where the use of $H'$ recovers well-known results.

Anomalous velocity
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In the presence of an electric field, charge carriers can acquire an anomalous velocity [@AV1; @AV2; @AV3] proportional to the Berry curvature [@AV4; @AV5; @ChioGoo1; @ChioGooRev]. We show that this result can be recovered via the Heisenberg equation of motion on Eq. \[Heff\]. Under the Heisenberg equation of motion, we have $\partial_t \vec{r} = -i  ( [\vec{r}, H_0]   + [\vec{r}, H_{MF}] = \nabla_{\vec{k}} (H_0 + H_{MF}) $ The first term is the usual velocity. We shall show that the second gives the usual Berry curvature anomalous contribution to the velocity. Taking the expectation value of the second with respect to the $i$th eigenstate of $H_0$, we have $$\begin{aligned}
    && -i \langle \epsilon_i | [x_b, H_{MF}]  |\epsilon_i \rangle \\
    &=& -i E_b \langle \epsilon_i| \partial_{k_b} \Big( \sum_{\substack{jk \\ j\neq k}} |\epsilon_j \rangle \langle  \partial_{k_a}\epsilon_j| \epsilon_k \rangle\langle\epsilon_k| \Big) |\epsilon_i \rangle \\
    &=&-i E_b \sum_j (\langle \epsilon_i |\partial_{k_b} \epsilon_j \rangle \langle \partial_{k_a}\epsilon_j|\epsilon_i\rangle + \langle \partial_{k_a} \epsilon_i | \epsilon_j\rangle\langle \partial_{k_b} \epsilon_j|\epsilon_i \rangle) \\
    &=& 2 E_b \mathrm{Im} ( \langle \partial_{k_a} \epsilon_i|\partial_{k_b} \epsilon_i \rangle).\end{aligned}$$

The third line is simply the expansion of the $\partial_{k_b}$ differential. Notice that the requirement that $j \neq k$ stemming 