Pomeranchuk instabilities of Fermi fluids in the spin channel

We study the Pomeranchuk instabilities of the Fermi surface in the spin channel. It is shown that the instabilities will lead to two classes of the ordered phases, the $\alpha$ and $\beta$-phases, named by analogy to the superfluid $^3$He-A and B-phases. The Fermi surfaces in the $\alpha$-phases exhibit spontaneous anisotropic distortions, while those in the $\beta$-phases remain circular with non-trivial spin configurations in momentum space. The low energy excitations of the ordered phases are studied by RPA approximation. In the $\alpha$-phases, the density excitations exhibit anisotropic overdamping and the spin density excitations are nearly isotropic and underdamped at small propagating wavevectors. The $\beta$-phases shows a Lifshitz- like instability in the $p$-wave channel, and will stabilize a chiral ground state inhomogeneity.