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We introduce the notion of transmission time to study the dynamics ofdisordered quantum spin chains and prove results relating its behavior tomany-body localization properties. We also study two versions of the so-calledLocal Integrals of Motion (LIOM) representation of spin chain Hamiltonians andtheir relation to dynamical many-body localization. We prove thatuniform-in-time dynamical localization expressed by a zero-velocityLieb-Robinson bound implies the existence of a LIOM representation of thedynamics as well as a weak converse of this statement. We also prove that for aclass of spin chains satisfying a form of exponential dynamical localization,sparse perturbations result in a dynamics in which transmission times divergeat least as a power law of distance, with a power for which we provide lowerbound that diverges with increasing sparseness of the perturbation.

Slow propagation in some disordered quantum spin chains