Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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3. Disorder<br />
Figure 3.4: Geometry <strong>of</strong> the scattering process.<br />
In terms <strong>of</strong> the momentum transfer<br />
q, the elasticity condition k 2 = k ′2 becomes<br />
q · (2k + q) = 0, geometrically recognized<br />
in the Thales circle.<br />
2k + q<br />
Altogether, the relative correction <strong>of</strong> the speed <strong>of</strong> sound at fixed particle<br />
number is given as<br />
ΛN(k) = Λµ(k) −<br />
σd<br />
2 + k2ξ2 � d d q<br />
(2π) d<br />
Cd(qσ) q2ξ2 (2 + q2ξ2 . (3.59)<br />
) 2<br />
Now, we can combine the above equation with the self-energy ReΣ(k, ɛk)<br />
(3.39) to a single integral over the disorder correlator<br />
� d d qσ<br />
ΛN(k) = P<br />
(2π) d<br />
Cd(qσ)<br />
(2 + q2ξ2 ) 2<br />
�<br />
2 k2ξ2 + q2ξ2 2 + k2 + h(kξ, qξ)<br />
ξ2 k ′<br />
q<br />
k<br />
�<br />
. (3.60)<br />
The first part in the brackets comes from the W (2) term and the transformation<br />
to the canonical ensemble. The other part is due to virtual scattering<br />
processes, which are momentum conserving but not energy conserving. The<br />
function h(kξ, qξ) collects the first-order envelope functions and the propagators<br />
from the ⊛ ⊛ part in (3.39)<br />
h(k, q) =<br />
num(k, q)<br />
. (3.61)<br />
den(k, q) + i0<br />
After some algebra, numerator and denominator are found as<br />
num(k, q) = 2(2 + k 2 )(2 + k 2 + q 2 + 2k · q)(k 2 + k · q) 2 /k 2<br />
+ 2(k 2 + q 2 + 2k · q)(k 2 + q 2 + k · q) 2<br />
− 4(2 + k 2 )(k 2 + q 2 + k · q)(k 2 + k · q) (3.62a)<br />
den(k, q) = (ɛ 2 k − ɛ 2 k+q)(2 + k 2 )<br />
= −q · (2k + q)(2 + 2k 2 + q 2 + 2k · q)(2 + k 2 ). (3.62b)<br />
The denominator exhibits an elastic-scattering pole at ɛk = ɛk+q. Geometrically,<br />
the first scalar product q · (2k + q) expresses the elasticity condition<br />
on the Thales circle, figure 3.4.<br />
The correction (3.60) to the dispersion relation ɛk in will be discussed in<br />
detail in chapter 4.<br />
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