Phonons in magic-angle twisted bilayer graphene

Magic-angle twisted bilayer graphene (TBG) has attracted significant interest recently due to the discoveries of diverse correlated and topological states in this system. Despite the extensive research on the electron-electron interaction effects and topological properties of the electrons, the phonons of magic-angle TBG are relatively less explored. In this work, we study the phonon properties in magic-angle TBG based on \textit{ab} \textit{initio} deep potential molecular dynamics. We have calculated phonon band structures and density of states at the magic angle, and have systematically analyzed the phonon eigenmodes at high-symmetry points in the moir\'e Brillouin zone. In particular, at the moir\'e $\Gamma$ point, we have discovered a number of soft modes which can exhibit dipolar-like, stripe-like, and octupolar-like vibrational patterns within the moir\'e supercell, as well as some "vortical" modes with nonzero curl in real space. At the moir\'e $K$/$K'$ points, there are time-reversal breaking chiral phonon modes with nonzero local phonon polarizations. We have further studied the phonon effects on the electronic structures by freezing certain soft phonon modes. We find that if a soft "stripe" phonon mode at moir\'e $\Gamma$ point is assumed to be frozen, the system would exhibit a charge order which naturally explains the recent observations from scanning tunnelling microscopy. Moreover, there are also low-frequency $C_{2z}$-breaking modes at moir\'e $\Gamma$ point, which would gap out the Dirac points at the charge neutrality point once these modes get frozen. This provides a new perspective to the origin of correlated insulator state at the charge neutrality point.Comment: 12 pages and 9 figure