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 Gf=G×Z₂ᶠ, where G is finite and Z₂ᶠ 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 Z₈ classification of 2D fermionic SPTs with G=Z₂.