Spin glass behavior in a random Coulomb antiferromagnet

We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.

Figure 1

Data for the spin glass susceptibility at small $\beta (\equiv 1/T)$ in two dimensions for a concentration $x=1/16$ for $N=16$ and 64. The curve is a parabolic fit to the data for $N=64$ and has the form $1+0.465\beta +0.0669{\beta }^2$. The behavior is clearly linear, not quadratic, at small $\beta$, which is a result of screening. The intercept at $\beta =0$ differs measurably from unity for very small sizes only because of the charge neutrality constraint [Eq. (4)].

Figure 2

Equilibration plot for $N=576,d=2$ with $x=1/16$ at the lowest temperature of $T=0.032$. The spin glass susceptibility ${\chi }_{SG}$ is plotted against Monte Carlo sweeps $t$. For each point, averaging is carried out over the last half of the sweeps. The inset shows an enlargement of the data points for the longest times.

Figure 3

Data for the Binder ratio in $d=2$.

Figure 4

Scaling plot for Binder ratio in $d=2$ according to Eq. (12) with $T_c=0$. We are not able to get all the data to collapse for any value of the correlation length exponent $\nu$. However, the data for the largest sizes collapse with $\nu \simeq 3.5$ (shown), consistent with the value in the EA model. However, there are big uncertainties in our estimate (see text).

Figure 5

The spin glass susceptibility in $d=2$ scaled according to Eq. (14) with $T_c=0$ and $\eta =0$. No value of $\nu$ succeeds in scaling all the data, though the largest sizes scale reasonably well with $\nu \simeq 2.7$ (shown), not very different from the value for the EA model of 3.4. However, there are big uncertainties in our estimate (see text).

Figure 6

Correlation length divided by system size in $d=2$.

Figure 7

Scaling plot of the correlation length divided by system size in $d=2$ assuming $\nu =5.0$.

Figure 8

A plot of the Binder ratio against the correlation length divided by (linear) system size $L$ for different values of $N$. Since both these quantities are dimensionless, the data should collapse in the absence of corrections to finite-size scaling. The results show that corrections are large for $N=64$, moderate for $N=144$, and quite small for the larger sizes.

Figure 9

Binder ratio for $d=3$.

Figure 10

Correlation length divided by system size for $d=3$.

Figure 11

A plot of the Binder ratio against the correlation length divided by (linear) system size $L$ for different values of $N$. Since both these quantities are dimensionless, the data should collapse in the absence of corrections to finite-size scaling. The results show that corrections are very large for $N=64$, moderate for $N=216$, and apparently quite small for the larger sizes.