Resilience of hidden order to symmetry-preserving disorder

We study the robustness of nonlocal string order in two paradigmatic disordered spin-chain models, a spin-1/2 cluster-Ising and a spin-1 XXZ Heisenberg chain. In the clean case, they both display a transition from antiferromagnetic to string order. Applying a disorder, which preserves the Hamiltonian symmetries, we find that the transition persists in both models. In the disordered cluster-Ising model, we can study the transition analytically-by applying the strongest coupling renormalization group -and numerically-by exploiting integrability to study the antiferromagnetic and string order parameters. We map the model into a quadratic fermion chain, where the transition appears as a change in the number of zero-energy edge modes. We also explore its zero-temperature-singularity behavior and find a transition from a nonsingular to a singular region, at a point that is different from the one separating nonlocal and local ordering. The disordered Heisenberg chain can be treated only numerically: by means of MPS-based simulations, we are able to locate the existence of a transition between antiferromagnetic and string-ordered phase, through the study of order parameters. Finally, we discuss possible connections of our findings with many-body localization.