Higher-Order Topological Acoustics

The discovery of topologically non-trivial phases has revolutionized our understanding of matter and materials. Topology is an important development of traditional band theories. As a previously unexplored degree of freedom, it describes the global properties of wavefunctions over an entire band and it is robust against local perturbation unless the perturbation is strong enough to close the gaps between bands. Such robustness of topological phases makes them fascinating and useful. Topological phases have their root in geometric phases, a universal concept arising from adiabatic evolution, which was first explored in optics and later generalized to all areas of physics. Topological matters were first investigated in condensed matter physics and expanded to electromagnetic waves, optics, and acoustic systems. The emergence of “topological acoustics” is a major breakthrough that has reinvigorated the study of phononic crystals, which are the acoustic analog of electronic bands in solids.

In conventional topological phases, an n-dimensional system can sustain (n-1)- dimensional topological edge modes. These edge modes are robust against local perturbations and in some cases can even be immune to back-scattering, making them scientifically fascinating and potentially useful as efficient conducting channels, or, in the case of light and sound, one-way waveguides. Recently, a new class of topological phases named higher-order topological insulators (HOTIs) have been proposed, in which n-dimensional system can have (n-m)-dimensional topological boundary states, with m>1. For example, 0D topological “corner modes” have been found in 2D systems. These modes are localized at corners where two or more edges meet. Similarly, 3D systems can support 1D “hinge modes.” In this proposal, we aim to study novel HOTIs in acoustic systems. First, we plan to use a theoretical approach that constructs finite-sized 2D HOTIs from 1D topological systems, which are much easier to analyze and construct. Specifically, we propose to use this approach to design exactly quantized topological dipole insulators and quadrupole insulators from a 1D Su-Shrieffer-Heeger model. We will also design an experimental platform to realize these 2D HOTIs and observe topological corner modes therein. Moreover, our approach allows us to analyze, from a transfer-matrix point of view, the evanescent coupling effect among corner modes. We hope to identify an experimentally observable effect that can be used to distinguish HOTIs with different physical origins. Lastly, we expect to extend our theoretical framework for the exploration of new topological phases by combining different classes of 1D topological systems.