The recent development of topological band theory has revealed the existence of higher-order topological insulators (HOTIs) [1–5]. An important hallmark of such HOTIs is the existence of ( D − n ) -dimensional topological boundary modes, where D is the dimensionality of the system, and n ∈ ( 1 ,   D ] is an integer. Thanks to the development of classical wave crystals, HOTIs have not only been observed in solid-state electronics but also in photonic crystals [6–12], sonic crystals [13–19], and elastic-wave crystals [20, 21].

For HOTIs with two real-space dimensions, the higher-order topological boundary modes are zero-dimensional modes localized at the corners of the lattice. These topological corner modes (TCMs) can be protected by a variety of topological invariants, such as quantized dipole moments [9, 10, 22], quantized quadrupole moments [1, 23, 24], combinations of first Chern numbers [17], etc. However, although their topological protection can be revealed by theoretical computation of the topological invariants, it is difficult to distinguish them from an observational point of view.

A previous theoretical study has analyzed the finite-size effect on neighboring TCMs in a 2D HOTI [25]. By comparing two different types of topologically nontrivial square-lattice HOTIs—a topological dipole insulator (TDI) wherein the dipole moments are quantized and a topological quadrupole insulator (TQI) with a quantized quadrupole moment, it was shown that the TCMs’ spectral responses split, and the line shapes are associated with the topological characteristics of the HOTI. As such, the spectral responses of the coupled TCMs are an observable effect, by which the underlying topological nature can be phenomenologically revealed.

For the sake of completeness, we first briefly summarize the important theoretical background. A complete theoretical analysis can be found in Ref. [25]. Here, the TDIs and TQIs are both based on the extensions of the 1D Su-Schrieffer-Heeger (SSH) model, and we show the unit-cell structures for the 2D TDI and TQI in Figure 1A,B, respectively. The strengths of the staggered nearest couplings along x and y directions are denoted as intracell λ (thin tubes) and intercell γ (thick tubes) hopping. For the TDI, all hopping coefficients are on the same sign so that the net magnetic flux in a plaquette is zero. For the TQI, the hopping coefficients can take opposite signs, as indicated by the red tubes in Figure 1B. The resultant net magnetic flux is π . When | λ / γ | < 1 , both systems are in the topologically nontrivial phase and have topological edge modes and TCMs, as shown in Figures 1C,D.

For a N × N TDI lattice, the corresponding Hamiltonian can be written as H TDI = I 2 N × 2 N ⊗ H x SSH + H y SSH ⊗ I 2 N × 2 N , (1) with H x SSH = H y SSH = H SSH = λ ∑ m = 1 N ( | m , B 〉 〈 m , A | + | m , A 〉 〈 m , B | ) + γ ∑ m = 1 N − 1 ( | m + 1 , A 〉 〈 m , B | + | m , B 〉 〈 m + 1 , A | ) , (2) where I 2 N × 2 N is a 2 N × 2 N identity matrix, ⊗ denotes Kronecker product, and | m , A 〉 and | m , B 〉 denote states of the left and right atoms, respectively, in the mth unit cell for a 1D SSH chain model. For a N × N TQI, the Hamiltonian is H TQI = I 2 N × 2 N ⊗ σ 3 ⊗ H x SSH + H y SSH ⊗ I 2 N × 2 N ⊗ σ 0 ,   (3) where σ 0 is a 2 × 2 identity matrix, and σ 3 is the z-component of the spin-1/2 Pauli matrices.

We plot the eigenvalues of Eq. 1, 3 in Figures 1C,D, respectively. The parameters are set as λ = 0.1 , γ = 0.5 , and N = 3 . The bulk, edge, and corner modes are marked by black, blue, and red points, respectively. In both cases, four TCMs are found. As seen in the insets, the four TCMs are not degenerate. This is because, in a finite-sized lattice, the edges can provide coupling to neighboring TCMs [25]. For the TDI, the TCMs split into three clusters, with the two modes in the middle being degenerate. The two degenerate TCMs are still pinned at zero energy because of chiral symmetry. For the TQI, it can be proved that all eigenstates, including bulk modes and TCMs, are at least doubly degenerate [25, 26]. Therefore, the TCMs are divided into two doubly degenerate clusters, which are symmetric about zero energy. The finite-sized coupling effect can be captured by a four-state effective Hamiltonian with the four TCMs as the basis, which reads H corner TDI = ( 0 t N t N 0 t N 0 0 t N t N 0 0 t N 0 t N t N 0 ) , (4) for the TDI, and H corner TQI = ( 0 t N t N 0 t N 0 0 t N t N 0 0 − t N 0 t N − t N 0 ) , (5) for the TQI. Here, t N = a 1 b N λ ( − λ / γ ) N − 1 , where a m ( b m ) denotes the strength of the eigenstate | m , A 〉 ( | m , B 〉 ) of H SSH . Since | λ / γ | < 1 , t N is vanishing for large N . These models are schematically shown in Figures 1E,F. Note that similar to their respective unit cells, there is a magnetic flux of 0 and π in the TDI and TQI-corner models, respectively. From Eqs. 4, 5, we can use a Green’s function to describe the spectral responses of the coupled TCMs G ^ ( E ) = ∑ j = 1 4 | ϕ j 〉 〈 ϕ j | E − E j + i η t N ,   (6) where E j is the eigenvalue and | ϕ j 〉 is the eigenvector, and η accounts for any dissipative effect in the system. When excited at corner | m 〉 , the response at the corner | n 〉 will be G m , n ( E ) = 〈 m | G ^ ( E ) | n 〉 . When excited at corner A, the responses at each corner are shown in Figure 1G for the TDI, and Figure 1H for TQI. It is seen that the spectral responses are different for the TDI and TQI. Particularly, the TDI responses can split into three peaks when measured at corners A and C, and the TQI response vanishes when measured at C. Such distinctions are an important manifestation of the quantized magnetic fluxes in the systems, which can be used as experimental evidence to distinguish the two classes of HOTIs.

