Few years ago, application of the mean field Bethe scheme on a given system was shown to produce a systematic change of the system intrinsic symmetry. For instance, once applied on a ferromagnet, individual spins are no more equivalent. Accordingly a new loopwise mean field theory was designed to both go beyond the one site Weiss approach and yet preserve the initial Hamitonian symmetry. This loopwise scheme is applied here to solve the triangular antiferromagnetic Ising model. It is found to yield Wannier's exact result of no ordering at non-zero temperature. No adjustable parameter is used. Simultaneously a non-zero critical temperature is obtained for the triangular Ising ferromagnet. This simple mean field scheme opens a new way to tackle random systems. PACS . 75.25.+z Spin arrangements in magnetically ordered materials (including neutron and spin-polarized electron studies, synchrotron-source X-ray scattering, etc.) -- 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) -- 75.50.-y Studies of specific magnetic materials 1
Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure
Z-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.
A new method of correlation reduction near the critical temperature is introduced for a classical spin system. This method is based on an assumption much less restrictive than the usual mean-field approximation. It permits an important parameter to be determined realistically. The transition temperature of the Ising ferromagnet is expressed very simply in a closed form. The values of the transition temperature evaluated with various approximations in this method are presented for three-dimensional Ising lattices.