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This paper develops a unified theory for magic angles in twisted bilayer graphene, explaining the emergence of flat bands via a chiral continuum model and a $2\times2$ Schrödinger operator in a non-Abelian $SU(2)$ pseudo-magnetic field. The theory derives an energy sum rule that differentiates the first magic angle from subsequent ones by analyzing the scaling of interlayer coupling channels. It also predicts an asymptotic rule for magic angle spacing and characterizes high-order zero modes as coherent Landau states, highlighting the necessity of the non-Abelian structure for the magic-angle sequence.
The non-Abelian gauge field structure in twisted bilayer graphene is revealed as the fundamental origin of magic angles and flat bands, not just a consequence.
At a discrete set of ``magic''twist angles, twisted bilayer graphene develops two bands at charge neutrality that are almost perfectly flat. We present a unified theory of this phenomenon within the chiral continuum model. The central object is the square of the chiral Hamiltonian, a $2\times2$ Schr\"odinger operator for an electron moving in a non-Abelian $SU(2)$ pseudo-magnetic field generated by the interlayer tunneling. From it we derive an energy sum rule in which the interlayer coupling splits into a bounded overlap channel and an unbounded current channel; their different scaling with the coupling $\alpha$ explains why the first magic angle is qualitatively different from the rest. The asymptotic rule $\alpha_{m+1}-\alpha_m\to3/2$ follows from a rescaling of the moir\'e potential together with its three-cell magnetic periodicity, with the two parity families of zero modes each repeating with period $3$ and interleaved with each other. Near the AA point the zero-mode problem becomes a lowest-Landau-level problem in an effective field $B_{\rm eff}=3\alpha$, so that the high-order zero modes are coherent Landau states of width $1/\sqrt{3\alpha}$ whose guiding centres converge to $\pm(\pi/3) q_\mu$, with kinetic and confinement energies in equipartition. Deforming the coupling into an Abelian one removes the magic-angle sequence, which shows that the non-Abelian structure is essential. Between consecutive magic angles, band inversions at $\Gamma$ and $M$ produce phases with Chern numbers $C=\pm2$, and the quantum metric reaches a sharp maximum at $\theta\approx0.43^\circ$, identifying a promising regime for correlated and topological phases beyond the first magic angle.