Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k\cdot p$ 2-Band model

B. Andrei Bernevig; Zhi-Da Song; Nicolas Regnault; Biao Lian

doi:10.1103/PhysRevB.103.205411

paper

Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k \cdot p$ 2-Band model

arXiv:2009.11301 · doi:10.1103/PhysRevB.103.205411

Abstract

We investigate the Twisted Bilayer Graphene (TBG) model to obtain an analytic understanding of its energetics and wavefunctions needed for many-body calculations. We provide an approximation scheme which first elucidates why the BM $K_{M}$ -point centered calculation containing only $4$ plane-waves provides a good analytical value for the first magic angle. The approximation scheme also elucidates why most 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 6 plane-waves, we analytically understand the small $Γ_{M}$ -point gap between the active and passive bands in the isotropic limit $w_{0} = w_{1}$ . Furthermore, we analytically calculate the group velocities of passive bands in the isotropic limit, and show that they are \emph{almost} doubly degenerate, while no symmetry forces them to be. Furthermore, away from $Γ_{M}$ 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 it is defined only by vanishing $K_{M}$ -point Dirac velocity. We derive analytically a connected "magic manifold" $w_{1} = 2 1 + w_{0}^{2} - 2 + 3 w_{0}^{2}$ , 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 $Γ_{M}$ - point gap between active and passive bands. Finally, perturbatively, we provide an analytic $Γ_{M}$ point $k \cdot p$ $2$ -band model that reproduces the TBG band structure and eigenstates in a certain $w_{0}, w_{1}$ parameter range. Further refinement of this model suggests a possible faithful $2$ -band $Γ_{M}$ point $k \cdot p$ model in the full $w_{0}, w_{1}$ parameter range.

arXiv comment25+21 pages, 13+7 figures. Published version

Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k \cdot p$ 2-Band model

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Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a k⋅p 2-Band model

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Twisted bilayer graphene I. Matrix elements, approximations, perturbation theory and a $k \cdot p$ 2-Band model