Variational methods for characterizing matrix product operator symmetries

We present a method of extracting information about topological order from the ground state strongly correlated two-dimensional system represented by an infinite projected entangled pair (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 we begin determining symmetries iPEPS matrix product operators (iMPO) that map between different transfer fixed points, to which apply fundamental theorem MPS find zipper tensors products iMPO's encode fusion properties anyons. The zippers can be combined extract $F$-symbols underlying category, unequivocally identify state. bring canonical gauge, as well compute Drinfeld center this unitary category $S$ $T$ matrices encoding mutual- self-statistics emergent algorithm is applied Abelian toric code, double semion twisted quantum $Z_3$, non-Abelian Fibonacci, Ising, $S_3$ ${\rm Rep}(S_3)$ string net models.