Tricritical point with fractional supersymmetry from a Fibonacci topological state

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.