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Canadian Journal of Physics

Generalized difference equation for the three-dimensional Ising ferromagnet

B. Frank, C. Y. Cheung

Published Jan 1, 1984• 7 citations
Publisher

Abstract

The difference equation method of Frank, Cheung, and Mouritsen (FCM) is made self-consistent by relaxing an assumption of the theory which pertains to the correlation of a spin with its nearest neighbour. The full set of A = 1 and I − δ equations and U equations is brought into play, with the U2 = T2 = (1/4) (1 − 1/F(1)) double equation replaced by a single relation between U2 and T2. The resulting ferromagnetic transition temperatures for the cubic lattices are consistent with the series results to within about 2%. The correlation functions are much closer to the series results than those of FCM.

DOI
10.1139/p84-005

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The difference equation method of Frank, Cheung, and Mouritsen (FCM) is made self-consistent by relaxing an assumption of the theory which pertains to the correlation of a spin with its nearest neighbour. The full set of A = 1 and I − δ equations and U equations is brought into play, with the U2 = T2 = (1/4) (1 − 1/F(1)) double equation replaced by a single relation between U2 and T2. The resulting ferromagnetic transition temperatures for the cubic lattices are consistent with the series results to within about 2%. The correlation functions are much closer to the series results than those of FCM.
An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.
The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and zzz nearest neighbors.
New relations, the 'i- delta relations', are presented for the evaluation of higher-order critical correlation functions for the three-dimensional spin-1/2 Ising model. The critical temperatures Tc for the cubic lattices are then calculated and are found to be within 1/2% of the results of series analysis. Monte Carlo calculations are performed on the simple cubic lattice, and are found to bear out the theoretical predictions for the correlation functions to within 1.3%, for the basic Tc-determining equations to within 0.5%, and for the i- delta relations to within 2.5%. An earlier theory, which uses the critical correlation function approximation, is critically examined both theoretically and by comparison with the Monte Carlo results.
A set of coupled integral equations for the correlation functions of the 3-D Ising ferromagnet is generalised to the case of a ferromagnet with the additional feature that beside the conventional two-spin interaction terms (J 2 S i z S j z ) it contains four-spin interaction terms (J 4 S i z S j z S k z S l z ) in its Hamiltonian. The second equation is linearised yielding the two-spin correlation function which is found to be the same as the two-spin correlation function for the pure two-spin interaction ferromagnet. The Ward identity is shown to be satisfied. The approximations which reproduce the Weiss and spherical results in a generalised form are indicated. Using a generalised non-uniform magnetic field method, a relation between the transition temperature for the combined two- and four-spin interaction ferromagnet and the transition temperature for the pure two-spin interaction is derived.
The Green function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.
Previous work on difference equations for the Ising model in d dimensions suggests that a useful ansatz for the k-space pair correlation function is G(k)∼1/[1−AJ(k)/J(0)]1−η/2, where A is the quantity that appears in the criticality equation A=1 at T=Tc, the transition temperature. This ansatz is in accord with the usual definition of η. The scaling relation γ=ν(2−η) follows automatically. Using the i−δ equations, the criticality equation and the U equations together with the above ansatz, the theory predicts, for the two-dimensional Ising model on the square lattice, η=0.2249, Tc/TMF=0.5854 and U2 (critical)=0.1731. The known values are, respectively, 0.2500, 0.5673 and 0.1768.