The dilute Bose gas in two dimensions.

The weakly interacting Bose gas in two dimensions is considered in the dilute limit n sup 1/2 a 1, where n is the particle density and a is the range of the potential. The standard many-body perturbation theory for this system suffers from two separate divergences: the first, associated with classical phase fluctuations is responsible for the vanishing of the long-range order; the second is quantum mechanical and is connected with the vanishing of the scattering t-matrix at long wavelengths and low energies. An earlier diagrammatic theory of Popov, which provides a consistent description of the system in the dilute limit, is rederived heuristically from a quasi-particle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is T sub c ~ 4pi (h sup 2 /2m)n/lnln(1/na sup 2), and the condition of validity of the dilute limit is lnln(1/na sup 2)>>1.