Non-universality of aging during phase separation of the two-dimensional long-range Ising model

Abstract

We investigate the aging properties of phase-separation kinetics following quenches from $T=\infty$ to a finite temperature below $T_c$ of the paradigmatic two-dimensional conserved Ising model with power-law decaying long-range interactions $\sim r^{-(2 + σ)}$. Physical aging with a power-law decay of the two-time autocorrelation function $C(t,t_w)\sim \left(t/t_w\right)^{-λ/z}$ is observed, displaying a complex dependence of the autocorrelation exponent $λ$ on $σ$. A value of $λ=3.500(26)$ for the corresponding nearest-neighbor model (which is recovered as the $σ\rightarrow \infty$ limes) is determined. The values of $λ$ in the long-range regime ($σ< 1$) are all compatible with $λ\approx 4$. In between, a continuous crossover is visible for $1 \lesssim σ\lesssim 2$ with non-universal, $σ$-dependent values of $λ$. The performed Metropolis Monte Carlo simulations are primarily enabled by our novel algorithm for long-range interacting systems.

Figures (4)

• Figure 1: Evolution of the local autocorrelation for $\sigma = 0.8$ and $L=1024$, each row for a different references time $t_w$ and each column for a constant value of the scaling variable $y=t/t_w$. The red (blue) spins point up (down) at times $t_w$and at $t$ and contribute with a positive sign to the autocorrelation function. The green (orange) spins point up (down) at times $t_w$but down (up) at $t$, thus, entering with a negative sign. The growth of the average domain sizes (size of the patches of red + orange and blue + green, respectively) as well as the decay of the autocorrelation (shrinking area of blue + red) with increasing time from left to right are clearly visible.
• Figure 2: Characteristic length $\ell(t)/t^\alpha$ together with the predicted growth law (dotted black lines) for all considered values of $\sigma$ (the system sizes differ among the different $\sigma$, see text). Error bars indicate twice the standard error of the mean. For $\sigma=0.6$ the observed growth matches the prediction for little less than a decade while for all other $\sigma$ the correspondence extends over $\approx 1.5-2$ decades.
• Figure 3: The autocorrelation function $C(t,t_w)$ for $\sigma=0.8$ and $L=2048$ for different waiting times $t_w$ shows an excellent data collapse when plotted against $t/t_w$ (main plot) while time-translation invariance is clearly violated (inset). The solid black line shows the best fit within the final chosen fitting range and the dotted black line its continuation to larger and smaller values of $y=t/t_w$ (see text for discussion).
• Figure 4: The autocorrelation exponent $\lambda$ plotted against the inverse of the exponent $\sigma$ of the power-law interaction. For large values of $\sigma$ we find $\lambda \approx 3.50$. In the long-range regime ($\sigma < 1$) the solid line represents the weighted average $4.038(56)$ with error bars indicated by the dash-dotted lines. The crossover happens continuously over an extended range $1 \lesssim\sigma\lesssim 2$. The lines refer to the nonequilibrium behavior while the shading of the background encodes the equilibrium critical behavior (see text for discussion).