arrows at each point in $k$ space indicate the direction of the Rashba SOI field there. The application of the electric field causes the momentum of the charge carrier to trace out the trajectory in $k$ space indicated by the dotted line. The spin of the charge carrier adiabatically follows the direction of the SOI field at each point in $k$ space. The rotation of the spin can be thought of as being due to an effective magnetization which both creates the torque necessary to rotate the spin as shown in the inset, as well as confers an out of plane spin accumulation. []{data-label="gA2SOIfield"}](A2SOIfield.eps)

We now proceed to a general description of the MF potential generalized to include other discrete degrees of freedom.

The Murakami-Fujita potential
=============================

Consider now a generic Hamiltonian $$H_0  = b_i(\vec{k})\kappa_i \label{H0}$$ where the $\kappa_i$s are finite sized matrices representing the discrete degrees of freedom. For example, in a spin 1/2 system with SOC, the 4 $\kappa_i$s are the Pauli matrices and the identity matrix. In the TI thin film Hamiltonian Eq. \[TIham1\] the $\kappa_i$s represent the 16 $\vec{\sigma}\otimes\vec{\tau}$ matrices.

In order to write the Hamiltonian Eq. \[H0\] as a numerical matrix, we need to express 