Growth kinetics of the two-dimensional Ising model with finite cooling rates

We investigate, via spin-flip kinetic Monte Carlo simulations, the entire growth kinetic process of the two-dimensional kinetic Ising model cooled to zero temperature with finite cooling rates. A new slow dynamics, suppressed, and unseen in the instantaneous quench emerges and eventually crosses over to the asymptotic standard coarsening behavior. We present a numerical observation of the excess defect number density that provides direct information on the generated defects due to the critical slowing down. We find that the excess defect density reveals the dynamics of defect generation, which is shown to be in good agreement with the scaling theory pioneered by Kibble and Zurek (KZ). The dynamic spin correlation function reveals that the impulse regime alluded by KZ is characterized by a unique critical coarsening process with the growth law dictated by the Kibble-Zurek mechanism. We determine a new dynamic exponent governing the growth kinetics at the onset times of the zero temperature. The proposed scaling scheme for the excess defect density leads to an analytic expression for this dynamic exponent involving the KZ exponents, indicating the extended influence of the KZ mechanism even down to the kinetics at the onset times of zero temperature. We also perform dynamic simulations of critical heating with finite rates from zero temperature where the power law relaxation (with respect to the inverse cooling rates) of the magnetization can be explained clearly by the KZ exponent.

Figure 1

The linear cooling protocol $T\left(t\right)=T_c\left(2-t/t_Q\right)$ used for the present simulations.

Figure 2

(a) Relaxation of the defect density $n_d\left(t\right)$ vs $t$ for different cooling rates in a double-logarithmic plot. The inverse cooling rate $t_Q$ increases from left to right. The dotted line represents that for the instantaneous zero-temperature quench. In (a)–(c), the blue circles and black triangles, respectively, denote the locations of the critical and the (onset) zero temperatures for different cooling rates. (b) The excess defect density $\delta n_d\left(t\right)$ vs $t$ for different cooling rates in a semilogarithmic plot. Passing through $T_c$, the excess defect density manifests that the generation of defects increases and culminates at the peak times $\widehat{t}$ (red squares). (c) The excess defect density $\delta n_d\left(t\right)$ vs $t$ for different cooling rates in a double-logarithmic plot. It shows that the excess defect density exhibits power-law relaxations at the critical temperature and at the peak times with the exponents in agreement with the KZ prediction $b=\nu /(1+\nu z)\simeq 0.316$. The onset zero-temperature relaxation shows another power-law decay with a new dynamic exponent (see the text). (d) The peak time $\widehat{t}$ (shifted by $t_Q$) vs $t_Q$ in a double-logarithmic plot. It shows a power-law growth $\widehat{t}\sim t_Q^{a_p}$ with $a_p\simeq 0.70\left(1\right)$, consistent with the KZ prediction $a=\nu z/(1+\nu z)\simeq 0.682$.

Figure 3

(a) Dynamic spin correlation functions $C(r,t_Q)$ at the critical temperature for various values of $t_Q$. The inverse cooling rate $t_Q$ increases from left to right. (b) A critical dynamic scaling $C(r,t_Q)=r^{-\eta }{F}_c[r/{\xi }_c\left(t_Q\right)]$ (with $\eta =1/4$) for $C(r,t_Q)$ shown in (a). The inset: the critical correlation length ${\xi }_c\left(t_Q\right)$ vs $t_Q$ in a double-logarithmic plot. It shows a power-law growth ${\xi }_c\left(t_Q\right)\sim t_Q^y$ with $y\simeq 0.316\left(1\right)$, in good agreement with the KZ exponent $b$.

Figure 4

(a) Scaling plot for the excess defect density, $\delta n_d\left(t\right)/\delta n_d\left(\widehat{t}\right)$ vs $t/\widehat{t}$. (b) Scaling plot for the excess defect density $\delta n_d\left(t\right)/\delta n_d\left(\widehat{t}\right)$ vs $t/\widehat{t}$ in a double-logarithmic plot. The filled circles represent the onset times of zero temperatures for $t_Q=1280,2560,5120,10240$ (MCS), respectively (from left to right).

Figure 5

(a) Dynamic spin correlation functions $C(r,2t_Q)$ at zero temperature for various cooling rates. The inverse cooling rate $t_Q$ increases from left to right. (b) A dynamic scaling $C(r,2t_Q)=F_c[r/{\xi }_0\left(2t_Q\right)]$ for the spin correlation function at zero temperature. Here, the zero-temperature correlation length ${\xi }_0$ is defined as $C({\xi }_0,2t_Q)=0.2$ for a given $t_Q$. The inset: the zero-temperature correlation length ${\xi }_0\left(2t_Q\right)$ vs $t_Q$ in a double-logarithmic plot. It shows a power-law growth ${\xi }_0\left(2t_Q\right)\sim t^{1/z_0}$ with $1/z_0\simeq 0.475\left(1\right)$.

Figure 6

(a) Dynamic spin correlation functions $C(r,t_Q)$ for the critical heating from $T=0$ to $T=T_c$ for various finite rates in a double-logarithmic plot. The inverse cooling rate $t_Q$ increases from top to bottom. It reveals the equilibrium critical decay $r^{-\eta }$ with $\eta =1/4$ (dotted line) and the NEQ relaxation of the magnetization $m^2\left(t_Q\right)=C(r\to \infty ,t_Q)$ for different rates. (b) Magnetization relaxation $m^2\left(t_Q\right)$ vs $t_Q$ for (a). It shows a power-law decay $m^2\left(t_Q\right)\sim t_Q^{-b\eta }$ with $b\eta \simeq 0.0794$ (red square), which is to be contrasted with that for the instantaneous critical heating $m_{\mathrm{inst}}^2\left(t\right)\sim t^{-\eta /z}$ with $\eta /z\simeq 0.116$ (blue circle).