Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

1. Interacting Bose Gases
Quantum heat transport
Chains of harmonic oscillators are a simple model for the heat conductivity
by phonons. Due to the integrability of the linear equations of motion, the
heat transport is ballistic and the temperature gradient vanishes. Apart
from the introduction of nonlinearities, disorder contributes to a finite heat
resistance in such systems, because the eigenstates of the chain become
localized. Indeed, in the disordered harmonic chain, a finite temperature
gradient is found [31], however, the heat resistance does not scale linearly
with the length of the chain as one would expect according to Fourier’s law.
1.3. Interaction
Interaction effects among the particles can dramatically change the properties
of physical systems. In contrast to fermions, bosons are not subject
to the Pauli exclusion principle. They can come much closer to each other,
such that interaction effects are more relevant.
Interaction and disorder
The problem of interacting disordered bosonic systems is known as the dirty
boson problem. Historically, it emerged in the context of superfluid Helium
in aerosol glasses (Vycor) [32] and has been subject of theoretical research
for a long time [2, 13]. In absence of interactions, bosons in disordered environments
will condense into the lowest-energy state, which is a localized
state. This phase is known as the Lifshits glass. A repulsive interaction
among the particles causes them to delocalize and finally leads to the disordered
BEC phase [33]. The non-interacting Bose gas is extremely sensitive
to external inhomogeneities like disorder. Thus, the non-interacting ground
state is not a good starting point for perturbation theory. For this reason,
the strategy in part I of this work is to start with a homogeneous interacting
system and then to introduce disorder perturbatively.
Interaction and lattices
The Hubbard model describes interacting fermions or bosons (Bose-Hubbard
model) on a lattice within the tight-binding approximation. The interaction
drives the transition from the superfluid to the Mott insulator. At integer
filling factor, i.e. with the same number of particles at each lattice site, the
interaction causes an energy gap that suppresses the motion of the particles.
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