Commuting-projector Hamiltonians for chiral topological phases built from parafermions

Abstract

We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve ℤ_3 parafermion zero modes residing in a parent fractional-quantum-Hall fluid. These commuting-projector models inherit nontrivial Hall conductance from the parent quantum-Hall states in which they are defined, and thus can describe chiral topological phases. 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 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 ℤ_3 parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with SU(2)_4 topological order, and can be accessed by gauging the ℤ_2 symmetry in the SET. Employing Levin-Wen string-net models with ℤ_2-graded structure, we generalize this picture to construct a large class of commuting-projector models for 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.