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.

We studied the robustness of a generalized Kitaev's toric code with Z N degrees of freedom in the presence of local perturbations. For N = 2, this model reduces to the conventional toric code in a uniform magnetic field. A quantitative analysis was performed for the perturbed Z 3 toric code by applying a combination of high-order series expansions and variational techniques. We found strong evidence for first-and second-order phase transitions between topologically ordered and polarized phases. Most interestingly, our results also indicate the existence of topological multi-critical points in the phase diagram. A. Series expansion in the limit |h Z | J for h X = h ⊥ = 0 23 Appendix B. Series expansion in the limit |h X |, |h Z | J for h ⊥ = 0 25 Appendix C. Series expansion in the limit 0 < h ⊥ J for h X = h Z = 0 26 References 26