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 $\mathcal{E}\mathrm{}\left(t\right)\mathrm{}$ 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 $\mathbf{P}\mathrm{}\left(t\right)\mathrm{}$ is given by a nonadiabatic version of the King-Smith–Vanderbilt geometric-phase formula. This leads to a computationally convenient form for the Schrödinger 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 $\mathcal{E}.$ 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 $\mathcal{E}\mathrm{}\left(t\right),$ displaying the Franz-Keldysh effect.