Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

4. Disorder—Results and Limiting Cases
MN
0
0.1
10 20
kσ
30 40 50
0.08
0.06
0.04
0.02
0
�0.02
1 2 3 4 5
σ/ξ
Figure 4.12.: Relative disorder correction MN = (ɛk − ɛk)µ/V 2
0 of the dispersion relation
at kξ = 10, V0 = 0.03µ in one dimension. Solid line: Full correction (3.60) for finite
kξ = 10. Dashed line: Limit (4.40) M (0)
N (σ/ξ) for kξ → ∞. Dots: Numerical integration
in a system of size L = 200σ, averaged over 50 realizations of disorder [Each realization
of disorder was normalized to mean value zero and rms value |V0|]. The red circle marks
kσ = 0.5, cf. figure 4.14
(4.38) is positive although the chemical potential has been lowered in order
to keep the particle number constant. At fixed µ, the correction would be
double, Λµ = 2ΛN.
For the 1D speckle potential, the integral (4.38) can be solved analytically
M (0)
N
= z
2
�
arctan
� 1
z
�
− z log
�
1 + 1
z2 ��
, z = σ
√ . (4.40)
2ξ
In figure 4.12, this limit is compared to the full correction at a finite value
kξ = 10.
Numerical test: Bogoliubov quasiparticles at kξ = 10. Analogously to
the procedure in section 4.3, we determine numerically the correction to the
dispersion relation in the particle regime kξ = 10, as function of the correlation
length σ. In figure 4.12, the numerical results are shown together with
the full Born prediction (3.60) and the limit (4.40), showing good agreement.
In contrast to the low-energy excitations, where the 1D correction
[(4.15) and (4.30)] is always negative, it is here found to be positive. Only
for σ/ξ � kξ, which is outside of the plot range, negative values are found.
The correction ΛN = MNµ/ɛk is much weaker than in the hydrodynamic
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