Scattering universality classes of side jump in the anomalous Hall effect

The anomalous Hall conductivity has an important extrinsic contribution known\nas side jump contribution, which is independent of both scattering strength and\ndisorder density. Nevertheless, we discover that side jump has strong\ndependence on the spin structure of the scattering potential. We propose three\nuniversality classes of scattering for the side jump contribution, having the\ncharacters of being spin-independent, spin-conserving and spin-flip\nrespectively. For each individual class, the side jump contribution takes a\ndifferent unique value. When two or more classes of scattering are present, the\nvalue of side jump is no longer fixed but varies as a function of their\nrelative disorder strength. As system control parameter such as temperature\nchanges, due to the competition between different classes of disorder\nscattering, the side jump Hall conductivity could flow from one class dominated\nlimit to another class dominated limit. Our result indicates that magnon\nscattering plays a role distinct from normal impurity scattering and phonon\nscattering in the anomalous Hall effect because they belong to different\nscattering classes.\n