Amorphous topological insulators constructed from random point sets

The discovery that the band structure of electronic insulators may be topologically non-trivial has revealed distinct phases of electronic matter with novel properties ^1,2 . Recently, mechanical lattices have been found to have similarly rich structure in their phononic excitations ^3,4, giving rise to protected unidirectional edge modes ^5-7 . In all of these cases, however, as well as in other topological metamaterials ^3,8, the underlying structure was finely tuned, be it through periodicity, quasi-periodicity or isostaticity. Here we show that amorphous Chern insulators can be readily constructed from arbitrary underlying structures, including hyperuniform, jammed, quasi-crystalline and uniformly random point sets. While our findings apply to mechanical and electronic systems alike, we focus on networks of interacting gyroscopes as a model system. Local decorations control the topology of the vibrational spectrum, endowing amorphous structures with protected edge modes - with a chirality of choice. Using a real-space generalization of the Chern number, we investigate the topology of our structures numerically, analytically and experimentally. The robustness of our approach enables the topological design and self-assembly of non-crystalline topological metamaterials on the micro and macro scale.