Exact Enumeration of Ground States in the Sherrington-Kirkpatrick Spin Glass

TL;DR

This paper presents an exact enumeration of all ground states in the Sherrington-Kirkpatrick spin glass with discrete bonds, revealing intricate patterns and a slow convergence towards a Gaussian distribution of ground state probabilities.

Contribution

It provides the first exact enumeration of ground states for the SK model with discrete bonds, uncovering detailed patterns and probabilistic behaviors.

Findings

Emergence of intricate ground state patterns at small system sizes

Ground state probability distribution evolves logarithmically slow towards Gaussian

Exact enumeration feasible for systems up to N=9

Abstract

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 $N\leq9$, 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 suggests that its highly skewed appearance for finite $N$ evolves logarithmically slow towards a Gaussian distribution.