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Journal of Physics C Solid State Physics

Spin-operator reduced relations in a new representation for the Ising ferromagnet

B. Frank

Published Sep 28, 1978• 8 citations
Publisher

Abstract

A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).

DOI
10.1088/0022-3719/11/18/010

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A stringent test is applied to a set of approximations used, in an earlier work (CCFA) see Frank et al. (ibid., vol.11, p.208 (1978) to calculate T c for the spin- 1 / 2 Ising ferromagnet. In CCFA, all cumulant averages M 2n (2n>or=6) were approximated, for all the cubic lattices, and justified a posteriori. Here, a spin operator reduction relation allows one, for the SC lattice, to approximate only M 6 , and moreover, to understand the approximation used. The extreme closeness at every stage (within 0.01%) of the present T c values to those of CCFA gives one confidence method of approximating the M 2n for the BCC and FCC lattices as well. Here also, as in CCFA, the M 60 value of T c is within 0.5% of the series approximation value of Sykes et al. (1972).
In a new theory for the spin one-half Ising ferromagnet, the fourth-order equation is used to provide expressions for the higher-order site-independent correlation functions M 2n (n=3,4,5,...) in terms of the two- and four-spin correlation functions M 2 and M 4 , at the critical point. As an example, the expression for M 6 is M 4 2 /M 2 -9M 2 M 4 -6M 2 3 . These expressions (for up to M 60 ) are applied to the calculation of T c for the SC, BCC, and FCC lattices. The results converge to, at most, 0.755, 0.799, and 0.819 respectively, which are within 2 / 3 % of the accepted results 0.752, 0.794, and 0.816 of Sykes et al. (1972).
For pt.I see ibid., vol.14, p.57 (1981). The recently developed theory of correlation reduction method near transition temperature is generalised to the Ising model with arbitrary spin. The generalisation from the case of spin 1 / 2 is natural and straightforward, although some non-trivial technicalities are involved for the detailed applications. The transition temperatures for the spin 1 and spin 3 / 2 Ising ferromagnets are obtained explicitly, and are shown to be consistent with the general trend exhibited by other calculations.
A generalised non-uniform magnetic field method is used to derive, from the linearisation assumption of a new theory for the Ising ferromagnet, a set of critical correlation-function approximations (CCFA) which express the higher-order correlations in terms of the second- and fourth-order correlations. This set is found to coincide at least for the first five relations, with an earlier CCFA set which was suggested by a term-by-term comparison of the second- and fourth-order equations of the theory, and which was justified a posteriori by the accurate results for the transition temperature to which it leads.
An infinite set of coupled integral equations is found, with the use of a spectral theorem for the correlation functions of the spin-1/2 three-dimensional Ising model. The second equation is linearised to yield the two spin correlation function. The Ward identity is then satisfied. Those approximations which give the Weiss and spherical model results are indicated. The transition temperature is calculated for SC, BCC and FCC lattices using a product average decomposition of the higher-order correlation functions. The scheme is seen to converge toward the respective values 0.738, 0.789 and 0.823 of 4/ beta c J(0) which are within 2% of the exact values 0.752, 0.794 and 0.816.