Aging following a zero-temperature quench in the $d=3$ Ising model

Aging in phase-ordering kinetics of the $d=3$ Ising model following a quench from infinite to zero temperature is studied by means of Monte Carlo simulations. In this model the two-time spin-spin autocorrelator $C_{\text{ag}}$ is expected to obey dynamical scaling and to follow asymptotically a power-law decay with the autocorrelation exponent $\lambda$. Previous work indicated that the lower Fisher-Huse bound of $\lambda \ge d/2=1.5$ is violated in this model. Using much larger systems than previously studied, the instantaneous exponent for $\lambda$ we obtain at late times does not disagree with this bound. By conducting systematic fits to the data of $C_{\text{ag}}$ using different Ansätze for the leading correction term, we find $\lambda =1.58\left(14\right)$, with most of the error attributed to the systematic uncertainty regarding the Ansätze. This result is in contrast to the recent report that below the roughening transition universality might be violated.

Figure 1

Characteristic length scale $\ell$ vs MC time $t$. Although power-law growth with exponent $1/2$ (solid line) is expected, at early times slower and at late times faster growth is observed.

Figure 2

Autocorrelation function $C_{\text{ag}}(t,t_w)$ plotted against $\ell \left(t\right)/{\ell }_w$. The dashed lines show the FH lower bound [6] and the solid lines the LM value [8]. (a) Results for different system sizes $L$ and (b) for different waiting times $t_w$ in units of MCS. The actual values for the mnemonic $t_w$ shown here and in subsequent figures are provided in Appendix pp1.

Figure 3

Instantaneous autocorrelation exponent ${\lambda }_i(t,t_w)$ vs ${[\ell \left(t\right)/{\ell }_w]}^{-1}$ (a) for different system sizes and (b) for different waiting times $t_w$. The solid black lines in (a) and (b) show an extrapolation in $1/x$ as performed in Ref. [17].

Figure 4

As in Fig. 2 but using $\sqrt{y}$ as the scaling variable instead: Autocorrelation function $C_{\text{ag}}(t,t_w)$ plotted against $\sqrt{y}$. The dashed lines show the FH lower bound [6] and solid lines the LM value [8]. (a) Results for different system sizes $L$ and (b) for different waiting times $t_w$. In both panels the upper abscissa shows $y$ (as opposed to $\sqrt{y}$ in the lower abscissa).

Figure 5

As in Fig. 3 but using $\sqrt{y}$ as the scaling variable instead: Instantaneous exponent ${[\lambda /z]}_i\times 2$ vs ${(t/t_w)}^{-1/2}$ (a) for different system sizes and (b) for different waiting times $t_w$. In both cases the solid lines are guide to the eye, connecting the data points to the LM value of 1.67 as $1/\sqrt{y}\to 0$. The value of the FH lower bound 1.5 is shown with dotted lines.

Figure 6

As in Fig. 5 but using $1/y$ as abscissa instead of $1/\sqrt{y}$: Instantaneous exponent ${[\lambda /z]}_i\times 2$ vs ${(t/t_w)}^{-1}$ (a) for different system sizes and (b) for different waiting times $t_w$. In both cases the solid lines are guide to the eye, connecting the data points to the FH lower bound 1.5 (shown as dotted lines) as $1/y\to 0$.

Figure 7

Correction-to-scaling fits on $C_{\text{ag}}(t,t_w)$ using Eqs. (7a) and (8a), denoted by “$1/\sqrt{y}$ fit” and “$1/y$ fit,” respectively, for $L=1536$. (a) ${\chi }^2$ per degree of freedom using $t_w=699$ and 1857 as a function of $y_{\text{min}}$ with $y_{\text{max}}=2\times {10}^5/t_w$. Colored shades correspond to error bars obtained by Jackknifing [29]. The dashed vertical line shows the $y_{\text{min}}$ chosen for subpanels (c) and (d). (b) Numerical value for $2\lambda /z$ obtained from the same fits as in (a). (c) Data and fitting functions for $C_{\text{ag}}(t,t_w)$ and $t_w=699$. The fitting range $y\in [76,290]$ is indicated by dashed vertical lines. The inset shows that the ratio Eq. (8a)/Eq. (7a) of the two fitting functions varies by less than 0.1% within the fitting range. (d) Instantaneous exponent as in Fig. 5. Solid lines show log-derivative of the fits from Eqs. (7a) and (8a), dashed lines are the graphical extrapolations in $1/\sqrt{y}$ and $1/y$ as shown as guide to the eye in Figs. 5 and 6, respectively.