Commuting-projector Hamiltonians for chiral topological phases built from parafermions

TL;DR

This paper constructs commuting-projector Hamiltonians involving parafermions to realize and analyze chiral topological phases, including symmetry-enriched and non-Abelian orders, providing new models with potential applications in topological quantum computing.

Contribution

It introduces a new family of commuting-projector Hamiltonians with parafermions that realize chiral topological phases, including the first such model for non-Abelian order.

Findings

Realization of a symmetry-enriched topological phase with parafermions.

Construction of a non-Abelian phase consistent with SU(2)_{4} topological order.

Development of a large class of models for Z_2 SETs and non-Abelian orders.

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.