Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
Abstract
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 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∞. Our results suggest that symmetries of the form Uq[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.