Universal critical behavior of the two-dimensional Ising spin glass

We use finite size scaling to study Ising spin glasses in two spatial dimensions. The issue of universality is addressed by comparing discrete and continuous probability distributions for the quenched random couplings. The sophisticated temperature dependency of the scaling fields is identified as the major obstacle that has impeded a complete analysis. Once temperature is relinquished in favor of the correlation length as the basic variable, we obtain a reliable estimation of the anomalous dimension and of the thermal critical exponent. Universality among binary and Gaussian couplings is confirmed to a high numerical accuracy.

Figure 1

Top: Binary model correlation length (in units of the system size) versus temperature. ${\xi }_L/L$ approaches its $T=0$ limit exponentially in $1/T$ (because of the existence of an energy gap). We have an inflection point at $T=T_{\inf }^{\left(L\right)}$ (obtained from a cubic spline interpolation of ${\xi }_L/L$) that we regard as a proxy for the crossover scale $T_L^{*}$ [33]. At low $T$ (discontinuous lines) we use less samples, see Appendix pp1. Inset: Size evolution of the inflection points $T_{\inf }^{\left(L\right)}$ (red full squares), compared to $T_{{\xi }_L/L=0.5}^{\left(L\right)}$ (open green circles). Data for binary model. As expected [34], the two temperature scales decouple for large $L$. Bottom: ${\xi }_L/L$ vs $T$ for the Gaussian model does not show any crossover.

Figure 2

Binder ratio $U_4$, Eq. (1), at $T$ where $\xi /L=0.3$ (top), 0.42 (center), and 0.54 (bottom) as a function of $L^{-\omega }$ for the two models. The large $L$ limit is model independent. The $\omega$ exponent and the solid lines were obtained from a joint fit to Eq. (9).

Figure 3

Order parameter $\overline{\langle q^2\rangle }$ computed at fixed values of ${\xi }_L/L$ vs ${\left[T_{{\xi }_L/L}^{\left(L\right)}\right]}^2$, for the binary (upper inset) and the Gaussian (lower inset) models. Main: Universal scaling function $g({\xi }_L/L)=F_{q^2}\left(0.4\right)/F_{q^2}({\xi }_L/L)$, Eq. (8), as computed for the Gaussian (empty symbols) and the binary (full symbols) models. The function $g(x={\xi }_L/L)$ scales as $1/x^2$ for small $x$ (dashed line).

Figure 4

Scaling field ${\left[{\widehat{u}}_h\left(T\right)\right]}^2$ [from Eq. (8)] vs ${\left[T_{{\xi }_L/L}^{\left(L\right)}\right]}^2$, as computed for the Gaussian (top) and the binary (bottom) models. The data collapses were obtained by multiplying the data in the two insets in Fig. 3 by the universal function $g({\xi }_L/L)$, depicted in the main panel of Fig. 3. The dots are for the extrapolation to $T^2=0$. The binary model data show the crossover between the $T=0$ (small $L$) and $T>0$ (large $L$) regimes (see Fig. 1 and Refs. [33, 34]).

Figure 5

Computing $\nu$ for the Gaussian model. Bare [top, see Eq. (11)] and renormalized [bottom, Eq. (12) with ${\widehat{u}}_3=-0.32$] temperature quotients at fixed ${\xi }_L/L(=0.3,0.42,0.54)$ as a function of $L^{-2/\nu }$. Continuous lines are our fits (see text), dotted lines are guides to eyes.

Figure 6

Effective value of $2-\eta$ as obtained from Eq. (C2) versus $x^{*}$ (which is the geometric mean of the two values of ${\xi }_L/L$ involved in the computation of $\eta$). We show estimations for several values of the minimal size included in the analysis, $L_{\min }$. Data for the binary model obtained with the same value of $L_{\min }$ are connected by dashed lines (continuous lines in the case of Gaussian distributed couplings). Inset: For the smallest argument $x^{*}$ that we reach in our simulations, we investigate the dependency of $2-\eta$ on $L_{\min }$.

Figure 7

The effective, size dependent critical exponents $\nu$ [top: binary model, $Q_T$ is defined in Eq. (D1)] and the anomalous dimension $\eta$ (middle: binary model, bottom: Gaussian model). The quotient $Q_{q^2}$ is defined in Eq. (D2) and analyzed in Eq. (D3).