We propose theoretically a reconfigurable two-dimensional (2D) hexagonal sonic crystal with higher-order topology protected by the six-fold, $C_6$, rotation symmetry. The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit-cell. In the nontrivial phase, the sonic crystal realizes the topological spin Hall effect in a higher-order fashion: (i) The edge states emerging in the bulk band gap exhibits partial spin-momentum locking and are gapped due to the reduced spatial symmetry at the edges. (ii) The gapped edge states, on the other hand, stabilize the topological corner states emerging in the edge band gap. The partial spin-momentum locking is manifested as pseudo-spin-polarization of edge states away from the time-reversal invariant momenta, where the pseudospin is emulated by the acoustic orbital angular momentum. We reveal the underlying topological mechanism using a corner topological index based on the symmetry representation of the acoustic Bloch bands.

中文翻译：

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声音的高阶拓扑自旋霍尔效应

我们在理论上提出了一种可重构二维 (2D) 六方声波晶体，其具有受六重 $C_6$ 旋转对称保护的高阶拓扑结构。可以通过旋转每个单元中的三角形散射体来控制声学带隙和带拓扑结构。在非平凡阶段，声波晶体以更高阶的方式实现拓扑自旋霍尔效应：（i）体带隙中出现的边缘态表现出部分自旋动量锁定，并且由于在边缘。(ii) 另一方面，有隙边缘状态稳定了边缘带隙中出现的拓扑角状态。部分自旋动量锁定表现为远离时间反转不变动量的边缘状态的伪自旋极化，其中赝自旋由声轨道角动量模拟。我们使用基于声学布洛赫带对称表示的角拓扑索引揭示了潜在的拓扑机制。