A ${\mathbb Z}_2$-index of symmetry protected topological phases with time reversal symmetry for quantum spin chains

TL;DR

This paper introduces a ${ m Z}_2$-index for classifying symmetry-protected topological phases in quantum spin chains with time reversal symmetry, providing a new invariant for gapped Hamiltonians.

Contribution

It defines a ${ m Z}_2$-index as an invariant for time reversal symmetric Hamiltonians, advancing the classification of topological phases in quantum spin chains.

Findings

The ${ m Z}_2$-index is invariant under $C^1$-classification.

The index distinguishes different topological phases with time reversal symmetry.

The index applies to Hamiltonians with unique gapped ground states.

Abstract

We introduce a ${\mathbb Z}_2$-index for time reversal invariant Hamiltonians with unique gapped ground state on quantum spin chains. We show this is an invariant of a $C^1$-classification of gapped Hamiltonians.