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