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

Area of Science:

• Statistical physics
• Complex systems
• Network science

Background:

• The collective behavior of interacting systems is crucial in physics.
• Ising models on networks exhibit diverse phase transition behaviors.
• Small-world networks possess unique topological properties influencing system dynamics.

Purpose of the Study:

• To investigate the phase transition in an Ising system on a small-world network.
• To analyze the effect of interaction decay rate (alpha) on long-range order.
• To determine the critical decay rate beyond which phase transitions are absent.

Main Methods:

• Utilizing Monte Carlo simulations to model the Ising system.
• Varying the interaction decay exponent (alpha) to study its impact.
• Analyzing the emergence of long-range order at different temperatures and alpha values.

Main Results:

• The system exhibits a mean-field phase transition for uniform interactions (alpha=0).
• No long-range order is observed at finite temperatures for short-range interactions (alpha→∞).
• The critical decay exponent alpha(c) was estimated to be 0, indicating a sharp transition.

Conclusions:

• The Ising system on a small-world network lacks a phase transition for any non-zero positive interaction decay rate (alpha).
• Algebraically decaying interactions (r^-alpha) prevent long-range order for alpha > 0.
• The findings contribute to understanding phase transitions in complex network systems.