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 challenging, particularly in systems beyond canonical examples, due to the absence of a generalized framework for identifying the governing degrees of freedom. The multiscale complexity formalism aims to address this challenge by employing information-theoretic indices of structure to identify the scales at which collective behaviors emerge. In this paper, we evaluate the complexity profile index as a tool for identifying such scales by applying it to two-dimensional ferromagnetic Ising systems. We demonstrate that these systems exhibit multiscale behavior in the critical region, and the emergence of macroscopic magnetization in the ordered phase corresponds to a non-monotonic behavior of the complexity profile at large scales. Additionally, we show that the pairwise complexity exhibits a maximum in the disordered phase that remains bounded in the thermodynamic limit.