content_image/371907.jpg content_image/371906.jpg content_image/371905.jpg
The last diagram of Fig. 2 reads 

<img src='content_image/371907.jpg'>

This contribution turns out to be smaller than Eq. (C11) by the order of ( ϵ$_{F}$ τ ) − $^{1}$. 

## Appendix D: Calculation of asymmetric β torque 

The torque generated by the spin-orbit interaction and uniform electric field when the spin structure is inhomogeneous was calculated in Ref. [10]. There, the angular averaging over the electron wave vectors was carried out assuming rotational symmetry on averaging the spin-orbit interaction, resulting in a symmetric torque called the β term. Here we carry out the calculation without such approximation, and show that asymmetric β term then arises. The torque reads (see Eqs. (19)(21)(35) of Ref. [10]) 

<img src='content_image/371906.jpg'>

where R$_{µν}$ ≡ 2 m$_{µ}$m$_{ν}$ − δ$_{µν}$ denotes a rotation matrix for the gauge transformation and A δ j ≡ ( m × ∇ $_{j}$m ) δ is the spin gauge field ( m ≡ (sin θ 2 cos φ, sin θ 2 sin φ, cos θ $_{2}$), where θ and φ are polar coordinates representing the magnetization direction) [10]. In Ref. [10], the term [( k × k $^{′}$)$_{ν}$ ] 2 arising from the spin-orbit interaction was averaged independent of the other factor, k$_{i}$k$_{j}$ , and thus the torque was proportional to 〈 k$_{i}$k$_{j}$ 〉 = k 2 3 δ$_{ij}$ (Eq. (37) of Ref. [10]). Here we carry out the averaging without assuming special symmetry. The averaging over the wave vectors in Eq. (D1) then reads 

<img src='content_image/371905.jpg'>

