Analytic formulas for the anomalous Hall effect in itinerant magnets

We provide analytical formulas to compute all the contributions to the intrinsic Hall conductivity in the presence of Kondo-coupled spins in any configuration and for any spin orbit coupling, and thereby clarify the origin of what is sometimes called the “topological anomalous Hall effect”. We also identify the relation between a momentum space quantity, the momentum space Berry curvature (which is in direct correspondence with the Hall conductivity — a global observable), and unit cell properties such as hopping parameters and spin configuration. More precisely, we find that the Berry curvature involves the scalar spin chirality on elementary unit cell triangles, χ i j k = S → i · ( S → j × S → k ) , but also contains scalar triple products of other quantities (such as hopping parameters with spin-orbit coupling t → i j ), t → i j · ( t → j k × t → k i ) , t → j k · ( S → i × S → j ) , ⋯ , and their dot products, S → i · S → j , t → i j · t → j k , t → i j · S → k , ⋯ The relative size of the different contributions depends on the strength of the Kondo coupling and our formula captures all regimes. We apply our method to the case of three-sublattice systems, and prove very generally that in the absence of spin-orbit couplin