从大自然的分形之美中寻找非凡物态

分形在大自然中无处不在，其具有自相似性、分数维度的性质。最近在分形晶格中的理论与实验研究表明，在分数维度中没有体的概念却可以存在拓扑绝缘体。分形中的拓扑态具有一些新奇的特性，比如具有压缩的拓扑相、拓扑边界态分布于不同代的分形几何中。这些与常规拓扑绝缘体不同的独特之处展现了一个审视空间维度与拓扑相变相互作用的新视角。文章简要回顾了分形体系中拓扑物态的发展历史，并重点介绍了人工微结构中的拓扑分形绝缘体。

Looking for the nontrivial states of matter from fractals

Fractals are ubiquitous in nature, characterized by the properties of selfsimilarity and fractional dimensions. Recent theoretical and experimental studies on fractal lattices have demonstrated the existence of topological insulators without the concept of “bulk” in the fractional dimension. Topological states in fractals exhibit novel properties, such as the squeezed topological phases and topological boundary states distributed in various generations of fractal geometry. These unique differences from conventional topological insulators provide a new perspective for examining the interaction between the spatial dimensions and topological phase transitions. This article briefly reviews the history of topological states in fractal systems and focuses on the topological fractal insulators in artificial microstructures.