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Europhysics Letters (EPL)

Quasi-Particle Amplitudes in the 1- and 2-Dimensional t - J Model

Jin‐Oh Song, James F. Annett

Published Mar 15, 1992• 6 citations
Publisher

Abstract

Using the t - J model of high- T c superconductors, we calculate the exact one-hole quasi-particle energies, E ( k ), and weights, Z ( k ), for clusters of up to 20 sites in size. We find that in all 2-dimensional clusters Z is relatively large (0.2 ÷ 0.4) at the Fermi energy, E F , and is a maximum there. These results are not consistent with the marginal or Luttinger Fermi liquid theories of high- t c superconductivity. Extrapolating to the thermodynamic limit is not possible with square lattices, however, we do find Z → 0 in N × 1 chains, as expected in a Luttinger liquid, and Z → constant in N × 2 chains implying a Landau Fermi liquid.

DOI
10.1209/0295-5075/18/6/013

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Using the t - J model of high- T c superconductors, we calculate the exact one-hole quasi-particle energies, E ( k ), and weights, Z ( k ), for clusters of up to 20 sites in size. We find that in all 2-dimensional clusters Z is relatively large (0.2 ÷ 0.4) at the Fermi energy, E F , and is a maximum there. These results are not consistent with the marginal or Luttinger Fermi liquid theories of high- t c superconductivity. Extrapolating to the thermodynamic limit is not possible with square lattices, however, we do find Z → 0 in N × 1 chains, as expected in a Luttinger liquid, and Z → constant in N × 2 chains implying a Landau Fermi liquid.
Several thermodynamic quantities within the planar $t\ensuremath{-}J$ model are calculated using the T>0T>0T>0 Lanczos method on clusters of up to 26 sites. The hole density ${c}_{h}(\ensuremath{\mu},T)$ shows a non-Fermi-liquid behavior as a function of TTT and suggests a transition from small to large Fermi surfaces at ${c}_{h}\ensuremath{\sim}0.15$. The specific heat reveals a maximum at the exchange energy scale up to ch=0.2{c}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2ch​=0.2, where it becomes almost TTT independent for $T\ensuremath{\gtrsim}0.15t$. At constant TTT the entropy has maximum for ${c}_{h}\ensuremath{\sim}0.15$ with large values at low TTT, consistent with experiments on cuprates. In the underdoped regime the spin susceptibility ${\ensuremath{\chi}}_{0}(T)$ exhibits a maximum at finite T=T∗{T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}^{*}T=T∗, with T∗{T}^{*}T∗ decreasing with doping and disappearing for ch>0.15{c}_{h}>0.15ch​>0.15.
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.
The quasi-particle spectral weight at the Fermi level (Z(F)) in the two-dimensional (2D) electron gas is determined as a function of density. A model local field correction factor (G(q)) possessing the correct features both in the small and large q limits is introduced and used to calculate the Z(F), by evaluating the electron self-energy within the GW approximation. It is found that the Z(F) calculated using the present G(q) is smaller than the one calculated using the Hubbard G(q) implying that the correlation interaction makes the system more localized. Our result also indicates that high T(c) superconductors, if described by a 2D electron gas system, will exhibit a small but nonzero (approximately 0.2) quasi-particle spectral weight at the Fermi level.
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.
We present an exact diagonalization study of the self-energy of the two-dimensional Hubbard model. To increase the range of available cluster sizes we use a corrected ttt-JJJ model to compute approximate Green's functions for the Hubbard model. This allows to obtain spectra for clusters with 18 and 20 sites. The self-energy has several ``bands'' of poles with strong dispersion and extended incoherent continua with kkk-dependent intensity. We fit the self-energy by a minimal model and use this to extrapolate the cluster results to the infinite lattice. The resulting Fermi surface shows a transition from hole pockets in the underdoped regime to a large Fermi surface in the overdoped regime. We demonstrate that hole pockets can be completely consistent with the Luttinger theorem. Introduction of next-nearest-neighbor hopping changes the self-energy strongly and the spectral function with nonvanishing next-nearest-neighbor hopping in the underdoped region is in good agreement with angle-resolved photoelectron spectroscopy in the cuprates.