First- and second-order transitions in the Potts model near four dimensions

Abstract

The continuum generalization of the $p$-state Potts model is analyzed in the ordered phase. Renormalization-group iterations in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are followed by an elimination of the transverse modes and a mapping onto an effective Ising model. This model is then used to show that the transition is first order for $p>{p}_{c}(d)$ and continuous for $p<{p}_{c}(d)$. We find that ${p}_{c}(d)=2$ for $d>4$ and ${p}_{c}(4\ensuremath{-}\ensuremath{\epsilon})=2+\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$.