Distinct trivial phases protected by a point-group symmetry in quantum spin chains

TL;DR

This paper identifies and characterizes new trivial phases in quantum spin chains that are protected by specific point-group symmetries, even when traditional topological protection is broken.

Contribution

It introduces the concept of symmetry-protected trivial phases in quantum spin chains and provides a theoretical and numerical framework to distinguish them.

Findings

Existence of distinct trivial phases protected by point-group symmetry.

A field-theoretical approach and numerical calculations demonstrate these phases.

A non-local order parameter and proof are provided using matrix-product states.

Abstract

The ground state of the $S=1$ antiferromagnetic Heisenberg chain belongs to the Haldane phase -- a well known example of symmetry-protected topological phase. A staggered field applied to the $S=1$ antiferromagnetic chain breaks all the symmetries that protect the Haldane phase as a topological phase, reducing it to a trivial phase. That is, the Haldane phase is then connected adiabatically to an antiferromagnetic product state. Nevertheless, as long as the symmetry under site-centered inversion combined with a spin rotation is preserved, the phase is still distinct from another trivial phase. We demonstrate the existence of such distinct symmetry-protected trivial phases using a field-theoretical approach and numerical calculations. Furthermore, a general proof and a non-local order parameter are given in terms of an matrix-product state formulation.