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Hall experiments in chiral magnets are often analyzed as the sum of ananomalous Hall effect, dominated by momentum-space Berry curvature, and atopological Hall effect, arising from the real-space Berry curvature in thepresence of skyrmions, in addition to the ordinary Hall resistivity. Thisraises the questions of how one can incorporate, on an equal footing, theeffects of the anomalous velocity and the real space winding of themagnetization, and when such a decomposition of the resistivity is justified.We provide definitive answers to these questions by including the effects ofall phase-space Berry curvatures in a semi-classical approach and by solvingthe Boltzmann equation in a weak spin-orbit coupling regime when themagnetization texture varies slowly on the scale of the mean free path. We showthat the Hall resistivity is then just the sum of the anomalous and topologicalcontributions, with negligible corrections from Berry curvature-independent andmixed curvature terms. We also use an exact Kubo formalism to numericallyinvestigate the opposite limit of infinite mean path, and show that the resultsare similar to the semi-classical results.

Unified Theory of the Anomalous and Topological Hall Effects with Phase Space Berry Curvatures