A self-consistent investigation of the topological state in the Ising superconductors.

By self-consistently solving the Bogoliubov-de Gennes equations, we investigate the superconductivity, the magnetization and the local density-of-state (DOS) of the Ising superconductor. The calculations show that the Ising spin orbital coupling is responsible for the anisotropic behaviors of the Ising superconductor in response to the external magnetic field. In the absence of the magnetic impurities, only opposite spin triplet superconducting (SC) pairing correlations can be induced from the intrinsic on-site singlet SC pairing. The equal spin triplet SC correlations can be induced when a magnetic impurity chain with in-plane spin direction is deposited on the Ising superconductor. The chain could support a pair of Majorana Fermions, which are originated from the induced 1-dimensional equal spin triplet SC pairing correlations. The local DOS for the Majorana Fermion states exhibits an excellent agreement with the experimental observations. The low energy spectrum as a function of the exchange coupling strength forms two hourglass shapes. The waist of the hourglass may serves as a boundary to separate the zero energy crossings of the two lowest energy levels into topologically trivial and non-trivial quantum states. These results not only account for the experimental observations in a realistic way, but also clarify several important questions in the studies of the Ising superconductors.