Coupled Wire description of the Correlated Physics in Twisted Bilayer Graphene

Abstract

Since the discovery of superconductivity and correlated insulator at fractional electron fillings in the twisted bilayer graphene, most theoretical efforts have been focused on describing this system in terms of an effective extended Hubbard model. However, it was recognized that an exact tight binding model on the Moir\'{e} superlattice which captures all the subtleties of the bands can be exceedingly complicated. Here we pursue an alternative coupled wire description of the system based on the observation that the lattice relaxation effect is strong at small twist angle, which substantially enlarges the AB and BA stacking domains. Under an out-of-plane electric field which can have multiple origins, the low energy physics of the system is dominated by interconnected wires with (approximately) gapless $1d$ conducting quantum valley hall domain wall states. We demonstrate that the Coulomb interaction likely renders the wires a $U(2)_2$ $(1+1)d$ conformal field theory with a tunable Luttinger parameter for the charge $U(1)$ sector. Spin triplet and singlet Cooper pair operator both have quasi-long range order in this CFT. The junction between the wires at the AA stacking islands can lead to either a two dimensional superconductor, or an insulator.