Exotic phases of matter, which fall beyond the Landau paradigm of phases and phase transitions, emerged as one of the main directions of research in the field of condensed matter physics in the last few decades. Among those exotic phases there are topologically ordered phases, the analysis of which is especially hard due to degeneracy of the ground state and no local order parameter. Apart from few exactly solvable models, the analysis of lattice Hamiltonians for the occurrence of topological order was considered a very hard problem. Meanwhile, the numerical method of choice in the study of strongly correlated two-dimensional systems, like topologically ordered systems, is projected entangled pair states (PEPS), as it allows analysing states which were not achievable by the state-of-the-art 2D DMRG algorithms due to long correlation length.
I will present numerical methods of analysing the optimized infinite PEPS (iPEPS), allowing to extract the information about the topological order. The key idea is to find the infinite matrix product operator (iMPO) symmetries of the iPEPS, whose existence is a necessary condition for the TN state to exhibit topological order. The iMPO symmetries can be later used to obtain topological S and T matrices, which (in most known cases) can be considered as a non-local order parameter of topologically ordered phases, in the sense that they give us unambiguous information about the model along with its excitations and their statistics. The method is immune to any small perturbations of the tensors, which had been a long feared problem due to numerical inaccuracies which may arise during the ground state optimization. Furthermore, finding iMPO symmetries enables an elegant description of the model in terms of the mathematical structure underlying the topologically ordered phases of matter – modular tensor category.