The Feynman paradox in a spherical axion insulator

TL;DR

This work analyzes a point charge near a spherical topological insulator within axion electrodynamics, showing that static fields carry angular momentum and can induce rotation of the TI body as the charge moves. Using an image-charge method adapted to axion boundary conditions, the authors derive both electric and magnetic image charges and line densities, enabling exact expressions for the electromagnetic angular momentum and the Hall-current–driven mechanical angular momentum on the surface. The key results include a rotation frequency $\omega = \frac{(N\alpha)^2}{I}\,\Upsilon(\epsilon, d/a)$ and an electromagnetic angular momentum $L_z=(N\alpha)^2\hbar\Upsilon(\epsilon, d/a)$, with a far-field limit $L_z^{\infty}=\frac{4(N\alpha)^2\hbar}{3(2+\epsilon)}(\frac{a}{d})^3$, and a surface electron velocity $v_e$ that scales as $(a/d)^2$, illustrating how angular momentum is transferred between the electromagnetic field and the TI. The analysis demonstrates a macroscopic realization of the Feynman paradox in a finite geometry and clarifies the role of image charges and Hall currents in TI versus trivial dielectrics, including limits to infinite-sphere and slab configurations.

Abstract

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 $ω=(Nα)^2Υ(ε,d/a)/I$, where $I$ is the moment of inertia, $α$ is the fine-structure constant, and the function $Υ$ depends on the dielectric constant $ε$ 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 $a/d$.

Figures (4)

• Figure 1: (a) Schematic representation of the problem of a spherical TI of radius $R=a$ in the presence of a point charge $q$ located at $\bm{R}_0=(0,0,d)$. (b) Due to the axion term, the point charge induces both electric and magnetic fields which can be thought to be generated by pairs of electric and magnetic point charges $(q_-,g_-)$ and $(q_+,g_+)$ located at $z=d$ (the so called Kelvin point) and $z=d_K=a^2/d$, respectively, and magnetic line charge densities $\mu_\pm(z)$ (shown in blue) in the intervals $0<z<d_K$ and $z>d$.
• Figure 2: Result of the summation in Eq. (\ref{['eq:LzSum']}) for the $L_z/(N\alpha)^2\hbar$ dependence on $d/a$ in log-log scale for $\epsilon=1 \text{ and } 20$. $N$ is taken to be 1.
• Figure 3: The $d$-dependence of the ratio of the electromagnetic angular momentum $L_z^\infty$ when $d\gg a$, to the angular momentum of the Hall current $L_{\text{Mech},z}$, with different values of $N$. The gray dashed line crosses points, where the ratio is equal to unity.
• Figure 4: Infinite magnetoelectric slab of thickness $L$ discussed in Appendix A.