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Journal of Physics A Mathematical and General

The critical exponent γ for the three-dimensional Ising model

David S. Gaunt, M F Sykes

Published Jan 1, 1979• 34 citations
Publisher

Abstract

Estimates for the critical exponent gamma for the initial susceptibility of the three-dimensional spin- 1 / 2 Ising model are summarised. There is a discrepancy between estimates based on high-temperature series expansions and those obtained using renormalisation group theory. High-temperature series estimates for gamma are reviewed and re-examined using some new data. It is tentatively concluded that a small discrepancy still appears to exist and that further work is needed to resolve it.

DOI
10.1088/0305-4470/12/1/007

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Journal of the Physical Society of Japan•Oct 1, 1972•1 citation
Estimates for the critical exponent gamma for the initial susceptibility of the three-dimensional spin- 1 / 2 Ising model are summarised. There is a discrepancy between estimates based on high-temperature series expansions and those obtained using renormalisation group theory. High-temperature series estimates for gamma are reviewed and re-examined using some new data. It is tentatively concluded that a small discrepancy still appears to exist and that further work is needed to resolve it.
New series expansion data for the three-dimensional Ising model are analysed. A special study of the diamond and face-centred cubic lattices is made. Below the critical temperature convergence is found to be slow. It is concluded that all available data are consistent with beta = lambda 5/16 (magnetization) and not inconsistent with the 'scaling' value gamma '=5/4 (susceptibility) and that with the data and methods available at present it is not possible to draw more precise conclusions.
The high-temperature, zero-field susceptibility series of the spin- 1 / 2 Ising model on the face centred cubic lattice is analysed assuming the asymptotic form A 1 t - gamma +A 2 t - gamma +1 +Bt - gamma + Delta 1 where t is the reduced temperature. Good convergence is obtained for gamma =1.241 and Delta 1 =0.496, the values predicted by renormalisation group theory. Attempts to fit the series coefficients with gamma =1.25 and B=0, do not prove as successful. The amplitudes A 2 /A 1 and B/A 1 are estimated.
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).
The susceptibility of the restricted-dimensional (one- or two-dimensional-like) Ising model is analyzed by means of the high temperature series expansion. The exchange interaction in a chain (or plane), and inter chains (or planes) are J and λ J , respectively. The first eight terms in the susceptibility series are explicitly obtained. The critical temperature T c and the critical exponent γ are obtained as a function of λ. It is found that the susceptibility shows restricted-dimensional behavior at high temperature ( T ≫ T c ), and shows three-dimensional behavior in the very vicinity of the critical temperature (T≳TcT{\gtrsim}T_{\text{c}}T≳Tc​) for small λ.