Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

3. Disorder
In this compact notation, the matrix multiplication implies the sum over
d � − the free momentum L 2
k ′. The matrix η is given as diag(1, −1) and
1 ′
kk = δkk ′.
Transforming to frequency space and multiplying by η from the left gives
a compact form amenable to perturbation theory
� −1
G0 − V� G = 1. (3.28)
The unperturbed generalized propagator G0 contains (3.22) and its conjugate
� 1
�ω−ɛk+i0
G0(k, ω) =
0
0
� �
G0(k, ω)
=
0
0
G∗ �
.
0(k, −ω)
(3.29)
1
−�ω−ɛk−i0
Again, the infinitesimal shifts ±i0 come from the convergence ensuring factors
in the Fourier transform, their sign being determined by the causality
factor Θ(t) in the definition of the propagators. The perturbations are
contained in the Bogoliubov scattering operator V = ( W Y
Y W ), which was
introduced in subsection 2.3.2.
3.3.2. The self-energy
With equation (3.28) we can now set up a usual diagrammatic perturbation
theory [16, 23, 120]. Equation (3.28) is solved for the full propagator G by
writing the inverse of operators in terms of a series expansion in powers of
V:
G = � G −1
0 − V� −1
= G0 [1 − VG0] −1
∞�
= G0 (VG0) n = G0 + G0VG0 + G0VG0VG0 + . . . (3.30)
n=0
In order to get meaningful results on a random system, the disorder has
to be averaged over. To this end, it is necessary to trace V back to the
speckle field amplitudes, because these are the basic quantities with known
statistical properties and a Gaussian probability distribution. Remember
subsection 2.3.3: The scattering operator depends nonlinearly on the external
potential V = V (1) + V (2) + V (3) + . . ., where V (n) contains the disorder
potential to the n-th power. At this point, the representation with Feynman
diagrams is again instructive. We start with the expansion (2.51) and draw
68
V = ⊛ + ⊛⊛ + ⊛⊛⊛ + . . . . (3.31)