  We study 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 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.
