Page 13 - Fall2021

FEATURED ARTICLE
Topological Acoustics
Andrea Alù, Chiara Daraio, Pierre A. Deymier, and Massimo Ruzzene
Introduction
The field of topology studies the properties of geometric objects that are preserved under continuous deforma- tions, for example, without cutting or gluing. A cup with a handle is topologically equivalent to a donut (or a bagel if you live in New York) because one shape can be deformed into the other while preserving their common invariant hole. Exotic topological shapes, such as vortices, knots, and mobius strips, can be globally analyzed using the mathematical tools offered by topology. The connection between topology and acoustics may appear far-fetched, yet recent developments in the field of condensed matter physics and quantum mechanics have been inspiring exciting opportunities to manipulate sound in new and unexpected ways based on topological concepts.
The field of topological acoustics has been inspired by the discovery in condensed matter of topologi- cal insulators, a class of materials that support highly unusual electrical conduction properties. Like con- ventional semiconductors, topological insulators are characterized by a gap in electron energy (bandgap) that separates their valence and the conduction bands. For electron energies within this bandgap, topological insulators are not electrically conductive in their bulk, hence their name. However, any finite sample of such materials necessarily supports conduction currents along its physical boundaries; the topological features of the valence and conduction bands ensure the exis- tence of these boundary currents. Therefore, these currents exist independent of the boundary shape or the presence of continuous defects and imperfections that do not affect the bandgap topology. Knowing this feature, we can predict the existence of conduction cur- rents flowing along the boundaries of any finite sample of such materials by simply analyzing the topological features of the bands of the infinite medium (Thouless et al., 1982; Haldane, 1988). As a result, these currents show an unusual robustness to defects and disorder. The electron spin plays a fundamental role in defining the topological response of these materials.
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In recent years, there has been a strong interest in exploring analogies for these topological concepts in other realms of physics, in particular, in the context of optics (Raghu and Haldane, 2008; Wang et al., 2009) and acoustics (Fleury et al., 2016; Zangeneh-Nejad et al., 2020). Given that sound does not possess an intrinsic spin, in this quest the role of the electron spin is replaced by the notion of acoustic pseudospins. These pseudospins include angular momen- tum (Fleury et al., 2014), geometrical asymmetries (Xiao et al., 2015; Ni et al., 2018), structured space- and time- dependent material properties (Trainiti et al., 2019; Darabi et al., 2020), and asymmetric nonlinearities (Boechler et al., 2011; Hadad et al., 2018).
These explorations have been enabling new opportuni- ties to route sound in novel and unintuitive ways. For example, topological sound can propagate only in one direction (forward, not backward), and it can take sharp turns following the arbitrary boundaries of an acoustic material just like the boundary currents of topological insulators. These exotic propagation modalities are unaf- fected by the presence of defects or imperfections that sound may encounter along the way, for example, in the form of localized scatterers or material heterogeneities.
Figure 1a shows one example of an acoustic topological insulator formed by an ordered array of subwavelength resonators whose properties are modulated in space and time with precise patterns to impart angular momentum (Fleury et al., 2016). As a result of the interplay between the array geometry and the angular momentum imparted by the modulation, topological sound propagation is achieved through the pressure fields that travel unidi- rectionally along the array boundaries (see Figure 1a).
In recent years, topological sound has expanded its realms, leading to the exploration of topological features not only in the bands of periodic structures, like the one in Figure 1a, but also in real space and parameter space. For example, Figure 1b shows the evolution of the eigen- values of a system as two generic degrees of freedom or
Volume 17, issue 3 | Fall 2021 • Acoustics Today 13