j (0) ,z 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 × [( f ( ω + Ω 2 ) − f ( ω − Ω 2 )) g r k − q $_{2}$,ω − Ω 2 $_{,σ}$g a k + q $_{2}$,ω + Ω 2 ,σ + f ( ω − Ω 2 ) g a k − q 2 ,ω − Ω 2 $_{,σ}$g a k + q $_{2}$,ω + Ω 2 ,σ − f ( ω + Ω 2 ) g r k − q 2 ,ω − Ω 2 $_{,σ}$g r k + q 2 ,ω + Ω 2 ,σ ] (C1) δj (0) ,z 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 × [ 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}$,ω,σ}$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}$,ω,σ}$) $^{2}$] ] . (C2) j A,α s ,i ( x , t ) = iδ$_{α,z}$ e m A$_{i}$ ( x , t ) ∫ dω 2 π ∑ k σ σg < k ωσ = iδ$_{α,z}$ e m A$_{i}$ ( x , t ) dω 2 π k σ σf ( ω )( g a k ωσ − g a k $_{ωσ}$) (C3) 19 Here we show first the derivation of the normal spin current without the vertex correction, i.e., the first term of Eq. (24). The leading contribution of Eq. (15) arises from the first diagram of Fig. 2. It reads Here the effect of spin-orbit interaction is included in the lifetime (selfenergy) of the Green’s functions. Estimating j (0) ,z s ,i to linear order in Ω and using f $^{′}$( ω ) = − δ ( ω ) at low tempera- tures, we see obviously that the first contribution in the square bracket reduces to the first term of Eq. (24). The remaining contribution, which we call δj (0) ,z s ,i , is calculated to linear order in Ω as On the other hand, the contribution from the gauge field, j A,α s ,i in Eq. (15), reads This contribution is proportional to A , and so is of the order of E / Ω. One can easily see that this contribution cancels with the first term of Eq. (C2) estimated at q = 0. We Appendix C: Calculation of the spin current 1. Normal spin current