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Topological materials hosting metallic edges characterized by integerquantized conductivity in an insulating bulk have revolutionized ourunderstanding of transport in matter. The topological protection of these edgestates is based on symmetries and dimensionality. However, onlyinteger-dimensional models have been classified, and the interplay of topologyand fractals, which may have a non-integer dimension, remained largelyunexplored. Quantum fractals have recently been engineered in metamaterials,but up to present no topological states were unveiled in fractals realized inreal materials. Here, we show theoretically and experimentally that topologicaledge and corner modes arise in fractals formed upon depositing thin layers ofbismuth on an indium antimonide substrate. Scanning tunneling microscopyreveals the appearance of (nearly) zero-energy modes at the corners ofSierpi\'nski triangles, as well as the formation of outer and inner edge modesat higher energies. Unexpectedly, a robust and sharp depleted mode appears atthe outer and inner edges of the samples at negative bias voltages. Theexperimental findings are corroborated by theoretical calculations in theframework of a continuum muffin-tin and a lattice tight-binding model. Thestability of the topological features to the introduction of a Rashbaspin-orbit coupling and disorder is discussed. This work opens the perspectiveto novel electronics in real materials at non-integer dimensions with robustand protected topological states.

Topological edge and corner states in Bi fractals on InSb