Topological band structure of surface acoustic waves on a periodically corrugated surface

Surface acoustic waves (SAWs) are elastic waves localized on a surface of an elastic body. We theoretically study topological edge modes of SAWs for a corrugated surface. We introduce a corrugation forming a triangular lattice on the surface of an elastic body. We treat the corrugation as a perturbation, and construct eigenmodes on a corrugated surface by superposing those for the flat surface at wave vectors which are mutually different by reciprocal-lattice vectors. We thereby show emergence of Dirac cones at the $K$ and $K^{\prime }$ points analytically. Moreover, by breaking the time-reversal symmetry, we show that the Dirac cones open a gap, and that the Chern number for the lowest band has a nonzero value. This indicates the existence of topological chiral edge modes of SAWs in the gap.

Figure 1

Dispersion relations of SAWs for the flat surface. The black line represents the bulk transverse wave mode $\omega =c_{t}k$ and the dashed black line represents the bulk longitudinal wave mode $\omega =c_{l}k$. The red line represents the dispersion relation of the SAWs $\omega =c_{R}k$. Here, we set $c_t=1$ and $c_l=1.2$.

Figure 2

Corrugated surface with a triangular lattice, and calculation of its band structure. (a) Profile of the surface corrugation of the elastic body described by Eq. (5). The figure shows $z=\zeta (x,y)$ with the values of the parameters $d=0.1$ and $a=4\pi /\sqrt{3}$. (b) Brillouin zone of the model. We call the six corners ${\mathit{K}}_j$ and ${\mathit{K}}_j^{\prime }$ ($j=1,2,3$). As reciprocal-lattice vectors, the three points ${\mathit{K}}_j$ are equivalent, meaning that their differences are reciprocal-lattice vectors, and are called $K$ point in the standard notation for special wave vectors in the Brillouin zone. Likewise, the three points ${\mathit{K}}_j^{\prime }$ are equivalent and are called $K^{\prime }$ point, but $K$ and $K^{\prime }$ are not equivalent. The $\mathrm{\Gamma }$ point denotes the origin $\mathit{k}=(0,0)$. (c) Schematic picture of the dispersion relation of SAWs when the surface is flat. It is shown in the first Brillouin zone for the triangular lattice. The dispersion relation of SAWs for the flat surface has a conical structure the apex of which is at the $\mathrm{\Gamma }$ point denoting the center of the first Brillouin zone. In the calculation of the dispersion of SAWs around ${\mathit{K}}_1$ for the corrugated surface, we consider only hybridization of the waves near the three $K$ points ${\mathit{K}}_1,{\mathit{K}}_2$, and ${\mathit{K}}_3$.

Figure 3

Schematic pictures of the band structures near the $K$ point, (a) when the time-reversal symmetry is preserved and (b) when the time-reversal symmetry is broken. In (b) the Dirac cone splits and has a gap $2|\delta {\omega }^{\left(D\right)}|$ at the $K$ point.

Figure 4

Schematic picture of the setup. The corrugated surface is within the $xy$ plane, and the system is rotated around the $z$ axis with the angular velocity $\mathrm{\Omega }$. As a result of breaking of time-reversal symmetry, the Dirac cones at $K$ and $K^{\prime }$ points will open a gap, and there will be topological edge modes going around the system in the counterclockwise way when the lowest bulk band has the Chern number ${\mathcal{C}}_{-}=-1$.

Figure 5

Calculation of the Chern number as an integral over the Brillouin zone. (a) When a single gauge is chosen for the entire Brillouin zone, the integral of the Berry curvature is rewritten as a contour integral along the boundary of the Brillouin zone, leading to ${\mathcal{C}}_n=0$. (b) In some systems, one should divide the Brillouin zone into two regions I and II, in each of which the gauge is defined smoothly.

Figure 6

Setup of Laughlin's gedanken experiment. (a) Bulk band structure. (b) Schematic picture representing the boundary condition imposed in the system. (c-1–c-3) Flows of the modes under the change of the phase twist $\varphi$ in the boundary conditions. The panels represent the cases (c-1) in the absence of modes in the gap, (c-2) in the presence of a branch with positive dispersion, and (c-3) in the presence of a branch with negative dispersion. In (c-1)–(c-3), the red dots represent the modes allowed by the boundary condition below ${\omega }_F$, and we put one particle per eigenmode below ${\omega }_F$, i.e., per red dot. By changing $\varphi$ the allowed modes flow along the bands towards the directions specified by red arrows. After one cycle of $\varphi$ from zero to $2\pi$, the set of the red dots returns to the original one, except for (c-2) where the number of allowed modes decreases by one, and for (c-3) where it increases by one. (d) Edge modes in the cylinder geometry, the existence of which is shown from the Laughlin gedanken experiment. (e) Edge mode in the open geometry.

Figure 7

Haldane model and its band structures. (a) Honeycomb lattice. The unit cell consists of two sublattice sites, A and B, represented by solid and open circles, respectively. (b, c) Band structures with zigzag-edge ribbon geometry for (b) $t_1=1,t_2=0.1$, $M=0.0166,\phi =1.57$ and (c) $t_1=1,t_2=0.1,M=0.7,\phi =1.57$. The Chern number for the lower band is (b) $\nu =1$ and (c) $\nu =0$. (d) Schematic picture of edge modes in (b). The edge modes shown in blue and red lines correspond to those in (b) with the same colors.