Coupled spin-charge dynamics in helical Fermi liquids beyond the random phase approximation

We consider a helical system of fermions with a generic spin- (or pseudo-spin-) orbit coupling. Using the equation of motion approach for the single-particle distribution functions and a mean-field decoupling of the higher-order distribution functions, we find a closed form for the charge- and spin-density fluctuations in terms of the charge- and spin-density linear-response functions. Approximating the nonlocal exchange term with a Hubbard-like local-field factor, we obtain coupled spin- and charge-density response matrices beyond the random phase approximation, whose poles give the dispersion of four collective spin-charge modes. We apply our generic technique to the well-explored two-dimensional system with Rashba spin-orbit coupling and illustrate how it gives results for the collective modes, Drude weight, and spin-Hall conductivity, which are in very good agreement with the results obtained from other more sophisticated approaches.