The relaxation torque reads where The spin relaxation torque depends on the definition of the spin current. For instance, if we redefine the spin current as j ′ s α ≡ j α s − C $^{α}$, where C is a vector, the continuity equation (1) becomes ˙ s $^{α}$+ ∇ · j ′ s α = T ′ $^{α}$, where the relaxation torque reads T ′ α ≡ T $^{α}$+ ∇ · C $^{α}$. This ambiguity of spin current definition of course does not affect physical quantities such as the total torque acting on the spin density, which is given by ˙ s . We calculate the spin relaxation torque as a linear response to the applied electric field. The uniform magnetization is chosen as along z axis. The spin-orbit interaction is included to the second order. The contributions to the relaxation torque, Eq. (17), are shown in Fig. FIG. 1: Feynman diagrams representing the relaxation torque. Solid lines represent the electron Green’s function with the lifetime ( τ$_{σ}$ ) included, v$_{so}$ represents the spin-orbit interaction (double dashed line) and dotted line represents the interaction with the gauge field ( A ). The first two diagrams are the contributions to T α so and the last diagram is the contribution to T A,α so . The vertex marked by cross represents the relaxation torque. 1. The leading contribution for small 1 / ( ϵ$_{F}$ τ ) and qℓ ( ℓ is the electron mean free path) turns A $^{v}$so $^{v}$so T α so ≡ i ∑ ijkl ϵ$_{ijk}$ϵ$_{αlk}$ ( ∇ $_{i}$v ( k ) so ) 〈 c $^{†}$σ$_{l}$ ↔ ∇ j c 〉 , (18) T A,α so ≡ 2 e ijkl ϵ$_{ijk}$ϵ$_{αlk}$ ( ∇ $_{i}$v ( k ) so ) A$_{j}$ 〈 c $^{†}$σ$_{l}$ c 〉 . (19) T α ≡ T α so + T A,α so , (17) III. SPIN RELAXATION TORQUE 7