CMSA Quantum Matter in Mathematics and Physics: Cluster state as a noninvertible SPT phase

We will point out a non-invertible symmetry of the ordinary 1+1d Z2xZ2 cluster Hamiltonian. Therefore, the cluster state can be viewed as a non-invertible SPT phase under this fusion category symmetry, which is Rep(D8). We furthermore find two new stabilizer states that are distinct from the cluster state as Rep(D8) SPT phases, but identical as Z2xZ2 SPT phases. Finally, we discuss the degenerate edge modes at the interfaces between these three Rep(D8) SPT phases.