Quenching along a gapless line: A different exponent for defect density

Abstract

We use a quenching scheme to study the dynamics of a one-dimensional anisotropic $XY$ spin-1/2 chain in the presence of a transverse field which alternates between the values $h+\ensuremath{\delta}$ and $h\ensuremath{-}\ensuremath{\delta}$ from site to site. In this quenching scheme, the parameter denoting the anisotropy of interaction $(\ensuremath{\gamma})$ is linearly quenched from $\ensuremath{-}\ensuremath{\infty}$ to $+\ensuremath{\infty}$ as $\ensuremath{\gamma}=t/\ensuremath{\tau}$, keeping the total strength of interaction $J$ fixed. The system traverses through a gapless phase when $\ensuremath{\gamma}$ is quenched along the critical surface ${h}^{2}={\ensuremath{\delta}}^{2}+{J}^{2}$ in the parameter space spanned by $h$, $\ensuremath{\delta}$, and $\ensuremath{\gamma}$. By mapping to an equivalent two-level Landau-Zener problem, we show that the defect density in the final state scales as $1/{\ensuremath{\tau}}^{1/3}$, a behavior that has not been observed in previous studies of quenching through a gapless phase. We also generalize the model incorporating additional alternations in the anisotropy or in the strength of the interaction and derive an identical result under a similar quenching. Based on the above results, we propose a general scaling of the defect density with the quenching rate $\ensuremath{\tau}$ for quenching along a gapless critical line.