Geometric properties of the complete-graph Ising model in the loop representation

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Computational Physics

Background:

• The Ising model on a complete graph (CG) offers mean-field insights into continuous phase transitions.
• Alternative formulations like the Fortuin-Kasteleyn random cluster and loop representations reveal geometric properties absent in spin representations.

Purpose of the Study:

• To investigate the CG-Ising model within the loop representation.
• To derive exact results for geometric quantities and scaling forms.
• To elucidate geometric phenomena observed in the random-cluster representation.

Main Methods:

• Utilized a lifted-worm irreversible algorithm to study the CG-Ising model in its loop representation.
• Performed theoretical and numerical analyses to derive exact results.
• Combined the loop representation with the loop-cluster algorithm.

Main Results:

• Obtained exact results for volume fractal dimensions and scaling forms of the CG-Ising model.
• Demonstrated how the loop representation intuitively explains geometric phenomena like two configuration sectors, two length scales, and two scaling windows.
• Provided a deeper understanding of the CG-Ising model's geometric properties.

Conclusions:

• The loop representation offers a powerful framework for analyzing the geometric aspects of the CG-Ising model.
• Exact results derived from the loop representation enhance the understanding of continuous phase transitions.
• The study bridges the gap between different representations of the Ising model, revealing rich geometric behaviors.