  A nonlocal string order parameter detecting topological order and
deconfinement has been proposed by Fredenhagen and Marcu (FM). However, due to
the lack of exact internal symmetries for lattice models and the nonlinear
dependence of the FM string order parameter on ground states, it is a priori
not guaranteed that it is a genuine order parameter for topological phase
transitions. In this work, we find that the FM string order parameter exhibits
universal scaling behavior near critical points of charge condensation
transitions, by directly evaluating the FM string order parameter in the
infinite string-length limit using infinite Projected Entangled Pair States
(iPEPS) for the toric code in a magnetic field. Our results thus demonstrate
that the FM string order parameter represents a quantitatively well-behaved
order parameter. We find that only in the presence of an emergent 1-form
symmetry the corresponding FM string order parameter can faithfully detect
topological transitions.
