Critical behavior of three-dimensional Ising spin glass models

We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size $L$: the $\pm J$ (Edwards-Anderson) Ising model for two values of the disorder parameter $p$, $p=0.5$ and $p=0.7$ (up to $L=28$ and $L=20$, respectively), and the bond-diluted bimodal model for bond-occupation probability $p_b=0.45$ (up to $L=16$). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\omega$ related to the leading irrelevant operator: $\omega =1.0\left(1\right)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at $p_b=0.7$ and $p_b=0.35$ (up to $L=10$) and of the Ising spin glass model with Gaussian bond distribution (up to $L=8$) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\nu =2.45\left(15\right)$ and $\eta =-0.375\left(10\right)$.

Figure 1

(Color online) Phase diagram of the three-dimensional $\pm J$ (Edwards-Anderson) Ising model in the $T\text{−}p$ plane for $p\ge 1/2$, i.e., $1-p\le 1/2$. The phase diagram is symmetric under $p\to 1-p$.

Figure 2

(Color online) Estimates of ${\overline{U}}_4\left(L\right)$ vs $L^{-1}$, for the $\pm J$ model at $p=0.5$ and $p=0.7$ and the BDBIM at $p_b=0.45$. The dotted lines are drawn to guide the eyes.

Figure 3

(Color online) Estimates of ${\overline{U}}_{22}\left(L\right)$ vs $L^{-1}$ for the $\pm J$ model at $p=0.5$ and $p=0.7$ and the BDBIM at $p_b=0.45$. The dotted lines are drawn to guide the eyes.

Figure 4

(Color online) Estimates of ${\overline{U}}_4\left(L\right)$ for the BDBIM for several values of $p_b$ and for the Ising spin glass model with Gaussian bond distribution. The dotted lines correspond to estimate (48), ${\overline{U}}_4^{\ast }=1.514\left(2\right)$. The dashed straight lines are extrapolations of the data at $p_b=0.7$ and $p_b=0.45$.

Figure 5

(Color online) Crossing point ${\beta }_{\text{cross}}\left(L\right)$ for the $\pm J$ Ising model at $p=0.5$, from several pairs $L,2L$, as obtained from $R_{\xi }$ and $U_4$, versus $L^{-1/\nu -\omega }\sim L^{-1.4}$. The dotted lines show the result of a fit to ${\beta }_c+c{L}^{-1.4}$, which takes into account only the data satisfying $L\ge L_{\text{min}}=8$; it corresponds to $L_{\text{min}}^{-1.4}\approx 0.054$. The bar along the $y$ axis corresponds to the final estimate ${\beta }_c=0.902\left(8\right)$.

Figure 6

(Color online) The values of the critical temperature $T_c$ versus $1-p$ for the various transitions of the $\pm J$ Ising model. Besides the estimates for the PG ISG transitions, we also show results for the Ising transition at $p=1$, the PF RDI transitions, the location of the Nishimori point (MNP) along the $N$ line, and the $T=0$ FG transition taken from Refs. 44, 45, 43, 56, respectively.

Figure 7

(Color online) Estimates of the exponent $\nu$ from the analysis of ${\overline{R}}_{\xi }^{\prime }$ at $R_{\xi }=0.63$ and $R_{\xi }=0.654$ for the $\pm J$ Ising model at $p=0.5$ obtained by fitting the data without scaling corrections (wsc) [Eq. (59)] and with scaling corrections [Eq. (60)]. We only show the results of the fits which satisfy ${\chi }^2/\text{DOF}\lesssim 1.5$, up to values of $L_{\text{min}}$ where the errors blow up. The dotted lines correspond to the final estimate $\nu =2.45\left(15\right)$.

Figure 8

(Color online) Estimates of the exponent $\nu$ from the analysis of ${\overline{U}}_4^{\prime }$ at $R_{\xi }=0.63$ and $R_{\xi }=0.654$ for the $\pm J$ Ising model at $p=0.5$ obtained by fits without scaling corrections (wsc) [Eq. (59)] and fits with scaling corrections [Eq. (60)]. We only show the results of the fits which satisfy ${\chi }^2/\text{DOF}\lesssim 1.5$, up to values of $L_{\text{min}}$ where the errors blow up. The dotted lines correspond to $\nu =2.45\left(15\right)$.

Figure 9

(Color online) Estimates of the exponent $\eta$ from the analysis of $\overline{\chi }\left(L,R_{\xi }\right)$ for $R_{\xi }=0.63$ and $R_{\xi }=0.654$ for the $\pm J$ Ising model at $p=0.5$. Only results of fits with ${\chi }^2/\text{DOF}<1.5$ are shown. The dotted lines correspond to the estimate $\eta =-0.375\left(10\right)$.

Figure 10

(Color online) Estimates of the critical exponent $\sigma \equiv 1/\nu +2-\eta$, as obtained by analyzing ${\overline{\chi }}^{\prime }$ at $R_{\xi }=0.654$. The dotted lines correspond to $\sigma =2.78\left(4\right)$.

Figure 11

(Color online) Plot of $\xi /L$ vs $\left({\beta }_c-\beta \right)L^{1/\nu }$ with ${\beta }_c=0.902$ and $\nu =2.45$. Data for the $\pm J$ model with $p=0.5$.

