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Journal of Physics Condensed Matter

Exact solution of the one-dimensional anisotropict-Jmodel: ground-state properties and the excitation spectrum

Ryuhei Sato

Published Oct 21, 1996• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/8/43/027

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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 one dimensional t-J Hamiltonian is diagonalized exactly for the supersymmetric case 2t=\ifmmode\pm\else\textpm\fi{}J using the Bethe ansatz. In this limit it is identical with models previously considered by Lai, Sutherland, and Schlottmann. In the present paper we discuss the ground-state properties and excitation spectrum in zero magnetic field. The ground state is a liquid of singlet bonds with varying spatial separation. Its most remarkable feature is the presence of bonds connecting particles at arbitrarily large distances. The ground-state energy is an analytic function of the band filling. There is no difference in the chemical potential for adding one or two particles and no evidence for the binding of holes. The low-lying part of the spectrum consists of two types of gapless excitations (charge and spin) with effective Fermi surfaces at 2kF{\mathit{k}}_{\mathit{F}}kF​ and kF{\mathit{k}}_{\mathit{F}}kF​, respectively. An interpretation of the energy spectrum in terms of spinons and holons is appropriate at low energies.
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 phase diagram, the Bethe-ansatz ground-state properties, including the chemical potential μ, the spin (magnetic) and charge susceptibilities, are calculated within the one-dimensional Hubbard model in entire range of interaction strength (-∞< U/t<+∞), magnetic field (h≥ 0) and all electron concentrations (0≤n≤1). The continuous and smooth variation of μ with n and h in the vicinity of n=1 points on the gapless character of charge excitations at U<0 and provides rigorous upper and lower bounds for μ. The spin (magnetic) susceptability χ at half-filling changes discontinuously as U→0 and is strongly enhanced by electron repulsion, comparing with that of the non-interactig case. The compressibility κ ch increases with n at U<0 and shows non-monotonous behavior with a dramatic increase at U>0. Variations of κ ch -1 in both repulsive and attractive cases qualitatively well reproduces corresponding behavior of charge stiffness.
[[abstract]]Cited: 0 References: 20 Citation Map Abstract: The phase diagram, the Bethe-ansatz ground-state properties, including the chemical potential mu, the spin (magnetic) and charge susceptibilities, are calculated within the one-dimensional Hubbard model in entire range of interaction strength (-infinity < U/t < + infinity), magnetic field (h greater than or equal to 0) and all electron concentrations (0 less than or equal to n less than or equal to 1). The continuous and smooth variation of mu with n and h in the vicinity of n = 1 points on the gapless character of charge excitations at U < <0 and provides rigorous upper and lower bounds for <mu>. The spin (magnetic) susceptibility chi at half-filling changes discontinuously as U --> 0 and is strongly enhanced by electron repulsion, comparing with that of the non-interacting case. The compressibility rc,h increases with n at U < 0 and shows non-monotonous behavior with a dramatic increase at U > 0. Variation of k(ch)(-1) in both repulsive and attractive cases qulitatively well reproduces corresponding behavior of charge stiffness.
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 $\ensuremath{\Delta}+1.$
Using the Bethe-Ansatz technique, we diagonalize exactly the one-dimensional t-J Hamiltonian for the supersymmetric case t=J. In this limit it is identical with models considered previously by Sutherland and by Schlottmann. The ground state is a liquid of singlet pairs of varying spatial separation for all band fillings. We find two types of gapless excitations with effective Fermi surfaces at 2kF{\mathit{k}}_{\mathit{F}}kF​ and kF{\mathit{k}}_{\mathit{F}}kF​ which we identify with the holon and the spinon excitations near half filling.