Mechanism of skyrmion condensation and pairing for twisted bilayer graphene

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