Recurrent lattices and self-consistent equations in the Ising model

• Smagin V.P.

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

• Semkin S.V.

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

In theoretical studies of the critical behavior of magnets, the Ising model is often used – a model with the simplest Hamiltonian that reflects the characteristic features of systems with collective interaction. However, even for simple lattices, the Ising model does not have an exact  solution;  therefore,  to  study  the  properties  of  this  model,  various  approximations  are used. Such approximations can only reflect certain features of the system, while the rest have to be neglected. In this case, as a rule, it is not known in advance which of the characteristic features of the system are the most important. For example, in real crystal lattices there always exists a minimal closed path that contains a certain number of atoms – its own for each lattice. But in known approximations, such as the mean field method or the Bethe approximation, the presence of such a minimum cycle is not taken into account. In this paper, we investigate the possibility of constructing an approximation that explicitly takes into account
the presence of such cycles. We have constructed a class of self-consistent equations that can be used for the approximate solution of the Ising model on various crystal lattices. A particular (and simplest) example of equations of this class is the well-known Bethe approximation,
and therefore our class of self-consistent equations can be viewed as a generalization of the Bethe approximation. As is well known, the Bethe approximation can be interpreted as a replacement of the real crystal lattice by the so-called Bethe lattice, which is the inner part of
the Kaylee tree. Similarly, the solutions of some of the self-consistent equations we propose can be interpreted as exact solutions of the Ising problem on specially constructed recursive lattices, which will be shown below. These recursive lattices are distinguished by the fact that
each node in them is included in a certain number of closed cycles. Using our self-consistent equations, we calculated Curie temperatures for simple lattices. It turned out that the inclusion of closed cycles leads to more accurate results.
Keywords: phase transitions, Ising model, recursive lattices.