Topological Aspects of Fermions on a Honeycomb Lattice

TL;DR

This paper explores the topological properties of fermions on a honeycomb lattice modeled after graphene, demonstrating the correspondence between zero modes and topological charge through numerical solutions and index theorem application.

Contribution

It introduces a numerical approach to analyze topological zero modes of Dirac operators on a honeycomb lattice, linking lattice results with continuum topological invariants.

Findings

Number of zero modes matches the Atiyah-Singer index theorem predictions.

Maximum measurable topological charge scales with lattice perimeter.

A gauge non-invariant chirality operator effectively distinguishes topological zero modes.

Abstract

We formulate a model of relativistic fermions moving in two Euclidean dimensions based on a tight-binding model of graphene. The eigenvalue spectrum of the resulting Dirac operator is solved numerically in smooth U(1) gauge field backgrounds carrying an integer-valued topological charge Q, and it is demonstrated that the resulting number of zero-eigenvalue modes is in accord with the Atiyah-Singer index theorem applied to two continuum flavors. A bilinear but gauge non-invariant chirality operator appropriate for distinguishing the topological zero modes is identified. When this operator is used to calculate Q, it is found that the maximum topological charge capable of being measured in this fashion scales with the perimeter of the lattice. Some concluding remarks compare these results to what is known for staggered lattice fermions.