Ising model on the Apollonian network with node dependent interactions

Details Ising model on the Apollonian network with node dependent interactions Ising model on the Apollonian network with node dependent interactions

2008-11-24

arxiv.org/abs/0811.3827v1

Physics

Condensed Matter

Statistical Mechanics

6 figures, 12 figure files

Scientific paper

10.1103/PhysRevE.79.036105

This work considers an Ising model on the Apollonian network, where the exchange constant $J_{i,j}\sim1/(k_ik_j)^\mu$ 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 $T=\infty$ 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.

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