Eur. Phys. J. B 72, 509-520 (2009)
https://doi.org/10.1140/epjb/e2009-00383-0

A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal

Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, 91405 Orsay, France

Corresponding author: ^a montambaux@lps.u-psud.fr

Received: 17 July 2009
Published online: 7 November 2009

Abstract

We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrödinger equation in a double well potential. The Landau levels obey the general scaling law ϵ_n ∝B^2/3 f_n(Δ/B^2/3), and they evolve continuously from a to a linear (n+1/2)B dependence, with a [(n+1/2)B]^2/3 dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameter is varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice.