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arXiv · 2603.29984

Pattern Expansion of Spin Glasses

Mutian Shen·Zohar Nussinov·Yang-Yu Liu
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Abstract

We introduce a systematic method for expanding general spin-glass Hamiltonians in terms of Mattis interactions, providing a novel perspective for understanding the fundamental differences between short-range Edwards-Anderson (EA) and mean-field Sherrington-Kirkpatrick (SK) spin glasses. By iteratively extracting patterns from the coupling matrix, we expand the original spin-glass system into a Hopfield-like model (a series of Mattis interactions) plus a residual system. Our analysis reveals profound distinctions between EA and SK models: while EA models in two and three dimensions break into isolated subconnected sections after expansion, the SK model exhibits remarkable self-similar behavior, with the residual system preserving the mean-field structure and Gaussian statistics throughout the expansion process. This self-similarity manifests in exponential decay of residual matrix norms and expansion coefficients, reflecting the inherent mean-field nature of the SK model. Furthermore, we demonstrate that pattern expansion can identify ultra-low energy excitations in EA models, revealing excitations with energies that decrease rapidly with expansion step. Through connected component analysis, we quantify the size-energy relationship of these independent excitation clusters, opening new avenues for understanding the low-energy landscape of spin glasses and providing insights into the nature of metastable states.

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Mutian Shen, Zohar Nussinov, Yang-Yu Liu. 2026-03-31. Pattern Expansion of Spin Glasses. https://arxiv.org/abs/2603.29984

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