Cluster State as a Noninvertible Symmetry-Protected Topological Phase

We show that the standard 1+1D Z₂×Z₂ cluster model has a noninvertible global symmetry, described by the fusion category Rep (D₈). Therefore, the cluster state is not only a Z₂×Z₂ symmetry protected topological (SPT) phase, but also a noninvertible SPT phase. We further find two new commuting Pauli Hamiltonians for the other two Rep (D₈) 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 noninvertible SPT phases. Finally, we show that there does not exist a symmetric entangler that maps between these distinct SPT states.