Dynamics of Berry-phase polarization in time-dependent electric fields

We consider the flow of polarization current $\mathbf{J}=d\mathbf{P}/dt$ produced by a homogeneous electric field $\mathsc{E}(t)$ or by rapidly varying some other parameter in the Hamiltonian of a solid. For an initially insulating system and a collisionless time evolution, the dynamic polarization $\mathbf{P}(t)$ is given by a nonadiabatic version of the King-Smith\char21{}Vanderbilt geometric-phase formula. This leads to a computationally convenient form for the Schr\"odinger equation where the electric field is described by a linear scalar potential handled on a discrete mesh in reciprocal space. Stationary solutions in sufficiently weak static fields are local minima of the energy functional of Nunes and Gonze. Such solutions only exist below a critical field that depends inversely on the density of k points. For higher fields they become long-lived resonances, which can be accessed dynamically by gradually increasing $\mathsc{E}.$ As an illustration the dielectric function in the presence of a dc bias field is computed for a tight-binding model from the polarization response to a step-function discontinuity in $\mathsc{E}(t),$ displaying the Franz-Keldysh effect.