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

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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 (J2SizSjz) it contains four-spin interaction terms (J4SizSjzSkzSlz) 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.