Exactly solvable model for a 4+1D beyond-cohomology symmetry-protected topological phase

We construct an exactly solvable commuting projector model for a (4 + 1)-dimensional Z(2)-symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall construction, with three-fermion Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective Z(2) symmetry on a (3 + 1)-dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton. A related property is that a codimension-two defect (for example, the termination of a Z(2) domain wall at a trivial boundary) will carry nontrivial chiral central charge 4 mod 8. We also construct a gapped symmetric topologically ordered boundary state for our model, which constitutes an anomalous symmetry-enriched topological phase outside of the classification of Chen and Hermele, and define a corresponding anomaly indicator.