# Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals

## TL;DR

This paper reports the design and experimental realization of a 3D higher-order topological insulator in a sonic crystal, revealing a hierarchy of boundary states across different dimensions, advancing topological wave control in three dimensions.

## Contribution

It introduces a 3D sonic crystal with a large topological band gap exhibiting multiple higher-order boundary states, surpassing previous 2D implementations.

## Key findings

- Observation of coexisting boundary states of different codimensions
- Demonstration of a 3D topological insulator with a large band gap
- Breaks chiral symmetry, enabling separation of bulk, surface, hinge, and corner states

## Abstract

Topological phases of matter have been extensively studied for their intriguing bulk and edge properties. Recently, higher-order topological insulators with boundary states that are two or more dimensions lower than the bulk states, have been proposed and investigated as novel states of matter. Previous implementations of higher-order topological insulators were based on two-dimensional (2D) systems in which 1D gapped edge states and 0D localized corner states were observed. Here we theoretically design and experimentally realize a 3D higher-order topological insulator in a sonic crystal with a large topological band gap. We observe the coexistence of third-, second- and first-order topological boundary states with codimension three, two and one, respectively, indicating a dimensional hierarchy of higher-order topological phenomena in 3D crystals. Our acoustic metamaterial goes beyond the descriptions of tight-binding model and possesses a band structure which automatically breaks the chiral symmetry, leading to the separation of bulk, surface, hinge and corner states. Our study opens a new route toward higher-order topological phenomena in three-dimensions and paves the way for topological wave trapping and manipulation in a hierarchy of dimensions in a single system.