Hybrid Monte Carlo simulation on the graphene hexagonal lattice

TL;DR

This paper develops a hybrid Monte Carlo simulation method directly on the graphene hexagonal lattice to study electron dynamics with Coulomb interactions, avoiding the sign problem and enabling precise numerical solutions.

Contribution

It introduces a novel lattice simulation approach for graphene directly on the hexagonal structure, improving accuracy over previous square lattice methods.

Findings

No sign problem in the simulation

Exact tight binding solutions can be approached numerically

Method tested on a single hexagonal cell

Abstract

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.