Critical behavior of the Fredenhagen-Marcu order parameter at topological phase transitions

A nonlocal string order parameter detecting topological phase transitions has been proposed by Fredenhagen and Marcu (FM). In this work, we find that the FM string order parameter for the toric code in a magnetic field exhibits universal scaling behavior near critical points by directly evaluating it in the infinite string-length limit using infinite Projected Entangled Pair States (iPEPS). Our results thus indicate that the FM string order parameter is a quantitatively good order parameter to characterize topological phase transitions. In the presence of the electric-magnetic duality symmetry, a novel critical exponent can be extracted from the FM string order parameter, which is consistent with the (2+1)D XY universality class. In addition, we argue that only in the regime with emergent 1-form Wilson loop symmetry, the corresponding FM string order parameter can faithfully detect the topological transition.