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

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 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.

18 pages, 15 figures

References in corpus (14)

• The electronic properties of graphene
• A tight-binding approach to uniaxial strain in graphene
• Tilted anisotropic Dirac cones in quinoid-type graphene and alpha-(BEDT-TTF)_2I_3
• Merging of Dirac points in a two-dimensional crystal
• Simulation and detection of Dirac fermions with cold atoms in an optical lattice
• A new magnetic field dependence of Landau levels on a graphene like structure
• Zero modes of tight binding electrons on the honeycomb lattice
• Tight-binding modeling and low-energy behavior of the semi-Dirac point
• Dirac-point engineering and topological phase transitions in honeycomb optical lattices
• BCS-BEC crossover on the two-dimensional honeycomb lattice
• Ultracold Fermions in a Graphene-Type Optical Lattice
• Quantum Hall effect and the topological number in graphene
• Massless Dirac Fermions in a Square Optical Lattice
• Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field

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