Ising antiferromagnet in the Curie – Weiss and Bethe approximations

• Semkin S.V.

  Sergey V. Semkin. Vladivostok State University. Vladivostok. Russia

• Smagin V.P.

  Viktor P. Smagin. Vladivostok State University. Vladivostok. Russia

In this paper we consider the antiferromagnetic Ising model in an external magnetic field. At present there is no exact analytical solution to this model (except for the one-dimensional case). Therefore, in the research the authors consider two approximate methods. The first of the methods considered has been known for quite a long time. This is the mean field method, also known as the Curie – Weiss approximation. Earlier, while studying ferromagnets, we constructed the form of the Bethe approximation based on the comparison of varisized clusters. In this paper we use this form of the Bethe approximation to develop self-consistent equations for an antiferromagnet. Our approach
allows us not only to obtain a new interpretation of the Bethe approximation, but also to combine it with the Curie – Weiss approximation within the framework of a unified theory of effective fields. The following main results are obtained in the research. Both the Curie – Weiss method and the Bethe method lead to the conclusion that at certain values of temperature and external field there occurs a second-order phase transition in the system. A homogeneous state of all atoms magnetizations being similar goes into a state of two magnetic sublattices standing out with different magnetizations. The boundary between the homogeneous and antiferromagnetic phases, called the line of critical field, has been constructed in both approximations considered. In addition, we plotted the full magnetization of the system as a function of temperature and external field.dition, we plotted the total magnetization of the system as a function of temperature and external field.
Keywords: phase transitions, Ising model, phase transitions, Ising model, antiferromagnet.