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