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

TL;DR

This paper develops a unified theoretical framework incorporating phase-space Berry curvatures to explain both anomalous and topological Hall effects in chiral magnets, clarifying their additive nature and regimes of validity.

Contribution

It introduces a comprehensive semi-classical approach including all phase-space Berry curvatures and confirms its validity with exact numerical calculations.

Findings

Hall resistivity is the sum of anomalous and topological contributions.

Negligible corrections from Berry curvature-independent and mixed curvature terms.

Semi-classical and numerical results are consistent across different regimes.

Abstract

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.