In this work we apply the quasiclassical formalism, an established tool in the context of superconducting heterostructures, to topological insulators and superconductors in one and two dimensions, with focus on the former.
We derive topological invariants in terms of the quasiclassical Green's function in the regions terminating the disordered one-dimensional wire geometries, and demonstrate the existence of edge modes in the corresponding topologically non-trivial phases.
A generalisation to two-dimensional geometries is established by the concepts of compactification and dimensional reduction.
The second part of this work is devoted to Majorana fermions in disordered topological quantum wires. We apply the quasiclassical approach developed in the first part of this work to a setup used in recent experiments, where the evidence for Majorana edge modes is drawn from zero-bias peaks in tunnelling experiments. Analytically we derive a formalism that lays the foundation for an efficient numerical method to calculate physical observables. Studying in particular the averaged local density of states, we show that effects arising from disorder may overshadow an unambiguous detection of Majorana edge modes in tunnelling experiments.
In the last part of this work we briefly discuss ongoing research on how disorder effects in one-dimensional quantum wires may actually lead to the formation of local topological domains and may stabilise these domains. Based on the numerical method introduced in the second part, we present results that point towards formation of such local domains
