Discrete spin structures and commuting projector models for 2d fermionic symmetry protected topological phases

We construct exactly solved commuting projector Hamiltonian lattice models for all known 2+1d fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group $G_f=G\times {\mathbb{Z}}_2^f$, where $G$ is finite and ${\mathbb{Z}}_2^f$ 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 $8$ members of the ${\mathbb{Z}}_8$ classification of 2d fermionic SPTs with $G={\mathbb{Z}}_2$.