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

Abstract

We study the phase diagram of the $S{O}_q(3)$ 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 analogue 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 two-fold 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 $q\to \infty$. Our results suggest that symmetries of the form $U_q[su(2)]$ 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 qSPT phase.

arXiv comment14 pages, 7 figures, v2: the entanglement scaling is analyzed for more values of the deformation parameter. Version as submitted to and accepted in Phys Rev B

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• Duality and hidden symmetry breaking in the q-deformed Affleck-Kennedy-Lieb-Tasaki model
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions