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We classify distinct types of quantum number fractionalization occurring intwo-dimensional topologically ordered phases, focusing in particular on phaseswith Z2 topological order, that is, on gapped Z2 spin liquids. We find that thefractionalization class of each anyon is an equivalence class of projectiverepresentations of the symmetry group, corresponding to elements of thecohomology group H^2(G,Z2). This result leads us to a symmetry classificationof gapped Z2 spin liquids, such that two phases in different symmetry classescannot be connected without breaking symmetry or crossing a phase transition.Symmetry classes are defined by specifying a fractionalization class for eachtype of anyon. The fusion rules of anyons play a crucial role in determiningthe symmetry classes. For translation and internal symmetry, braidingstatistics plays no role, but can affect the classification when point groupsymmetries are present. For square lattice space group, time reversal and SO(3)spin rotation symmetry, we find 2,098,176 ~ 2^21 distinct symmetry classes. Wegive an explicit construction of symmetry classes for square lattice spacegroup symmetry in the toric code model. Via simple examples, we illustrate howinformation about fractionalization classes can in principle be obtained fromthe spectrum and quantum numbers of excited states. Moreover, the symmetryclass can be partially determined from the quantum numbers of the fourdegenerate ground states on the torus. We also extend our results to arbitraryAbelian topological orders, and compare our classification with the relatedprojective symmetry group classification of parton mean-field theories. Ourresults provide a framework for understanding and probing the sharpdistinctions among symmetric Z2 spin liquids, and are a first step toward afull classification of symmetric topologically ordered phases.

Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions