Local Quench, Majorana Zero Modes, and Disturbance Propagation in the Ising chain

We study the generation and propagation of local perturbations in a quantum many-body spin system. In particular, we study the Ising model in transverse field in the presence of a local field defect at one edge. This system possesses a rich phase diagram with different regions characterized by the presence of one or two Majorana zero modes. We show that their localized character {\it i}) enables a characterization of the Ising phase transition through a local-only measurement performed on the edge spin, and {\it ii}) strongly affects the propagation of quasiparticles emitted after the sudden removal of the defect, so that the dynamics of the local magnetization show clear deviations from a ballistic behavior in presence of the Majorana fermions.