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.

14 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

References in corpus (13)

• The density-matrix renormalization group in the age of matrix product states
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• The iTEBD algorithm beyond unitary evolution
• Scaling of entanglement support for Matrix Product States
• A classification of symmetry enriched topological phases with exactly solvable models
• Applying matrix product operators to model systems with long-range interactions
• Spin nematic correlations in bilinear-biquadratic S=1 spin chains
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• On topological phases of spin chains
• Phase diagram and continuous pair-unbinding transition of the bilinear-biquadratic S=1 Heisenberg chain in a magnetic field
• Symmetry protected topological phases beyond groups: The q-deformed Affleck-Kennedy-Lieb-Tasaki model

Cited by in corpus (7)

• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Classifying phases protected by matrix product operator symmetries using matrix product states
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• 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