Nonlocal adiabatic response of a localized system to local manipulations

We examine the response of a system localized by disorder to a time-dependent local perturbation that varies smoothly with a characteristic timescale tau. We find that such a perturbation induces a nonlocal response, involving a rearrangement of conserved quantities over a length scale similar to ln tau. This effect lies beyond linear response, is absent in undisordered insulators and highlights the remarkable subtlety of localized phases. The effect is common to both single-particle and many-body localized phases. Our results have implications for numerous fields, including topological quantum computation in quantum Hall systems, quantum control in disordered environments, and time-dependent localized systems. For example, they indicate that attempts to braid quasiparticles in quantum Hall systems or Majorana nanowires will not succeed if the manipulations are performed asymptotically slowly, and thus using such platforms for topological quantum computation will require considerable engineering. They also establish that disorder-localized insulators suffer from a statistical orthogonality catastrophe.