Fragility of String Orders

TL;DR

This paper demonstrates that string order in one-dimensional gapped systems is fragile and decays with any weak inter-chain coupling, challenging the assumption of its robustness due to the energy gap.

Contribution

It reveals the inherent fragility of string order under weak inter-chain interactions using perturbation theory and bosonization techniques.

Findings

String order decays with any weak inter-chain coupling.

Fragility is due to non-local commutation relations.

Results apply to spin-one Haldane chains, band insulators, and the transverse-field Ising model.

Abstract

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.