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arXiv:0805.3328 · Physical Review B

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

Uma Divakaran, Amit Dutta, Diptiman Sen

Published Oct 17, 2008• 92 citations
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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.

DOI
10.1103/physrevb.78.144301

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We study the quenching dynamics of a one-dimensional spin-1/2 XYXYXY model in a transverse field when the transverse field $h(=t/\ensuremath{\tau})$ is quenched repeatedly between $\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$. A single passage from $h\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ to $h\ensuremath{\rightarrow}+\ensuremath{\infty}$ or the other way around is referred to as a half period of quenching. For an even number of half periods, the transverse field is brought back to the initial value of $\ensuremath{-}\ensuremath{\infty}$; in the case of an odd number of half periods, the dynamics is stopped at $h\ensuremath{\rightarrow}+\ensuremath{\infty}$. The density of defects produced due to the nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$; however, the magnitude is found to depend on the number of half periods of quenching. For two successive half periods, the defect density is found to decrease in comparison to a single half period, suggesting the existence of a corrective mechanism in the reverse path. A similar behavior of the density of defects and the local entropy is observed for repeated quenching. The defect density decays as $1/\sqrt{\ensuremath{\tau}}$ for large $\ensuremath{\tau}$ for any number of half periods, and shows an increase in kink density for small $\ensuremath{\tau}$ for an even number; the entropy shows qualitatively the same behavior for any number of half periods. The probability of nonadiabatic transitions and the local entropy saturate to 1/2 and ln 2\text{ln}\text{ }2ln 2, respectively, for a large number of repeated quenching.
It is well known that the dynamics of a quantum system is always nonadiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system. To illustrate the general argument, we study the defect generation in the quantum critical dynamics of a spin-1/2 anisotropic quantum XY spin chain with three spin interactions and a linearly driven staggered magnetic field.
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We study quench dynamics and defect production in the Kitaev and the extended Kitaev models. For the Kitaev model in one dimension, we show that in the limit of slow quench rate, the defect density $n\ensuremath{\sim}1/\sqrt{\ensuremath{\tau}}$, where $1/\ensuremath{\tau}$ is the quench rate. We also compute the defect correlation function by providing an exact calculation of all independent nonzero spin correlation functions of the model. In two dimensions, where the quench dynamics takes the system across a critical line, we elaborate on the results of earlier work [K. Sengupta, D. Sen, and S. Mondal, Phys. Rev. Lett. 100, 077204 (2008)] to discuss the unconventional scaling of the defect density with the quench rate. In this context, we outline a general proof that for a ddd-dimensional quantum model, where the quench takes the system through a $d\ensuremath{-}m$ dimensional gapless (critical) surface characterized by correlation length exponent $\ensuremath{\nu}$ and dynamical critical exponent zzz, the defect density $n\ensuremath{\sim}1/{\ensuremath{\tau}}^{m\ensuremath{\nu}/(z\ensuremath{\nu}+1)}$. We also discuss the variation of the shape and spatial extent of the defect correlation function with both the rate of quench and the model parameters and compute the entropy generated during such a quenching process. Finally, we study the defect scaling law, entropy generation and defect correlation function of the two-dimensional extended Kitaev model.
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