Ballistic propagation of a local impact in the one-dimensional $XY$ model

Light-cone-like propagation of information is a universal phenomenon of nonequilibrium dynamics of integrable spin systems. In this paper, we investigate propagation of a local impact in the one-dimensional $XY$ model with the anisotropy $γ$ in a magnetic field $h$ by calculating the magnetization profile. Applying a local and instantaneous unitary operation to the ground state, which we refer to as the local-impact protocol, we numerically observe various types of light-cone-like propagation in the parameter region $0\leqγ\leq1$ and $0\leq h \leq2$ of the model. By combining numerical integration with an asymptotic analysis, we find the following: (i) for $|h|\geq|1-γ^{2}|$ except for the case on the line $h=1$ with $0<γ<\sqrt{3}/2$, a wave front propagates with the maximum group velocity of quasiparticles, except for the case $γ=1$ and $0<h<1$, in which there is no clear wave front; (ii) for $|h|<|1-γ^{2}|$ as well as on the line $h=1$ with $0<γ<\sqrt{3}/2$, a second wave front appears owing to multiple local extrema of the group velocity; (iii) for $|h|=|1-γ^{2}|$, edges of the second wave front collapses at the origin, and as a result, the magnetization profile exhibits a ridge at the impacted site. Furthermore, we find by an asymptotic analysis that the height of the wave front decays in a power law in time $t$ with various exponents depending on the model parameters: the wave fronts exhibit a power-law decay $t^{-2/3}$ except for the line $h=1$, on which the decay can be given by either $\sim t^{-3/5}$ or $\sim t^{-1}$; the ridge at the impacted site for $|h|=|1-γ^{2}|$ shows the decay $t^{-1/2}$ as opposed to the decay $t^{-1}$ in other cases.