Detection of Symmetry Protected Topological Phases in 1D

TL;DR

This paper presents a straightforward method to identify and distinguish symmetry-protected topological phases in one-dimensional systems using projective representations and non-local order parameters, enhancing understanding of topological matter.

Contribution

It introduces a simple approach to determine the matrices of projective representations and derive non-local order parameters for various symmetries in 1D topological phases, including cases previously not accessible.

Findings

Method to determine projective representation matrices directly

Derivation of non-local order parameters for multiple symmetries

Relation of string order to phase transitions and identification of complex symmetries

Abstract

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped phases in 1D systems can be completely characterized using tools related to projective representations of the symmetry groups. We show how to determine the matrices of these representations in a simple way in order to distinguish between different phases directly. From these matrices we also point out how to derive several different types of non-local order parameters for time reversal, inversion symmetry and $Z_2 \times Z_2$ symmetry, as well as some more general cases (some of which have been obtained before by other methods). Using these concepts, the ordinary string order for the Haldane phase can be related to a selection rule that changes at the critical point. We furthermore point out an example of a more complicated internal symmetry for which the ordinary string order cannot be applied.