Topology in One-Dimensional Quantum Systems

Abstract: States of matter which are described by concepts in topology are of great interest in the community of condensed matter physics. Both from a pure theoretical perspective, where many non-intuitive and mathematically rich results can be found from seemingly simple models, and also since recent experimental developments allows for explicit construction of devices where the underlying topology has observable consequences. This thesis provides a conceptual introduction to the framework describing such topological states of matter (TSM). It starts by explaining ordinary band theory which is followed by a discussion of quantum mechanical symmetries. These two concepts provides a method of classifying all non-interacting fermionic TSM. Following are explicit applications to a few models in one spatial dimension. In the accompanying paper, we study one-dimensional superconductors with Josephson-junctions allowing phase shifts of ?. We make a detailed analysis of the junction bound states and their properties in some different settings and compare the behaviour of trivial and topological superconducting junctions. In addition, we provide a phenomenological topological field theory in the low energy limit.