Topological phase transition and ${\mathbb{Z}}_2$ index for $S=1$ quantum spin chains

Abstract

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.

6 pages, no figures, minor changes have been made

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