  We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose
gas in a trap. We prove that there is a breaking of the rotational symmetry in
the ground state; more precisely, for any value of the angular velocity and for
large enough values of the interaction strength, the ground state of the
functional is not an eigenfunction of the angular momentum. This has
interesting consequences on the Bose gas with spin; in particular, the ground
state energy depends non-trivially on the number of spin components, and the
different components do not have the same wave function. For the special case
of a harmonic trap potential, we give explicit upper and lower bounds on the
critical coupling constant for symmetry breaking.
