  Neutrinos emitted deep within a supernova explosion experience a self-induced
index of refraction. In the stationary, one-dimensional (1D) supernova "bulb
model", this self-induced refraction can lead to a collective flavor
transformation which is coherent among different neutrino momentum modes. Such
collective oscillations can produce partial swaps of the energy spectra of
different neutrino species as the neutrinos stream away from the proto-neutron
star. However, it has been demonstrated that the spatial symmetries (such as
the spherical symmetry in the bulb model) can be broken spontaneously by
collective neutrino oscillations in multi-dimensional models. Using a
stationary, 2D neutrino ring model we demonstrate that there exist two limiting
scenarios where collective oscillations may occur. In one limit, the collective
flavor transformation begins at a radius with relatively high neutrino
densities and develops small-scale flavor structures. The loss of the spatial
correlation in the neutrino flavor field results in similar (average) energy
spectra for the anti-neutrinos of almost all energies and the neutrinos of
relatively high energies. In the other limit, the flavor transformation starts
at a radius where the neutrino densities are smaller (e.g., due to the
suppression of the high matter density near the proto-neutron star). Although
the spatial symmetry is broken initially, it is restored as the neutrino
densities decrease, and the neutrinos of different flavors partially swap their
energy spectra as in the 1D bulb model. This finding may have interesting
ramifications in other aspects of supernova physics.
