Universality of the complete-graph Potts model with $0< q \leq 2$

TL;DR

This paper demonstrates that the critical exponents of the complete-graph Potts model are universal for all $0< q extless 2$, showing a hyper-universality in the mean-field limit and revealing $q$-dependent finite-size scaling properties.

Contribution

It provides the first numerical evidence of hyper-universality of critical exponents for the CG Potts model across $0< q extless 2$, and explores the $q$-dependence of finite-size scaling properties.

Findings

Critical exponents are identical for all $0< q extless 2$.

The Ising case ($q=2$) shows unique geometric critical properties.

Finite-size scaling functions depend on $q$, indicating non-universality in these properties.

Abstract

Universality is a fundamental concept in modern physics. For the $q$-state Potts model, the critical exponents are merely determined by the order-parameter symmetry $S_q$, spatial dimensionality and interaction range, independent of microscopic details. In a simplest and mean-field treatment--i.e., the Potts model on complete graph (CG), the phase transition is further established to be of percolation universality for the range of $0 < q <2$. By simulating the CG Potts model in the random-cluster representation, we numerically demonstrate such a hyper-universality that the critical exponents are the same for $0< q <2$ and, moreover, the Ising system ($q = 2$) exhibits a variety of critical geometric properties in percolation universality. On the other hand, many other universal properties in the finite-size scaling (FSS) theory, including Binder-like ratios and distribution function of the order parameter, are observed to be $q$-dependent. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.