Improving approximate solutions of the Ising model using many-particle spin correlations

• 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

The Ising model is often used for the theoretical analysis of phase transitions in magnetic systems. This model can, in many cases, in itself serve as a fairly accurate description of real systems. In addition, the principle of universality allows one to extend many of the results
obtained for simple lattice Ising models to more complex systems. However, there are practically no exact solutions for the Ising model. In fact, the only exact solution is Onsager's solution for a square lattice. There are, of course, approximate solution methods,
but they have fundamental drawbacks. Namely, approximate methods give overestimated estimates of the Curie temperature and incorrectly describe the features of the behavior of the system near the phase transition point. However, as shown in this work, there are ways
to improve virtually any approximate methods. It turns out that by averaging over the exchange fields it is possible (in some cases) to find a relationship between the spontaneous magnetization and the average products of three or more neighboring spins. Using these
connections, one can construct an algorithm for improving approximate solutions. Expressions are found for the mean values of the products of three neighboring spins in the Ising model on lattices with coordination numbers 3 and 4 as a function 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 for improving the approximate methods is proposed, applicable, in particular, to
the Bethe approximation, and leading to more accurate values of the critical temperature and to a change in the critical exponent of the temperature dependence of the spontaneous magnetization.

Keywords: phase transitions, Ising model, сritical exponents.