Topology in the Sierpiński-Hofstadter problem

Abstract

Using the Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet and gasket, we investigate whether fractal lattices embedded in two-dimensional space can support topological phases when subjected to a homogeneous external magnetic field. To this end, we study the localization property of eigenstates, the Chern number, and the evolution of energy level statistics when disorder is introduced. Combining these theoretical tools, we identify regions in the phase diagram of both the carpet and the gasket, for which the systems exhibit properties normally associated with gapless topological phases with a mobility edge.