Topological aspects of quantum Hall effect in graphene

We study the recently observed quantum Hall effect(QHE) in graphene from a theoretical viewpoint of topological nature of the QHE to pose questions: (i) The zero-mass Dirac dispersion, which is the origin of the anomalous QHE, exists only around the zero gap, so a natural question is what happens to the QHE topological numbers over the entire energy spectrum. (ii) How the property that the bulk QHE topological number is equal to the edge QHE topological number, shown for the ordinary QHE, applies to the honeycomb lattice. We have shown that (a) the anomalous QHE proportional to (2N + 1) persists, surprisingly, all the way up to the van-Hove singularities, at which the normal behaviour abruptly takes over. (b) The edge-bulk correspondence persists as shown from the result for finite systems. All these properties hold for the entire sequence of lattice Hamiltonians that interpolate between square <-> honeycomb <->pi-flux lattices, so the anomalous QHE is on a quantum critical line.