Ising model in scale-free networks: A Monte Carlo simulation

TL;DR

This study uses Monte Carlo simulations to analyze the phase transition behavior of the Ising model on uncorrelated scale-free networks, revealing size-dependent crossover temperatures and deviations from earlier theoretical predictions.

Contribution

It provides new numerical insights into the critical behavior of the Ising model on scale-free networks, especially for $oldsymbol{ ext{γ} extless 3}$, challenging previous analytical results.

Findings

For $ ext{γ} > 3$, results agree with earlier analytical phase transition predictions.

For $ ext{γ} extless 3$, a size-dependent crossover temperature is observed.

The crossover temperature scales logarithmically with system size at $ ext{γ} = 3$ and as a power law for $2 < ext{γ} < 3$, with a lower exponent than predicted.

Abstract

The Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a degree (or connectivity) distribution $P(k) \sim k^{-\gamma}$. The ferromagnetic-paramagnetic transition temperature has been studied as a function of the parameter $\gamma$. For $\gamma > 3$ our results agree with earlier analytical calculations, which found a phase transition at a temperature $T_c(\gamma)$ in the thermodynamic limit. For $\gamma \leq 3$, a ferromagnetic-paramagnetic crossover occurs at a size-dependent temperature $T_{co}$, and the system is in the ordered ferromagnetic state at any temperature for a system size $N \to \infty$. For $\gamma = 3$ and large enough $N$, the crossover temperature is found to be $T_{co} \approx A \ln N$, with a prefactor $A$ proportional to the mean degree. For $2 < \gamma < 3$, we obtain $T_{co} \sim < k > N^z$, with an exponent $z$ that decreases as $\gamma$ increases. This exponent is found to be lower than predicted by earlier calculations.