|\epsilon_j \rangle}{E_i - E_j},\end{aligned}$$ we can rewrite $$\begin{aligned}
    H_{MF} &=& -i E_i \sum_{a \neq b} |\epsilon_a \rangle \langle \partial_{k_i} \epsilon_a|\epsilon_b \rangle \langle \epsilon_b|. \\
    &=&-i \sum_{a \neq b} |\epsilon_a\rangle  \frac{ \langle \epsilon_a| \partial_{k_i} H_0 |\epsilon_b \rangle}{E_a - E_b} \langle \epsilon_b|.\end{aligned}$$

A common form of the Kubo formula is $$\delta \langle O \rangle \propto \sum_{\vec{k}} \sum_{a= \neq  b} \frac{n(E_a)-n(E_b)}{(E_a-E_b)^2} \mathrm{Im} ( \langle a|O|b\rangle \langle b| (\partial_k H_0)|a \rangle \label{kuboPaper}.$$

Substituting this back into Eq. \[c1oe\] gives a result similar to the Kubo expression for the change in an expectation value of $O$ under an electric field – $$\delta \langle i| O|i \rangle  = 2 \mathrm{Im} \sum_j \vec{E}\cdot\frac{  \langle \epsilon_i |O| \epsilon_j \rangle \langle \epsilon_j |\nabla_{\vec{k}} H_0 |\epsilon_i\rangle}{(E_i - E_j)^2}. \label{c2oe}$$

Our result Eq. \[c2oe\] corresponds to Eq. \[kuboPaper\] with the occupancy factor $n$ set to 1 for the $i$th state we are interested in and 0 for the other states, and without a second summation over all states.

Having established the link between the MF potential and the Kubo formula, we now proceed to use Eq. \[c2oe\] to study the exemplary system of a topological insulator thin film system.

TI thin films
=============

The effective Hamiltonian for the surface states of 