# Progress in Topological Mechanics
## 5 Topological Defect States  
Topological lattice defects [27] have been extensively studied in electronic systems [138; 139; 140] and classical systems [141; 142; 143; 144; 145; 146; 147; 148] over the past decade. The classification of topological lattice defects depends on the holonomy along a closed path around the defect core [25; 149]. At present, the common achievements of topological defects are dislocations [138; 149; 150; 151; 152] and disclinations [25; 139; 153]. The dislocations and disclinations are defined as a gauge flux for translational symmetry and rotational symmetry, respectively [28]. Dislocations and disclinations are mainly investigated on honeycomb lattices and square lattices [149], and it has been demonstrated that they can serve as a bulk probe to detect the topological nature which exhibits topological states [139; 146; 147; 148; 154; 155; 156; 157; 158] and/or fractional charges [144; 145; 157; 158] in TCIs protected by crystal symmetries. More recently, topological defect states have been proposed in elastic materials [87; 159; 160; 161], and then we will discuss the related papers. In 2021, [PERSON] et al. experimentally realized valley-polarized edge states by introducing disclinations into elastic phononic plates [160]. The original honeycomb lattices elastic plate with two same cylindrical magnets attached to the top and bottom nodes of sublattices A and B is protected by \(C_{6v}\) symmetry, as shown in Figure 9a. Hence, the band could degenerate at the K point of the Brillouin zone and produce a Dirac cone state. Then changing the masses of sublattices A and B, the mirror symmetry of the unit cell will be broken, and a gap will appear to arise from the lifting of the Dirac cone. In addition, the modes at K point are inversed with each other. Furthermore, cutting a \(2\pi/6\) sector and gluing two boundary beams, a pentagonal elastic plate is shown in Figure 9b. Of note is the fact that the masses of the sublattices A and B above or below the interface is inversed. The tight-binding Hamiltonians [141] of this pentagonal elastic plate are \(H=-iv(\tau\sigma_{1}\partial^{\prime}_{1}+\sigma_{2}\partial^{\prime}_{2})-m \sigma_{3}\), where \(v\) is the Dirac velocity, \(\tau\) and \(\sigma_{i}\) signal valley Pauli matrices and sublattice Pauli matrices, respectively, \(m\) is the sublattice detuning and \(\partial^{\prime}_{i}\) denote the sublattice distortion. Consequently, a topological edge state will be accessed at the interface attributed to the opposite valley polarizations of the two types of unit cells as shown in Figure 9c. Moreover, we constructed a more complex structure with a straight waveguide by inducing a disclination dipole on honeycomb lattices, as shown in Figure 9d. As such, a topological edge state could be observed in this phononic plate, as shown in Figure 9e. In the same year, [PERSON] et al. experimentally demonstrated the topological bound states in elastic phononic plates by inducing disclinations [161]. The pentagonal plate is designed by cutting a \(\pi/3\) sector and reattaching the remaining boundary sides; the deformation process is shown in Figure 10a-c. The topological features can be presented by modulating the intensity of the intracell and intercell couplings. It was revealed that the pentagonal plate with shrunken unit cells has topological nontrivial states, as shown in Figure 10d. The distribution of the Wannier centers around the TCIs depend on the tight-binding models and crystal symmetries [19]. The schematics of the distribution of Wannier centers in topological nontrivial or trivial (expanded unit cells) plates are shown in Figure 10e,f. Moreover, \(1/2\) fractional charge is trapped at the pentagonal core in topological nontrivial elastic phononic plates. As expected, the localized states immunize against the finite sizes and the moderate structural defects. Figure 10: Bulk–disclination correspondence in elastic phononic plates. (**a**) The hexagonal lattices.