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Journal of Physics A Mathematical and General

The Ising model on the tetrahedron lattice. III. Four-spin interactions

J Ho-Ting-Hun, J. Oitmaa

Published Dec 1, 1976• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/9/12/016

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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.
For pt.I see ibid., vol.8, p.1920 of 1975. The authors have carried out a further analysis of the recently derived 16 term high temperature susceptibility series for the spin 1/2 Ising model on the tetrahedron lattice (crystobalite or B-site spinel lattice). They have found convincing evidence for the asymptotic form chi =C 0 (1-K/K c -5 4/+C 1 (1-K/K c -1 4/+X 0 +C 2 (1-K/K c ) 3 4/+... and have obtained estimates for the amplitudes C 0 , C 1 , C 2 and X 0 . They have also used recent lattice-lattice scaling hypotheses to calculate the specific heat and magnetization amplitudes.
The authors have calculated the first 16 terms of the series expansion for the high-temperature susceptibility of the Ising model for a lattice made up of the B sites of the crystobalite or spinel structures (the 'tetrahedron' lattice). By means of a transformation technique which allows the enumeration of graphs to be carried out on the simpler diamond lattice six new terms have been added to previously published results. They estimate that the critical temperature is given by nu c =0.23300+or-0.00001 and that the exponent gamma is gamma =1.250+or-0.001, a result which gives further confirmation of the universality hypothesis.
For Pt.1 see ibid., vol.8, no.8, p.1236 of 1975. Amplitudes for a number of three-dimensional Ising model series have been estimated from recently extended series expansions. The amplitudes have then been used to re-estimate the field and temperature scaling parameters that are fundamental to the Generalized Law of Corresponding States. These new scaling parameters have been used (i) to estimate the amplitudes of the next most singular term in the high-temperature susceptibility for the four common three-dimensional lattices, and (ii) to show that an earlier conjecture, that the field scaling parameter n x is the same for the Ising and spherical models, is unlikely to hold.