Spin accumulation without spin current

The spin Hall (SH) effect is a phenomenon in which the current flows perpendicular to an applied electric field and causes accumulation at boundaries. However, presence of spin-orbit couplings, not well defined. Here, we calculate response gradient, naturally appears We derive generic formula using Bloch wave functions phenomenological relaxation time. also for Rashba model with $\delta$-function nonmagnetic disorder within first-order Born approximation corresponding vertex corrections. find nonzero accumulation, although SH conductivity exactly vanishes.