Commuting-projector Hamiltonians for 2D time-reversal-invariant fermionic topological phases and many-body topological invariants

Inspired by a recently constructed commuting-projector Hamiltonian for a 2D time-reversal-invariant topological superconductor [Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D time-reversal-invariant topological insulator. We explicitly show that the edge properties of our model, both gapped and gapless, are consistent with those of band-theoretic or weakly interacting quantum spin Hall systems. Additionally, the models for both topological insulators and superconductors can be defined on non-orientable spatial manifolds while retaining the commuting-projector Hamiltonian structure. Ground-state wavefunctions of these models on non-orientable manifolds provide intuitive pictures of many-body topological invariants of time-reversal-invariant fermionic topological phases.