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

Appendix B
Linear Response
Applying linear response to first order with a perturbation W(t) we get for the
time dependent expectation value of an operator Â
[ ] 〈Â〉t = Tr ρ 0 Â −i
∫ ∞
−∞
dt ′ θ(t−t ′ )
〈 〉
[ÂD(t),ŴD(t ′ )] , (B.1)
0
where the index D indicates the Dirac picture. The source of the perturbation is an electric
field E(t) = E 0 e i(ω+iη)t with η being an infinitesimally small positive value. The perturbation
is then given by W(t) = −ˆP · E(t), with the dipole operator ˆP = ∑ i q iˆr i and the
charge q i at the positions r i .
The response to the applied electric field is a current J µ = σ µν E ν . For the next steps we
keep the definition of the current general. Later on we can relate it to the spin Hall current
J z x = −σz xy E y, with the spin Hall conductivity σ z xy ≡ σ SH.
〈J µ 〉 t
= i ∑ ν
∫ ∞
−∞
dtθ(t−t ′ ) 〈 [J µ D (0),P νD(t ′ −t)] 〉 e i(ω+iη)(t′ −t) E 0ν e −i(ω+iη)t
} {{ }
E ν(t)
(B.2)
the spin Hall conductivity (in the following the subscript D will be left out)
σ µν (ω) = i
∫ ∞
−∞
dtθ(−t)〈[J µ (0),P νD (t)]〉e −i(ω+iη)t
(B.3)
In the next step we want to replace ˆP by its time derivative. This can be accomplished by
applying the Kubo identity
[A(t),ρ 0 ] = −iρ 0
∫ β
0
dx ˙ A(t−ix)
(B.4)
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