therefore obtain Since we consider a slowly varying gauge field, it is sufficient for our purpose to estimate this expression at q = 0 in the Green’s functions (the adiabatic limit). The result is which turns out after k -summation to be negligibly small (smaller by a factor of 1 / ( ϵ$_{F}$ τ ) than the dominant contribution). We therefore obtain Eq. (24) as the dominant normal spin current at small Ω and q . The contribution from the third diagram in Fig. 2, which arises from the normal current and the spin-orbit interaction including the gauge field, reads Taking the lesser component, the dominant contribution at small Ω and q turns out to be 20 j (3) ,α s ,i = − ie m ∫ dω 2 π ∫ d Ω 2 π ∑ kk ′ ∑ q ϵ$_{jkβ}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( β ) so ( p ) v ( γ ) so ( − p ) 〉 i A$_{m}$ ( q , Ω) k$_{i}$ ( k − k $^{′}$)$_{n}$ × [ ( k − q 2 ) j ( k ′ − q 2 ) k tr[ σ $^{α}$g$_{k}$$_{−}$ q $_{$_{2}$,ω}$σ$_{β}$ g$_{k}$ $^{′}$− q $_{$_{2}$,ω}$σ$_{γ}$ g$_{k}$$_{+}$ q $_{2}$,ω $_{+Ω}$] + k + q 2 j k ′ + q 2 k tr[ σ $^{α}$g$_{k}$$_{−}$ q $_{$_{2}$,ω}$σ$_{γ}$ g$_{k}$ $^{′}$+ q $_{2}$,ω $_{+Ω}$σ$_{β}$ g$_{k}$$_{+}$ q $_{2}$,ω $_{+Ω}$] ] < p = k $^{′}$− k . (C6) j (3) ,α s ,i = ie m ∫ dω 2 π ∫ d Ω 2 π ∑ kk ′ ∑ q ϵ$_{jkβ}$ϵ$_{mnγ}$e − i q · $^{x}$e i Ω t 〈 v ( β ) so ( k ′ − k ) v ( γ ) so ( k − k $^{′}$) 〉 i f $^{′}$( ω )Ω A$_{m}$ ( q , Ω) k$_{i}$k$_{j}$k ′ $_{k}$k ′ n × tr[ σ $^{α}$g r $_{k}$( σ$_{β}$g r k ′ σ$_{γ}$ + σ$_{γ}$g a k ′ σ$_{β}$ ) g a $_{k}$] . (C7) δj (0) ,z s ,i + j A,α s ,i = i e m 2 ∑ kq e − i q · x ∫ d Ω 2 π ∫ dω 2 π e i Ω t ∑ j k$_{i}$k$_{j}$A$_{j}$ ( q , Ω) ∑ σ σ × ( − Ω 2 f $^{′}$( ω ) ) [( g a k $_{,ω,σ}$) 2 + ( g r k $_{,ω,σ}$) $^{2}$] , (C5) δj (0) ,z s ,i + j A,α s ,i = i e m 2 ∑ kq e − i q · x ∫ d Ω 2 π ∫ dω 2 π e i Ω t ∑ j k$_{i}$k$_{j}$A$_{j}$ ( q , Ω) ∑ σ σ × [ − Ω 2 f $^{′}$( ω ) [ g a k − q $_{$_{2}$,ω,σ}$g a k + q $_{2}$,ω,σ + g r k − q $_{$_{2}$,ω,σ}$g r k + q $_{$_{2}$,ω,σ}$] + Ω 2 f ( ω ) [( g a k − q $_{$_{2}$,ω,σ}$) $^{2}$ga k + q $_{2}$,ω,σ − g a k − q $_{$_{2}$,ω,σ}$( g a k + q $_{$_{2}$,ω,σ}$) 2 − ( g r k − q $_{$_{2}$,ω,σ}$) $^{2}$gr k + q $_{2}$,ω,σ + g r k − q $_{$_{2}$,ω,σ}$( g r k + q $_{$_{2}$,ω,σ}$) $^{2}$] ] + O ( q ) . (C4) 2. Calculation of the third diagram in Fig. 2