Hierarchies of Hofstadter butterflies in 2D covalent-organic frameworks

TL;DR

This paper explores the potential to experimentally observe Hofstadter butterflies in 2D covalent-organic frameworks, leveraging their large pore structures to access magnetic fields needed for revealing fractal quantum phenomena.

Contribution

It proposes that 2D covalent-organic frameworks can enable the first experimental verification of Hofstadter butterflies in single-layer materials due to their unique pore structures.

Findings

Large pore structures make Hofstadter features accessible with current magnetic field strengths.

Hierarchical pore sizes introduce additional complexity to the Hofstadter butterfly patterns.

Potential for new physical observations 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.