Energy landscapes of small SK spin glasses

TL;DR

This work investigates the zero-temperature structure of the ±J Sherrington-Kirkpatrick spin glass for small systems (up to $N=9$) by classifying all coupling realizations into isospectral gauge- and permutation-equivalence classes and computing the corresponding energy spectra. It leverages gauge fixing and signed-graph concepts to enumerate representatives and determine spectra via $-\tfrac{1}{2}S^TJS$, enabling a detailed map of ground- and low-lying-state structures and the emergence of hierarchical, ultrametric-like organization at surprisingly small sizes. The authors present comprehensive results for $N=5$ and $N=6$, including counts of frustrated cycles and exact energy degeneracies, and they visualize energy maps showing ground-state basins and the onset of fragmentation. They also report rich higher-energy features such as isolated clusters and a toroidal 60-cluster at $N=8$, illustrating the non-convex landscape even in tiny SK systems. A web resource extends the results to $N=7,8,9$, highlighting the rapid growth of isospectral classes and the persistent influence of symmetries on the spectrum. The study connects finite-size spin-glass behavior to broader themes in disordered systems and signed-graph theory, offering benchmarks and insights for understanding complex energy landscapes in small, non-self-averaging ensembles.

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.

Figures (10)

• Figure 1: Classes for $N=4$
• Figure 2: Classes for $N=5$.
• Figure 3: Classes for $N=6$.
• Figure 4: Phase space maps for $N=3$ (black: ground state, blue: 1st excited state), (a) the ferromagnet, (b) the antiferromagnet. The dots represent the spin states, the links connect states that are next neighbors in phase space, which means they differ in the flip of a single spin.
• Figure 5: Phase space maps for $N=4$ (black: ground state, blue: 1st excited state, green: 2nd excited state, yellow: 3rd excited state, orange: 4th excited state).