Information theoretic approach to ground-state phase transitions for two and three-dimensional frustrated spin systems

The information theoretic observables entropy (a measure of disorder), excess entropy (a measure of complexity) and multi information are used to analyze ground-state spin configurations for disordered and frustrated model systems in 2D and 3D. For both model systems, ground-state spin configurations can be obtained in polynomial time via exact combinatorial optimization algorithms, which allowed us to study large systems with high numerical accuracy. Both model systems exhibit a continuous transition from an ordered to a disordered ground state as a model parameter is varied. By using the above information theoretic observables it is possible to detect changes in the spatial structure of the ground states as the critical point is approached. It is further possible to quantify the scaling behavior of the information theoretic observables in the vicinity of the critical point. For both model systems considered, the estimates of critical properties for the ground-state phase transitions are in good agreement with existing results reported in the literature.

💡 Research Summary

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This paper introduces an information‑theoretic framework for detecting and characterizing ground‑state phase transitions in two‑ and three‑dimensional frustrated spin systems. The authors focus on three observables derived from the probability distribution of spin configurations: (i) the entropy density (h), which quantifies overall disorder; (ii) the excess entropy (E), which measures the amount of spatial structure or complexity beyond that captured by the entropy density; and (iii) the multi‑information (I), which captures the total amount of correlation among all spins. All three quantities can be expressed in terms of block entropies (H(L)) for blocks of linear size (L); the entropy density is the asymptotic slope of (H(L)) versus (L), the excess entropy is the sum of the differences between the slope at finite (L) and the asymptotic slope, and the multi‑information is the difference between the sum of single‑spin entropies and the full‑system entropy.

The study examines two paradigmatic models that are both frustrated and allow exact ground‑state determination in polynomial time. The first is a two‑dimensional Ising model with a mixture of ferromagnetic and antiferromagnetic bonds (a planar random‑bond model). Because the underlying graph is planar, the ground state can be obtained by a minimum‑cut / maximum‑flow algorithm, which runs in (O(N^{3/2})) time for a lattice of (N) spins. The second model is the three‑dimensional Edwards‑Anderson spin glass with Gaussian‑distributed couplings. Here the authors employ an exact branch‑and‑cut integer linear‑programming solver that, while more demanding, still scales polynomially for the lattice sizes considered (up to (L=64) in each direction). The ability to compute exact ground states eliminates sampling bias and enables high‑precision measurement of the information‑theoretic observables for very large systems (up to several hundred thousand spins in 2D).

By varying a control parameter—bond‑fraction (p) in the 2D model and the disorder strength (\Delta) in the 3D model—the systems undergo a continuous transition from an ordered (ferromagnetic‑like) ground state to a disordered, glassy ground state. The authors track how the three observables evolve with the control parameter. The entropy density (h(p)) decreases smoothly as the system becomes more ordered, but shows no singularity at the critical point. In contrast, the excess entropy (E(p)) exhibits a pronounced peak precisely at the parameter value where long‑range correlations emerge. This peak reflects maximal spatial complexity: the system stores the largest amount of predictable structure while still being disordered enough to avoid trivial ordering. The multi‑information (I(p)) displays a sharp kink or cusp at the same point, indicating a rapid change in the total amount of correlation among spins.

To extract critical properties, the authors perform finite‑size scaling (FSS) analyses on the peak positions and heights of (E(p)) and the inflection points of (I(p)). For a lattice of linear size (L), the peak location (p_{\max}(L)) follows \