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Journal of Physics C Solid State Physics

Transition temperature for the three-dimensional Ising ferromagnet with two- and four-spin interaction

O Mitran

Published Feb 14, 1979• 20 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/12/3/022

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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.
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.
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.
The rational values beta = 5 / 16 , delta =5, alpha = 1 / 8 , nu = 5 / 8 , and eta =0 of the critical exponents for the three-dimensional spin 1 / 2 Ising ferromagnet are shown to be consistent with a recently-derived linearised equation for the Ising two-spin correlation function. This equation arises from an exact integral representation which previously, on application of a straightforward approximation for higher-order cumulant averages, was shown to yield transition temperatures T c to within 2% of their correct values for the cubic lattices. In the search for beta , the requirement is imposed that the correction terms for the magnetisation and for M 2 , a quantity closely related to the system internal energy, be analytic in some power of T c -T (which turns out to be 1 / 8 ) when the external magnetic field h is zero.
Two improvements to a previous calculation of the transition temperature for the three-dimensional Ising ferromagnet with combined two- and four-spin interaction are presented. The first improvement follows from the complete implementation of a refined product average decomposition (PAD) method which distinguishes between nearest-neighbour and non-nearest-neighbour pairs. The second improvement comes from a comparison of the value for the ordinary two-spin interaction average of order four calculated using the PAD method, with its more accurate value calculated using a critical correlation-function approximation method. The comparison with the available values of the transition temperature for the FCC lattice obtained from low-temperature expansions shows that the errors are below 0.1% at alpha = infinity ( alpha =J 4 /J 2 ), below 5.8% for alpha <or=2, and below 2.4% for alpha <or=1. At alpha =0 the errors are less than 0.3%. For intermediate values of alpha (4<or= alpha <or=8, for example) the errors are about 11%.
Making use of the Stratonovich linearization procedure for bilinear exponents and certain eigenvalue properties of the generalised Jacobi matrices, an exact expression has been obtained, in a closed form, for the partition function of the d (=3) dimensional, nearest neighbour coupled, Ising model for finite continuous spins(=D) in the presence of an external field. A Gaussian measure has been chosen for integration over the spin space. The method admits complete generalisation with respect to the dimensionality, the range of interaction and the sign of the interaction constant J (ferro- or antiferromagnetic). All thermodynamical quantities of interest can be obtained by taking suitable partial derivatives of the partition function. The critical indices can be ascertained for the cases of d=3 for which the theory predicts a second-order phase transition. For the ferromagnetic case(J>0), the spontaneous magnetisation vanishes identically the temperature T>T (the transition temperarture) and the magnetic susceptibility follows essentially the Curie Weiss Law.
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.