Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.4. Deriving physical quantities from the self-energy
3.4. Deriving physical quantities from the
self-energy
The self-energy derived in the previous section is the central result of the
present chapter 3 “Disorder”. It allows deriving many physical quantities.
3.4.1. The physical meaning of the self-energy
In order to understand the meaning of the self-energy, it is useful to define
the spectral function S(k, ω) = −2ImG(k, ω) [16], which contains all
information about the frequency and lifetime of the excitations. In the
unperturbed system, the spectral function is given by a Dirac δ-function
S0(k, ω) = −2ImG0(k, ω) = 2πδ(�ω − ɛk). (3.41)
In presence of disorder, this function may get modified, but in any case it
will stay normalized �
S(k, E)
dE = 1, (3.42)
2π
such that the spectral function is the energy distribution of a quasiparticle
with wave vector k. Let us express the spectral function in terms of the
self-energy ReΣ + i ImΣ. First, the Dyson equation (3.36) has to be solved
for the average propagator, i.e. the 2×2 matrix [G −1
0 −Σ] has to be inverted.
One finds
�
G(k, ω) = G0(ω) −1 − Σ11(k, ω) − Σ12(k, ω)Σ21(k, ω)
G∗ 0 (−ω)−1 �−1 . (3.43)
− Σ22(k, ω)
The numerator of the last term is quadratic in the self-energy, i.e. fourth
order in V0/µ, whereas G ∗ 0(−ω) −1 = −�ω − ɛk ≈ 2ɛk is finite. In the scope
of the Born approximation, i.e. to second order in V0/µ, this term has to be
dropped. Consequently equation (3.43) simplifies
with Σ (2)
11 = ΣB . This leads to
S B (k, ω) =
G(k, ω) = � G0(k, ω) −1 − Σ B (k, ω) � −1 , (3.44)
−2ImΣ B (k, ω)
[�ω − (ɛk + ReΣ B (k, ω))] 2 + [ImΣ B (k, ω)] 2.
(3.45)
The peak of the spectral function is shifted by the real part of the self-energy
and broadened by the imaginary part (which will turn out to be negative,
such that the spectral function is positive and fulfills (3.42)).
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