  When smooth, zero-on-average, periodic magnetic and electric fields are
applied to a carbon mono-layer (graphene), a gap between the valence and
conduction band is introduced. Here this gapped state is studied analytically.
It is found that it does not correspond to a band insulator: a constant
electric field induces a quantized Hall current even though the magnetic flux
through the sample is zero and there are no Landau levels. The phenomenon is of
the same type as that discovered by Haldane for a graphene sample in a periodic
magnetic field that is not smooth, i.e. varies rapidly on the scale of the
graphene lattice constant. The effect can be explained in terms of the
topological theory of Thouless, Kohmoto, Nightingale and den Nijs. For the
system studied in this paper, an explanation in terms of simple physical
principles is also presented. Thus some of the mystery is taken out of the
apparently strange phenomenon of a Hall effect without magnetic flux.
Furthermore, Haldane's model requires control over external magnetic fields on
length scales less than an angstrom and is therefore hard to realize
experimentally. For the model studied here, control over external fields on
length scales that are larger by two orders of magnitude or more is sufficient.
The model is therefore more amenable to experimental realization.
