Time evolution within a comoving window: Scaling of signal fronts and magnetization plateaus after a local quench in quantum spin chains

Abstract

We present a modification of Matrix Product State time evolution to simulate the propagation of signal fronts on infinite one-dimensional systems. We restrict the calculation to a window moving along with a signal, which by the Lieb-Robinson bound is contained within a light cone. Signal fronts can be studied unperturbed and with high precision for much longer times than on finite systems. Entanglement inside the window is naturally small, greatly lowering computational effort. We investigate the time evolution of the transverse field Ising (TFI) model and of the S = 1/2 XXZ antiferromagnet in their symmetry broken phases after several different local quantum quenches. In both models, we observe distinct magnetisation plateaus at the signal front for very large times, resembling those previously observed for the particle density of tight binding (TB) fermions. We show that the normalised difference to the magnetisation of the ground state exhibits similar scaling behaviour as the density of TB fermions. In the XXZ model there is an additional internal structure of the signal front due to pairing, and wider plateaus with tight binding scaling exponents for the normalised excess magnetisation. We also observe parameter dependent interaction effects between individual plateaus, resulting in a slight spatial compression of the plateau widths. In the TFI model, we additionally find that for an initial Jordan-Wigner domain wall state, the complete time evolution of the normalised excess longitudinal magnetisation agrees exactly with the particle density of TB fermions.