# Progress in Topological Mechanics
## 3 Mechanical Higher-Order Topological Insulators
### C\({}_{n}\)-Symmetric HOTIs
Consequently, the trivial corner states locate at the acute-angled corners of \(\pi/3\) in the case of \(N=0\), while the nontrivial corner states locate at the obtuse-angled corners of \(2\pi/3\) in the case of \(N\
eq 0\). In 2020, [PERSON] et al. proposed a pseudospin-valley-coupled HOTI and observed topological higher-states [109]. The topological higher-states are induced by combining the coupling of lattice deformation and reflection symmetry breaking. The lattice deformation can be induced by expanding or shrinking the composite unit cell, which will lead to the pseudospin Hall effect. The reflection symmetry can be broken by varying the adjacent additional masses in a composite unit cell, which will lead to the valley Hall effect. The two perturbative composite unit cells are shown in Figure 6a. Here, the additional masses on the A and B nodes are obtained by the formulas \(m_{A}=m_{0}+\Delta m\) and \(m_{B}=m_{0}-\Delta m\). The length of the intracell and intercell beam is \(l_{intra}=(1-\Delta\gamma)l\) and \(l_{inter}=(1+2\Delta\gamma)l\) respectively. The elastic phononic crystal sample with the two kinds of structural perturbations is shown in Figure 6b. Specifically, a valley-polarized edge state will emerge when the value is \(\Delta m\
eq 0\), and a pseudospin-dependent edge state will emerge when the value \(\Delta\gamma\
eq 0\). The edge states and corner states will appear at domain walls and the central position, respectively, arising from the topological phases transition. The diagram of the corner state is shown in Figure 6c. Figure 5: Elastic hexagon-shaped Tis [106]. **(a)** A composite unit cell of the hexagonal lattice. (**b–d**) Hexagon-shaped structures consisting of expanded unit cells (**b**), shrunken unit cells (**c**) and shrunken unit cells with defects (**d**), respectively. The defects are realized through adding a magnet on six nodes, respectively, at the upper side of 12 composite unit cells (marked by the red dashed box). **(e)** The schematic of the honeycomb lattice model. The red (blue) beams whose lengths are denoted by \(l_{intra}<l_{inter}\) representing the intracell (intercell) hoppings. **(f–h)** The schematics of the simulated state number responding to (**b–d**), respectively. **(i–l)** The simulated modes of bulk state at 1750.2 Hz (**i**), topological corner state at 1555.8 Hz (**j**), trivial corner state at 1529.1 Hz (**k**) and gapped edge state at 1600.1 Hz (**l**). In the same year, [PERSON] et al. designed a HOTI with 2D elastic phononic plates protected by \(C_{4}\) symmetry [110]. The unit cell consists of stainless steel prisms (blue), and a squared aluminum plate (red), as shown in Figure 6d. This structure could generate much stronger scattering because of the different acoustic impedance between stainless steel and aluminum. The key to realizing higher-order states is to produce gapped edge states, and the emergence of the bandgap is controlled by the rotation angle \(\theta\) of atoms arising from the glide and time-reversal symmetries. In this context, they designed a squared structure formed by two parts of phononic crystals with \(\theta=-25^{\circ}\) and \(\theta=50^{\circ}\), respectively, as shown in Figure 6e. As expected, the 0D corner states and 1D edge states appeared at the interface because the glide symmetry is broken at the interface. The diagram of the corner state is shown in Figure 6f. In 2021, there are also a few achievements on implementing higher-order states in mechanics [107, 111, 112, 117]. Recently, [PERSON] et al. experimentally demonstrated that higher-order topological states for elastic waves at MHz could emerge in an on-chip micromechanical metastructure [111]. The sample chip is constructed with two partial structures referring to a HOTI surrounded by a normal insulator (NI), as shown in Figure 6g. The unit cell with four elliptic pillars in a square lattice is shown in Figure 6h. Manipulating the  
Figure 6: Other experimental realizations of \(C_{n}\)-symmetric HOTIs.