Mechanism for the failure of the Edwards hypothesis in the Sherrington-Kirkpatrick spin glass

The dynamics of the Sherrington-Kirkpatrick model at $T=0$ starting from random spin configurations is considered. The metastable states reached by such dynamics are atypical of such states as a whole, in that the probability density of site energies, $p\left(\lambda \right)$, is small at $\lambda =0$. Since virtually all metastable states have a much larger $p\left(0\right)$, this behavior demonstrates a qualitative failure of the Edwards hypothesis. We look for its origins by modeling the changes in the site energies during the dynamics as a Markov process. We show how the small $p\left(0\right)$ arises from features of the Markov process that have a clear physical basis in the spin glass, and hence explain the failure of the Edwards hypothesis.

Figure 1

Discrepancies between simulations of sequential spin-glass dynamics on a system of 5000 spins and the analytical predictions based on flat-measure assumptions. Left panel, curve (left axis): Complexity of metastable states of energy $E$, $\sum \left(E\right)=(1∕N)\ln N_s\left(E\right)$ for the SK model, where $N_s\left(E\right)$ is the mean number of metastable states with energy $E$. Bar (right axis): Histogram of the converged energies for 65 runs of the sequential spin-glass dynamics. Right panel, curve: Average $p\left(\lambda \right)$ in the metastable states of energy $-0.7$. Histogram: Average $p\left(\lambda \right)$ over the final states of the spin-glass dynamics.

Figure 2

Total changes in the site energies, ${\sum }_i\Delta {\lambda }_{j;i}$ of unflipped spins when all spins with site energies in the ranges $-1.0<{\lambda }_i<-0.5$ are flipped, in configurations generated by 100 (left panel, 671 flipped spins), and 500 (right panel, 420 flipped spins) steps of the greedy algorithm on a system of 5000 spins. This algorithm converged after 2465 flips. The straight lines show linear fits to the data.

Figure 3

The solution to (3), with a Gaussian initial condition and $c=4$ (top three panels) and 8 (lowest panel), for ${\sigma }^2=0$ (top panel), 1 (middle panel), and 4 (lower two panels). Curves are plotted at time intervals of 0.05. The bold curves in the lower two figures are the earliest at which $p(0,t)\approx 0.03$. They agree with the histograms obtained by direct simulation of the population dynamics model with 5000 fields, shown for $c=8$ (crosses).