Ferromagnetic Ising spin systems on the growing random tree

TL;DR

This paper investigates the ferromagnetic Ising model on a growing scale-free tree network, deriving a relation for the divergence temperature of susceptibility and supporting it with exact solutions and numerical analysis.

Contribution

It provides a new estimate for the divergence temperature in the Ising model on a growing scale-free tree, linking it explicitly to the network's attachment parameter.

Findings

Divergence temperature $T_s$ depends on the attachment parameter $eta$ as $ anh(J/T_s)=eta/[2(eta+1)]$.

Exact solutions and numerical calculations support the derived estimate.

The model reveals how network growth influences magnetic susceptibility divergence.

Abstract

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.