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arXiv:1004.2264 · Physical Review Letters

Time-Dependent Density-Functional Theory Meets Dynamical Mean-Field Theory: Real-Time Dynamics for the 3D Hubbard Model

Daniel Karlsson, A. Privitera, Claudio Verdozzi

Published Mar 15, 2011• 65 citations
arXivHTMLPDF

Abstract

We introduce a new class of exchange-correlation potentials for a static and time-dependent density-functional theory of strongly correlated systems in 3D. The potentials are obtained via dynamical mean-field theory and, for strong enough interactions, exhibit a discontinuity at half-filling density, a signature of the Mott transition. For time-dependent perturbations, the dynamics is described in the adiabatic local density approximation. Results from the new scheme compare very favorably to exact ones in clusters. As an application, we study Bloch oscillations in the 3D Hubbard model.

DOI
10.1103/physrevlett.106.116401

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We introduce a new class of exchange-correlation potentials for a static and time-dependent density-functional theory of strongly correlated systems in 3D. The potentials are obtained via dynamical mean-field theory and, for strong enough interactions, exhibit a discontinuity at half-filling density, a signature of the Mott transition. For time-dependent perturbations, the dynamics is described in the adiabatic local density approximation. Results from the new scheme compare very favorably to exact ones in clusters. As an application, we study Bloch oscillations in the 3D Hubbard model.
We illustrate the scope of time-dependent density-functional theory for strongly correlated (lattice) models out of equilibrium. Using the exact many-body time evolution, we reverse engineer the exact exchange correlation (xc) potential v_(xc) for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation to v_(xc) 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.
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 vxcv_{xc}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 vxcv_{xc}vxc​ for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation (ALDA) to vxcv_{xc}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.
We present some details of our recently-proposed Time-Dependent Density-Functional Theory (TDDFT) for strongly-correlated materials in which the exchange-correlation (XC) kernel is derived from the charge susceptibility obtained using Dynamical Mean-Field Theory (the TDDFT + DMFT approach). We proceed with deriving the expression for the XC kernel for the one-band Hubbard model by solving DMFT equations via two approaches, the Hirsch–Fye Quantum Monte Carlo (HF-QMC) and an approximate low-cost perturbation theory approach, and demonstrate that the latter gives results that are comparable to the exact HF-QMC solution. Furthermore, through a variety of applications, we propose a simple analytical formula for the XC kernel. Additionally, we use the exact and approximate kernels to examine the nonhomogeneous ultrafast response of two systems: a one-band Hubbard model and a Mott insulator YTiO3. We show that the frequency dependence of the kernel, i.e., memory effects, is important for dynamics at the femtosecond timescale. We also conclude that strong correlations lead to the presence of beats in the time-dependent electric conductivity in YTiO3, a feature that could be tested experimentally and that could help validate the few approximations used in our formulation. We conclude by proposing an algorithm for the generalization of the theory to non-linear response.
The physics of interacting many-body systems is a very complex and challenging subject. Therefore one in general needs to use simplified models and approximate methods for studying these systems theoretically. In this work we study the exchange-correlation (xc) potential for the Hubbard-Holstein (HH) model for the application of Density Functional Theory (DFT) to the same. The Hohenberg-Kohn theorem and the two sets of Kohn-Sham equations are derived for the HH-model extending a procedure outlined in previous work. The xc-potentials for the fermions and bosons respectively are then derived analytically as functions of the density nnn and phononic displacement xxx for a single HH-site in contact with a heat bath, generalizing the procedure for the pure Hubbard model. The potential for the HH-site is found to have a smaller gap at half filling than for the Hubbard model, due to a rescaling of the effective electron-electron interaction. It also has an additional term linear in (n−1)(n-1)(n−1). The xc-potentials are used to study two simple systems, an Andersson impurity dimer and a four-site chain of HH-sites. The results are compared to those of an exact solution and to the Hartree-Fock (HF) solution. For the dimer the DFT-method performs consistently better than the HF-method, while the results for the chain are more ambiguous. However, the central result of this thesis is to produce xc-potentials for an infinitely coordinated Bethe lattice, using Dynamical Mean Field Theory (DMFT). The potentials for a small but representative set of interaction parameters are presented and compared to the analytical potential. The potentials are similar in that they do not exhibit much structure, and they behave very similarly far from half filling. Close to half filling they show more differences, as the analytic potential is always discontinuous around half filling while the potential attained via DMFT is only discontinuous for large electron-electron interactions.
In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.