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

TL;DR

This study investigates the phase transition behavior of an Ising model on a small-world network with distance-dependent interactions, finding no phase transition for any positive decay exponent, thus extending understanding of long-range order in such systems.

Contribution

The paper provides the first comprehensive analysis of the Ising model with algebraically decaying interactions on small-world networks, establishing the absence of phase transitions for all positive decay exponents.

Findings

No phase transition for any positive alpha

Critical alpha_c estimated to be zero

Long-range order only at alpha=0

Abstract

We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) \propto r^{-\alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $\alpha = 0$ 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 $(\alpha\to\infty)$ does not exhibit long-range order at any finite temperature. 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 zero. 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 $\alpha$.