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arXiv:cond-mat/0112428 · Physical Review Letters

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

Neemias Alves de Lima, M. F. Silva, L. Nunes de Oliveira, Klaus Capelle

Published Apr 10, 2003• 168 citations
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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 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.

DOI
10.1103/physrevlett.90.146402

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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.
We study the Mott insulating phase of the one-dimensional Hubbard model using a local-density approximation (LDA) that is based on the Bethe Ansatz (BA).Unlike conventional functionals, the BA-LDA has an explicit derivative discontinuity.We demonstrate that, as a consequence of this discontinuity, the BA-LDA yields the correct Mott gap, independently of the strength of the correlations.A convenient analytical formula for the Mott gap in the thermodynamic limit is also derived.We find that in one-dimensional quantum systems the contribution of the discontinuity to the full gap is more important than that of the band-structure gap, and discuss some consequences this finding has for electronic-structure calculations.
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.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
We calculate the correlation functions and the dc conductivity of Luttinger liquid superlattices, modelled by a repeated pattern of interacting and free Luttinger liquids. In a specific realization, where the interacting subsystem is a Hubbard chain, the system exhibits a rich phase diagram with four different phases: two metals and two compressible insulators. In general, we find that the effective low-energy description amalgamates features of both types of liquids in proportion to their spatial extent, suggesting the interesting possibility of `engineered' Luttinger liquids. In recent years, new experimental techniques have made it possible to grow nanostructures which are topologically one-dimensional, such as quantum wires and Carbon nanotubes. However, care must be taken when invoking existing models to discuss their electronic properties, since inhomogeneities must be taken into account in a fundamental way. Consider, for instance, the Luttinger liquid (LL), which is the standard model for low-energy phenomena involving interacting electrons in one dimension [1]. The absence of conductance renormalization in long high-mobility GaAs wires [2] has been explained in terms of a usual LL (representing the wire) in contact with a non-interacting LL at each of its ends (representing the Fermi liquid leads); that is, overall, the system can be thought of as an inhomogeneous Luttinger Liquid (ILL). LLs with different inhomogeneity profiles have also been used in the context of the fractional quantum Hall effect (FQHE), to describe transitions between edge states at different fillings [3, 4], or between an edge state connected to a Fermi liquid [5]. A different class of inhomogeneous systems is represented by superlattices (SLs) and multilayers. By varying the relative th...
We study a model of a quantum Hall system with the electrons confined to a narrow, linear channel. The system is mapped to a one-dimensional (1D) system which in the low-energy approximation has the form of a Luttinger liquid with different interactions between particles of equal and of opposite chiralities. In a previous paper (Part I) [M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195321 (2007)], we studied this mapping at the microscopic level, and examined how the interaction in the two-dimensional (2D) system renormalizes 1D Luttinger liquid parameters. We follow up this study here and derive the low-energy form of the 2D electron correlation function by applying the mapping from the 1D correlation function. The derived expression has a natural separation in a bulk part, which is not modified by interactions, and an edge part, which is modified. We study the effect of interactions on the electron density and on the asymptotic form of the correlation function. The asymptotic parameter is found to be consistent with the expected value from the Luttinger liquid theory, but deviations from the asymptotic form are found in a rather wide intermediate interval. We suggest some questions of interest for further study.