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We study square lattice space group symmetry fractionalization in a family ofexactly solvable models with $\mathbb{Z}_2$ topological order in twodimensions. In particular, we have obtained a complete understanding of whichdistinct types of symmetry fractionalization (symmetry classes) can be realizedwithin this class of models, which are generalizations of Kitaev's$\mathbb{Z}_2$ toric code to arbitrary lattices. This question is motivated byearlier work of A. M. Essin and one of us (M. H.), where the idea of symmetryclassification was laid out, and which, for square lattice symmetry, produces2080 symmetry classes consistent with the fusion rules of $\mathbb{Z}_2$topological order. This approach does not produce a physical model for eachsymmetry class, and indeed there are reasons to believe that some symmetryclasses may not be realizable in strictly two-dimensional systems, thus raisingthe question of which classes are in fact possible. While our understanding islimited to a restricted class of models, it is complete in the sense that foreach of the 2080 possible symmetry classes, we either prove rigorously that theclass cannot be realized in our family of models, or we give an explicit modelrealizing the class. We thus find that exactly 487 symmetry classes arerealized in the family of models considered. With a more restrictive type ofsymmetry action, where space group operations act trivially in the internalHilbert space of each spin degree of freedom, we find that exactly 82 symmetryclasses are realized. In addition, we present a single model that realizes all$2^6 = 64$ types of symmetry fractionalization allowed for a single anyonspecies ($\mathbb{Z}_2$ charge excitation), as the parameters in theHamiltonian are varied. The paper concludes with a summary and a discussion oftwo results pertaining to more general bosonic models.

Space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order