Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4. Disorder—Results and Limiting Cases
contributions in (4.27). The positive part comes from the W (2) contribution,
whereas the negative parts enter via (4.26) and stems from the VG0V part.
The first step in evaluating equation (4.27) is to compute the angular integral
over cos 2 (β), where β is taken as polar angle in d-dimensional spherical
coordinates. The angular average of cos 2 (β) decreases with the dimension
and is found as 1/d, cf. discussion before (4.12). Thus, in higher dimensions
the negative contribution to the integral (4.27) loses weight with respect to
the positive contribution. The remaining radial integral reads
�
� 2 2 q ξ
�
. (4.28)
ΛN = 2 Sd
(2π) d
dq q d−1 d Cd(qσ)
σ
(2 + q2ξ2 ) 2
2 + q2 1
−
ξ2 d
This result is plotted in figure 4.7(a) for speckle disorder in d = 1, 2, 3
dimensions.
Limits
The limit ( σ
ξ → ∞)k=0 touches the limit (kσ → 0)ξ=0 of the previous section
in the lower right corner ξ ≪ σ ≪ k −1 of the parameter space represented
in figure 4.1. Thus, the result (4.19a) is recovered.
In the other limit of δ-correlated disorder, new results are found: For
ξ ≫ σ, the smoothing factor (2 + q 2 ξ 2 ) −2 in (4.28) is sharply peaked at
q = 0 with a width of ξ −1 , so that the correlator Cd(qσ) can be evaluated
as Cd(0). Substituting y = qξ and integrating the first part in the bracket
of (4.28) by parts, we find
ΛN(k = 0, σ = 0) = 2 Sd
(2π) d
σdCd(0) ξd = σd Cd(0)
ξ d
� d
4
� � ∞
1
−
d 0
, d = 1
dy
⎧
⎪⎨ −
×
⎪⎩
3
16 √ 2
0, d = 2
, d = 3.
+ 5
48 √ 2π
y d−1
(2 + y 2 ) 2
(4.29)
Remarkably, the correction is negative in one dimension, vanishes in two
dimensions and is positive in three dimensions. In the scaling with V0/µ
fixed, used so far, the correction vanishes for σ → 0, figure 4.7(a). The limit
of uncorrelated disorder becomes well-defined, when Pd(0) = V 2
0 σ d Cd(0) is
kept fixed while σ is decreased, figure 4.7(b). The 1D result will be studied in
detail by means of a numerical integration of the Gross-Pitaevskii equation
in section 4.3. As expected, the 3D result coincides exactly with the findings
of Giorgini, Pitaevskii and Stringari [78], which have been confirmed by
Lopatin and Vinokur [75] and by Falco et al. [80], but have been contradicted
by Yukalov et al. [82].
94