Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.3. Effective medium and diagrammatic perturbation theory
Here, we have taken into account that each speckle potential Vk contributes
the electrical fields Eq and E ∗ k−q , which are drawn as and ∗ , respectively.
Together with the free propagator G0 = , all contributions from (3.30)
can be written as diagrams, for example
V (1) G0V (1) = ⊛ ⊛ , V (1) G0V (1) G0V (1) = ⊛ ⊛ ⊛ , V (1) G0V (2) = ⊛ ⊛⊛ .
(3.32)
Now, equation (3.30) is averaged over the disorder, which yields the so-called
Born series
G = G0 + G0VG0 + G0VG0VG0 + . . . . (3.33)
According to the Gaussian moment theorem [121], averages over a product of
several Gauss-distributed random variables separate into averages of pairs,
the so-called contractions. Each summand on the right hand side is obtained
as the sum over its contractions, i.e. the fields (dotted lines) are pairwise
connected using their pair correlation function (3.7), see also box 3.1 on
page 70. Let us have a look at the disorder average of (3.31):
V (1) = ⊛ = 0 V (2) = ⊛⊛ =: ⊛⊛ �= 0 V (3) = ⊛⊛⊛ �= 0 (3.34)
The first-order term vanishes, because the potential V (r) has been shifted
such that its average vanishes [equation (3.9)]. Any other diagram, where a
field correlator forms a simple loop ⊛ , i.e. returns to the same sub-vertex,
vanishes as well. The other diagrams in (3.34), however, are non-zero. In
the second-order term, two field correlations (dotted lines) were combined
to a single potential correlation (dashed line). In the non-zero diagram of
V (3) , the field correlators form a ring. Similarly, the disorder averages of
(3.32) translate into the following diagrams
V (1) G0V (1) = ⊛ ⊛ , V (1) G0V (1) G0V (1) = ⊛ ⊛ ⊛ , V (1) G0V (2) = ⊛ ⊛⊛ .
(3.35)
These averages consist of only one diagram each. Many diagrams of higher
order are reducible, i.e. they separate into independent factors when a single
propagator is removed. The forth-order diagram ⊛ ⊛ ⊛ ⊛, for example,
separates into twice the first diagram in (3.35), connected with a free propagator.
The redundant information of the reducible diagrams in G can be
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