Ising model on the Apollonian network with node dependent interactions

This work considers an Ising model on the Apollonian network, where the exchange constant $J_{i,j}\sim 1\mathrm{/}(k_i{k}_j{)}^{\mu }$ 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 spins models on scale-free networks, where the node distribution $P(k)\sim k^{-\gamma }$, with node dependent interacting constants. We observe that, by increasing $\mu$, the critical behavior of the model changes, from a phase transition at $T=\mathrm{\infty }$ for a uniform system $(\mu =0)$, to a T=0 phase transition when $\mu =1$: in the thermodynamic limit, the system shows no exactly critical behavior at a finite temperature. The magnetization and magnetic susceptibility are found to present non-critical scaling properties.