  For systems of lattice anyons like Majorana and parafermions, the
unconventional quantum statistics determines a set of global symmetries (e.g.,
fermion parity for Majoranas) admitting no relevant perturbations. Any operator
that breaks these symmetries explicitly would violate locality if added to the
the Hamiltonian. As a consequence, the associated quasi-degeneracy of
topologically non-trivial phases is protected, at least partially, by locality
via the symmetries singled out by quantum statistics. We show that it is
possible to bypass this type of protection by way of specifically engineered
gauge fields, in order to modify the topological structure of the edge of the
system without destroying the topological order completely. To illustrate our
ideas in a concrete setting, we focus on the \(\Z_6\) parafermion chain.
Starting in the topological phase of the chain (sixfold ground degeneracy), we
show that a gauge field with restricted dynamics acts as a relevant
perturbation, driving a transition to a phase with threefold degeneracy and
\(\Z_3\) parafermion edge modes. The transition from the \(\Z_3\) to the
topologically trivial phase occurs on a critical line in the three-state Potts
universality class. We also investigate numerically the emergence of Majorana
edge modes when the \(\Z_6\) chain is coupled to a differently restricted gauge
field.
