Universality in three-dimensional Ising spin glasses: A Monte Carlo study

We study universality in three-dimensional Ising spin glasses by large-scale Monte Carlo simulations of the Edwards-Anderson Ising spin glass for several choices of bond distributions, with particular emphasis on Gaussian and bimodal interactions. A finite-size scaling analysis suggests that three-dimensional Ising spin glasses obey universality.

Figure 1

(Color online) Graphical representation of different estimates of the critical exponents $\eta$ and $\nu$ (taken from Table ). The centers of the ellipses represent the different estimates, and the major axes represent the error bars of the estimates. The data show a considerable spread. The thick lines represent the estimates from the current study. Note that the data from the current study for bimodal (solid lines) and Gaussian (dashed lines) distributions agree within error bars thus providing evidence for universality. The dotted line represents data for a random-anisotropy Heisenberg model in the strong-anisotropy limit (Ref. 12) which is expected to be in the same universality class as the Ising spin glass.

Figure 2

(Color online) Finite-size correlation length ${\xi }_L∕L$ as a function of inverse temperature $\beta$ for the three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder for several system sizes $L$. The data cross at ${\beta }_{\mathrm{c}}^{-1}\simeq 0.951$. The dashed lines represent the optimal values obtained from a finite-size scaling for ${\beta }_{\mathrm{c}}$ and ${\xi }_L\left({\beta }_c\right)∕L$.

Figure 3

(Color online) Finite-size scaling analysis of ${\xi }_L∕L$ for the Ising spin glass with Gaussian bonds according to Eq. (12). The sizes are $6\le L\le 16$. The solid line is the third-order polynomial used in the fit, see Sec. 4.

Figure 4

(Color online) Binder ratio $g$, defined in Eq. (7), as a function of inverse temperature $\beta$ for the three-dimensional Edwards-Anderson Ising spin glass with Gaussian bonds. The data cross at ${\beta }_c^{-1}\simeq 0.931$. The dashed lines represent the optimal values obtained from a finite-size scaling for ${\beta }_c$ and $g\left({\beta }_c\right)$.

Figure 5

(Color online) Finite-size scaling analysis of the data for the Binder ratio $g$ according to Eq. (9) for the three-dimensional Ising spin glass with Gaussian disorder. The scaling analysis is performed for $L\ge 6$ and the solid line represents the best fit to the data from the finite-size scaling analysis.

Figure 6

(Color online) Finite-size scaling of the spin-glass susceptibility ${\chi }_{\mathrm{SG}}$ according to Eq. (19) for Gaussian disorder. In fact, the method of scaling worked better when using $T$ rather than $\beta$ and the fit parameters shown are for the $T$ fit. However, we show the resulting scaling plot using $\beta$ for consistency with the other plots. The scaling analysis is performed for $L\ge 6$.

Figure 7

(Color online) Finite-size correlation length ${\xi }_L∕L$ as a function of inverse temperature $\beta$ for the three-dimensional Edwards-Anderson Ising spin glass with $\pm J$ bonds for several system sizes $L$. The data cross at ${\beta }_{\mathrm{c}}^{-1}\simeq 1.12$. The dashed lines represent the optimal values obtained from a finite-size scaling for ${\beta }_{\mathrm{c}}$ and ${\xi }_L\left({\beta }_c\right)∕L$.

Figure 8

(Color online) Finite-size scaling analysis of ${\xi }_L∕L$ for the Ising spin glass with $\pm J$ bonds according to Eq. (12). The scaling analysis is performed for $L\ge 6$.

Figure 9

(Color online) Binder ratio $g$ as a function of inverse temperature $\beta$ for the three-dimensional Edwards-Anderson Ising spin glass with $\pm J$ bonds for several system sizes $L$. The data cross at ${\beta }_{\mathrm{c}}^{-1}\simeq 1.088$. The dashed lines represent the optimal values obtained from a finite-size scaling for ${\beta }_c$ and $g\left({\beta }_c\right)∕L$.

Figure 10

(Color online) Finite-size scaling analysis of the data for the Binder ratio $g$ according to Eq. (9) for the three-dimensional Ising spin glass with $\pm J$ bonds. The sizes used are $6\le L\le 24$.

Figure 11

(Color online) Finite-size scaling of the spin-glass susceptibility ${\chi }_{\mathrm{SG}}$ according to Eq. (19) for the three-dimensional Ising spin glass with $\pm J$ bonds. As for the corresponding plot for the Gaussian distribution, Fig. 6, the fit is actually done in $T$, but the plot is given as a function of $\beta$ for consistency with the other plots. The scaling analysis is performed for $L\ge 6$.

Figure 12

(Color online) Binder ratio $g$ as a function of the finite-size correlation length ${\xi }_L∕L$ for several system sizes $L\ge 6$ and for all models described in Sec. 2. All data collapse onto a universal curve, thus providing clear evidence that spin glasses obey universality. Different system sizes are labeled with different colors, and different models use different types of symbols as indicated.