Projected entangled pair states (PEPS) provide a natural ansatz for theground states of gapped, local Hamiltonians in which global characteristics ofa quantum state are encoded in properties of local tensors. We develop aframework to describe on-site symmetries, as occurring in systems exhibitingsymmetry-protected topological (SPT) quantum order, in terms of virtualsymmetries of the local tensors expressed as a set of matrix product operators(MPOs) labeled by distinct group elements. These MPOs describe the possiblyanomalous symmetry of the edge theory, whose local degrees of freedom areconcretely identified in a PEPS. A classification of SPT phases is obtained bystudying the obstructions to continuously deforming one set of MPOs intoanother, recovering the results derived for fixed-point models [X. Chen et al.,Phys. Rev. B 87, 155114 (2013)]. Our formalism accommodates perturbations awayfrom fixed point models, opening the possibility of studying phase transitionsbetween different SPT phases. We also demonstrate that applying the recentlydeveloped quantum state gauging procedure to a SPT PEPS yields a PEPS withtopological order determined by the initial symmetry MPOs. The MPO frameworkthus unifies the different approaches to classifying SPT phases, viafixed-points models, boundary anomalies, or gauging the symmetry, into thesingle problem of classifying inequivalent sets of matrix product operatorsymmetries that are defined purely in terms of a PEPS.
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