Hybrid Monte Carlo Simulations of Graphene

Abstract

We give a discussion on the Hybrid Monte Carlo simulations of the tight binding model with Coulomb interactions, for the electronic properties of Graphene. The aim of our simulations is to determine the phase transition in which Graphene goes from a semimetal to an insulator through spontaneous breaking of sublattice symmetry. This is done non-perturbatively using lattice field theory techniques, working in the path-integral formalism with discrete euclidean time. We explain the method by [Brower, Rebbi and Schaich, arXiv:1101.5131v1, 26 Jan 2011] through which the theory of Graphene can be written in a form independent of a sign problem. An implementation of this method has been made in Julia, and can be found in [https://github.com/Rik-Breebaart/GrapheneHMC.git]. Our measurements show no clear phase-transition in the order parameters for this sublattice symmetry breaking. However, the system sizes on which the simulation have been performed where to small to make conclusive prediction on the presence and critical value of the phase-transition.