Topological aspects of graphene - Dirac fermions and the bulk-edge correspondence in magnetic fields

Topological aspects of the electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: ( i) Presence of the pair of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect ( QHE), is not accidental to honeycomb, but is generic for a class of two-dimensional lattices that interpolate between square and pi-flux lattices. Topological stability guarantees persistence of the peculiar QHE. ( ii) While we have the massless Dirac dispersion only around E = 0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behaviour accompanied by huge jumps in the QHE with a sign change. ( iii) We establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is another topological effect. We have also explicitly shown that the E = 0 edge states in honeycomb in zero magnetic field persist in magnetic field. ( iv) We have also identified a topological origin of the fermion doubling in terms of the chiral symmetry.