String order in spin liquid phases of spin ladders

Two-leg spin ladders have a rich phase diagram if rung, diagonal, and plaquette couplings are allowed for. Among the possible phases there are two Haldane-type spin liquid phases without local order parameter, which differ, however, in the topology of the short range valence bonds. We show that these phases can be distinguished numerically by two different string order parameters. We also point out that long-range string and dimer orders can coexist.