Field induced positional shift of Bloch electrons and its dynamical implications

We derive the field correction to the Berry curvature of Bloch electrons, which can be traced back to a positional shift due to the interband mixing induced by external electromagnetic fields. The resulting semiclassical dynamics is accurate to second order in the fields, in the same form as before, provided that the wave packet energy is derived up to the same order. As applications, we discuss the orbital magnetoelectric polarizability and predict nonlinear anomalous Hall effects.