Critical behavior of Fredenhagen-Marcu string order parameters at topological phase transitions with emergent higher-form symmetries

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.