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

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 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.