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

Recent scanning tunneling microscopy experiments in twisted bilayer [K. P. Nuckolls et al., Nature (London) 620, 525 (2023)0028-083610.1038/s41586-023-06226-x] and trilayer [H. Kim et al., Nature (London) 623, 942 (2023)0028-083610.1038/s41586-023-06663-8] graphene have revealed the ubiquity of Kekulé charge-density wave order in magic-Angle graphene. Most samples are moderately strained and show "incommensurate Kekulé spiral"(IKS) order involving a graphene-scale charge density distortion uniaxially modulated on the scale of the moiré superlattice, in accord with theoretical predictions. However, ultralow strain bilayer samples instead show graphene-scale Kekulé charge order that is uniform on the moiré scale. This order, especially prominent near filling factor ν=-2, is unanticipated by theory which predicts a time-reversal breaking Kekulé current order at low strain. We show that including the coupling of moiré electrons to graphene-scale optical zone-corner (ZC) phonons stabilizes a uniform Kekulé charge ordered state at |ν|=2 with a quantized topological (spin or anomalous Hall) response. Our work clarifies how this phonon-driven selection of electronic order emerges in the strong-coupling regime of moiré graphene.