Symmetry Classfication of Two-dimensional Topologically Ordered Phases Under Exchange of Anyons

Symmetry classification of 2d topologically ordered phases, such as topological spin liquids, has received much attention in the past. Previous works have focused on the trivial group action, where symmetries (spatial or internal) do not exchange the anyons. In this work, we extend classification to cases where the group action is nontrivial so that a subset of symmetry elements exchanges the anyon types. We formulate a general framework, based on the theory of group extensions, and apply it to the Z₂ topological phases considering the 17 2d space groups under all possible nontrivial group actions. We demonstrate that the symmetry localization is no longer possible, meaning that two-particle symmetry operations cannot be written as a direct product of the one-particle actions. Moreover, because the anyons are exchanged by the symmetry group, it is no longer possible to discuss symmetry fractionalization that would assign a symmetry class for each anyon independently, and instead projective representations of different anyon types become intertwined. We show that the symmetry classification can nevertheless be formulated in a mathematical rigorous way, and prove its application to the Wen plaquette model on a square lattice as a special case.