  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 letter [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
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 byproduct, 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.
