Chiral Majorana hinge modes on curved surface with magnetic impurities

We discuss possible physical realizations of higher-order odd-parity topological superconductors (TSCs) starting with a time-reversal-invariant TSC (class DIII) on a three-dimensional sphere geometry. We construct such phases by introducing time-reversal and inversion symmetry breaking mass terms, that can be engineered with a magnetic field and magnetic impurities on the surface. We show that the RKKY interaction between magnetic impurities on a curved geometry promotes a ferromagnetic radial order due to the Ising properties of Majorana surface fermions, that competes with the external magnetic field. Using the RKKY interaction with dense enough magnetic impurities, we determine the optimal magnetic field to minimize orbital effects and enable the mass term which is odd under inversion and energetically favorable compared to a spherically symmetric hedgehog configuration. The resulting mass term will gap out the surface leaving a gapless one-dimensional chiral Majorana mode along inversion symmetric line on the surface, thus implementing a second-order TSC.