We show that zero-dimensional (0D) systems can host nontrivial topology analogous to macroscopic topological materials in higher dimensions. Unlike macroscopic periodic systems with translational symmetry, zero-dimensional materials such as molecules, clusters, and quantum dots can exhibit discrete rotation symmetry. The eigenstates can thus be grouped into discrete bands and Bloch-like wave functions. Since the symmetry is discrete, the Berry phase and the topological indices must be defined by discrete Wilson polygons. Here, we demonstrate nontrivial ${\mathcal{Z}}_2$ orders in two representative 0D molecules, [m]-Cycloparaphenylene and [m]-iso-thianaphthene, where topological transitions occur when modifying the coupling between the repeating units. Similar to macroscopic topological systems in higher dimensions, localized boundary states emerge in composite nanohoops formed by segments that are topologically distinct. This opens up the possibility of nontrivial topological phases in 0D systems.

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• Topological phases of matter
• 0-dimensional systems
• Topological materials