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:2309.06489 · Physical review. E

q -state Potts model from the nonperturbative renormalization group

Carlos A. Sánchez-Villalobos, Bertrand Delamotte, Nicolás Wschebor

Published Dec 14, 2023• 9 citations
arXivHTMLPDF

Abstract

We study the $q$-state Potts model for $q$ and the space dimension $d$ arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and ${\mathrm{LPA}}^{\ensuremath{'}}$). We determine the curve ${q}_{c}(d)$ separating the first $[q>{q}_{c}(d)]$ and second $[q<{q}_{c}(d)]$ -order phase transition regions for $2.8<d\ensuremath{\le}4$. At small $\ensuremath{\epsilon}=4\ensuremath{-}d$ and $\ensuremath{\delta}=q\ensuremath{-}2$ the calculation is performed in a double expansion in these parameters, and we find ${q}_{c}(d)=2+a{\ensuremath{\epsilon}}^{2}$ with $a\ensuremath{\simeq}0.1$. For finite values of $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$, we obtain this curve by integrating the LPA and ${\mathrm{LPA}}^{\ensuremath{'}}$ flow equations. We find that ${q}_{c}(d=3)=2.11(7)$, which confirms that the transition is of first order in $d=3$ for the three-state Potts model.

DOI
10.1103/physreve.108.064120

Related papers

Q -state Potts model by Wilson's exact renormalization-group equation

Kathie E. Neuman, Eberhard K. Riedel, Shunichi Muto

Critical properties of the $Q$-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of $Q$ that annihilate as $Q$ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For $Q>{Q}_{c}$, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients $a$ and $b$ cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, $a=6.52$ and $b=0.065$. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, $Q=1$, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.

Physical review. B, Condensed matter•Jan 1, 1984•33 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

q -state Potts model in general dimension

Bernard Nienhuis, Eberhard K. Riedel, Michael Schick C

Critical properties of the $q$-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for ${q}_{c}$, the largest value of $q$ for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.

Physical review. B, Condensed matter•Jun 1, 1981•77 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

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

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 study the qqq-state Potts model for qqq and the space dimension ddd arbitrary real numbers using the derivative expansion of the nonperturbative renormalization group at its leading order, the local potential approximation (LPA and LPA\ensuremath′{\mathrm{LPA}}^{\ensuremath{'}}LPA\ensuremath′). We determine the curve qc(d){q}_{c}(d)qc​(d) separating the first [q>qc(d)][q>{q}_{c}(d)][q>qc​(d)] and second [q<qc(d)][q<{q}_{c}(d)][q<qc​(d)] -order phase transition regions for 2.8<d\ensuremath≤42.8<d\ensuremath{\le}42.8<d\ensuremath≤4. At small \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d and \ensuremathδ=q\ensuremath−2\ensuremath{\delta}=q\ensuremath{-}2\ensuremathδ=q\ensuremath−2 the calculation is performed in a double expansion in these parameters, and we find qc(d)=2+a\ensuremathϵ2{q}_{c}(d)=2+a{\ensuremath{\epsilon}}^{2}qc​(d)=2+a\ensuremathϵ2 with a\ensuremath≃0.1a\ensuremath{\simeq}0.1a\ensuremath≃0.1. For finite values of \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ and \ensuremathδ\ensuremath{\delta}\ensuremathδ, we obtain this curve by integrating the LPA and LPA\ensuremath′{\mathrm{LPA}}^{\ensuremath{'}}LPA\ensuremath′ flow equations. We find that qc(d=3)=2.11(7){q}_{c}(d=3)=2.11(7)qc​(d=3)=2.11(7), which confirms that the transition is of first order in d=3d=3d=3 for the three-state Potts model.
Critical properties of the QQQ-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of QQQ that annihilate as QQQ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For Q>QcQ>{Q}_{c}Q>Qc​, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients aaa and bbb cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, a=6.52a=6.52a=6.52 and b=0.065b=0.065b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1Q=1Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
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.
Critical properties of the qqq-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for qc{q}_{c}qc​, the largest value of qqq for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.
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.