Hybrid Monte Carlo simulation on the graphene hexagonal lattice

One of the many remarkable properties of graphene is that in the low energy limit the dynamics of its electrons can be effectively described by the massless Dirac equation. This has prompted investigations of graphene based on the lattice simulation of a system of 2-dimensional fermions on a square staggered lattice. We demonstrate here how to construct the path integral for graphene working directly on the graphene hexagonal lattice. For the nearest neighbor tight binding model interacting with a long range Coulomb interaction between the electrons, this leads to the hybrid Monte Carlo algorithm with no sign problem. The only approximation is the discretization of the Euclidean time. So as we extrapolate to the time continuum limit, the exact tight binding solution maybe found numerically to arbitrary precession on a finite hexagonal lattice. The potential for this approach is tested on a single hexagonal cell.