# A Drude-Lorentz dielectric in the presence of a magnetic current density

Pedro D. S. Silvaa<EMAIL_ADDRESS><EMAIL_ADDRESS>M. J.
Nevesb<EMAIL_ADDRESS>Manoel M. Ferreira Jr.a,c<EMAIL_ADDRESS><EMAIL_ADDRESS>aPrograma de Pós-graduação em Física, Universidade
Federal do Maranhão, Campus Universitário do Bacanga, São Luís (MA),
65080-805, Brazil  bDepartamento de Física, Universidade Federal Rural do Rio
de Janeiro, BR 465-07, Seropédica (RJ), 23890-971, Brazil  cDepartamento de
Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, São
Luís (MA), 65080-805, Brazil

###### Abstract

The dispersive propagation and absorption in the domain of a Drude-Lorentz
dielectric modified by a magnetic current are classically investigated, taking
as the starting point the Drude electric permittivity, in connection with the
chiral magnetic effect. Using standard electromagnetic methods, the dispersion
relations and refractive indices are evaluated and discussed, as well as the
group and the energy velocities. We consider three configurations for the
magnetic conductivity tensor: isotropic, antisymmetric, and symmetric. In all
these cases, we carry out the group velocity and the energy velocity, for
investigating the properties of signal propagation. We observe that the
isotropic magnetic conductivity enhances the signal propagation at low and
intermediary frequencies, while the symmetric and antisymmetric conductivity
severely constrains the wave propagation in these frequency windows.

Drude-Lorentz model, Magnetic current density, optical effects.

###### pacs:

11.15.-q; 11.10.Ef; 11.10.Nx

##  I Introduction

The Maxwell’s electrodynamics a general framework to describe the
electromagnetic interaction in both classical and quantum scales. The
classical electrodynamics explains the interaction of charges in motion via
solutions of Maxwell’s equations [1, 2, 3]. The quantum electrodynamics (QED)
describes with excellent precision the interaction of the photons with leptons
at the MeV-scale [4]. The electrodynamics applied to the material physics is
one of the basic tools of Condensed Matter Physics (CMP). In theoretical CMP,
many models are developed to study the electronic properties of solids. One of
the examples is the Drude model, initially used to explain the properties of
the electrons transported through the metals [5, 6], providing a complex and
dispersive conductivity and refractive index associated with the absorption of
electromagnetic waves in metals [7]. The Drude model is also applied to
describe the optical propagation in dielectric media [7, 8, 9], in which the
electromagnetic waves interact with vibrating atoms, considered as classical
dipoles. This kind of approach allows us to obtain optical properties of
solids, including dispersion, absorption and optical activity.

In some specific continuous media, there appears a magnetic current density
linear in the magnetic field, originated from an asymmetry between the density
of the left- and right-handed chiral fermions [10, 11, 12]. This is known as a
macroscopical manifestation of the quantum Chiral Magnetic Effect (CME), being
investigated in different research areas in physics, as Cosmology, quark-gluon
plasm, electroweak interactions and others [13, 14, 15, 16, 17]. In CMP, the
CME is associated with the Weyl semimetals (WSM) in which massless fermions
acquire velocity along the magnetic field whose direction is driven by the
chirality of these fermions [18]. In WSM, the magnetic current density may
present a general form, ${\bf{J}}=\alpha\,(\bf{E}\cdot\bf{B})\,{\bf{B}}$ [19,
20, 21], with $\alpha$ being a constant, in such a way the magnetic current,

$\displaystyle{\bf{J}}=\sigma^{B}\cdot{\bf{B}}\;,$ (1)

is written in terms of an anisotropic magnetic conductivity tensor denoted by
$\sigma^{B}_{ij}$, which depends on the material properties. Classical optical
properties of a dielectric medium endowed magnetic current were already
investigated, supposing the medium is described by usual constitutive
relations, $\mathbf{D}={\epsilon}\,\mathbf{E}$ and
$\mathbf{B}={\mu}\,\mathbf{H}$ [22]. The refractive indices, propagating
modes, and optical effects were examined for three configurations of the
magnetic conductivity tensor,

${\sigma}_{ij}^{B}=\Sigma\,\delta_{ij}\;,\quad{\sigma}_{ij}^{B}={\epsilon}_{ijk}\,b_{k}\;,\quad\sigma_{ij}^{B}=\frac{1}{2}\left(a_{i}\,c_{j}+a_{j}\,c_{i}\right)\;,$
(2)

associated with isotropic, antisymmetric [24] and symmetric currents, given as

$\displaystyle{\bf{J}}=\Sigma\,{\bf{B}}\;,\quad{\bf{J}}={\bf{b}}\times{\bf{B}}\;,\quad{\bf{J}}=\frac{1}{2}\left[(\bf{a}\cdot\bf{B})\bf{c}+(\bf{c}\cdot\bf{B})\bf{a}\right]\;,$
(3)

where $\bf{a}$, $\bf{b}$, $\bf{c}$ are constant vectors in space.

The electrodynamics of a continuous medium is described by the Lagrangian
density,

$\displaystyle{\cal L}=-\frac{1}{4}\,G_{\mu\nu}F^{\mu\nu}-J_{\mu}A^{\mu}\;,$
(4)

where $F^{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and
$G^{\mu\nu}=\chi^{\mu\nu\rho\lambda}F_{\rho\lambda}/2$ are the field strength
electromagnetic tensors in vacuum and matter, and $J^{\mu}=(\rho,{\bf J})$ is
the source four-current111We adopt natural units $\hbar=c=1$, and the
Minkowski metric $\eta^{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$ throughout this
paper. The constitive tensor $\chi^{\mu\nu\rho\lambda}$ satisfies the symmetry
properties $\chi^{\mu\nu\rho\lambda}=-\chi^{\nu\mu\rho\lambda}$,
$\chi^{\mu\nu\rho\lambda}=-\chi^{\mu\nu\lambda\rho}$ and
$\chi^{\mu\nu\rho\lambda}=\chi^{\rho\lambda\mu\nu}$. The action principle
applied to the Lagrangian (4) yields the field equation

$\displaystyle\partial_{\mu}G^{\mu\nu}=J^{\nu}\;,$ (5)

while the EM strength field tensor satisfies the Bianchi identity,
$\partial_{\mu}\widetilde{F}^{\mu\nu}=0$, where
$\widetilde{F}^{\mu\nu}=\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}/2$ is the
dual tensor. Explicitly, the motion equations can be written in the standard
form

$\displaystyle\nabla\cdot{\bf D}\\!$ $\displaystyle=$ $\displaystyle\\!0\hskip
8.5359pt,\hskip 8.5359pt\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial
t}={\bf 0}\;,$ (6a) $\displaystyle\nabla\cdot{\bf B}\\!$ $\displaystyle=$
$\displaystyle\\!0\hskip 8.5359pt,\hskip 8.5359pt\nabla\times{\bf
H}-\frac{\partial{\bf D}}{\partial t}={\bf J}\;,$ (6b)

where ${\bf D}$ and ${\bf H}$ are the fields in the matter represented by the
components $G^{0i}=-D^{i}$ and $H^{i}=\epsilon^{ijk}G^{jk}/2$. In this paper,
we use the constitutive relations,

