Electro- and magnetostatics of topological insulators as modeled by planar, spherical, and cylindrical θ boundaries: Green’s function approach
Urrutia, L. F.
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The Green's function method is used to analyze the boundary effects produced by a Chern-Simons extension to electrodynamics. We consider the electromagnetic field coupled to a theta term that is piecewise constant in different regions of space, separated by a common interface Sigma, the theta boundary, model which we will refer to as theta electrodynamics. This model provides a correct low-energy effective action for describing topological insulators. Features arising due to the presence of the boundary, such as magnetoelectric effects, are already known in Chern-Simons extended electrodynamics, and solutions for some experimental setups have been found with a specific configuration of sources. In this work we construct the static Green's function in theta electrodynamics for different geometrical configurations of the theta boundary, namely, planar, spherical and cylindrical theta-interfaces. Also, we adapt the standard Green's theorem to include the effects of the theta boundary. These are the most important results of our work, since they allow one to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well-defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for theta boundaries are provided. On the one hand, the adapted Green's theorem is illustrated by studying the problem of a pointlike electric charge interacting with a planar topological insulator with prescribed boundary conditions. On the other hand, we calculate the electric and magnetic static fields produced by the following sources: (i) a pointlike electric charge near a spherical theta boundary, (ii) an infinitely straight current-carrying wire near a cylindrical theta boundary and (iii) an infinitely straight uniformly charged wire near a cylindrical theta boundary. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the method of images.