Consider a spinless fermionic system consisting of a local level coupled to an electron band. The system is described by the following Hamiltonian

Theoretical Condensed Matter Physics

PD A. Komnik, Universit¨ at Heidelberg, SS07 9. Set of Exercises: 17.07.07

15. Finite lifetime of a localised state coupled to a continuum:

Consider a spinless fermionic system consisting of a local level coupled to an electron band. The system is described by the following Hamiltonian

H = H

[ψ] + E

d

d + γψ

(0)d + γ

d

ψ (0) ,

where d, d

describe the localised fermionic state with energy E

. Tunnelling from/to the continuum, which is described by the fields ψ (x) and ψ

(x), is local at x = 0 and γ, γ

are the corresponding tunnelling amplitudes. H

[ψ] is identical with the Hamiltonian of Problem 12a.

a) Identify the self–energy contribution to the Green’s function of the localised state, which is defined by

D

(t − t

) = −iθ(t − t

)hd(t)d

(t

) + d

(t

)d(t)i .

b) Calculate D

(t − t

) using the corresponding Dyson’s equation and show that it is given by

D

(t − t

) = 1

ω − E

+ i/τ

, where

τ

= 1/Γ = 2

|γ|

ν

is the lifetime of the electron in the localised state.

c) Calculate the spectral function A(ω) and show that the sum rule

Z

dω

2π A(ω) = 1

holds.

16. Ferromagnetism of the interacting electron gas:

Consider electrons interacting via a contact potential U (r − r

) = U δ(r − r

) .

From the Kubo formalism it is known that the magnetic susceptibility is given by a free loop diagram. The simplest way to include interactions is then to consider the following diagram series

a) Calculate the magnetic susceptibility and show it to be given by χ

= −2µ

Π

1 + UΠ

, (1)

where Π

is the diagram in the zeroth order of the interaction (first picture in the figure).

b) From the lectures one knows that Π

= −ν

, where ν

is the density of states

at the Fermi edge. Eq. (??) then implies a divergent susceptibility as U approaches

1/ν

. What happens to the system in this situation?