Normal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$: an almost localized Fermi liquid

The Hubbard model is used to calculate static properties of normal-liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ at $T=0\mathrm{}$. For this, Gutzwiller's variational approach to that model is employed. The work is based on an observation by Anderson and Brinkman that the results of this method, obtained by Brinkman and Rice for the metal-insulator transition in the case of one particle per site, appear to be in qualitative agreement with the experimentally measured properties of that liquid. In this sense normal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ can be understood to be close to a localization transition of the particles where their effective mass diverges. The incipient localization is found to determine the properties of that liquid. Hence ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is "almost localized" rather than "almost ferromagnetic," as often claimed by paramagnon theory. The author further investigates this motion. Discussing Gutzwiller's approach to the Hubbard model, he shows that it is well suited for a description of a liquid system like ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The approach and its physical implications are investigated by means of the reformulation of the solution due to Ogawa et al. It is shown explicitly that Gutzwiller's results can be placed into the concepts of Landau-Fermi-liquid theory and that within this model the Landau parameters $F_0^s$ and $F_0^a$ are related. Furthermore, the author identifies two different kinds of spin-fluctuation processes inherent to the model, one of which is shown to be responsible for the largeness of $F_0^s$. Going beyond these qualitative aspects, the author evaluates $F_0^a$ and $F_0^s$ quantitatively, finding that $F_0^a$ agrees very well with the experimentally determined values at all pressures, with $F_0^a\to -\frac{3}{4}p$ at high pressures, where $p$ is always close to unity. Hence the system is never close to a ferromagnetic transition. By means of the forward scattering sum rule for $l<\mathrm{}2\mathrm{}$ an analytic expression for $F_1^a$ is obtained. Finally, the author extends the analysis to large magnetic fields, finding that in the case of normal ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ the magnetization increases very rapidly with the magnetic field. This is due to the large zero-field effective mass. There is a line of critical values for the interaction and the magnetic field where a fully magnetized state is formed via a first-order transition. Calculating the drop in melting pressure due to the magnetic field, the author finds that it essentially removes the minimum in the melting curve. Thus the melting pressure even of fully polarized ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ is larger than zero, in agreement with arguments by Castaing and Nozières.