Majorana fermions in topological superconductors with spin-orbit interaction

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 faulttolerant 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. 1. What is Majorana fermion First, I would like to explain briefly what a Majorana fermion is. The definition of a Majorana fermion is very simple. A Majorana fermion is a Dirac fermion which satisfies the Majorana condition. Its Hamiltonian is given by the Dirac Hamiltonian, H(k) = σ · k (2D) or H(kx) = ckx (1D), (1) and its wavefunction Ψ satisfies the Majorana condition