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Physics of the Solid State

Construction of an approximate solution for an Ising magnet in an external magnetic field

S. V. Semkin, Виктор Павлович Смагин, Elena G. Lavrushina

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.21883/pss.2023.06.56114.51

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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.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
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 paper presents new method for calculating the low-temperature asymptotics of free energy of the 3D Ising model in external magnetic field (H = 0). The results obtained are valid in the wide range of temperature and magnetic field values fulfilling the condition: [1 − tanh(h/2)] ∼ ε, for ε ≪ 1, where h = βH, β- the inverse temperature and H- external magnetic field. For this purpose the method of transfer-matrix, and generalized Jordan-Wigner transformations, in the form introduced by the author in [1], are applied. PACS number(s): 05.50. +q Typeset using REVTEX 1 I. FORMULATION OF THE PROBLEM As is well known, till now an exact solution for the 2D Ising model in external magnetic field (H = 0) was not found. In the case of the 3D Ising model there does not exist an exact solution forvanishing magnetic field (H = 0), without even mentioning the case with magnetic field. Despite great successes in investigations of the Ising model reachedusing
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.
We consider a ferromagnet in an external magnetic field with $T\ensuremath{\le}{T}_{c}$. Using a variational approximation based on the zero-field solution, we obtain an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. Explicit numerical calculations have been carried out for the two-dimensional Ising model, and we find that our results provide a good approximation to the results of series expansions throughout the region $T\ensuremath{\le}{T}_{c}$. We also apply our procedure to the eight-vertex model. Metastability can be discussed within our approximation in a manner analogous to the way it is dealt with in mean-field theory. In our treatment, the critical exponents have nonclassical values, but the pseudocritical exponents all represent square-root singularities, which is the same result as found within mean-field theory.