Microscopic Theory of Nonlinear Hall Effect in Three-dimesional Mangetic Systems

The nonlinear Hall effect (NLHE) has been detected in various of condensed matter systems. Unlike linear Hall effect, NLHE may exist in physical systems with broken inversion symmetry in the crystal. On the other hand, real space spin texture may also break inversion symmetry and result in NLHE. In this letter, we employ the Feynman diagramatic technique to calculate nonlinear Hall conductivity (NLHC) in three-dimensional magnetic systems. The results connect NLHE with the physical quantity of emergent electrodynamics which oringates from the magnetic texture. The leading order contribution of NLHC $\chi_{abb}$ is proportional to the emergent toroidal moment $\mathcal{T}_{a}^{e}$ which reflects how the spin textures wind in three dimension.