content_image/359183.jpg content_image/359182.jpg content_image/359181.jpg content_image/359180.jpg content_image/359179.jpg content_image/359178.jpg
the spin, and [ ] < represents the lesser components, and 〈 〉 i represents the averaging over the random spin-orbit impurities. Taking the lesser component, we obtain 

<img src='content_image/359183.jpg'>

The retarded and advanced electron Green’s functions at zero frequency are represented by g r and g $^{a}$, respectively. 

The leading contribution for small 1 / ( ϵ$_{F}$ τ ) is given by 

<img src='content_image/359182.jpg'>

Expanding q in the Green’s function, we obtain the leading contribution as 

<img src='content_image/359181.jpg'>

Using the rotational symmetry, i.e., 〈 k$_{i}$k$_{j}$ 〉 = k 2 3 δ$_{ij}$ and 〈 k$_{i}$k$_{j}$k$_{l}$k$_{k}$ 〉 = k 4 $_{15}$( δ$_{ij}$δ$_{kl}$ + δ$_{ik}$δ$_{jl}$ + δ$_{il}$δ$_{jk}$ ) (the average here denotes the angular average), we obtain 

<img src='content_image/359180.jpg'>

We carry out the average over the impurity spin-orbit interaction as Eq. (10) to obtain 

<img src='content_image/359179.jpg'>

The asymmetric part of the trace is calculated by use of ( A ( ≡ 1 $_{2}$( A$_{+}$ + A$_{−}$ + σ$_{z}$ ( A$_{+}$ − A$_{−}$ ))) and B ( ≡ 1 $_{2}$( B$_{+}$ + B$_{−}$ + σ$_{z}$ ( B$_{+}$ − B$_{−}$ ))) are any diagonal 2 × 2 matrices) 

<img src='content_image/359178.jpg'>

