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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Systematic approximation method for the critical properties of lattice spin systems

James L. Monroe

Published Jun 26, 2001• 26 citations
Publisher

Abstract

We present an extension of past approximations such as the Bethe approximation where we look at generalized Husimi trees. In doing so we obtain a sequence of ever more accurate approximations. The sequence is very systematic, and sequences of as few as three points can be extrapolated to give critical temperature estimates to within 0.1% or better of the exact value for our test cases. In the case of the square lattice, Ising ferromagnet, where five levels of approximation are obtained, estimates within 0.003% are obtained. Critical exponents can also be approximated but with much less accuracy. Much of our attention is devoted to the critical line of the phase diagram of the antiferromagnetic Ising model on the square lattice, where our results compare very favorably with other approximations. The method is general enough to be applicable to any spin lattice system where the spin takes on a discrete set of values.

DOI
10.1103/physreve.64.016126

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We present an extension of past approximations such as the Bethe approximation where we look at generalized Husimi trees. In doing so we obtain a sequence of ever more accurate approximations. The sequence is very systematic, and sequences of as few as three points can be extrapolated to give critical temperature estimates to within 0.1% or better of the exact value for our test cases. In the case of the square lattice, Ising ferromagnet, where five levels of approximation are obtained, estimates within 0.003% are obtained. Critical exponents can also be approximated but with much less accuracy. Much of our attention is devoted to the critical line of the phase diagram of the antiferromagnetic Ising model on the square lattice, where our results compare very favorably with other approximations. The method is general enough to be applicable to any spin lattice system where the spin takes on a discrete set of values.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
Two properties of the Ising model are considered. Firstly, by extrapolating appropriate series expansions, the spontaneous magnetization of a ferromagnet for three-dimensional lattices is obtained. From the results accurate values for the critical temperature are found. Secondly, the high temperature susceptibility of an antiferromagnet is derived from the approximation of Kikuchi. The resulting susceptibility curves show that for loose-packed lattices a maximum value occurs at a temperature slightly above the critical point, while for the close-packed lattices no such effect is obtained.
We obtain an approximation of the phase diagram of an Ising model with both nearest-neighbor and next-nearest-neighbor interactions on the square lattice. We use the Fisher zeros of the partition functions of a sequence of finite-sized systems along with various extrapolation methods to obtain phase transition points. In addition, we obtain an approximate value of the correlation length critical exponent nu. Our results are compared to previous results for this system using a wide variety of other approaches.
Abstract The expressions for average products of three neighboring spins in the Ising model have been found on lattices with coordination numbers 3 and 4 as functions of temperature and spontaneous magnetization. These expressions are used to compare the exact solution for the Ising model on a square lattice and the solutions found by approximate methods. A method of improving approximate methods is proposed for applying, in particular, to the Bethe approximation and to a change in the critical index of the temperature dependence of the spontaneous magnetization.
Abstract We compute the exact partition function for the 3-state Potts model on square lattices of several sizes larger than previously accessible. Making comparison with the exactly solved Ising model we show that, for aspects of the analytic structure close to the ferromagnetic transition point, these lattices are large enough to approach the thermodynamic limit. Subject to certain assumptions this allows for computation of estimates for the specific heat critical exponent. We thus obtain an estimate for this exponent. The estimate is consistent with the known result, thus demonstrating the potential use of this method for other models. We also discuss the antiferromagnetic transition.