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

TL;DR

This paper investigates how local impacts propagate in the one-dimensional XY spin model, revealing various wave front behaviors and decay laws depending on model parameters through numerical and asymptotic analysis.

Contribution

It provides a detailed characterization of light-cone-like propagation and decay dynamics in the XY model with anisotropy and magnetic field, including new insights into wave front formation and decay exponents.

Findings

Wave fronts propagate with maximum group velocity in certain parameter regimes.

Multiple wave fronts can appear due to local extrema of group velocity.

Decay of wave front height follows specific power laws depending on parameters.

Abstract

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 $\gamma$ 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 $0\leq\gamma\leq1$ and $0\leq h \leq2$ of the model. By combining numerical integration with an asymptotic analysis, we find the following: (i) for $|h|\geq|1-\gamma^{2}|$ except for the case on the line $h=1$ with $0<\gamma<\sqrt{3}/2$, a wave front propagates with the maximum group velocity of quasiparticles, except for the case $\gamma=1$ and $0<h<1$, in which there is no clear wave front; (ii) for $|h|<|1-\gamma^{2}|$ as well as on the line $h=1$ with $0<\gamma<\sqrt{3}/2$, a second wave front appears owing to multiple local extrema of the group velocity; (iii) for $|h|=|1-\gamma^{2}|$, 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 $|h|=|1-\gamma^{2}|$ shows the decay $t^{-1/2}$ as opposed to the decay $t^{-1}$ in other cases.