I consider several examples of 2D crystals where the low energy properties can be described with a 2 x 2 Hamiltonian with a spectrum exhibiting several Dirac cones. These cones are characterized by a linear dispersion relation, but more importantly by a topological number, or “charge”, related to a Berry phase associated to the spinorial structure of the wave function. For example, the graphene spectrum has a pair of Dirac cones with opposite Berry phases (±π). We study under which conditions these Dirac cones can be manipulated, created or suppressed, through a modification of band parameters, under the condition of conservation of the total “charge”.

We have found two universal scenarios:  [1] The merging of Dirac points with opposite “charges” constitutes a topological transition between a semi-metallic phase and a band insulator, with a remarkable “semi-Dirac” spectrum at the transition: it is  linear in one direction and quadratic in the other.[2] A pair of Dirac points with same charge can also merge into a single point with double charge 2π, characterized by a gapless quadratic spectrum. This transition occurs in twisted graphene bilayers. For both types of merging, we derive a universal Hamiltonian that describes continuously the coupling between valleys. We find a general scaling law for the Landau energy levels which the magnetic field, we calculate the number of zero energy modes and we compare the exact spectrum with its semiclassical approximation. The case of strained bilayer is particularly interesting since the spectrum exhibits a pair of four Dirac points can be manipulated under strain. Different types of merging can occur.

Finally, I will briefly discuss a recent experiment in cold atoms which has recently detected the merging of Dirac points and whose results agree perfectly with our predictions. 

[1] A universal Hamiltonian for the motion and the merging of Dirac cones in a two-dimensional crystal, G. Montambaux, F. Piéchon, J.-N. Fuchs and  M.O. Goerbig, Eur. Phys. J. B  72, 509 (2009)

[2] Merging of Dirac points in a two-dimensional crystal, G. Montambaux, F. Piéchon, J.-N. Fuchs and  M.O. Goerbig, Phys. Rev. B 80, 153412 (2009)