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 dimensi
on 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.
