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Phase Transitions

General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization

E. Jurčišinová, Marian Jurčisin

Published Nov 7, 2021• 3 citations
Publisher

Abstract

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.

DOI
10.1080/01411594.2021.1998495

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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.
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.
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.
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.
The two-state Ising model with nearest-neighbour bilinear exchange constant J (2) , three-spin coupling constant J (3) and four-spin coupling constant J (4) is studied using the effective-field approximation. The temperature variation in spontaneous magnetization is studied for different coordination numbers z and for various values of the parameters alpha =J (3) /J (2) and alpha '=J (4) /J (2) . It is seen that, in the absence of three-spin coupling, there exists a critical value of alpha ' (say, alpha ' c ) above which the magnetization becomes double valued at and beyond the Curie temperature T c , forming a bulge near T c which indicates the occurrence of a first-order transition. The expressions for alpha ' c for z=3, 4, 6 are derived and it is found that alpha ' c increases as z increases. In the absence of four-spin coupling it is found that first-order transitions appear for alpha >or=0. The magnetization curves are also studied for the case when all the interactions are present. The results are discussed with reference to those obtained from the molecular-field approximation.
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.
Abstract A method in Ising models is introduced by using the effective field theory (EFT) with correlations. In the present method, for the spin‐1 Ising ferromagnetic systems with three coordination numbers q (q = 3, 4 and 6), we have derived a set of linear equations by taking as a basis the thermal averages of a central spin and a perimeter spin at the site i, which is defined within spin identities and differential operator technique. By solving numerically the set of linear equations derived individually for the Ising systems, we have evaluated the phase diagrams of both the spin‐1 Ising ferromagnetic systems with a single‐ion anisotropy and the spin‐1 transverse Ising ferromagnetic systems with a transverse field. The present results of critical transverse field for the spin‐1 transverse Ising systems improve those obtained by the pair approximation. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)