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We study the phase diagram of the $SO_q(3)$ quantum group invariant spin-1bilinear-biquadratic spin chain for real values of $q>1$. Numericalcomputations suggest that the chain has at least three clearly distinguishedphases: A chiral analogue of the Haldane phase, a dimerized phase and aferromagnetic phase. In contrast, the counterpart of the extended criticalregion that is known to exist for $q=1$ remains elusive. Our results show thatthe Haldane phase fails to exhibit a two-fold degeneracy in the entanglementspectrum but that the degeneracy is restored upon a suitable $q$-deformation ofthe entanglement Hamiltonian which can be interpreted as a Zeeman field. Thestructure of the phase diagram is confirmed through analytical calculations inthe extreme anisotropic limit $q\to\infty$. Our results suggest that symmetriesof the form $U_q[su(2)]$ for distinct choices of $q$ should be interpreted asone single family instead of separate symmetries when defining SPT phases,leading naturally to the notion of a qSPT phase.

Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain