Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4. Disorder—Results and Limiting Cases
1D speckle potential
In the 1D speckle case, the correlation function (3.12a) is piecewisely linear,
and the integral (4.14) yields [115, 126]
�
1 kσ
Λ = − +
2 8 ln
� �
�
�
1 − kσ �
�
�1
+ kσ � − k2σ2 8 ln
�
�
�
1 − k
�
2σ2 k2σ2 ��
�
�
�
= − 1
�
1 +
2
kσ � �
2kσ ln kσ − (kσ + 1) ln(kσ + 1) − (kσ − 1) ln |kσ − 1|
4
�
.
(4.15)
From the last term in the second form, it is apparent that Λ is non-analytic
at kσ = 1, the value beyond which elastic backscattering is suppressed. The
correction (4.15) is a negative correction to the dispersion relation, as shown
schematically in figure 4.4(a).
Limits in d dimensions
In higher dimensions, the integral (4.14) over the correlation functions gets
more complicated, so that analytical solutions like (4.15) are not available
in general. But in all cases, the principle-value integral (4.14) can be evaluated
numerically. figure 4.4(b) shows the corresponding curves. Short-range
correlated potentials (kσ ≪ 1) affect low dimensions more than high dimensions
and vice versa.
For smooth potentials with k-space correlators Cd(kσ) that decrease sufficiently
fast, the limits kσ → 0 and kσ → ∞ can be calculated analytically
as follows. It is useful to rewrite equation (4.14) in terms of η = 1/kσ as
Λ = − 1
2 P
� d d q
(2π) d
Cd(q) [1 + ηq cos β] 2
2ηq cos β + η2q2 . (4.16)
Denoting the angular part of the integral by Ad(ηq), one arrives at the
radial integral � ∞
0 dq qd−1Cd(q)Ad(ηq). In the limit kσ ≪ 1, the parameter
ηq tends to infinity nearly everywhere under the integral. Then
�
dΩd
Ad(∞) =
(2π) d(cos β)2 = Sd
(2π) dd−1 , (4.17)
and with � d d q
(2π) dCd(q) = Cd(r = 0) = 1, we arrive at Λ = −1/2d. In the limit
kσ → ∞ we proceed similarly with η → 0. The angular integrand reduces
to 1 + � 2η cos β + η 2� −1 , whose principle-value integral evaluates after some
algebra to
88
Ad(0) = Sd
(2π) d
d + 2
, (4.18)
4