We investigate the collective modes in the spin-orbit coupled Fermi gas with repulsive s-wave interaction. The interplay between spin-orbit coupling and atom-atom interactions plays the crucial role in the collective behaviors of Fermi gas. In contrast with ordinary Fermi liquid, spin-orbit coupled Fermi gas has strongly correlated spin and density excitations. Within the scheme of random phase approximation, we classify collective modes based on the symmetry group and determine their properties via the poles of corresponding correlation functions. Besides, the particle-hole continuum is obtained, where the imaginary part of these correlation functions become non-vanishing. We also propose an experimental protocol for detecting these collective modes and discuss corresponding experimental signatures in the ultracold Fermi gases experiment.

Magnetic phenomena in itinerant electron systems has been at the forefront in material science. Here we show that the Weyl spin-orbit couplings (SOC) in 3 dimensional repulsively interacting itinerant Fermi systems opens up a platform to host new itinerant magnetic phases, excitations and phase transitions. A putative Ferromagnetic state (FM) is always unstable against a stripe Spiral spin density wave (S-SDW) or a stripe Longitudinal-SDW at small or large SOC strengths respectively. The stripe ordering wavevector is given by the nesting momentum of the two SOC split Fermi surfaces with the same or opposite helicities at small or large SOC strengths respectively. The LSDW is accompanied by a charge density wave (CDW) with half of its pitch. The transition from the paramagnet to the SSDW or LSDW+CDW is described by quantum Lifshitz type actions, in sharp contrast to the Hertz-Millis types for itinerant electron systems without SOC. The collective excitations and FS re-constructions inside the SSDW and LSDW+CDW are also studied. The effects of a harmonic trap in cold atom experiments are briefly discussed. In view of recent groundbreaking experimental advances in generating 2d SOC in cold atoms, these phenomena can be observed in current or near future cold atom experiments even at very weak interactions. They may also be relevant to some itinerant magnetic materials with a strong SOC.