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

Hall experiments in chiral magnets are often analyzed as the sum of an anomalous Hall effect, dominated by momentum-space Berry curvature, and a topological Hall effect, arising from the real-space Berry curvature in the presence of skyrmions, in addition to the ordinary Hall resistivity. This raises the questions of how one can incorporate, on an equal footing, the effects of the anomalous velocity and the real space winding of the magnetization, and when such a decomposition of the resistivity is justified. We provide definitive answers to these questions by including the effects of all phase-space Berry curvatures in a semi-classical approach and by solving the Boltzmann equation in a weak spin-orbit coupling regime when the magnetization texture varies slowly on the scale of the mean free path. We show that the Hall resistivity is then just the sum of the anomalous and topological contributions, with negligible corrections from Berry curvature-independent and mixed curvature terms. We also use an exact Kubo formalism to numerically investigate the opposite limit of infinite mean path, and show that the results are similar to the semi-classical results.