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
0[ψ] + E
0d
†d + γψ
†(0)d + γ
∗d
†ψ (0) ,
where d, d
†describe the localised fermionic state with energy E
0. 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
0[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
R(t − t
0) = −iθ(t − t
0)hd(t)d
†(t
0) + d
†(t
0)d(t)i .
b) Calculate D
R(t − t
0) using the corresponding Dyson’s equation and show that it is given by
D
R(t − t
0) = 1
ω − E
0+ i/τ
0, where
τ
0= 1/Γ = 2
|γ|
2ν
Fis the lifetime of the electron in the localised state.
c) Calculate the spectral function A(ω) and show that the sum rule
Z