Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3.3. Effective medium and diagrammatic perturbation theory
For the Fourier transform to frequency domain, we introduce a convergence
ensuring parameter η > 0, which appears as an infinitesimal shift of the pole
G0(k, ω) = lim
η→0
1
�ω − ɛk + iη =:
1
. (3.22)
�ω − ɛk + i0
In the disordered problem, the equation of motion (2.43) for ˆγk mixes
with ˆγ †
−k ′. Thus, we define the anomalous Green function [49]
F ′
kk := 1 �� †
ˆγ −k (t), ˆγ†
i�
k ′(0) �� Θ(t). (3.23)
The equations of motion for the Green functions are coupled
i� d
dt Gkk ′(t) = ɛkGkk ′(t) + 1
L d
�
2
k ′′
[Wkk ′′G ′′ ′
k k + Ykk ′′F ′′
k k ′] + δ(t)δ ′
k k
i� d
dt Fkk ′(t) = −ɛkFkk ′(t) − 1
L d
�
[Wkk ′′F ′′ ′
k k + Ykk ′′G ′′
k k ′] . (3.24)
2
k ′′
Here, the commutator (3.18) and ∂tΘ(t) = δ(t) were used. In absence
of disorder, the anomalous Green function F vanishes, at least within the
Bogoliubov approximation, i.e. to order O(ˆγ 2 ).
It is useful to combine these two coupled equations of motion to one
matrix valued equation. This can be achieved by setting up the equations
of motion for the Hermitian conjugate of the propagators
G †
kk ′(t) = 1 �� †
ˆγ k i�
(t), ˆγ k ′(0) �� Θ(t), F †
kk ′(t) = 1 ��
ˆγ−k(t), ˆγ k ′(0)
i�
�� Θ(t)
(3.25)
and combining them to the generalized propagator
�
† G F
G =
F G †
�
. (3.26)
The Hermitian conjugates of the propagators are closely related to the advanced
propagators. In frequency domain, the relation G †
kk ′(ω) = GA k ′ k (−ω)
holds. Similar pseudo-spinor structures appear also in other field of physics.
Examples are the Nambu formalism of the BCS theory [16, Chapter 18.5],
where electrons and holes are combined in a spinor, or applications in particle
physics [119].
The equation of motion for the generalized propagator reads
i� d
��� � � �� � � �
ɛ 0 W Y
1 0
G = η
+
G + δ(t) . (3.27)
dt 0 ɛ Y W 0 1
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