The electronic spectrum on the spherical surface of a topological insulator reflects an active property of the helical surface state that stems from a constraint on its spin on a curved surface. The induced effective vector potential (spin connection) can be interpreted as an effective vector potential associated with a fictitious magnetic monopole induced at the center of the sphere. The strength of the induced magnetic monopole is found to be g=2pi, -2pi, being the smallest finite (absolute) value compatible with the Dirac quantization condition. We have established an explicit correspondence between the bulk Hamiltonian and the effective Dirac operator on the curved spherical surface. An explicit construction of the surface spinor wave functions implies a rich spin texture possibly realized on the surface of topological insulator nanoparticles. The electronic spectrum inferred by the obtained effective surface Dirac theory, confirmed also by the bulk tight-binding calculation, suggests a specific photo absorption/emission spectrum of such nanoparticles.
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