Pairing function for Bose fluids

Abstract

An elaborate cluster analysis of the pair occupancy ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ measures the simultaneous occupation of single-particle orbitals $\stackrel{^}{q}$, $\stackrel{^}{p}$ and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion ${\ensuremath{\chi}}_{q}^{2}$ for pairs of particles with equal and opposite momenta, $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\ensuremath{\chi}(r)$ which is defined as the Fourier inverse of quantity ${\ensuremath{\chi}}_{q}$ are displayed. It is further demonstrated that a close kinship exists between the function $\ensuremath{\chi}(r)$ and the one-particle density matrix $n(r)$. It is also shown that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlation factor implies a singular behavior $\ensuremath{\sim}{q}^{\ensuremath{-}1}$ of the function ${\ensuremath{\chi}}_{q}$ for small momenta $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$. The pairing function $\ensuremath{\chi}(r)$ is calculated numerically for the ground state of liquid $^{4}\mathrm{He}$ at equilibrium density adopting the familiar short-ranged correlation factor $f(r)$ employed by Schiff and Verlet and others.