Majorana fermions in topological superconductors with spin-orbit interaction

Abstract

When a topological order is realized in a certain class of superconductors, it supports Majorana fermions on the edge or in the vortex core. Vortices with Majorana fermion modes are neither fermions nor bosons but non-Abelian anyons, obeying the non-Abelian statistics for which the exchange operations of particles are not commutative. Because of this remarkable feature, Majorana fermions plays an important role for the realization of fault-tolerant topological quantum computation.Here I would like to report our recent work on Majorana fermions and non-Abelian anyons in topological superconductors with spin-orbit interaction [1, 2, 3, 4, 5, 6]. Until recently, only spin-triplet superconducting states, such as such ν = 5/2 fractional quantum Hall states or Sr2RuO4, had been known to exhibit Majorana fermions [7, 8, 9, 10, 11]. However, now it is revealed that spin-singlet superconductors also support Majorana fermions if we take into account the spin-orbit interaction. While a possible realization of non-Abelian anyon in an s-wave superconducting state was first discussed in the context of cosmology [1], by using a zero mode solution in a vortex by Jackiw and Rossi [12] (or its non-gauged version [13]), now there are many examples of such systems in condensed matter physics. In particular, a scheme using the spin-orbit interaction and Zeeman one, which was proposed in [2, 3, 4], is powerful to realize Majorana fermions in various systems. It is shown that Majorana fermions are realized in full-gapped s-wave superfluids/superconductors [3, 4, 14, 15]. Also, Majorana fermions is shown to possible even in nodal superconductors such as high-Tc cuprates [5].We also show that a peculiar Majorana fermion is obtained in a certain non-centrosymmetric superconductor [6, 16, 17]. In sharp contrast to ordinary Majorana fermions, the novel Majorana fermion has a flat dispersion and preserves the time-reversal invariance.