Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors

The non-Abelian topological order for superconductors is characterized by the existence of zero-energy Majorana fermions in edges of systems and in a vortex of a macroscopic condensate, which obey the non-Abelian statistics. This paper is devoted to an extensive study on the non-Abelian topological phase of spin-singlet superconductors with the Rashba spin-orbit interaction proposed in our previous paper [M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. 103, 020401 (2009)]. We mainly consider the $s$-wave pairing state and the $d+id$ pairing state. In the case of $d+id$-wave pairing, Majorana fermions appear in almost all parameter regions of the mixed state under an applied magnetic field, provided that the Fermi level crosses $k$ points in the vicinity of the $\ensuremath{\Gamma}$ point or the M point in the Brillouin zone while in the case of $s$-wave pairing, a strong magnetic field, the Zeeman energy of which is larger than the superconducting gap is required to realize the topological phase. We clarify that Majorana fermions in Rashba spin-singlet superconductors are much more stable than those realized in spin-triplet $p+ip$ superconductors in certain parameter regions. We also investigate the topological number which ensures the topological stability of the phase in detail. Furthermore, as a by-product, we found that topological order is also realized in conventional spin (or charge)-density wave states with the Rashba spin-orbit interaction, for which massless Dirac fermions appear in the edge of the systems and charge fractionalization occurs.