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Journal of Physics A Mathematical and General

The high-temperature susceptibility and spin-spin correlation function of the three-dimensional Ising model

Anthony John Guttmann

Published May 11, 1987• 39 citations

Abstract

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.

DOI: 10.1088/0305-4470/20/7/030

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Journal of Physics A Mathematical and General•Dec 1, 1976•5 citations

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.

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.

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.

For pt. I see ibid., vol. 5, no. 5, 624. Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K for the simple cubic lattice to v 17 , the body-centred cubic lattice to v 15 , and the face-centred cubic lattice to v 12 , inclusive. The coefficients are analysed by the ratio method and it is estimated that, subject to the implicit assumptions thereof, the critical temperatures correspond to v f =0.21813+or-0.00001 (SC), v f =0.15612+or-0.00003 (BCC) and v f =0.10174+or-0.00001 (FCC). The symptotic behaviour of the coefficients is studied in detail; for the loose-packed lattices the decay of the even-odd oscillation in the ratios is found to be consistent with the assumption that the critical index of the high temperature specific heat is very close to 1/8.

Nickel has generated 21 term series for the susceptibility and the correlation length on the B.C.C. lattice, for various values of the spin. We analyse them here using a variant of the ratio method. Our results fully support Nickel's conclusions i.e. that is no apparent disagreement any longer for the exponents γ and ν between the renormalization group and high temperature series estimates. We believe that these results shed serious doubts about any claim, based on shorter series, of evidence for hyperscaling violations.

For pt.II see ibid., vol.9, p.479 (1976). An earlier study of the conventional nearest-neighbour Ising model on the tetrahedron lattice is extended to include four-spin interactions about each tetrahedron. The problem of deriving high-temperature susceptibility series can be converted into a form for which previous graphical techniques can be readily applied. Analysis of the series by Pade approximants and the ratio method suggests that the exponent gamma has the constant value of 1.25 independent of the strength of the four-spin coupling, in agreement with universality.