It is shown that a trilinear symmetry breaking which destroys the equivalence of the states in a continuum generalization of the $p$ -state Potts model yields first-order phase transitions for all $p>1$ , in contrast to the results of the symmetric theory where there is a second-order transition for $p<2$ and a first-order transition for $p>2$ , in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, and in spite of mean-field---theory predictions. A calculation in renormalized perturbation theory, to one-loop order, on a three-trilinear-coupling theory yields two new accessible and partly stable asymmetric fixed points beyond the symmetric one, except for the three-state Potts model, one for ${2.2}^{\ensuremath{-}}\ensuremath{\le}p<\ensuremath{\infty}$ and the other for $1<p<\frac{13}{3}$ . There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when $p>\frac{13}{3}$ , which are interpreted as the usual first-order transitions. For the indicated ranges of $p$ the transitions are of first order near a spinodal point, with uniaxial ordering in the first case and transverse ordering in the second. Critical exponents that could describe the approach to the spinodal points are explicitly calculated.It is shown that a trilinear symmetry breaking which destroys the equivalence of the states in a continuum generalization of the $p$-state Potts model yields first-order phase transitions for all $p>1$, in contrast to the results of the symmetric theory where there is a second-order transition for $p<2$ and a first-order transition for $p>2$, in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, and in spite of mean-field---theory predictions. A calculation in renormalized perturbation theory, to one-loop order, on a three-trilinear-coupling theory yields two new accessible and partly stable asymmetric fixed points beyond the symmetric one, except for the three-state Potts model, one for ${2.2}^{\ensuremath{-}}\ensuremath{\le}p<\ensuremath{\infty}$ and the other for $1<p<\frac{13}{3}$. There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when $p>\frac{13}{3}$, which are interpreted as the usual first-order transitions. For the indicated ranges of $p$ the transitions are of first order near a spinodal point, with uniaxial ordering in the first case and transverse ordering in the second. Critical exponents that could describe the approach to the spinodal points are explicitly calculated.

Physical review. B, Condensed matterJun 1, 19835 citations

The Phase Transition in the Three-State Potts Model: An Analytical Analysis

Ya-Li Song, Jin Min Yang, Gong Ru Lu

We investigate the phase transition of the three-state Potts model in an analytical approach—the generalized cumulant expansion with the effective mean field hypothesis. We find a first order phase transition in the three-dimensional three-state Potts model with ferromagnetic nearest neighbor coupling. For the model with antiferromagnetic next-to-nearest neighbor coupling, we find a first order transition when the relative strength of the -coupling is fixed to . The critical values given by this method are also in agreement with the recent high statistics Monte Carlo results.We investigate the phase transition of the three-state Potts model in an analytical approach—the generalized cumulant expansion with the effective mean field hypothesis. We find a first order phase transition in the three-dimensional three-state Potts model with ferromagnetic nearest neighbor coupling. For the model with antiferromagnetic next-to-nearest neighbor coupling, we find a first order transition when the relative strength of the -coupling is fixed to . The critical values given by this method are also in agreement with the recent high statistics Monte Carlo results.

Communications in Theoretical PhysicsDec 1, 19920 citations

Phase Transition in Three-State Three-Dimensional Potts Model.

Jing-Dong Wang

Jan 1, 19920 citations

Phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction

Alessandro Campa, Vahan Hovhannisyan, Stefano Ruffo +1 more

Abstract We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-1 Blume–Emery–Griffiths model and solving it by using a Hubbard–Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling K and the temperature T . We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase transition lines. We also prove that, when the nearest-neighbour coupling K is large and negative, the first-order transition temperature becomes asymptotically independent of K .Abstract We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-1 Blume–Emery–Griffiths model and solving it by using a Hubbard–Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling K and the temperature T . We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase transition lines. We also prove that, when the nearest-neighbour coupling K is large and negative, the first-order transition temperature becomes asymptotically independent of K .

arXiv:2512.02930Journal of Statistical Mechanics Theory and ExperimentFeb 1, 20260 citations

Phase transition properties of 3D Potts models

Alexei Bazavov, Bernd A. Berg, Santosh Dubey

arXiv:0804.1402Nuclear Physics BApr 28, 200839 citations

It is shown that a trilinear symmetry breaking which destroys the equivalence of the states in a continuum generalization of the

p

-state Potts model yields first-order phase transitions for all

p>1

, in contrast to the results of the symmetric theory where there is a second-order transition for

p<2

and a first-order transition for

p>2

, in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, and in spite of mean-field---theory predictions. A calculation in renormalized perturbation theory, to one-loop order, on a three-trilinear-coupling theory yields two new accessible and partly stable asymmetric fixed points beyond the symmetric one, except for the three-state Potts model, one for ${2.2}^{\ensuremath{-}}\ensuremath{\le}p<\ensuremath{\infty}$ and the other for

1<p<\frac{13}{3}

. There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when

p>\frac{13}{3}

, which are interpreted as the usual first-order transitions. For the indicated ranges of

p

the transitions are of first order near a spinodal point, with uniaxial ordering in the first case and transverse ordering in the second. Critical exponents that could describe the approach to the spinodal points are explicitly calculated.