Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4.1. Hydrodynamic limit I: ξ = 0
The free Green function � G0 = (2π) d δ(k − k ′ ) � G0(k, ω) is diagonal in k,
�G0(k, ω) = � ω 2 − c 2 k 2 + i0 � −1 . (4.7)
The full Green function � G is expanded in the same way as in equation (3.30),
�G = � G0 + � G0 � V � G0 + . . .. Again, we take the disorder average and find
�G(k, ω) =
�
�G0(k, ω) −1 − � � −1
Σ(k, ω) . (4.8)
The poles of this average Green function at ω 2 = c 2 k 2 + � Σ(k, ω) now determine
the effective dispersion relation. In the Born approximation, the
self-energy � Σ(k, ω) evaluates as
�Σ(k, ω) = V 2
0 σ d
� d ′ d k
(2π) dCd(|k − k ′ |σ) (k′ ·k) 2
m2 �G0(k ′ , ω). (4.9)
The self-energy corrects the quadratic dispersion relation (�ω) 2 = c2k2 +
�Σ(k, ω). To leading order, we can use the on-shell approximation and replace
�Σ(k, ω) with � Σ(k, ck), such that ω = ck � 1 + � Σ(k, ck)/(2ck) � . This converts
to the inverse mean free path (kls) −1 = −Im� Σ(k, ck)/(c2k2 ) and the relative
correction of the speed of sound Λ = Re� Σ(k, ck)µ 2 /(2c2k2V 2
0 ).
Before discussing these results in detail, we verify that these results are
in agreement with the result (3.39) from section 3.4 in the limit ξ → 0. The
envelope functions reduce to w (2) = O(kξ), w (1)
k ′ k = −k′ ·k ξ/ √ 2kk ′ = −y (1)
k ′ k .
With this, we indeed find
lim
ξ→0 ΣB (k, ɛk) = � � Σ(k, ck)
. (4.10)
2ck
Next, let us consider the central results inverse mean free path and correction
to the speed of sound.
4.1.2. Transport length scales
Mean free path
The imaginary part of the self-energy � Σ leads to the inverse mean free path
1
kls
= V 2
0
4µ 2(kσ)d
�
dΩd
(2π) d−1Cd
� � 2
2kσ sin θ/2 cos (θ). (4.11)
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