An order parameter for symmetry-protected phases in 1D spin chains

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.