An order parameter for symmetry-protected phases in one dimension

Understand

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.