二维声子晶体中简单旋转操作导致的拓扑相变

构建了一种简单的二维声子晶体:由两个横截面为三角形的钢柱所组成的复式元胞按三角点阵的形式排列在空气中,等效地形成了一个蜂巢点阵结构.当三角形钢柱的取向与三角点阵的高对称方向一致时,整个体系具有C_(6v)对称性.研究发现:在保持钢柱填充率不变的条件下,只需要将所有三角柱绕着自己的中心旋转180°,就可实现二重简并的p态和d态在布里渊区中心Γ点处的频率反转,且该能带反转过程实质上是一个拓扑相变过程.通过利用Γ点的P态和d态的空间旋转对称性,构造了一个赝时反演对称性,并在声学系统中实现了类似于电子系统中量子自旋霍尔效应的赝自旋态.随后通过k·p微扰法导出了Γ点附近的有效哈密顿量,并分别计算了拓扑平庸和非平庸系统的自旋陈数,揭示了能带反转和拓扑相变的内在联系.最后通过数值模拟演示了受到拓扑不变量保护的声波边界态的单向传输行为和对缺陷的背向散射抑制.文中所研究的声波体系,尽管材料普通常见,但其拓扑带隙的相对宽度超过21%,比已报道的类似体系的带隙都要宽,且工作原理涵盖从次声波到超声波的很大频率范围,从而在实际应用上具有较大的优势和潜力.

Topological phase transitions caused by a simple rotational operation in two-dimensional acoustic crystals

We design a two-dimensional acoustic crystal (AC) to obtain topologically protected edge states for sound waves. The AC is composed of a triangular array of a complex unit cell consisting of two identical triangle-shaped steel rods arranged in air. The steel rods are placed on the vertices of the hexagonal unit cell so that the whole lattice possesses the C_6v symmetry. We show that by simply rotating all triangular rods around their respective centers by 180 degrees, a topological phase transition can be achieved, and more importantly, such a transition is accomplished with no need of changing the fill ratios or changing the positions of the rods. Interestingly, the achieved topologically nontrivial band gap has a very large frequency width, which is really beneficial to future applications. The topological properties of the AC are rooted in the spatial symmetries of the eigenstates. It is well known that there are two doubly-degenerate eigenstates at the Γ point for a C_6v point group, and they are usually called the p and d states in electronic system. By utilizing the spatial symmetries of the p and d states in the AC, we can construct the pseudo-time reversal symmetry which renders the Kramers doubling in this classical system. We find pseudospin states in the interface between topologically trivial and nontrivial ACs, where anticlockwise (clockwise) rotational behaviors of time-averaged Poynting vectors correspond to the pseudospin-up (pseudospin-down) orientations of the edge states, respectively. These phenomena are very similar to the real spin states of quantum spin Hall effect in electronic systems. We also develop an effective Hamiltonian for the associated bands to characterize the topological properties of the AC around the Brillouin zone center by the k·p perturbation method. We calculate the spin Chern numbers of the ACs, and reveal the inherent link between the band inversion and the topological phase transition. With full-wave simulations, we demonstrate the one-way propagation of sound waves along the interface between topologically distinct ACs, and demonstrate the robustness of the edge states against different types of defects including bends, cavity and disorder. Our design provides a new way to realize acoustic topological effects in a wide frequency range spanning from infrasound to ultrasound. Potential applications and acoustic devices based on our design are expected, so that people can manipulate and transport sound waves in a more efficient way.