Topological materials hosting metallic edges characterized by integer-quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. While integer-dimensional effects on topological properties have been studied extensively, the interplay of topology and fractals, which may have a non-integer dimension, remains largely unexplored. Here we demonstrate that topological edge and corner modes arise in fractals formed upon depositing thin layers of bismuth on an indium antimonide substrate. Our scanning tunnelling microscopy results and theoretical calculations reveal the appearance and stability of nearly zero-energy modes at the corners of Sierpiński triangles, as well as the formation of outer and inner edge modes at higher energies. This work opens the perspective to extend electronic device applications in real materials at non-integer dimensions with robust and protected topological states.

具有金属边缘的拓扑材料以绝缘体中的整数量子化电导率为特征，彻底改变了我们对物质传输的理解。这些边缘态的拓扑保护基于对称性和维度。虽然整数维对拓扑性质的影响已被广泛研究，但拓扑和分形（可能具有非整数维）的相互作用仍然很大程度上未被探索。在这里，我们证明了在锑化铟衬底上沉积薄层铋时形成的分形中出现拓扑边缘和角模式。我们的扫描隧道显微镜结果和理论计算揭示了谢尔宾斯基三角形拐角处近零能量模式的出现和稳定性，以及较高能量下外边缘和内边缘模式的形成。这项工作开辟了以鲁棒且受保护的拓扑状态在非整数维度的真实材料中扩展电子设备应用的前景。