New renormalization group study of the 3-state Potts model and related statistical models

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 $\varepsilon$-expansion can fail.