Hofstadter Butterflies and Metal/Insulator Transitions for Moiré Heterostructures

We consider a tight-binding model recently introduced by Timmel and Mele (Phys Rev Lett 125:166803, 2020) for strained moiré heterostructures. We consider two honeycomb lattices to which layer antisymmetric shear strain is applied to periodically modulate the tunneling between the lattices in one distinguished direction. This effectively reduces the model to one spatial dimension and makes it amenable to the theory of matrix-valued quasi-periodic operators. We then study the charge transport and spectral properties of this system, explaining the appearance of a Hofstadter-type butterfly and the occurrence of metal/insulator transitions that have recently been experimentally verified for non-interacting moiré systems (Wang et al. in Nature 577:42–46, 2020). For sufficiently incommensurable moiré lengths, described by a diophantine condition, as well as strong coupling between the lattices, which can be tuned by applying physical pressure, this leads to the occurrence of localization phenomena.