterize mechanical  topological insulators. The formalism of Kane and Lubensky  will be applied to the continuum elasticity model used by ref [1] as well as to a lattice model that provides access to the full Brillouin zone. In the second part of the project we will attempt to develop an alternative formalism to characterize topological insulators based on the the non-Hermitian dynamical matrix which is the mechanical analog of the Hamiltonian for electronic systems. This alternative formalism will require a generalization of the concept of Berry’s phase to non-Hermitian matrices.