content_image/347227.jpg content_image/347228.jpg content_image/347226.jpg
2 and 3, which correspond to the field-induced contribution and the effect of the diffusive electron motion, respectively. 

<img src='content_image/347227.jpg'>

<img src='content_image/347228.jpg'>

We first estimate the normal part of the spin current, j (n) ,z s ,i (Eq. (15)), shown in the first two diagrams in Fig. 2. The contribution j (n) ,z s ,i is defined including the anomalous contribution from the electromagnetic gauge field, j A,z s ,i . Its dominant contribution in the limit of small Ω and small q is calculated as (see Appendix C for detail) 

<img src='content_image/347226.jpg'>

In Eq. (24), the first term is the contribution shown in the left of Fig. 2, and the second term is the contribution from the vertex correction (Fig. 3). 

