Topological edge modes by smart patterning

Abstract

The research in topological materials and meta-materials has reached maturity and is gradually entering the phase of practical applications and devices. However, scaling down experimental demonstrations presents a major challenge. In this work, we study identical coupled mechanical resonators whose collective dynamics are fully determined by the pattern in which they are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using Ktheory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.

Topological bands and localized vibration modes in quasiperiodic beams

“…The black regions define ranges of frequency populated by the bulk eigenvalues, while the white areas correspond to frequency ranges where no states exist and identify bandgaps. The spectrum has features similar to the Hofstadter butterfly spectrum encountered in quantum mechanics for lattices under a magnetic field [13] and in discrete mechanical QP lattices [16,25]. One can observe the existence of a low frequency bandgap starting at zero, due to the presence of the ground springs, and a number of other gaps associated with Bragg scattering.…”

“…In particular, η i are determined by imposing that the weighted error ò integrated over the domain  is zero, which may be written as Figure 2. (a) Generation of a 1D patterns through projections from a circle [25]. (b) Periodic and QP sequences corresponding to three different values of the parameter θ: θ=0 and θ=0.4=2/5 define two periodic sequences (top and middle) of unit cell highlighted by the dashed rectangle, while q = 1 2 identifies a QP pattern with no spatial translational symmetry.…”

“…The ground springs are located according to patterns generated through the projection approach described in [25]. Specifically, the springs location x s are obtained from an underlying periodic lattice of period a, and by projecting from a manifold, which here is a circle of radius R<a/2 centered at the periodic sites sa,  Î s ( figure 2(a)).…”

“…The number of unit cells included in the computations depends on the objective of the investigations, which is two-fold. We first investigate structures that are defined by sufficiently large domains and are representative of the bulk properties of infinite periodic and QP domains( [25]). This analysis reveals a fractal frequency spectrum where the associated integrated density of states (IDS) define topological invariants that predict the existence of localized modes.…”

“…Thus, we can investigate the spectral properties of infinite beams as a function of the projection parameter θ by discretizing its range in steps Δ θ=1/S, which leads to an infinitely dense subset of [0, 1] as  ¥ S . In this framework, the bulk spectrum for irrational θ values is not explicitly computed, but the discretization identifies all commensurate, or periodic, rings defined by rational values of θ, whose vibrational spectrum approximates the bulk spectrum evaluated for all θä[0, 1] [25,32].…”