Recently there has been great progress in realizing platforms for topological quantum computation, with mounting evidence of the experimental observation of Majorana zero modes. However, braiding such zero modes does not yield a set of transformations sufficient to perform universal, fault tolerant computation. One way forward is to engineer systems realizing Z3 parafermion zero modes, which generalize Majorana zero modes. Coupled Z3 parafermions could hybridize into a phase supporting bulk Fibonacci anyons, a type of non-Abelian anyon that does have universal braiding statistics.
Using the density matrix renormalization group (DMRG), we study a two-dimensional model of coupled Z3 parafermions. By working close to the weakly-coupled chain limit, we are able to identify the Fibonacci phase on cylinders as small as four sites in circumference then track its evolution, finding it survives even to the isotropic limit of our model on larger cylinders. We examine the extent of this phase and the wider phase diagram of our model, which turns out to harbor a second topological phase.