$\tau_x$ term then represents the coupling between the two surfaces of the film due to the finite thickness.

This paper is organized as follows. We first revisit the emergence of the MF potential in a spin 1/2 SOI system. We then generalize the MF potential to include other discrete DoFs, and provide three evidences to support our claim that the MF potential accounts for the effects of the electric field in the sense that an effective Hamiltonian can be constructed by replacing the $\vec{E}\cdot\vec{r}$ term in the original Hamiltonian can be replaced by the position-*independent* MF potential.

We first show that taking the momentum derivative of the effective Hamiltonian in the Heisenberg equation of motion for the position operator reproduces the usual Berry curvature expression for the anomalous Hall velocity. As part of their paper on the microscopic origin of spin torque [@ChengRan], Cheng Ran and Niu Qian had extended the original Sundaram-Niu wavepacket formalism [@ChioGoo1], which gave only the time variation of the position and momentum expectation values, to now include the time variation of the spin expectation values. We show that Ran and Niu’s expressions for the time evolution of the spin expectation values can be readily extended 