  Mean-field methods are a very powerful tool for investigating weakly
interacting many-body systems in many branches of physics. In particular, they
describe with excellent accuracy trapped Bose-Einstein condensates. A generic,
but difficult question concerns the relation between the symmetry properties of
the true many-body state and its mean-field approximation. Here, we address
this question by considering, theoretically, vortex nucleation in a rotating
Bose-Einstein condensate. A slow sweep of the rotation frequency changes the
state of the system from being at rest to the one containing one vortex. Within
the mean-field framework, the jump in symmetry occurs through a turbulent phase
around a certain critical frequency. The exact many-body ground state at the
critical frequency exhibits strong correlations and entanglement. We believe
that this constitutes a paradigm example of symmetry breaking in - or change of
the order parameter of - quantum many-body systems in the course of adiabatic
evolution.
