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We study a simple model of symmetry-enriched topological order obtained bydecorating a toric code model with lower-dimensional symmetry-protectedtopological states. We show that the symmetry fractionalization in this modelcan be characterized by string order parameters, and that these signatures arerobust under the effects of external fields and interactions, up to the phasetransition point. This extends the recent proposal of [New Journal of Physics21, 113016 (2019)] beyond the setting of fixed-point tensor network states, andsolidifies string order parameters as a useful tool to characterize and detectsymmetry fractionalization. In addition to this, we observe how thecondensation of an anyon that fractionalizes a symmetry forces that symmetry tospontaneously break, and we give a proof of this in the framework of projectedentangled pair states. This phenomenon leads to a notable change in the phasediagram of the toric code in parallel magnetic fields.

String order parameters for symmetry fractionalization in an enriched toric code