$\displaystyle{\bf{D}}$
$\displaystyle=\epsilon(\omega)\,{\bf{E}}\;,\quad{\bf{H}}=\frac{1}{\mu}\,{\bf{B}}\;,$
(7)

applied to a Drude-Lorentz dielectric, whose complex permittivity

$\displaystyle\epsilon(\omega)=\epsilon_{0}\left[1+\frac{\omega_{p}^{2}}{\omega_{0}^{2}-\omega^{2}-i\,\gamma\,\omega}\right]\;,$
(8)

takes into account the electron’s damped oscillation. Here, $\omega_{0}$ is
the free electron frequency, $\gamma$ is the damping factor and
$\omega_{p}=\sqrt{nq^{2}/\epsilon_{0}m}$ is the plasma frequency. From the
Drude-Lorentz model, $q=-e$ is the electron charge, $m$ is the electron mass,
and $n$ is the number density. Equivalently, one can write Eq. (8) as

$\displaystyle\epsilon(\omega)$
$\displaystyle=\epsilon^{\prime}(\omega)+i\epsilon^{\prime\prime}(\omega)\;,$
(9)

where

$\displaystyle\epsilon^{\prime}(\omega)$
$\displaystyle=\epsilon_{0}\left[1+\frac{\omega_{p}^{2}\,(\omega_{0}^{2}-\omega^{2})}{(\omega_{0}^{2}-\omega^{2})^{2}+\gamma^{2}\omega^{2}}\right]\;,$
(10a) $\displaystyle\epsilon^{\prime\prime}(\omega)$
$\displaystyle=\epsilon_{0}\left[\frac{\gamma\,\omega_{p}^{2}\,\omega}{(\omega_{0}^{2}-\omega^{2})^{2}+\gamma^{2}\omega^{2}}\right]\;.$
(10b)

This Drude dielectric, described by the usual constitutive relations (7), will
be examined in the presence of the non conventional current constitutive
relation,

${J}_{i}={\sigma}_{ij}^{B}\,{B}_{j}\;.$ (11)

Substituting the plane wave solutions, ${\bf E}({\bf r},t)={\bf E}\,e^{i({\bf
k}\cdot{\bf r}-\omega t)}$ and ${\bf B}({\bf r},t)={\bf B}\,e^{i({\bf
k}\cdot{\bf r}-\omega t)}$ in the Maxwell relations, and using the current
density (11), we obtain the following equation for the electric field ${\bf
E}$:

$\displaystyle\left[\,{\bf
k}^{2}\,\delta_{ij}-k_{i}\,k_{j}-\omega^{2}\,\mu\,\overline{\epsilon}_{ij}(\omega)\,\right]E^{j}=0\;,$
(12)

with the effective permittivity,

$\displaystyle\overline{\epsilon}_{ij}(\omega)=\epsilon(\omega)\,\delta_{ij}-\frac{i}{\omega^{2}}\,\sigma^{B}_{ia}\epsilon_{abj}\,k_{b}\;,$
(13)

where $\sigma^{B}_{ij}$ is the magnetic conductivity tensor, and
$\epsilon(\omega)$ is the dielectric permittivity. Equation (12) can be
written in terms of the refractive index components $n^{i}=k^{i}/\omega$, in
which the refractive index of the medium is defined by
$n=+\sqrt{n_{i}\,n_{i}}$. Thereby, (12) can be cast into the form

$\displaystyle M_{ij}E^{j}=0\;,$ (14)

in which the tensor $M_{ij}$ reads

$\displaystyle
M_{ij}=n^{2}\,\delta_{ij}-n_{i}\,n_{j}-\mu\,\overline{\epsilon}_{ij}(\omega)\;.$
(15)

The nontrivial solutions for $E^{j}$ in Eq. (14) are obtained by requiring
$\mathrm{det}[M_{ij}]=0$, which provides the dispersion relations of the model
and all the useful information to obtain the group and energy velocities. In
dispersive media the electromagnetic fields may be complex, ${\bf E}={\bf
E}^{\prime}+i\,{\bf E}^{\prime\prime}$, ${\bf B}={\bf B}^{\prime}+i\,{\bf
B}^{\prime\prime}$. Additionally, in the presence of absorption the wave
vector is complex,

${\bf k}={\bf k}^{\prime}+i\,{\bf k}^{\prime\prime}\;,$ (16)

with the energy balance being ruled by the Poynting’s theorem [1], whose
Poynting vector and electromagnetic energy density (stored in the field) are

$\displaystyle{\bf S}\\!$ $\displaystyle=$ $\displaystyle\left({\bf
E}\times{\bf H}^{\ast}\right)=\frac{1}{2\mu^{\prime}}\left({\bf E}\times{\bf
B}^{\ast}\right)\;,$ (17a) $\displaystyle u_{EM}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{2}\frac{\partial[\omega\epsilon^{\prime}]}{\partial\omega}\left({\bf{E}}\cdot{\bf{E}}^{*}\right)+\frac{1}{2\mu^{\prime}}\left({\bf{B}}\cdot{\bf{B}}^{*}\right)\;.$
(17b)

In the absence of sources $(\rho=0)$, the Gauss’s law is compatible with the
attainment of transversal modes,

$\displaystyle({\bf{k}}^{\prime}\cdot{\bf{E}}^{\prime})=({\bf{k}}^{\prime\prime}\cdot{\bf{E}}^{\prime\prime})=0\;,$
(18a)
$\displaystyle({\bf{k}}^{\prime}\cdot{\bf{E}}^{\prime\prime})=({\bf{k}}^{\prime\prime}\cdot{\bf{E}}^{\prime})=0\;,$
(18b)

for which the time-averaged quantities read

$\displaystyle\left\langle{\bf{S}}\right\rangle$
$\displaystyle=\frac{{\bf{E}}^{2}}{2\omega\mu^{\prime}}\,{\bf k}^{\prime}\;,$
(19a) $\displaystyle\left\langle u_{EM}\right\rangle$
$\displaystyle=\frac{1}{4}\left[\epsilon^{\prime}+\omega\,\frac{\partial\epsilon^{\prime}}{\partial\omega}+\frac{|{\bf{k}}|^{2}}{\mu\omega^{2}}\right]{\bf{E}}^{2}\;.$
(19b)

In lossy media, the ratio of these quantities defines the energy velocity,

$\displaystyle{\bf{v}}_{E}$
$\displaystyle=\frac{\left\langle{\bf{S}}\right\rangle}{\left\langle
u_{EM}\right\rangle}\;,$ (20)

suitable to describe the energy transport in the presence of absorption.

In a non-absorbing dieletric ($\epsilon^{\prime\prime}=0$), the group velocity

$\displaystyle{\bf{v}}_{g}$
$\displaystyle=\frac{d\omega}{dk}\,\hat{\bf{k}}\;,$ (21)

is a real quantity suitable for representing the signal propagation. However,
in an absorbing medium, the group velocity becomes complex, as
${\bf{v}}_{g}=\mathrm{Re}[d\omega/dk]+i\,\mathrm{Im}[d\omega/dk]$, no longer
describing energy transport, being replaced by the energy velocity [27, 28,
29, 30, 31, 32] in this respect.

