Fractal dimension of interfaces in Edwards-Anderson spin glasses for up to six space dimensions

The fractal dimension of domain walls produced by changing the boundary conditions from periodic to antiperiodic in one spatial direction is studied using both the strong-disorder renormalization group algorithm and the greedy algorithm for the Edwards-Anderson Ising spin-glass model for up to six space dimensions. We find that for five or fewer space dimensions, the fractal dimension is lower than the space dimension. This means that interfaces are not space filling, thus implying that replica symmetry breaking is absent in space dimensions fewer than six. However, the fractal dimension approaches the space dimension in six dimensions, indicating that replica symmetry breaking occurs above six dimensions. In two space dimensions, the strong-disorder renormalization group results for the fractal dimension are in good agreement with essentially exact numerical results, but the small difference is significant. We discuss the origin of this close agreement. For the greedy algorithm there is analytical expectation that the fractal dimension is equal to the space dimension in six dimensions and our numerical results are consistent with this expectation.

Figure 1

Representative evolution of ${\mathrm{\Omega }}_i$ of the decimated spin as a function of the RG step, which corresponds to the number of spins which have been decimated for the EA model for (a) $d=2$ and $L=100$ and (b) $d=6$ and $L=6$. Over most of the iteration range for $d=2, {\mathrm{\Omega }}_i$ is positive. The SDRG estimate for the exponent $d_{\mathrm{s}}$ is also quite accurate in this case. As $d$ increases, the values of ${\mathrm{\Omega }}_i$ turn negative after a decreasing number of iterations, suggesting that the SDRG becomes less accurate in higher dimensions, as can be seen for $d=6$ [in (b)]. Note the different horizontal scales.

Figure 2

Plot of $ln\mathrm{\Gamma }$ for various space dimensions $d$ as a function of $lnL$ computed using the SDRG algorithm. Note that $\mathrm{\Gamma }\sim L^{d_{\mathrm{s}}-d}$. Our estimate of $d_{\mathrm{s}}$ is determined by the slope of the straight lines drawn through the points at large-$L$ values. Note how the data for $d=6$ level off, i.e., $d_{\mathrm{s}}\to d$. (See Fig. 3 for an enlarged figure in six dimensions.) Error bars are smaller than the symbols.

Figure 3

Plot of $ln\mathrm{\Gamma }$ for $d=6$ as a function of $lnL$ computed using the SDRG and GA algorithms. Our estimate of $d_{\mathrm{s}}$ is determined by the slope of the straight lines drawn through the points at large-$L$ values. Using $\mathrm{\Gamma }\sim L^{d_{\mathrm{s}}-d}$, the leveling off of the lines at the larger values of $L$ implies that $d_{\mathrm{s}}\to d$ in six dimensions. Error bars are smaller than the symbols.

Figure 4

The bifurcation of a tree is a self-similar fractal. The four figures are measurements of its length using square domains whose linear size is reduced at each step of the renormalization. For a self-similar fractal, like the ponderosa pine depicted here, the scaling dimension $d_{\mathrm{s}}$ is the same no matter what length scale is used to determine it. Panel (a) shows the coarsest measurements which are successively refined by reducing the size of the squares in (b)–(d). Note that the domains are smaller than the image resolution in (d). The fractal dimension of the ponderosa pine is approximately 1.88. One could in principle obtain the correct fractal dimension by studies at the coarsest length scales, which is why we suspect that the SDRG, which works better on the coarsest length scales, is capable of getting accurate answers for $d_{\mathrm{s}}$.

Figure 5

Plot of $ln\mathrm{\Gamma }$ for various space dimensions $d$ for the EA model as a function of $lnL$ computed using the GA. Note that $\mathrm{\Gamma }\sim L^{d_{\mathrm{s}}-d}$. Our estimate of $d_{\mathrm{s}}$ is determined by the slope of the straight lines drawn through the points at large-$L$ values. Error bars are smaller than the symbols.

Figure 6

Greedy algorithm results (blue pentagons) compared with strong-disorder renormalization group results (red squares) for $d=2$, 3, 4, 5, and 6. The upper bound $d_{\mathrm{s}}-d+1$ at unity is marked by a horizontal blue line, while the lower bound at zero is marked with a horizontal red line. The value $d_{\mathrm{s}}=0$ for $d=1$ is exact and given by both methods. Only statistical errors are included and error bars are smaller than the symbols. Numerical values are summarized in Table 2.