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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>q</a:mi> <a:mo>=</a:mo> <a:mi>N</a:mi> <a:mi>e</a:mi> </a:mrow> </a:math> , where <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>N</b:mi> </b:math> is a positive integer and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>e</c:mi> </c:math> is the elementary electric charge, we obtain that the rotation frequency is given by <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>ω</d:mi> <d:mo>=</d:mo> <d:msup> <d:mrow> <d:mo>(</d:mo> <d:mi>N</d:mi> <d:mi>α</d:mi> <d:mo>)</d:mo> </d:mrow> <d:mn>2</d:mn> </d:msup> <d:mi mathvariant="normal">Υ</d:mi> <d:mrow> <d:mo>(</d:mo> <d:mi>ε</d:mi> <d:mo>,</d:mo> <d:mi>d</d:mi> <d:mo>/</d:mo> <d:mi>a</d:mi> <d:mo>)</d:mo> </d:mrow> <d:mo>/</d:mo> <d:mi>I</d:mi> </d:mrow> </d:math> , where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mi>I</f:mi> </f:math> is the moment of inertia, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>α</g:mi> </g:math> is the fine-structure constant, and the function <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi mathvariant="normal">Υ</h:mi> </h:math> depends on the dielectric constant <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>ε</j:mi> </j:math> and the relative distance <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>d</k:mi> <k:mo>/</k:mo> <k:mi>a</k:mi> </k:mrow> </k:math> of the charge from the center of the sphere of radius <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>a</l:mi> </l:math> . Since the point charge also induces Hall currents on the surface, we also compute their associated angular momentum. This allows us to derive an exact expression for the electronic velocity on the surface as a function of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>/</m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> .