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Physical review. B, Condensed matter

Spin correlations in theS=1XXZchain

Kenn Kubo

Published Jul 1, 1992• 27 citations
Publisher

Abstract

The spin correlation functions as well as the string correlation functions introduced by den Nijs and Rommelse in the S=1 XXZ chain have been calculated by using the quantum transfer-matrix method. The temperatuare variation of the correlation lengths and the susceptibilities has been examined and found to differ according to the phase to which the ground state belongs. The excitation gap is estimated from the temperature dependence of these quantities.

DOI
10.1103/physrevb.46.866

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The spin correlation functions as well as the string correlation functions introduced by den Nijs and Rommelse in the S=1 XXZ chain have been calculated by using the quantum transfer-matrix method. The temperatuare variation of the correlation lengths and the susceptibilities has been examined and found to differ according to the phase to which the ground state belongs. The excitation gap is estimated from the temperature dependence of these quantities.
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.
We study the string correlation functions proposed by den Nijs and Rommelse for the S=1 XXZ Hamiltonian by an exact diagonalization method. Using a finite size analysis, we estimate values of the string order parameters in the thermodynamic limit. Using the extrapolated values of the string order parameter we investigate the phase boundary between the two expected phases of the model, namely, the massive Haldane phase and the x-y--like massless phase. In the massless phase the exponent of the string correlation functions on the periodic system seems to be given by dimensions, which normally occur in the operator content of the chain when antiperiodic boundary condition is imposed.
Thermodynamic properties of the quantum Heisenberg spin chains with S=1/2,S=1/2,S=1/2, 1, and 3/23/23/2 are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.
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.