Averaging over interaction fields and spin correlations in the Ising model

• Semkin S.V.

  S. V. Semkin. Vladivostok State University of Economics and Service. Vladivostok. Russia

• Smagin V.P.

  V. P. Smagin. Vladivostok State University of Economics and Service. Vladivostok. Russia

In 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 par-
ticles 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.
Keywords:  phase transitions, Ising model, exact solution.