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

We study the phase diagram of the ${\text{SO}}_q(3)$ quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of $q>1$. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for $q=1$ remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable $q$ deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form ${\mathcal{U}}_q\left[su(2)\right]$ for distinct choices of $q$ should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a $q$-SPT phase.