  A surprising consequence of non-linear flavor evolution is the spontaneous
breaking of the initial symmetries of the neutrino gas propagating in a dense
astrophysical environment.
  We explore the flavor conversion physics by taking into account the polar and
azimuthal angular distributions of neutrinos and present the very first example
of spontaneous symmetry breaking in the context of fast flavor mixing in the
nonlinear regime. Intriguingly, we find that fast flavor mixing does not always
develop in the proximity of the angular regions with vanishing electron lepton
number, as commonly assumed, and large flavor mixing can rapidly spread through
all neutrino modes. Such behavior cannot be predicted from the linear regime of
the flavor evolution. These results can have major consequences on the physics
of compact astrophysical objects.
