Three-state Heisenberg model on the Bethe lattice

• Semkin S.V.

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

• Smagin V.P.

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

The Ising, Potts, Heisenberg, and other lattice models are used for the theoretical description of many objects and phenomena in condensed matter physics and nuclear physics. As a rule, in theoretical studies of the critical behavior of magnetic materials, the Ising model is used —
a model with the simplest Hamiltonian. This is explained by the universality hypothesis, according to which this critical behavior is determined only by the symmetry of the Hamiltonian of the system and does not depend on its details. That is, the same critical behavior (for ex
ample, critical exponents) is not characteristic of each specific Hamiltonian, but refers to a whole class of Hamiltonians with the same symmetry. However, the universality hypothesis does not in itself contain any means of determining to which universality class each particular
Hamiltonian belongs. Therefore, consideration of more complex lattice models, such as the Heisenberg model, is not without meaning.
In this paper, we considered the Heisenberg model with three states on the Bethe lattice. The task is to find the equilibrium probabilities of these states at a given temperature and external  field.  This problem can be solved  exactly by creating a  system  of recurrent  equations,
which is done in this paper. However, our main goal was not even to solve the problem itself for the Heisenberg model. Regarding the Ising model, it is known that its solution on the Bethe lattice can be interpreted as a fixed-scale renormalization group transformation in a constant effective field. In this paper, we investigated the possibility of a similar interpretation for the Heisenberg model. It turned out that it is impossible for the original Heisenberg model, but it turns out to be possible for a model with a more general form of the Hamiltonian.

Keywords: phase transitions, Heisenberg model, Bethe lattice.