Hierarchies of Hofstadter butterflies in 2D covalent-organic frameworks

Abstract

The Hofstadter butterfly is one of the first and most fascinating examples of the fractal and self-similar quantum nature of free electrons in a lattice pierced by a perpendicular magnetic field. However, the direct experimental verification of this effect on single-layer materials is still missing as very strong and inaccessible magnetic fields are necessary. For this reason, its indirect experimental verification has only been realized in artificial periodic 2D systems, like moir\'e lattices. The only recently synthesized 2D covalent-organic frameworks might circumvent this limitation: Due to their large pore structures, magnetic fields needed to detect most features of the Hofstadter butterfly are indeed accessible with today's technology. This work opens the door to making this exotic and theoretical issue from the 70s measurable and might solve the quest for the experimental verification of the Hofstadter butterfly in single-layer materials. Moreover, the intrinsic hierarchy of different pore sizes in a 2D covalent-organic framework adds additional complexity and beauty to the original butterflies and leads to a directly accessible playground for new physical observations