Electro and magneto statics of topological insulators as modeled by planar, spherical and cylindrical $\theta$ boundaries: Green function approach

TL;DR

This paper develops a Green function method to analyze static electromagnetic fields in topological insulators modeled by $ heta$ boundaries with planar, spherical, and cylindrical geometries, enabling precise field calculations for various sources.

Contribution

It constructs the static Green function for $ heta$ electrodynamics in multiple geometries, generalizing Green theorem to include $ heta$ boundary effects, facilitating analytical and numerical solutions.

Findings

Explicit Green functions for different geometries are derived.

The method accurately reproduces known results using image techniques.

The approach enables calculation of fields for arbitrary sources and boundary conditions.

Abstract

The Green function (GF) method is used to analyze the boundary effects produced by a Chern Simons (CS) 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 ($\theta$ ED). This model provides a correct low energy effective action for describing topological insulators (TI). In this work we construct the static GF in $\theta$ ED for different geometrical configurations of the $\theta$ boundary, namely: planar, spherical and cylindrical $\theta$ interfaces. Also we adapt the standard Green theorem to include the effects of the $\theta$ boundary. These are the most important results of our work, since they allow 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. The adapted Green theorem is illustrated by studying the problem of a point like electric charge interacting with a planar TI with prescribed boundary conditions. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the methods of images.