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

摘要

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 K_M-point centered original calculation containing only four plane waves provides a good analytical value for the first magic angle (θ_M ≈ 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 Γ_M-point centered model of six plane waves, we analytically understand the reason for the small Γ_M-point gap between the active and passive bands in the isotropic limit ω_0 = ω_1. 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 Γ_M point, where no symmetry forces them to be. Furthermore, moving away from the Γ_M and K_M 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 K_M-point Dirac velocity. We derive analytically a connected "magic manifold" ω_1 = 2(1+ω_0~2)~(1/2) - (2 + 3ω_0~2)~(1/2) on which the bands remain extremely flat as ω_0 is tuned between the isotropic (ω_0 = ω_1) and chiral (ω_0 = 0) limits. We analytically show why going away from the isotropic limit by making ω_0 less (but not larger) than ω_1 increases the Γ_M-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic Γ_M point k · p two-band model that reproduces the TBG band structure and eigenstates within a certain ω_0, ω_1 parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band Γ_M point k · p model in the full ω_0, ω_1 parameter range.