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Physical review. B, Solid state

Susceptibility amplitudes for the two-dimensional Ising model

Anthony John Guttmann

Published Jun 1, 1974• 14 citations
Publisher

Abstract

The recent critical amplitude calculation of Barouch, McCoy, and Wu for the susceptibility of the Ising model on a square lattice is extended to yield amplitudes for the triangular, honeycomb, and kagom\'e lattices. The extension assumes the validity of the generalized law of corresponding states---as first postulated by Betts, Guttmann, and Joyce---for the two-dimensional Ising model.

DOI
10.1103/physrevb.9.4991

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Journal of physics. A, Proceedings of the Physical Society. General•May 1, 1972•90 citations
The recent critical amplitude calculation of Barouch, McCoy, and Wu for the susceptibility of the Ising model on a square lattice is extended to yield amplitudes for the triangular, honeycomb, and kagom'e lattices. The extension assumes the validity of the generalized law of corresponding states---as first postulated by Betts, Guttmann, and Joyce---for the two-dimensional Ising model.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.
For the study of Ising models of general spin S on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann, and Jensen. We have computed various critical parameters and improved the estimates of others. Moreover, the properties of hyperscaling and of universality (spin-S independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin-S Ising model on the triangular, honeycomb, and kagom\'e lattices.
An extension of the critical-phenomena universality hypothesis assumed previously to be valid for less singular terms is shown to be invalid. A reformulated extension of the universality hypothesis is found to be valid exactly for the spin-1/2 Ising model on the triangular, square, and honeycomb lattices, and, further support is supplied by high- and low-temperature expansions for the susceptibility. The extension fails however for the Ising model on the kagome\r{} lattice and for the spherical model. At present there are insufficient data to test the extended principle for other models or for experimental systems.
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.
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 triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.