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

TLDR

Non‑Abelian topological order in superconductors is defined by zero‑energy Majorana fermions localized at edges and vortices, which obey non‑Abelian statistics. This work investigates the non‑Abelian topological phase of spin‑singlet superconductors with Rashba spin‑orbit coupling, extending the authors’ earlier proposal. The authors analyze s‑wave and d+id pairing states, compute topological invariants, and examine the stability of Majorana modes under magnetic fields. They find that d+id pairing hosts Majorana fermions across almost all parameter regimes when the Fermi level lies near the Γ or M points, while s‑wave pairing requires a Zeeman energy exceeding the superconducting gap; moreover, Rashba spin‑singlet superconductors support more robust Majorana modes than spin‑triplet p+ip systems, and topological order also emerges in Rashba‑coupled spin or charge density waves, producing edge Dirac fermions and charge fractionalization. The study builds on the authors’ previous Phys.

Abstract

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.