Topological and nematic superconductivity mediated by ferro-SU(4) fluctuations in twisted bilayer graphene

We propose an SU(4) spin-valley-fermion model to investigate the superconducting instabilities of twisted bilayer graphene (TBG). In this approach, bosonic fluctuations associated with an emergent SU(4) symmetry, corresponding to combined rotations in valley and spin spaces, couple to the low-energy fermions that comprise the flat bands. These fluctuations are peaked at zero wave vector, reflecting the “ferromagnetic-like” SU(4) ground state recently found in strong-coupling solutions of microscopic models for TBG. Focusing on electronic states related to symmetry-imposed points of the Fermi surface, dubbed here “valley hot-spots” and “van Hove hot-spots,” we find that the coupling to the itinerant electrons partially lifts the huge degeneracy of the ferro-SU(4) ground state manifold, favoring intervalley order, spin-valley coupled order, ferromagnetic order, spin-current order, and valley-polarized order, depending on details of the band structure. These fluctuations, in turn, promote attractive pairing interactions in a variety of closely competing channels, including a nodeless $f$-wave state, a nodal $i$-wave state, and topological $d+id$ and $p+ip$ states with unusual Chern numbers 2 and 4, respectively. Nematic superconductivity, although not realized as a primary instability of the system, appears as a consequence of the near-degeneracy of superconducting order parameters that transform as one-dimensional and two-dimensional irreducible representations of the point group $D_6$.

Figure 1

Schematic phase diagram of superconductivity mediated by fluctuations associated with a correlated state that displays “ferromagnetic-like” SU(4) order. As we show in this paper, the types of ferro-SU(4) order favored by the coupling to itinerant fermions are intervalley order, spin-valley coupled order, ferromagnetic order, spin-current order, and valley-polarized order. As shown in Table 1, these fluctuations promote a rich landscape of pairing states, such as a nodeless $f$-wave state, a nodal $i$-wave state, and topological $d+id$ and $p+ip$ states with unusual Chern numbers 2 and 4, respectively.

Figure 2

Schematics of the Fermi surface. Blue and red correspond to different valleys. Regardless of band structure details, these Fermi surfaces display two prominent features: the symmetry-imposed intersection points between the Fermi surfaces from the two valleys [the valley hot-spots, highlighted with black dots in (a)], and the proximity to a van Hove singularity for appropriate doping levels [the van Hove hot-spots, highlighted with grey dots in (b)]. Valley hot-spots are located along the $\overline{\mathrm{\Gamma }}\text{−}\overline{K}$ and $\overline{\mathrm{\Gamma }}\text{−}{\overline{K}}^{\prime }$ lines, whereas the van Hove hot-spots are located along the $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ lines of the moiré Brillouin zone.

Figure 3

Illustration of the $E_1$ ($p\pm ip$) (a) and $E_2$ ($d\pm id$) (b) valley-triplet pairing states mediated by intervalley fluctuations.

Figure 4

Illustration of the $A_2$ ($i$-wave) (a) and $E_2$ ($d\pm id$) (b) valley-singlet pairing states mediated by spin-valley fluctuations.

Figure 5

The structure of ${\mathrm{\Delta }}_o^{\prime }$ for the case of valley-triplet, spin-triplet $p+ip$ pairing mediated by ferromagnetic fluctuations. Note the sign change between a pair of van Hove hot-spots.

Figure 6

Illustration of the $E_1$ ($p\pm ip$) (a) and $B_2$ ($f$-wave) (b) spin-triplet pairing states mediated by the ferromagnetic fluctuations.

Figure 7

Illustration of $E_2$ ($d\pm id$) (a) and the $A_1$ ($s$-wave) (b) spin-singlet pairing states mediated by spin-current fluctuations.