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.