Current density in anomalous Hall effect regime under weak scattering

A finite equilibrium current density arises in the anomalous Hall effect (AHE) as a result of time-reversal symmetry breaking, affecting both the differential current density and total current. In this paper, we illustrate the equilibrium current density in a ribbon-shaped system within the AHE regime, consisting of two sets of counterpropagating channels arranged in a zebra stripes pattern. While the middle channels are susceptible to scattering, the edge channels remain relatively robust. Despite this difference, all channels exhibit the same differential current density when subjected to a differential voltage across the two ends of the ribbon. When a differential voltage is applied to both sides of the ribbon, it results in a snaking pattern of differential current density forming across it. Furthermore, in a four-terminal device comprising the ribbon and two normal leads, it is found that Hall conductance is independent of ribbon width within certain scattering strengths due to the differences in robustness between middle and edge channels. These findings disclose the details of the AHE transport in a finite-sized system under weak scattering.