Robustness and Stability of Spin Glass Ground States to Perturbed Interactions

TL;DR

This paper studies the stability of spin glass ground states under interaction perturbations, revealing large, highly connected sets of configurations with robustness properties similar to biological and computational systems, indicating potential universal principles.

Contribution

It demonstrates that spin glass ground states exhibit high robustness with topological properties akin to biological networks, suggesting a universal scaling law across different complex systems.

Findings

Large sets of bond configurations produce the same ground state.

Robustness scales logarithmically with subgraph size.

Similarity to biological and computational network scaling laws.

Abstract

Across many scientific and engineering disciplines, it is important to consider how much the output of a given system changes due to perturbations of the input. Here, we investigate the glassy phase of $\pm J$ spin glasses at zero temperature by calculating the robustness of the ground states to flips in the sign of single interactions. For random graphs and the Sherrington-Kirkpatrick model, we find relatively large sets of bond configurations that generate the same ground state. These sets can themselves be analyzed as subgraphs of the interaction domain, and we compute many of their topological properties. In particular, we find that the robustness, equivalent to the average degree, of these subgraphs is much higher than one would expect from a random model. Most notably, it scales in the same logarithmic way with the size of the subgraph as has been found in genotype-phenotype maps for RNA secondary structure folding, protein quaternary structure, gene regulatory networks, as well as for models for genetic programming. The similarity between these disparate systems suggests that this scaling may have a more universal origin.