  The non-abelian topological phase with Fibonacci anyons minimally supports
universal quantum computation. In order to investigate the possible phase
transitions out of the Fibonacci topological phase, we propose a generic
quantum-net wavefunction with two tuning parameters dual with each other, and
the norm can be exactly mapped into a partition function of the two-coupled
$\phi^{2}$-state Potts models, where $\phi =(\sqrt{5}+1)/2$ is the golden
ratio. By developing the tensor network representation of this wavefunction on
a square lattice, we can accurately calculate the full phase diagram with the
numerical methods of tensor networks. More importantly, it is found that the
non-abelian Fibonacci topological phase is enclosed by three distinct
non-topological phases and their dual phases of a single $\phi^{2}$-state Potts
model: the gapped dilute net phase, critical dense net phase, and spontaneous
translation symmetry breaking gapped phase. We also determine the critical
properties of the phase transitions among the Fibonacci topological phase and
those non-topological phases.
