  It has recently been suggested that black holes may be described as
condensates of weakly interacting gravitons at a critical point, exhibiting
strong quantum effects. In this paper, we study a model system of attractive
bosons in one spatial dimension which is known to undergo a quantum phase
transition. We demonstrate explicitly that indeed quantum effects are important
at the critical point, even if the number of particles is macroscopic. Most
prominently, we evaluate the entropy of entanglement between different momentum
modes and observe it to become maximal at the critical point. Furthermore, we
explicitly see that the leading entanglement is between long wavelength modes
and is hence a feature independent of ultraviolet physics. If applicable to
black holes, our findings substantiate the conjectured breakdown of
semiclassical physics even for large black holes. This can resolve long
standing mysteries, such as the information paradox and the no-hair theorem.
