  One-dimensional gapped systems are often characterized by a 'hidden'
non-local order parameter, the so-called string order. Due to the gap,
thermodynamic properties are robust against a weak higher-dimensional coupling
between such chains or ladders. To the contrary, we find that the string order
is not stable and decays for arbitrary weak inter-chain or inter-ladder
coupling. We investigate the vanishing of the order for three different
systems: spin-one Haldane chains, band insulators, and the transverse-field
Ising model. Using perturbation theory and bosonization, we show that the
fragility of the string order arises from non-local commutation relations
between the non-local order parameter and the perturbation.
