# Progress in Topological Mechanics
## 1 Introduction  
Over the last decade, topological mechanics has rapidly attracted extensive research. This field mainly investigates topological wave propagation, which is immune to defects and backscattering in mechanical systems, originally discovered in quantum systems. In this review, we aim to cover the main achievements of topological mechanics, including the fundamental concepts of topological phases of matter, theories and experimental realizations. Over the past 40 years, topological phases of matter appeared as a leading field of research. This field of research originated from the discovery of the quantum Hall effect (QHE). In 1980, [PERSON] et al. found that the Hall conductance equal to integer multiples of the fundamental constant with a two-dimensional electron gas sample in a low temperature and strong magnetic environment [1]. In 1982, [PERSON] et al. explored the mechanism of the integer quantum Hall effect and elucidated the relationship between the integer in the Hall conductance and a topological invariant in the QHE system [2]. Importantly, they proposed the TKNN (Thouless-Kohmoto-Nightingale-den Nijs) theory to define the integer, namely the Chern number. The Chern number characterizes the topological phases of matter in momentum space over the Brillouin zone. Specifically, combining the bulk-boundary correspondence [3], there exist edge states in the system with a non-zero Chern number, while the ordinary insulators and vacuum with a zero Chern number are trivial. However, realizing QHE requires breaking time-reversal symmetry, which is not beneficial to practical application. About 20 years later, in 2005, the quantum spin Hall effect (QSHE) or topological insulators(TIs) were proposed in graphene [4], conventional semiconductors with a strain gradient [5] and HgTe quantum wells [6]. Realizing QSHE exploits the spin-orbit interactions and time-reversal symmetry. In these systems, a Chern number of zero arose from the existence of conjugate electronic spins, hence the topological nature is characterized by a \(Z_{2}\) topological invariant or the spin Chern number [7, 8]. Phenomenologically, TIs have gapless edge or surface states when the spin Chern numbers are non-zero. Shortly afterwards, the quantum valley Hall effect (QVHE) was discovered in 2007 [9; 10]. Valleys refer to the extrema of the band structure in momentum space, and the sign of the valley Chern number at \(K\) and \(K^{\prime}\) is opposite. Of note is the fact that the total Chern number remains zero. In addition, the edge or surface states protected by the time-reversal symmetry could be observed at the interface between the two lattices with opposite valley Chern numbers. Over the past five years, a new type of topological phase called higher-order topological insulators (HOTIs) has attracted rapidly growing attention, a phase which does not apply the bulk-boundary correspondence [11; 12; 13]. Phenomenologically, a \(d\)-dimensions \(n\)-th order HOTI hosts gapped (\(d-1\))-, (\(d-2\))-, \(\dots\), (\(d-n+1\))-dimensional gapped boundaries and (\(d-n\))-dimensional gapless modes [14]. For example, a two-dimensional (2D) HOTI has gapped one-dimensional (1D) boundaries and zero-dimensional (0D) corner states. A three-dimensional (3D) HOTI has 2D gapped surface states and 1D gapless hinge states, or 2D gapped surface states, 1D gapped hinge states and 0D corner states. The topological phases of HOTIs are characterized by new topological invariants [15]. Specifically, the topological invariants for the quantized multipole insulators are quantized multipole moments, which can be calculated by the nested Wilson loop approach [11; 12] or the many-body multipole operators [16; 17]. Specifically, for the nested Wilson loop approach, the gapped Wannier bands carrying their own topological invariants can be evaluated by this approach. One could start by constructing Wilson loop operators \(W_{x,k}\) and \(W_{y,k}\) in the \(x\) and \(y\) directions, and \(k=(k_{x},k_{y})\) is the starting point of the loop.