Soft modes in magic angle twisted bilayer graphene

We present a systematic study of the low-energy collective modes for different insulating states at integer fillings in twisted bilayer graphene. In particular, we provide a simple counting rule for the total number of soft modes, and analyze their energies and symmetry quantum numbers in detail. We find two different types of low-energy modes - (i) approximate Goldstone modes associated with breaking an enlarged U(4)xU(4) symmetry and, surprisingly, a second branch (ii) of nematic modes with non-zero angular momentum under three-fold rotation. While the modes of type (ii) are always gapped, we show that their gap depends sensitively on the distribution of Berry curvature, decreasing as the Berry curvature grows more concentrated. For realistic parameter values, the gapped soft modes of both types are found to have comparable gaps of only a few meV, and lie completely inside the mean-field bandgap. Finally, we present a general analysis of the symmetry representations of the soft modes for all possible insulating Slater determinant states, independent of the energetic details. The resulting soft mode degeneracies and symmetry quantum numbers provide a fingerprint of the different insulting states enabling their experimental identification from a measurement of their soft modes.