Density matrix and spin-dependent correlations of normal liquidHe3

The influence of spin-dependent correlations on the one-body density matrix of a Fermi system is explored, for a normal-state wave function in which differing Jastrow factors are assigned to pairs in spin-projection states ↑↑ and ↑↓. The structural results of Ristig and Clark for state-independent Jastrow correlations are generalized by appealing to the commutativity of the assumed correlation operators and the topological properties of the diagrammatic representations of cluster expansions of the density matrix and corresponding occupation probability. Application to liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ (or the electron gas) calls for suitable spin-dependent spatial distribution functions. Such inputs may be supplied by Fermi hypernetted-chain theory adapted to ${\sigma }_z$- dependent correlations. The required extension is carried out for the Krotscheck-Ristig version of the theory, resulting in two coupled nonlinear integral equations for theoretical determination of the experimentally accessible spin-dependent radial distribution functions $g^{\uparrow \uparrow }\mathrm{}\left(r\right)\mathrm{}$ and $g^{\uparrow \downarrow }\mathrm{}\left(r\right)\mathrm{}$. Preliminary to a full implementation of this approach for liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, the spin-dependent structure functions associated with a Jastrow factor of Schiff-Verlet type are determined.