Anomalous Hall effect in ultraclean electronic channels

We develop a theory of anomalous Hall effect in ultraclean channels with two-dimensional electron gas. In ultraclean systems, the electron mean free path due to static disorder and phonons is considered to be much larger compared to the channel width. The electron momentum relaxation and conductivity are mainly determined by the diffusive scattering at the channel edges and electron-electron collisions making electron magnetotransport highly nontrivial. The anomalous Hall electric field and Hall voltage are generated owing to the skew scattering of electrons, side-jump, and anomalous velocity effects that appear as a result of the spin-orbit coupling. We study both ballistic and hydrodynamic transport regimes which are realized depending on the relation between the electron-electron mean free path and the channel width. Compact analytical expressions for the anomalous Hall field and voltage are derived. We demonstrate that both in the ballistic and hydrodynamic regime the electron momentum relaxation in the bulk of the channel due to impurities or phonons is required for the anomalous Hall effect in contrast to the normal Hall effect and conductivity that can appear due to the electron-electron collisions combined with momentum relaxation at the channels edge scattering.