Gapped state of a carbon monolayer in periodic magnetic and electric fields

When smooth zero-on-average periodic magnetic and electric fields are applied to a carbon monolayer (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. The phenomenon is of the same type as in Haldane's model for a quantum Hall effect without Landau levels, although there is the following important difference between the two models: 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 several orders of magnitude is sufficient. The Hall effect is explained in terms of the topological theory of Thouless, Kohmoto, Nightingale, and den Nijs. For the system studied in this paper, a complementary explanation in terms of simple physical principles is also presented.