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arXiv · 2501.16930

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

Zirui Peng·Sheng Fang·Hao Hu·Youjin Deng
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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.

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Zirui Peng, Sheng Fang, Hao Hu, Youjin Deng. 2025-01-28. Universality of the complete-graph Potts model with $0< q \leq 2$. https://doi.org/10.1103/physreve.111.054134

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