Witten effect in a crystalline topological insulator

TL;DR

The paper addresses the Witten effect in a solid-state axion medium by realizing axion electrodynamics in a strong topological insulator (STI). Using a minimal lattice model, it demonstrates that the STI's axion angle is θ = π and that a bulk magnetic monopole binds a fractional charge Q = -e/2 (mod e), consistent with Witten's prediction, with the bound charge remaining robust under weak perturbations. It further proposes experimental routes to test the effect, including emergent monopoles in spin ice and a planar monopole in a thin-film STI realized via an exciton condensate, and provides numerical evidence for the charge localization and its dependence on disorder and Zeeman coupling. These results offer a concrete, numerically validated pathway to tabletop tests of a fundamental high-energy physics concept and shed light on spin-charge separation in 3D topological quantum matter.

Abstract

It has been noted a long time ago that a term of the form theta (e^2/2πh) B dot E may be added to the standard Maxwell Lagrangian without modifying the familiar laws of electricity and magnetism. theta is known to particle physicists as the 'axion' field and whether or not it has a nonzero expectation value in vacuum remains a fundamental open question of the Standard Model. A key manifestation of the axion term is the Witten effect: a unit magnetic monopole placed inside a medium with non-zero theta is predicted to bind a (generally fractional) electric charge -e(theta/2 pi+n) with n integer. Here we conduct a first test of the Witten effect, based on the recently established fact that the axion term with theta=pi emerges naturally in the description of the electromagnetic response of a new class of crystalline solids called topological insulators - materials distinguished by strong spin-orbit coupling and non-trivial band structure. Using a simple physical model for a topological insulator, we demonstrate the existence of a fractional charge bound to a monopole by an explicit numerical calculation. We also propose a scheme for generating an 'artificial' magnetic monopole in a topological insulator film, that may be used to facilitate the first experimental test of Witten's prediction.

Figures (2)

• Figure 1: (Color online) Charge density in our model TI on the cube-shaped lattice with $20^{3}$ sites with a unit monopole at its center, with parameters $t=\lambda$, $\epsilon=4t$, leading to a bulk gap $\Delta=4t$. (a) Charge density $\delta\rho$ of the three closest layers below the monopole, for $g=0$. (b) The excess charge $\delta Q(r)$ (in units of $e$) for different Zeeman coupling $g$. The knee feature seen at $r=10$ corresponds to the radius at which the sphere used to calculate $\delta Q(r)$ first touches the system boundary. (c) Log-log plot of $\delta Q_{g}-\delta Q_{0}$ showing the power-law approach $\sim r^{-\alpha}$ of the accumulated charge to its assymptotic value of $1/2$. The least-square fit yields exponents $\alpha=2.85,3.04,2.79$ for $g=2,6,10$, respectively. We attribute the deviations of the numerically determined exponent $\alpha$ from the expected value of 3 to the finite size effect.
• Figure 2: (Color online) A cubic sample of a TI including disorder with a planar unit monopole at its center, size $L=14$ and parameters as in Fig. \ref{['fig_monopole']}. In all cases shown we use the same disorder realization but vary its overall strength parametrized by $\mu$. Panels (a) and (b) show charge density $2-\rho_{1}$ for the layer just below the planar monopole for weak disorder $\mu=0.05\Delta$, and (c) larger disorder $\mu=0.20\Delta$. Panels (e) and (f) show the difference in charge density $\delta\rho=\rho_{0}-\rho_{1}$ for $\mu/\Delta=0.05,0.20$, respectively, for the same layer. (d) The excess charge $\delta Q(r)$ (in units of $e$) for different values of disorder strength $\mu$. The inset shows a close up of the saturation. At this scale a small deviation from the expected asymptotic value $1/2$ that increases with the disorder strength becomes visible. We attribute this deviation to the finite-size effect in our numerical calculation. This identification is supported by the fact that the deviations grow more pronounced for smaller system sizes and close to the surface. Also, it is consistent with the notion that the bound charge is localized on the length scale $\xi\sim 1/\Delta$ which increases as the disorder reduces the spectral gap.