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Dirac cones for a mean-field model of graphene

Jean Cazalis

Vol. 6 (2024), No. 1, 129–185
DOI: 10.2140/paa.2024.6.129
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 = Δ + V L acting on L2(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 , the low-lying spectral bands of HL 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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Keywords
periodic Schrödinger operators, Hartree–Fock, graphene, nonlinear analysis, Dirac points
Mathematical Subject Classification
Primary: 35J10
Secondary: 35P30, 35Q40, 81Q10, 81V99
Milestones
Received: 21 July 2022
Revised: 18 August 2022
Accepted: 4 February 2023
Published: 22 February 2024
Authors
Jean Cazalis
CNRS and CEREMADE
University of Paris-Dauphine, PSL University
Paris
France