The effects of linear and quadratic symmetry-breaking perturbations on the continuous version of the three-state Potts model are analysed, using both Landau's theory of phase transitions and renormalisation-group techniques. Variation of the strength of the perturbations produces many different types of phase diagrams, featuring lambda lines, first-order lines, critical, tricritical and triple points and other complexities. Universal amplitude ratios characterising the multicritical points are calculated to first order in epsilon =4-d.
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.
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<313. There is a fixed-point runaway for the first one when $p<{2.2}^{\ensuremath{-}}$ and for the second when
p>313, 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.
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∗, where
p∗\ensuremath{\lesssim}2 but not with a second-order transition for p
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.
The Q-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general special choices of these magnetic fields produce effective models with smaller symmetry . The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behaviour, for finite values of the symmetry-breaking fields, in the cases where Q > 4 is also analysed. Our results indicate that for effective models with symmetry the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.
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.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (
n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.