Second order quantum phase transition of a homogeneous Bose gas with attractive interactions

We consider a homogeneous Bose gas of particles with an attractive interaction. Mean field theory predicts for this system a spontaneous symmetry breaking at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze {\it quantum fluctuations} in the system even when the Bogoliubov approach breaks down.