Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model

Abstract

We investigate the twisted bilayer graphene (TBG) model of Bistritzer and MacDonald (BM) [Bistritzer and MacDonald, Proc. Natl. Acad. Sci. 108, 12233 (2011)] to obtain an analytic understanding of its energetics and wave functions needed for many-body calculations. We provide an approximation scheme for the wave functions of the BM model, which first elucidates why the BM KM-point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θM≈1∘). The approximation scheme also elucidates why most of the many-body matrix elements in the Coulomb Hamiltonian projected to the active bands can be neglected. By applying our approximation scheme at the first magic angle to a ΓM-point centered model of six plane waves, we analytically understand the reason for the small ΓM-point gap between the active and passive bands in the isotropic limit w0=w1. Furthermore, we analytically calculate the group velocities of the passive bands in the isotropic limit, and show that they are almost doubly degenerate, even away from the ΓM point, where no symmetry forces them to be. Furthermore, moving away from the ΓM and KM points, we provide an explicit analytical perturbative understanding as to why the TBG bands are flat at the first magic angle, despite the first magic angle is defined by only requiring a vanishing KM-point Dirac velocity. We derive analytically a connected “magic manifold” w1=2√1+w20−√2+3w20, on which the bands remain extremely flat as w0 is tuned between the isotropic (w0=w1) and chiral (w0=0) limits. We analytically show why going away from the isotropic limit by making w0 less (but not larger) than w1 increases the ΓM-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic ΓM point k⋅p two-band model that reproduces the TBG band structure and eigenstates within a certain w0, w1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band ΓM point k⋅p model in the full w0, w1 parameter range.