Real-space Picture of the Intrinsic Anomalous Hall Effect

The intrinsic anomalous Hall effect is commonly expressed as the integral of Berry curvature in the reciprocal space. It has been shown that this intrinsic effect can also be computed in real space for systems which lack periodicity (e.g finite or open systems). Motivated by this, we study a tight-binding toy model to show that the intrinsic anomalous Hall effect could be interpreted in terms of an electric field-induced change in the charge quadropole moment. We additionally present a first-principle calculation of bcc Fe and demonstrate how the charge quadrupole moments changes in the presence of electric field. We also consider a real space interpretation of intrinsic anti-damping torques in a ferromagnet/heavy metal bilayer system.