Mechanism of skyrmion condensation and pairing for twisted bilayer graphene

When quantum flavor Hall insulator phases of itinerant fermions are disordered by strong fluctuations, the condensation skyrmion textures order parameter fields can lead to superconductivity. In this work, we address mechanism considering scattering between $(2+1)$-dimensional Weyl and hedgehog-type tunneling configurations parameters that violate skyrmion-number conservation law. We show quantized, conductivity $({\ensuremath{\sigma}}_{xy}^{f})$ controls degeneracy topologically protected, fermion zero-modes, localized on hedgehogs. The overlap zero-mode eigenfunctions or 't Hooft vertex is shown control nature paired states. Employing formalism for $N=2$ model twisted bilayer graphene, describe competition among orders, charge $4{e}^{\ensuremath{-}}$ superconductivity, various $2{e}^{\ensuremath{-}}$ states in BCS paired-density-wave channels. At neutrality, insulators be captured $SO(9)$ nonlinear sigma models. If topological pairing dominate over conventional mechanism, our work predicts flavor-symmetry-preserving superconductivity as a natural candidate state vicinity neutral point.