Exact enumeration of ground states in the Sherrington-Kirkpatrick spin glass

Using the discrete $\pm J$ bond distribution for the Sherrington-Kirkpatrick spin glass, all ground states for the entire ensemble of the bond disorder are enumerated. Although the combinatorial complexity of the enumeration severely restricts attainable system sizes, here $N⩽9$, some remarkably intricate patterns found in previous studies already emerge. The analysis of the exact ground-state frequencies suggests a direct construction of their probability density function. Against expectations, the result indicates that its highly skewed appearance for finite $N$ evolves logarithmically slow towards a Gaussian distribution.

Figure 1

Plot of the rescaled probability density function of ground state energies using $\pm J$ bonds. Shown are the numerical data for the SK model obtained with the extremal optimization (EO) heuristic and their fit by Eq. (3) with $m=5.4$, properly rescaled, from Ref. 8, and the exact results for $N=8,9$ based on Table . Remarkably, there is very little variation between the exact data at $N=8,9$ and the numerical data obtained for $N=127–511$.