标题： 威尔逊多边形与零维系统的拓扑 显示英文标题

标题： Wilson polygons and the topology of zero-dimensional systems

摘要： 我们展示了零维（0-D）系统可以承载类似于更高维度的宏观拓扑材料的非平凡拓扑。与具有平移对称性的宏观周期系统不同，零维材料如分子、团簇和量子点可以表现出离散旋转对称性。因此，本征态可以被分组成离散的能带和Bloch-like波函数。由于对称性是离散的，Berry相位和拓扑指标必须通过离散Wilson多边形来定义。在这里，我们演示了两个代表性0-D分子中的非平凡Z2序，即[m]-环对二苯并和[m]-异硫代萘并，其中当修改重复单元之间的耦合时会发生拓扑转变。类似于更高维度的宏观拓扑系统，局域边界态出现在由拓扑上不同的片段组成的复合纳米环中。这开启了0-D系统中非平凡拓扑相的可能性。 显示英文摘要

摘要： We show that zero-dimensional (0-D) systems can host non-trivial topology analogous to macroscopic topological materials in greater 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 non-trivial Z2 orders in two representative 0-D 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 greater dimensions, localized boundary states emerge in composite nanohoops formed by segments that are topologically distinct. This opens up the possibility of non-trivial topological phases in 0-D systems.