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We investigate the Twisted Bilayer Graphene (TBG) model to obtain an analyticunderstanding of its energetics and wavefunctions needed for many-bodycalculations. We provide an approximation scheme which first elucidates why theBM $K_M$-point centered calculation containing only $4$ plane-waves provides agood analytical value for the first magic angle. The approximation scheme alsoelucidates why most many-body matrix elements in the Coulomb Hamiltonianprojected to the active bands can be neglected. By applying our approximationscheme at the first magic angle to a $\Gamma_M$-point centered model of 6plane-waves, we analytically understand the small $\Gamma_M$-point gap betweenthe active and passive bands in the isotropic limit $w_0=w_1$. Furthermore, weanalytically calculate the group velocities of passive bands in the isotropiclimit, and show that they are \emph{almost} doubly degenerate, while nosymmetry forces them to be. Furthermore, away from $\Gamma_M$ and $K_M$ points,we provide an explicit analytical perturbative understanding as to why the TBGbands are flat at the first magic angle, despite it is defined only byvanishing $K_M$-point Dirac velocity. We derive analytically a connected "magicmanifold" $w_1=2\sqrt{1+w_0^2}-\sqrt{2+3w_0^2}$, on which the bands remainextremely 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 limitby making $w_0$ less (but not larger) than $w_1$ increases the $\Gamma_M$-point gap between active and passive bands. Finally, perturbatively, we providean analytic $\Gamma_M$ point $k\cdot p$ $2$-band model that reproduces the TBGband structure and eigenstates in a certain $w_0,w_1$ parameter range. Furtherrefinement 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.

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