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Progress of Theoretical Physics

Spin-Pair Correlations for the Low-Dimensional Quantum XXZ Model

T. Kawabe

Published May 1, 1989• 2 citations
Publisher

Abstract

Nearest neighbour spin-pair correlations for the S = 1/2 Ising-like and XY-like system of a linear chain and for the XY-like system of a square lattice are obtained as a function of temperature by the second-order Green's function method. The transverse correlation and the specific heat for the XY model of a linear chain are in good agreement with rigorous results except for quite low temperatures. The longitudinal correlation for the XY model of a square lattice is given by -0.07311 in the ground state in fair agreement with -0.076 ±0.002 by the finite lattice method and with -0.0714 by the variational method. For the XY-like system of a square lattice the longitudinal nearest neighbour correlation in the ground state becomes always negative. The possibility of a transition of a certain kind is predicted near the Heisenberg limit for a square lattice but not for a linear chain.

DOI
10.1143/ptp.81.976

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•Jan 1, 2008•1 citation
Nearest neighbour spin-pair correlations for the S = 1/2 Ising-like and XY-like system of a linear chain and for the XY-like system of a square lattice are obtained as a function of temperature by the second-order Green's function method. The transverse correlation and the specific heat for the XY model of a linear chain are in good agreement with rigorous results except for quite low temperatures. The longitudinal correlation for the XY model of a square lattice is given by -0.07311 in the ground state in fair agreement with -0.076 ±0.002 by the finite lattice method and with -0.0714 by the variational method. For the XY-like system of a square lattice the longitudinal nearest neighbour correlation in the ground state becomes always negative. The possibility of a transition of a certain kind is predicted near the Heisenberg limit for a square lattice but not for a linear chain.
The transverse and longitudinal spin-pair correlations are obtained as a function of temperature by the second-order Green's function method. The longitudinal N. N. spin-pair correlation at the ground state becomes negative for anisotropy parameter η between the X Y model (η=0) and quite near the Heisenberg model (η∼1.0). It is predicted from these results that the first-order phase transition occurs near the Heisenberg limit. For η=0.9, it occurs at a temperature τ c restricted by 2.4<τ<6.3.
The calculation of two- and four-spin correlations of the [Formula: see text] ferromagnet has been extended to an N = 21 site triangular lattice. By fitting a quadratic in 1/N to the nearest neighbour transverse pair correlations, we have estimated the ground-state energy per bond on the infinite lattice to be E0/3NJ = −0.5326 ± 0.003. We conjecture that the square of the magnetization per site vanishes very sharply as N−0.06. The nearest neighbour longitudinal correlation per bond [Formula: see text] for all two-dimensional lattices.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
We study the long-range behavior of spin correlations in low-dimensional spin-1/2 antiferromagnets with both Ising anisotropy and spatial anisotropy. We represent the spin operators in terms of spin-density operators and spin-phase operators. Then for spin-1/2 antiferromagnets, within a simple harmonic approximation, we obtain expressions for the spin correlations in the ground state. For the one-dimensional linear chain, we find that the two leading terms in spin correlation are (-1)r{)}^{\mathit{r}})rc/${\mathit{r}}^{\mathrm{\ensuremath{\eta}}}$-c1{\mathit{c}}_{1}c1​/r2{\mathit{r}}^{2}r2 for large distance r. The power \ensuremath{\eta} depends on the Ising anisotropy, and \ensuremath{\eta}=0.405 at XY point and \ensuremath{\eta}=1.062 at Heisenberg point. For the two-dimensional square lattice, there is long-range staggered magnetization, and the dependence of magnetization on Ising anisotropy is given. For the crossover from one-dimensional linear chain to two-dimensional square lattice, we find that the disorder-order transition occurs as long as the interchain coupling is introduced. We believe that our method gives a unified approach to the long-range behavior of spin correlations in low-dimensional spin-1/2 antiferromagnets.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.
In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half XY model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half XY model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half XY model, and the two formalisms are then compared and contrasted. In this paper we consider the T = 0 properties of the quantum spin system known as the spin-half XY model, described by the Hamiltonian