Ising model on a $restricted$ scale-free network

Abstract

The Ising model on a $restricted$ scale-free network (SFN) has been studied employing Monte Carlo simulations. This network is described by a power-law degree distribution in the form $P(k)~k^{-\alpha}$, and is called restricted, because independently of the network size, we always have fixed the maximum $k_{m}$ and a minimum $k_{0}$ degree on distribution, being that for it, we only limit the minimum network size of the system. We calculated the thermodynamic quantities of the system, such as, the magnetization per spin $\textrm{m}_{\textrm{L}}$, the magnetic susceptibility $\chi_{\textrm{L}}$, and the reduced fourth-order Binder cumulant $\textrm{U}_{\textrm{L}}$, as a function of temperature $T$ for several values of lattice size $N$ and exponent $1\le\alpha\le5$. For the values of $\alpha$, we have obtained the finite critical points due to we also have finite second and fourth moments in the degree distribution, and the phase diagram was constructed for the equilibrium states of the model in the plane $T$ versus $k_{0}$, $k_{m}$, and $\alpha$, showing a transition between the ferromagnetic $F$ to paramagnetic $P$ phases. Using the finite-size scaling (FSS) theory, we also have obtained the critical exponents for the system, and a mean-field critical behavior is observed.