Exactly solvable model for a 4 +1 D beyond-cohomology symmetry-protected topological phase - Fidkowski, Lukasz; Haah, Jeongwan; Hastings, Matthew (Physical Review B, 2020)
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 (QCA). 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 arXiv:1602.00187, and define a corresponding anomaly indicator.