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

Journal of Applied Physics

First order phase transitions and the three state Potts model

Henk W. J. Blöte, Robert H. Swendsen

Published Nov 1, 1979• 117 citations
Publisher

Abstract

We have used the Monte Carlo renormalization group method to study the order-disorder phase transition of the three and four dimensional, three-state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free-energy branches, no additional singularities occur at the transition.

DOI
10.1063/1.326900

Related papers

First-Order Phase Transitions and the Three-State Potts Model

Henk W. J. Blöte, Robert H. Swendsen

We have used the Monte Carlo renormalization-group method to study the three-state Potts model in three and four dimensions. In both cases, we find a first-order transition without an associated discontinuity fixed point. The transition in three dimensions is "almost second order" in the sense that some evidence was found for the existence of second-order fixed points associated with singularities in the metastable region just beyond the first-order transition.

Physical Review Letters•Sep 10, 1979•173 citations

Renormalization-group calculation of first- and second-order phase transitions in the Potts model

Erling Pytte

A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for $p>2$. For $2<p<\frac{10}{3}$ the transition is first order in spite of the fact that there is an accessible stable fixed point for all $p<\frac{10}{3}$. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature ${T}_{c}$, due to instantons, which is of the same form as that of the Ising model at $H=0$.

Physical review. B, Condensed matter•Nov 1, 1980•22 citations

Is the Phase Transition of the Three-State Potts Model Continuous in Three Dimensions?

Svend J. Knak Jensen, Ole G. Mouritsen

The internal energy and the order parameter of the three-state Potts model in a simple cubic lattice are calculated using the Monte Carlo technique. Both properties exhibit metastabilities in a small temperature region demonstrating that the phase transition is of first order. This result is in agreement with the prediction of the $\ensuremath{\epsilon}$ expansion but in contrast to the position-space renormalization-group calculations which lead to a continuous transition.

Physical Review Letters•Dec 3, 1979•101 citations

First- and second-order phase transitions in Potts models: Competing mechanisms (invited)

A. Nihat Berker, David Andelman

Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.

Journal of Applied Physics•Nov 1, 1982•24 citations

New renormalization group study of the 3-state Potts model and related statistical models

José Gaite

The critical behavior of three-state statistical models invariant under the full symmetry group S 3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S 3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10 / 3 , which extends continuously to d = 3 . It does not correspond to a second-order phase transition of the 3-state Potts model , but is interesting in its own right. In particular, it shows how the ε -expansion can fail.

arXiv:2407.15497•Nuclear Physics B•Jan 15, 2025•3 citations

New renormalization group study of the 3-state Potts model and related statistical models

José Gaite

The critical behavior of three-state statistical models invariant under the full symmetry group S3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10/3, which extends continuously to d = 3 . It does not correspond to a second- order phase transition of the 3-state Potts model, but is interesting in its own right. In particular, it shows how the E-expansion can fail.

•Jan 16, 2025•0 citations
We have used the Monte Carlo renormalization group method to study the order-disorder phase transition of the three and four dimensional, three-state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free-energy branches, no additional singularities occur at the transition.
A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for p>2p>2p>2. For 2<p<1032<p<\frac{10}{3}2<p<310​ the transition is first order in spite of the fact that there is an accessible stable fixed point for all p<103p<\frac{10}{3}p<310​. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature Tc{T}_{c}Tc​, due to instantons, which is of the same form as that of the Ising model at H=0H=0H=0.
Many condensed matter systems, ranging from adsorbed surfaces to bulk magnets, are microscopically modelled by interacting q-state Potts spins, arrayed in d dimensions. A changeover from second-order phase transitions at q≤qc(d) to first-order transitions at q≳qc can be understood as a condensation of effect vacancies, which are patches of local disorder favored by entropy. Accordingly, the renormalization-group treatment of Potts models is within context of Potts-lattice-gas models, where critical and tricritical fixed points occur at low q, but merge and annihilate at qc. This picture has led to exact tricritical exponents in two dimensions. It is also consistent with recent experimental results on intercalated systems in three dimensions. Effective vacancies in pure Potts models have also been studied by Monte Carlo simulation. Their effective chemical potential can be controlled by a four-point interaction, which proved useful in Monte Carlo renormalization-group studies.
The critical behavior of three-state statistical models invariant under the full symmetry group S 3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S 3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10 / 3 , which extends continuously to d = 3 . It does not correspond to a second-order phase transition of the 3-state Potts model , but is interesting in its own right. In particular, it shows how the ε -expansion can fail.
The critical behavior of three-state statistical models invariant under the full symmetry group S3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10/3, which extends continuously to d = 3 . It does not correspond to a second- order phase transition of the 3-state Potts model, but is interesting in its own right. In particular, it shows how the E-expansion can fail.