Universality of the complete-graph Potts model with 0<q≤2

Area of Science:

• Statistical Physics
• Critical Phenomena

Background:

• Universality dictates critical exponents depend on symmetry, dimensionality, and range, not microscopic details.
• The complete graph (CG) Potts model simplifies phase transitions, linking to percolation universality for 0

Purpose of the Study:

• To numerically demonstrate hyperuniversality in the CG Potts model.
• To investigate the critical geometric properties of the Ising system (q=2) within percolation universality.
• To analyze the q-dependence of finite-size scaling (FSS) properties.

Main Methods:

• Simulations of the complete graph Potts model using the random-cluster representation.
• Development of improved Monte Carlo algorithms for efficient simulation.
• Analysis of critical exponents and finite-size scaling behaviors.

Main Results:

• Numerical confirmation of hyperuniversality: critical exponents are identical for 0
• The Ising system (q=2) displays critical geometric properties consistent with percolation universality.
• Finite-size scaling theory reveals q-dependent universal properties, including Binder-like ratios and order parameter distributions.

Conclusions:

• The study confirms hyperuniversality in the CG Potts model for a range of q.
• Findings offer insights into critical phenomena in finite dimensions, especially concerning FSS theory.
• Improved algorithms enhance the simulation efficiency of the CG Potts model.