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We study quasiparticle states on a surface of a topological insulator (TI)with proximity-induced superconductivity under an external magnetic field. Anapplied magnetic field creates two Majorana bound states: a vortex Majoranastate localized inside a vortex core and an exterior Majorana state localizedalong a circle centered at the vortex core. We calculate the spin-resolvedlocal density of states (LDOS) and demonstrate that the shrinking of the radiusof the exterior Majorana state, predicted in Ref. [R. S. Akzyanov et al., Phys.Rev. B 94, 125428 (2016)], under a strong magnetic field can be seen in LDOSwithout smeared out by non-zero-energy states. The spin-resolved LDOS furtherreveals that the spin of the exterior Majorana state is strongly polarized.Accordingly, the induced odd-frequency spin-triplet pairs are found to bespin-polarized as well. In order to detect the exterior Majorana states,however, the Fermi energy should be closed to the Dirac point to avoidcontributions from continuum levels. We also study a different two-dimensionaltopological-superconducting system where a two-dimensional electron gas withthe spin-orbit coupling is sandwiched between an s-wave superconductor and aferromagnetic insulator. We show that the radius of an exterior Majorana statecan be tuned by an applied magnetic field. However, on the contrary to theresults at a TI surface, neither the exterior Majorana state nor the inducedodd-frequency spin-triplet pairs are spin-polarized. We conclude that thespin-polarization of the Majorana state is attributed to the spin-polarizedLandau level which is characteristic for systems with the Dirac-likedispersion.

Local density of states in two-dimensional topological superconductors under a magnetic field: signature of an exterior Majorana bound state