In recent years there has been a lot of interest in topological phases of matter. Unlike
conventional phases of matter, topological phases are not distinguished by symmetries,
but by so-called topological invariants which have more subtle physical implications. It
comes therefore as no surprise that for a long time only a few topological phases were
studied and those that were, were not studied in the full topological context, which is
only known now. One of the topological phases that has been know for a very long
time is the quantum Hall eect. The quantum Hall eect is a topological phase in
two-dimensions without any symmetries. Even though the bulk of a quantum Hall system
is insulating, it exhibits gapless edge modes. It is therefore dierent from other insulating
two-dimensional materials. It was soon realized after the discovery of the quantum Hall
eect, that there is a quantized invariant (topological invariant) associated with the
quantum Hall eect [48], but only much later such invariants were found and studied in
other systems.
By now other topological systems are also being studied from an experimental and
theoretical point of view [7]. There exist topological phases in any number of dimensions.
One of the topological phases that received a lot of attention in recent years, is the
one-dimensional topological superconducting phase, without time-reversal symmetry [5].
Similar to the quantum Hall eect, this phase exhibits edge excitations, which are zerodimensional
for one-dimensional systems. For this particular phase the edge excitations
are called Majorana bound states and they are interesting in themselves. There has been
a lot of eort in detecting Majorana bound states in the lab. One reason is that these
excitations provide evidence that a system is indeed in a topological phase. It is therefore
required to have unambiguous experimental evidence for the presence Majorana bound
states, which in turn requires a good theoretical understanding of the physics associated
with Majorana bound states. In particular for the most common experimental methods
that are used to study them, the signature of Majorana bound states in the measurement
still has to be understood better. And example would be the frequently performed
tunnel probe measurement on Majorana bound states [26, 40, 41]. A second reason why
Majorana bound states are interesting is their potential application to a certain quantum
computation scheme. This scheme, called topological quantum computation, relies on the
braiding of so-called non-abelian anyons in order to perform computations [18]. Majorana
bound states are the simplest example of such non-abelian anyons. No other non-abelian
anyons have been realized experimentally yet, which puts further focus on the study of
Majorana bound states. Additionally to probing Majorana bound states, their use in
topological quantum computation also requires them to be manipulated. This also poses
an interesting problem for both experimentalists and theorists [25, 27].
We can summarize the challenges presented so far as being related to generating
a topological phase, probing Majorana bound states and manipulating them. These
challenges are actually important beyond the intensively studied topological phase of
one-dimensional superconducting systems without time-reversal symmetry. In particular
they are very important for the closely related phase of one-dimensional topological
superconductors with time-reversal symmetry. This phase also exhibits Majorana bound
states, and we will study some of its aspects in this thesis. We will discuss some issues
related to obtaining this topological phase and how electron-electron interactions may
help in achieving this. We will also discuss issues related to using this phase and its edge
states for topological quantum computation, by calculating the result of an exchange of
two such edge states. Finally we will return to the broken time-reversal-symmetry case
and discuss aspects related to tunnel probing Majorana bound states.