  Understanding and control of spin degrees of freedom on the surfaces of
topological materials are key to future applications as well as for realizing
novel physics such as the axion electrodynamics associated with time-reversal
(TR) symmetry breaking on the surface. We experimentally demonstrate
magnetically induced spin reorientation phenomena simultaneous with a
Dirac-metal to gapped-insulator transition on the surfaces of manganese-doped
Bi2Se3 thin films. The resulting electronic groundstate exhibits unique
hedgehog-like spin textures at low energies, which directly demonstrate the
mechanics of TR symmetry breaking on the surface. We further show that an
insulating gap induced by quantum tunnelling between surfaces exhibits spin
texture modulation at low energies but respects TR invariance. These spin
phenomena and the control of their Fermi surface geometrical phase first
demonstrated in our experiments pave the way for the future realization of many
predicted exotic magnetic phenomena of topological origin.
