Relativistic response and novel spin-charge plasmon at the Tl/Si(111) surface

We present a comprehensive \textit{ab initio} analysis of the spin-charge correlations at the Tl/Si(111) surface, where the spin-orbit interaction is so strong that a detailed treatment of the non-collinear electron spin appears decisive for the correct description of the response properties. The relativistic limit enforces a unified treatment of the spin and charge densities as a four-vector, and the response function acquires then a 4$\times$4 tensor structure. Our all-electron implementation allows to resolve the real space structure of the possible collective modes, and demonstrates the emergence of a novel collective excitation combining transverse-spin and ordinary charge oscillations of a similar order of magnitude, whose spin character is strongly enhanced as we approach the $q$$\rightarrow$0 momentum limit.