  Collective neutrino oscillations play a crucial role in transporting lepton
flavor in astrophysical settings, such as supernovae, where the neutrino
density is large. In this regime, neutrino-neutrino interactions are important
and simulations in mean-field approximations show evidence for collective
oscillations occurring at time scales much larger than those associated with
vacuum oscillations. In this work, we study the out-of-equilibrium dynamics of
a corresponding spin model using Matrix Product States and show how collective
bipolar oscillations can be triggered by quantum fluctuations if appropriate
initial conditions are present. The origin of these flavor oscillations, absent
in the mean-field, can be traced to the presence of a dynamical phase
transition, which drastically modifies the real-time evolution of the
entanglement entropy. We find entanglement entropies scaling at most
logarithmically in the system size, suggesting that classical tensor network
methods could be efficient in describing collective neutrino dynamics more
generally.
