Phase transition of the Cayley tree with Ising interaction

Abstract

The properties of a spin system are discussed in which the interactions show a Cayley-tree structure. A nonlinear recursion equation for the partition function is derived, which is solved exactly for H=0 and numerically for arbitrary uniform H not=0. Although the zero-field partition function is analytic for all temperatures, the zero-field susceptibility diverges at a critical temperature Tc not=0. This is a critical temperature determined by the surface of the Cayley tree which gives the dominant contribution in the thermodynamic limit. Below Tc, the magnetization is nonanalytic in H at H=0, but a spontaneous magnetization does not exist.