Extracting thermodynamic behavior of spin glasses from the overlap function

The nature of equilibrium states in disordered materials is often studied using an overlap function $P\left(q\right)$, the probability of two configurations having similarity $q$. Exact sampling simulations of a two-dimensional proxy for three-dimensional spin glasses indicate that common measures of $P\left(q\right)$ in smaller samples do not decide between theoretical pictures. Strong corrections result from $P\left(q\right)$ being an average over many scales, as seen in a toy droplet model. However, the median $\tilde{I}\left(q\right)$ of the integrals of sample-dependent $P\left(q\right)$ curves shows promise for deciding the thermodynamic behavior.

Figure 1

Numerically sampled spin overlap distributions $P_{\mathcal{J}}\left(q\right)$ for six realizations $\mathcal{J}$ for the two-dimensional $\pm J$ spin glass at zero temperature for (a) $L=16$ and (b) $L=128$. The peaks at small $q$ have comparable height in (a) and (b), but become narrower and so have less total area for larger $L$.

Figure 2

Plots of the sample-averaged $P\left(q\right)$ for system sizes $L=8,...,256$. With increasing $L$, the peaks sharpen and the values at small $q$ decrease, though slowly at smaller $L$. The error bars show 90$\%$ confidence intervals.

Figure 3

A plot of $I\left(0.2\right)$, the cumulative distribution through $q_0=0.2$, vs $L$, for sampled ground states of the $\pm J$ Ising model (error bars show 90$\%$ confidence intervals) and toy model simulations. The expected $I(q_0<q_{\mathrm{EA}})\sim L^{-\theta }$ holds only for large $L$ in the simulations and the toy model.

Figure 4

Plot of the median $\tilde{I}\left(q\right)$ over realizations $\mathcal{J}$ of the cumulative distribution $I_{\mathcal{J}}\left(q\right)$ for ground states of the 2D $\pm J$ Ising model. The values of $L$ range from $L=8$ for the broadest curve to $L=256$ for the narrowest curve. The error bars show 90$\%$ confidence intervals. The small values of $\tilde{I}\left(q\right)$ at small $q$ do not agree with a many-states picture for this model. Inset: Toy model results for the same statistic.

Figure 5

Plots of $\Delta (q_0,\kappa ,L)$, the probability that $P\left(q\right)$ exceeds $\kappa$ in the range $0\le q\le q_0$, vs $L$ for the 2D $\pm J$ Ising spin-glass ground states, with $\kappa =0.5$ and 2 and $q_0=0.0625$, $0.125$, and $0.25$. For fixed $q_0$ and $\kappa$, this statistic can rise or decline with increasing $L$, due to the competition between the diminishing weight of peaks and the narrowing of the peaks.