Correlation length of the two-dimensional Ising spin glass with Gaussian interactions

We study the correlation length of the two-dimensional Ising spin glass with a Gaussian distribution of interactions, using an efficient Monte Carlo algorithm, proposed by Houdayer, that allows larger sizes and lower temperatures to be studied than was possible before. We find that the “effective” value of the bulk correlation length exponent $\nu$ increases as the temperature is lowered, and, at low temperatures, apparently approaches $-1∕\theta$, where $\theta \simeq -0.29$ is the stiffness exponent obtained at zero temperature. This means scaling is satisfied and earlier results at higher temperatures that find a smaller value for $\nu$ are affected by corrections to scaling.

Figure 1

A log–log plot of the finite size correlation length ${\xi }_L$ for different system sizes and temperatures. The data labeled $64\to \infty$ and $128\to \infty$ come from an extrapolation to the thermodynamic limit. The slope of the extrapolated data gives $-\nu \simeq -3.45$, which is consistent with $\theta \equiv -1∕\nu \simeq -0.287$ found in zero-temperature domain-wall calculations (Ref. 24).

Figure 2

Scaling plot of the data for the correlation length ${\xi }_L$ according to Eq. (5), with $\theta =-1∕\nu =-0.29$. The dashed line, which fits the data at low $T$, has slope $-1∕2$.

Figure 3

Scaling plot of the data for the correlation length ${\xi }_L$ according to Eq. (5), with $\theta =-1∕\nu =-0.45$.