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) (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 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). In this section, we show details of calculation of spin relaxation torque. The first contri- bution in Fig. 1 reads 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 16 Appendix B: Calculation of spin relaxation torque T z so = i e m ∫ dω 2 π ∫ d Ω 2 π ∑ kk ′ ∑ q ∑ ijkl ∑ mnβγ ϵ$_{ijk}$ϵ$_{zβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k − k $^{′}$) v ( γ ) so ( k ′ − k ) 〉 i × A$_{l}$ ( k − k $^{′}$)$_{i}$ ( k + k ′ + q )$_{j}$ k ′ $_{m}$k$_{n}$ ( k ′ + q 2 ) $_{l}$tr[ σ$_{β}$g$_{k}$$_{,ω}$σ$_{γ}$g$_{k}$ ′ $_{,ω}$g$_{k}$ $^{′}$+ q ,ω $_{+Ω}$] < + c . c . (B1) ∂$_{t}$s α ≡ −∇ · ( j (n) ,α s + j so ,α s ) + T α so + T A,α so , (A7) ˆ j (n) ,α s ,i ≡ − i 2 m c $^{†}$σ α ↔ ∇ i c − e m A$_{i}$c $^{†}$σ$_{α}$ c ˆ j so ,α s ,i ≡ − ∑ j ϵ$_{ijα}$ ( ∇ $_{j}$v ( α ) so ) c $^{†}$c. (A6) ˆ j α s ≡ ˆ j (n) ,α s + ˆ j so ,α s , (A5) ∂$_{t}$ ˆ s α = i 2 m ∇ · ( c $^{†}$σ$_{α}$ ↔ ∇ c ) − ∑ ij ϵ$_{ijα}$ ( ∇ $_{i}$v ( α ) so ) ∇ $_{j}$( c $^{†}$c ) + e m ∑ i A$_{i}$ ∇ $_{i}$( c $^{†}$σ$_{α}$ c ) + i ∑ ijkl ϵ$_{ijk}$ϵ$_{kαl}$ ( ∇ $_{i}$v ( k ) so )( c $^{†}$σ$_{l}$ ↔ ∇ j c ) + 2 e ∑ ijkl ϵ$_{ijk}$ϵ$_{kαl}$ ( ∇ $_{i}$v ( k ) so ) A$_{j}$ ( c $^{†}$σ$_{l}$ c ) . (A4)