Symmetry reduction induced by anyon condensation: a tensor network approach

Topological ordered phases are related to changes in the properties of their quasi-particle excitations (anyons). We study these relations in the framework of projected entanglement pair states (\textsf{PEPS}) and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well, and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states (\textsf{MPS}). The case of $\mathbb{Z}_2$ on-site symmetry is studied in detail.