Energy landscapes of small SK spin glasses

TL;DR

This paper investigates the energy landscapes of small SK spin glasses with up to 9 spins, analyzing their energy spectra and hierarchical organization of states to understand the structure of these complex systems.

Contribution

It provides a detailed analysis of the energy spectra and landscape organization of small SK spin glasses, highlighting the emergence of hierarchical structures at small system sizes.

Findings

Hierarchical organization of ground states appears at N as small as 6

Energy spectra are classified by symmetry equivalence classes

Energy landscapes show early signs of complex structure in small systems

Abstract

We study the $\pm J$ SK model for small $N$'s up to $N=9$. We sort the $2^{N(N-1)/2}$ possible realizations of the coupling matrix into equivalence classes according to the gauge symmetry and permutation symmetry and determine the energy spectra for each of these classes. We also study the energy landscape in these small systems and find that the elements of the hierarchic organization of ground states %and higher energy local minima start to appear in some samples already for $N$'s as small as 6.