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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

An application of Padé approximants to Heisenberg ferromagnetism and antiferromagnetism

J. Tinka Gammel, W. Marshall, L. Morgan

Published Sep 24, 1963• 84 citations
Publisher

Abstract

Abstract The method of successive Padé approximants is applied to the high-temperature series for the susceptibility of Heisenberg systems with nearest-neighbour interactions only. It is concluded that the susceptibility of a ferromagnet diverges near the critical temperature with a law (T —Tc)-r, where r is either exactly 4/3 or indistinguishable from 4/3 by the method used with the power series at present available. The susceptibility of an antiferromagnet is also discussed. The specific heat near the critical point is considered but the results are inconclusive.

DOI
10.1098/rspa.1963.0169

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The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.

Proceedings of the Physical Society•Feb 1, 1967•90 citations
Abstract The method of successive Padé approximants is applied to the high-temperature series for the susceptibility of Heisenberg systems with nearest-neighbour interactions only. It is concluded that the susceptibility of a ferromagnet diverges near the critical temperature with a law (T —Tc)-r, where r is either exactly 4/3 or indistinguishable from 4/3 by the method used with the power series at present available. The susceptibility of an antiferromagnet is also discussed. The specific heat near the critical point is considered but the results are inconclusive.
The method of Padé approximants is applied to the high-temperature series for the susceptibility of the Heisenberg and pair-product models. It is concluded that in both cases the susceptibilities diverge as χ 0 ∝|T-T c | -γ , where γ is nearly 43\frac{4}{3}34​ in three dimensions.
To avoid the difficulties with the usual type of (canonical) high-temperature series which arise when T<Tc and H=O, it is suggested that a high-temperature expansion be accomplished using the microcanonical ensembles of systems defined as involving only those states having values of the order parameter equal to the temperature-averaged one. The method is applied for the Ising and Heisenberg model utilising the molecular-field picture, and for the Heisenberg model using the spin-wave picture. In the latter case, the spin-wave energy is expanded according to the high-temperature parameter. It is shown that (m,m+1)-type Pade approximants are good at all temperatures. It is found that the longitudinal susceptibility chi L, diverges like H-1/2 at T<Tc; consequently, the critical exponent gamma ' no longer holds. A new critical exponent, gamma is defined as scaling the reduced susceptibility, chi 0=H1/2chi L. Deriving the scaling-law relating gamma with beta and delta , gamma is estimated as 0.18.
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.
Pad\'e-approximant techniques are applied to the high-temperature series for the temperature derivative of the specific heat of the Ising model. This method indicates a critical exponent $\ensuremath{\alpha}$ in the neighborhood of 18\frac{1}{8}81​ for those Pad\'e approximants which yield an estimate of the critical temperature in substantial agreement with that determined from the susceptibility series.
The temperature dependence of the paramagnetic zero-field susceptibility chi is calculated for the second-neighbour S=1/2 Heisenberg model and for the Baxter-Wu model, from Pade approximants to existing high-temperature series expansions and by Monte Carlo simulations. The data are well represented by a power law chi T=(t')-y using the critical susceptibility exponent gamma over the whole paramagnetic temperature range when a nonlinear scaling variable t' is used; namely t'=1-Tc/T for the second-neighbour models and t'=1-(Tc/T)2 for the Baxter-Wu model.
The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.