Quench dynamics and defect production in the Kitaev and extended Kitaev models

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 $n\sim 1/\sqrt{\tau }$, where $1/\tau$ 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 $d$-dimensional quantum model, where the quench takes the system through a $d-m$ dimensional gapless (critical) surface characterized by correlation length exponent $\nu$ and dynamical critical exponent $z$, the defect density $n\sim 1/{\tau }^{m\nu /\left(z\nu +1\right)}$. 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.

Figure 1

(Color online) Schematic representation of the Kitaev model on a honeycomb lattice showings the bonds $J_1$, $J_2$ and $J_3$. Schematic pictures of the ground states, which correspond to pairs of spins on vertical bonds locked parallel (antiparallel) to each other in the limit of large negative (positive) $J_3$, are shown at one bond on the left (right) edge respectively. ${\vec{M}}_1$ and ${\vec{M}}_2$ are spanning vectors of the lattice, and $a$ and $b$ represent inequivalent sites.

Figure 2

Defect density produced by quenching $J_1$ in $d=1$.

Figure 3

(Color online) Plot of correlation function $⟨O_r⟩$ sans the $\delta$-function peak as a function of $r$ for $J_2\tau =10$ (red circles and red solid line) and $J_2\tau =1$ (black squares and black dashed line). $⟨O_r⟩$ shows a damped oscillatory behavior as a function of $r$.

Figure 4

(Color online) Plot of defect density $n$ as a function of the quench time $J\tau$ and $\alpha ={\text{tan}}^{-1}\left(J_2/J_1\right)$. The density of defects is maximum at $J_1=J_2$.

Figure 5

(Color online) Plot of $O_{\vec{r}}^{2D}$ sans the $\delta$-function peak at the origin for $J_1=J_2=J$ and $J\tau =5$ as a function of $n_1$ and $n_2$ (see text for details). The spatial anisotropy of the defect correlation function is clearly evident even for $J_1=J_2$.

Figure 6

(Color online) Plot of $⟨O_r^{2D}⟩$ sans the $\delta$-function peak at the origin as a function of $\vec{r}$ for several representative values of $J_2/J$ for $J_1=J$ and $J\tau =5$. The plot displays the change in the shape of defect correlation function as a function of $J_2/J_1$ (see text for details).

Figure 7

(Color online) Plot of $⟨O_r^{2D}⟩$ (sans the $\delta$-function peak) at representative points $\left(-1,0\right)$ on the $x$ axis (black solid line) $\left(0,2\right)$ on the $y$ axis (blue dotted line) and $\left(2,-2\right)$ along $-45°$ in the $n_1-n_2$ plane (red dashed line) as a function of $\alpha ={\text{tan}}^{-1}\left(J_2/J_1\right)$ for fixed $J^2=1$.

Figure 8

(Color online) Plot of the entropy density $s$ as a function of $J\tau$ and $\alpha ={\text{tan}}^{-1}\left(J_2/J_1\right)$. The entropy density peaks when $⟨O_{\vec{0}}⟩$ crosses from $-1$ to 1 as discussed in the text.

Figure 9

(Color online) Plot of the defect density as a function of $\eta =J_3/J_1$ and $J\tau$ for $J_1=J_2=J$.

Figure 10

(Color online) Plot of the defect correlation function [sans the first term with the $\delta$-function peak in Eq. (47)] with $J\tau =3$ and $J_1=J_2=J$ for several representative values of $\eta =J_3/J_1$. See text for details.

Figure 11

(Color online) Plot of the entropy density $s$ as a function of quench time $\tau$ and $\eta =J_3/J_1$.