Reduction of the twisted bilayer graphene chiral Hamiltonian into a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> matrix operator and physical origin of flat bands at magic angles

The chiral Hamiltonian for twisted graphene bilayers is written as a $2\ifmmode\times\else\texttimes\fi{}2$ matrix operator by renormalization of the that takes into account particle-hole symmetry. This results in an effective terms Pauli matrices with three contributions: kinetic term, confinement potential, and non-Abelian gauge field. action proposed maps zero-mode flat bands ground states. On each layer, modes near zero energy have antibonding nature triangular lattice. leads to phase-frustration effect associated massive degeneration makes flat-band similar confined observed other bipartite lattices. At magic angles, it shown intralayer frustration exactly zero. Surprisingly, suggests at angles are akin floppy-mode flexible crystals or glasses, making unexpected connection between rigidity topological theory two-dimensional superconductor systems.