Strongly Correlated Superconductivity in Twisted Bilayer Graphene: A Gutzwiller Study

We study strongly correlated superconductivity in magic-angle twisted bilayer graphene (MATBG) using variational Gutzwiller wavefunction where the Gutzwiller projector $\hat{P}_{R}$ is allowed to break charge U(1) symmetry to accommodate superconducting (SC) order. The ground state energy is evaluated via the Gutzwiller Approximation applied to an 8-band model consisting of correlated f-orbitals and uncorrelated c-orbitals, with interactions including onsite Coulomb repulsion $U$, phonon-mediated anti-Hund's coupling $\hat{H}_{J_A}$, and intra-orbital Hund's coupling $\hat{H}_{J_H}$. At filling $\nu=2.5$, we map out the phase diagram as a function of $U$ and $J_A$, finding a dome-shaped Fermi liquid (FL) phase that separates a weakly correlated BCS-like SC (BCS-SC) at small $U$ from a strongly correlated SC (SC-SC) at large $U$. A nematic SC state, stabilized over a large region of the phase diagram including the realistic parameter regime of MATBG, acquires a nodal gap structure with V-shaped density of states at large $U$ via interaction-driven SC gap reconstruction. In the SC-SC regime, the off-diagonal (charge-U(1)-breaking) components of $\hat{P}_{R}$ strongly suppress $f$-orbital charge fluctuations while maintaining finite pairing order and a sizeable quasiparticle weight $Z$, distinguishing it from a conventional Mott insulator. We further identify a novel small Fermi liquid (sFL) state with effective Fermi surface volume $=\nu+2$. Interestingly, in the intermediate- ($U \lesssim 40$ meV) and large-$U$ ($U \gtrsim 40$ meV) regimes, the conventional FL and the sFL are the lowest-energy normal phases, respectively, potentially serve as the parent states of the SC-SC phase. These results illuminate the interplay between strong correlations and unconventional pairing in MATBG, and establish a versatile Gutzwiller framework applicable to other strongly correlated superconductors.