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Physical review. B, Condensed matter

Correlation length and free energy of theS=1/2XXZchain in a magnetic field

Minoru Takahashi

Published Dec 1, 1991• 46 citations
Publisher

Abstract

The correlation length and the free energy of the S=1/2 XXZ chain in a magnetic field in the z direction are calculated with use of a set of equations for a series of infinite numbers. These are derived by the Bethe-ansatz method for the quantum transfer matrix. The equations for free energy should be equivalent to the thermodynamic Bethe-ansatz nonlinear integral equations. A numerical method that calculates precisely the free energy and the correlation length is given.

DOI
10.1103/physrevb.44.12382

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The correlation length and the free energy of the S=1/2 XXZ chain in a magnetic field in the z direction are calculated with use of a set of equations for a series of infinite numbers. These are derived by the Bethe-ansatz method for the quantum transfer matrix. The equations for free energy should be equivalent to the thermodynamic Bethe-ansatz nonlinear integral equations. A numerical method that calculates precisely the free energy and the correlation length is given.
A set of equations that allows one to calculate the correlation length and the free energy of the S=1/2 XYZ chain at a given temperature is obtained. It contains infinite unknown numbers and is derived by the largest and the second-largest eigenvalues of the quantum transfer matrix in the limit of infinite Trotter number. The numerical solution of this set of equations gives very accurate values of the free energy and the correlation length at arbitrary temperature. The energy gaps that appear in the correlation length and the free energy at low temperature are discussed.
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.
Using the quantum transfer-matrix method we have calculated the spin correlation functions of the S =3/2 XXZ chain and examined the temperature dependence of the mass (≡ the inverse of the correlation length), susceptibility χ x ( Q ) and the magnetic structure factor S z ( Q ). The result shows that the mass vanishes at T =0 for the planar and isotropic cases (-1≦ Δ ≦1) in agreement with the theoretical prediction. At a low temperature the peak of S z ( Q ) at Q =π has been observed to persist even in the ferromagnetic region (0< Δ <0.8). The susceptibility χ z ( Q =0) at T =0 has been obtained with use of the extrapolation. The critical indices have been also examined.