Assembling Fibonacci Anyons From a $\mathbb{Z}_3$ Parafermion Lattice Model

Recent concrete proposals suggest it is possible to engineer a two-dimensional bulk phase supporting non-Abelian Fibonacci anyons out of Abelian fractional quantum Hall systems. The low-energy degrees of freedom of such setups can be modeled as $\mathbb{Z}_3$ parafermions "hopping" on a two-dimensional lattice. We use the density matrix renormalization group to study a model of this type interpolating between the decoupled-chain, triangular-lattice, and square-lattice limits. The results show clear evidence of the Fibonacci phase over a wide region of the phase diagram, most notably including the isotropic triangular lattice point. We also study the broader phase diagram of this model and show that elsewhere it supports an Abelian state with semionic excitations.