Discrete spin structures and commuting projector models for two-dimensional fermionic symmetry-protected topological phases

We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group G(f) = G x Z(2)(f), where G is finite and Z(2)(f) is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface M on which our model is defined, namely a Kasteleyn orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the Z(8) classification of 2D fermionic SPTs with G = Z(2).