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Majorana zero modes in condensed matter systems have been the subject of muchinterest in recent years. Their non-Abelian exchange statistics, making them aunique state of matter, and their potential applications in topological quantumcomputation, earned them attention from both theorists and experimentalists. Itis generally understood that in order to form Majorana zero modes inquasi-one-dimensional topological insulators, time-reversal symmetry must bebroken. The straightforward mechanisms for doing so -- applying magnetic fieldsor coupling to ferromagnets -- turned out to have many unwanted side effects,such as degradation of superconductivity and the formation of sub-gap states,which is part of the reason Majorana zero modes have been eluding directexperimental detection for a long time. Here we review several proposal thatrely on controlling the phase of the superconducting order parameter, either asthe sole mechanism for time-reversal-symmetry breaking, or as an additionalhandy knob used to reduce the applied magnetic field. These proposals holdpractical promise to improve Majorana formation, and they shed light on thephysics underlying the formation of the topological superconducting state.

Majorana zero modes induced by superconducting phase bias