  We investigate the self-dual three-state quantum chain with nearest-neighbor
interactions and $S_3$, time-reversal, and parity symmetries. We find a rich
phase diagram including gapped phases with order-disorder coexistence,
integrable critical points with U(1) symmetry, and ferromagnetic and
antiferromagnetic critical regions described by three-state Potts and
free-boson conformal field theories respectively. We also find an unusual
critical phase which appears to be described by combining two conformal field
theories with distinct "Fermi velocities". The order-disorder coexistence phase
has an emergent fractional supersymmetry, and we find lattice analogs of its
generators.