For the Drude dielectric permittivity, Eq. (19b) is rewritten as

$\displaystyle\langle
u_{EM}\rangle=\frac{\epsilon_{0}|{\bf{E}}|^{2}}{4}\left[1+\frac{\omega_{p}^{2}(\omega_{0}^{2}-\omega^{2})}{\Gamma_{+}(\omega)}+\frac{2\omega^{2}\omega_{p}^{2}\Gamma_{-}}{\Gamma_{+}^{2}(\omega)}+\frac{|{\bf
k}|^{2}}{\mu\epsilon_{0}\omega^{2}}\right],$ (22)

with
$\Gamma_{\pm}(\omega)=(\omega_{0}^{2}-\omega^{2})^{2}\pm\gamma^{2}\omega^{2}$,
in such a way the energy velocity reads

$\displaystyle{\bf v}_{E}=\frac{{\bf
k}^{\prime}}{\mu\epsilon_{0}\omega}\left[1+\frac{\omega_{p}^{2}(\omega_{0}^{2}-\omega^{2})}{\Gamma_{+}(\omega)}+\frac{2\omega^{2}\omega_{p}^{2}\Gamma_{-}}{\Gamma_{+}^{2}(\omega)}+\frac{|{\bf
k}|^{2}}{\mu\epsilon_{0}\omega^{2}}\right]^{-1},$ (23)

By definition, ${\bf v}_{E}$ points along the $\hat{{\bf
k}}^{\prime}$-direction, also depending on the dispersion relation of
$k(\omega)$. In general non absorbing dielectrics, energy velocity and group
velocity are the same one. But such an equality that is lost in lossy media,
where it holds ${\bf v}_{E}\neq{\bf v}_{g}$. Such results have been recently
verified for dispersive, absorbing and lossless hyperbolic metamaterial (HMM)
[33], as far as general bi-anisotropic media [34]. In accordance with the
definition (20), the energy velocity depends on the way the energy density and
the dissipated energy density in a lossy medium are written [35, 36], which
constitute an active line of research actually [37, 38].

An alternative procedure to investigate group and energy velocities in
dissipative dynamical systems [39] was used recently to investigate the group
and energy propagation in a dielectric medium endowed with magnetic
conductivity [40]. Such a study was performed considering
$\epsilon^{\prime}=cte$ and $\epsilon^{\prime\prime}=0$ or
$\epsilon^{\prime}=cte$ and $\epsilon^{\prime\prime}=\sigma\omega^{-1}$,
comparing group velocity and energy velocity for the cases of isotropic,
symmetric and antisymmetric magnetic conductivity shown in Eq. (2).

In this paper, we investigate the properties of propagation and absorption in
a realistic Drude dielectric (DD) endowed with magnetic conductivity following
the procedures of Ref. [40]. Initially, we obtain the solutions of refractive
indices, the propagating modes for the scenarios of an isotropic conductivity
magnetic, including evaluation of the group and the energy velocities.
Posteriorly, it is done for the case of anisotropic conductivity magnetic
tensor, antisymmetric and a symmetric tensor. The features of the refractive
indices are discussed having as comparison basis the DD ones. The paper is
organized as follows. Section II is dedicated to the case the isotropic tensor
$\sigma^{B}_{ij}$. In Section III we obtain the results for an antisymmetric
magnetic conductivity tensor parametrized by one constant vector. Section IV
is devoted to examinining the symmetric conductivity tensor configuration
parametrized by two constant vectors. All these sections are divided in two
subsections in which the optical properties and the group/energy velocities
aspects are addressed. For end, in the section V, we present the conclusions
and the final remarks.

##  II The case of the isotropic magnetic conductivity tensor

The magnetic current density with a diagonal current, the isotropic
conductivity magnetic tensor is

$\displaystyle\sigma^{B}_{ij}=\Sigma\,\delta_{ij}\;,$ (24)

where $\Sigma$ is a real and positive constant, in which the bar permittivity
tensor is

$\displaystyle\overline{\epsilon}_{ij}(\omega)=\epsilon(\omega)\,\delta_{ij}+\frac{i\,\Sigma}{\omega}\,\epsilon_{ijk}\,n_{k}\;.$
(25)

The null determinant of the $M_{ij}$ matrix implies into the $n$-polynomial
equation

$\displaystyle\left[\,n^{2}-\mu\,\epsilon(\omega)\,\right]^{2}-n^{2}\,\frac{\mu^{2}\Sigma^{2}}{\omega^{2}}=0\;.$
(26)

The solutions of the equation (26) are given by

$\displaystyle n_{\pm}$
$\displaystyle=\sqrt{\mu(\epsilon^{\prime}(\omega)+i\epsilon^{\prime\prime}(\omega))+\left(\frac{\mu\Sigma}{2\omega}\right)^{2}}\pm\frac{\mu\Sigma}{2\omega}\;.$
(27)

Substituting the dielectric permittivity (10), both solutions can be split
into their real and imaginary pieces,

$\displaystyle n_{\pm}$ $\displaystyle=\Re[n_{\pm}]+i\,\Im[n_{\pm}]\;,$ (28)

where

$\displaystyle\Re[n_{\pm}]$
$\displaystyle=\sqrt{\frac{|h(\omega)|}{2}}\,I_{+}\pm\frac{\mu\Sigma}{2\omega}\;,$
(29a) $\displaystyle\Im[n_{\pm}]$
$\displaystyle=\sqrt{\frac{|h(\omega)|}{2}}\,I_{-}\;,$ (29b)

with

$\displaystyle I_{\pm}$
$\displaystyle\equiv\sqrt{\sqrt{1+\mu^{2}\,\left[\frac{\epsilon^{\prime\prime}(\omega)}{h(\omega)}\right]^{2}}\pm\mbox{sgn}[h(\omega)]}\;,$
(30a) $\displaystyle h(\omega)$
$\displaystyle=\mu\epsilon^{\prime}(\omega)+\left(\frac{\mu\Sigma}{2\omega}\right)^{2}\;,$
(30b)

where sgn represents the signal function of $h(\omega)$. Notice that
$\epsilon^{\prime}(\omega)$ from (10a) can assume negative values for a given
range of $\omega$-frequency. Consequently, the $h(\omega)$-function can also
be negative.

The general behavior of $n_{\pm}$ real and imaginary pieces, written in Eqs.
(29a) and (29b), is illustrated in Figs. 1 and 2 in terms of the dimensionless
variable $\omega/\omega_{0}$. We note that both $\Re[n_{\pm}]$ indicate
propagation regime for the range $\omega<\omega_{0}$. Near the origin the real
piece of $n_{+}$ tends to infinity, an analog behavior of magnetized plasmas
[41]. At the resonance frequency, $\omega=\omega_{0}$, there occurs an
absorption peak which coincides with the beginning of the window for
$\Im[n_{\pm}]\neq 0$. Furthermore, immediately after $\omega=\omega_{0}$ the
values of $\Re[n_{\pm}]$ rapidly decrease (anomalous dispersion), being
followed by a short increasing region for the index $n_{+}$, which in the
sequel exhibits another anomalous region (extending to the asymptotic limit);
see Fig. 1. As for the index $n_{-}$, however, the anomalous dispersion region
is followed by a normal dispersion one; see Fig. 2. For both indices, the
absorption peak occurs in the region $\omega_{1}^{i}<\omega<\omega_{2}^{i}$,
with $\omega_{1}^{i}\approx\omega_{0}$, which defines the absorption window.
Note that near the origin, the real piece of $n_{-}$ is null, which is
distinct from the conventional DD behavior.

