Yang-Mills flows for multilayered graphene

We clarify the origin of magic angles in twisted multilayered graphene using Yang-Mills flows in two dimensions. We relate the effective Hamiltonian describing the electrons in the multilayered graphene to the ${\bar\partial}_{A}$ operator on a two dimensional torus coupled to an $SU(N)$ gauge field. Despite the absence of a characteristic class such as $c_{1}$ relevant for the quantum Hall effect, we show that there are topological invariants associated with the zero modes occuring in a family of Hamiltonians. The flatbands in the spectrum of the effective Hamiltonian are associated with Yang-Mills connections, studied by M.Atiyah and R.Bott long time ago. The emergent $U(1)$ magnetic field with nonzero flux is presumably responsible for the observed Hall effect in the absence of (external) magnetic field. We provide a numeric algorithm transforming the original single-particle Hamiltonian to the direct sum of ${\bar\partial}_{A}$ operators coupled to abelian gauge fields with non-zero $c_1$'s. Our perspective gives a simple bound for magic angles: if the gauge field $A({\alpha})$ is such that the YM energy $\Vert F_{A({\alpha})} \Vert^2$ is smaller than that of $U(1)$ magnetic flux embedded into $SU(2)$, then $\alpha$ is not magic.