Dirac lattices, zero-range potentials and self adjoint extension

TL;DR

This paper analyzes electromagnetic fields interacting with polarizable point dipoles modeled by delta potentials, comparing methods, and explores the behavior of lattice limits, revealing smooth and singular transitions depending on polarization and regularization.

Contribution

It provides a detailed comparison of self-adjoint extension, regularization, and zero-range potential methods for dipole interactions and investigates the lattice spacing limit effects on electromagnetic scattering and plasmons.

Findings

Smooth transition for scalar and TE polarization in the zero lattice spacing limit.

Singular transition for TM polarization, especially with parallel polarizability.

Regularization and renormalization are necessary for accurate modeling of dipole lattices.

Abstract

We consider the electromagnetic field in the presence of polarizable point dipoles. In the corresponding effective Maxwell equation these dipoles are described by three dimensional delta function potentials. We review the approaches handling these: the selfadjoint extension, regularization/renormalisation and the zero range potential methods. Their close interrelations are discussed in detail and compared with the electrostatic approach which drops the contributions from the self fields. For a homogeneous two dimensional lattice of dipoles we write down the complete solutions, which allow, for example, for an easy numerical treatment of the scattering of the electromagnetic field on the lattice or for investigating plasmons. Using these formulas, we consider the limiting case of vanishing lattice spacing. For a scalar field and for the TE polarization of the electromagnetic field this transition is smooth and results in the results known from the continuous sheet. Especially for the TE polarization, we reproduce the results known from the hydrodynamic model describing a two dimensional electron gas. For the TM polarization, for polarizability parallel and perpendicular to the lattice, in both cases, the transition is singular. For the parallel polarizability this is surprising and different from the hydrodynamic model. For perpendicular polarizability this is what was known in literature. We also investigate the case when the transition is done with dipoles described by smeared delta function, i.e., keeping a regularization. Here, for TM polarization for parallel polarizability, when subsequently doing the limit of vanishing lattice spacing, we reproduce the result known from the hydrodynamic model. In case of perpendicular polarizability we need an additional renormalization to reproduce the result obtained previously by stepping back from the dipole approximation.