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

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 nontopological structures. For the SO(3) symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying Z2 × Z2 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 (Z2 x Z2)S symmetry, yielding the 2S-fold topological degeneracy. The topological degeneracy and associated protecting symmetry can be identified by a nonlocal unitary transformation, which changes the topological order of the VBS states into conventional ferromagnetic order. [ABSTRACT FROM AUTHOR]