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The chiral Hamiltonian for twisted graphene bilayers is written as a$2\times2$ matrix operator by a renormalization of the Hamiltonian that takesinto account the particle-hole symmetry. This results in an effectiveHamiltonian with an average field plus and effective non-Abelian gaugepotential. The action of the proposed renormalization maps the zero-mode regioninto the ground state. Modes near zero energy have an antibonding nature in atriangular lattice. This leads to a phase-frustration effect associated withmassive degeneration, and makes flat-bands modes similar to confined modesobserved in other bipartite lattices. Suprisingly, the proposed Hamiltonianrenormalization suggests that flat-bands at magic angles are akin tofloppy-mode bands in flexible crystals or glasses, making an unexpectedconnection between rigidity topological theory and magic angle twistedtwo-dimensional heterostructures physics.

Reduction of the Twisted Bilayer Graphene Chiral Hamiltonian into a $2\times2$ matrix operator and physical origin of flat-bands at magic angles