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 [Formula presented] where r represents the Euclidean distance between two nodes. In the case of [Formula presented] 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 [Formula presented] does not exhibit long-range order at any finite temperature. The Monte Carlo simulations are performed at various values of [Formula presented] and the critical value [Formula presented] 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 [Formula presented] for any nonzero positive value of [Formula presented]