We develop a theory for the Hall, Nernst, and thermal Hall effects in magnetic materials that harbor topological spin textures such as skyrmions. In addition to the ordinary response to an external magnetic field, there are two other contributions that have been described by two distinct theories: an intrinsic anomalous response arising from momentum-space Berry curvature and a topological response from the real-space Berry curvature related to the topological charge density of the spin texture. We develop here a unified semiclassical theory that incorporates the effects of all phase-space Berry curvatures on an equal footing within a controlled calculation. We analyze the electrical and thermal currents carried by electrons with arbitrary dispersion and spin-orbit coupling (SOC) of strength $\lambda$ interacting with arbitrary three-dimensional spin textures via an exchange coupling $J$. For small $\lambda /J$, we show that all linear response conductivities—electrical, thermoelectric, and thermal Hall—can be written as the sum of three contributions: ordinary, anomalous, and topological. We derive various general relations including the Weidemann-Franz, Kelvin, and Mott relations. We show that there is a topological phase transition as a function of $\lambda /J$ at which the semiclassical approach breaks down and the large $\lambda /J$ regime exhibits qualitatively different behavior with a vanishing topological Hall response.

Close

PrevNext

View figure in article

• Anomalous Hall effect
• Skyrmions
• Topological Hall effect
• Topological materials
• Boltzmann theory