8. Green’s function in the energy–coordinate representation:

Theoretical Condensed Matter Physics PD A. Komnik, Universit¨ at Heidelberg, SS07

5. Set of Exercises: 29.05.07

8. Green’s function in the energy–coordinate representation:

a) Show that the Green’s function of the electron band in the energy–coordinate repre- sentation is given by

G(ω, r) = − m

2πr e

(1)

where κ =

2m(E

+ ω).

b) Expand the above result for small energies |ω|/E

1.

9. Ruderman–Kittel effect:

Consider a localised spin-1/2 impurity which interacts with conductance band electrons.

While the spin density σ

(r), i = x, y, z is uniform in the clean system, it develops spacial oscillations as soon as the impurity spin is introduced. The interaction between the con- ductance band electrons and the localised spin can be very good described by the following term,

V

= J S

σ ˆ

(r = 0) , (2)

where S

is the spin operator of the impurity located at the coordinate origin r = 0, ˆ

σ

(r) = ψ

(r) σ

ψ

(r)

is the spin density operator for the band electrons, α, β =↓, ↑, σ

are the Pauli matrices and J is a coupling constant.

a) Show that the spin density σ

(r) = hˆ σ

(r)i is related to the Green’s function in the following way,

σ

(r) = −i lim

t⁰→t+δ,r=r⁰

σ

G

(r, t; r

, t

) , where δ is a positive infinitesimal.

b) Calculate the spin density σ

(r) in the lowest order in interaction. Evaluate its long-

range behaviour for rp

1, where r is the distance from the impurity. Hint: Use the

expansion from Problem 8 b) instead of the exact formula (??).