The examination of supposedly well-known condensed matter systems through the prism of topology has led to the discovery of new quantum phenomena that were previously overlooked. Just like insulators can present topological phases characterized by Dirac edge states, superconductors can exhibit topological phases characterized by Majorana edge states. In particular, one-dimensional topological superconductors are predicted to host zero energy Majorana fermions at their extremities. Zero bias anomalies localized at the edge of proximity induced superconducting wires were interpreted as fingerprints of the emergence of topological superconductivity [1,2].

By contrast, two-dimensional (2D) superconductors have a one-dimensional boundary which would naturally lead to propagating edge states characterized by a Dirac-like dispersion. In addition to dispersive edge states, 2D topological superconductors are also supposed to support localized Majorana bound states in their vortex cores. We will review different options that were investigated in order to obtain two-dimensional topological superconductors, either by using hybrid magnetic-superconducting system or by proximity effect to the surface state of a topological insulator. The former approach allowed to measure dispersive Majorana edge states in 2D systems [3,4] while the latter allowed exhibiting zero energy Majorana states in vortex cores [5,6].

In addition to zero energy states one usually gets also a very large amount of Caroli-Matricon de Gennes states that fill the gap in topological vortex cores except when the Fermi energy is comparable to the energy gap [5,6]. However we will show that a new kind of vortex like phase defects can induce zero energy Majorana bound states without additional low energy states. In particular, a phase defect in the spin-orbit field or a skyrmion like magnetic defect can lead to the formation of energetically isolated pairs of Majorana zero modes in a hard gap of a 2D topological superconductor [7].

References
[1] V. Mourik et al., Science 336, 1003 (2012)
[2] S. Nadj-Perge, et al., Science 346, 602 (2014)
[3] G. C. Ménard et al., Nature Communications 8, 2040 (2017)
[4] A. Palacio-Morales et al., Science Advance 5, eaav6600 (2019)
[5] D. Wang et al., Science 362, 333 (2018)
[6] Q. Liu, et al., Phys. Rev. X 8, 041056 (2018)
[7] G. C. Ménard et al., Nature Communications 10, 2587 (2019)