We explore critical quantum phase transitions of the
celebrated Z2-toric-code model and its Z3 generalization in
the presence of a magnetic field. The zero-temperature phase
diagram is determined by combining strong-coupling
expansions (pCUTs) and variational methods (iPEPS).
Interestingly, we find a multi-critical line with exotic
properties for the Z2 case. For the generalized
Z3-toric-code we find critical lines in parameter space
which fall in the 3d xy universality class. The latter can
be rigorously shown for special cases using exact mappings
to the antiferromagnetic Potts model.

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