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arXiv:1209.3145 · Physical Review B

Lattice density functional theory at finite temperature with strongly density-dependent exchange-correlation potentials

Gao Xianlong, Ahai Chen, I. V. Tokatly, Stefan Kurth

Published Dec 26, 2012• 32 citations
arXivHTMLPDF

Abstract

The derivative discontinuity of the exchange-correlation (xc) energy at an integer particle number is a property of the exact, unknown xc functional of density functional theory (DFT) which is absent in many popular local and semilocal approximations. In lattice DFT, approximations exist which exhibit a discontinuity in the xc potential at half-filling. However, due to convergence problems of the Kohn-Sham (KS) self-consistency cycle, the use of these functionals is mostly restricted to situations where the local density is away from half-filling. Here a numerical scheme for the self-consistent solution of the lattice KS Hamiltonian with a local xc potential with rapid (or quasidiscontinuous) density dependence is suggested. The problem is formulated in terms of finite-temperature DFT where the discontinuity in the xc potential emerges naturally in the limit of zero temperature. A simple parametrization is suggested for the xc potential of the uniform one-dimensional (1D) Hubbard model at finite temperature which is obtained from the solution of the thermodynamic Bethe ansatz. The feasibility of the numerical scheme is demonstrated by application to a model of fermionic atoms in a harmonic trap. The corresponding density profile exhibits a plateau of integer occupation at low temperatures which melts away for higher temperatures.

DOI
10.1103/physrevb.86.235139

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The derivative discontinuity of the exchange-correlation (xc) energy at an integer particle number is a property of the exact, unknown xc functional of density functional theory (DFT) which is absent in many popular local and semilocal approximations. In lattice DFT, approximations exist which exhibit a discontinuity in the xc potential at half-filling. However, due to convergence problems of the Kohn-Sham (KS) self-consistency cycle, the use of these functionals is mostly restricted to situations where the local density is away from half-filling. Here a numerical scheme for the self-consistent solution of the lattice KS Hamiltonian with a local xc potential with rapid (or quasidiscontinuous) density dependence is suggested. The problem is formulated in terms of finite-temperature DFT where the discontinuity in the xc potential emerges naturally in the limit of zero temperature. A simple parametrization is suggested for the xc potential of the uniform one-dimensional (1D) Hubbard model at finite temperature which is obtained from the solution of the thermodynamic Bethe ansatz. The feasibility of the numerical scheme is demonstrated by application to a model of fermionic atoms in a harmonic trap. The corresponding density profile exhibits a plateau of integer occupation at low temperatures which melts away for higher temperatures.
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.
We study a lattice version of the local density approximation (LDA) based on Bethe ansatz (BALDA). Contrary to what happens in density functional theory in the continuum and despite its name, BALDA displays some very nonlocal features and it has a discontinuous functional derivative. The same features prevent the convergence of the self-consistent Kohn-Sham cycle thus hindering the study of BALDA solutions close to a Mott phase or in the Coulomb blockade regime. Here, we propose a numerical approach which, differently from previous works, does not introduce ad hoc parameters to smear out the singularity. Our results are relevant for all lattice models where BALDA is applied ranging from Kondo systems to harmonically trapped Hubbard fermions. As an example, we apply the method to the study of a one-dimensional lattice model with Hubbard interaction and a staggered potential which can be driven from an ionic to a Mott-insulating state. In the Mott regime, the presence of a ``vacuum'' allows us to calculate the different contribution to the gap and to highlight an ultranonlocality of BALDA.
We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on strongly interacting systems, this functional yields exact features crucial for important applications such as quantum transport. In particular, the steplike structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accurate quantitative agreement with exact many-body chemical potentials. While it can be shown that a sharp derivative discontinuity is present only in the infinitely strongly correlated limit, at finite correlation regimes we observe a slightly smoothened discontinuity, with qualitative and quantitative features that improve with increasing correlation. From the fundamental point of view, our results obtain the derivative discontinuity without making the assumptions used in its standard derivation, offering independent support for its existence.
Thermal density functional theory calculations often use the Mermin-Kohn-Sham scheme, but employ ground-state approximations to the exchange-correlation (XC) free energy. In the simplest solvable nontrivial model, an asymmetric Hubbard dimer, we calculate the exact many-body energies and the exact Mermin-Kohn-Sham functionals for this system and extract the exact XC free energy. For moderate temperatures and weak correlation, we find this approximation to be excellent. We extract various exact free-energy correlation components and the exact adiabatic connection formula.
The question of whether the exact exchange-correlation (XC) potential in density functional theory can have discontinuities as a function of the number of electrons is discussed. For a very simple model the exact XC functional is constructed analytically using Levy's constrained-search method and shown to have a discontinuity in its derivative. Exact numerical calculations for the XC potential for a simple finite one-dimensional 'semiconductor' chain are performed. An extrapolation to infinite chain length shows the occurrence of a sizable discontinuity in certain parameter ranges, but the model has a small discontinuity for 'realistic' parameters. The question of how to calculate the value of the discontinuity for a general system is addressed using many-body theory. A formula proposed recently by other researchers is shown to be incorrect.
In this work, nonadiabatic exchange-correlation (XC) potentials for time-dependent density-functional theory (TDDFT) for strongly correlated materials are derived in the limits of strong and weak correlations. After summarizing some essentials of the available dynamical mean-field theory (DMFT) XC potentials valid for these systems, we present details of the Sham–Schluter equation approach that we use to obtain, in principle, an exact XC potential from a many-body theory solution for the nonequilibrium electron self-energy. We derive the XC potentials for the one-band Hubbard model in the limits of weak and strong on-site Coulomb repulsion. To test the accuracy of the obtained potentials, we compare the TDDFT results obtained with these potentials with the corresponding nonequilibrium DMFT solution for the one-band Hubbard model and find that the agreement between the solutions is rather good. We also discuss possible directions to obtain a universal XC potential that would be appropriate for the case of intermediate interaction strengths, i.e., a nonadiabatic potential that can be used to perform TDDFT analysis of nonequilibrium phenomena, such as transport and other ultrafast properties of materials with any strength of electron correlation at any value in the applied perturbing field.