Topological superconductivity, ferromagnetism, and valley-polarized phases in moiré systems: Renormalization group analysis for twisted double bilayer graphene

Recent experiments have observed possible spin- and valley-polarized insulators and spin-triplet superconductivity in twisted double bilayer graphene, a moiré structure consisting of a pair of Bernal-stacked bilayer graphene. Besides the continuously tunable bandwidths controlled by an applied displacement field and twist angle, these moiré bands also possess Van Hove singularities near the Fermi surface and a field-dependent nesting which is far from perfect. Here we carry out a perturbative renormalization group analysis to unbiasedly study the competition among all possible instabilities in twisted double bilayer graphene and related systems with a similar Van Hove fermiology in the presence of weak but finite repulsive interactions. Our key finding is that there are several competing magnetic, valley, charge, and superconducting instabilities arising from interactions in twisted double bilayer graphene, which can be tuned by controlling the displacement field and the twist angle. In particular, we show that spin- or valley-polarized uniform instabilities generically dominate under moderate interactions smaller than the bandwidth, whereas $p$-wave spin-triplet topological superconductivity and exotic spin-singlet modulated paired state become important as the interactions decrease. Realization of our findings in general moiré systems with a similar Van Hove fermiology should open up new opportunities for manipulating topological superconductivity and spin- or valley-polarized states in highly tunable platforms.

Figure 1

(a) Energy contour plot for the first moiré conduction band in the $+K$ valley of TDBG with a twist angle $\theta =1.{24}^{\circ }$ and a layer-dependent potential $U=50$ meV that is generated by an out-of-plane displacement field. See Ref. [46] for details of the moiré Hamiltonian that leads to the band structure. (b) Similar as (a) but for $U=35$ meV. The yellow lines in (a) and (b) mark the Fermi surfaces at the Van Hove energy, which are tunable by the displacement field. (c) The calculated density of states as a function of filling factor for the bands shown in (a) and (b). The VHS we consider correspond to the largest peak in each curve. (d) Schematics of the patch model we consider for the representative TDBG Van Hove fermiology in (a). The hexagon, red points, and blue points represent the moiré Brillouin zone, the patch centers from the $+K$ valley, and those from the $-K$ valley, respectively. The arrows represent the characteristic momenta connecting the inter- and intravalley patches.

Figure 2

(a) Schematics for all the momentum-preserving inter- and intrapatch interactions. The hexagon, red dots, and blue dots represent the moiré Brillouin zone, the VH points from the $K$ valley, and those from the $-K$ valley, respectively. Diagrammatic expressions for (b) the test vertices in the particle-hole and particle-particle channels.

Figure 3

The phase diagram in the absence of intervalley scattering ($B=0$).

Figure 4

Representative RG flows of intra- and interpatch interactions in the absence of intervalley scattering ($B=0$) in the regimes where the dominant instabilities are (a) intrapatch PDW, (b) $p/d$-wave uniform SC, (c) intravalley CDW, (d) ferromagnetism with the $s/f$-wave form factor, and (e) valley-polarized uniform charge order, respectively.

Figure 5

The phase diagram in the presence of intervalley scattering ($B=A/4$).

Figure 6

Representative RG flows of intra- and interpatch interactions in the presence of intervalley scattering ($B=A/4$) in the regimes where the dominant instabilities are (a) $p$-wave uniform SC, (b) $d$-wave uniform SC, and (c) ferromagnetism, respectively.