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

Appendix B: Linear Response 135
Proof. We start by writing down the Green’s function with finite complex self-energy Σ
1
G R/A (E) =
E −(H 0 −µ+Σ R/A )
= x∓iI
x 2 +I 2
(B.22)
(B.23)
with x = E − (H 0 − µ + RΣ) and ∓I = IΣ R/A , where µ is the chemical potential. To
calculate G R (E)G R (E), and accordingly for the pair of advanced Green’s functions, we
write the expression in terms of x, I and the spectral function
This yields
S = 1 π
I
x 2 +I 2.
(B.24)
G R G R = (x−iI)2
(x 2 +I 2 ) 2 (B.25)
=
1
(x 2 +I 2 ) − 2iIx
(x 2 +I 2 ) 2 − 2I 2
(x 2 +I 2 ) 2 (B.26)
= π S I −2π2 i S2
I x−2π2 S 2 .
(B.27)
Assuming the weak disorder limit, i.e. the impurity density n imp → 0, it follows that
due to τ ∝ n −1
imp, with τ −1 ≡ −2IΣ R , the spectral function becomes a delta distribution.
Using[Mah00]
we end up with
In the last step we applied again Eq.(B.28).
( ) S
lim
I→0 I −2πS2 = 0 (B.28)
G R G R = lim
I→0
−2π 2 i S2
I x
= −iπ 1
I 2xδ(x)
(B.29)
(B.30)
= 0. (B.31)
In contrast to this result we get for the retarded-advanced-pair of Green’s functions in the
weak disorder limit
which is divergent and therefore significant.
G R G A = lim
I→0
1
x 2 +I 2
= lim
I→0
π S I ,
(B.32)
(B.33)