Figure 12

(Color online) Estimates of $\nu$ (top) and ${\beta }_c$ (bottom) for the $\pm J$ model at $p=0.5$. We report estimates obtained by fitting $\xi /L$ to Eq. (71) (fit a), to Eq. (71) with $x\equiv \left[\left({\beta }_c-\beta \right)+b{\left({\beta }_c-\beta \right)}^2\right]L^{1/\nu }$ (fit b), to Eq. (72) with $\omega =1$ (fit c), and to extended-scaling Ansatz (86) (fit d). The dotted lines correspond to the estimates $\nu =2.45\left(15\right)$ (top) and ${\beta }_c=0.902\left(8\right)$ (bottom).

Figure 13

(Color online) Plots of $\xi /L$ vs $\left({\beta }_c-\beta \right)L^{1/\nu }$ for the $\pm J$ model at $p=0.7$ (top) and for the BDBIM at $p_b=0.45$ (bottom). We use $\nu =2.45$, ${\beta }_c=0.87$ (top), and ${\beta }_c=1.54$ (bottom). The dashed line is the curve $R_{\xi }\left(cx\right)$, where $R_{\xi }\left(x\right)$ is function (76), which is estimated in fits of $\xi /L$ for the $\pm J$ model at $p=0.5$, and $c$ is a model-dependent constant: $c\approx 1.026$ for the $\pm J$ model at $p=0.7$ and $c\approx 0.5641$ for the BDBIM at $p_b=0.45$.

Figure 14

(Color online) Plot of $U_4$ vs $\xi /L$ for the $\pm J$ model at $p=0.5$ and 0.7 and the BDBIM at $p_b=0.45$.

Figure 15

(Color online) Plots of $\chi {\xi }^{\eta -2}$ (upper panel) and of $\chi {\xi }^{\eta -2}{\tilde{u}}_h^{-2}$ (lower panel) vs $\xi /L$ with $\eta =-0.375$. The dashed line in the upper panel is the universal curve $\tilde{C}\left(\xi /L\right)$, as estimated in fits of $\chi$ to Eq. (80). Data for the $\pm J$ model at $p=0.5$.

Figure 16

(Color online) Estimates of $\eta$ for the $\pm J$ model at $p=0.5$. We report estimates obtained by fitting $\chi$ to Ansatz (80) (fit a) and Ansatz (81) (fit b) for several values of $L_{\text{min}}$ and ${\beta }_{\text{min}}$. The dotted lines correspond to the estimate $\eta =-0.375\left(10\right)$.

Figure 17

(Color online) Scaling-field function ${\tilde{u}}_h\left(\beta \right)$ for the $\pm J$ model at $p=0.5$, as determined in the fits of $\chi$ to Ansatz (80), for different values of ${\beta }_{\text{min}}$ and $L_{\text{min}}$. It is normalized such that ${\tilde{u}}_h\left({\beta }_c\right)=1$. We also report the approximation ${\tilde{u}}_h\left(\beta \right)\approx {\left(\beta /{\beta }_c\right)}^{1-\eta /2}$, which is used in the extended-scaling scheme (see the discussion in Sec. 7C).

Figure 18

(Color online) Plot of $\chi {\xi }^{\eta -2}{\tilde{u}}_h^{-2}$ vs $\xi /L$ for the $\pm J$ model at $p=0.7$ with $\eta =-0.375$ (upper panel) and for the BDBIM at $p_b=0.45$ with $\eta =-0.375$ (lower panel). In each panel we report the curve $b\Gamma \left(\xi /L\right)$, where $\Gamma \left(\xi /L\right)$ is defined in Eq. (84) and $b$ is a model-dependent constant. We use $b=1.99346$ and $2.12794$ in the upper and lower panels, respectively. They are determined by requiring a perfect fit for $\xi /L\le 0.2$.

Figure 19

(Color online) Extended-scaling results for the $\pm J$ model at $p=0.5$. In the upper panel we report ${\beta }^{2-\eta }\chi {\xi }^{\eta -2}$ vs $\xi /L$. The dashed line is the universal curve $\tilde{C}\left(\xi /L\right)$, as estimated in fits of $\chi$ to Eq. (80). In the lower panel we report estimates of $\eta$ obtained in three different fits for several values of $L_{\text{min}}$ and ${\beta }_{\text{min}}$. Fit c uses Ansatz (89) and fits d and e are defined in text below Eq. (89). The dotted lines correspond to the estimate $\eta =-0.375\left(10\right)$.

Figure 20

(Color online) Estimates of $G_4\left(L,\beta \right)$ (top) and of $G_{22}\left(L,\beta \right)$ (bottom) versus $L/\xi$ for $\beta =0.55$ and $\beta =0.625$, corresponding to the infinite-volume correlation lengths ${\xi }_{\infty }\approx 1.79$ and ${\xi }_{\infty }\approx 3.27$. Results for the $\pm J$ model at $p=0.5$. The dotted lines correspond to the estimates $G_4^{\ast }=90.3\left(5\right)$ (lower panel) and $G_{22}^{\ast }=-17\left(1\right)$ (lower panel).