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Search for Majorana fermions in superconductors

This is a colloquium-style introduction to the midgap excitations in superconductors known as Majorana fermions. These elusive particles, equal to their own antiparticle, may or may not exist in Nature as elementary building blocks, but in condensed matter they can be constructed out of electron and hole excitations. What is needed is a superconductor to hide the charge difference, and a topological (Berry) phase to eliminate the energy difference from zero-point motion. A pair of widely separated Majorana fermions, bound to magnetic or electrostatic defects, has non-Abelian exchange statistics. A qubit encoded in this Majorana pair is expected to have an unusually long coherence time. We discuss strategies to detect Majorana fermions in a topological superconductor, as well as possible applications in a quantum computer. The status of the experimental search is reviewed. Contents: I. What Are They? (Their origin in particle physics; Their emergence in superconductors; Their potential for quantum computing) II. How to Make Them (Shockley mechanism; Chiral p-wave superconductors; Topological insulators; Semiconductor heterostructures) III. How to Detect Them (Half-integer conductance quantization; Nonlocal tunneling; 4π-periodic Josephson effect; Thermal metal-insulator transition) IV. How to Use Them (Topological qubits; Read out; Braiding) V. Outlook on the Experimental Progress [scheduled for vol. 4 of Annual Review of Condensed Matter Physics]