\langle \lambda_c | \\
    &=& -i |\epsilon_a \rangle \langle \partial_k \epsilon_a|   \end{aligned}$$

The second line gives the numerical values of the laboratory frame $ac$th matrix elements, and the third line the simplification using a resolution of identity. Notice that we have the combination $|\epsilon_b \rangle \langle \partial_k \epsilon_b|$ with the same state index $b$ occurring together so that any phase factor $\exp(i \phi_b)$ introduced in $|\epsilon_b\rangle \rightarrow |\epsilon_b \rangle\exp(i \phi_b)$ cancels out.

Returning now to the diagonal elements of the rotated frame $i U\partial_k U^\dagger$, we see that they correspond to $i |\epsilon_i \rangle \langle \epsilon_i|\partial_k |\epsilon_i\rangle\langle \epsilon_i|$. Subtracting them off from $-i U^\dagger\partial_k U$ gives the MF potential $H_{MF}$ where $$H_{MF} = -i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_k \epsilon_a|\epsilon_b \rangle \langle \epsilon_b| E.$$

We argue that, at least for the purposes of calculating currents and spin$\otimes$pseudospin accumulations the effects of the electric field $E_i$ to the first order in $E$ can be incorporated by replacing $E_i x_i$ with $H_{MF}$ so that the effective Hamiltonian reads $$H' = H_0 + H_{MF} = b_i(\vec{k})\kappa_i - i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_{k_i} \epsilon_a|\epsilon_b \rangle \langle \epsilon_b| {E_i}. \label{Heff}$$

In order to support our claim, we list three examples 