  Employing the recently developed momentum-space quantum Monte Carlo scheme,
we study the dynamic response of single-particle and collective excitations in
realistic continuum models of twisted bilayer graphene. At charge neutrality,
this unbiased numerical method reveals strong competition between different
symmetry breaking channels with a leading instability towards the intervalley
coherent state. Single-particle spectra indicate that repulsive interactions
push the fermion spectral weight away from the Fermi energy and open up an
insulating gap. The spectra of collective excitations suggest an approximate
valley $SU(2)$ symmetry. At low-energy, long-lived valley waves are observed,
which resemble spin waves of Heisenberg ferromagnetism. At high-energy, these
sharp modes quickly become over-damped, when their energy reaches the fermion
particle-hole continuum.
