Topology in the Sierpi\'nski-Hofstadter problem

TL;DR

This paper explores the potential for topological phases in fractal lattices like the Sierpiński carpet and gasket under magnetic fields, analyzing eigenstates, Chern numbers, and disorder effects.

Contribution

It introduces a novel investigation of topological phases in fractal lattices, combining multiple theoretical tools to identify topological regions in their phase diagrams.

Findings

Fractal lattices can support topological phases with gapless properties.

Identification of phase diagram regions exhibiting topological characteristics.

Evidence of mobility edges in fractal systems under magnetic fields.

Abstract

Using the Sierpi\'nski 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 to gapless topological phases with a mobility edge.