A central theme in condensed matter physics is the classification and characterization of states of matter. In the recent decades, it has become evident that there exists a large class of quantum mechanical systems that must be classified according to properties deeply rooted in the mathematical field of topology, rather than in terms of which symmetries they break. Together with rapid technological developments, such topological states of matter pose a promising path for a fundamentally new generation of quantum devices and exotically engineered materials, with applications in quantum metrology, quantum sensing, and quantum computations.

This doctoral thesis comprises a study of a general theoretical framework describing topological states of matter, followed by applications in systems of low dimension. Together, these two parts form the foundation for the following accompanying papers:

PAPERS I-II concerns Majorana zero modes in various Josephson junction setups of one-dimensional topological superconductors. In addition to determining the mathematical conditions for the existence of such exotic states, the papers also provide experimental proposals for their detection. 

PAPER III deals with the nature of a widely used model of a synthetically engineered one-dimensional topological superconductor. It is shown that in a certain parameter limit, the superconducting order parameter obtains a geometric contribution, which originates from the directional nature of the Rashba spin-orbit coupling. This geometrical dependence is argued to manifest itself in various Josephson junction setups, and in particular as a direct connection between the charge current density and the local curvature.

PAPER IV proposes a method of constructing non-local order parameters for two-dimensional Chern insulators, and describes how such operators can be used to distinguish between different topological sectors. 

We investigate the sub gap properties of a three terminal Josephson T-junction composed of topologically superconducting wires connected by a normal metal region. This system naturally hosts zero energy Andreev bound states which are of self-conjugate Majorana nature and we show that they are, in contrast to ordinary Majorana zero modes, spatially extended in the normal metal region. If the T-junction respects time-reversal symmetry, we show that a zero mode is distributed only in two out of three arms in the junction and tuning the superconducting phases allows for transfer of the mode between the junction arms. We further provide tunneling conductance calculations showing that these features can be detected in experiments. Our findings suggest an experimental platform for studying the nature of spatially extended Majorana zero modes.

We theoretically investigate Josephson junctions with a phase shift of pi in various proximity induced one-dimensional superconductor models. One of the salient experimental signatures of topological superconductors, namely the fractionalized 4 pi periodic Josephson effect, is closely related to the occurrence of a characteristic zero energy bound state in such junctions. We make a detailed analysis of a more general type of pi-junctions coined 'phase winding junctions' where the phase of the order parameter rotates by an angle pi while its absolute value is kept finite. Such junctions have different properties, also from a topological viewpoint, and there are no protected zero energy modes. We compare the phenomenology of such junctions in topological (p-wave) and trivial (s-wave) superconducting wires, and briefly discuss possible experimental probes. Furthermore, we propose a topological field theory that gives a minimal description of a wire with defects corresponding to p-junctions. This effective theory is a one-dimensional version of similar theories describing Majorana bound states in half-vortices of two-dimensional topological superconductors.

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 π. 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.