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

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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 $p$-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.