Polarization dependence of Landau parameters for normal Fermi liquids in two-dimensions

Landau's formulation of his normal Fermi liquid theory was a key development in condensed matter and nuclear physics permitting one to describe the behavior and properties of a class of strongly interacting fermions with just a handful of microscopic parameters. We shall examine an application of normal Fermi liquid theory to two-dimensional fermion systems with a finite polarization. Two dimensional systems have practical importance as thin films, devices, and possibly models for certain high T$_{c}$ superconductors. Two dimensions offers one important theoretical advantage: the constant density of states permits calculations to be done exactly which in three dimensions would not be possible. In a low density approach we shall calculate exact, analytic expressions to quadratic order in the s-wave and p-wave interaction parameters for the basic Landau parameters, $f^{\uparrow \uparrow},f^{\uparrow \downarrow },f^{\downarrow \downarrow }$. This will enable us to study the polarization dependence of the state-dependent effective masses, the spin susceptibility, the compressibility, zero sound and spin-zero sound. Application of these results is made by studying and predicting the polarization properties of thin ${ }^3$He films.