An order parameter for symmetry-protected phases in one dimension

TL;DR

This paper introduces a symmetry-dependent order parameter for one-dimensional symmetry-protected phases that can identify different phases, including gapped and gapless ones, through numerical simulations and potential experimental measurements.

Contribution

It presents a novel order parameter based on string-like operators that depends solely on symmetry, enabling direct identification of symmetry-protected phases in 1D systems.

Findings

Effective in distinguishing dimerized and Haldane phases

Returns distinct signatures for gapless phases

Potential for experimental measurement in cold atom systems

Abstract

We introduce an order parameter for symmetry-protected phases in one dimension which allows to directly identify those phases. The order parameter consists of string-like operators and swaps, but differs from conventional string order operators in that it only depends on the symmetry but not on the state. We verify our framework through numerical simulations for the SO(3) invariant spin-1 bilinear-biquadratic model which exhibits a dimerized and a Haldane phase, and find that the order parameter not only works very well for the dimerized and the Haldane phase, but it also returns a distinct signature for gapless phases. Finally, we discuss possible ways to measure the order parameter in experiments with cold atoms.