Dirac cones for a mean-field model of graphene

TL;DR

This paper demonstrates that the reduced Hartree-Fock model of graphene exhibits Dirac points at the Brillouin zone vertices, with the Fermi level precisely at the cone intersections, linking spectral properties to the tight-binding approximation.

Contribution

It establishes the presence of Dirac points in the mean-field model of graphene under certain conditions, connecting spectral analysis with the tight-binding approximation.

Findings

Dirac points are present at Brillouin zone vertices.

Fermi level coincides with the Dirac cones.

Spectral bands are approximated by the tight-binding model in the large lattice limit.

Abstract

In this article, we show that, in the dissociation regime and under a non-degeneracy assumption, the reduced Hartree-Fock theory of graphene presents Dirac points at the vertices of the first Brillouin zone and that the Fermi level is exactly at the coincidence point of the cones. For this purpose, we first consider a general Schr\"odinger operator $H=-\Delta+V_L$ acting on $L^2(\mathbb{R}^2)$ with a potential $V_L$ which is assumed to be periodic with respect to some lattice with length scale $L$. Under some assumptions which covers periodic reduced Hartree-Fock theory, we show that, in the limit $L\to\infty$, the low-lying spectral bands of $H_L$ are given to leading order by the tight-binding model. For the hexagonal lattice of graphene, the latter presents singularities at the vertices of the Brillouin zone. In addition, the shape of the Bloch bands is so that the Fermi level is exactly on the cones.