Optimal Jastrow correlations for Fermi liquids: Application to liquidHe3

Abstract

A trial wave function of the Jastrow product form is used to describe the ground state of a normal Fermi liquid. The Euler-Lagrange equation is obtained which determines the optimal pair-correlation function from the space of Jastrow functions. The way in which the Pauli exclusion principle determines the long-range behavior of the correlation function is discussed. A form of the hypernetted-chain approximation which allows correctly for the exclusion principle is described and introduced into the equations for the purposes of numerical solution. For liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ the Euler-Lagrange equations are shown to have solutions in two distinct regions: a low-density region which is dominated by the Fermi statistics and an intermediate-density region which is dominated by the interparticle interaction. The low-density region may be relevant to the study of dilute mixtures of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The static structure function at the experimental equilibrium density is in good agreement with the x-ray-scattering data and with the predictions of Landau theory. The theoretical static structure function has a slight shoulder for $k<\mathrm{}0.6\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$. The spin-dependent part of the static structure function is also evaluated and compared with Landau theory. Finally, it is argued that the Jastrow trial wave function does not give a good description of the (virtual) single-particle excited states, and an improved trial wave function is discussed which should permit the evaluation of Landau parameters.