Ising Model with General Spin and Planar Rotator Model on the Bethe Lattice

Abstract

In the Bethe lattice in which the Bethe approximation is exact, the Curie temperatures of the Ising model with spin \(S=\frac{1}{2}\), 1,…, \(\frac{5}{2}\) are obtained exactly by generalizing the method of Katsura and Takizawa for the system with \(S=\frac{1}{2}\). This method is applied to the planar rotator model on the infinite Cayley tree, and the critical temperature and the susceptibility are rigorously calculated. It is pointed out that the susceptibility diverges below the critical temperature, on the other hand the spontaneous magnetization vanishes as the ground state is degenerated infinitely.