  We consider the many-body system of neutrinos interacting with each other
through neutral current weak force. Emerging many-body effects in such a system
could play important roles in some astrophysical sites such as the core
collapse supernovae. In the literature this many-body system is usually treated
within the mean field approximation which is an effective one-body description
based on omitting entangled neutrino states. In this paper, we consider the
original many-body system in an effective two flavor mixing scenario under the
single angle approximation and present a solution without using the mean field
approximation. Our solution is formulated around a special class of many-body
eigenstates which do not undergo any level crossings as the neutrino self
interaction rate decreases while the neutrinos radiate from the supernova. In
particular, an initial state which consists of electron neutrinos and
antineutrinos of an orthogonal flavor can be entirely decomposed in terms of
those eigenstates. Assuming that the conditions are perfectly adiabatic so that
the evolution of these eigenstates follow their variation with the interaction
rate, we show that this initial state develops a spectral split at exactly the
same energy predicted by the mean field formulation.

High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics;Quantum Gases;Nuclear Theory