Physics of the Solid State

Method of averaging over interaction fields for constructing the fixed-scale renormalization group transformation

S. V. Semkin, Виктор Павлович Смагин

Published May 1, 2013• 4 citations

Abstract

The application of the method of averaging over exchange interaction fields to clusters consisting of two magnetic atoms in the Ising model using a simple lattice has been considered. A version of the fixed-scale renormalization group transformation has been constructed. The application of these methods to the square-lattice Ising magnet with anisotropic interaction has been discussed.

DOI: 10.1134/s1063783413050302

Related papers

Ising model of a dilute ferromagnet in the self-consistent field approximation

S. V. Semkin, Виктор Павлович Смагин

The method of averaging over the exchange interaction fields has been considered as applied to clusters of one and two magnetic atoms in the Ising model of a dilute ferromagnet, and a variant of fixed-scale group renormalization on the basis of this method has been constructed. It has been shown that the approximation obtained makes it possible to distinguish models with the site and bond dilutions.

Physics of the Solid StateJun 1, 20145 citations

Averaging over interaction fields and spin correlations

S. V. Semkin, Victor Smagin

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.

Территория новых возможностей. Вестник Владивостокского государственного университета экономики и сервисаJan 1, 20200 citations

Anisotropic triangular Ising model in the extended mean-field renormalization-group approach

Christos N. Likos

The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsFeb 1, 19976 citations

Renormalization-group calculation for the equation of state of an Ising ferromagnet

Bernard Nienhuis, Michael Nauenberg

The renormalization-group series expansion for the free energy of the Ising model is applied to obtain the equation of state in an external magnetic field. Numerical calculations for a square lattice in a four-cell-cluster approximation are compared in the scaling limit with results of high- and low-temperature series expansions.

Physical review. B, Solid stateJun 1, 197553 citations

Kadanoff's variational renormalisation group method: the Ising model on the square and triangular lattices

B. W. Southern

The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.

Journal of Physics A Mathematical and GeneralJan 1, 19785 citations

Correlation functions of pure and diluted Ising magnets in the mean-field approximation

S. V. Semkin, Виктор Павлович Смагин

A mean-field method, which is a variant of the fixed-scale renormalization group transformation and is applied to both pure and diluted magnets, has been considered. It has been shown that, for pure magnets, the method is equivalent to the Bethe approximation. This method has been used to calculate the magnetization and correlation functions of both pure and bond-diluted Ising magnets.

Physics of the Solid StateJul 1, 20141 citation