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 electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported. 71.15.Mb, 71.10.Pm, 71.10.Fd, 71.27.+a Density-functional theory (DFT) [1] is the basis of almost all of todays electronic-structure theory, and much of materials science and quantum chemistry. Many-body effects enter DFT via the exchange-correlation (xc) functional, which is commonly approximated by the localdensity approximation (LDA) [1]. The essence of the LDA is to locally approximate the xc energy of the inhomogeneous system under study by that of the homogeneous electron gas. This electron gas plays the role of a reference system, whose correlations are transfered by the LDA into the DFT description of the inhomogeneous system. The most popular improvement upon the LDA are generalized gradient approximations [2], whose basic philosophy is to abandon the requirement of homogeneity of the reference system. This system, however, is normally still the interacting electron gas [2].In the present paper we propose to explore a different paradigm for the construction of novel density functionals: instead of sticking to the electron gas as a reference system, and abandoning homogeneity, it may sometimes be advantageous to do the reverse: stick to homogeneity (and thus to the LDA) but abandon the electron gas as a reference system. The new reference system is chosen such that it accounts for the correlations present in the inhomogeneous system under study.The only requirement for the reference system is that in the absence of any spatially varying external potential its xc energy must be known exactly or to a high degree of numerical precision. Besides the electron gas (or Jellium model) there are many other physically interesting model systems that satisfy this criterium. Most notably among these is a large class of low-dimensional models which can be solved exactly by Bethe Ansatz (BA) techniques or bosonisation (in one dimension, e.g., the repulsive and the attractive Hubbard model, the hard-core Fermi and Bose gases, the Heisenberg, the supersymmetric t-J, and the Tomonaga-Luttinger model [3,4]). The solutions to these models in the homogeneous case can be used instead of the electron gas to construct LDA functionals that can then be applied to study these models also in inhomogeneous situations. The main advantage offered by a DFT treatment of such models is the gain in simplicity that arises from mapping the inhomogeneous interacting many-body system onto a noninteracting auxiliary system, which is diagonalized much more easily...
We construct and apply an exchange-correlation functional for the one-dimensional Hubbard model. This functional has built into it the Luttinger-liquid and Mott-insulator correlations, present in the Hubbard model, in the same way in which the usual ab initio local-density approximation (LDA) has built into it the Fermi-liquid correlations present in the electron gas. An accurate expression for the exchange-correlation energy of the homogeneous Hubbard model, based on the Bethe Ansatz (BA), is given and the resulting LDA functional is applied to a variety of inhomogeneous Hubbard models. These include finite-size Hubbard chains and rings, various types of impurities in the Hubbard model, spin-density waves, and Mott insulators. For small systems, for which numerically exact diagonalization is feasible, we compare the results obtained from our BA-LDA with the exact ones, finding very satisfactory agreement. In the opposite limit, large and complex systems, the BA-LDA allows to investigate systems and parameter regimes that are inaccessible by traditional methods.
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