Zero- and Low-Temperature Behavior of the Two-Dimensional $\pm J$ Ising Spin Glass

Scaling arguments and precise simulations are used to study the square lattice $\pm J$ Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.

Figure 1

An $L=128$ sample. (a) Nearest-neighbor correlations: Black lines are bonds that are rigidly correlated at $T=0$. (b) Relative domain wall. Black lines are bonds whose correlations are not affected by the change $\mathrm{P}\to \mathrm{AP}$ in the horizontal boundary conditions. The shaded bonds indicate the change in the correlations on a logarithmic scale, with light colors for bonds with $|\delta ⟨s_i{s}_j{⟩}_0|\sim 1$ and the darkest colors indicating $|\delta ⟨s_i{s}_j{⟩}_0|\sim e^{-20}$. This “diffuse” domain wall appears to have a multifractal geometry [22].

Figure 2

(a) Correlation functions at low temperatures for $L=32,64,128,256$ plotted versus $r^{-{\theta }_S}$, using ${\theta }_S=0.50$. Droplet theory based on the entropy of the droplets predicts a linear curve at small values of $r^{-{\theta }_S}$ and large $L$, as is seen. (b) Sample scaling collapse for correlation function $G_T(\overrightarrow{r})$. Using nonzero-temperature scaling, we expect $L^{{\theta }_S}[G_T(\overrightarrow{r})-G_0(\infty )]\approx g(\overrightarrow{r}/L,T{L}^{{\theta }_S})$. Adjusting $G_0(\infty )$ for the best collapse, we find $G_0(\infty )=0.395\pm 0.010$.

Figure 3

Scaling plot for the low-$T$ specific heat $C$. For $T<L^{{\theta }_S}$, there are no active droplets, so $C$ is exponentially suppressed. The low-$T$ scaling behavior in large-enough systems is seen when $C{T}^{1-2/{\theta }_S}$ is flat at $T{L}^{{\theta }_S}>1$. The value [28] of ${\theta }_S=0.50$ is independently set by the scaling of the entropy of domain walls.