Ferromagnetic Ising spin systems on the growing random tree

We analyze the ferromagnetic Ising model on a scale-free tree; the growing random network model with the linear attachment kernel $A_k=k+\alpha$ introduced by [Krapivsky et al.: Phys. Rev. Lett. {\bf 85} (2000) 4629-4632]. 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 support the validity of this estimate.