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

TL;DR

This paper introduces a $Z_2$ index for $S=1$ quantum spin chains, proving its invariance under smooth deformations and establishing the existence of a topological phase transition, confirming the AKLT state as a symmetry-protected topological phase.

Contribution

It provides the first rigorous proof of a topological phase transition in $S=1$ chains using a new $Z_2$ index, advancing understanding of symmetry-protected topological phases.

Findings

Defined a $Z_2$ index for gapped ground states

Proved the index's invariance under smooth deformation

Established the topological phase transition between AKLT and trivial states

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.