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Journal of Physics C Solid State Physics

Low-temperature thermodynamics and correlation functions for a classical Heisenberg chain with two anisotropies

Hans C. Fogedby, Per Hedegård, A. Svane

Published Jul 10, 1984• 9 citations
Publisher

Abstract

The authors derive the thermodynamics and the static correlation functions for the classical Heisenberg chain with the anisotropies within a low-temperature steepest-descent approach. As regards the thermodynamics, they find agreement with transfer matrix and 'ideal-gas' phenomenological results. For the longitudinal correlation function the authors find an antiferromagnetic-type behaviour with a correlation length inversely proportional to the domain wall density. The transverse in-plane correlation function exhibits a damped oscillatory behaviour due to domain-wall-spin interference. The transverse out-of-plane correlations, on the other hand, are exponentially damped with the correlation length increasing as the easy-axis limit is approached.

DOI
10.1088/0022-3719/17/19/021

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For pt. I, see ibid, vol.8, no.3, 347 (1975). In the first of this pair of papers exact expressions were obtained for the pair correlation functions and susceptibilities of the classical Heisenberg model in terms of ellipsoidal wavefunctions. Here the author examines the low temperature behaviour of these quantities. This necessitates an asymptotic expression for the ellipsoidal functions. The derivation of such an expression follows the method of Langer and is to leading order only. This asymptotic form allows explicit calculation of the eigenvalues of the transfer matrix (to leading order) and hence a discussion of the low temperature behaviour of the correlation functions and susceptibilities.

Journal of Physics A Mathematical and General•Mar 1, 1975•3 citations
The authors derive the thermodynamics and the static correlation functions for the classical Heisenberg chain with the anisotropies within a low-temperature steepest-descent approach. As regards the thermodynamics, they find agreement with transfer matrix and 'ideal-gas' phenomenological results. For the longitudinal correlation function the authors find an antiferromagnetic-type behaviour with a correlation length inversely proportional to the domain wall density. The transverse in-plane correlation function exhibits a damped oscillatory behaviour due to domain-wall-spin interference. The transverse out-of-plane correlations, on the other hand, are exponentially damped with the correlation length increasing as the easy-axis limit is approached.
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 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.
A theoretical study of the low-temperature dynamics of the antiferromagnetic Heisenberg chain in an external field is presented. The time- and frequency-dependent transverse and longitudinal spin fluctuations and energy fluctuations are calculated by means of a continued fraction representation. The frequency moments are evaluated analytically by a computer program. The transfer-operator technique is used to calculate all static correlation functions. For modest magnetic fields, a two-peak structure in the dynamic form factor of the transverse spin fluctuations is observed. Apart from the usual spin-wave excitations, the longitudinal-spin-fluctuation spectra show a broad second resonance for some temperatures and fields. The energy fluctuations display a dominant oscillatory time dependence.
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.
For pt. I, see ibid, vol.8, no.3, 347 (1975). In the first of this pair of papers exact expressions were obtained for the pair correlation functions and susceptibilities of the classical Heisenberg model in terms of ellipsoidal wavefunctions. Here the author examines the low temperature behaviour of these quantities. This necessitates an asymptotic expression for the ellipsoidal functions. The derivation of such an expression follows the method of Langer and is to leading order only. This asymptotic form allows explicit calculation of the eigenvalues of the transfer matrix (to leading order) and hence a discussion of the low temperature behaviour of the correlation functions and susceptibilities.