Nonequilibrium evolution of window overlaps in spin glasses

We investigate numerically the time dependence of “window” overlaps in a three-dimensional Ising spin glass below its transition temperature after a rapid quench. Using an efficient GPU implementation, we are able to study large systems up to lateral length $L=128$ and up to long times of $t={10}^8$ sweeps. We find that the data scales according to the ratio of the window size $W$ to the nonequilibrium coherence length $\xi \left(t\right)$. We also show a substantial change in behavior if the system is run for long enough that it globally equilibrates, i.e., $\xi \left(t\right)\approx L/2$, where $L$ is the lattice size. This indicates that the local behavior of a spin glass depends on the spin configurations (and presumably also the bonds) far away. We compare with similar simulations for the Ising ferromagnet. Based on these results, we speculate on a connection between the nonequilibrium dynamics discussed here and averages computed theoretically using the “metastate.”

Figure 1

Length scales that are needed to discuss the Aizenman-Wehr (AW) metastate. The overall size of the system is $L$, which is assumed to be very large and has periodic or free boundary conditions. We consider an outer region, of size between $M$ and $L$, shown shaded, where we average over different bond configurations, and an inner region, unshaded, where we consider just a single set of bonds. Spin correlations will be studied in a window of size $W$, less than $M$. In the metastate, we require that the different length all ultimately tend to infinity such that $L\gg M\gg W\gg 1$. In our simulations, the length scales are, of course, finite (actually quite small) but we shall still view the situation in the simulations as analogous to the theoretical discussion, in which, as for the metastate, $L$ is the size of the system (with periodic boundary conditions) and $W$ is a small region where we measure correlations, but now $M$, the intermediate scale, is the nonequilibrium coherence length $\xi \left(t\right)$, the scale to which correlations have developed after a quench at time $t=0$. In the simulations we can run for long enough, and take sufficiently small window sizes, to get data in the region where $\xi \left(t\right)>W$ as shown.

Figure 2

Representative set of results for the window overlap distribution, for window size $W=4$ for lattice size $L=128$ at $T=0.8$. Data is shown for times $t={10}^k$, where $k=1,2,...,8$.

Figure 3

Results for the weight in the window overlap distribution at $q=0$ for different window sizes as a function of time. The lattice size is $L=128$ and $T=0.8$.

Figure 4

Main figure shows a scaling plot of $P_W\left(0\right)/W^{\alpha /2}$ against $\xi /W$, in which we used the results for $\xi \left(t\right)$ shown in the inset, which are obtained in Ref. [9], and took the value $\alpha =0.438$ also from Ref. [9]. The data collapse is excellent. The data is for system size $L=128$ and $T=0.8$.

Figure 5

Data for $P_W\left(0\right)$ for different values of $W$ for size $L=16$. At time in the range ${10}^7–{10}^9$ this rather small system fully equilibrates leading to a decrease in the data which is quite rapid on this log scale. The dashed line shows the equilibrium value of the bulk order parameter distribution for $L=16$, i.e., $W=L=16$. One sees that the equilibrium values of $P_W\left(0\right)$ for $W<L$ are very similar to that of the bulk overlap.

Figure 6

Data for $P_4\left(0\right)$ for different system sizes at $T=0.8$. For the three smaller sizes, $L=12,16$, and 20, the system equilibrates, leading to a pronounced drop in the data, at a time which increases with $L$. The inset plots the same data against $\xi /L$ where for $\xi$ we use a fit to the data in the inset of Fig. 4. The collapse of the data in the region where it decreases, which is quite rapid on this log scale, shows that the decrease occurs when $\xi$ is a fixed fraction (about $1/2$) of the system size, indicating full equilibration.

Figure 7

Data for $P_6\left(0\right)$ for different system sizes at $T=3.6\simeq 0.80T_c$ for the ferromagnet. The plateau at intermediate times becomes longer for larger $L$. The explanation is that a fraction of the runs gets stuck in a state with two big domains for a long time, thus contributing a certain number of overlaps with $q\simeq 0$. The inset shows the probability that two large domains coexist.

Figure 8

Plot of $P_W\left(0\right)$ multiplied by the ratio of the system size $L$ to the window size $W$ for different values of $W$ and $L$ at $T=3.6\simeq 0.80T_c$ for the ferromagnet. On this plot the height of the plateau at intermediate times seems to be independent of $L$ and $W$ showing that $P_W\left(0\right)$ itself is proportional to $W/L$, which we interpret as the probability that a straight domain wall goes through the window. The dashed horizontal line is a guide to the eye.