Features of Fermi Systems near $\ell$=0 Pomeranchuk Instabilities: A Crossing Symmetric Approach

In Fermi systems, interactions can cause symmetry-breaking deformations of the Fermi surface, called Pomeranchuk instabilities. In Fermi liquid (FL) language, this occurs when one of the Landau harmonics F$_{\ell}^{a,s}$ $\rightarrow$ -(2$\ell$ + 1); e.g. F$_{0}^{a,s}$ = -1 are related to ferromagnetic (a), and density instabilities (s) resepctively. The corresponding point in parameter space may be viewed as a quantum critical point (QCP). Using graphical and numerical methods to solve coupled non-linear integral equations of a crossing symmetric equation (TSCE) scheme, we study the behavior of spin/density excitations; effective mass; ferromagnetic, spin density wave, phase separation, and pairing transitions near $\ell$=0 Pomeranchuk instabilities in a 3D Fermi system. Considering momentum dependence of the renormalized FL interactions, we find a number of results for repulsive and attractive couplings of arbitrary strengths; viz. attraction in both singlet and triplet pairing amplitudes (though singlet pairing is primarily favored); possibility of a second ferromagnetic transition due to spin waves, and possibility of phase separation with and without ferromagnetic transition. Some of our results may apply to ferromagnetic superconductors, such as UGe$_{2}$ and UIr.