Multiscale complexity of two-dimensional Ising systems with short-range, ferromagnetic interactions

TL;DR

This work introduces and applies the multiscale complexity framework, focusing on the complexity profile (CP) and scale-specific information to a finite-range 2D Ising model across hexagonal, square, and triangular lattices. By expressing subsystem probabilities through marginal distributions and leveraging Markov random-field properties, the authors compute CP measures that reveal how structure emerges across scales near the critical region, including domain formation and magnetization. The study finds an area-law-like conservation for CP across scales, identifies a subcritical peak in pairwise complexity in the disordered phase, and demonstrates universal-like scaling in the derivatives of CP-related quantities with system size. These results provide an information-theoretic lens on emergent phenomena in lattice spin systems, offering a complementary perspective to traditional thermodynamic observables and suggesting broader applicability to other finite-range, equilibrium systems.

Abstract

Complex systems exhibit macroscopic behaviors that emerge from the coordinated interactions of their individual components. Understanding the microscopic origins of these emergent properties remains a significant challenge, especially in less-understood systems, due to the absence of a generalized framework for identifying the governing degrees of freedom. The multiscale complexity formalism, developed to address this challenge, consists of a set of information-theoretic indices designed to identify the scales at which collective behaviors emerge. In this article, we evaluate one such index, the complexity profile, by applying it to the two-dimensional Ising model with finite-range interactions. In particular, we show that the complexity profile captures the transition between the disordered and ordered phases by detecting the emergence of multiscale structure 1 exclusively in the critical region, and therefore offering insights into the formation of magnetic domains from an information-theoretic perspective. Additionally, we show that the pairwise complexity exhibits a maximum in the disordered phase that remains bounded in the thermodynamic limit. These results highlight the potential of the multiscale complexity formalism to probe emergent behaviors and detect hidden features of critical phenomena in interacting systems beyond the classical characterization of correlations.

Figures (10)

• Figure 1: The hexagonal, square and triangular lattices with periodic boundary conditions upon which we define the Ising model. The labeling of the spin sites is for the proper enumeration of the spin subsystems and correlation functions.
• Figure 2: The reduction in the number of DoF in the conditional probability $p(\sigma^{(u)}|\sigma^{(u^{\prime})})$ using the Markov field property of conditional independence. The highlighted spin subsystem $\boldsymbol{\partial}^{(u^{\prime})}$ is a subset of $\boldsymbol{\sigma}^{(u^{\prime})}$ and is the boundary that separates subsystem $\boldsymbol{\sigma}^{(u)}$ from the rest of the system.
• Figure 3: The complexities as functions of $\beta$ for the ferromagnetic (a) hexagonal, (b) square and (c) triangular lattices with $N=18$ spins. The curves are labeled by scale from $k=2$ (light) to $k=18$ (dark). (d) The finest-scale complexity $C(1)$ for the three lattices, which corresponds to the joint Shannon entropy of the whole system $H(\boldsymbol{s})$.
• Figure 4: (a) The scale-specific shared information for the ferromagnetic (a) hexagonal, (b) square and (c) triangular lattices with $N=18$ spins (the insets show the global behavior at scales $k\geq2$). (d) A comparison between the behavior of the extrema of $D(k)$ of each lattice as a function of scale. Note that these extrema correspond to different $\beta$ as shown in (a)-(c).
• Figure 5: Venn diagram representing the dependency space of spin system $\boldsymbol{s}=\{s_1, s_2, s_3\}$.