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Journal of Physics C Solid State Physics

Static correlations in one-dimensional classical easy-plane magnets

Marta Z. Cieplak, M. Gutowski, Alf Sjölander

Published Sep 20, 1980• 6 citations
Publisher

Abstract

The static two- and four-spin correlations for the one-dimensional easy-plane Heisenberg spin system with a ferromagnetic coupling, like CsNiF 3 , are investigated in the classical limit by means of the transfer matrix method. They are found to consist essentially of two components which both depend exponentially on the distance between sites. Both in the high- and low-temperature regions, only one component remains and this component still dominates at intermediate temperatures. Analytic forms for the correlation functions are given for asymptotically low temperatures. These are obtained by calculating the second-frequency moment of the dynamical structure factor in three different ways, and by requiring the T=0 limit of the moment to coincide with the square of the spin wave frequency. Generalisations of the Tomita-Mashiyama recursion relations (1972) for the isotropic case are given, valid for low temperatures, Monte Carlo calculations of the correlation functions have also been performed and some results of these are presented as well.

DOI
10.1088/0022-3719/13/26/020

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Brazilian Journal of Physics•Dec 1, 1998•0 citations
The static two- and four-spin correlations for the one-dimensional easy-plane Heisenberg spin system with a ferromagnetic coupling, like CsNiF 3 , are investigated in the classical limit by means of the transfer matrix method. They are found to consist essentially of two components which both depend exponentially on the distance between sites. Both in the high- and low-temperature regions, only one component remains and this component still dominates at intermediate temperatures. Analytic forms for the correlation functions are given for asymptotically low temperatures. These are obtained by calculating the second-frequency moment of the dynamical structure factor in three different ways, and by requiring the T=0 limit of the moment to coincide with the square of the spin wave frequency. Generalisations of the Tomita-Mashiyama recursion relations (1972) for the isotropic case are given, valid for low temperatures, Monte Carlo calculations of the correlation functions have also been performed and some results of these are presented as well.
The static two- and four-spin correlations for one-dimensional easy-axis Heisenberg systems are investigated in the classical limit by means of the transfer matrix method. Both in the high- and low-temperature regions they are found to depend exponentially on the distance between sites. Thus generalisations of the Tomita-Mashiyama recursion relations for the isotropic case can be given. Analytic forms for several types of the correlation functions are obtained for asymptotically low temperatures. These are derived by discussing properties of the second frequency moment of the dynamical structure factor. The dynamical structure factor for the axial components, unlike in the easy-plane systems, should be of non-vanishing width at T=0.
The exact analytical forms of the low-temperature thermodynamic quantities and correlation functions are obtained for classical one-dimensional Heisenberg ferromagnets with single-site anisotropy including both the systems with an easy axis and with an easy plane. The problem has been managed on the basis of the functional integral method. The calculated results have a few leading terms in series-expansion with respect to the reduced temperature, exhibiting the characteristic points borne out in the recent numerical studies. In particular, the longitudinal correlation function and susceptibility of the system with an easy axis show Ising-like behaviours and the transverse counterparts of the system with an easy plane show isotropic XY- or Heisenberg-like behaviours. Physical implications of the calculated results are also discussed.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
In this paper we pay attention to the classical one-dimensional planar spin system and, in particular, to the dynamics of such a model. We use the Monte Carlo method to calculate the static correlation functions, needed to determine the relaxation functions completely. We are then able to give the results for the in- and out-of-plane component of the relaxation function. At low temperatures the excitations of the out-of-plane component behave as free particles while the excitations of the in-plane component are very similar to the excitations in the Heisenberg chain.
Ground-state and thermodynamic properties of the spin-1/2 two-dimensional easy-plane XXZ model are investigated by both a Green's-function approach and by Lanczos diagonalization on lattices with up to 36 sites. We calculate the spatial and temperature dependences of various spin correlation functions, as well as the wave-vector dependence of the spin susceptibility for all anisotropy parameters delta. In the easy plane ferromagnetic region (-1 < delta < 0), the longitudinal correlators of spins at distance r change sign at a nite temperature T0 (delta, r). This transition, observed in the 2D case for the first time, can be interpreted as a quantum to classical crossover.
The spin dynamics of two dimensional XY ferromagnet are reexamined at low temperatures in the framework of the Mori continued fraction formalism using the Gaussian approximation. In this formalism, the terms on denominators for the Laplace transform of the relaxation function R(t) are related to the frequency moments of the relaxation shape function R(q,w). In the Gaussian approximation scheme, we truncate the continued fraction for R(t) on the second stage. Adopting this approximation, we calculated up to the sixth moment. The moments are writing in terms the static spin correlation functions. In the estimation of the fourth and sixth moment at finite temperature, the four and six-spin correlation functions may be approximated by a sum of products of appropriate pair correlation functions(mode-mode decoupling). In this work we calculated the static spin correlation functions for the expressions to the fourth and sixth moments, needed in the study of the dynamics.