Emergent properties and the multiscale characterization challenge in condensed matter, from crystals to complex materials: a Review

Abstract: The complexity of condensed matter arises from emergent behaviors that cannot be understood by analyzing individual constituents in isolation. While traditional condensed-matter approaches-developed primarily for ideal crystalline solids-have provided deep insights into symmetry, order, and electronic structure, they fall short in describing the rich, multiscale organization of hierarchical and soft materials. These systems exhibit structural correlations across multiple length and time scales, often governed by nonlinear interactions that span from molecular to macroscopic domains. This review explores how the convergence of emerging experimental and computational strategies are redefining our ability to characterize and model such systems. We examine how multimodal techniques-combining scattering, imaging, and spectroscopy-can map structural order and dynamics across scales, with methods like small-angle scattering tensor tomography, dark-field imaging, and ultrafast spectroscopies providing unprecedented spatiotemporal resolution. On the computational front, machine learning approaches such as graph neural networks, neural operators, and physics-informed models offer powerful tools to connect disparate scales and uncover hidden correlations in high-dimensional data. These advancements have the potential to close the gap between structure and function in complex materials, thereby addressing one of the grand challenges of contemporary material science: understanding and engineering multiscale architectures, whose emergent properties underpin the behavior of next-generation functional materials, biological systems, and adaptive technologies.