  We consider a generic Fibonacci topological wave function on a square
lattice, and the norm of this wave function can be mapped into the partition
function of two-coupled $\phi ^{2}$-state Potts models with $\phi
=(\sqrt{5}+1)/2$ as the golden ratio. A global phase diagram is thus
established to display non-abelian topological phase transitions. The Fibonacci
topological phase corresponds to an emergent new phase of the two-coupled Potts
models, and continuously change into two non-topological phases separately,
which are dual each other and divided by a first-order phase transition line.
Under the self-duality, the Fibonacci topological state enters into the
first-order transition state at a quantum tricritical point, where two
continuous quantum phase transitions bifurcate. All the topological phase
transitions are driven by condensation of anyonic bosons consisting of
Fibonacci anyon and its conjugate. However, a fractional supersymmetry is
displayed at the quantum tricritical point, characterized by a coset conformal
field theory.
