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

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