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Journal of Physics C Solid State Physics

Exponents far from Tc-a reanalysis of the high-temperature susceptibility of some model systems

M. Fähnle, Jean Souletie

Published Jun 30, 1984• 45 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/17/18/005

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Journal de physique•Jan 1, 1979•86 citations
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.
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.
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 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.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.
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.
High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).