Pattern Expansion of Spin Glasses

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.

Figures (5)

• Figure 1: Pattern expansion for different spin glass systems. Panels (a)--(c) use simplified models to demonstrate the mapping process from original spin configurations to pattern representations. Panel (d) shows how the energy function approximates the true energy landscape as the number of patterns increases: more patterns lead to finer energy descriptions, while model complexity also increases. From top to bottom in (a)--(c) are the 2D EA model (a), 3D EA model (b), and the SK model (c). After a certain number of expansion steps, it is evident that different systems exhibit distinct behaviors in their residual systems post-expansion. In the cases of 2D and 3D, the residual system often consists of several independent subconnected sections, which are typically single bonds or frustrated plaquettes composed of four bonds. However, for the SK system, the situation is quite different. The SK model actually demonstrates a self-similar behavior, whereby the residual system after each step of expansion is another SK model. This self-similarity is prominently evident in the pattern expansion process. The existence of this self-similarity may be attributed to the mean-field nature of the SK model. In Figure \ref{['fig:matrix_norm_alphas']}, we will provide a more detailed explanation of the presence of this self-similarity.
• Figure 2: Residual coupling statistics and norms under pattern expansion in EA and SK spin glasses. (a) Cumulative distribution function (CDF) of normalized residual coupling matrix elements $J_{ij}^{(\kappa)} / \sigma^{(\kappa)}$ for $\kappa = 10$ (solid lines) and $\kappa = 100$ (dashed lines) in 2D EA (blue), 3D EA (orange), and SK (green) models. The black line shows the standard Gaussian CDF for comparison. For the SK model, the CDF remains close to Gaussian at all expansion steps, demonstrating self-similarity: the residual system preserves the statistical properties of the original SK model. In contrast, EA models deviate from Gaussian behavior, particularly at larger $\kappa$ values, indicating the breakdown of self-similarity. (b) Normalized residual matrix norm $\left[ \|\mathbf{J}^{(\kappa)}\|/\|\mathbf{J}^{(0)}\| \right]$ as a function of expansion step $\kappa$ for 2D EA (triangles), 3D EA (crosses), and SK (inverted triangles) models. For SK model, quantities are normalized by $\sqrt{N}$ to account for mean-field scaling. SK shows near-exponential decay, while EA decay slows down as the system fractures into isolated islands. Additional details are shown in Supplementary Fig. \ref{['fig:si_matrix_norm_alphas']}. Error bars represent standard errors across multiple samples.
• Figure 3: Frustration density and thermodynamic behavior under pattern expansion in three-dimensional EA spin glasses. (a) At different expansion orders $\kappa$, the proportion of frustrated plaquettes $\rho$ is statistically analyzed. This quantity serves as an indicator of the "roughness" of the energy landscape. For typical spin glasses, the proportion of frustrated plaquettes is close to 0.5. (b) Binder cumulant $g$ of the spin overlap versus temperature $T$ for three-dimensional Edwards--Anderson systems with $L=4$, computed by population annealing with thermal boundary conditions (Supplementary Sec. \ref{['sec:supp_pa']}), for the original coupling matrix and for cumulative matrices at $\kappa=1,2,3,4,5,10$. All coupling matrices are normalized so that the standard deviation of bond strengths $J_{ij}$ (over nonzero couplings) equals unity. Vertical lines mark $T_{\mathrm{c}}$ of the three-dimensional ferromagnetic Ising model $T_{\mathrm{c}}^{\mathrm{FM}}\approx 4.511$ferrenberg1991critical and of the three-dimensional Edwards--Anderson spin glass $T_{\mathrm{c}}^{\mathrm{SG}}\approx 0.96$wangPopulationAnnealingMonte2015. For the ferromagnet, the inflection near $T_{\mathrm{c}}^{\mathrm{FM}}$ that drives $g$ below zero is also seen in similar calculationslundowIsingSpinGlasses2017.
• Figure 4: Ultra-low energy excitations revealed by pattern expansion in two- and three-dimensional EA spin glasses. In two-dimensional spin glasses, the energy changes caused by single-spin flips are analyzed. The spin difference is defined as a measure of local energy response. Panels (a) and (b) show 2D illustrations, while (c) and (d) show 3D generalizations; the method is consistent, with only the spatial dimension differing. (a) and (c) show energy and spin differences as functions of expansion step $\kappa$ for 2D EA (a) and 3D EA (c) models. The left y-axis shows the energy difference $\Delta E^{(\kappa)}$, while the right y-axis shows the normalized spin difference $\mathcal{V}^{(\kappa)}/N$. For 2D and 3D cases, multiple excitations often exist, so $\Delta E^{(\kappa)}$ and $\mathcal{V}^{(\kappa)}/N$ are computed in the sense of summing over all excitations. (b) and (d) show scatter plots of "energy difference vs. size" for independent excitation clusters in 2D (b) and 3D (d) EA models. Each point corresponds to a connected excitation cluster, with the horizontal axis showing the normalized size $\mathcal{V}/N$ and the vertical axis showing the cluster energy difference $\Delta E_{\text{cluster}}$. The correlation coefficient is also shown in the figure to characterize whether there is a significant correlation between energy and size.
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