Itinerant Spin Dynamics in Structures of ... - Jacobs University

134 Appendix B: Linear Response
Finally we perform the t-integration and rewrite the formula for currents j:
= i V
∑
m,n
(f(E m )−f(E n ))〈m|j ν |n〉〈n|j µ |m〉
E n −E m E n −E m +ω +iη .
(B.16)
B.1 Kubo Formula for Weak Disorder
In the following we are interested in the real part of Eq.(B.16) for the longitudinal
conductivity at zero frequency. We rewrite the real part to a form which is suitable for the
diagrammatic perturbation. The first step is
2π
σ x,x (ω = 0) = lim
ω→0 V
= 2π
V
e 2
m 2 e m,n
e 2 ∫ ∞
m 2 e 0
∑ f(E m )−f(E n )
〈m|p x |n〉〈n|p x |m〉δ(E m −E n +ω) (B.17)
E n −E m
(
dE − ∂f(E) ) ∑ f(E m )−f(E n )
·
∂E E
m,n n −E m
·〈m|p x |n〉〈n|p x |m〉δ(E −E m )δ(E −E n ).
(B.18)
Including impurities we average over all configurations, writing the sum as a trace
σ impx,x (0) = 2π
V
e 2 ∫ ∞
m 2 e 0
dE
(
− ∂f(E) )
〈Tr[δ(E −H 0 )p x δ(E −H 0 )p x ]〉
∂E
imp
.
Now we chose the basis in momentum space {|k〉}. After applying orthogonality relation
we get
σ impx,x (0) = 1 e 2 ∫ ∞
2πV m 2 dE ∑ 〈
〉
k x k x
′ k
1
∣
e 0 E −H 0 −iη − 1
E −H 0 +iη∣ k′ ·
k,k
〈 ∣ ′ 〉
∣∣∣
· k ′ 1
E −H 0 −iη − 1
E −H 0 +iη∣ k
= 1
2πV
e 2 ∫ ∞
m 2 e 0
dE ∑ k,k ′ k x k ′ x·
(B.19)
·〈2G R E(k,k ′ )G A E(k ′ ,k)−G R E(k,k ′ )G R E(k ′ k)−G A E(k,k ′ )G A E(k ′ ,k) 〉 imp
(B.20)
with the definition
G R/A
E (k′ ,k) =
〈 ∣ ∣∣∣
k ′ 1
E −H 0 ∓iη
〉
∣ k . (B.21)
This can be simplified by noticing that the averages 〈 G R G R〉 and 〈 G A G A〉 are small
imp imp
compared with the other terms