Feynman paradox in a spherical axion insulator

We show that a small charged probe near a spherical topological insulator causes the latter to rotate around a symmetry axis defined by the center of the sphere and the position of the charge outside the latter. The rotation occurs when the distance from the charge to the center of the sphere is changed. This phenomenon occurs due to induced static fields and is a consequence of the axion electrodynamics underlying the electromagnetic response of a topological insulator. Assuming a regime where the charged probe can be regarded as a point charge $q=Ne$, where $N$ is a positive integer and $e$ is the elementary electric charge, we obtain that the rotation frequency is given by $\omega=(N\alpha)^2\Upsilon(\epsilon,d/a)/I$, where $I$ is the moment of inertia, $\alpha$ is the fine-structure constant, and the function $\Upsilon$ depends on the dielectric constant $\epsilon$ and the relative distance $d/a$ of the charge from the center of the sphere of radius $a$. Since the point charge also induces Hall currents on the surface, we compute also their associated angular momentum. This allows us to derive an exact expression for the electronic velocity on the surface as a function of $a/d$.