Topological Aspects of Fermions on a Honeycomb Lattice

Understand

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.