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 $\left({\sigma }_{xy}^f\right)$ 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 $4e^{-}$ superconductivity, and various charge $2e^{-}$ paired states in BCS and paired-density-wave channels. At charge neutrality, we show that the competition between flavor Hall insulators and charge $2e^{-}$ states can be captured by $SO\left(9\right)$ nonlinear sigma models. If the topological pairing mechanism can dominate over the conventional pairing mechanism, our work predicts the flavor-symmetry-preserving charge $4e^{-}$ superconductivity as a natural candidate for the paired state in the vicinity of the charge neutral point.

Figure 1

Illustrations of (a) static skyrmion textures in two-dimensional space with $W_{sk}=-1$ [see Eq. (3)], (b) radial hedgehog singularity in (2$+$1)-dimensional Euclidean space-time with topological invariant $q=+1$ [see Eq. (7)], and (c) a dipole configuration due to a pair of hedgehog singularities in (2$+$1)-dimensional Euclidean space-time with $q_h=\pm 1$. If the hedgehogs are separated along the imaginary time axis, all spatial planes, lying between (outside) the hedgehogs, will (not) exhibit skyrmion textures. Such planes possess “magnetic flux” of unit vector field $\widehat{\mathit{\Omega }}$, obtained by integrating the skyrmion number density $J_0^{sk}$. If the hedgehogs are separated along any spatial direction, the orthogonal space-time planes will display skyrmion textures. Such planes support “electric flux” of $\widehat{\mathit{\Omega }}$, obtained by integrating the spatial components of skyrmion current density $J_i^{sk}$, with $i=1,2$.

Figure 2

Summary of the main results for twisted bilayer graphene (TBLG with $N=2$), with correlated insulator at $\nu =0$ being described by the quantum spin Hall (QSH) order. The most favorable paired state at $\nu =0$ or its immediate vicinity is a $SU\left(8\right)$ flavor symmetry preserving charge $4e^{-}$ phase. Inside the superconducting state, the $SO\left(6\right)=SU\left(4\right)/{\mathbb{Z}}_2$ symmetry in the combined valley and mini-valley space can be spontaneously broken down to $SO\left(5\right)$ $=USp\left(4\right)/{\mathbb{Z}}_2$, giving rise to charge $2e^{-}$ BCS, or pair-density-waves, or Moiŕe-pair-density waves (see Table 1), and five Goldstone modes. The other two types of quantum flavor Hall orders support the same charge $4e^{-}$ state. But the fragmentation of the $4e^{-}$ state through spontaneous symmetry breaking will involve charge $2e^{-}$ spin-singlet and spin-triplet pairings. Whether the charge $4e^{-}$ or $2e^{-}$ states are realized can be detected from the presence of $hc/4e$ or $hc/2e$ vortices.

Figure 3

The degeneracy of fermion zero modes, localized on hedgehog configurations of quantum spin Hall order parameter and the nature of paired state. Here $E_1$, $E_2$, and $E_3$ represent three different types of flavor symmetry breaking chemical potentials.

Figure 4

Summary of the main results for monolayer graphene (MLG) at charge neutrality and the twisted bilayer graphene in the vicinity of $\nu =\pm 2,\pm 3$, if the correlated insulating states are described by quantum spin Hall order. The presence of flavor symmetry breaking, high-temperature order described by flavor chemical potentials in TBLG give rise to $N=1$ type low-energy Dirac theories. As discussed in Sec. 7 and illustrated in Fig. 3, the form of charge $2e^{-}$ pairing is determined by the type of flavor chemical potentials. If the correlated insulator is described by quantum valley or mini-valley Hall orders, the charge $2e^{-}$ spin-triplet BCS-type pairing become important.