Complete inverse design to customize two-dimensional dispersion relation via nonlocal phononic crystals

We report a new method to tailor the entire two-dimensional (2D) dispersion relation based on nonlocal phononic crystals, where beyond-nearest-neighbor (BNN) interactions are used to achieve precise sculpting of the band surface. Focusing on square lattices, we demonstrate unconventional band structures such as multifold symmetries, roton/maxon/saddle-type critical points, anisotropy-to-isotropy transitions, and other exotic band morphologies. We anticipate that the design protocols in this study can be extended to all other two-dimensional lattices and multiband customization.