We next present the designs of phononic crystals to realize both the TDI and TQI. The unit cells are shown in Figures 2A,B, respectively. The gray blocks denote the acoustic cavities, whose first-order resonance fulfills the role of the on-site orbital. The cavities have a height of H = 80 mm and a width of L = 35 mm , and they are coupled by tubes that facilitate the hopping terms. For the TDI, the widths of the intracell and intercell coupling tubes are w = 17 mm and W = 30 mm , respectively. They are connected at a vertical position with the height being the same h = 21 mm . The lattice constant is a = 150 mm . The design of the TQI is different because we need to realize hopping terms with a negative sign. To achieve this, we connect the top of one cavity to the bottom of the designated neighbor using a bent tube (red in Figure 2B). The blue tubes which facilitate positive hopping are also bent in the same manner so that all tubes have the same length. The positions of the cavities are staggeredly elevated so that the lengths of the intracell or intercell coupling tubes are the same. We use COMSOL Multiphysics to compute the band structures of the two types of unit cells. The medium inside the cavity and coupling tubes is air with a mass density of 1.23   kg / m 2 and a sound speed of 343 × ( 1 + 0.005 i )   m / s , where the imaginary part denotes losses. The results are respectively shown in Figures 2C,D, where four bands are seen for both cases.

Based on these two designs, we have fabricated the phononic crystals. The cavities are machined from aluminum alloy and the coupling tubes are 3D printed using photosensitive resin. The photographs of the TDI and TQI configurations are shown in Figure 3A, 4A, respectively. Both lattices are 3 × 3 in size, containing a total of 36 cavities. At the top of the cavities, we drilled a small hole (covered with a small white plug), where a sound signal is sent or a probe can detect the acoustic signal inside the cavity. We excite corner A with a loudspeaker, as shown in Figure 3A, 4A. Then, we used a microphone to obtain the spectral response field in every cavity. For the TDI lattice, the responses measured at corners A, B, C, D are shown in Figure 3B. In the predicted frequency regime, i. e., 2,050–2,150 Hz, a three-peak response line shape is seen for both the spectra measured at corners A and C. And two-peak line shape is seen for corners B and D. These results agree well with the theoretical prediction by the tight-binding model (Figures 1C,G). To confirm that these responses are due to the coupled TCMs, we have measured the pressure responses in all cavities at the frequencies of the response peaks. The results are shown in Figures 3D–F. Clearly, the spatial distributions are strongly localized at the corners, which is a signature characteristic of TCMs. We have further verified the responses in numerical simulations. The results in Figure 3C,G–I also show excellent agreement with the experiment.

Similar experiments were performed for the TQI lattice. In Figure 4B, two response peaks are identified in the bulk gap (2,000–2,150 Hz) for the spectra measured at corners A, B, D. Also, the response at corner C is significantly weaker. These observations again align with the prediction (Figures 1D,H) and simulations (Figure 4C). We further confirm in the measured (Figures 4D,E) and simulated (Figures 4F,G) spatial maps that the response peaks are indeed due to the TCMs.

In summary, we have experimentally observed the coupling effects of TCMs in two different types of acoustic HOTIs. The measured line shapes of the corner responses agree excellently with a previous theoretical study, which confirms that the topological properties of the HOTIs can indeed influence the coupling effects of TCMs. Therefore, the corner responses can serve as an experimental hallmark to separate HOTIs of different classes. It is interesting to further the study by investigating the coupling effects in other types of HOTIs such as a Kagome lattice [4], honeycomb lattice [20], etc. On the other hand, the coupled higher-order topological modes can be a useful starting point for higher-order non-Hermitian physics [27, 28]. They may also find applications such as topological wave and light confinement [13, 29] and topological lasing [30, 31].