Exact Density-Functional Potentials for Time-Dependent Quasiparticles

We calculate the exact Kohn-Sham potential that describes, within time-dependent density-functional theory, the propagation of an electron quasiparticle wave packet of nonzero crystal momentum added to a ground-state model semiconductor. The potential is observed to have a highly nonlocal functional dependence on the charge density, in both space and time, giving rise to features entirely lacking in local or adiabatic approximations. The dependence of the nonequilibrium part of the Kohn-Sham electric field on the local current and charge density is identified as a key element of the correct Kohn-Sham functional.

Figure 1

The charge density additional to the ground state (top) and the current density (bottom) of both the model and Kohn-Sham systems at 1 time step (solid red lines), 75 time steps (dashed green lines), and 150 time steps (dotted blue lines) of $\Delta t=4\times {10}^{-3}\mathrm{a}.\mathrm{u}.$ The quasiparticle (solid black line) and Kohn-Sham (dashed blue line) band structures are inset, with the gray shaded region marking the range of pure quasiparticle Bloch states included in the electron wave packet.

Figure 2

The reverse-engineering algorithm. A self-consistent calculation of the exact (in the limit of $\Delta t\to 0$) time-dependent Kohn-Sham potential which is repeated for every time step in the quasiparticle system history. The algorithm makes incremental corrections to the Kohn-Sham current density, with the correct charge density arising due to the continuity equation. Here, $v_{\mathrm{KS}}$ is the Kohn-Sham potential, $\Psi$ is the Kohn-Sham Slater determinant, $\Delta j$ is the current density error, and ${\widehat{U}}_{t,t-\Delta t}$ is the unitary evolution operator which time evolves $\Psi$ from time $t-\Delta t$ to time $t$.

Figure 3

The additional (relative to the ground state) time-dependent Kohn-Sham potential $\Delta v_{\mathrm{KS}}$ (top) and electric field $\Delta {\mathcal{E}}_{\mathrm{KS}}=-\partial v_{\mathrm{KS}}/\partial x$ (bottom) after 1 time step (solid red lines), 75 time steps (dashed green lines), and 150 time steps (dotted blue lines) of $\Delta t=4\times {10}^{-3}\mathrm{a}.\mathrm{u}.$ The gradient is localized and more slowly varying, making it more amenable to approximation than the potential itself.