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(r)∝r^-α, where r represents the Euclidean distance between two nodes. In the case of α=0 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 (α⃗∞) does not exhibit long-range order at any finite temperature. The Monte Carlo simulations are performed at various values of α, and the critical value α_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^-α for any nonzero positive value of α.