and . A caveat is warranted though: Despite the close similarity between a graphene system with a real magnetic field and one with a deformed lattice, there is a fundamental difference. The gapped state produced by deformations is not anomalous. The reason is that for deformation induced fields, the anomalous behavior of the two valleys cancel each other, while for real magnetic fields, the behavior is the same in both valleys and there is no cancellation.

Several other publications deal with systems similar to that studied here. In Refs. and the two-dimensional non-relativistic Schrödinger equation with a periodic magnetic field that is zero on average was studied. It is interesting to compare this system to the graphene system in the limit of strong magnetic fields, when the magnetic length is much smaller than the periodicity of the fields. In this regime the anomalous Hall effect[@foot1] was found to disappear in the case of the Schrödinger equation.[@Tai08] It remains present in graphene as is shown here. Another relevant study[@Brey09] dealt with graphene in a periodic electric field alone. It was found that a strong electric field can induce additional touching points between graphene’s valence and conduction bands. In the present work, 