一维$XY$模型中局部碰撞的弹道传播
Ballistic propagation of a local impact in the one-dimensional $XY$ model
摘要：
类光锥信息传播是可积自旋系统非平衡动力学的普遍现象。在本文中，我们研究的传播的局部碰撞在一维$XY$模型与各向异性$γ$在磁场$h$通过计算磁化剖面。应用本地和瞬时幺正操作的基态，我们称之为本地的影响协议，我们数值观察各种类型的光锥状传播的参数区域$0\leqγ\leq1$和$0\leq h \leq2$的模型。通过数值积分与渐近分析相结合，我们发现：（1）对于$|h|\geq|1-γ^{2}|$，除了在$h=1$与$0<γ<\sqrt{3}/2$的直线上的情形外，波前以准粒子的最大群速度传播，除了$γ=1$和$0<h<1$没有清晰的波前外;（ii）对于$|h|<|1-γ^{2}|$以及在$h=1$与$0<γ<\sqrt{3}/2$的连线上，由于群速度的多个局部极值，出现了第二波阵面;（iii）对于$|h|=|1-γ^{2}|$，第二波阵面的边缘在原点处塌陷，结果，磁化曲线在受影响部位呈现脊。此外，我们还通过渐近分析发现，波前高度随时间按幂律衰减，其指数随模型参数的变化而变化：除了谱线$h=1$外，波前呈现幂律衰减$t^{-2/3}$，在谱线$h=1$上的衰减可以由$\sim t^{-3/5}$或$\sim t^{-1}$给出; $|h|=|1-γ^{2}|$的受影响部位的脊显示出衰变$t^{-1/2}$，而在其它情况下是衰变$t^{-1}$。
英文摘要：
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.
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