content_image/331531.jpg content_image/331530.jpg content_image/331529.jpg content_image/331528.jpg content_image/331527.jpg
The last term gives rise to asymmetric torques. Evaluating the trace in the spin space, the torque reads 

<img src='content_image/331531.jpg'>

Symmetric contribution (contribution from the E · A δ term) to the first line in the square bracket was calculated in Ref. [10] by use of 

<img src='content_image/331530.jpg'>

where n denotes the magnetization direction. ( The second and third terms in the square bracket in Eq. (D3) lead to a correction to the spin Berry phase (proportional to A $^{z}$) and the standard spin transfer torque (proportional to ∇ n $^{α}$), respectively. ) As for the asymmetric contribution (terms proportional to 1 $_{2}$E$_{ν}$ A δ ν in Eq. (D3)), summation over ν cannot taken independent of the electric field, resulting in complicated torques. We here look into only the correction to the β term (the first line in the square bracket), which is obtained as (neglecting irrelevant terms) 

<img src='content_image/331529.jpg'>

The result of the β term is thus 

<img src='content_image/331528.jpg'>

<img src='content_image/331527.jpg'>

( σ$_{B}$ is the Boltzmann conductivity) and δβ/β = − 1 / 5 in the present model. Thus the β term arising from the spin-orbit interaction has asymmetric contribution, in contrast to the 

