# Progress in Topological Mechanics
## 3 Mechanical Higher-Order Topological Insulators
### Quantized Multipole Insulators  
[PERSON] et al. proposed quantized multipole insulators (QMIs) with generalized berry phases in 2017 and mainly discussed the quantized quadrupole moment in a 2D crystal and the quantized octupole moment in a 3D crystal (see Figure 4a,b) [11; 12]. Specifically, a 2D quadrupole TI has gapped first-order edge states and second-order corner states in the gap; a 3D octupole TI has gapped first-order surface states, second-order hinge states and third-order corner states in the gap. Importantly, there must be reflection and inversion symmetries to realize QMIs, and there must be at least two or four occupied bands to make the dipole moments or quadrupole moments vanish. In the following, we will clearly illustrate the advances of QMIs in elastic wave systems.  
In 2018, [PERSON] et al. proposed a mechanical configuration to realize a quadrupole topological insulator [105]. A tight-binding model with positive and negative couplings is shown in Figure 4c. The unit cell and experimental photo of the setup are, respectively, shown in Figure 4d,e. The quantized quadrupole TIs can be realized by manipulating the positive or negative couplings and the amplitudes of the intracell or intercell couplings. Specifically, the amplitudes of the intracell hoppings \(\gamma\) or intercell hoppings \(\lambda\) are controlled by the distance between the given beam and the nodal line, and the positive or negative hoppings depend on which sides of the nodal lines are connected by the beams. The amplitudes of the intracell or intercell hoppings controlling the emergence of the bandgap between two pairs of degenerate bands. The positive or negative couplings emulate a magnetic \(\pi\) flux per plaquette. The gapped edge states and in-gap corner states appear at the single plate for \(\gamma<\lambda\). The corresponding spectrum is shown in Figure 4f. Here, two pivotal features of the quantized quadrupole phase should be focused on. The first feature is that the response is located at around 72.92 kHz and 74.89 kHz, which represents the gapped edge modes. The second feature is that there are strong corner resonances in the middle of the gap, which is indicative of the in-gap corner states. To further confirm the TIs  
Figure 4: Quantized multipole insulators. (**a**) Bulk quadrupole moment \(q_{xy}\), edge dipole moments \(p_{i}\) and corner charges Q [11]. (**b**) Bulk octupole moment \(O_{xyz}\), surface quadrupole moments \(q_{ij}\), hinge dipole moments \(p_{i}\), and corner charges Q [11]. (**c**) A 2D tight-binding model of quadrupole insulators. Red (black) lines represent negative (positive) hopping terms. Thin (thick) lines indicate intracell (intercell) hopping terms [105]. (**d**) Metamaterial structure. Dashed white lines represent the two nodal lines of the out-of-plane modes which are coupled based on the bent beams [105]. (**e**) Photo of the experimental sample [105]. (**f**) The spectrum for the quadrupole insulator [105]. (**g**) The schematic of the Wannier bands [105].  
is quadrupole, the Wannier bands are calculated as shown in Figure 4g. The bulk-induced edge polarization could be calculated using the eigenvectors of the Wilson loops since the Wannier bands \(v_{\pm}(k_{y})\) and \(v_{\pm}(k_{x})\) are gapped. The value of the polarizations are \((p_{x}^{v-},p_{y}^{v-})=(0.5,0.56)\), which suggest the existence of in-gap states.