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

Details A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal

2009-07-02

arxiv.org/abs/0907.0500v1

Eur. Phys. J. B 72, 509 (2009)

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

18 pages, 15 figures

Scientific paper

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 Schrodinger equation in a double well potential. They obey the general scaling law e_n \propto B^{2/3} f_n(Delta /B^{2/3}. They evolve continuously from a sqrt{n B} 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.

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