  In astrophysical environments with intense neutrino fluxes, neutrino-neutrino
forward scattering contributes both diagonal and off-diagonal potentials to the
flavor-basis Hamiltonian that governs neutrino flavor evolution. We examine a
special case where adiabatic flavor evolution can produce an off-diagonal
potential from neutrino-neutrino forward scattering that dominates over both
the corresponding diagonal term and the potential from neutrino-matter forward
scattering. In this case, we find a solution that, unlike the ordinary
Mikeyhev-Smirnov-Wolfenstein scenario, has both neutrinos and antineutrinos
maximally mixed in medium over appreciable ranges of neutrino and antineutrino
energy. Employing the measured solar and atmospheric neutrino mass-squared
differences, we identify the conditions on neutrino fluxes that are required
for this solution to exist deep in the supernova environment, where it could
affect the neutrino signal, heavy-element nucleosynthesis, and even the revival
of the supernova shock. We speculate on how this solution might or might not be
attained in realistic supernova evolution. Though this solution is ephemeral in
time and/or space in supernovae, it may signal the onset of subsequent
appreciable flavor mixing for both neutrinos and antineutrinos. A similar
solution may also exist in an early universe with significant net
neutrino-lepton numbers.
