An Index Theorem for Graphene

TL;DR

This paper derives an index theorem linking zero energy modes in graphene to lattice topology, applicable to various geometries including nanotubes and complex molecules, with implications for topological quantum computing.

Contribution

It introduces a novel index theorem for graphene that connects zero energy modes with lattice topology, extending previous results to complex molecular structures.

Findings

The index theorem matches analytical and numerical results for known graphene structures.

The theorem applies to arbitrary geometries, including folded sheets and nanotubes.

Potential applications in topological quantum computation are discussed.

Abstract

We consider a graphene sheet folded in an arbitrary geometry, compact or with nanotube-like open boundaries. In the continuous limit, the Hamiltonian takes the form of the Dirac operator, which provides a good description of the low energy spectrum of the lattice system. We derive an index theorem that relates the zero energy modes of the graphene sheet with the topology of the lattice. The result coincides with analytical and numerical studies for the known cases of fullerene molecules and carbon nanotubes and it extend to more complicated molecules. Potential applications to topological quantum computation are discussed.