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Majorana Fermions in Semiconductor Nanowires: Fundamentals, Modeling, and Experiment

After a recent series of rapid and exciting developments, the long search for the Majorana fermion - the elusive quantum entity at the border between particles and antiparticles - has produced the first positive experimental results, but is not over yet. Originally proposed by E. Majorana in the context of particle physics, Majorana fermions have a condensed matter analog in the zero-energy bound states emerging in topological superconductors. A promising route to engineering topological superconductors capable of hosting Majorana zero modes consists of proximity coupling semiconductor thin films or nanowires with strong spin-orbit interaction to conventional s-wave superconductors in the presence of an external Zeeman field. The Majorana zero mode is predicted to emerge above a certain critical Zeeman field as a zero-energy state localized near the order parameter defects, viz., vortices for thin films and wire-ends for the nanowire. These Majorana bound states are expected to manifest non--Abelian quantum statistics, which makes them ideal building blocks for fault--tolerant topological quantum computation. This review provides an update on current status of the search for Majorana fermions in semiconductor nanowires by focusing on the recent developments, in particular the period following the first reports of experimental signatures consistent with the realization of Majorana bound states in semiconductor nanowire--superconductor hybrid structures. We start with a discussion of the fundamental aspects of the subject, followed by considerations on the realistic modeling which is a critical bridge between theoretical predictions based on idealized conditions and the real world, as probed experimentally. The last part is dedicated to a few intriguing issues that were brought to the fore by the recent encouraging experimental advances.