The $q-$state Potts model from the Nonperturbative Renormalization Group

We study the $q$-state Potts model for $q$ and the space dimension $d$ arbitrary real numbers using the Derivative Expansion of the Nonperturbative Renormalization Group at its leading order, the local potential approximation (LPA and LPA'). We determine the curve $q_c(d)$ separating the first ($q>q_c(d)$) and second ($q<q_c(d)$) order phase transition regions for $2.8<d\leq 4$. At small $\epsilon=4-d$ and $\delta=q-2$ the calculation is performed in a double expansion in these parameters and we find $q_c(d)=2+a \epsilon^2$ with $a\simeq 0.1$. For finite values of $\epsilon$ and $\delta$, we obtain this curve by integrating the LPA and LPA' flow equations. We find that $q_c(d=3)=2.11(7)$ which confirms that the transition is of first order in $d=3$ for the three-state Potts model.