Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

5. Conclusions and Outlook (Part I)
Figure 5.1: Density of states in three dimensions
(compare to figure 3.5(c)). For sufficiently
strong disorder, the Bogoliubov excitations in
the crossover region at kξ ≈ 1 are expected
to localize (gray).
ρ
0.25 0.5 0.75 1 1.25 1.5 1.75 2
¯hω/µ
are now combined to the Bogoliubov dispersion relation (2.35), figure 2.3.
Depending on the disorder strength, two regimes are possible:
1. Weak disorder µ < ω ∗ , E ∗ < µ. Phonon and particle regime are
combined without any localized states.
2. Strong disorder ω ∗ < µ < E ∗ . There is a range of localized states
around the crossing from phonon to particle excitations.
The characterization of these localized states between phonons and particles
is an interesting topic for further research. A possible approach is
a diagrammatic perturbation theory analogous to Vollhardt’s and Wölfle’s
approach [123], which allows computing the weak-localization to the diffusion
constant. This approach has already been applied to non-interacting
cold atoms in speckle potentials [22, 128]. With the block-matrix notation
employed in subsection 3.3.1, the Bogoliubov propagator follows equivalent
equations of motion. It should be possible to compute the weak-localization
correction to the diffusion constant analogously.
Finite temperature
With the Bogoliubov formalism set up in this work, it is in principle possible,
to consider finite-temperature problems. Similar to the zero-temperature
Green functions in the block-matrix formalism of subsection 3.3.2, also Matsubara
Green functions could be set up and computed. However, the Bogoliubov
formalism relies on the macroscopic occupation of the condensate mode
and neglects the interactions between the Bogoliubov excitations. At higher
temperatures, these interaction effects become relevant and one should take
beyond-Bogoliubov terms into account. These interactions appear as additional
perturbations and might be accessible with methods similar to those
used by Hugenholtz and Pines [76]. However, two perturbations at once,
disorder and the interaction among quasiparticles should make the problem
extremely complicated.
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