Topological Hall Effects in Magnetic Metals with Chiral Spin Textures

We derive a Boltzmann transport equation for conduction electron transport in magnetic metals with chiral spin textures (skyrmions) starting from the Keldysh formulation of Dyson's equation for metals with an s-d exchange interaction. We show that the anomalous electrical and thermal Hall effects, electric polarizations induced by the electric field and temperature gradient, and conductivity due to the steady-state motion of skyrmions depend on the scalar spin chirality, S · (∂_xS × ∂_yS), and spatial average of these transport coefficients are proportional to the net topological charge, N_2D = (4πS)^-1 ∫dxdy S · (∂_xS × ∂_yS). In addition, we find that the spin Hall effect, spin Nernst effect, Edelstein effect, and temperature-gradient-induced magnetization are determined by the vector spin chirality, S × ∂_yS and ∂_xS × ∂_yS. Finally, we show that conduction electron scattering off moving skyrmions leads to violation of the Wiedemann-Franz law.