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Modern Physics Letters B

IMPROVED MEAN-FIELD TRANSFER MATRIX MODEL FOR HYPERCUBIC ISING SYSTEMS

Tuncer Kaya

Published May 31, 2012• 1 citation
Publisher

Abstract

A mean-field method for Ising systems is introduced generalizing the reduced transfer matrix method introduced by Kaya and Arık. The important assumption of the method is that the neighboring spins of a central spin can be replaced by the average magnetization 〈σ〉 of the system. The approximation is same as the previously introduced method. However, the derivative of average magnetization with respect to external magnetic field is defined without further approximation. In addition, the previously introduced model is extended to all hypercubic lattices. The critical coupling strength Kc for hypercubic lattices is evaluated self-consistently. Obviously, this assumption leads to some significant deviation from the exact treatment. However, the model introduced here is an improvement on most self-consistent mean-field-type approximations.

DOI
10.1142/s0217984912501114

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A mean-field method for Ising systems is introduced generalizing the reduced transfer matrix method introduced by Kaya and Arık. The important assumption of the method is that the neighboring spins of a central spin can be replaced by the average magnetization 〈σ〉 of the system. The approximation is same as the previously introduced method. However, the derivative of average magnetization with respect to external magnetic field is defined without further approximation. In addition, the previously introduced model is extended to all hypercubic lattices. The critical coupling strength Kc for hypercubic lattices is evaluated self-consistently. Obviously, this assumption leads to some significant deviation from the exact treatment. However, the model introduced here is an improvement on most self-consistent mean-field-type approximations.
A mean-field method for the hypercubic nearest-neighbor Ising system is introduced and applications to the method are demonstrated. The main idea of this work is to combine the Kadanoff’s mean-field approach with the model presented by one of us previously. The mean-field approximation is introduced with the replacement of the central spin in Ising Hamiltonian with an average value of particular spin configuration, i.e, the approximation is taken into account within each configuration. This approximation is used in two different mean-field-type approaches. The first consideration is a pure-mean-field-type treatment in which all the neighboring spins are replaced with the assumed configurational average. The second consideration is introduced by the reduced transfer matrix method. The estimations of critical coupling values of the systems are evaluated both numerically and also analytically by the using of saddle point approximation. The analytic estimation of critical values in the first and second considerations are [Formula: see text] and [Formula: see text] respectively. Obviously, both of the considerations have some significant deviation from the exact treatment. In this work, we conclude that the method introduced here is more appropriate physical picture than self-consistent mean-field-type models, because the method introduced here does not presume the presence of the phase transition from the outset. Consequently, the introduced approach potentially makes our research very valuable mean-field-type picture for phase transition treatment.
In this paper we present a simple approximate transfer matrix method for 2D and 3D hyper cubic nearest neighbor Ising models with various coordination number z to calculate the corresponding critical coupling strengths Kc. The critical coupling strengths of the Ising ferromagnets are obtained quite accurately by simple improvements over the self-consistent correlated field (SCCF) approximation. The important physical effect included in this work is some of the fluctuation effects of the systems by the help of a reduced transfer matrix method. When used in combination with the accuracy of the average magnetization obtained from the SCCF approximation, this reduced transfer matrix method leads to estimate of Kc more accurate than those obtained from the Bethe–Peierls–Weiss approximation and also SCCF approximation. Therefore, we believe that the approach we refer to as the correlated reduced transfer matrix method is potentially very useful scheme for obtaining approximate values of the critical coupling strengths of the Ising models with a mathematically easy meaner.
In this work, we present a simple approximate transfer matrix method for 2D and 3D Ising ferromagnet to calculate spontaneous magnetization of the system. The critical coupling strength Kc of 2D and 3D Ising models in reduced transfer matrix approximation is obtained quite accurately by simple improvements over the mean field theory. The important physical effect we include is the some of the fluctuations effects of the systems directly with help of this method. We predict from the spontaneous magnetization curve that the critical coupling strength Kc=J/kBT = 0.401 and 0.245 for two-dimensional (2D) and three-dimensional (3D) systems, respectively.
The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inverse Ising problem without running the Susceptibility Propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in presence of an external field.
An effective mean-field theory based on cumulant expansion is expanded to deal with the first-order transition of the Potts model on general d-dimensional hypercubic lattice in this paper. It is a generalized mean-field method developed from the cumulant expansion of average free energy. As we shall see, it gives the traditional mean-field result at its first-order approximation and the effective mean field introduced by this method will be amended with the approximation's order increasing. For a full view of this method, we have carried out calculations up to the seventh-order approximation on a square (d=2)(d=2)(d=2) lattice since some known exact quantites of this case are terminals that should be approached. The results show this method has the potential to be a useful tool to calculate thermodynamical observables of spin systems.
Ising systems are approximated by a generalized mean-field approximation in which only some of the interactions are replaced by their expectation values. If all the interactions are positive then it is shown that a self-consistent solution exists for the magnetization in this model and that it gives an upper bound for the true magnetization. Using this mean-field approximation, a generalization of the random-phase approximation is obtained. This is shown to give an upper bound for two-spin correlation functions.