  It is widely believed that mean-field theory is exact for a wide-range of
classical long-range interacting systems. Is this also true once quantum
fluctuations have been accounted for? As a test case we study the Hamiltonian
Mean Field (HMF) model for a system of indistinguishable bosons which is
predicted (according to mean-field theory) to undergo a second-order quantum
phase transition at zero temperature. The ordered phase is characterized by a
spontaneously broken $O(2)$ symmetry, which, despite occurring in a
one-dimensional model, is not ruled out by the Mermin-Wagner theorem due to the
presence of long-range interactions. Nevertheless, a spontaneously broken
symmetry implies gapless Goldstone modes whose large fluctuations can restore
broken symmetries. In this work, we study the influence of quantum fluctuations
by projecting the Hamiltonian onto the continuous subspace of symmetry breaking
mean-field states. We find that the energetic cost of gradients in the center
of mass wavefunction inhibit the breaking of the $O(2)$ symmetry, but that the
energetic cost is very small --- scaling as $\mathcal{O}(1/N^2)$. Nevertheless,
for any finite $N$, no matter how large, this implies that the ground state has
a restored $O(2)$ symmetry. Implications for the finite temperature phases, and
classical limit, of the HMF model are discussed.
