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

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 hyperuniversality 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. Meanwhile, we have made improvements to the Monte Carlo algorithms for efficiently simulating the CG Potts model. Our finding provides valuable insights for the study of critical phenomena in finite spatial dimensions, particularly when the FSS theory is utilized.