Majorana vortex modes in spin-singlet chiral superconductors with noncollinear spin ordering: Local density of states study

 https://doi.org/10.1103/PhysRevB.107.144513

In this study, topologically nontrivial edge and vortex bound states are described in the coexistence phase of chiral spin-singlet superconductivity and noncollinear spin ordering on a triangular lattice in the presence of few (up to four) vortices. We consider the topological phase transition induced by the magnetic order between the phase hosting Majorana modes and the initial phase of the chiral $d$-wave superconductivity supporting non-Majorana modes which is also topologically nontrivial. The change of the excitation spectrum at the critical point is obtained in both cases of open and periodic boundary conditions in the presence of vortices. It is proved that zero-energy Majorana modes localized at vortex cores are caused by noncollinear long-range magnetic ordering. Even though nearby excitation energies of subgap states including the edge-localized and vortex-localized states are very close to each other, the energy difference between different vortex bound states is an order of magnitude higher. This difference determines the energy gap for Majorana vortex modes separating them from other vortex bound states. It is found that even in the presence of noncollinear spin ordering its value can be estimated from the excitation energy of vortex bound states in the pure chiral $d$-wave state for the nonmagnetic case. By studying local density of states near the vortex cores the possibility to experimentally detect the described Majorana vortex modes by scanning tunneling microscopy is discussed. It is demonstrated that Majorana vortex modes and Majorana antivortex modes induced by noncollinear magnetism have different features in energy and spatially resolved density of states due to the chiral symmetry on the superconducting order parameter.