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

Analysis of PDEs