Classifying Potts critical lines

We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points invariant under the permutational symmetry $S_q$ in two dimensions, and we show how one of these scattering solutions describes the ferromagnetic and square lattice antiferromagnetic critical lines of the $q$-state Potts model. Other solutions we determine should correspond to new critical lines. In particular, we obtain that a $S_q$-invariant fixed point can be found up to the maximal value $q=(7+\sqrt{17})/2$. This is larger than the usually assumed maximal value 4 and leaves room for a second-order antiferromagnetic transition at $q=5$.

Figure 1

Amplitudes $S_0,S_1,S_2$, and $S_3$ of the $S_q$-invariant theory. Different letters correspond to different colors.

Figure 2

Pictorial representations associated to the unitarity Eqs. (13), (14), (15), and (16), respectively. The amplitude for the lower crossing multiplies the complex conjugate of the amplitude for the upper crossing, and sum over $\varepsilon$ is implied.