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

We study the phase diagram of the SOq(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 -> infinity. 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 q-SPT phase.

Automated summary

The study investigates the phase diagram of the SOq(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of q > 1, revealing three distinct phases. The Haldane phase was found to lack twofold degeneracy in the entanglement spectrum, restored through q deformation. The analytical calculations in the limit q -> infinity confirmed the structure of the phase diagram.