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Bulletin of the Korean Chemical Society

Generalization of the Curie-Weiss Model to the D-dimensional Spin System

Hyung-june Woo, Eun Kyung Lee, Eok-Kyun Lee

Published Aug 1, 1993• 0 citations
OpenAlex

Abstract

The critical behavior of the classical D-dimensional spin model (D2), which is intermediate model that link up the Ising (D = 1) and the spherical model (D = ), is studied for the case of constant coupling interaction independent of the spin-spin distance (Curie-Weiss model). Analytical results show that the critical behavior of the present model is in quantitative agreement with the prediction of the phenomenological mean-field theory independent of D. Critical temperature is calculated to be T=k/JD. This gives a quantitative explanation of the relationship between the spin degree of freedom and the critical temperature.

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The critical behavior of the classical D-dimensional spin model (D2), which is intermediate model that link up the Ising (D = 1) and the spherical model (D = ), is studied for the case of constant coupling interaction independent of the spin-spin distance (Curie-Weiss model). Analytical results show that the critical behavior of the present model is in quantitative agreement with the prediction of the phenomenological mean-field theory independent of D. Critical temperature is calculated to be T=k/JD. This gives a quantitative explanation of the relationship between the spin degree of freedom and the critical temperature.
The Curie–Weiss model is an exactly solvable model of ferromagnetism that allows one to study thermodynamic functions in detail, in particular their properties near the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and the correctness of at least some of its predictions, the Curie–Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin-glass models, and usually only general features of the Curie–Weiss model are presented. In this paper, we discuss the properties of this model in a rather detailed way. We present the exact, approximate and numerical results for this particular model. The exact expression for the limiting magnetic field is derived.
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.
A mathematical model, the spherical model, of a ferromagnet is described. The model is a generalization of the Ising model; and one-, two-, and three-dimensional lattices of infinite extent can be extensively discussed. A three-dimensional lattice shows ferromagnetic behavior and provides a statistical model of the Weiss phenomenological theory. The limiting free energy appears in a form which contains two of the essential features of the exactly known Ising model results in one and two dimensions. This suggests the probable form of the limiting free energy for the three-dimensional Ising model. A simplified model, the Gaussian model, is briefly discussed because this model also contains some of the significant features of the Ising model. However, the Gaussian model, unlike the spherical model, is not defined for all temperatures.
ABSTRACT. This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the decay of the order parameter in the approach to the critical curve and gives useful stability properties of a variational problem associated with the representation. 1.
The critical temperatures Kc and the nearest-neighbour correlation functions Uc have been calculated for the d-dimensional hypercubic Ising ferromagnet for d=4 to d=30, by generalizing a recent theory reported for d=3. The results are of relevance for Monte Carlo Renormalization Group work involving critical properties in random fields, which calls for a knowledge of Kc in the limit of zero field. As d becomes large, Kc and Uc are found to approach the spherical model values. Polynomial expressions in 1/d are fitted to the calculated values, using the analogous spherical model expressions as guide. Comparison is made with the corresponding expressions of Fisher and Gaunt. Uc/(1−Kc), independent of d for the spherical model, is investigated; it is seen to be approximately independent of lattice structure for d=2 and for d=3.