Quasitopological rotational waves in mechanical granular graphene

Granular crystals are periodic structures of elastic beads arranged in crystal lattices. One important feature of granular crystals is that the interactions between beads can take place via noncentral contact forces, leading to the propagation of rotational and coupled rotational-translational waves. Here, we theoretically demonstrate the topological properties of these mechanical rotational waves in a granular graphene, a two-dimensional monolayer honeycomb granular crystal with Dirac dispersion at the center of the Brillouin zone. Around the Dirac point, effective spin, helicity, and effective spin-orbit coupling are illustrated in the mechanical granular system. Finally, quasitopological transport, where the rotational edge waves are nearly topologically protected, is observed on the interface.

Figure 1

(a) Granular graphene: The red dashed box marks the unit cell containing two particles, $\mathrm{A}$ and $\mathrm{B}$. Movements of individual particles in (b) can generate shear forces (c), torsional forces (d), and bending forces (e). The band structures for ${\eta }_b={\eta }_t=0.5$ and $\mathrm{P}=1.55$ are shown when perturbation ${\eta }^{\prime }=0$ in (f) and ${\eta }^{\prime }=+0.1$ in (g). The red branches in (f) are dominated by $\mathit{d}$ modes, and the blue ones by $\mathit{p}$ modes. The black (red) bonds in the insets of (f) [(g)] indicate that the bending and torsional rigidities between particles are unmodified (modified).

Figure 2

(a) Possible mechanical spin states in the GG. The left (right) circular polarization of rotation is the spin $\uparrow (\downarrow )$ state. The red (blue) arrow represents the rotation vector of particle $A\left(B\right)$. Projection of helical modes with perturbation (b) ${\eta }^{\prime }=0$ and (c) ${\eta }^{\prime }=0.1$ on $\mathit{k}$ space. The arrow denotes the axis of spin projection. The “currents” in the upper branches are outgoing (right handed) while in the lower branches they are incoming (left handed). The color in the background represents the eigenfrequency. The eigenmodes of the $\mathrm{\Gamma }$ point are also shown in (b) when ${\eta }^{\prime }=0$ and in (c) when ${\eta }^{\prime }=0.1$.

Figure 3

(a) Dispersion of the edge wave along the $y$ direction. The gray lines correspond to the bulk modes and color lines to the edge modes. (b) Zooms of different edge branches. The green lines represent topologically trivial edge modes. (c) Schematics to reveal the spin nature of modes in the cyan and magenta branches in (b). The arrow in red (blue) in the unit cell represents the rotation direction of particle $A\left(B\right)$. Time evolution simulations for the $S_u$ edge wave in (d) and the $S_d$ edge wave in (e). The green stars highlight the positions of source. (f) The same configuration as (c) except that there is a hole close to the interface. The color scale is the magnitude of the total rotation normalized to the maximum. The movement of a bead marked by the circle is monitored to compare with the case in (e). (g) Rotation signals at the monitored bead.

Figure 4

Robustness of the quasitopological rotational wave against defects: a zigzag path containing two corners. (a) $S_d$ edge wave is launched from the position marked by the green star. For comparison, a normal edge wave is excited from the same positions in (b).