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 Phys. Rev. B 101, 041108 (2020) and 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 MPS 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 identify the topological order of the ground state. We bring the F -symbols to the canonical gauge, as well as compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutualand self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, double semion and twisted quantum double of Z3, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3 and Rep(S3) string net models.