Hypernetted-chain theory of the momentum distribution for Bose systems with mixture formalism

Abstract

The hypernetted-chain equations for the momentum distribution $n(k)$ of the $N$-particle Bose system are derived by relating its Fourier transform, the one-particle density matrix $n(r)$, to one of the three radial distribution functions, ${g}_{\ensuremath{\beta}\ensuremath{\beta}}(r)$, in a formally equivalent system of a two-component mixture of ${N}_{\ensuremath{\beta}}=2$ and ${N}_{\ensuremath{\alpha}}=N\ensuremath{-}1$ particles in the zero-concentration limit ${\ensuremath{\rho}}_{\ensuremath{\beta}}\ensuremath{\approx}0$. The Bose-Einstein condensate fraction is shown to be proportional to the inverse of ${g}_{\ensuremath{\beta}\ensuremath{\beta}}(r)$ at the origin, ${n}_{c}={g}_{\ensuremath{\beta}\ensuremath{\beta}}^{\ensuremath{-}1}(0)$. When the theory is used to calculate $n(k)$ in the ground state of liquid $^{4}\mathrm{He}$, it is found that the kinetic-energy sum rule required to be satisfied by $n(k)$ is strongly violated unless the elementary diagrams are taken into account. This is done using a scaling approximation for the lowest-order elementary diagrams by determining scaling factors so that different procedures for calculating the kinetic energy give the same result.