Theory of Magic Angles in Twisted Bilayer Graphene: from Non-Abelian Gauge Fields to Flat Bands

Abstract: 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 $2\times2$ Schrödinger 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 $α$ explains why the first magic angle is qualitatively different from the rest. The asymptotic rule $α<em>{m+1}-α_m\to3/2$ follows from a rescaling of the moiré 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 $B</em>{\rm eff}=3α$, so that the high-order zero modes are coherent Landau states of width $1/\sqrt{3α}$ whose guiding centres converge to $\pm(π/3) q_μ$, 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 $Γ$ and $M$ produce phases with Chern numbers $C=\pm2$, and the quantum metric reaches a sharp maximum at $θ\approx0.43<sup>\circ$, identifying a promising regime for correlated and topological phases beyond the first magic angle.