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We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in theregion of a U(1)-invariant self-dual multicritical point. We find four distinctproximate gapped phases. One has three-state Potts order, corresponding totopological order in a parafermionic formulation. Also nearby is a phase with"representation" symmetry-protected topological (RSPT) order. Its dual exhibitsan unusual "not-A" order, where the spins prefer to align in two of the threedirections. Within each of the four phases, we find a frustration-free pointwith exact ground state(s). The exact RSPT ground state is similar to that ofAffleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-A phase areproduct states, each an equal-amplitude sum over all states where one of thethree spin states on each site is absent. A field-theory analysis shows thatall transitions are in the universality class of the critical three-state Pottsmodel. They provide a lattice realization of a flow from a free-boson fieldtheory to the Potts conformal field theory.

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