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Journal of Physics A Mathematical Nuclear and General

Estimation of critical indices for the three-dimensional Ising model

David S. Gaunt, M F Sykes

Published Oct 1, 1973• 49 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/6/10/010

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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.
We demonstrate the applicability of the εεε-convergence algorithm in extracting the critical temperatures and critical exponents of three-dimensional Ising models. We analyze the low temperature magnetization as well as high temperature susceptibility series of simple cubic, body-centered cubic, face-centered cubic and diamond lattices, using two different variables for the inverse critical temperature. In the case of simple cubic lattices, the magnetization series was modified to deduce accurate values of the critical temperatures. The alternate variable for dimensionless inverse temperature suggested by Guttmann and Thompson has also been employed for the estimation of the critical parameters.
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.
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 series analysis protocol developed in a preceding paper (ibid., vol.20, p.1839, 1987) is applied to the susceptibility and correlation function of the three-dimensional Ising model. For the spin-1/2 Ising data, the author argues that the protocol needs modification to take account of trends in the data, which trends arise from the nature of the series. In this way the author obtains gamma =1.239+or-0.003 for all cubic lattices and for a variety of spin values. The diamond lattice data lies outside this range, and it is argued that this series is effectively far shorter than the others. The author also estimates nu =0.632 +0.002 -0.003 for the BCC lattice and obtains critical point estimates for all lattice and spin models studied.