Commuting-projector Hamiltonians for Chiral Topological Phases Built from Parafermions*

We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve Z₃ 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. These characteristics indicate that the first model realizes a symmetry-enriched topological phase (SET) with an anyon-permuting Z₂ symmetry action, while the ground state in the second model is consistent with non-Abelian SU(2)₄ topological order. The non-Abelian phase can be accessed by gauging the Z₂ symmetry in the SET. Employing Levin-Wen string-net models with Z₂-graded structure, we generalize this picture to construct a large class of commuting-projector models for Z₂ SET's 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.