Figure 1: Refractive index $n_{+}$ of Eq. (28). Solid (dotted) curves
represent the real (imaginary) pieces of $n_{+}$. The black lines represent
the usual case, where $\Sigma=0$. The red ones represent the index for
$\Sigma\neq 0$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and
$\Sigma/\omega_{0}=3$. The frequency $\omega_{2}^{i}$ marks the end of
absorption zone, which is now shortened by the presence of the magnetic
conductivity.

It is worth mentioning that negative refraction occurs for $n_{-}$ in the zone
$\omega_{1}^{i}<\omega<\omega^{\prime\prime}$, which also contains the
absorption frequency window. This negative refraction is a consequence of the
isotropic magnetic conductivity. For $n_{\pm}$, the frequency $\omega_{2}^{i}$
defines the absorption zone limit, which is reduced in the presence of the
magnetic conductivity. It is also interesting to note that, besides the
absorption width reduction, the indices $n_{\pm}$ also develop a non-null real
piece, enabling energy transport in this region, not allowed in the
correspondent usual DD one. For $\omega>\omega_{2}^{i}$, the absorption is
null and attenuation-free propagation occurs. For very high-frequencies
$(\omega\gg\omega_{0})$, one obtains
$\Re[n_{\pm}]\rightarrow\sqrt{\mu\epsilon_{0}}$.

Figure 2: Refractive index $n_{-}$ of Eq. (28). Solid (dotted) curves
represent the real (imaginary) pieces of $n_{-}$. Black curves indicate the
usual case, where $\Sigma=0$. The blue ones represent the index for
$\Sigma\neq 0$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and
$\Sigma/\omega_{0}=3$. The absorption window narrowing becomes evident for
$\Sigma\neq 0$.

For very low-frequencies ($\omega\ll\omega_{0}$), the refractive indices
behave as

$\displaystyle
n_{\pm}\simeq\sqrt{\mu\epsilon_{0}\left(1+\frac{\omega_{p}^{2}}{\omega_{0}^{2}}\right)+\left(\frac{\mu\Sigma}{2\omega}\right)^{2}}\pm\frac{\mu\Sigma}{2\omega}\;,$
(31)

which near the origin yields $n_{-}=0$ and $n_{+}\simeq\mu\Sigma/\omega$, thus
explaining the $n_{+}$ divergence at $\omega\rightarrow 0$ in Fig. 1. It is
worth to point out that such low-frequency behavior,
$\Re[n_{+}]\rightarrow\infty$, $\Re[n_{-}]\rightarrow 0$, is already present
in the case of a simple dielectric with constant magnetic conductivity
$\epsilon^{\prime}=cte$, $\epsilon^{\prime\prime}=\sigma/\omega$,
$\Sigma=cte$. See Ref. [40]. As a consistency check, one notes that when
$\Sigma\rightarrow 0$, the refractive indices of Eq. (31) yield a finite and
non-null value in the origin, recovering the standard scenario. See the black
continuous lines in Figs. 1 and 2.

The peak of the imaginary part is located at $\omega\approx\omega_{0}$. Using
Eq. (29b) and expressions (10a), (10b), the peak amplitude is

$\displaystyle\left.\frac{}{}\Im(n_{\pm})\right|_{\omega=\omega_{0}}$
$\displaystyle=\sqrt{\frac{\mu\epsilon_{0}}{2}+\left(\frac{\mu\Sigma}{2\sqrt{2}\omega_{0}}\right)^{2}}\,\mathcal{I}\;,$
(32)

with

$\displaystyle\mathcal{I}$
$\displaystyle=\sqrt{\sqrt{1+\left[\frac{4\omega_{0}\omega_{p}^{2}\epsilon_{0}}{\gamma\left(4\omega_{0}^{2}\epsilon_{0}+\mu\Sigma^{2}\right)}\right]^{2}}-1}\;,$
(33)

revealing that the magnetic conductivity also modifies the absorption peak
magnitude. In the next section, we will discuss the consequence of $\Sigma$ on
the imaginary part of $n_{\pm}$.

###  II.1 Propagation modes and optical effects

Let us consider propagation along the ${\cal Z}$-axis, ${\bf{n}}=(0,0,n)$,
such that the tensor $M_{ij}$ becomes

$\displaystyle[M_{ij}]$
$\displaystyle=\left[\begin{array}[]{ccc}n^{2}-\mu\epsilon(\omega)&+i\mu\frac{\Sigma}{\omega}n&0\\\
\\\ -i\mu\frac{\Sigma}{\omega}n&n^{2}-\mu\epsilon(\omega)&0\\\ \\\
0&0&-\mu\epsilon(\omega)\end{array}\right]\;.$ (39)

In order to find the propagating modes, we solve the equation $M_{ij}E^{j}=0$,
including in the matrix (39) the solution of Eq. (26), that is,

$\displaystyle n^{2}-\mu\epsilon(\omega)=\pm\,n\,\frac{\mu\Sigma}{\omega}\;,$
(40)

providing the following electric fields:

$\displaystyle{\bf{E}}_{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ \pm\,i\\\
0\end{pmatrix}\;,$ (41)

which represent left-handed (LCP) and right-handed (RCP) circularly polarized
vectors, respectively. The two different real refractive indices associated
with the modes (41) imply circular birefringence, ascribed to the isotropic
magnetic conductivity, $\Sigma$, and expressed in terms of the rotatory power
(RP),

$\displaystyle\delta$
$\displaystyle=-\frac{\omega}{2}\left[\Re(n_{+})-\Re(n_{-})\right]\;,$ (42)

specialized for the indices (28) as

$\displaystyle\delta$ $\displaystyle=-\frac{\mu\Sigma}{2}\;.$ (43)

It is a nondispersive RP that depends linearly on the magnetic conductivity,
being equal to the one that appears in nondispersive dielectrics
($\epsilon^{\prime}=cte$) endowed with isotropic magnetic conductivity [22].
It is interesting to mention that the absorption peak amplitude decreases with
an increasing magnitude of the magnetic conductivity $\Sigma$. This is in
agreement with Eq. (32), which provides $\Im(n_{\pm})\rightarrow 0$ in the
limit of very large conductivity $(\Sigma\rightarrow\infty)$. In this limit,
therefore, this Drude-Lorentz-like dielectric exhibits a nearly null
absorption. Such an effect is represented by the different peak amplitudes in
Fig. 3.

In lossy systems, the dichroism takes place when one propagating mode is more
strongly absorbed than the other, which is directly associated to the
imaginary piece of the refractive indices. Dichroism represents a tool to
distinguish between Dirac and Weyl semimetals [42], being used also for
enantiomeric discrimination [43, 44] and for developing graphene-based devices
at terahertz frequencies [45]. For circularly polarized modes, the difference
of absorption between left and right-handed circularly polarized modes [46] is
given by the circular dichroism coefficient,

${\delta}_{\mathrm{d}}=-\frac{\omega}{2}[\mathrm{Im}(n_{+})-\mathrm{Im}(n_{-})]\,.$
(44)

Concerning the modes (41) associated with the refractive indices (28), the
dichroism coefficient is null,

${\delta}_{\mathrm{d}}=0,$ (45)

since these indices have equal imaginary parts, see Eq. (29b). This is the
same result obtained for the case of a simple dielectric with constant
magnetic conductivity $\epsilon^{\prime}=cte$, $\Sigma=cte$ [40]. In this
sense, the Drude permittivity, although being complex and involved, does not
modify either the RP or the dichroism coefficient in relation to the optical
panorama of a simpler dielectric.

