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Journal of the Physical Society of Japan

The Spin Correlations of theS=3/2XXZChain

Kenn Kubo, Kazuo Saitoh, Satoshi Takada

Published May 15, 1988• 1 citation
Publisher

Abstract

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.

DOI
10.1143/jpsj.57.1601

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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.
The S x - S x correlation function of S =1 XXZ chain is calculated at low temperatures by means of the transfer matrix with use of the Trotter formula. Critical properties of the correlation function, such as critical indices γ, η and ν and the mass identified by the inverse correlation length at T =0, are obtained. We conclude that a groundstate with an excitation gap appears for the anisotropy parameter Δ<Δ 1 ≃ -0.3 including the isotropic antiferromagnetic point (Δ=-1). We have obtained η≃1/4 at Δ=Δ 1 in accord with the theoretical prediction by Haldane and den Nijs. Our result, however, shows the monotonic increase of the excitation gap with decreasing Δ for Δ<Δ 1 contrary to Haldane's prediction.
The S x S x correlation function of S =1/2 XXZ chain is numericaly obtained at low temperatures by means of the transfer matrix with use of the Trotter approximation. The inplane susceptibility and the correlation length are extrapolated to N =∞ where N is the number of the Trotter slicings. It is shown that the extrapolation works quite effectively and makes the estimation of the critical indices ν, γ and η possible for the anisotropy parameter -0.75≤Δ≤0.5. The obtained values of ν, γ and η satisfy the scaling relation and agree quite well with the analytic expression derived by Luther and Peschel.
We present a Green's-function theory of magnetic short-range order in the S=1/2S=1/2S=1/2 easy-plane XXZ\mathrm{XXZ}XXZ chain based on the projection method for the dynamic spin susceptibility and a decoupling of three-spin operator products introducing vertex parameters. The longitudinal and transverse static susceptibilities and two-point correlation functions of arbitrary range are calculated self-consistently for all wave numbers, temperatures, and anisotropy parameters $\ensuremath{-}1<~\ensuremath{\Delta}<~1.$ In the easy-plane ferromagnetic region $(\ensuremath{\Delta}<0),$ the longitudinal correlators of spins at distance n change sign at a finite temperature ${T}_{0}(n,\ensuremath{\Delta}),$ in reasonable agreement with recent data obtained by finite-chain diagonalizations. The temperature dependence of the uniform static susceptibilities exhibits a maximum which is explained as an effect of magnetic short-range order which decreases with increasing temperature.
A crossover phenomenon of the longitudinal spin–spin correlation function is investigated on the spin-1/2 XXZ chain in a magnetic field at finite temperatures. A commensurate–incommensurate transition of the correlation function in the asymptotic behaviour has been pointed out by Klumper et al in the framework of the quantum transfer matrix method. We show the field dependence and the anisotropy-constant dependence of the critical temperature by the quantum transfer matrix method. Using a quantum Monte Carlo method with the continuous time loop algorithm, we directly observe the commensurate–incommensurate transition. Detailed quantum Monte Carlo analyses for the longitudinal spin–spin correlation in both the high and low temperature regions are presented. Especially in the low temperature region, evaluating the precise distance dependence of the correlation function, we prove that the incommensurate oscillation has detectable amplitude, which has not been estimated before. We also estimate the phase factor of the incommensurate oscillation, which accurately coincides with the one from the quantum transfer matrix method.