Variational methods for characterizing matrix product operator symmetries

We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga ibid. 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the F symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z(3), as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S-3 and Rep(S-3) string-net models