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We consider a case where electric field E ( x , t )( ≡ − ˙ A ( x , t )) is position and time dependent. The electromagnetic interaction is written as 

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where Ω is the frequency of the electric field. We will consider the limit of small Ω and small q . The scattering by the normal impurities is represented by 

<img src='content_image/344586.jpg'>

where v$_{i}$ represents the strength of the impurity potential, R$_{i}$ represents the position of random impurities, N$_{imp}$ is the number of impurities, and N ≡ V/a 3 is number of sites. To estimate physical quantities, we take the random average over impurity positions in a standard manner [9]. 

To derive the spin continuity equation, Eq. (1), we derive the equation of motion for the spin density, s ( x , t ) ≡ 〈 c $^{†}$( x , t ) σ c ( x , t ) 〉 ( 〈 〉 represents the quantum average). The time development of the spin density reads 

where H is the total Hamiltonian of the system. The commutators are calculated as in Appendix A, and Eq. (13) turns out to be Eq. (1), namely 

with the spin current given as 

where 

and 

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<img src='content_image/344583.jpg'>

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