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 which have 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 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 AKLT ground state and trivial ground states. This rigorously establishes that the AKLT model is in a nontrivial symmetry protected topological phase.