The result is which is Eq. (20). Since the summation over k and k is dominated by the imaginary part of the Green’s functions, we use g r k ,σ = − i 2 τ$_{σ}$ | g r k ,σ | $^{2}$+ o ( ℏ ϵ$_{F}$ $_{τ}$) and g r k $^{′}$, − $_{σ}$( g a k $^{′}$, − $_{σ}$) 2 = i 2 τ $^{−}$σ | g r k $^{′}$, − σ | $^{4}$+ o ( ℏ ϵ$_{F}$ $_{τ}$), and obtain the leading contribution as Here spin-dependent density of states and the Fermi wavelength are represented by ν$_{σ}$ and k$_{F σ}$ , respectively. Another contribution to the relaxation torque, T A,α so of Eq. (19), represented by the third diagram of Fig. 1, reads As is seen, T A,α so is an odd function of k or k $^{′}$, and thus it vanishes. The second diagram in Fig .1 (we denote it as δ T α $_{so}$) reads Here, the wave vectors of the two Green’s function, k and k $^{′}$, are independent, and thus the magnitude of δ T α so turns out to be smaller than T z so by the order of 1 / ( ϵ$_{F}$ τ ). We therefore see that the dominant contribution to the relaxation torque arises from the first process in Fig. 1, which is estimated to be Eq. (B9) (Eq. (20)). 18 δ T α so = − e ∫ dω 2 π ∫ d Ω 2 π ∑ kk ′ ∑ q Ω ∑ ijk ∑ mnβγ ϵ$_{ijk}$ϵ$_{αβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( k ′ − k − q ) v ( γ ) so ( k − k ′ + q ) 〉 i × A$_{l}$ ( q , Ω)( k ′ − k − q )$_{i}$( k ′ − k − q )$_{n}$( k + k $^{′}$)$_{j}$ tr[ σ$_{β}$g$_{k}$$_{,ω}$σ$_{γ}$g$_{k}$ $^{′}$,ω $_{+Ω}$] $^{<}$. (B11) T A,α so = ie ∫ dω 2 π ∫ d Ω 2 π ∑ kp ∑ q Ω ∑ ijk ∑ mnβγ ϵ$_{ijk}$ϵ$_{αβk}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( k ) so ( p ) v ( γ ) so ( − p ) 〉 i × A$_{l}$ ( q , Ω)( k − k $^{′}$)$_{i}$ ( k − k $^{′}$)$_{m}$ ( k + k $^{′}$)$_{n}$ tr[ σ$_{β}$g$_{k}$$_{,ω}$σ$_{γ}$g$_{k}$ ′ $_{,ω}$] < p ≡ k $^{′}$− $_{k}$. (B10) T z so = − 2 π 45 n$_{so}$λ$_{so}$ 2 e m 2 ∫ d Ω 2 π ∑ q e − i q · $^{x}$e i Ω $^{t}$Ω(3 q · A + q$_{z}$A$_{z}$ ) ν$_{+}$ν$_{−}$k 2 F $_{+}$k 2 F $_{−}$( k 2 F $_{+}$τ 2 + − k 2 F $_{−}$τ 2 $_{−}$) . (B9) T z so = 2 45 π n$_{so}$λ$_{so}$ 2 e m 2 ∫ d Ω 2 π ∑ kk ′ ∑ q e − i q · $^{x}$e i Ω $^{t}$Ω(3 q · A + q$_{z}$A$_{z}$ ) k $^{2}$( k $^{′}$) 4 ∑ σ σg r k $_{,σ}$g r k $^{′}$, − $_{σ}$( g a k $^{′}$, − $_{σ}$) 2 + c . c ., (B8)