Dynamical Majorana edge modes in a broad class of topological mechanical systems

Abstract

Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion centre and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behaviour of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, and hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics.

Mechanical Analogue of a Majorana Bound State

“…Topological modes at an extended boundary are propagating, but this is not generic. Prominent examples of bound states have been reported in Maxwell lattices for static floppy modes, 1D spring‐connected dimer chains, and photonic Majorana zero‐mode crystals . Other notable examples of 0D states emerge in higher‐order topological insulators that are characterized by a quantized nontrivial bulk polarization .…”

“…[3,4] Other phononic metamaterials have been constructed to enable mechanical versions of the quantum spin Hall effect [5][6][7] and the valley degree of freedom [8] for unidirectional phononic signal guiding.Topological modes at an extended boundary are propagating, but this is not generic. Prominent examples of bound states have been reported in Maxwell lattices for static floppy modes, [2,9] 1D spring-connected dimer chains, [10] and photonic Majorana zero-mode crystals. [11,12] Other notable examples of 0D states emerge in higher-order topological insulators that are characterized by a quantized nontrivial bulk polarization.…”

Topological modes in radiofrequency resonator arrays

“…This discovery has not only led to observations of new properties of periodic systems observed in nature, but also to new materials in the quantum and classical regimes. Many of these systems are relatively simple, and include chains of massspring oscillators [3], magnetically coupled spinners [2], plasmonic nanoparticles [4,5], dielectric resonator chains [6,7], planar metasurface architectures [8], microwave arrays [9], and acoustic systems [10].…”

Directional mechanical response in the bulk of topological metamaterials

“…Kane and Lubensky mapped their systems onto topological insulators in the BDI class and considerations of symmetry and dimension show that other classes of topological insulators also have mechanical analogs [25]. Because topology is robust against disorder, does not require fine-tuning and controls mechanical response, it has been suggested as a mechanism that arises in natural biological structures [26][27][28]. Related topological states with nontrivial edge modes arise in a variety of other classical systems, including photonics [29], magnetic systems [30][31][32], classical electrical circuits [33,34], acoustic resonators [35], active fluid [36,37] and nonequilibrium stochastic biological systems [38].…”