Observation of an acoustic octupole topological insulator

Haoran Xue,Yong Ge,Qiang Wang,Hong-Xiang Sun,Yidong Chong,Baile Zhang,Yi-Jun Guan,Shou-Qi Yuan,Ding Jia

doi:10.1038/s41467-020-16350-1

Haoran Xue, Yong Ge + Show 7 more

Open Access

https://doi.org/10.1038/s41467-020-16350-1

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Abstract

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. Recent theories have generalised the concept of quantised dipoles to multipoles, resulting in the discovery of multipole topological insulators which exhibit a hierarchy of multipole topology: a quantised octupole moment in a three-dimensional bulk induces quantised quadrupole moments on its two-dimensional surfaces, which in turn induce quantised dipole moments on one-dimensional hinges. Here, we report on the realisation of an octupole topological insulator in a three-dimensional acoustic metamaterial. We observe zero-dimensional topological corner states, one-dimensional gapped hinge states, two-dimensional gapped surface states, and three-dimensional gapped bulk states, representing the hierarchy of octupole, quadrupole and dipole moments. Conditions for forming a nontrivial octupole moment are demonstrated by comparisons with two different lattice configurations having trivial octupole moments. Our work establishes the multipole topology and its full hierarchy in three-dimensional geometries.

Highlights

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators
In the one-dimensional (1D) case, the quantisation of the electric dipole moment is associated with the Berry phase for a parallel transport of the ground state in momentum space[3,4], which leads to the concept of a 1D topological insulator (TI)[5]
Take the quantised quadrupole insulator as an example. 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, 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 TIs7–19; topological corner states in both 2D9–14,20,21 and three-dimensional (3D)[22,23,24] systems have been observed, arising from quantised multipole moments[9,10,11,12,13,14] and from quantised dipole moments[20,21,22,23,24]

Summary

Introduction

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. 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 TIs7–19; topological corner states in both 2D9–14,20,21 and three-dimensional (3D)[22,23,24] systems have been observed, arising from quantised multipole moments[9,10,11,12,13,14] and from quantised dipole moments[20,21,22,23,24]. This work demonstrates the hierarchical nature of multipole topology—from an octupole moment to quadrupole and dipole moments—which is a distinctive and unique feature of a 3D quantised octupole TI

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