Chiral limits and effect of light on the Hofstadter butterfly in twisted bilayer graphene

We study the magnetic field induced Hofstadter butterfly in twisted bilayer graphene (TBG) in various kinds of situations. First, we study the equilibrium case and identify the interlayer hopping processes that are most crucial for the appearance of a Hofstadter butterfly. Surprisingly, the hopping processes that are important for the appearance of the Hofstadter butterfly can be categorized as AA stacking type, that is, interlayer hoppings between equivalent sublattices. This is in contrast to AB/BA-type hoppings that are important for the appearance of flat bands in magic angle TBG and were discussed by G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath [Phys. Rev. Lett. 122, 106405 (2019)]. We also find that if AB-type interlayer-hopping processes are turned off, the resulting model is chiral but differs from the model discussed mentioned above. Therefore TBG has two separate chiral limits: One of them is important to understand the formation of flat bands and the other for the Hofstadter butterfly. Taking this as motivation we discuss how the role of AA-type hoppings in combination with lattice relaxation effects can make individual Landau levels slightly harder to resolve in an experimental setting than one would expect from a nonrelaxed lattice setting. Finally, we consider the impact of different forms of light on the fractal structure of the butterfly spectrum. Particularly, we study the impact of circularly polarized light and longitudinal light originating from a waveguide. As the system is exposed to circularly polarized light we find butterflies with increasingly pronounced asymmetry with respect to energy E = 0. This is due to the introduction of a gap term that breaks the chiral symmetries for both of the two chiral limits mentioned above. Lastly, we study the effect of longitudinal light that can be produced at the exit of a waveguide, in a slightly simplified model. Here, we find that no additional terms that break chiral symmetry are introduced. Therefore it is found to lead to no increase in asymmetry of the energy spectrum. In fact, we identify specific experimentally accessible driving regimes in which the TBG achieves any of the two chiral limits.