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.