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arXiv:cond-mat/9706263 · Journal of Physics A Mathematical and General

On the second-neighbour correlator in 1D XXX quantum antiferromagnetic spin chain

J. Dittrich, V. I. Inozemtsev

Published Sep 21, 1997• 15 citations

Abstract

We have calculated the energy per site for the ground state of the antiferromagnetic quantum spin chain with variable range exchange in the framework of the asymptotic Bethe ansatz. By expanding it in powers of , we have confirmed the value of the second-neighbour correlator for the model with nearest-neighbour exchange obtained earlier in the atomic limit of the Hubbard chain.

DOI: 10.1088/0305-4470/30/18/003

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We have calculated the energy per site for the ground state of the antiferromagnetic quantum spin chain with variable range exchange in the framework of the asymptotic Bethe ansatz. By expanding it in powers of , we have confirmed the value of the second-neighbour correlator for the model with nearest-neighbour exchange obtained earlier in the atomic limit of the Hubbard chain.

We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.

An exact correspondence between the staggered biquadratic spin-1 antiferromagnetic chain and the quantum Hamiltonian limit of the nine-state Potts model is established for finite chains with free ends. For uniform interactions this equivalence is used (via a further exact mapping to a Bethe ansatz soluble XXZ model) to calculate exactly the infinite lattice values of the ground-state energy per site and the (nonzero) gap to the lowest-energy excited state. Periodic boundary conditions and the nature of the ground state as a function of bond alternation are also discussed.

The ground-state energy of the XY antiferromagnetic chains with spin-1/2 and spin-1 are obtained by a real space renormalization group method. For S=1/2 a large spin cluster is used in the mapping to improve the accuracy and a boundary theory is applied to extrapolate to the result which is in excellent agreement with the exact Bethe Ansatz result. For S=1 the ground-state energy of the model is obtained by using the renormalization group method and a finite chain calculation for lattices up to 16 sites. The two results are in very good agreement.

We calculate the ground-state energy and correlation length of the spin-1 antiferromagnetic biquadratic chain, as well as the excitation energy of the two lowest-lying states. We use the equivalence of this model to the spin-1/2 XXZ chain for a specific anisotropy parameter. The XXZ model is solvable by the Bethe ansatz. We solve the Bethe ansatz equations numerically and obtain the correlation length from the ground-state energy of finite chains by use of a finite-size scaling ansatz. We find a surprisingly large correlation length. We also perform quantum Monte Carlo simulations which show clearly that the biquadratic chain dimerizes.