Introduction to topological superconductivity and Majorana fermions

This short review article provides a pedagogical introduction to the rapidly growing research field of Majorana fermions in topological superconductors. We first discuss in some details the simplest "toy model" in which Majoranas appear, namely a one-dimensional tight-binding representation of a p-wave superconductor, introduced more than ten years ago by Kitaev. We then give a general introduction to the remarkable properties of Majorana fermions in condensed matter systems, such as their intrinsically non-local nature and exotic exchange statistics, and explain why these quasiparticles are suspected to be especially well suited for low-decoherence quantum information processing. We also discuss the experimentally promising (and perhaps already successfully realized) possibility of creating topological superconductors using semiconductors with strong spin-orbit coupling, proximity-coupled to standard s-wave superconductors and exposed to a magnetic field. The goal is to provide an introduction to the subject for experimentalists or theorists who are new to the field, focusing on the aspects which are most important for understanding the basic physics. The text should be accessible for readers with a basic understanding of quantum mechanics and second quantization, and does not require knowledge of quantum field theory or topological states of matter.