Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four 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 an anomalous boundary $\mathbb{Z}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $\pi$ statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.