Rigorous upper bound on the critical temperature of dilute Bose gases

We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a2ρ is small and the temperature T satisfies T>4πρ/ln|ln(a2ρ)|. Here, a is the scattering length of the repulsive interaction potential and ρ is the density. To the leading order in a2ρ, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when T−Tc(0)/Tc(0)>5√aρ1/3, where Tc(0) is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation