The Bogoliubov Inequality and the Nature of Bose-Einstein Condensates for Interacting Atoms in Spatial Dimensions D = 2

We consider the restriction placed by the Bogoliubov inequality on the nature of the BoseEinstein condensates (BECs) for interacting atoms in a spatial dimension D = 2 and in the presenceof an external arbitrary potential, which may be a confining “box”, a periodic, or a disorderedpotential. The atom-atom interaction gives rise to a (gauge invariance) symmetry-breaking term thatplaces further restrictions on BECs in the form of a consistency proviso. The necessary condition forthe existence of a BEC in D = 2 in all cases is macroscopic occupation of many single-particlemomenta states with the origin a limit point (or accumulation point) of condensates. It is shown thatthe nature of BECs for noninteracting atoms in a disordered potential is precisely the same as that ofBECs for interacting atoms in the absence of an external potential