Theory of Hall effects with chiral magnetic textures and spin-orbit coupling*

Chiral magnetic materials offer a rich landscape of spin textures, including spirals and skyrmion phases, which influence the dynamics of itinerant electrons in the presence of spin-orbit coupling (SOC).  Experiments are often analyzed as the sum of an anomalous Hall effect (AHE), dominated by momentum-space Berry curvature, and a topological Hall effect (THE) arising from the real-space Berry curvature of skyrmions, in addition to the ordinary Hall resistivity proportional to field. Here we present a theory that treats both momentum- and real-space Berry curvatures on equal footing.  First, we will present results from a semi-classical Boltzmann approach with generalized phase space Berry curvatures. Apart from the AHE and THE, we find an additional contribution that depends on the SOC and on first-order gradients of the magnetic texture and can be non-zero even in the absence of all phase-space Berry curvatures.  We discuss the relative strengths of the various contributions and find conditions under which the new Hall contribution vanishes. We also present results obtained from a complementary approach, valid in the regime of very long mean paths, using the Kubo formula and exact diagonalization in the magnetic unit cell. Finally, we comment on the question of using the Hall signal for detection of skyrmions.