Density Functionals Not Based on the Electron Gas: Local-Density Approximation for a Luttinger Liquid

Abstract

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...

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

“…This functional is based on the thermodynamic Bethe ansatz (TBA) solution of the uniform Hubbard model in one dimension 25 and is thus an extension of the corresponding work at zero temperature 4,8 .…”

“…Typically, these models are studied with techniques different from DFT. However, DFT can be a useful tool for the investigation of these models as well, especially when one wants to take into account the effects of non-uniform external potentials [3][4][5][6] . For example, cold atoms in optical lattices confined in an harmonic trap may very well be modelled by a lattice model with confining external potential where the particles interact through a Hubbardtype interaction 7 .…”

“…As with usual DFT, the applicability and success of lattice DFT hinges on the availability of approximations to the unknown exchange-correlation (xc) functional. Capelle and coworkers proposed a local functional for the xc energy per site based on the Betheansatz solution of the uniform 1D Hubbard model at zero temperature 4,9 . Thus, for one-dimensional lattice models, the 1D Hubbard model takes the role of the uniform electron gas in continuum formulation of DFT as an exactly solvable model system which provides the essential input for the construction of the local approximation.…”

Nonadiabatic exchange-correlation kernel for strongly correlated materials

“…In particular, Refs. [20][21][22][23][24] have proposed adiabatic (static DFT) XC potentials based on the BA solution. For studying timedependent phenomena, adiabatic TDDFT XC potentials based on the results for the BA 22 and DMFT 25 XC energies for the one-band Hubbard model have also been proposed.…”

Current density functional theory for one‐dimensional fermions

“…For recent applications of Bethe ansatz LDA, see also, for example, Refs. [6][7][8][9][10][11][12].In addition, the time-dependent version of DFT has been developed and applied [13,14], in particular, the current density version [15]; a recent review [16] and book [17] provide excellent overviews, including the relation to standard many-body Green's function approaches.In this article, we focus on the one-dimensional spinless fermion model with nearest-neighbor interaction, which is exactly solvable in the homogeneous case [19,20]. We extend our recent DFT-LDA approach [18] to current den-…”