The "not-A", SPT and Potts phases in an $S_3$-invariant chain.

We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in the region of a $U(1)$-invariant self-dual multicritical point. We show that there are four proximate gapped phases, all distinct. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Less common are phases with symmetry-protected topological (SPT) order and its dual, with an unusual "not-$A$" order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). In the SPT phase, the exact ground state is similar to that of Affleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-$A$ phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. By a field-theory analysis, we show that the phases are separated by four transition lines in the universality class of the critical three-state Potts model. These thus provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.