Logarithmic lightcones in the multiparticle Anderson model with sparse interactions

We prove that the dynamics of the one-dimensional $XY$ model with random magnetic field perturbed by a sparse set of $ZZ$ terms with a large coupling constant $\Delta$ gives rise to Lieb-Robinson (L-R) bounds with a logarithmic lightcone and amplitude proportional to $\Delta^{-1}$. These spin systems are equivalent to a set of spinless lattice fermions subjected to a random on site potential and sparse density-density interactions. In the absence of the random magnetic field we also obtain a suppression of the L-R bounds as $\Delta^{-1}$. These results follow from the application of a general theorem about the L-R bound of a generic local time-dependent one-dimensional spin system with local time-dependent perturbations. Adopting the interaction picture of the dynamics, the large and sparse $ZZ$ perturbations of the $XY$ model, with or without disorder, are mapped into high-frequency periodic perturbations. All our results are non-perturbative.