Breakdown of a perturbed Z_N topological phase

We study the robustness of a generalized Kitaevs 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 is performed for the perturbed Z_3 toric code by applying a combination of high-order series expansions and variational techniques. We provide strong evidences 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.