Topological Phase Transition and ${\mathbb{Z}}_2$ Index for $S=1$ Quantum Spin Chains

We study $S=1$ quantum spin systems on the infinite chain with short ranged Hamiltonians that have a certain rotational and discrete symmetry. We define a ${\mathbb{Z}}_2$ index for any gapped unique ground state, and prove that it is invariant under a smooth deformation. By using the index, we provide the first rigorous proof of the existence of a “topological” phase transition, which cannot be characterized by any conventional order parameters, between the Affleck-Kennedy-Lieb-Tasaki (AKLT) model and trivial models. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.