Phase transition of the Cayley tree with Ising interaction

Susceptibilities for the Müller-Hartmann-Zitartz countable infinity of phase transitions on a Cayley tree

We obtain explicit susceptibilities for the countable infinity of phase transition temperatures of Müller-Hartmann-Zitartz on a Cayley tree. The susceptibilities are a product of the zeroth spin with the sum of an appropriate set of averages of spins on the outermost layer of the tree. A clear physical understanding for these strange phase transitions emerges naturally. In the thermodynamic limit, the susceptibilities tend to zero above the transition and to infinity below it.

Ferromagnetic Ising spin systems on the growing random tree

We analyze the ferromagnetic Ising model on a scale-free tree; the growing random tree model with the linear attachment kernel A(k) = k + alpha . We derive an estimate of the divergent temperature T(s) below which the zero-field susceptibility of the system diverges. Our result shows that T(s) is related to alpha as tanh(J/T(s)) = alpha/[2(alpha+1)] , where J is the ferromagnetic interaction. An analysis of exactly solvable limit for the model and numerical calculation supports the validity of this estimate.

Ising model on the scale-free network with a Cayley-tree-like structure

We derive an exact expression for the magnetization and the zero-field susceptibility of the Ising model on a random graph with degree distribution P(k) proportional, k-gamma and with a boundary consisting of leaves, that is, vertices whose degree is 1. The system has no magnetization at any finite temperature, and the susceptibility diverges below a certain temperature Ts depending on the exponent gamma. In particular, Ts reaches infinity for gamma<or=4. These results are completely different from those of the case having no boundary, indicating the nontrivial roles of the leaves in the networks.