Nonequilibrium Spin-Glass Dynamics from Picoseconds to a Tenth of a Second

We study numerically the nonequilibrium dynamics of the Ising spin glass, for a time spanning 11 orders of magnitude, thus approaching the experimentally relevant scale (i.e., seconds). We introduce novel analysis techniques to compute the coherence length in a model-independent way. We present strong evidence for a replicon correlator and for overlap equivalence. The emerging picture is compatible with noncoarsening behavior.

Figure 1

Fit parameters, $A$ and $\alpha$ ($C(t,t_w)=A(t_w)(1+t/t_w{)}^{-1/\alpha (t_w)}$) vs $t_w$ for temperatures below $T_c$ ($T=0.6$ line: fit, for $t_w>{10}^5$, to $\alpha (t_w)={\alpha }_0+{\alpha }_1\log t_w+{\alpha }_2{log}^2{t}_w$, ${\alpha }_0=6.35795$, ${\alpha }_1=0.18605$, ${\alpha }_2=-0.00351835$, diagonal ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.=66.26/63$). Oscillations are due to strong correlations of $\alpha (t_w)$ at neighboring times (the fit and ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ do not change if we bin data in blocks of 5 consecutive $t_w$).

Figure 2

Left: SG coherence length ${\xi }_{1,2}$ vs waiting time, for $T\le T_c$. Right: ${\xi }_{1,2}$ vs $I_1$, (${\xi }_{1,2}\propto I_1^{1/(2-a)}$). Also shown data for the site-diluted Ising model (${\xi }_{1,2}$ and $I_1$ rescaled by 2). Full lines: Ising (coarsening, $a=0$) and SG, $a(T_c)=0.616$ [31]. Inset: $[{\xi }_{1,2}^L(t_w)-{\xi }_{1,2}^{\infty }(t_w)]/L$ vs ${\xi }_{1,2}^{\infty }(t_w)/L$ for $T=0.8$ and $L=24$, 40 and 80 (${\xi }_{1,2}^{\infty }(t_w)$ from a fit ${\xi }_{1,2}(t_w)=A(T)t_w^{1/z(T)}$ for $L=80$ in the range $3<{\xi }_{1,2}<10$).

Figure 3

For appropriate $t_w$ and $L$, the nonequilibrium $C_{\mathrm{link}}(t,t_w)$ vs $C^2(t,t_w)$ at $T=0.7$, coincides with equilibrium $Q_{\mathrm{link}}{|}_q$ vs $q^2$ (full lines, equilibrium data from [40] at $T=0.7$, see text). From the length-time dictionary up to $L=20$ we predict the equilibrium curve for $L=33$.

Figure 4

Bottom: $C_{\mathrm{link}}(t,t_w)$ vs $C^2(t,t_w)$ for $T=0.6$ and some of our largest $t_w$ (vertical line: $q_{\mathrm{EA}}^2$ from [24]). We also show our extrapolation of the $C_{\mathrm{link}}$ vs $C^2$ curve to $t_w={10}^{14}$ ($\sim 100\mathrm{s}$, see text). Top: Derivative of $C_{\mathrm{link}}$ with respect to $C^2$ for $T=0.6$. The horizontal line corresponds to the slope of a linear fit of $t_w={10}^{14}$ extrapolations (the line width equals twice the error). Inset: As in bottom panel, for the ferromagnetic site-diluted $D=2$ Ising model (same simulation of Fig. 2).