Two-body density matrix of a normal Fermi fluid

Abstract

The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}^{\ensuremath{'}}$) initiated for uniform Bose fluids in an earlier paper is continued for the Fermi case. We present formal results on the structure of the generalized momentum distribution n(p,q)=${\mathrm{\ensuremath{\Sigma}}}_{\mathbf{k}\mathbf{\^{}}}$〈\ensuremath{\Psi}|${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}+\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}\mathrm{\ensuremath{-}}\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}}$${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}}$|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}$)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), based on a variational ground-state wave function of Jastrow-Slater form. The structural relations are inferred from the cluster expansions of these objects, from the asymptotic condition relating ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), and from formal diagrammatic connections with the Bose problem. The two-body density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) are thereby expressed in closed form in terms of certain sums of irreducible cluster diagrams. Some of these diagram sums are familiar from the analogous theory of the one-body density matrix; all can be evaluated quantitatively by solving a set of Fermi-hypernetted-chain (FHNC) equations. Upon invoking the sequential relation between ${\mathrm{\ensuremath{\rho}}}_{2}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{2}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(${\mathbf{r}}_{1}$,${\mathbf{r}}_{1}^{\ensuremath{'}}$), the corresponding result for the generalized momentum distribution n(p,q) effects a resolution into contributions from various scattering processes occurring in the many-body medium, specified by form factors that are susceptible to FHNC evaluation. This decomposition is comparable to that derived earlier for the Bose-fluid ground state but is complicated by contributions from exchange scattering and by a dynamically dressed Pauli kinematic correction. Silver has proposed a simple expression for the generalized momentum distribution n(p,q), a function which plays an essential role in his theory of final-state effects in deep-inelastic neutron scattering from the helium liquids. Based on the present microscopic treatment, the quality of Silver's estimate is assessed for the case of normal liquid ${\mathrm{He}}^{3}$, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.