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 ${J}_{i,j}\ensuremath{\sim}1/{({k}_{i}{k}_{j})}^{\ensuremath{\mu}}$ 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 spin models on scale-free networks, where the node distribution $P(k)\ensuremath{\sim}{k}^{\ensuremath{-}\ensuremath{\gamma}}$, with node-dependent interacting constants. We observe that, by increasing $\ensuremath{\mu}$, the critical behavior of the model changes from a phase transition at $T=\ensuremath{\infty}$ for a uniform system $(\ensuremath{\mu}=0)$ to a $T=0$ phase transition when $\ensuremath{\mu}=1$: in the thermodynamic limit, the system shows no true critical behavior at a finite temperature for the whole $\ensuremath{\mu}\ensuremath{\ge}0$ interval. The magnetization and magnetic susceptibility are found to present noncritical scaling properties.

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• Physical Review E - View
• arXiv (Cornell University) - View - PDF
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