Theory of Emergent Josephson Lattice in Neutral Twisted Bilayer Graphene (Moi\'re is Different)

Abstract

`More is Different' (Anderson, 1972) in graphene. A bilayer and a twist spring surprises. Recently discovered superconductivity (T$_c\approx$ 1.7 K) at an ultra low doping density $\sim 10^{11}$cm${}^{-2}$ has alerted the community to look for an electron-electron interaction based mechanism, as phonon-induced attraction seems inadequate. We suggest a mechanism of superconductivity, where an important role is played by the dense (density $\approx$ 2 $\times$ 10${}^{15}$cm$^{-2}$) $\pi$-electron fluid of graphene layers. This fluid bears off-shell resonating valence bond correlations (RVB) at the carbon-carbon bond scale. A commensurate twist $\theta\approx 1.1^\circ$, creates charge neutral carrier puddles (size $\sim$ 50 \AA) and forms a triangular Moir\'e lattice of local AA registry. AA registry dopes equal numbers of electrons and holes via interlayer tunneling, whereas AB registry does not. Carriers inside the charge neutral puddles form equal numbers of -2e and +2e Cooper pairs, using on-shell RVB correlations. A Josephson-Moir\'e lattice emerges. Coulomb blockade competes with pair tunneling and creates a Bose Mott insulator. Gate doping dopes the Bose Hubbard model and creates superconductivity. Our message is that RVB correlations, which remain dormant in (carrierless) neutral graphene become on-shell for two added electrons, as they are indistinguishable from electrons that make the background $\pi$-fluid in graphene.