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Flat band electronic modes are responsible for superconductivity in twistedbilayer graphene (TBG) rotated at magic angles. From there other magic anglescan be found for any multilayered twisted graphene systems. Eventually, thislead to the discovery of the highest ever known electron-electron correlatedmaterial. Moreover, the quantum phase diagram of TBG is akin to those observedamong high-$T_{c}$ superconductors and thus there is a huge research effort tounderstand TBG in the hope of clarifying the physics behind such strongcorrelations. A particularity of the TBG is the coexistence ofsuperconductivity and the fractional Quantum Hall effect, yet this relationshipis not understood. In this work, a simple $2\times 2$ matrix model for TBG isintroduced. It contains the magic angles and due to the intrinsic chiralsymmetry in TBG, a lowest energy level related to the quantum Hall effect. Thenon-Abelian properties of this Hamiltonian play a central role in theelectronic localization to produce the flat bands and here it is proved thatthe squared Hamiltonian of the chiral TBG model is equivalent to a singleelectron Hamiltonian inside of a non-Abelian pseudo-magnetic field produced byelectrons in other layers. Therefore, the basic and fundamental elements in thephysics of magic angles are determined. In particular, a study is made on thesefundamental energy contributions at the $\Gamma$-point due to its relation tothe recurrence of magic angles and its relationship with the Quantum Halleffect.

Flat bands, quantum Hall effect and superconductivity in twisted bilayer graphene at magic angles