Commuting-projector Hamiltonians for chiral topological phases built from parafermions

Abstract

We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve ${\mathbb{Z}}_3$ parafermion zero modes residing in a parent fractional-quantum-Hall fluid. The two simplest models in this family emerge from dressing Ising-paramagnet and toric-code spin models with parafermions; we study their edge properties, anyonic excitations, and ground-state degeneracy. We show that the first model realizes a symmetry-enriched topological phase (SET) for which ${\mathbb{Z}}_2$ spin-flip symmetry from the Ising paramagnet permutes the anyons. Interestingly, the interface between this SET and the parent quantum-Hall phase realizes symmetry-enforced ${\mathbb{Z}}_3$ parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with $\text{SU}(2{)}_4$ topological order, and can be accessed by gauging the ${\mathbb{Z}}_2$ symmetry in the SET. Employing Levin-Wen string-net models with ${\mathbb{Z}}_2$-graded structure, we generalize this picture to construct a large class of commuting-projector models for ${\mathbb{Z}}_2$ SETs and non-Abelian topological orders exhibiting the same relation. Our construction provides the first commuting-projector-Hamiltonian realization of chiral bosonic non-Abelian topological order.

29+18 pages, 25 figures

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