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arXiv:cond-mat/9606128 · Journal of the Physical Society of Japan

Partially Solvable Anisotropict-J Model with Long-Range Interactions

Yasuhiro Saiga, Yusuke Kato, Yoshio Kuramoto

Published Aug 15, 1996• 7 citations
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Abstract

A new anisotropic t-J model in one dimension is proposed which has long-range hopping and exchange. This t-J model is only partially solvable in contrast to known integrable models with long-range interaction. In the high-density limit the model reduces to the XXZ chain with the long-range exchange. Some exact eigenfunctions are shown to be of Jastrow-type if certain conditions for an anisotropy parameter are satisfied. The ground state as well as the excitation spectrum for various cases of the anisotropy parameter and filling are derived numerically. It is found that the Jastrow-type wave function is an excellent trial function for any value of the anisotropy parameter.

DOI
10.1143/jpsj.65.2361

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A new anisotropic t-J model in one dimension is proposed which has long-range hopping and exchange. This t-J model is only partially solvable in contrast to known integrable models with long-range interaction. In the high-density limit the model reduces to the XXZ chain with the long-range exchange. Some exact eigenfunctions are shown to be of Jastrow-type if certain conditions for an anisotropy parameter are satisfied. The ground state as well as the excitation spectrum for various cases of the anisotropy parameter and filling are derived numerically. It is found that the Jastrow-type wave function is an excellent trial function for any value of the anisotropy parameter.
The one-dimensional anisotropic t - J model was exactly solved using the Bethe ansatz . We first discuss the effect of anisotropy on the ground-state properties. We find a continuous phase transition induced by a magnetic field. The low-lying (spin and charge) excitation spectrum is obtained for various fillings. According to the anisotropy, the model has massive or massless modes for spin excitations. At half-filling, the model reduces to the spin- antiferromagnetic XXZ -model. We calculate the correct excitation continuum whose lowest mode is lower than that obtained by des Cloizeaux and Gaudin. Charge excitations are massless for any value of the anisotropy.
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.
In this work, we continue the study of the supersymmetric t-J model with 1/r2{\mathit{r}}^{2}r2 hopping and exchange without translational invariance. A set of Jastrow eigenfunctions are obtained for the system, with eigenenergies explicitly calculated. The ground state of the t-J model is included in this set of wave functions. The spectrum of this t-J model consists of equidistant energy levels which are highly degenerate.
A multicomponent anisotropic $t\ensuremath{-}J$ model with a hard-core potential is formulated and solved by the Bethe ansatz method in one dimension for an arbitrary core radius $(\ensuremath{\Delta}+1)/2.$ The ground-state Bethe equations are analyzed and solved numerically for an arbitrary band filling, several values of color components, and the coupling constants. The ground-state energy, the Fermi velocity, and the critical exponents of the correlation functions have been calculated numerically for an arbitrary density of electrons. We discuss the effect of the hard-core potential. The Fermi velocity has a maximum as a function of band filling, vanishing for the empty and full bands. In comparison to the traditional anisotropic $t\ensuremath{-}J$ model in which $\ensuremath{\Delta}=0,$ the Fermi velocity is strongly enhanced in general. In the high-density limit, there is no dynamics of charges, the model exhibits a metal-insulator phase transition, and reduces to an antiferromagnetic spin chain with a new spacing parameter $\ensuremath{\Delta}+1.$
The momentum distribution and the spin-correlation functions are studied for the one-dimensional t-J model, in which the exchange interaction J is anisotropic. The discussion is focused on the nontrivial limit that J is much smaller than t. In this limit the exact ground-state wave function is a product of the spinless-fermion Slater determinant and the spin-wave function of the s=1/2 anisotropic Heisenberg antiferromagnet. With this wave function, correlation functions are calculated explicitly for finite but large systems and the nature of their singularities is studied. It is found that for the XY-like anisotropy the critical behavior of correlation functions can be described by the Tomonaga-Luttinger model, while for the Ising-like anisotropy the model seems to belong to the universality class of the Luther-Emery model of the one-dimensional electron gas.