  Motivated by recent experiments on Bose-Einstein condensed atoms which rotate
in annular/toroidal traps we study the effect of the finiteness of the atom
number $N$ on the states of lowest energy for a fixed expectation value of the
angular momentum, under periodic boundary conditions. To attack this problem,
we develop a general strategy, considering a linear superposition of the
eigenstates of the many-body Hamiltonian, with amplitudes that we extract from
the mean field approximation. This many-body state breaks the symmetry of the
Hamiltonian, it has the same energy to leading order in $N$ as the mean-field
state and the corresponding eigenstate of the Hamiltonian, however it has a
lower energy to subleading order in $N$ and thus it is energetically favorable.
