Amorphous topological insulators constructed from random point sets

Abstract

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][6][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.Condensed-matter science has traditionally focused on systems with underlying spatial order, as many natural systems spontaneously aggregate into crystals. The behaviour of amorphous materials, such as glasses, has remained more challenging 9 . In particular, our understanding of common concepts such as bandgaps and topological behaviour in amorphous materials is still in its infancy when compared with crystalline counterparts. This is not only a fundamental problem; advances in modern engineering, both of metamaterials and of quantum systems, have opened the door for the creation of materials with arbitrary structure, including amorphous materials. This prompts a search for principles that can apply to a wide range of amorphous systems, from interacting atoms to mechanical metamaterials.In the exploration of topological insulators, conceptual advances have proven to carry across between disparate physical realizations, from quantum systems 10 , to photonic waveguides 11 , to acoustical resonators 12,13 , to hinged or geared mechanical structures 3,14 . One promising model system is a class of mechanical insulators consisting of gyroscopes suspended from a plate. Appropriate crystalline arrangements of such gyroscopes break time-reversal symmetry, opening topological phononic bandgaps and supporting robust chiral edge modes 5,6 . Unlike trivial insulators, whose electronic states can be thought of as a sum of independent local insulating states, topological insulators require the existence of delocalized states in each non-trivial band and prevent a description in terms of a basis of localized Wannier states [15][16][17] . It is natural, therefore, to assume that some regularity over long distances may be key to top...

Topological Elasticity of Nonorientable Ribbons

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“…Studies of gyroscopic metamaterials [11,13,14] showed that perturbations migrate to the edge of the sample and propagate around the edge in one direction. In our system, for very weak damping or small α m , it would be possible to perturb skyrmions in the bulk of the pinfree channel and obtain motion that is localized on the edges of the channel, producing a transient current which eventually damps away.…”

Topological valley transport under long-range deformations

“…Another emerging topic is topological edge modes existing at the interface of two different amorphous materials. Recently, amorphous systems without time-reversal symmetry have been reported with unidirectional edge modes along the interface of two different amorphous bulks (35,36). It is natural, therefore, to further find topological transport between different amorphous structures without breaking time-reversal symmetry.…”