We  investigate a microscopic model whose degrees of freedom form  a class of non-abelian anyons, so-called Fibonacci anyons.

We find that  wide parts of the phase diagram are covered by non-abelian topological phases, and  reveal the  role of  topology in determining the essential properties of these phases. Furthermore, we observe  a phase transition between two distinct  topological phases.  Numerically, we establish that this critical point can be described by a conformal field theory with central charge c=14/15.