Dilute Bose gas in two dimensions

The weakly interacting Bose gas in two dimensions is considered in the dilute limit $n^{1/2}$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 has 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 quasiparticle picture, and also using the renormalization group. It is shown that the superfluid transition temperature is $T_c$≊4π(${ħ}^2$/2m)n/[ln ln(1/${\mathrm{na}}^2$)], and the condition of validity of the dilute limit is ln ln(1/${\mathrm{na}}^2$)≫1. The connection to the dilute Bose gas in dimensions d>2 and the universal behavior beyond the extreme asymptotic low-density domain are also discussed.