Pairing function for Bose fluids

Abstract

An elaborate cluster analysis of the pair occupancy $P_{\hat{q}\hat{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity $P_{\hat{q}\hat{p}}$ measures the simultaneous occupation of single-particle orbitals $\hat{q}$, $\hat{p}$ 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 ${\chi }_q^2$ for pairs of particles with equal and opposite momenta, $\hslash \overrightarrow{\mathrm{q}}+\hslash \overrightarrow{\mathrm{p}}=0\mathrm{}$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\chi \mathrm{}\left(r\right)\mathrm{}$ which is defined as the Fourier inverse of quantity ${\chi }_q$ are displayed. It is further demonstrated that a close kinship exists between the function $\chi \mathrm{}\left(r\right)\mathrm{}$ and the one-particle density matrix $n\left(r\right)\mathrm{}$. It is also shown that the customary $r^{-2\mathrm{}}$ long-range behavior of the two-body correlation factor implies a singular behavior $\sim q^{-1\mathrm{}}$ of the function ${\chi }_q$ for small momenta $\hslash \overrightarrow{\mathrm{q}}$. The pairing function $\chi \mathrm{}\left(r\right)\mathrm{}$ is calculated numerically for the ground state of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at equilibrium density adopting the familiar short-ranged correlation factor $f\left(r\right)\mathrm{}$ employed by Schiff and Verlet and others.