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arXiv:1201.3015

An improved effective field theory formulation of spin-1 Ising systems with arbitrary coordination number z

Ümit Akıncı

Published Jan 14, 2012• 0 citations
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Abstract

An improved unified formulation based on the effective field theory is introduced for a spin-1 Ising model with nearest neighbor interactions with arbitrary coordination number z. Present formulation is capable of calculating all the multi-spin correlations systematically in a representative manner, as well as its single site counterparts in the system and gives much better results for critical temperature, in comparison with the other works in the literature. The formulation can be easily used to various kinds of spin-1 Ising models, as long as the system contains only the nearest neighbor interactions as spin-spin interactions.

DOI
10.48550/arxiv.1201.3015

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An improved unified formulation based on the effective field theory is introduced for a spin-1 Ising model with nearest neighbor interactions with arbitrary coordination number z. Present formulation is capable of calculating all the multi-spin correlations systematically in a representative manner, as well as its single site counterparts in the system and gives much better results for critical temperature, in comparison with the other works in the literature. The formulation can be easily used to various kinds of spin-1 Ising models, as long as the system contains only the nearest neighbor interactions as spin-spin interactions.
An improved unified formulation based on the effective field theory is introduced for a spin-1/2 Ising model with nearest neighbor interactions with arbitrary coordination number z. Present formulation is capable of calculating all the multi-spin correlations systematically in a representative manner, as well as its single site counterparts in the system and gives results for the critical temperature of the system much better than those of the other works in the literature. The formulation can be easily applied to various extensions of s-1/2 Ising models, as long as the system contains only the nearest neighbor interactions as spin-spin interactions.
Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with zzz nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.
The system of two polynomial equations is derived for direct simultaneous determination of the critical temperatures of the spin-1 Ising model on lattices with arbitrary values of the coordination number in the single-site cluster effective field theory approximation. Analysis of the critical temperatures allows to determine simple linear function 2z/3−1/2 that approximates the critical temperatures of the model with very high precision for all values of the coordination number z. It is demonstrated that the system of two equations for the critical temperatures of the model can be reduced to the corresponding single equation in the cases with z=3,4, and 5. In addition, the system of entirely algebraic polynomial equations for the determination of the spontaneous magnetization of the model simultaneously for all values of the coordination number is also derived and discussed.
A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where zzz is the number of nearest neighbors.
A unified effective-field renormalization-group framework (EFRG) for both quenched bond- and site-diluted Ising models is herein developed by extending recent works. The method, as in the previous works, follows up the same strategy of the mean-field renormalization-group scheme (MFRG), and is achieved by introducing an alternative way for constructing classical effective-field equations of state, based on rigorous Ising spin identities. The concentration dependence of the critical temperature, Tc(p), and the critical concentrations of magnetic atoms, pc, at which the transition temperature goes to zero, are evaluated for several two- and three-dimensional lattice structures. The obtained values of Tc and pc and the resulting phase diagrams for both bond and site cases are much more accurate than those estimated by the standard MFRG approach. Although preserving the same level of simplicity as the MFRG, it is shown that the present EFRG method, even by considering its simplest size-cluster version, provides results that correctly distinguishes those lattices that have the same coordination number, but differ in dimensionality or geometry.