from the removal of the diagonal terms of the rotated frame $-i U^\dagger \partial_k U$ results in the absence of terms $\langle \partial_{k_a}\partial_{k_b} \epsilon_i|\epsilon_i\rangle$ and $\langle \partial_{k_b} \epsilon_i|\partial_{k_a} \epsilon_i\rangle$ due to the $\partial_{k_b}$ acting on the second and third terms in the big bracket in the second line. The last line is the usual Berry curvature term for the anomalous velocity.

Spin and other discrete DoFs
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As part of their paper on explaining the microscopic origin of spin torque, Cheng and Niu extended the original Sundaram-Niu formalism, which described only the spatial evolution of position and velocity, to now cover the time evolution of spin 1/2 as well. Their formalism can be easily extended to cover the time evolution of operators with finite discrete spectra. We describe the extension in the appendix, and simply state the end result here.

For a state $$|\psi\rangle = \sum_i |\epsilon_i \rangle \eta_i$$ where the summation $i$ runs over the discrete DoFs (e.g spin up / down for spin 1/2, and the upper / lower surfaces for a TI thin film) and the continuous quantum numbers (e.g. $\vec{k}$ in SOI systems) and the $\eta_i$s are the weightages of the $i$th basis state, we show 