Crossover scaling in a dilute bose superfluid near zero temperature

The behavior of a dilute Bose superfluid at low temperatures, $T<<T_{\lambda }$, is considered with emphasis on the superfluid density, ${\rho }_s\mathrm{}\left(T\right)\mathrm{}$. Many of the predictions of the Bogoliubov model are exact in this regime and are shown to lead to a scaling description of the crossover from ideal to interacting Bose gas behavior as the density is reduced. Combined with previous work on scaling near criticality, $T\simeq T_{\lambda }$, these results complete the crossover description of the dilute superfluid over the full temperature range. Diagrammatic perturbation theory in the superfluid phase is outlined and previous results from the helium literature are summarized. The various divergences encountered have previously been shown to arise from nonanalyticities in the self-energy functions at zero momentum, and these are related to the spin-wave nature of the superfluid phase. A full renormalization-group fixed-point picture is presented summarizing the zero-temperature and finite-temperature regimes and the flows connecting them.