  We examine the zero-temperature phase diagram of the two-dimensional
Levin-Wen string-net model with Fibonacci anyons in the presence of competing
interactions. Combining high-order series expansions around three exactly
solvable points and exact diagonalizations, we find that the non-Abelian
doubled Fibonacci topological phase is separated from two nontopological phases
by different second-order quantum critical points, the positions of which are
computed accurately. These trivial phases are separated by a first-order
transition occurring at a fourth exactly solvable point where the ground-state
manifold is infinitely many degenerate. The evaluation of critical exponents
suggests unusual universality classes.
