The Landau Level of Fragile Topology

Biao Lian, Fang Xie, B. Andrei Bernevig

Published 2018-11-28Version 1

We study the Landau levels and Hofstadter butterfly of the flat bands in twisted bilayer graphene (TBG). We predict that the nontrivial fragile topology of the two flat bands near the charge neutral point leads to two topological LLs of Chern number $C=-1$ separating away from the flat bands at magnetic fields above $15$T, and the Hofstadter butterfly of the flat bands is energetically unbounded. This is an experimentally testable consequence of the predicted but yet not confirmed fragile topology of the TBG. We also develop two new methods in analyzing Hofstadter bands: an open-shell method for computing LL edge states without a real space boundary, and the Peierls substitution for charge densities not located at the Wannier center. We further show the TBG band theory with Zeeman splitting being the most sizable splitting could result in Landau fans at the charge neutral point and half fillings near the magic angle, and we predict their changes under a fixed in-plane magnetic field.