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

TL;DR

This paper analytically investigates twisted bilayer graphene's band structure, providing approximation schemes, understanding of flat bands, and a simplified 2-band model to facilitate many-body calculations.

Contribution

It introduces an analytical approximation scheme and a $k ext{-}p$ 2-band model for understanding TBG's energetics, wavefunctions, and flat band conditions.

Findings

Approximate calculation of the first magic angle using 4 plane-waves.

Analytical understanding of the small gap at $ ext{Gamma}_M$ point.

Derivation of a connected

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 $\Gamma_M$-point centered model of 6 plane-waves, we analytically understand 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 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 $\Gamma_M$ 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 it is defined only by vanishing $K_M$-point Dirac velocity. We derive analytically a connected "magic manifold" $w_1=2\sqrt{1+w_0^2}-\sqrt{2+3w_0^2}$, 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 active and passive bands. Finally, perturbatively, we provide an analytic $\Gamma_M$ 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 $\Gamma_M$ point $k\cdot p$ model in the full $w_0, w_1$ parameter range.