  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. To
study the soft mode spectra, we employ time dependent Hartree-Fock whose
results are reproduced analytically via an effective sigma model description.
We find two different types of low-energy modes - (i) approximate Goldstone
modes associated with breaking an enlarged U(4)$\times$U(4) symmetry and,
surprisingly, a set of (ii) nematic modes with non-zero angular momentum under
three-fold rotation. The modes of type (i) include true gapless Goldstone modes
associated with exact symmetries in addition to gapped "pseudo-Goldstone" modes
associated with approximate symmetries. While the modes of type (ii) are always
gapped, we show that their gap decreases as the Berry curvature grows more
concentrated. For realistic parameter values, the gapped soft modes of both
types have comparable gaps of only a few meV, and lie completely inside the
mean-field bandgap. The entire set of soft modes emerge as Goldstone modes of a
different idealized model in which Berry flux is limited to a solenoid, which
enjoys an enlarged U(8) symmetry. Furthermore, we discuss the number of
Goldstone modes for each symmetry-broken state, distinguishing the linearly vs
quadratically dispersing modes. Finally, we present a general symmetry analysis
of the soft modes for all possible insulating Slater determinant states at
integer fillings that preserve translation symmetry, 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.
