Bloch dynamics with second order Berry phase correction

We derive the semiclassical Bloch dynamics with second-order Berry phase correction in presence of slow-varying scalar potential as perturbation. Our mathematical derivation is based on a two-scale WKB asymptotic analysis. For uniform external electric field, bi-characteristics system after positional shift introduced by connections agrees recent result previous works. Moreover, for case linear we show that extra terms arising are also gauge independent.