Chern band insulators in a magnetic field.

The effect of a magnetic field on a two-dimensional Chern band insulator is discussed. It is shown that, unlike the trivial insulator, an anomalous Hall insulator with Chern number C becomes a metal when a magnetic field is applied at constant particle density, for any C > 0. For a time-reversal invariant topological insulator with a spin Chern resolved number, C↑ = −C↓ = C, the magnetic field induces a spin polarized spin Hall insulator. We consider also the effect of a superlattice potential and extend previous results for the quantization of the Hall conductance of filled Hofstadter bands to this problem.