Figure 3:  Imaginary part of $n_{\pm}$ of Eq. (28) in terms of
$\omega/\omega_{0}$. Here, we have set $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and
$\Sigma/\omega_{0}=2$ (blue), $\Sigma/\omega_{0}=10$ (red), and
$\Sigma/\omega_{0}=20$ (black).

###  II.2 The phase, group, and the energy velocities

Implementing $n=\sqrt{{\bf{k}}^{2}}/\omega$ in Eq. (40), we obtain

$\displaystyle k^{2}\mp\mu\Sigma\,k-\mu\,\omega^{2}\,\epsilon(\omega)=0\;,$
(46)

whose the solutions read

$\displaystyle k_{\pm}$
$\displaystyle=\omega\,\sqrt{\mu\epsilon^{\prime}(\omega)+i\mu\epsilon^{\prime\prime}(\omega)+\frac{\mu^{2}\Sigma^{2}}{4\omega^{2}}}\pm\frac{\mu\Sigma}{2}\;,$
(47)

with $\epsilon^{\prime}$ and $\epsilon^{\prime\prime}$ of Eqs. (10a) and
(10b), respectively. These solutions can also be written as
$k_{\pm}=k_{\pm}^{\prime}+i\,k_{\pm}^{\prime\prime}$, where

$\displaystyle k_{\pm}^{\prime}$ $\displaystyle=$
$\displaystyle\omega\,\sqrt{\frac{|h(\omega)|}{2}}\,I_{+}\pm\frac{\mu\Sigma}{2}\;,$
(48a) $\displaystyle k_{\pm}^{\prime\prime}$ $\displaystyle=$
$\displaystyle\omega\,\sqrt{\frac{|h(\omega)|}{2}}\,I_{-}\;,$ (48b)

and the functions $I_{\pm}$ and $h(\omega)$ are given in Eqs. (30a) and (30b).

For the propagation analysis, we consider the real part Eq. (48a), which
represents the two distinct modes, $k_{+}$ and $k_{-}$. The phase velocities
are

$\displaystyle v_{\mathrm{ph}}^{(\pm)}$
$\displaystyle=\frac{\omega}{k^{\prime}_{\pm}}=\frac{1}{\Re[n_{\pm}]}\;,$ (49)

yielding the phase velocity difference, $\Delta
v_{\mathrm{ph}}=v_{\mathrm{ph}}^{(-)}-v_{\mathrm{ph}}^{(+)}$, namely,

$\displaystyle\Delta v_{\mathrm{ph}}$
$\displaystyle=\frac{4\omega\,\mu\,\Sigma}{2\omega^{2}\,|h(\omega)|\,I_{+}^{2}-\mu^{2}\,\Sigma^{2}}\;,$
(50)

indicating that the magnetic conductivity $\Sigma$ engenders distinct modes
propagating with different phase velocities and birefringence. The group
velocity is given in this case by

$\displaystyle v_{g}^{(\pm)}$
$\displaystyle=\frac{1}{\mu}\frac{k_{\pm}/\omega\mp\mu\Sigma/(2\omega)}{\epsilon(\omega)+(\partial_{\omega}\epsilon)\omega/2}\;,$
(51)

with $\partial_{\omega}\epsilon=\partial\epsilon/\partial\omega$. Considering
the expression (47), the group velocity takes on the form,

$\displaystyle
v_{g}^{\pm}=\frac{1}{\mu}\frac{\displaystyle\sqrt{\mu\epsilon^{\prime}(\omega)+i\mu\epsilon^{\prime\prime}(\omega)+\frac{\mu^{2}\Sigma^{2}}{4\omega^{2}}}}{\displaystyle\epsilon^{\prime}(\omega)+i\epsilon^{\prime\prime}(\omega)+\frac{\omega}{2}(\partial_{\omega}{\epsilon})},$
(52)

which can be rewritten by separating its real and imaginary pieces as follows:

$\displaystyle\Re[v_{g}^{(\pm)}]\\!$ $\displaystyle=$
$\displaystyle\\!\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|h(\omega)|}{2}}\left(\,I_{+}\,C+\,I_{-}\,D\right)\;,\;\;$
(53a) $\displaystyle\Im[v_{g}^{(\pm)}]\\!$ $\displaystyle=$
$\displaystyle\\!\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|h(\omega)|}{2}}\left(\,I_{-}\,C-I_{+}\,D\,\right)\;,$
(53b)

where

$\displaystyle C$
$\displaystyle=\epsilon^{\prime}(\omega)+\frac{\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}\,,$
(54a) $\displaystyle D$
$\displaystyle=\epsilon^{\prime\prime}(\omega)+\frac{\omega}{2}\frac{\partial\epsilon^{\prime\prime}}{\partial\omega}\;.$
(54b)

For the real part of the group velocity, see Eq. (53a), it goes to infinity
for very low-frequencies $(\omega\rightarrow 0)$, losing physical meaning.
Figure 4 illustrates the behavior of $v_{g}$. In the high-frequency regime
$(\omega\rightarrow\infty)$, one finds $\Re[v_{g}^{(\pm)}]\rightarrow
1/\sqrt{\mu\epsilon_{0}}$ and $\Im[v_{g}^{(\pm)}]\rightarrow 0$. This
asymptotic behavior is the same one attained in the limit $\Sigma\rightarrow
0$ and $\omega_{p}\rightarrow 0$, which implies $\epsilon^{\prime}\rightarrow
1$ and $\epsilon^{\prime\prime}\rightarrow 0$, recovering a non-absorbing
dielectric.

Figure 4:  Group velocity of Eq. (51) or Eq. (52) as function of the
dimensionless parameter $\omega/\omega_{0}$. Solid (dashed) lines represent
the real (imaginary) parts of $v_{g}$, respectively. Here, we have used:
$\mu=1$, $\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$,
$\gamma/\omega_{0}=0.01$, and $\Sigma/\omega_{0}=3$ (blue), and $\Sigma=0$
(black, usual case). The vertical dashed lines indicate
$\omega/\omega_{0}\approx\\{1,1.44,1.8\\}$, from left to right, accordingly
with Eq. (II.2).

