Disentangling topological degeneracy in the entanglement spectrum of one-dimensional symmetry protected topological phases

Abstract

One-dimensional valence bond solid (VBS) states represent the simplest symmetry protected topological phases. We show that their ground state entanglement spectrum contains both topological and non-topological structures. For the $SO(3)$ symmetric VBS states with odd-integer spins, the two-fold topological degeneracy is associated with an underlying $Z_2\times Z_2$ symmetry that protects the corresponding topological phase. In general, for the $SO(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the $(Z_2\times Z_2{)}^S$ symmetry, yielding the $2^S$-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a non-local unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order.

6 pages, 1 figure, 1 table

References in corpus (7)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
• On topological phases of spin chains
• Exact Analysis of Entanglement in Gapped Quantum Spin Chains
• String order and hidden topological symmetry in the SO(2n+1) symmetric matrix product states
• Spin-quadrupole ordering of spin-3/2 ultracold fermionic atoms in optical lattices in the one-band Hubbard model
• Mott insulating phases and quantum phase transitions of interacting spin-3/2 fermionic cold atoms in optical lattices at half filling

Cited by in corpus (4)

• Non-Invertible Duality Transformation Between SPT and SSB Phases
• Critical entanglement spectrum of one-dimensional symmetry protected topological phases
• Duality, Criticality, Anomaly, and Topology in Quantum Spin-1 Chains
• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities