Abstract

We show that, in the dissociation regime and under a nondegeneracy 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. To this end, we first consider a general Schrödinger operator $H=-\mathrm{\Delta }+V_L$ acting on $L^2({ℝ}^2)$ with a potential $V_L$ which is assumed to be periodic with respect to some lattice with length scale $L$. Under some assumptions which cover 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.

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