In scenarios where the group velocity is complex, the sign propagation is
better represented by the energy velocity, $V_{E}$, which is here obtained by
substituting the solutions (48a) and (48b) in expression (23), yielding

$\displaystyle V_{E}^{\pm}$
$\displaystyle=\frac{\displaystyle\sqrt{2|h(\omega)|}\omega
I_{+}\pm\mu\Sigma}{2\omega U^{I}_{\pm}}\;,$ (55a) associated with
$k_{+}^{\prime}$ and $k_{-}^{\prime}$, respectively, with $\displaystyle
U^{I}_{\pm}$
$\displaystyle=\frac{\mu\epsilon^{\prime}}{2}+\frac{\mu\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}+\frac{|h(\omega)|}{4}(I_{+}^{2}+I_{-}^{2})+\frac{\mu^{2}\Sigma^{2}}{8\omega^{2}}+$
$\displaystyle\phantom{=}\pm\frac{\mu\Sigma}{2\omega}\,\sqrt{\frac{|h(\omega)|}{2}}\,I_{-}\;.$
(55b)

Figure 5 illustrates the energy velocities as function of the dimensionless
parameter $\omega/\omega_{0}$. The characteristic frequencies
$\omega^{i}_{1,2}$ are given by the real non-negative solutions of the sixth-
order equation in $\omega$,

$\displaystyle\gamma^{2}\left(\omega^{4}+\frac{\mu\Sigma^{2}}{4\epsilon_{0}}\omega_{0}^{2}\right)+(\omega_{0}^{2}-\omega^{2})\times$
$\displaystyle\times\left[\omega^{2}\left(\omega_{0}^{2}-\omega^{2}+\omega_{p}^{2}\right)+\frac{\mu\Sigma^{2}}{4\epsilon_{0}}\left(\omega_{0}^{2}-\omega^{2}-\gamma^{2}\right)\right]=0\;,$

and $\omega^{i}_{1,2}$ define the limits of the absorption zone for the
propagating modes. This absorption effect modifies the behavior of
$V_{E}^{\pm}$ with the frequency, as one can see. The frequency
$\omega^{\prime\prime}$ marks the end of the absorption window for a usual DD,
where the energy velocity is vanishing. Note, however, that the energy
velocity becomes non-null in the absorption zone of the DD modified by the
magnetic conductivity since the refractive indices develop a finite real part.
It implies the reduction of the lossy window, favoring the signal propagation.

It is also relevant to note the much more involved energy velocity of Fig. 5
in relation to the correspondent one of Ref. [40], obviously a consequence of
the intricate Drude permittivity here considered.

Figure 5:  Energy velocities of Eq. (55a) in terms of the dimensionless
parameter $\omega/\omega_{0}$. Red (blue) lines represent $V_{E}^{\pm}$,
respectively. The black dashed curve illustrates the usual case (23), where
$\Sigma=0$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and
$\Sigma/\omega_{0}=3$. The vertical dashed lines indicate
$\omega/\omega_{0}\approx\\{1,1.44,1.8\\}$, from left to right, accordingly
with Eq. (II.2).

The real non-negative solutions of Eq. (II.2) yield the characteristic
frequencies $\omega^{i}_{1,2}$ when $\Sigma\neq 0$, and also the frequencies
$\omega^{\prime}$, $\omega^{\prime\prime}$ for $\Sigma=0$. In the numerical
examples we have considered, one obtains
$\omega^{\prime}\approx\omega^{i}_{1}$. This is the reason only
$\omega^{i}_{1}$ appears in the plots. Furthermore, we observe that:

  * •

For $0<\omega<\omega^{i}_{1}$, both modes propagates without attenuation. In
this regime, the energy velocity $V_{E}^{\pm}$ increases slowly with
frequency, starting from zero. It decreases as long as it approaches the
resonance frequency, $\omega\rightarrow\omega_{0}\approx\omega^{i}_{1}$, where
there is absorption peaks indicated by the dotted lines in Figs. 1 and 2. At
this frequency, the energy velocities $V_{E}^{+}$ and $V_{E}^{-}$ tend to
zero. As for $V_{E}^{-}$, it exactly holds $V_{E}^{-}(\omega^{i}_{1})=0$.

  * •

For $\omega^{i}_{1}<\omega<\omega^{i}_{2}$, there occurs propagation and
absorption simultaneously (non-null real and imaginary pieces of $n_{\pm}$),
since the energy velocities assume non-null values. In fact, $V_{E}^{+}$ is
positive in this interval (see the red curve in Fig. 5. On the other hand, the
velocity $V_{E}^{-}$ is negative in this zone, indicating negative refraction.
This latter feature is in accordance with Fig. 2, where the real piece of
$n_{-}$ (blue solid curve) is negative in
$\omega^{i}_{1}<\omega<\omega^{\prime\prime}$. The frequency $\omega^{i}_{2}$
defines the end of the absorption band, whose width is shortened by the
presence of magnetic conductivity. At the same time, the real part of the
indices becomes finite (positive or negative), allowing the signal
propagation, $V_{E}^{\pm}\neq 0$ (forbidden in this interval for a usual Drude
dielectric).

  * •

For $\omega^{i}_{2}<\omega<\omega^{\prime\prime}$, there is no absorption as
one can see in the dotted curves of Figs. 1 and 2. Consequently, the energy
velocities $V_{E}^{\pm}$ start to increase progressively to their asymptotic
values (which coincide with the usual case ones). Furthermore, in this range
one still has $V_{E}^{-}<0$, which is in agreement with the negative
refraction of $n_{-}$ that occurs for
$\omega^{i}_{1}<\omega<\omega^{\prime\prime}$.

  * •

For $\omega>\omega^{i}_{2}$: it happens propagation without attenuation for
both modes $n_{\pm}$.

One last comment, yet very subtle, concerns the similarity between the black
solid curves in Figs. 4 and 5. These curves represent the real part of the
group velocity and the energy velocity, both for $\Sigma=0$, respectively.
Equations (55) and (53a) for $\Sigma=0$ read

$\displaystyle V_{E}^{\pm}$
$\displaystyle=\frac{\displaystyle\sqrt{\frac{|h(\omega)|}{2}}I_{+}}{\displaystyle\mu
C+\frac{\mu}{2}\sqrt{\epsilon^{\prime}(\omega)^{2}+\epsilon^{\prime\prime}(\omega)^{2}}-\mu\epsilon^{\prime}(\omega)/2}\;,$
(57a) $\displaystyle\Re[v_{g}^{\pm}]$
$\displaystyle=\frac{\displaystyle\sqrt{\frac{|h(\omega)|}{2}}\left(I_{+}C+I_{-}D\right)}{\displaystyle\mu\epsilon_{0}\left(C^{2}+D^{2}\right)}\;.$
(57b)

These velocities are only equivalent when there is no absorption, $\gamma=0$.
Hence, applying this condition, one finds $\epsilon^{\prime\prime}=0$ and
$D=0$, such that

$\displaystyle V_{E}^{\pm}-\Re[v_{g}^{\pm}]$
$\displaystyle=-\frac{\sqrt{|\epsilon^{\prime}(\omega)|+\epsilon^{\prime}(\omega)}}{C\sqrt{2\mu\epsilon_{0}}}\,\frac{|\epsilon^{\prime}(\omega)|-\epsilon^{\prime}(\omega)}{2C+|\epsilon^{\prime}(\omega)|-\epsilon^{\prime}(\omega)}\;,$
(58)

where we have used

$\displaystyle I^{\prime}_{+}$
$\displaystyle=\sqrt{1+\mathrm{sgn}[\mu\epsilon^{\prime}(\omega)]}\;.$ (59)

Obviously, it holds

$\displaystyle V_{E}^{\pm}-\Re[v_{g}^{\pm}]$ $\displaystyle=0\;,$ (60)

for $\epsilon^{\prime}(\omega)>0$ or $\epsilon^{\prime}(\omega)<0$. Note that
one has: i) $\epsilon^{\prime}(\omega)>0$ for $\omega<\omega_{0}$, or
$\omega>\sqrt{\omega_{0}^{2}+\omega_{p}^{2}}$; ii)
$\epsilon^{\prime}(\omega)<0$, when
$\omega_{0}<\omega<\sqrt{\omega_{0}^{2}+\omega_{p}^{2}}$.

