Density matrix and momentum distribution of helium liquids and nuclear matter

The one-body density matrix $n\left(r\right)\mathrm{}$ and the associated momentum distribution $n\left(k\right)\mathrm{}$ (for given spin-isospin projection) are calculated for the ground states of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, and a simple model of nuclear matter, based on suitable Jastrow wave functions $\psi ={\Pi }_{i<j}f\left(r_0\right)\Phi$. The model function $\Phi$ is taken as the ground-state wave function of a system of noninteracting bosons or fermions, as appropriate. The evaluations are performed within the framework of the diagrammatic formalism developed recently by Ristig and Clark. For the most part, the correlations incorporated via the two-body factor $f\left(r\right)\mathrm{}$ are assumed to be of short range. In the case of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, the asymptotic value ${\rho }^{-1\mathrm{}}n(r\to \infty )$, which may be identified with the condensate fraction, is found to be 0.119 at the experimental equilibrium value of the particle density $\rho$. In the cases of ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and nuclear matter, there is no long-range order, i.e., $n(r\to \infty )=0\mathrm{}$. On the other hand, for these Fermi systems the momentum distribution $n\left(k\right)\mathrm{}$ exhibits a discontinuity $z_{k_F}$ at the Fermi wave number $k_F$, which measures the strength of the quasiparticle pole. The calculation yields $z_{k_F}=0.350\mathrm{}$ for liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and $z_{k_F}=0.747\mathrm{}$ for nuclear matter, at the respective equilibrium densities. The results presented for the helium liquids are of considerable interest in connection with proposed scattering experiments using intense neutron beams.