content_image/345212.jpg content_image/345211.jpg content_image/345209.jpg content_image/345210.jpg
therefore obtain 

<img src='content_image/345212.jpg'>

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 

<img src='content_image/345211.jpg'>

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 . 

2. Calculation of the third diagram in Fig. 2 

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 

<img src='content_image/345209.jpg'>

Taking the lesser component, the dominant contribution at small Ω and q turns out to be 

<img src='content_image/345210.jpg'>

