Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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3.3. Effective medium and diagrammatic perturbation theory<br />
For the Fourier transform to frequency domain, we introduce a convergence<br />
ensuring parameter η > 0, which appears as an infinitesimal shift <strong>of</strong> the pole<br />
G0(k, ω) = lim<br />
η→0<br />
1<br />
�ω − ɛk + iη =:<br />
1<br />
. (3.22)<br />
�ω − ɛk + i0<br />
In the disordered problem, the equation <strong>of</strong> motion (2.43) for ˆγk mixes<br />
with ˆγ †<br />
−k ′. Thus, we define the anomalous Green function [49]<br />
F ′<br />
kk := 1 �� †<br />
ˆγ −k (t), ˆγ†<br />
i�<br />
k ′(0) �� Θ(t). (3.23)<br />
The equations <strong>of</strong> motion for the Green functions are coupled<br />
i� d<br />
dt Gkk ′(t) = ɛkGkk ′(t) + 1<br />
L d<br />
�<br />
2<br />
k ′′<br />
[Wkk ′′G ′′ ′<br />
k k + Ykk ′′F ′′<br />
k k ′] + δ(t)δ ′<br />
k k<br />
i� d<br />
dt Fkk ′(t) = −ɛkFkk ′(t) − 1<br />
L d<br />
�<br />
[Wkk ′′F ′′ ′<br />
k k + Ykk ′′G ′′<br />
k k ′] . (3.24)<br />
2<br />
k ′′<br />
Here, the commutator (3.18) and ∂tΘ(t) = δ(t) were used. In absence<br />
<strong>of</strong> disorder, the anomalous Green function F vanishes, at least within the<br />
<strong>Bogoliubov</strong> approximation, i.e. to order O(ˆγ 2 ).<br />
It is useful to combine these two coupled equations <strong>of</strong> motion to one<br />
matrix valued equation. This can be achieved by setting up the equations<br />
<strong>of</strong> motion for the Hermitian conjugate <strong>of</strong> the propagators<br />
G †<br />
kk ′(t) = 1 �� †<br />
ˆγ k i�<br />
(t), ˆγ k ′(0) �� Θ(t), F †<br />
kk ′(t) = 1 ��<br />
ˆγ−k(t), ˆγ k ′(0)<br />
i�<br />
�� Θ(t)<br />
(3.25)<br />
and combining them to the generalized propagator<br />
�<br />
† G F<br />
G =<br />
F G †<br />
�<br />
. (3.26)<br />
The Hermitian conjugates <strong>of</strong> the propagators are closely related to the advanced<br />
propagators. In frequency domain, the relation G †<br />
kk ′(ω) = GA k ′ k (−ω)<br />
holds. Similar pseudo-spinor structures appear also in other field <strong>of</strong> physics.<br />
Examples are the Nambu formalism <strong>of</strong> the BCS theory [16, Chapter 18.5],<br />
where electrons and holes are combined in a spinor, or applications in particle<br />
physics [119].<br />
The equation <strong>of</strong> motion for the generalized propagator reads<br />
i� d<br />
��� � � �� � � �<br />
ɛ 0 W Y<br />
1 0<br />
G = η<br />
+<br />
G + δ(t) . (3.27)<br />
dt 0 ɛ Y W 0 1<br />
67