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Physics of the Solid State

Approximate Accounting of Spin Correlations in the Ising Model

S. V. Semkin, Виктор Павлович Смагин

Published Sep 1, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.1134/s1063783421080278

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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.
Averaging over interaction fields and spin correlations in the Ising modelIn the theory of phase transitions in magnetic systems, there are a small number of exact results.One of them is the well-known Onsager solution for the Ising model on a square lattice.The problem of phase transitions in magnets is a special case of the problem of many bodies.There are many approaches and methods in this area, including the method of averaging over exchange fields.The application of the distribution function over interaction fields to the study of the properties of a system of many interacting particles has been used for a long time.In previous works of the authors, the method of averaging over exchange fields was applied to the analysis of the magnetic properties of pure and diluted magnets.In this paper, we formulate and prove the relations on which the method of averaging over exchange fields can be based on clusters of several spins.It turns out that based only on the possibility of representing thermodynamic means as means of the distribution function of the fields, some nontrivial exact relations can be obtained.The paper considers the Ising model on simple lattices with coordination numbers of three and four.Exact expressions are found that connect the average products of three spins with spontaneous magnetization in the system.Inequalities are obtained that limit the limiting value of the Curie temperature and the maximum value of spontaneous magnetization at a given temperature.
Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision.
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.
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 method of approximate calculation of the magnetization of an Ising magnet located in an external magnetic field is proposed. This method is based on using an exact or approximate expression for the spontaneous magnetization of an Ising magnet on the same lattice. The method uses the ratio of effective fields for clusters of one and two lattice nodes. Using the proposed method for a magnet located in an external field, the dependence of magnetization on temperature and the magnitude of the external field is calculated. The proposed method is applied in the work to the solution in the mean field approximation, to the solution in the Beta approximation and to the exact solution on a square lattice; for all these solutions, critical indices characterizing the behavior of a magnet in an external field are calculated. Keywords: the Ising model, phase transitions, the influence of an external field.
We use an effective field model which explicitly incorporates correlations to determine the critical temperatures for Ising ferromagnets on the common lattices. We specifically retain only the first nontrivial odd correlation function, i.e., the triplet correlation function, while neglecting higher order correlations. Within this approximation, assuming the equivalence of triplet correlations involving both central and noncentral sites in the nearest neighbor cluster yields values for the critical temperature below the Bethe approximations for z⩾4. Likewise, assuming the inequivalence of triplet correlations involving the central and noncentral sites yields values of the critical temperature only slightly higher than the Bethe approximation for z⩾4. For lattices with four- and six-fold coordination this equivalence and inequivalence of triplet correlation functions is sufficient to cause the critical temperatures to approach their two- and three-dimensional values. For instance, for quadratic lattices we obtain values for the critical temperature of 2.5284 and 2.9796 as compared to the exact and series results of 2.2692 and 2.7044 for the simple quadratic and diamond lattices, respectively, and the Bethe approximation of 2.8854.