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Advanced series in mathematical physics

Exactly Soluble Supersymmetric t-J-Type Model with Long-Range Exchange and Transfer

Yoshio Kuramoto, H. Yokoyama

Published May 1, 1994• 1 citation
Publisher

Abstract

The Gutzwiller wave function is shown to be the exact solution of a supersymmetric t-J-type model. The model realizes a Fermi-liquid state in one dimension with a discontinuity in the momentum distribution. Analytic results are obtained for spin and charge susceptibilities, and the specific-heat coefficient with the help of the Luttinger-liquid theory. In the high-density limit the model exhibits a Mott-Hubbard gap and reduces to an antiferromagnetic spin chain with long-range exchange solved by Haldane and Shastry.

DOI
10.1142/9789812798268_0042

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International Journal of Modern Physics B•Feb 20, 1997•143 citations
The Gutzwiller wave function is shown to be the exact solution of a supersymmetric t-J-type model. The model realizes a Fermi-liquid state in one dimension with a discontinuity in the momentum distribution. Analytic results are obtained for spin and charge susceptibilities, and the specific-heat coefficient with the help of the Luttinger-liquid theory. In the high-density limit the model exhibits a Mott-Hubbard gap and reduces to an antiferromagnetic spin chain with long-range exchange solved by Haldane and Shastry.
The Gutzwiller wave function is shown to be the exact solution of a supersymmetric t-J-type model. The model realizes a Fermi-liquid state in one dimension with a discontinuity in the momentum distribution. Analytic results are obtained for spin and charge susceptibilities, and the specific-heat coefficient with the help of the Luttinger-liquid theory. In the high-density limit the model exhibits a Mott-Hubbard gap and reduces to an antiferromagnetic spin chain with long-range exchange solved by Haldane and Shastry.
A set of generalized Gutzwiller-type wave functions with spin and charge currents and/or spin polarization are shown to be exact eigenstates of the supersymmetric t - J -type model with long-range interactions. The increment of energy in the presence of currents is proportional to the square of the currents without any higher-order corrections for arbitrary size of the system. The exact excited states are identified by the absence of statistical fluctuations in the variational Monte Carlo calculation. The spin susceptibility is found to be independent of the size, as in the case of the charge susceptibility.
The density matrix---i.e., the Fourier transform of the momentum distribution---is obtained analytically in closed form for the Gutzwiller wave function with exclusion of double occupancy per site. The density matrix for the majority spin is obtained for all magnetizations including the singlet case. Since the wave function gives the ground state of the supersymmetric $t\text{\ensuremath{-}}J$ model with the 1∕r21∕{r}^{2}1∕r2 exchange and transfer, the result gives the exact density matrix of the model at zero temperature. From the oscillating behavior of the density matrix, the discontinuity of the momentum distribution at the Fermi momentum kF{k}_{F}kF​ is identified. The form of the weaker singularity at 3kF3{k}_{F}3kF​ is also obtained; there is a discontinuity in the second derivative of the momentum distribution, whose magnitude is calculated analytically. The momentum distribution over the whole Brillouin zone is obtained numerically from the analytic solution of the density matrix. The result is in excellent agreement with previous results derived by different methods.
Luttinger liquids are characterized by the critical exponent Θ of the momentum distribution around the Fermi momentum k F . Typically Θ⩽1/2, signalling a singularity characterizing a residual Fermi surface. Results of photoemission experiments can be interpreted in terms strongly interacting Luttinger liquids with Θ⩾1 with the residual Fermi surface disappearing. We construct integrable models with such behaviour - models given by the SU(ν) t - J interaction with a hard-core repulsive potential between electrons at distances less than or equal to Δ. The models exhibit both weakly and strongly interacting Luttinger behaviours with Θ varying continuously in the range 0⩽Θ⩽½(1 + Δ-1/[ν(1 + Δ)]) 2 depending on the electron density. In the extreme high-density limit the model exhibits a Mott-Hubbard gap and reduces to an isotropic Heisenberg chain with a new spacing parameter Δ + 1.
The one-dimensional (1D) t-J model is investigated by using a Gutzwiller-Jastrow-type variation method and the exact diagonalization of small systems. Variational expectation values are estimated by the variational Monte Carlo method with sufficient accuracy. First, we give detailed descriptions of the preceding paper [Phys. Rev. Lett. 67, 3610 (1991)], where we discussed the properties of the Fermi-liquid-type Jastrow wave function as well as the Gutzwiller wave function. Secondly, these wave functions are compared with the Tomonaga-Luttinger-liquid-type wave function proposed by Hellberg and Mele. It is found that the correlation factors in short distances control bulk quantities like energy and the magnitude of the correlation functions, while the long-range part of the correlation factors determines the critical behavior of correlation functions. Finally, using these functions, charge, spin susceptibilities, and magnetization curve are estimated, which agree with the exact results. It is shown that the Mott transition in the 1D t-J model is quite different from the Brinkman-Rice transition. \textcopyright{} 1996 The American Physical Society.
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.