Network science Ising states of matter

TL;DR

The paper introduces IsingNets—networks built from Monte Carlo snapshots of the 2D Ising model—to study phase transitions with network science. It employs two complementary analyses: a distance-filtration approach that leverages percolation, persistent homology, MST visualization, and embeddings to reveal topology and geometry across $T$, and a fixed-threshold network analysis that uncovers degree/strength distributions, correlations, $K$-core structures, and spectral properties. The results show nonrandom organization, including two giant components below $T_c$, topological signatures in persistent diagrams that peak near criticality, and distinctive spectral and correlation patterns that distinguish ferromagnetic from paramagnetic phases. The framework is interpretable, broadly applicable to numerical and experimental data, and offers unsupervised indicators of critical points with potential extensions to other models and to quantum wave-function snapshots.

Abstract

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the $2$D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the $2$D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.

Figures (18)

• Figure 1: The percolation properties of the IsingNet (left panels) generated from $2$D Ising model Monte Carlo simulations on spin systems of linear size $L=40$ at temperature $T=2.12<T_c$ are shown as a function of filtration parameter $r$. Nodes are connected if their distance is less than $r$. Five quantities are measured: the fraction of nodes in the largest connected component (the first row, blue line) and the fraction of nodes in the second largest connected component (the first row, orange line), the average size of components that are smaller than the second largest component $\langle s \rangle$ (the second row), the number of components $n_s$ (the third row) and the inverse participation ratio $Y$ (the fourth row). The results are compared with these quantities obtained from corresponding percolation properties obtained from a randomly permuted distance matrix (right panels). The number of nodes of the IsingNets is $N=6000.$
• Figure 2: Same as Figure \ref{['fig:1']} but with IsingNets obtained from $2$D Ising model simulations at $T=2.25$.
• Figure 3: Same as Figure \ref{['fig:1']} but with IsingNets obtained from $2$D Ising model simulations at $T=2.50$.
• Figure 4: A schematic illustration of the filtration process. The barcodes are used to show the appearance and disappearance of topological features corresponding to different homology classes as the filtration parameter $r$ is increased. The filtration process ends at $r=r_{\max}$ when all $N$ nodes are fully connected and forms a $N$-simplex. Simplices of different dimensions are indicated by different colors.
• Figure 5: The persistent diagram corresponding to homology classes in $H_0$, $H_1$, $H_2$, and $H_3$ of the IsingNet clique complexes are plotted as a function of the filtration parameter $r$. Panels (a), (b), and (c) show the persistent diagrams of IsingNets obtained from the spin system of linear size $L=40$ at $T=2.12$ (a), $T=2.25$ (b), and $T=2.50$ (c). Panels (d), (e), and (f) show the persistent diagram of corresponding randomized null models obtained at $T=2.12$ (d), $T=2.25$ (e), and $T=2.50$ (f). The networks are formed by $N=100$ nodes.