We illustrate the potential of Time Dependent Density Functional Theory (TDDFT) for describing the nonequilibrium behavior of strongly correlated (lattice) models. Starting from an exact time-evolution of the many-body wavefunction, we determine, via reverse engineering, the exact exchange correlation (xc) potential
vxc for small Hubbard chains of different lengths and electron fillings and exposed to time-dependent perturbations. We compare some of the exact results to those of adiabatic-like treatments, in order to extract some of the properties that approximate xc potentials should have. Finally, we provide details of work in progress and future directions.
We illustrate the scope of Time Dependent Density Functional Theory (TDDFT) for strongly correlated (lattice) models out of equilibrium. Using the exact many body time evolution, we reverse engineer the exact exchange correlation (xc) potential
vxc for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation (ALDA) to
vxc for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.
I describe the time-dependent superfluid local density approximation, which is an adiabatic extension of the density functional theory to superfluid Fermi systems and their real-time dynamics. This new theoretical framework has been used to describe several phenomena in cold atomic gases and nuclear collective motion: excitation of the Higgs modes in strongly interacting Fermi superfluids, generation of quantized vortices, crossing and reconnection of vortices, excitation of the superflow at velocities above the critical velocity, excitation of quantum shock waves and domain walls in the collisions of superfluid atomic clouds, and excitation of collective states in nuclei.
In this comparative study we benchmark a recently developed nonadiabatic exchange-correlation potential within time-dependent density-functional theory (TDDFT) [Phys. Rev. Lett. 120, 157701 (2018)] by (a) validating the transient dynamics using a numerically exact density-matrix renormalization-group approach as well as by (b) comparing the
RC time, a typical linear-response quantity, to up to second-order perturbation theory results. As a test bed we use the dynamics of the single-impurity Anderson model. These benchmarks show that the nonadiabatic potential yields quantitatively accurate results for the transient dynamics for temperatures of the order of the hybridization strength, while the TDDFT
RC times quantitatively agree with those from second-order perturbation theory for temperatures which are large compared to the hybridization strength. Both results are particularly intriguing given the relatively low numerical cost of a TDDFT calculation.
We introduce a method of evaluating the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method [G\{o}rling and Levy, Phys. Rev. A {\bf 50}, 196 (1994), G\{o}rling, ibid. {\bf 55}, 2630 (1997)], from which the expansion of the density matrix in powers of the interaction constant
λ naturally arises. By this, we obtain the one-density-matrix $\rho_\lambda(\rv,\rv',t)$, which, by construction, has the
λ-independent diagonal elements $\rho_\lambda(\rv,\rv,t)=n(\rv,t)$, where $n(\rv,t)$ is the particle density. The off-diagonal elements of $\rho_\lambda(\rv,\rv',t)$ importantly contribute to the processes beyond the reach of TDDFT, of which we consider the momentum-resolved photoemission, doing this to the first order in
λ (exact exchange). In an illustrative calculation of photoemission from the quasi-2D electron gas with {\it one filled subband} we find strong deviations from the independent-particle Fermi golden rule formula.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.