Strongly Correlated Superconductivity in Twisted Bilayer Graphene: a Gutzwiller Study

Abstract

We study strongly correlated superconductivity in magic-angle twisted bilayer graphene (MATBG) using a variational Gutzwiller wavefunction $\ket{Ψ_G} = \prod_{\vb{R}} \hat{P}_{\vb{R}} \ket{Φ_0}$, where the Gutzwiller projector $\hat{P}_{\vb{R}}$ is allowed to break charge U(1) symmetry to accommodate superconducting (SC) order. The ground state energy is evaluated via the \textit{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 $J_A$, and reveal a strongly correlated SC (SC-SC) phase dominates at large $U$, wherethe strong on-site interaction U strongly suppress the $f$-orbital charge fluctuations while maintaining finite pairing order and a sizeable quasiparticle weight Z, distinguishing it from a conventional Mott insulator. For a range of $J_{\mathrm{A}}$, SC-SC transitions to FL as $U$ decreases, until the weakly correlated BCS-like SC (BCS-SC) re-enters as $U\to 0$. We further identify a novel small Fermi liquid (sFL) state with effective Fermi surface formed by $c$-orbitals, which is essentially different with the normal Fermi liquid. 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.