Infinite Susceptibility without Spontaneous Magnetization: Exact Properties of the Ising Model on the Cayley Tree

The nearest-neighbor coupling (coupling constant J) ferromagnetic Ising model on the finite Cayley tree is studied in the limit of large size. It is proved that the spontaneous magnetization per spin vanishes at any temperature. On the other hand, the susceptibility per spin diverges if and only if the temperature T satisfies the relation th(J/kT) ≥√1/(z - 1), where z is the number of nearest neighbors. The absence of spontaneous magnetization per spin of the Ising model on a wider class of reducible lattices is pointed out.