(3+1)D topological order with gravitational anomaly and exactly solvable lattice models for beyond group cohomology SPT phases

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order 2 and 4 in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has anomalous boundary Z2 topological order with fermion particle and fermionic loop excitation that have mutual π statistics. We argue the construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order 2. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary Z2 symmetry in (4+1)D.