ordered Green’s function of the Fermi liquid can be brought to the form

Theoretical Condensed Matter Physics

PD A. Komnik, Universit¨ at Heidelberg, SS07 10. Set of Exercises: 24.07.07

17. Momentum Distribution function in the Fermi liquid:

Similar to the discussion of the retarded Green’s function it can be shown that the time–

ordered Green’s function of the Fermi liquid can be brought to the form

G(~ p, ω) = Z

ω − ξ

+ iδsgn(ξ

) + g(~ p, ω) ,

where Z is the quasiparticle weight, δ represents a positive infinitesimal increment and g(~ p, ω) is analytic in the vicinity of the Fermi edge. Using the Fourier transform of Eq. (143) [see lecture notes] show that at T = 0 the momentum dependent distribution function n

has a discontinuity at the Fermi edge

n

− n

= Z , for q → 0 . How does n

look like? Draw a picture!

18. Cooper instability:

In order to demonstrate the instability of the Fermi sea against electron pairing Cooper introduced a simplified version of the Fr¨ ohlich problem. Consider | ~ ki as an electron pair state with opposite momenta ~ k and − ~ k and moreover |ψi =

α( ~ k)| ~ ki as a collective state of such pairs. The Hamiltonian of the system H = H

+ V

consists of the free part H

| ~ ki = 2ξ

| ~ ki and the interaction of the form

h ~ k

|V

| ~ ki =

−V for 0 < ξ

, ξ

k

< ω

0 otherwise.

It is attractive for all states with energies smaller than the Debye cut–off ω

and zero otherwise. Calculate the energy of the Cooper pair

E ≈ 2E

− 2ω

e

,

where E

is the Fermi energy and ν

is the density of states at the fermi edge under the

assumption that ν

V 1.