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arXiv:0802.2490 · Physical Review B

Successes and failures of Bethe ansatz density functional theory

Stefan Schenk, Michael Dzierzawa, Peter F. H. Schwab, Ulrich Eckern

Published Oct 2, 2008• 28 citations
arXivHTMLPDF

Abstract

The response of a one-dimensional fermion system is investigated using density functional theory (DFT) within the local-density approximation (LDA) and compared with exact results. It is shown that DFT-LDA reproduces surprisingly well some of the characteristic features of the Luttinger liquid, for example, the dispersion of the collective charge excitations. On the other hand, the approximation fails even qualitatively for details of the dynamic response and for quantities for which backscattering is important, i.e., those quantities which are crucial for an accurate description of transport. In particular, the Drude weight in the presence of a single impurity is discussed.

DOI
10.1103/physrevb.78.165102

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The response of a one-dimensional fermion system is investigated using density functional theory (DFT) within the local-density approximation (LDA) and compared with exact results. It is shown that DFT-LDA reproduces surprisingly well some of the characteristic features of the Luttinger liquid, for example, the dispersion of the collective charge excitations. On the other hand, the approximation fails even qualitatively for details of the dynamic response and for quantities for which backscattering is important, i.e., those quantities which are crucial for an accurate description of transport. In particular, the Drude weight in the presence of a single impurity is discussed.
Abstract The frequency‐dependent response of a one‐dimensional fermion system is investigated using Current Density Functional Theory (CDFT) within the local approximation (LDA). DFT‐LDA, and in particular CDFT‐LDA, reproduces very well the dispersion of the collective excitations. Unsurprisingly, however, the approximation fails for details of the dynamic response for large wavevectors. In particular, we introduce CDFT for the one‐dimensional spinless fermion model with nearest‐neighbor interaction, and use CDFT‐LDA plus exact (Bethe ansatz) results for the groundstate energy as function of particle density and boundary phase to determine the linear response. The successes and failures of this approach are discussed in detail (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We present a theoretical study on the response properties to an external electric field of strongly correlated one-dimensional metals. Our investigation is based on the recently developed Bethe ansatz local-density approximation (BALDA) to the density functional theory formulation of the Hubbard model. This is capable of describing both Luttinger liquid and Mott-insulator correlations. The BALDA calculated values for the static linear polarizability are compared with those obtained by numerically accurate methods, such as exact (Lanczos) diagonalization and the density-matrix renormalization group, over a broad range of parameters. In general BALDA linear polarizabilities are in good agreement with the exact results. The response of the exact exchange and correlation potential is found to point in the same direction of the perturbing potential. This is well reproduced by the BALDA approach, although the fine details depend on the specific parameterization for the local approximation. Finally we provide a numerical proof for the nonlocality of the exact exchange and correlation functional.
A density-functional theory (DFT) of lattice fermion models is presented, which uses the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ as basic variable. A simple, explicit approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ of the Hubbard model is derived from exact dimer results, scaling properties of $W[\ensuremath{\gamma}]$ and known limits. Systematic tests on the one-dimensional chain show a remarkable agreement with the Bethe-ansatz exact solution for all interaction regimes and band fillings. New results are obtained for the ground-state energy and charge-excitation gap in two dimensions. A successful description of strong electron correlations within DFT is achieved.
Most of the basic ideas of the density-functional theory (DFT) are shown to be unrelated to the fact that the particle density is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, we discuss various approaches to density-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-density approximation (LDA) are compared to the exact results for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approximation is needed to formulate the LDA.
By shifting the reference system for the local-density approximation (LDA) from the electron gas to other model systems, one obtains a new class of density functionals, which by design account for the correlations present in the chosen reference system. This strategy is illustrated by constructing an explicit LDA for the one-dimensional Hubbard model. While the traditional ab initio LDA is based on a Fermi liquid (the three-dimensional interacting electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported.
The Bethe ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compare favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates the value of CLDFT in describing the transport properties of one-dimensional correlated electron systems. As its computational overheads are rather modest, we propose this method as a tool for studying problems where both disorder and interaction are present.