The Ground State of the S=1 Antiferromagnetic Heisenberg Chain is Topologically Nontrivial if Gapped

TL;DR

This paper proves that the ground state of the S=1 antiferromagnetic Heisenberg chain is a nontrivial symmetry-protected topological phase, assuming a unique gapped ground state exists.

Contribution

It rigorously establishes the topological nature of the ground state in the model under common assumptions, ruling out trivial phases.

Findings

The ground state has a nontrivial topological index.

Presence of gapless edge excitations in the half-infinite chain.

Existence of a topological phase transition between the Heisenberg and trivial models.

Abstract

Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological (SPT) phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.