Realization of an Acoustic Third-Order Topological Insulator

Abstract

The recent discovery of higher-order topological insulators (TIs) [1-5] has opened new possibilities in the search for novel topological materials and metamaterials. Secondorder TIs have been implemented in two-dimensional (2D) systems [6-19] exhibiting topological 'corner states', as well as three-dimensional (3D) systems having one-dimensional (1D) topological 'hinge states' [20]. Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamondlattice acoustic metamaterial. The bulk acoustic bandstructure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedron-like structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.Higher-order TIs are a new class of topological materials supporting a generalization of the bulk-boundary correspondence principle, in which topological states are guaranteed to exist along boundaries two or more dimensions smaller than that of the bulk [1][2][3][4][5]. In standard TIs, topological edge states occur at one lower dimension than the bulk [21,22]; for instance, a quantum Hall insulator has a 2D bulk and topological states on 1D edges. By contrast, a 2D second-order TI supports zero-dimensional (0D) topological 'corner states'. Such a lattice was first devised based on quantized quadrupole moments [1,2] and quickly realized in mechanical [6], electromagnetic [7], and electrical [9] metamaterials. Later, another type of 2D second-order TIs based on quantized Wannier centers, was proposed [23][24][25] and demonstrated in acoustic metamaterials [10,11]. In 3D materials, second-order TI behavior has also been observed in the form of 1D topological 'hinge states' in bismuth [20].According to theoretical predictions, TIs of arbitrarily high order are possible. However, in real materials the bulk is at most 3D. Thus, barring the use of 'synthetic' dimensions [26,27], the only remaining class of high-order TI is a third-order TI with 3D bulk and 0D corner states.As of this writing, no such material has been reported in the literature, although there exists a theoretical proposal based on quantized octupole moments [1,2].Here, we realize a third-order TI in a 3D acoustic metamaterial, observing topological states at the corners of a rhombohedron-like sample. This third-order TI is based on the extension of Wannier-type second-order TIs to 3D [23,25], and can be regarded as a 3D generalization of the classic 1D Su-Schrieffer-Heeger (SSH) model [28]. Just as in the SSH case, the eigenmode polarizations are quantized by lattice symmetries, and the Wannier centers are pinned to highsymmetry points; the mismatch between the Wannier centers and lattice truncations gives rise to charge fractionalization and hence lower-dimensio...

Fragile topologically protected perfect reflection for acoustic waves

“…That is, if the defect is placed just near the AC, the AC eigenfrequency would be changed and therefore, the TPPR will move to the new AC eigenfrequency as long as it still falls into the topological band gap. This can be overcome by replacing the present AC with the topological cavity with eigenfrequency robust against disorders, for example, corner states in high-order topological insulators [65][66][67][88][89][90][91][92][93].…”

Octupole corner state in a three-dimensional topological circuit

“…Thus far, HOTIs are mostly studied in 2D systems that host a quadrupole corner state, such as 2D microwave circuits 19 , lowfrequency electrical circuits 20 , photonic crystals [21][22][23][24][25] , mechanical systems 26 , and acoustic systems 27 . The 3D topological corner mode has been demonstrated very recently [28][29][30][31] ; however, some of these modes result from the nontrivial Zak phase of 3D bulk states 28 , which is of a very different origin than octupole modes.…”

Observation of an acoustic octupole topological insulator

“…It has been shown, both theoretically and in recent experiments [9][10][11][12][13][14] , that a twodimensional (2D) quantised quadrupole TI exhibits topological states at "boundaries of boundaries": it lacks 1D topological edge states (unlike standard 2D TIs), but instead hosts topologically protected zero-dimensional (0D) corner states. This generalised bulk-boundary correspondence principle has opened the door to the pursuit of higher-order TIs [7][8][9][10][11][12][13][14][15][16][17][18][19] ; topological corner states in both 2D [9][10][11][12][13][14]20,21 and three-dimensional (3D) [22][23][24] systems have been observed, arising from not only quantised multipole moments [9][10][11][12][13][14] but also from quantised dipole moments [20][21][22][23][24] .…”