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We present a modification of Matrix Product State time evolution to simulatethe propagation of signal fronts on infinite one-dimensional systems. Werestrict the calculation to a window moving along with a signal, which by theLieb-Robinson bound is contained within a light cone. Signal fronts can bestudied unperturbed and with high precision for much longer times than onfinite systems. Entanglement inside the window is naturally small, greatlylowering computational effort. We investigate the time evolution of thetransverse field Ising (TFI) model and of the S=1/2 XXZ antiferromagnet intheir symmetry broken phases after several different local quantum quenches.  In both models, we observe distinct magnetization plateaus at the signalfront for very large times, resembling those previously observed for theparticle density of tight binding (TB) fermions. We show that the normalizeddifference to the magnetization of the ground state exhibits similar scalingbehaviour as the density of TB fermions. In the XXZ model there is anadditional internal structure of the signal front due to pairing, and widerplateaus with tight binding scaling exponents for the normalized excessmagnetization. We also observe parameter dependent interaction effects betweenindividual plateaus, resulting in a slight spatial compression of the plateauwidths.  In the TFI model, we additionally find that for an initial Jordan-Wignerdomain wall state, the complete time evolution of the normalized excesslongitudinal magnetization agrees exactly with the particle density of TBfermions.

Time Evolution within a Comoving Window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains