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

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 (theta(M) approximate to 1 degrees). 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 Gamma(M)-point centered model of six plane waves, we analytically understand the reason for the small Gamma(M)-point gap between the active and passive bands in the isotropic limit w(0) = w(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 Gamma(M) point, where no symmetry forces them to be. Furthermore, moving away from the Gamma(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 w(1) = 2 root 1 + w(0)(2) - root 2 + 3w(0)(2), on which the bands remain extremely flat as w(0) is tuned between the isotropic (w(0) = w(1)) and chiral (w(0) = 0) limits. We analytically show why going away from the isotropic limit by making w(0) less (but not larger) than w(1) increases the Gamma(M)-point gap between the active and the passive bands. Finally, by perturbation theory, we provide an analytic Gamma(M) point k . p two-band model that reproduces the TBG band structure and eigenstates within a certain w(0), w(1) parameter range. Further refinement of this model are discussed, which suggest a possible faithful representation of the TBG bands by a two-band Gamma(M) point k . p model in the full w(0), w(1) parameter range.

Automated summary

This study examines the twisted bilayer graphene model proposed by Bistritzer and MacDonald, providing an analytical understanding of its energetics and wave functions for many-body calculations. Approximation schemes for wave functions are provided, revealing insights into the accuracy of the original calculations and the neglect of certain matrix elements in the Coulomb Hamiltonian. Analytical calculations are also conducted to understand band properties at magic angles and in isotropic limits, shedding light on the band structure and symmetry considerations in TBG.