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Topological phases exhibit unconventional order that cannot be detected byany local order parameter. In the framework of Projected Entangled PairStates(PEPS), topological order is characterized by an entanglement symmetry ofthe local tensor which describes the model. This symmetry can take the form ofa tensor product of group representations, or in the more general case acorrelated symmetry action in the form of a Matrix Product Operator(MPO), whichencompasses all string-net models. Among other things, these entanglementsymmetries allow for the description of ground states and anyon excitations.Recently, the idea has been put forward to use those symmetries and the anyonicobjects they describe as order parameters for probing topological phasetransitions, and the applicability of this idea has been demonstrated forAbelian groups. In this paper, we extend this construction to the domain ofnon-Abelian models with MPO symmetries, and use it to study the breakdown oftopological order in the double Fibonacci (DFib) string-net and its Galoisconjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start byshowing how to construct topological order parameters for condensation anddeconfinement of anyons using the MPO symmetries. Subsequently, we set upinterpolations from the DFib and the DYL model to the trivial phase, and showthat these can be mapped to certain restricted solid on solid(RSOS) models,which are equivalent to the $((5\pm\sqrt{5})/2)$-state Potts model,respectively. The known exact solutions of the statistical models allow us tolocate the critical points, and to predict the critical exponents for the orderparameters. We complement this by numerical study of the phase transitions,which fully confirms our theoretical predictions; remarkably, we find that bothmodels exhibit a duality between the order parameters for condensation anddeconfinement.

physical review. b./physical review. b

Characterization of topological phase transitions from a non-Abelian  topological state and its Galois conjugate through condensation and  confinement order parameters