honeycomb lattice also supports a fundamentally different kind of gapped state which I will call anomalous. A signature of the anomalous state is a Hall effect without Landau levels and in a zero average magnetic field. The resulting Hall conductivity is a topological invariant.[@Hal04]

In Haldane’s model, the anomalous state is realized by the combined effect of next nearest neighbor hopping together with a magnetic field that has the same periodicity as the honeycomb lattice and zero flux through a lattice unit cell. Because of the very short length scales ($\sim1$Å) on which this magnetic field must be controlled, the experimental realization of Haldane’s model does not seem feasible. Recently, Kane and Mele [@Kane05] pointed out that the spin orbit interaction can generate an anomalous gapped state in graphene without the need for any external magnetic field. In this case, since time-reversal symmetry is not broken, no Hall effect with electric charge current is produced. Rather a spin-Hall effect occurs. However, the effect may be hard to observe because of the smallness of the spin orbit interaction in graphene.

A gapped state can also occur when periodic magnetic and electric fields that are smooth on the scale of the graphene 