The method of cyclic clusters in the Ising model of a dilute magnet

• Smagin V.P.

  Doctor of physics and mathematics, professor, Head of the Laboratory of fundamental and applied physics
  Vladivostok State University of Economics and Service
  Vladivostok. Russia

• Semkin S.V.

  Candidate of physical and mathematical sciences, associate professor of Chair of information technology and system
  Vladivostok State University of Economics and Service
  Vladivostok. Russia

The Ising model with nonmagnetic dilution is used for the theoretical description of many objects and phenomena in the physics of condensed matter and nuclear physics. The effect of nonmagnetic dilution on the critical behavior of magnets, including those described by the Ising model, is of considerable scientific interest. In this paper we obtain approximate values of the critical temperature for simple lattices and calculate the percolation thresholds for sites and bonds using the method of cyclic clusters.

For the Ising model with nonmagnetic dilution, it is not possible to construct an exact solution for any crystal lattice. The properties of this model are investigated either numerically or in one or another approximation. Approximate methods, proposed and investigated earlier by the authors of this paper, considered clusters of different sizes, allocated on a lattice. It was shown that if the clusters in question do not contain closed paths, the approximations obtained with their help are one or another generalized version of the Bethe approximation, which agrees with the fact that the Bethe approximation can be regarded as an exact solution on a lattice without closed paths (the Cayley tree). With the help of this method, it is possible to construct approximate concentration dependences of the magnetization, the critical temperature, and to find approximate values of the percolation thresholds for the sites and for the bonds. The results obtained in this way are, as shown in the paper, more accurate than those obtained by the Bethe method or its modifications.

Keywords: phase transitions, Ising model, dilute magnet.