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

Abstract

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 find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Also nearby is a phase with "representation" symmetry-protected topological (RSPT) order. Its dual exhibits 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). The exact RSPT 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. A field-theory analysis shows that all transitions are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.

arXiv comment14 pages, 5 figures. v2: refined the SPT discussion, and added one letter to the title. v3: further refinement, n.b. RSPT in title is spelled out in journal

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Cited by in corpus (14)

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• Integrable Quantum Circuits from the Star-Triangle Relation
• Self-dual $S_3$-invariant quantum chains
• Quantum criticality of a ${\mathbb{Z}}_3$ symmetric spin chain with long-range interactions
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Pivoting through the chiral-clock family
• Symmetry-Enforced Fermi Surfaces
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Interrelations among frustration-free models via Witten's conjugation
• Spurious Strange Correlators in Symmetry-Protected Topological Phases
• Phase diagram of an extended parafermion chain