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Physica C Superconductivity

Unambiguous relationship between the Hubbard, t–J and d–p models in one-dimension based on the Luttinger liquid theory

Kazuhiro Sano, Yoshiaki Ōno

Published May 1, 2003• 0 citations
Publisher
DOI
10.1016/s0921-4534(02)02662-x

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We examine effective interactions of the 1D d-p model mapped to a typical one-dimensional(1D) model such as the Hubbard model or the t-J model by the numerical diagonalization method combined with the Luttinger liquid theory. Analysis of the effective coupling constants of the mapped model gives the relationship between these models without ambiguity. We find that the renormalized interaction is relatively weak, though the repulsion of the d-p model at Cu-sites is very strong. We also find that the d-p model corresponds to not the t-J model but the Hubbard model for wide range of the parameters. It suggests that the Hubbard model is more favorable than the t-J model as a effective model of the d-p model in the low energy limit.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.