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We study the topological magnetoelectric effect on a conical topologicalinsulator when a point charge $q$ is near the cone apex. The Hall currentinduced on the cone surface and the image charge configuration are determined.We also study a kind of gravitational Aharonov-Bohm effect in this geometry andrealize a phase diference betwen the components of the wavefunctions (spinors)upon closed parallel transport around the (singular) cone tip. Concretely, anet current flowing towards cone apex (or botton) shows up, yielding electricpolarization of the conical topological insulator. Such an effect may bedetected, for instance, by means of the net accumulated Hall charge near theapex. Once it depends only on the geometry of the material (essetially, thecone apperture angle) this may be faced as a microscopic scale realization of(2+1)-dimensional Einstein gravity.

Geometrically induced electric polarization in conical topological insulators