Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.16930 · Physical review. E

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

Zirui Peng, Sheng Fang, Hao Hu, Youjin Deng

Published May 23, 2025• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/physreve.111.054134

Related papers

Spontaneous edge order and geometric aspects of two-dimensional Potts models

Youjin Deng, Henk W. J. Blöte

Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.

Physical Review E•Sep 30, 2004•8 citations

Emergent O( n ) symmetry in a series of three-dimensional Potts models

Chengxiang Ding, Henk W. J. Blöte, Youjin Deng

Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-${T}_{c}$ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of $q$-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O($n$) symmetry with $n$=$q$-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O($n$=$q$-1) universality classes.

arXiv:1508.04538•Physical review. B./Physical review. B•Sep 1, 2016•13 citations

Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

Yunqing Ouyang, Youjin Deng, Henk W. J. Blöte

We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number $z$ of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite $z$ are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for $q=3$ and $q=4$ Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.

arXiv:1808.05812•Physical review. E•Dec 3, 2018•22 citations

Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

Sheng Fang, Zongzheng Zhou, Youjin Deng

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.

arXiv:2008.07256•Physical review. E•Jan 4, 2021•8 citations

Randomness-induced second-order transition in the two-dimensional eight-state Potts model: A Monte Carlo study

S. Chen, Alan M. Ferrenberg, D. P. Landau

We have studied the effect of quenched, bond randomness on the nature of the phase transition in the two-dimensional eight-state Potts model. Through extensive Monte Carlo simulations, we confirm that the phase transition changes from first order to second order. A finite-size-scaling analysis of several thermodynamic quantities strongly suggests that the critical exponents fall into the universality class of the two-dimensional Ising model.

Physical Review Letters•Aug 24, 1992•116 citations

Dilute Potts model in two dimensions

Xiaofeng Qian, Youjin Deng, Henk W. J. Blöte

We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.

Physical Review E•Nov 29, 2005•46 citations
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.
Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.
Scaling, universality, and renormalization are three pillars of modern critical phenomena. According to the hypothesis of universality, continuous phase transitions fall into classes mainly determined by spatial dimensionality and symmetry of the order parameter. The latter is usually reflected by the degeneracy of the ground state of the Hamiltonian. However, for certain systems at criticality, a higher symmetry emerges in the order parameter, and the associated critical behavior may become very rich. Such emergent symmetry has been found in spin ice systems, deconfined quantum critical points, high-Tc{T}_{c}Tc​ superconductors, and so forth, and are often accompanied by very interesting critical phenomena. New results presented here are based on Monte Carlo simulations and finite-size scaling of a series of qqq-state Potts models on the simple cubic lattice with ferromagnetic interactions in one lattice direction and antiferromagnetic interactions in the other two directions. The staggered magnetization appears to display an emergent continuous O(nnn) symmetry with nnn=qqq-1, as illustrated by two-dimensional intersections of the distribution functions. Also the estimated critical exponents are consistent with the O(nnn=qqq-1) universality classes.
We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number zzz of equivalent neighbors. We also consider the $z\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite zzz are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for q=3q=3q=3 and q=4q=4q=4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as ${y}_{r}\ensuremath{\approx}2/3$. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.
The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q=2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q=2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q=1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q=1 instead of q=2. This demonstrates the percolation effects in the FK Ising model on the complete graph.
We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.