Electron-phonon coupling and competing Kekulé orders in twisted bilayer graphene

Magic-angle twisted bilayer graphene has attracted significant attention for its strongly correlated phenomena. Recent scanning tunneling experiments have measured graphene-scale Kekul\'e charge ordering in some of the correlated insulators and nearby metallic phases. While most of these can be explained by the strain-induced incommensurate Kekul\'e spiral phase, the presence of Kekul\'e ordering in ultralow strain devices is unexpected within the existing theory. Using Hartree-Fock and strong-coupling calculations, the authors show that such observations can be rationalized by invoking electron-phonon coupling to zone-corner graphene phonons.