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Journal of the Physical Society of Japan

An Application of Padé Approximants to Heisenberg Ferromagnetism and Pair-Product Model

Masuo Suzuki, Chikao Kawabata

Published Jun 1, 1966• 3 citations
Publisher

Abstract

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 (\frac{4}{3}) in three dimensions.
DOI
10.1143/jpsj.21.1063

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Journal of Physics Condensed Matter•Jan 29, 1990•1 citation
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.
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 authors reanalyse the Pade approximants for the susceptibility of different Ising and Heisenberg systems using the non-linear scaling variable t'=(T-T c )/T instead of t=(T-T c )/T c . The susceptibility data are well represented by the modified power law chi approximately (1/T)(t') - gamma up to high temperatures, whereas the conventional relation chi approximately t - gamma is valid only in a small temperature range. The trend of the deviations from the modified power law in terms of temperature, spin and lattice dimensionality looks quite sensible and physical. Altogether the domain where scaling ideas are useful appears to be much more extended than generally suspected.
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.
Series expansions for certain thermodynamic variables have been computed for various standard model lattices. The high-temperature series for the magnetic susceptibility of the Ising model of three-dimensional SC, BCC and FCC lattices allows estimates of the critical index gamma of each series to be made. Dramatic progress in this direction was made by the Pade Dlog method, and here a similar technique is introduced. Estimates of gamma from row sequences of generalised inverse vector-valued Pade approximants are made. The results point to the conclusion that gamma =1.2406+or-0.0017 is the critical index common to all three lattices.
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.