Yang-Mills flows for multilayered graphene

A bstract 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 $$ {\overline{\partial}}_A $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>∂</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>A</mml:mi> </mml:msub> </mml:math> 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 $$ {\overline{\partial}}_A $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>∂</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>A</mml:mi> </mml:msub> </mml:math> operators coupled to abelian gauge fields with non-zero c 1 ’s. Our gradient flow perspective gives a simple bound for magic angles: if the gauge field A ( α ) is such that the YM energy $$ {\int}_{{\mathbbm{T}}^2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> </mml:math> tr $$ {F}_{A\left(\alpha \right)}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mrow> <mml:mi>A</mml:mi> <mml:mfenced> <mml:mi>α</mml:mi> </mml:mfenced> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> is smaller than that of U(1) magnetic flux embedded into SU(2), then α is not magic.