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

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 $\gamma \le 3$, 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.

5 pages, 5 figures

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