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Physical review. B, Condensed matter

Asymptotic correlation functions in the one-dimensional Hubbard model with applications to high- Tc superconductivity

Yong Ren, P. W. Anderson

Published Dec 1, 1993• 46 citations
Publisher

Abstract

The one-dimensional Hubbard model with repulsive interaction is studied. We calculate the asymptotic correlation functions in the less-than-half-filled band case. In particular, our approach centers on the large-U wave function and emphasizes the finite phase shifts on the Fermi surface. This is important since the concept of finite phase shifts can be easily applied to higher dimensions. We then discuss the non-Fermi-liquid behavior of this model and the relevance to the understanding of the two-dimensional Hubbard model, which is believed to give a good description of the high-${\mathit{T}}_{\mathit{c}}$ superconductors.

DOI
10.1103/physrevb.48.16662

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The one-dimensional Hubbard model with repulsive interaction is studied. We calculate the asymptotic correlation functions in the less-than-half-filled band case. In particular, our approach centers on the large-U wave function and emphasizes the finite phase shifts on the Fermi surface. This is important since the concept of finite phase shifts can be easily applied to higher dimensions. We then discuss the non-Fermi-liquid behavior of this model and the relevance to the understanding of the two-dimensional Hubbard model, which is believed to give a good description of the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.
An exact description is given of the long-distance behaviour of the one-dimensional t-J model at t=J. The authors employ the Bathe ansatz method and the finite-size scaling technique in conformal field theory. The charge and spin degrees of freedom are separated, and described by two independent c=1 conformal theories. The critical exponents for the charge, spin, electron and superconducting correlation functions are obtained for arbitrary band filling. The authors then make detailed comparison of the t-J model with the repulsive Hubbard model with emphasis on the Luttinger liquid properties. Analysing the electron filling dependence they observe the enhancement of the superconducting correlations compared with the highly correlated Hubbard model. The effect of the external magnetic field at and near half-filling is also discussed.
An exact description is given of the long-distance behaviour of the one-dimensional t-J model at t=J. The authors employ the Bathe ansatz method and the finite-size scaling technique in conformal field theory. The charge and spin degrees of freedom are separated, and described by two independent c=1 conformal theories. The critical exponents for the charge, spin, electron and superconducting correlation functions are obtained for arbitrary band filling. The authors then make detailed comparison of the t-J model with the repulsive Hubbard model with emphasis on the Luttinger liquid properties. Analysing the electron filling dependence they observe the enhancement of the superconducting correlations compared with the highly correlated Hubbard model. The effect of the external magnetic field at and near half-filling is also discussed.
The two-dimensional Hubbard model near half-filling is studied using a configuration-interaction method. Superconducting order-parameter and spin-spin correlation functions are calculated as functions of distance to probe for superconductivity and antiferromagnetism. Consistent with our previous results, these more extensive calculations point toward d-wave superconductivity in the ground state of the two-dimensional repulsive Hubbard model. The d-wave correlations tend to increase with interaction strength, size of the state space used in the calculation, and doping away from half-filling. The superconducting correlations on the Fermi surface are interpreted as d-wave pairing between spin density-wave quasiparticles.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
We study the phase diagram of a half-filled one-dimensional superlattice Hubbard model with an alternating orbital energy ± ϵ/2 and on-site Coulomb repulsion U(A) and U(B) using the density-matrix renormalization group and finite-size scaling analysis. For U(A) >> U(B), we observe an incommensurate charge correlation phase. For U(B) = 0, we find a transition from the band insulating phase to the correlated metallic phase at the critical point ϵ = U(A)/2. For U(B) > 0, we find three insulating phases as in the ionic Hubbard model. In the gapless region, the correlation functions exhibit the asymptotic behavior of power-law decay as in the Tomonaga-Luttinger liquid.