The last diagram of Fig. 2 reads This contribution turns out to be smaller than Eq. (C11) by the order of ( ϵ$_{F}$ τ ) − $^{1}$. 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]) 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 〈 k$_{i}$k$_{j}$ [( k × k $^{′}$)$_{ν}$ ] $^{2}$〉 = 2 45 k $^{4}$( k $^{′}$) $^{2}$(2 δ$_{ij}$ − δ$_{iν}$δ$_{jν}$ ) . (D2) T α so = − e πm $_{2}$n$_{so}$ λ$_{so}$ 2 ∑ µνβγij ϵ$_{αµν}$R$_{µβ}$R$_{νγ}$E$_{i}$A δ j ∑ kk ′ k$_{i}$k$_{j}$ [( k × k $^{′}$)$_{ν}$ ] 2 × tr[( σ$_{β}$g r k ′ σ$_{γ}$ + σ$_{γ}$g a k ′ σ$_{β}$ )( | g r k | $^{2}$σ$_{δ}$ g a k + g r $_{k}$σ$_{δ}$ | g r k | $^{2}$)] , (D1) j (5) ,α s ,i = δ$_{αz}$ ie 2 π n$_{so}$λ$_{so}$ $^{2}$ϵ$_{ijz}$ ϵ$_{klz}$ ˙ A$_{l}$ ∑ kk ′ ( k − k $^{′}$)$_{j}$ ( k − k $^{′}$)$_{k}$ tr[ g r $_{k}$σ$_{z}$ g a k ′ ] . (C12) Appendix D: Calculation of asymmetric β torque 22