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Phase transitions of antiferromagnetic Potts models are investigated by cactus approximations and Monte Carlo simulations. These models show phase transitions associated with highly degenerate ground states. By the simulations for systems on a simple cubic lattice, a Kosterlitz-Thouless phase has been found below kT/ IJI::::e 1.25 for the three-state Potts model, while a longrange ordering consisting of two sublattices has appeared below k T/ If I : : : : e 0. 7 for the four-state Potts model. by guest on March 23, 2015 http://ptps.oxfordjournals.org/ Downloaded from

The Creen function method is applied to the Ising model of ferromagnetism. It is essential in this paper that we have solved the equations for the Green functions in terms of the correlation functions without using the approximation of decoupling the Green functions as is usually done. Although we get several rigorous relations between the many-spin correlation functions, we do not succeed in solving for the correlation functions, except in the case of a linear chain. We introduce the approximation of eliminating some of the correlation functions using the high temperature expansion, so as to reduce the number of the independent correlation functions. The Curie points of a square lattice and a simple cubic lattice are calculated.Bogolyubov and Tjablikov 1 ) have introduced the two-time, temperaturedependent Green functions into statistical theory, and this method has been applied to the Heisenberg model of ferromagnetism by Tjablikov, 2 ) who obtained the temperature dependence of the magnetization over the whole range of temperature with fairly good accuracy. However, the chain of equations for the various Green functions have to be decoupled somewhere along the hierarchy in the Heisenberg model of ferromagnetism. This decoupling scheme is one of the most difficult and ambiguous points in the Green function method. The situation is quite different in the Ising model of ferromagnetism, in which the chain of equations for the Green functions is closed. Thus we can solve for the Green functions rigorously without introducing any decoupling.In this paper we shall obtain the coupled equations for the Green functions in the Ising model, and then solve for the Green functions in terms of the various spin correlation functions. Using spectral theorem, which gives the relationship between the Green functions and the correlation functions, we shall get several equations connecting the correlation functions. The number of these equations is less than that of the correlation functions, so that we cannot solve for the correlation functions. In § 2 the case of a linear chain will be treated, and we shall get a difference equation connecting the two-spin correlation functions. Consequently the correlation function will be obtained rigorously. In § 3 we shall consider a honeycomb lattice. This lattice is topologically the second simplest lattice next to a linear chain. For this case we shall get a at