Motivated by the recent theoretical studies on a two-dimensional (2D) chiral Hamiltonian based on the Su-Schrieffer-Heeger chains [, , and , Phys. Rev. B 99, 041117(R) (2019)], we experimentally and computationally demonstrate that topological flat frequency bands can occur at open edges of 2D planar metamaterials and at antiphase boundary seams of ring-shaped or tubular metamaterials. Specifically, using mechanical systems made of magnetically coupled spinners, we reveal that the presence of the edge or seam bands that are flat in the entire projected reciprocal space follows the predictions based on topological winding numbers. The edge-to-edge distance sensitively controls the flatness of the edge bands and the localization of excitations, consistent with the theoretical analysis. The analog of the fractional charge state is observed. Possible realizations of flat bands in a large class of metamaterials, including photonic crystals and electronic metamaterials, are discussed.

中文翻译：

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拓扑机械超材料中开口边缘和反相边界缝处平坦频带的观察

受最近基于Su-Schrieffer-Heeger链的二维（2D）手性哈密顿量理论研究的启发[，和，Phys。B版 99，041117（R）（2019）]，我们通过实验和计算证明了在2D平面超材料的开放边缘以及环形或管状超材料的反相边界接缝处可能出现拓扑平坦的频带。具体而言，使用由磁耦合微调器制成的机械系统，我们揭示出在整个投影互易空间中平坦的边缘或接缝带的存在遵循了基于拓扑绕组数的预测。边缘到边缘的距离可以灵敏地控制边缘带的平坦度和激发的位置，这与理论分析一致。观察到分数电荷状态的类似物。讨论了包括光子晶体和电子超材料在内的一大类超材料中平坦带的可能实现。