Realization of a topological phase transition in a gyroscopic lattice

Abstract

Topological metamaterials exhibit unusual behaviors at their boundaries, such as unidirectional chiral waves, that are protected by a topological feature of their band structure. The ability to tune such a material through a topological phase transition in real time could enable the use of protected waves for information storage and readout. Here we dynamically tune through a topological phase transition by breaking inversion symmetry in a metamaterial composed of interacting gyroscopes. Through the transition, we track the divergence of the edge modes' localization length and the change in Chern number characterizing the topology of the material's band structure. This work provides a new axis with which to tune the response of mechanical topological metamaterials.A central challenge in physics is understanding and controlling the transport of energy and information. Topological materials have proven an exceptional tool for this purpose, since topological excitations pass around impurities and defects and are immune to back-scattering at sharp corners [1,2]. Furthermore, topological edge modes are robust against weak disorder, such as variations in the pinning energy of each lattice site, in contrast to typical edge waves [3,4].Mechanical topological insulators represent a rapidlygrowing class of materials that exhibit topologicallynontrivial phononic band structure [2,5]. An important subset of these materials send finite-frequency waves around their perimeter, in a direction determined by the band topology [4,[6][7][8][9]. Just as in electronic materials, these edge waves are immune to scattering either into the bulk or in the reverse direction along the edge. Here we present a method for reversibly passing through a topological phase transition, which allows us to switch chiral edge modes on and off in real time.Our system consists of rapidly-spinning gyroscopes hanging from a plate (Figure 1). If displaced from equilibrium, a single gyroscope will precess: its tip moves in a circular orbit about the equilibrium position as a result of the torques from gravity and the spring suspension. To induce repulsive interactions between gyroscopes, we place a magnet in each gyroscope with the dipole moments aligned.A honeycomb lattice of such gyroscopes behaves as a Chern insulator, exhibiting robust chiral edge waves that pass around corners uninhibited. The phononic spectrum has a band gap, and shaking a boundary site at a frequency in the gap generates a wavepacket that travels clockwise along the edge. Figure 1b shows such a wavepacket as seen from below.The origin of these chiral edge modes is broken time reversal symmetry, which arises from a combination of lattice structure and spinning components [7]. As in [7,9], an effective time reversal operation both reverses time (t → −t) and reflects one component of each gyroscope's displacement (ψ → ψ * , where ψ = δx + iδy is the displacement of a gyroscope). Breaking effective time re-motor magnet coil spring suspension FIG. 1. Modulating the magnetic field at each la...

Tunable Band Topology in Gyroscopic Lattices

“…4A. We note that this feature was absent in the gyroscopic honeycomb lattice previously studied [4,5], whose topology was unaffected by changes in Ω k and Ω g . This allowed the topology to be continuously connected to the electronic Haldane model, unlike in the spindle lattice.…”

“…To date, the only ordered lattices that have been considered in this framework are the honeycomb lattice and simple distortions thereof [4,5]. A slightly different manifestation of gyroscopic metamaterials considered in [7] found that by including staggered sublattice precession frequencies and bond strengths, time reversal symmetry could be effectively broken in lattices with square and honeycomb symmetries.…”

“…In mechanical systems, these chiral edge waves have recently been demonstrated using structures composed of coupled gyroscopes [4][5][6][7]. To date, studies of topological gyroscopic materials have focused on a handful of simple lattices that possess a nonzero Chern numbers in their phononic band structure.…”

Chiral edge currents for ac-driven skyrmions in confined pinning geometries

“…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.…”

Inverse design of discrete mechanical metamaterials

“…Topological mechanics offers a powerful framework for the control of phononic excitations. Topologically pro- tected phonons localized at the edges of gapped mechanical materials have been studied extensively in special lattices [19,20,25,42,43] and in random networks [24,44].…”