Spin-valley Silin modes in graphene with substrate-induced spin-orbit coupling

In the presence of an external magnetic field the Fermi-liquid state supports oscillatory spin modes known as Silin modes. We predict the existence of the generalized Silin modes in a multivalley system, monolayer graphene. Along with the spin, the generalized Silin modes may involve the valley degree of freedom.  A gauge- and Berry-gauge- invariant kinetic equation for a multivalley Fermi liquid is developed and applied to the case of graphene with extrinsic spin-orbit coupling (SOC). The interplay of SOC and Berry curvature allows for the excitation of generalized Silin modes in the spin and valley-staggered-spin channels via an AC electric field. The resonant contributions from these modes to the optical conductivity are calculated.