Diamagnetism and the dispersion of the magnetic permeability

TL;DR

This paper explores the compatibility of diamagnetism with Kramers--Kronig relations by analyzing the dispersion of magnetic permeability, showing that diamagnetic materials can satisfy these relations under certain conditions involving spatial dispersion.

Contribution

It demonstrates that diamagnetism can be compatible with Kramers--Kronig relations when considering spatial dispersion and high-frequency asymptotics, challenging previous assumptions.

Findings

Diamagnetism does not necessarily violate Kramers--Kronig relations.

Permeability can tend to a value different from 1 at high frequencies due to spatial dispersion.

Examples of diamagnetic media with non-trivial high-frequency behavior are provided.

Abstract

It is well known that the usual Kramers--Kronig relations for the relative permeability function $\mu(\omega)$ are not compatible with diamagnetism ($\mu(0)<1$) and a positive imaginary part ($\text{Im}\,\mu(\omega)>0$ for $\omega>0$). We demonstrate that a certain physical meaning can be attributed to $\mu$ for all frequencies, and that in the presence of spatial dispersion, $\mu$ does not necessarily tend to 1 for high frequencies $\omega$ and fixed wavenumber $\mathbf k$. Taking the asymptotic behavior into account, diamagnetism can be compatible with Kramers--Kronig relations even if the imaginary part of the permeability is positive. We provide several examples of diamagnetic media and metamaterials for which $\mu(\omega,\mathbf k)\not\to 1$ as $\omega\to\infty$.