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

TL;DR

This paper reveals how Ising spin-orbital coupling causes local breaking of spin and sublattice degeneracies in vortex states of Ising superconductors, inducing antiphase ferromagnetic order and affecting local density of states.

Contribution

It demonstrates the local breaking of degeneracies and induced magnetic orders in vortex cores due to Ising spin-orbital coupling using self-consistent BdG calculations.

Findings

Induced antiphase ferromagnetic order in vortex cores

Splitting and shifting of in-gap states by Ising coupling

Opposite shifts of LDOS peaks on different sublattices

Abstract

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