# Progress in Topological Mechanics
## 1 Introduction
Here \(W_{x,k}=F_{x,k+N_{x}}\Delta k_{x}\dots F_{x,k+\Delta k_{x}}F_{x,k}\), and \(W_{y,k}\) is a similar form, where \([F_{x,k}]^{nm}=\left\langle\mu_{k+\Delta k_{x}}^{m}\left|\mu_{k}^{n}\right\rangle\right\rangle\), \(\mu_{k}\) are the occupied Bloch functions. For the many-body multipole operators, the bulk multipoles in crystalline systems could be expressed in terms of ground-state expectation values of many-body operators, and this approach has a wider range of applications that could be used in interacting quantum many-body systems. The topological invariants for the \(C_{n}\)-symmetric HOTIs are the dipole polarizations (extended Zak phase), which are defined as an integration of the Berry phase vector potential \(\mathbf{A}\) over the first Brillouin zone(BZ): \(\mathbf{P}=\frac{1}{2\pi}\int dk_{x}dk_{y}Tr\left[A(k_{x},k_{y})\right]\)[18; 19]. An interesting exploration direction, topological pump, was firstly proposed in 1983 by [PERSON] [20]. This work showed the relationship between the 1D quantization of particle transport and 2D QHE. Recently, a topological pump was connected to 6D QHE [21; 22] via synthetic dimensions. More recently, a topological pump has been observed in higher-order topological systems [23]. The topological phases of 2D QHE are characterized by the first Chern number, which is defined as the integration of the Berry curvature over the generalized 2D Brillouin zone, and 4D QHE, characterized by the second Chern number, which is defined as the integration of the Berry curvature over the generalized 4D Brillouin zone [24]. As we all know, topological crystalline insulators can be identified by the robust spectral features and quantized fractional charges on boundaries or corners in experiments. Moreover, topological lattice defects could serve as a bulk probe to characterize the topological crystalline insulators [25]. Topological lattice defects, such as dislocations and disclinations, break the local crystal symmetry, which can be constructed by a Volterra process [25; 26; 27; 28]. In general, topological defects are characterized by their holonomy [25] or the Burgers vector [19; 29], and the topological nature is characterized by the topological index (the polarizations) [19]. The abovementioned distinct topological states could also be realized in classical systems, such as photonics, acoustics, mechanics, electric circuits and so on since the topological nature is unrelated to the quantum characteristics and depends on the wave characteristics [30]. In this context, there are extensive efforts to realize topological states in various elastic systems, leading to the emerging research field of topological mechanics. Elastic solids support both longitudinal and transverse waves, which are highly hybridized, bringing a great challenge to realize topological waveguides. However, elastic materials are still a good platform for implementing topological band structures because their large scale makes the control of fabrication and the experimental process more flexible. Moreover, elastic systems are not restricted by the Fermi levels therefore spectrum signature can be measured directly at any region. Furthermore, there are a lot of potential applications containing sensors, low loss devices, actuation, signal processing and so on [31]. The framework of this review is as follows. In Section 2, we introduce the main achievements of elastic topological edge states. In Section 3, we present the main studies of elastic HOTIs. In Section 4, we show the main researches of elastic topological pumps. In Section 5, we discuss the main realization of elastic topological defect states. Finally, some of the future perspectives in topological mechanical systems are outlined in Section 6.