  The chiral Hamiltonian for twisted graphene bilayers is written as a
$2\times2$ matrix operator by a renormalization of the Hamiltonian that takes
into account the particle-hole symmetry. This results in an effective
Hamiltonian with an average field plus and effective non-Abelian gauge
potential. The action of the proposed renormalization maps the zero-mode region
into the ground state. Modes near zero energy have an antibonding nature in a
triangular lattice. This leads to a phase-frustration effect associated with
massive degeneration, and makes flat-bands modes similar to confined modes
observed in other bipartite lattices. Suprisingly, the proposed Hamiltonian
renormalization suggests that flat-bands at magic angles are akin to
floppy-mode bands in flexible crystals or glasses, making an unexpected
connection between rigidity topological theory and magic angle twisted
two-dimensional heterostructures physics.
