Slow relaxation in the Ising model on a small-world network with strong long-range interactions

TL;DR

This study investigates how long-range interactions affect phase transitions and relaxation in the Ising model on small-world networks, revealing slow relaxation with strong interactions and proposing an improved simulation algorithm.

Contribution

It introduces a modified Monte Carlo updating algorithm to accelerate equilibration in the Ising model with strong long-range interactions on small-world networks.

Findings

Critical temperature increases with long-range interaction strength.

Strong long-range interactions cause very slow relaxation.

Modified algorithm improves convergence to equilibrium.

Abstract

We consider the Ising model on a small-world network, where the long-range interaction strength $J_2$ is in general different from the local interaction strength $J_1$, and examine its relaxation behaviors as well as phase transitions. As $J_2/J_1$ is raised from zero, the critical temperature also increases, manifesting contributions of long-range interactions to ordering. However, it becomes saturated eventually at large values of $J_2/J_1$ and the system is found to display very slow relaxation, revealing that ordering dynamics is inhibited rather than facilitated by strong long-range interactions. To circumvent this problem, we propose a modified updating algorithm in Monte Carlo simulations, assisting the system to reach equilibrium quickly.