Design and characterization of all 2D fragile topological bands.

Designing topological materials with specific topological indices is a complex inverse problem, traditionally tackled through manual, intuition-driven methods that are neither scalable nor efficient for exploring the vast space of possible material configurations. In this work, we develop an algorithm that leverages the covariance matrix adaptation evolution strategy to optimize the Fourier representation of the periodic functions shaping the designer material's characteristics. This includes mass profiles or dielectric tensors for phononic and photonic crystals, respectively, as much as synthetic potentials applicable to ultra-cold atomic systems. We demonstrate our methodology with a detailed characterization of a class of topological bands known as "fragile topological," showcasing the algorithm's capability to address both topological characteristics and spectral quality, and demonstrating the experimental feasibility of realizing all of the classified fragile topological phases. This automation not only streamlines the design process but also significantly expands the potential for identifying and constructing high quality designer materials across the wide range of platforms, and is readily extendable to other setups, including higher-dimensional and nonlinear systems.