Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
1. Interacting <strong>Bose</strong> Gases<br />
Quantum heat transport<br />
Chains <strong>of</strong> harmonic oscillators are a simple model for the heat conductivity<br />
by phonons. Due to the integrability <strong>of</strong> the linear equations <strong>of</strong> motion, the<br />
heat transport is ballistic and the temperature gradient vanishes. Apart<br />
from the introduction <strong>of</strong> nonlinearities, disorder contributes to a finite heat<br />
resistance in such systems, because the eigenstates <strong>of</strong> the chain become<br />
localized. Indeed, in the disordered harmonic chain, a finite temperature<br />
gradient is found [31], however, the heat resistance does not scale linearly<br />
with the length <strong>of</strong> the chain as one would expect according to Fourier’s law.<br />
1.3. Interaction<br />
Interaction effects among the particles can dramatically change the properties<br />
<strong>of</strong> physical systems. In contrast to fermions, bosons are not subject<br />
to the Pauli exclusion principle. They can come much closer to each other,<br />
such that interaction effects are more relevant.<br />
Interaction and disorder<br />
The problem <strong>of</strong> interacting disordered bosonic systems is known as the dirty<br />
boson problem. Historically, it emerged in the context <strong>of</strong> superfluid Helium<br />
in aerosol glasses (Vycor) [32] and has been subject <strong>of</strong> theoretical research<br />
for a long time [2, 13]. In absence <strong>of</strong> interactions, bosons in disordered environments<br />
will condense into the lowest-energy state, which is a localized<br />
state. This phase is known as the Lifshits glass. A repulsive interaction<br />
among the particles causes them to delocalize and finally leads to the disordered<br />
BEC phase [33]. The non-interacting <strong>Bose</strong> gas is extremely sensitive<br />
to external inhomogeneities like disorder. Thus, the non-interacting ground<br />
state is not a good starting point for perturbation theory. For this reason,<br />
the strategy in part I <strong>of</strong> this work is to start with a homogeneous interacting<br />
system and then to introduce disorder perturbatively.<br />
Interaction and lattices<br />
The Hubbard model describes interacting fermions or bosons (<strong>Bose</strong>-Hubbard<br />
model) on a lattice within the tight-binding approximation. The interaction<br />
drives the transition from the superfluid to the Mott insulator. At integer<br />
filling factor, i.e. with the same number <strong>of</strong> particles at each lattice site, the<br />
interaction causes an energy gap that suppresses the motion <strong>of</strong> the particles.<br />
4