Renormalization-group analysis of the ground-state properties of dilute Bose systems in d spatial dimensions

A low-density system of Bose particles of mass m and density n interacting through a short-range potential with range a is considered in d dimensions using a renormalization approach. The expansion parameter of the ground-state energy E per particle for d<2 is (${\mathit{na}}^{\mathit{d}}$${)}^{\left(2\mathrm{}\mathrm{-}\mathit{d}\right)/\mathit{d}}$. For some special cases the leading universal term of the expansion E=${\mathit{C}}_{\mathit{d}}$${\mathrm{ħ}}^2$${\mathit{n}}^{2/\mathit{d}}$/m+..., where ${\mathit{C}}_{\mathit{d}}$ is a universal number, gives an exact expression for the ground-state energy. For d>2 the energy expansion scales as E=(1/2)nt[1+const(${\mathit{nf}}^{\mathit{d}}$${)}^{\left(\mathit{d}\mathrm{-}2\right)/2}$], where t is the pseudopotential connected to the bare potential and f is the corresponding scattering length. This is a generalization of the generally accepted results (for d=3) of Lee and Yang [Phys. Rev. 105, 1119 (1957)], but a stronger condition for validity is found: ${\mathit{nf}}^{\mathit{d}}$≪(f/a${)}^2$≤1. The possibility of the formation of quantum crystal phases is considered. We argue that for d<2, crystal phases and melting transitions in the traditional sense are impossible. For d>2 the liquid phase is stable relative to the solid one. The corresponding energy difference diminishes for d→2+0, and for d=2 it is much less than was suggested by Nelson [J. Stat. Phys. 57, 511 (1989)]; here the liquid phase is stable only in the extremely dilute regime. For d>2, attractive Bose systems exhibit a second-order phase transition connected to the formation of the bound state. For d≤2 only the bound state can occur.