content_image/372114.jpg content_image/372112.jpg content_image/372113.jpg content_image/372111.jpg content_image/372110.jpg
The commutator for the creation operator is given by [ H, c $^{†}$] = − [ H, c ] $^{†}$. Equation (13) in the operator form thus reads (ˆ s α ≡ c $^{†}$σ$_{α}$ c is the electron spin operator) 

<img src='content_image/372114.jpg'>

(The contribution from the impurity scattering, H$_{imp}$ , vanishes.) The first three terms on the right-hand side of Eq. (A4) are written as divergence (choosing the electromagnetic vector potential as divergenceless, ∇ · A = 0). From Eq. A4, it is useful to define the spin current operators as 

where 

<img src='content_image/372112.jpg'>

<img src='content_image/372113.jpg'>

Taking the average, the equation of motion for the spin density operator, Eq. (A4), results in the spin continuity equation, 

where each terms are given by Eqs. (15)(18). 

## Appendix B: Calculation of spin relaxation torque 

<img src='content_image/372111.jpg'>

In this section, we show details of calculation of spin relaxation torque. The first contri- bution in Fig. 1 reads 

<img src='content_image/372110.jpg'>

Here g$_{k}$$_{ω}$ represents the contour ordered electron Green’s function with the wave vector k and the frequency ω ( g$_{k}$$_{ω}$ is a 2 × 2 diagonal matrix in the spin space), tr is the trace over 