##  III The case of antisymmetric magnetic conductivity tensor

In this case, the antisymmetric conductivity tensor is parameterized in terms
of a vector ${\bf{b}}$,

$\displaystyle\sigma_{ij}^{B}=\epsilon_{ijk}\,b_{k}\;,$ (61)

implying an antisymmetric magnetic current density,

$\displaystyle{\bf{J}}={\bf{b}}\times{\bf{B}}\;,$ (62)

where $\bf{b}$ is a constant vector in space. Such an antisymmetric magnetic
current was investigated in the context of Weyl semimetals [24], where
$\hat{\bf{b}}$ defines a direction in space. This kind of allowed transverse
chiral magnetic photocurrent (orthogonal to the $\bf{B}$ field) is induced by
linearly polarized light in WSMs in the presence of parity violation. The
effective permittivity tensor is

$\displaystyle\overline{\epsilon}_{ij}(\omega)=\left[\,\epsilon(\omega)-\frac{i}{\omega}({\bf
b}\cdot{\bf n})\,\right]\delta_{ij}+\frac{i}{\omega}\,n_{i}\,b_{j}\;,$ (63)

which replaced in Eq. (14) yields the $n$-polynomial equation

$\displaystyle{\bf n}^{2}+\frac{i\,\mu}{\omega}\,({\bf b}\cdot{\bf
n})-\mu\,\epsilon(\omega)=0\;,$ (64)

whose solution provides the dispersive refractive index

$\displaystyle n(\omega)=-\frac{i\,\mu}{2\omega}\,(\hat{{\bf k}}\cdot{\bf
b})+\sqrt{\mu\epsilon(\omega)-\frac{\mu^{2}}{4\omega^{2}}(\hat{{\bf
k}}\cdot{\bf b})^{2}}\;,$ (65)

where $\hat{\bf{k}}$ is the direction of propagation.

Substituting the dielectric permittivity (8), the refractive index (65) can be
rewritten as

$\displaystyle n(\omega)=\Re[n(\omega)]+i\,\Im[n(\omega)]\;,$ (66a) where the
real and imaginary parts are $\displaystyle\Re[n]$
$\displaystyle=\sqrt{\frac{|f(\omega)|}{2}}\,A_{+}\;,$ (66b)
$\displaystyle\Im[n]$
$\displaystyle=\sqrt{\frac{|f(\omega)|}{2}}\,A_{-}-\frac{\mu}{2\omega}(\hat{\bf{k}}\cdot{\bf{b}})\;,$
(66c) with $\epsilon^{\prime}$ and $\epsilon^{\prime\prime}$ of Eqs. (10a) and
(10b), respectively, and $\displaystyle A_{\pm}$
$\displaystyle=\sqrt{\sqrt{1+\mu^{2}\epsilon_{0}^{2}\left[\frac{\epsilon^{\prime\prime}(\omega)}{f(\omega)}\right]^{2}}\pm\mbox{sgn}[f(\omega)]}\;,$
(66d)

where $f(\omega)=\mu\epsilon^{\prime}(\omega)-\mu^{2}({\bf b}\cdot\hat{{\bf
k}})^{2}/(2\omega)^{2}$.

The general behaviors of $\Re[n]$ and $\Im[n]$ are represented in Figs. 6 and
7, considering two distinct configurations of the vector $\bf{b}$ relative to
the direction of propagation, $\cos\theta=\pm 1$, defining propagation
parallel and anti-parallel to ${\bf{b}}$, respectively, with
$\hat{\bf{k}}\cdot{\bf{b}}=b\cos\theta$. The case $\theta=\pi/2$ is
effectively equivalent to the situation of a dielectric stripped of magnetic
conductivity ($b=0$), which is represented by a black line in Figs. 6 and 7.

Figure 6: Refractive index $n$ of Eq. (66a). Solid (dotted) curves represent
the real (imaginary) pieces of $n$. Here, we have used: $\mu=1$,
$\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$,
$b/\omega_{0}=2.5$, $\cos\theta=1$ (red). The black lines indicate the usual
case. The frequency $\omega^{a}_{2}$ defines the end of the modified
absorption zone, enlarged in relation to the one of a usual DD. Figure 7:
Refractive index $n$ of Eq. (66a). Solid (dotted) curves represent the real
(imaginary) pieces of $n$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, $b/\omega_{0}=2.5$,
$\cos\theta=-1$ (blue). The black lines indicate the usual case. Here one also
notes the enlargment of the absorption zone in relation to the one of a usual
DD.

Near the origin, the real piece of $n$ goes to infinity for $\cos\theta=1$
whereas it goes to zero for the antiparallel propagation ($\cos\theta=-1$), a
different behavior in comparison with the conventional DD, as already
mentioned. We notice in Figs. 6 and 7 that around $\omega/\omega_{0}=1$ the
real parts of $n$ for parallel and anti-parallel propagation $(\cos\theta=\pm
1)$ become equivalent and nearly overlap to the black line. This superposition
also occurs in the very high-frequencies regime.

In the very high-frequency regime, the real part goes as
$\Re[n](\omega\gg\omega_{0})\rightarrow\sqrt{\mu\epsilon_{0}}$. For the very
low-frequency limit, $\omega\rightarrow 0$, the refractive index becomes
purely imaginary,

$\displaystyle n$ $\displaystyle\simeq\frac{i\mu
b}{2\omega}\left(-\cos\theta+|\cos\theta|\right)\;,$ (67)

establishing $\Re[n]=0$ near the origin. Further, it yields $\Im[n]=0$ for
$\cos\theta=1$ (see dashed line in Fig. 6) and a diverging $n$ for
$\cos\theta=-1$ (see dashed line in Fig. 7).

Differently from the standard Drude-Lorentz model, where the absorption occurs
around the resonance frequency $\omega_{0}$, the presence of a non-null
magnetic conductivity, given by $b=|{\bf{b}}|$, enables absorption even in the
small frequency regime, $\omega\ll\omega_{0}$, accordingly to Eq. (67).
Therefore, in this antisymmetric scenario for the magnetic conductivity,
absorption effects can occur in regions outside the neighborhood of the
resonance frequency $\omega_{0}$. On the other hand, such an effect does not
happen by means of absorption peak as it occurs in the DD.

The peaks of the real and imaginary parts are given by

$\displaystyle\Re[n](\omega=\omega_{0})$
$\displaystyle=\sqrt{\frac{|f(\omega_{0})|}{2}}A_{+}(\omega_{0})\;,$ (68a)
$\displaystyle\Im[n](\omega=\omega_{0})$
$\displaystyle=\sqrt{\frac{|f(\omega_{0})|}{2}}A_{-}(\omega_{0})-\frac{\mu}{2\omega_{0}}(\hat{\bf{k}}\cdot{\bf{b}})\;,$
(68b)

where $f(\omega_{0})=\mu\epsilon_{0}-\mu^{2}({\bf b}\cdot\hat{\bf
k})^{2}/(2\omega_{0})^{2}$. A very subtle difference between the absorption
peaks (imaginary part of $n$) can be observed by comparing Figs. 6 and 7. One
notices that $\Im[n]|_{\theta=0}<\Im[n]|_{\theta=\pi}$, a direct consequence
of the last term in Eq. (68b), which reveals an observable distinction
regarding the propagation direction relative to the magnetic conductivity
vector ${\bf{b}}$.

