Density matrix of quantum fluids

Abstract

An elaborate cluster analysis of the single-particle occupation probability ${n}_{\stackrel{^}{q}}$ and associated one-body density matrix $n(r)$ is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that ${n}_{\stackrel{^}{q}}$ may be written as $n[N(q)+{N}_{1}(q)]$, where $n$ is a strength factor independent of wave number $q$ and the quantities $N(q)$ and ${N}_{1}(q)$ may be expressed as series of irreducible cluster contributions. The strength factor $n$ has the form $n={e}^{Q}$, where $Q$ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for $Q$ in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of ${n}_{\stackrel{^}{q}}$, it is further demonstrated that $n(r)$ may be cast in the form $\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)$, where $\ensuremath{\rho}$ is the particle density and the functions ${N}_{1}(r)$, ${N}_{2}(r)$, and $\mathcal{Q}(r)$ are all given by irreducible cluster series. Massive partial summations are executed in the $\mathcal{Q}(r)$ series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has $\mathcal{Q}(0)=Q$. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid $^{3}\mathrm{He}$ and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid $^{4}\mathrm{He}$ to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}$ being just the strength factor $n$. It may also be shown (under certain reasonable assumptions) that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlations implies a singular behavior ${n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}$ of the Bose momentum distribution for small $q$.