We obtain the exact Kohn-Sham potentials $V_{\mathrm{KS}}$ of time-dependent density-functional theory for one-dimensional Hubbard chains, driven by a dc external field, using the time-dependent electron density and current density obtained from exact many-body time evolution. The exact $V_{\mathrm{xc}}$ is compared to the adiabatically exact $V_{\mathrm{xc}}^{\mathrm{ad}}$ and the “instantaneous ground state” $V_{\mathrm{xc}}^{\mathrm{igs}}$. The latter is shown to work effectively in some cases when the former fails. Approximations for the exchange-correlation potential $V_{\mathrm{xc}}$ and its gradient, based on the local density and on the local current density, are also considered and both physical quantities are observed to be far outside the reach of any possible local approximation. Insight into the respective roles of ground-state and excited-state correlation in the time-dependent system, as reflected in the potentials, is provided by the pair correlation function.

• Received 7 January 2014
• Revised 9 April 2014

DOI:https://doi.org/10.1103/PhysRevB.89.195114