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Physical review. B, Condensed matter

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

Erling Pytte

Published Nov 1, 1980• 22 citations
Publisher

Abstract

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 $H=0$.

DOI
10.1103/physrevb.22.4450

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A renormalization-group calculation has been carried out in d=6\ensuremath−\ensuremathϵd=6\ensuremath{-}\ensuremath{\epsilon}d=6\ensuremath−\ensuremathϵ dimensions to lowest order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, 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−2p\ensuremath{-}2p\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.
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.
We report the numerical study, using Wilson's $\ensuremath{\eta}=0$ approximate renormalization-group recursion formula, of a continuous-spin generalization of the three-component Potts model. Previous numerical studies, based on series expansions of the partition function, have indicated strongly that the model has a phase transition of second order. This is in conflict with Landau theory which predicts that the transition be of first order. We find, in agreement with Landau theory, that our model exhibits a first-order phase transition having finite zero-field spontaneous magnetization at the transition point.
It is shown that a trilinear symmetry breaking which destroys the equivalence of the states in a continuum generalization of the ppp-state Potts model yields first-order phase transitions for all p>1p>1p>1, in contrast to the results of the symmetric theory where there is a second-order transition for p<2p<2p<2 and a first-order transition for p>2p>2p>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<1331<p<\frac{13}{3}1<p<313​. There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when p>133p>\frac{13}{3}p>313​, which are interpreted as the usual first-order transitions. For the indicated ranges of ppp 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.
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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 $3\ensuremath{\le}d\ensuremath{\le}6$.