Cluster state as a non-invertible symmetry protected topological phase

We show that the standard 1+1d $\mathbb{Z}_2\times \mathbb{Z}_2$ cluster model has a non-invertible global symmetry, described by the fusion category Rep(D$_8$). Therefore, the cluster state is not only a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry protected topological (SPT) phase, but also a non-invertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep(D$_8$) SPT phases on a tensor product Hilbert space of qubits, matching the classification in field theory and mathematics. We identify the edge modes and the local projective algebras at the interfaces between these non-invertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.