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Physical review. B, Condensed matter

First-order phase transitions in the Potts model with trilinear symmetry breaking

W. K. Theumann

Published Jun 1, 1983• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.27.6941

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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−\ensuremathϵd=6\ensuremath{-}\ensuremath{\epsilon}d=6\ensuremath−\ensuremathϵ 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≤p<\ensuremath∞{2.2}^{\ensuremath{-}}\ensuremath{\le}p<\ensuremath{\infty}2.2\ensuremath−\ensuremath≤p<\ensuremath∞ 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−p<{2.2}^{\ensuremath{-}}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.
Renormalized perturbation theory with generalized minimal subtraction is used as an appropriate renormalization-group procedure for the study of crossover behavior in the continuum version of the p-state Potts model with quadratic and trilinear symmetry breaking, within the representation of Priest and Lubensky, by means of a two-loop-order calculation in d=6-\ensuremath{\epsilon} dimensions. The boundary between first- and second-order phase transitions is studied for longitudinal and transverse ordering as a function of p. A fixed-point runaway for longitudinal ordering is consistent with a mean-field interpretation of a first-order transition for p>p∗{p}^{\mathrm{*}}p∗, where p∗{p}^{\mathrm{*}}p∗\ensuremath{\lesssim}2 but not with a second-order transition for pp∗{p}^{\mathrm{*}}p∗. Finite and stable fixed points are obtained for transverse ordering, one that follows by crossover from the symmetric fixed point for 2(13/3), in consistency with the usual mean-field interpretation of a second-order transition for 2p3.
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.
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 effect of quadratic symmetry breaking (QSB) on two representations for the Potts vectors in a continuum-field model are studied to two-loop order in renormalized perturbation theory in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, in extension of an earlier group-theoretical analysis by Wallace and Young. The explicit dependence of the crossover exponent $\ensuremath{\varphi}$ that corresponds to QSB that destroys the equivalence between pairs of Potts vectors is obtained as a function of ddd and nnn for the ppp-state model with p=n+1p=n+1p=n+1. It is shown that this exponent follows from the calculation of vertex functions in a representation due to Wallace and Young, whereas a second crossover exponent $\overline{\ensuremath{\varphi}}$, that can be identified with the critical exponent $\ensuremath{\beta}$, and which corresponds to QSB that favors a single Potts vector against the others, follows from a calculation using the representation of the Potts vectors due to Priest and Lubensky.
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.