The First Magic Angle Beyond the Chiral Limit in Twisted Bilayer Graphene

We develop a squared-Hamiltonian description of twisted bilayer graphene beyond the chiral limit to explain why the first magic angle remains robust under lattice relaxation, while higher-order magic angles are strongly destabilized. Starting from the non-chiral Bistritzer--MacDonald model with finite same-sublattice tunneling, we show that lattice relaxation reshapes the effective confinement landscape rather than acting as a simple perturbation of the chiral theory. A central result is that the realistic relaxation-renormalized tunneling ratio lies close to a special confinement point where the oscillatory part of the symmetric confinement potential nearly cancels. This places realistic twisted bilayer graphene near a nearly uniform confinement regime. At the same time, finite same-sublattice tunneling activates an additional inter-sublattice current-like channel that competes with the chiral orbital channel. The first magic angle survives because these confinement and current-like contributions remain balanced, whereas higher-order magic angles lose this balance through stronger remote-band hybridization and enhanced real-space localization around AA regions. Our results provide a single-particle mechanism for the breakdown of the chiral magic-angle hierarchy and clarify why the experimentally relevant first magic angle remains the most stable remnant of the chiral flat-band structure.