Phase transition in the Ising model on a small-world network with distance-dependent interactions

We study the collective behavior of an Ising system on a small-world network with the interaction $J\left(r\right)\mathrm{}\propto r^{-\alpha },$ where r represents the Euclidean distance between two nodes. In the case of $\alpha =0\mathrm{}$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $\left(\overrightarrow{\alpha }\infty \right)$ does not exhibit long-range order at any finite temperature. The Monte Carlo simulations are performed at various values of $\alpha ,$ and the critical value ${\alpha }_c$ beyond which the long-range order does not emerge is estimated to be 0. Thus, concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-\alpha }$ for any nonzero positive value of $\alpha .$