Demonstration of a quantized acoustic octupole topological insulator

Recently introduced quantized multipole topological insulators (QMTIs) reveal new types of gapped boundary states, which themselves represent lower-dimensional topological phases and host symmetry protected zero-dimensional corner states. Inspired by these predictions, tremendous efforts have been devoted to the experimental observation of quantized quadrupole topological phase. However, due to stringent requirements of anti-commuting reflection symmetries, it is challenging to achieve higher-order quantized multipole moments, such as octupole moments, in a three-dimensional structure. Here, we overcome this challenge, and experimentally realize the acoustic analogue of a quantized octupole topological insulator using negatively coupled resonators. We confirm by first-principle studies that our design possesses a quantized octupole topological phase, and experimentally demonstrate spectroscopic evidence of a hierarchy of boundary modes, observing 3rd order topological corner states. Furthermore, we reveal topological phase transitions from higher- to lower-order multipole moments. Our work offers a pathway to explore higher-order topological states in 3D classical platforms.

where is discretized in steps, and = 2 / 0 . The projection operator into the occupied bands can be represented as The position operator projected into the occupied bands becomes ̂ô̂o = ∑ , ′ ( + , ) , ′ , where , ′ ( + , ) = ∑ + , * , ′ is not unitary. However, this problem can be solved by singular value decomposition (SVD) method Redefine the unitary operator = † such that ̂ô̂o | 〉 = ( + , )| + 〉 ̂ô̂o | + 〉 = ( + 2 , + )| +2 〉 … ̂ô̂o | +2 − 〉 = ( + 2 , + 2 − )| +2 〉, Since | +2 〉 = | 〉, and we define 1 st order Wilson loop over 1D Brillouin zone as 2 + ← = (2 + , 2 + − ) … ( + 2 , + ) ( + , ), Thus, Supplementary Eq. (6) becomes 2 + ← | 〉 = (̂ô̂o) | 〉, and the eigenvalue problem for Wilson loop operator is where is the eigenvalue of projected position operator, = 2 since Wilson loop is unitary. is the Wannier center of energy band indexed by , and dipole moment is the summation of over the occupied bands. Similarly, Wilson loop eigenvalue problem in three dimensional case can be generalized as 2 + ← , | , ⟩ = 2 , | , ⟩, where = ( , , ), , , = 0, , … , ( − 1) . = 1,2, … , . , is the Wannier band of the occupied energy band indexed by , which describes the average position of the electron from the center of the unit cell in the direction of at momentum vector . The polarization along is obtained through the Wannier bands, in which , ± , is the 2 nd order Wannier center for Wannier sector + . The polarization over the Wannier sector ± is given by the equation, The quadrupole moment of a specific surface , is defined as and it is quantized as either 1/2 or 0 because of the constraint by reflection symmetries. The physical consequence of the nontrivial quadrupole moment of the 2D crystal is the presence of corner charges robustly localized at the corners of the surface, and of quadrupole-induced edge polarization which has its own topology.
The reflection symmetries in the octupole TI have anti-commuting relation among each other, which not only determine the degeneracy of energy bands, but also play an important role in the properties of 1 st order Wilson loop over occupied bands, of 2 nd order Wilson loop over 1 st order Wannier-sector as well as of 3 nd Wilson loop over 2 nd order Wannier-sector. In the following text, we'll discuss these contexts case by case.

Chiral symmetry plays a role in determining energy bands of the octupole TI as well, and the
Hamiltonian under its operation obeys It is easily concluded from Supplementary Eq. (27) that the (bulk, surface, and hinge) energy spectra are symmetric at zero energy = 0 . Furthermore, if a zero energy state exists ̂( )| 〉 = 0, which is independent of , then Γ 0 ′ −1 | 〉 is also zero energy state. Since each sublattice has four degrees of freedom, eight-fold degeneracy zero energy states are found, and they are revealed to be 3 rd order corner states demonstrated in the main text and protected by the chiral symmetry.

Supplementary note 3 -Symmetries constraints over 1 st order Wannier bands
In general, the constraint of the symmetries over the Wilson loop satisfies the following relation is the path in the direction of in Brillouin zone, and ⊥ = ( , ) is the momentum vector perpendicular to .
Thus, the symmetry operator ̂ enforces Wannier bands ,( , ) to be quantized as 0, 1/2 or to be a pair ( , − ). As a consequence, the polarization over the occupied energy bands in direction is quantized as = 0 or = 1/2. The symmetry operator ̂ quantizes Wannier bands  Therefore, two-fold degeneracy Wilson eigenvalues occur everywhere in the Brillouin zone.
Next, we assume the eigenvalue of ̂ to be positive for one of the eigenstates at TRIM points s, which indicates that ̂| ( )〉 = −| ( )〉 , thus, the pair of inversion eigenvalues (+, −) is guaranteed for occupied states, implying that their corresponding 1 st order Wannier bands are gapped in the Brillouin zone. To this end, we have proven the gapped 1 st order Wannier bands have two-fold degeneracy for their respective bands, as seen in Fig.2a.

Supplementary note 4 -Symmetries constraints over 2 nd order Wannier bands
Following the similar steps as before, we obtain Reflection operator ̂ imposes the 2 nd order Wannier bands to be quantized as 0, 1/2 or a pair ( , − ). As a consequence, the polarization over the Wannier section + in the direction is quantized as + = 0 or + = 1/2 , which is true for arbitrary order , , . Thus, the total quadrupole moment vanishes because of conservation of inversion symmetry.
To facilitate the study of 2 nd order Wannier bands, eigenvalues of projected inversion operator ̂⊥ is discussed. Assume the eigenvalue of ̂⊥ to be positive for one of the eigenstates at projected TRIM points ⊥ , ̂⊥ | , ⊥ 〉 = +| , ⊥ 〉, therefore, it satisfies −| , ⊥ 〉 =̂⊥ 2 | , ⊥ 〉 =̂⊥| , ⊥ 〉, Thus ̂⊥ | , ⊥ 〉 = −| , ⊥ 〉 has to be true. Therefore, a pair of projected inversion operator eigenvalues at ⊥ s (+, −) is guaranteed by the condition ̂⊥ 2 = −I 4×4 . It can be inferred from the eigenvalues of projected inversion symmetry that the 2 nd order Wannier bands are gapped in the projected Brillouin zone, as shown in Fig. 2b, which is the precondition of defining octupole moment of the 2 nd order Wannier bands.