Strong correlations and $d+\mathit{id}$ superconductivity in twisted bilayer graphene

We compute the phase diagram of twisted bilayer graphene near the magic angle where the occurrence of flat bands enhances the effects of electron-electron interactions and thus unleashes strongly correlated phenomena. Most importantly, we find a crossover between $d+\mathit{id}$ superconductivity and antiferromagnetic insulating behavior near half filling of the lowest electron band when the temperature is increased. This is consistent with recent experiments. Our results are obtained using unbiased many-body renormalization group techniques combined with a mean-field analysis of the effective couplings. We provide a qualitative understanding by considering the competition between Fermi-surface nesting and van Hove singularities.

Figure 1

Phase diagram of twisted bilayer graphene near the magic angle as a function of the temperature and chemical potential. Interactions $U/t=2$ drive the system into various competing, strongly correlated phases: Near half filling of the lower band ($\mu \sim -t$), we observe a crossover between $d+\mathit{id}$ superconductivity and AFM insulating behavior as $T$ is increased. Near charge neutrality ($\mu =0$), the system is driven towards an AFM insulator, which, however, features a much higher critical interaction and becomes more pronounced only as $U/t$ is increased (which can be achieved experimentally by tuning the twist angle).

Figure 2

Noninteracting dispersion relation of the lower band of twisted bilayer graphene as predicted by the Yuan-Fu model (1) [30]. The first Brillouin zone is marked by green dots. Near half filling of the lower band ($\mu \sim -t$), the Fermi surface (red squares) is both highly nested and contains a van Hove singularity in the density of states $\rho$ (shown as the inset).

Figure 3

Main panel: Maximum value of the effective coupling constant ${\mathrm{\Gamma }}_2^{\mathrm{\Lambda }}$ at the end of the RG flow (fixed ${\mathrm{\Lambda }}_{f}inal={10}^{-4}t$) as a function of the chemical potential for a bare interaction $U/t=2$. We observe a strong enhancement around $\mu \sim -t$, and the corresponding momentum structure of ${\mathrm{\Gamma }}_2$ along the Fermi surface (left inset) suggests $d$-wave SC in this region (see the main text for details). Near $\mu =0$, we find a momentum structure that indicates AFM insulating behavior (right inset). The enhancement of ${\mathrm{\Gamma }}_2$ is much smaller than for the SC but becomes more pronounced when $U/t$ is increased.

Figure 4

Main panel: Fit of the effective coupling constant ${\mathrm{\Gamma }}_2$ near $\mu =-t$ along the dominant diagonal ${\vec{k}}_1+{\vec{k}}_2=0$ to the form factors $d_{x^2-y^2}$ and $d_{xy}$. Inset: Absolute value of the gap function (modulo an overall factor) for the mean-field solution that minimizes the minimal grand canonical potential. The gap is large at the van Hove points (which minimizes the grand free energy).