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

TL;DR

This paper introduces a universal Hamiltonian model that captures the motion, merging, and topological transition of Dirac points in 2D materials, with applications to graphene and related systems.

Contribution

It proposes a simple, universal Hamiltonian describing Dirac cone merging and topological transitions in 2D crystals, extending understanding of electronic spectra near such critical points.

Findings

The model accurately describes the density of states and specific heat near the transition.

Spectrum evolution from sqrt(nB) to linear dependence is characterized and matches numerical results.

Reproduces the low-field Rammal-Hofstadter spectrum for honeycomb lattices.

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.