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

TL;DR

This paper develops exactly solvable lattice models for all known 2+1d fermionic SPT phases with finite symmetry groups, using discrete spin structures and extending beyond previous supercohomology approaches.

Contribution

It introduces a new class of commuting projector models for 2+1d fermionic SPTs that incorporate discrete spin structures, covering all known phases including the $ ext{Z}_8$ classification.

Findings

Constructed models for all 2+1d fermionic SPTs with finite symmetry groups.

Extended beyond supercohomology models to include all known phases.

Provided explicit models for the $ ext{Z}_8$ classification of fermionic SPTs.

Abstract

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 $G = \mathbb{Z}_2$.