Anomalous Hall response of topological insulators

The anomalous Hall effect due to the surface conduction band of three-dimensional topological insulators with an out-of-plane magnetization is always dominated by an intrinsic topological term of the order of the conductivity quantum. We determine the contributions due to the band structure, skew scattering, side jump, and magnetic impurities on the same footing, demonstrating that the topological term, renormalized due to disorder, overwhelms all other terms, providing an unmistakable signature of $Z_2$ topological order. Uncharacteristically, skew scattering contributes in the Born approximation as well as in the third order in the scattering potential, while in addition to the side-jump scattering term we identify a different intrinsic side-jump term of a similar magnitude. These, however, never overwhelm the topological contribution.