Low-temperature behavior of the statistics of the overlap distribution in Ising spin-glass models

Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the “replica symmetry breaking” picture of the spin-glass phase from the “droplet picture,” finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.

Figure 1

Absolute difference between $q_l$ and the quantity $q_l^{\prime }\left(U\right)\equiv (2T/z)U+1$, obtained from the equilibrium relationship of Eq. (8), for the LR model with $N=1024$, $\sigma =0.896$ (top panel) and the 3D EA model with $L=8$ (lower panel) as a function of Monte Carlo sweeps ($t$) on a log-linear scale. At large times the difference is zero, but the simulation continues well beyond this point to ensure that good statistics are obtained for all samples. Error bars are smaller than the symbols. Both panels have the same horizontal scale.

Figure 2

Plots of several observables obtained from the overlap distribution, defined in Sec. 4, vs the number of Monte Carlo sweeps for the largest size studied, $N=1024$, for the long-range model at the lowest temperature simulated for each value of $\sigma$. (See Table 1.)

Figure 3

$\Delta (q_0,\kappa )$ as a function of system size $N$ for the long-range models and the SK model for all available values of $\sigma$ and various values of the window $q_0$ and threshold $\kappa$. In all cases the temperature is $0.4T_c$. All panels have the same horizontal scale, and all panels in a row have the same vertical scale.

Figure 4

$\Delta (q_0,\kappa )$ as a function of system size $N$ for the short-range models and the SK model for several values of the window $q_0$ and threshold value $\kappa$. The points connected by solid lines are for the EA models, while those connected by dashed lines are for the SK model. The temperatures are $0.4T_c$ for the 3D data and $0.5T_c$ for the 4D data. All panels in a column have the same horizontal scale and all panels in a row have the same vertical scale.

Figure 5

Mean and median over samples of the integrated distribution $I_{\mathcal{J}}\left(q\right)$ for the long-range models and the SK model. In all cases the temperature is close to $0.4T_c$. For both the SK model and the 1D models, the median shows a relatively strong size dependence compared with the mean, this difference being the least pronounced for $\sigma =0.896$. The “theory” curve for the SK data [Eq. (19)] is expected to be valid for small $q$ only. The theory expression can be multiplied by an (unknown) constant which has been set to unity. All panels have the same horizontal and vertical scales.

Figure 6

Log-linear plot of $I^{\mathrm{med}}\left(q\right)$ and $I^{\mathrm{av}}\left(q\right)$ vs $q$ plot for the 3D EA model at $T\simeq 0.42$ (upper panel) and for the 4D EA model at $T\simeq 0.90$ (lower panel). Both panels have the same horizontal scale.

Figure 7

Log-linear $P^{\mathrm{typ}}\left(q\right)$ plot for the SK model for $N=2048$ showing the strong dependence on the zero-replacement value $\epsilon /k$. See the main text for details.