Landau Levels as a Probe for Band Topology in Graphene Moiré Superlattices

Flat bands in twisted multilayer systems are often characterized by non-trivial topology due to the pseudo-gauge field caused by the moiré potential. In this work, we will show that Landau levels are a good probe for the topological character of electronic bands in two-dimensional moiré superlattices. Twisted double bilayer graphene (TDBG), a system constructed by twisting two AB-stacked bilayer graphene (BLG) counterparts placed on top of each other, is considered as an example. TDBG has AB-AB and AB-BA stacking configurations, that have different valley Chern numbers of the flat bands, yet have very similar band structures. Different valley Chern numbers in these two configurations of TDBG manifest as different Landau level sequences in the Hofstadter butterfly. The results are explained from the point of view of the distribution of orbital magnetization in momentum space that is governed by the rotational C2 and time-reversal T symmetries. Our results can be readily extended to other twisted graphene multilayers and h-BN/graphene heterostructures thus establishing the Hofstadter butterfly spectra as a powerful tool for detecting the non-trivial valley band topology.
[1] arXiv:2005.10620 (2020)
[2] Nano Lett., 20, 2410 (2020)