Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.3. Effective medium and diagrammatic perturbation theory
avoided by reorganizing the Born series (3.33). The diagrams are sorted by
the number of irreducible sub-diagrams they contain, and not by their order
in V/µ. The result is the Dyson equation
G = G0 + G0ΣG, (3.36)
where the self-energy Σ contains all irreducible contributions
Σ = ⊛⊛ + ⊛ ⊛
� �� �
Σ (2)
�
+ 2 ⊛ ⊛ ⊛ + 2 ⊛ ⊛⊛ + ⊛⊛⊛
� �� �
Σ (3)
�
+ . . . (3.37)
Due to the disorder average, each block of Σ is diagonal in k. The contributions
to Σ are then sorted by their order in V/µ. The first two terms
are second order in the disorder potential, the others are of higher order.
In principle, any desired order of the above series can be determined in a
systematic way, see box 3.1. In practice, however, the complexity and the
number of diagrams grows very rapidly.
Born approximation
In the following, we will restrict ourselves to the leading order V 2 /µ 2 , i.e.
Σ (2) = ⊛⊛ + ⊛ ⊛ = V (2) + V (1) G0V (1) , (3.38)
which is known as the Born approximation. Note that this leading-order
approximation in the self-energy still implies an infinite number of diagrams
in the correction to the averaged propagator G via the recursive formula
(3.36). This allows a shift of the pole of the propagator (3.22), which would
be impossible in leading-order perturbation theory directly for G.
3.3.3. Computing the self-energy in the Born
approximation
In the Born approximation (3.38), the two field correlators in each diagram
= � ddq ′ γ(q − q ′ )γ(q ′ ) =
are combined to one intensity correlator ⊛ ⊛
q
71