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Twisted bilayer graphene (TBG) is known for exhibiting highly correlatedphases at magic angles due to the emergence of flat bands that enhanceelectron-electron interactions. In the TBG chiral model, electronic wavefunction properties depend on a single parameter ($\alpha$), inverselyproportional to the relative twist angle between the two graphene layers. Inprevious studies, as the twist angles approached small values, strongconfinement, and convergence to coherent Landau states were observed. This workexplores flat-band electronic modes, revealing that flat band states exhibitself-duality; they are coherent Landau states in reciprocal space and exhibitminimal dispersion, with standard deviation $\sigma_k=\sqrt{3\alpha/2\pi}$ as$\alpha$ approaches infinity. Subsequently, by symmetrizing the wave functionsand considering the squared TBG Hamiltonian, the strong confinement observed inthe $\alpha\rightarrow\infty$ limit is explained. This confinement arises fromthe combination of the symmetrized squared norm of the moir\'e potential andthe quantized orbital motion of electrons, effectively creating a quantum well.The ground state of this well, located at defined spots, corresponds to Landaulevels with energy determined by the magic angle. Furthermore, we demonstratethat the problem is physically analogous to an electron attached to anon-Abelian $SU(2)$ gauge field with an underlying $C_3$ symmetry. In regionsof strong confinement, the system can be considered as Abelian. This allows todefine a magnetic energy in which the important role of the wave functionparity and gap closing at non-magic angles is revealed. Finally, we investigatethe transition from the original non-Abelian nature to an Abelian state byartificially changing the pseudo-magnetic vector components from an $SU(2)$ toa $U(1)$ field, which alters the sequence of magic angles.

Self-duality properties and localization centers of the electronic wave functions at high magic angles in twisted bilayer graphene