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Ising spin-orbital coupling is usually easy to identify in the Isingsuperconductors via an in-plane critical field enhancement, but we show thatthe Ising spin-orbital coupling also manifests in the vortex physics forperpendicular magnetic fields. By self-consistently solving the Bogoliubov-deGennes equations of a model Hamiltonian built on the honeycomb lattice with theIsing spin-orbital coupling pertinent to the transition metal dichalcogenides,we numerically investigate the local breaking of the spin and sublatticedegeneracies in the presence of a perpendicular magnetic field. It is revealedthat the ferromagnetic orders are induced inside the vortex core region by theIsing spin-orbital coupling. The induced magnetic orders are antiphase in termsof their opposite polarizations inside the two nearest-neighbor vortices withone of the two polarizations coming dominantly from one sublattice sites,implying the local breaking of the spin and sublattice degeneracies. Thefinite-energy peaks of the local-density-of-states for spin-up and spin-downin-gap states are split and shifted oppositely by the Ising spin-orbitalcoupling, and the relative shifts of them on sublattices $A$ and $B$ are alsoof opposite algebraic sign. The calculated results and the proposed scenariomay not only serve as experimental signatures for identifying the Isingspin-orbital coupling in the Ising superconductors, but also be prospective inmanipulation of electron spins in motion through the orbital effect in thesuperconducting vortex states.

Local breaking of the spin degeneracy in the vortex states of Ising superconductors: Induced antiphase ferromagnetic order