Ising model on the Apollonian network with node-dependent interactions

Abstract

This work considers an Ising model on the Apollonian network, where the exchange constant Ji,j∼1/(kikj)μ between two neighboring spins (i,j) is a function of the degree k of both spins. Using the exact geometrical construction rule for the network, the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit. The results can be compared to the predictions of a general framework for spin models on scale-free networks, where the node distribution P(k)∼k−γ, with node-dependent interacting constants. We observe that, by increasing μ, the critical behavior of the model changes from a phase transition at T=∞ for a uniform system (μ=0) to a T=0 phase transition when μ=1: in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole μ≥0 interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.