Ising model on the Apollonian network with node dependent interactions

TL;DR

This paper investigates an Ising model on the Apollonian network with node-dependent interactions, revealing how varying the interaction parameter $$ alters the critical behavior from infinite temperature phase transition to zero temperature transition.

Contribution

It introduces an exact iterative method to analyze the thermodynamics of the Ising model with node-dependent interactions on the Apollonian network, extending understanding of spin models on scale-free networks.

Findings

Critical behavior shifts from T= for =0 to T=0 for =1.

No finite-temperature critical point exists for >0.

Magnetization and susceptibility exhibit non-critical scaling.

Abstract

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.