Nontrivial magnetic field dependence of the anomalous Hall conductivity due to spatially correlated inhomogeneities

We consider the problem of finding the effective anomalous Hall conductivity (AHC) of an inhomogeneous two-dimensional (2D) magnetic conductor governed by classical transport equations. Using homogenization theory, we proved that the effective AHC typically cannot exceed the bounds of local AHC. Conversely, relaxing the conditions needed for the proof allows one to potentially overcome the bounds. As an example, we explored alternative mechanisms than could lead to the hump or dip features in the magnetic-field dependence of the AHC, usually ascribed to the topological Hall effect. Through perturbation calculations and numerically solving the transport equation with random coefficients in a finite system, we found that such features could be caused by inhomogeneous saturated anomalous Hall resistivity correlated with magnetization domain profile, or by the anisotropic resistance associated with magnetic domain walls.