Spin-hall conductivity in electron-phonon coupled systems

We derive the ac spin-Hall conductivity sigma(sH)(omega) of two-dimensional spin-orbit coupled systems interacting with dispersionless phonons of frequency omega(0). For the linear Rashba model, we show that the electron-phonon contribution to the spin-vertex corrections breaks the universality of sigma(sH)(omega) at low frequencies and provides a nontrivial renormalization of the interband resonance. On the contrary, in a generalized Rashba model for which the spin-vertex contributions are absent, the coupling to the phonons enters only through the self-energy, leaving the low-frequency behavior of sigma(sH)(omega) unaffected by the electron-phonon interaction.