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

TL;DR

This paper investigates the phase diagram of a q-deformed spin-1 chain with $SO_q(3)$ symmetry, revealing new phases and insights into symmetry-protected topological phases beyond traditional group symmetries.

Contribution

It introduces the concept of qSPT phases, showing how q-deformation affects the entanglement spectrum and phase structure of the spin chain.

Findings

Identification of three distinct phases: chiral Haldane, dimerized, and ferromagnetic.

Restoration of entanglement spectrum degeneracy via q-deformation of the entanglement Hamiltonian.

Confirmation of phase diagram structure through analytical calculations in the limit $q oinity$.

Abstract

We study the phase diagram of the $SO_q(3)$ 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 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.