Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model

We argue that the $q$-deformed spin-1 AKLT Hamiltonian should be regarded as a representative of a symmetry protected topological phase. Even though it fails to exhibit any of the standard symmetries known to protect the Haldane phase it still displays all characteristics of this phase: Fractionalized spin-$\frac{1}{2}$ boundary spins, non-trivial string order and - when using an appropriate definition - a two-fold degeneracy in the entanglement spectrum. We trace these properties back to the existence of an $SO_q(3)$ quantum group symmetry and speculate about potential links to discrete duality symmetries. We expect our findings and methods to be relevant for the identification, characterization and classification of other symmetry-protected topological phases with non-standard symmetries.

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