Classification of topological phonons in linear mechanical metamaterials

Topological phononic crystals, alike their electronic counterparts, are characterized by a bulk-edge correspondence where the interior of a material dictates the existence of stable surface or boundary modes. In the mechanical setup, such surface modes can be used for various applications such as wave guiding, vibration isolation, or the design of static properties such as stable floppy modes where parts of a system move freely. Here, we provide a classification scheme of topological phonons based on local symmetries. We import and adapt the classification of noninteracting electron systems and embed it into the mechanical setup. Moreover, we provide an extensive set of examples that illustrate our scheme and can be used to generate models in unexplored symmetry classes. Our work unifies the vast recent literature on topological phonons and paves the way to future applications of topological surface modes in mechanical metamaterials.