Aging in the Long-Range Ising Model

The current understanding of aging phenomena is mainly confined to the study of systems with short-ranged interactions. Little is known about the aging of long-ranged systems. Here, the aging in the phase-ordering kinetics of the two-dimensional Ising model with power-law long-range interactions is studied via Monte Carlo simulations. The dynamical scaling of the two-time spin-spin autocorrelator is well described by simple aging for all interaction ranges studied. The autocorrelation exponents are consistent with $\lambda =1.25$ in the effectively short-range regime, while for stronger long-range interactions the data are consistent with $\lambda =d/2=1$. For very long-ranged interactions, strong finite-size effects are observed. We discuss whether such finite-size effects could be misinterpreted phenomenologically as subaging.

Figure 1

Characteristic length scale $\ell (t)$ versus time $t$ for the 2D LRIM with $\sigma =0.6$ on $L\times L$ lattices quenched to $T=0.1T_c$. The solid line depicts the theoretical prediction in Eq. (2). The snapshots are obtained from a single run for $L=4096$, with spins pointing up marked in blue.

Figure 2

Double-log plot of the order-parameter autocorrelation function $C(y{t}_w,t_w$) against the scaling variable $y=t/t_w$ for the 2D LRIM with (a) $\sigma =1.5$ with $L=2048$ and (b) $\sigma =0.8$ and (c) $\sigma =0.6$ with $L=4096$ quenched to $T=0.1T_c$. The solid lines are fits using Eq. (5). In the insets, we plot the same data on a linear scale but divide out the asymptotic behavior $y^{-\lambda /z}$.

Figure 3

Autocorrelator scaling function for $\sigma =0.6$ and $L=4096$. (a) $C(y{t}_w,t_w)$ versus $y$, omitting the finite-size affected data for $t>t_{\times }=1000$. The straight line shows the asymptotic power law $y^{-\lambda /z}$ with the fitted $\lambda =0.995$. The inset shows a finite-size scaling plot of $\ell (t)$ by plotting $\ell (t)/t^{1/z}$ versus $t^{1/z}/L$ for different $L$. (b) Plot against $1/y$ of the data for $t_w=20$, dividing out the asymptotically expected behavior of $y^{-\lambda /z}$. The assumed value of $\lambda$ is varied and the solid line is the expected behavior assuming the correction form in Eq. (5).

Figure 4

Untruncated $C(y{t}_w,t_w)$ for $\sigma =0.6$ and $L=4096$ plotted against scaling variable $h(t)/h(t_w)$ with $\mu =0.976$, indicating subaging.