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Recent theoretical insights into the possibility of non-Abelian phases in$\nu=2/3$ fractional quantum Hall states revived the interest in the numericalphase diagram of the problem. We investigate the effect of various kinds oftwo-body interlayer couplings on the $(330)$ bilayer state and exactly solvethe Hamiltonian for up to $14$ electrons on sphere and torus geometries. Weconsider interlayer tunneling, short-ranged repulsive/attractivepseudopotential interactions and Coulomb repulsion. We find a 6-foldground-state degeneracy on the torus when the interlayer hollow-coreinteraction is dominant. To identify the topological nature of this phase wemeasure the orbital-cut entanglement spectrum, quasihole counting, topologicalentanglement entropy, and wave-function overlap. Comparing the numericalresults to the theoretical predictions, we interpret this 6-fold ground-statedegeneracy phase to be the non-Abelian bilayer Fibonacci state.

Non-Abelian phases in two-component $\nu=2/3$ fractional quantum Hall states: Emergence of Fibonacci anyons