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 nontopological structures. For the $\text{SO}(3)$ symmetric VBS states with odd-integer spins, the twofold topological degeneracy is associated with an underlying ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry that protects the corresponding topological phase. In general, for the $\text{SO}(2S+1)$ symmetric VBS states with integer spins $S$, the corresponding protecting symmetry is identified as the ${({Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2})}^{S}$ symmetry, yielding the ${2}^{S}$-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.