# Progress in Topological Mechanics
## 2 Mechanical Topological Edge Sates
### Analogue Hall Insulators  
As we all know, realizing the QHE must break time-reversal symmetry. QHE analogs have been realized in photonics using gyromagnetic material in a magnetic field [33] and acoustics by introducing moving airflow [43]. Elastic waves, such as sound waves, hardly couple with the magnetic field, and therefore the interaction between elastic materials and magnetic fields cannot be exploited to break time-reversal symmetry. Hence, breaking time-reversal symmetry is quite challenging for mechanical systems. So far, the analogs of QHE have been verified for elastic waves by using Coriolis force [54, 55, 80] in the non-inertial reference frame or inducing gyroscopic inertial effects [56, 57]. In 2015, [PERSON] et al. designed a mechanical system (see Figure 1a) that exhibits topologically nontrivial one-way edge states, numerically demonstrating the topological nature [54]. Specifically, the structure consists of a mass-spring honeycomb lattice, whose time-reversal symmetry is broken by placing it in a rotating frame, as shown in Figure 1b. Here, the objects only have Coriolis force since the centrifugal force can be neglected for enough small angular frequencies. The nontrivial topological orders can be confirmed by calculating the Chern numbers for each band. The Chern numbers from the first band to the fourth band are \(\{-1,0,0,1\}\) when the constant angular velocity (z direction) \(\Omega=4\)Hz, while the Chern numbers are \(\{-1,1,-1,1\}\) when \(\Omega\) reaches \(\Omega_{c}=\sqrt{3C/8M}\sim 12.247\)Hz. In 2015, [PERSON] et al. constructed a new type of elastic phononic crystal by inducing the gyroscopic inertial effects [56]. The ordinary unit cell with equal masses (\(m_{1}=m_{2}\)) connected by linear springs is shown in Figure 1d. The corresponding band structure of the ordinary phononic crystal is shown in Figure 1g. The gap between the second and third bands is topologically trivial. To acquire the nontrivial band gaps, how to break the time-reversal symmetry needs to be solved. Consequently, the topological unit cell for the gyroscopic phononic crystal is designed with each mass attached to the tip of the rotational axis of a gyroscope, as shown in Figure 1e. The time-reversal symmetry is broken by modulating gyroscopic coupling. The corresponding band structure of the  
Figure 1: Analogue Hall insulators. (**a**) The mass-spring system is arranged in honeycomb lattices [54]. (**b**) The non-inertial system [54]. (**c**) The hexagonal gyroscopic lattice [57]. (**d**) The ordinary (nongyroscopic) unit cell [56]. (**e**) The gyroscopic unit cell [56]. (**f**) The experimental system [57]. (**g**) The dispersion of the ordinary (nongyroscopic) phononic crystal [56]. (**h**) The dispersion of the gyroscopic phononic crystal with the Chern numbers labeled on the bulk bands [56]. (**i**) One period of the unidirectional waveguide mode at \(t=3.2s\), the ellipses denote the shape of each orbit, and the colors denote the phase at a fixed time [57]. gyroscopic phononic crystal with the spinner constants of the gyroscopes \(\alpha_{1}=\alpha_{2}=0.3m_{1}\) is shown in Figure 1h. The two band gaps are nontrivial according to the non-zero Chern numbers labeled on the bands. There are gapless one-way edge states in the two topologically nontrivial gap frequency ranges. In these frequency ranges, the edge modes can unidirectionally propagate without any reflection under topological protection even in the presence of the sharp corner and the line defect. In the same year, [PERSON] et al. experimentally and theoretically demonstrated that a coupled system of gyroscopes could generate topological states [57]. The honeycomb lattices consisting of gyroscope-spring networks are shown in Figure 1c, and the experimental system is shown in Figure 1f. The experimental system consists of 54 coupling gyroscopes on a honeycomb lattice; the gyroscopes are composed of small dc motors spinning cylindrical masses, which are coupled by placing a small neodymium magnet in each mass with its moment aligned vertically.