###  III.1 Propagation modes

The propagation modes are found for ${\bf{n}}$ along the ${\cal Z}$-axis, and
the vector $\mathbf{b}$ written as

$\mathbf{n}=(0,0,n)\,,\quad\mathbf{b}=b\left(0,\sin\theta,\cos\theta\right)\equiv(0,b_{2},b_{3})\;,$
(69)

where $\theta$ is the angle between $\mathbf{n}$ and $\mathbf{b}$. The matrix
$M_{ij}$ of Eq. (15) is

$[M_{ij}]=\left[\begin{array}[]{ccc}n^{2}-\mu\epsilon+n\frac{i\mu}{\omega}b_{3}&0&0\\\
\\\ 0&n^{2}-\mu\epsilon+n\frac{i\mu}{\omega}b_{3}&0\\\ \\\
0&-n\frac{i\mu}{\omega}b_{2}&-\mu\epsilon\\\ \end{array}\right]\;,$ (70)

with $\mathbf{n\cdot b}=n\,b_{3}$. Replacing the index relation (64) in the
matrix (70), the condition $M_{ij}E_{j}=0$ provides two mutually orthogonal
propagation modes

$\mathbf{E}_{\pm}=\frac{1}{\sqrt{2(1+|Q|^{2})}}\begin{pmatrix}\pm\sqrt{1+|Q|^{2}}\\\
-1\\\ \mathrm{i}Qe^{i\alpha}\\\ \end{pmatrix}\;,$ (71)

with $Q=b_{2}N/(\omega\epsilon)$, endowed however with a longitudinal
component. Here, we have rewritten the complex refractive index as
$n=Ne^{i\alpha}$, with

$\displaystyle N$
$\displaystyle=\sqrt{\left(\sqrt{\frac{|f(\omega)|}{2}}A_{+}\right)^{2}+\left(\sqrt{\frac{|f(\omega)|}{2}}A_{-}-\frac{\mu
b\cos\theta}{2\omega}\right)^{2}}\;,$ (72)

and

$\displaystyle\tan\alpha$ $\displaystyle=\frac{A_{-}}{A_{+}}-\frac{\mu
b\cos\theta}{\omega A_{+}\sqrt{2|f(\omega)|}}\;.$ (73)

For the particular case where $b_{2}=0$, the solutions in Eq. (71) simplify as

$\displaystyle{\mathbf{E}}_{\pm}=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm 1\\\
-1\\\ 0\end{pmatrix}\;,$ (74)

representing linearly polarized vectors, orthogonal to the propagation
direction. As the refractive index of Eq. (66a) has an imaginary piece, there
is absorption for both modes given in Eq. (74) (in equal magnitude), measured
in terms of the absorption coefficient [1],
$\tilde{\alpha}=2\omega\,\mathrm{Im}(n)=\mu\,b_{3}$.

###  III.2 The group and energy velocities of the antisymmetric case

Rewritting Eq. (64) in terms of ${\bf k}$, one has

$\displaystyle{\bf k}^{2}+i\,\mu\left({\bf b}\cdot{\bf
k}\right)-\mu\epsilon(\omega)\,\omega^{2}=0\;,$ (75)

whose solution yields the dispersion relation

$\displaystyle k(\omega)=-\frac{i\mu}{2}({\bf b}\cdot\hat{{\bf
k}})+\sqrt{\mu\epsilon(\omega)\,\omega^{2}-\frac{\mu^{2}}{4}({\bf
b}\cdot\hat{{\bf k}})^{2}}\;,$ (76)

with real and imaginary parts given as, respectively,

$\displaystyle k^{\prime}(\omega)$
$\displaystyle=\omega\,\sqrt{\frac{|f(\omega)|}{2}}\,A_{+}\;,$ (77a)
$\displaystyle k^{\prime\prime}(\omega)$
$\displaystyle=\omega\,\sqrt{\frac{|f(\omega)|}{2}}\,A_{-}-\frac{\mu}{2}(\hat{\bf{k}}\cdot{\bf{b}})\;.$
(77b)

Taking the derivative in relation to $k^{i}$-component in (75), the $i$th-
component of the group velocity is

$\displaystyle
v_{g}^{i}=\frac{k^{i}+i\,\mu\,b^{i}/2}{\mu\omega\left[\epsilon(\omega)+f_{\epsilon}(\omega)\,\omega/2\right]}\;.$
(78)

Using now $k^{i}=k^{i\prime}+ik^{i\prime\prime}$, one finds

$\displaystyle{\bf v}_{g}$ $\displaystyle=\frac{{\bf
k}^{\prime}/(\mu\omega)+i\,\omega\epsilon^{\prime\prime}\hat{{\bf
k}}^{\prime}/{(2k^{\prime})}}{C+iD}\;,$ (79)

with $C$ and $D$ defined by Eq. (54). The latter also indicates that the group
velocity is defined along the ${\bf{k}}^{\prime}$ direction. Substituting the
solution (77), the group velocity is written as a function of $\omega$ :

$\displaystyle\Re[v_{g}]$
$\displaystyle=\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|f(\omega)|}{2}}\left(A_{+}C+A_{-}D\right)\;,$
(80a) $\displaystyle\Im[v_{g}]$
$\displaystyle=\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|f(\omega)|}{2}}\left(A_{-}C-A_{+}D\right)\;.$
(80b)

The group velocity behavior as function of the dimensionless parameter
$\omega/\omega_{0}$ is shown in Fig. 8. For high-frequencies, it holds
$\Re[v_{g}]=1/\sqrt{\mu_{0}\epsilon_{0}}=1$ (for $\mu=\mu_{0}$), whereas the
imaginary part goes to zero, $\Im[v_{g}]=0$. At the origin, its real piece is
well behaved whereas the imaginary part goes to infinity. This latter behavior
is also observed in a dielectric with constant magnetic conductivity,
$\epsilon^{\prime}=cte$, $\epsilon^{\prime\prime}=\sigma/\omega$,
$\Sigma=cte$, as seen in Ref. [40].

Figure 8:  Group velocity of Eq. (80) for $\cos\theta=\pm 1$. The solid
(dashed) lines represent the real (imaginary) parts of $v_{g}$. Here, we have
used: $\mu=1$, $\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$,
$\gamma/\omega_{0}=0.01$, $b/\omega_{0}=2.5$ (blue), $b=0$ (black, the usual
case). The vertical dashed lines indicate
$\omega/\omega_{0}\approx\\{0.59,1,1.8,2.11\\}$ from left to right,
accordingly with Eq. (82).

Assessing now the energy propagation signal, we implement (77a) in (23), in
such a way the energy velocity is

$\displaystyle V_{E}$
$\displaystyle=\sqrt{\frac{|f(\omega)|}{2}}\frac{A_{+}}{U^{A}}\;,$ (81a) with
$\displaystyle U^{A}$
$\displaystyle=\frac{\mu\epsilon^{\prime}}{2}+\frac{\mu\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}+\frac{|f(\omega)|}{4}(A_{+}^{2}+A_{-}^{2})+\frac{\mu^{2}b^{2}}{8\omega^{2}}\cos^{2}\theta+$
$\displaystyle\phantom{=}-\frac{\mu
b\cos\theta}{2\omega}A_{-}\sqrt{\frac{|f(\omega)|}{2}}\;.$ (81b)

The behavior of the energy velocity is depicted in Fig. 9 in terms of
$\omega/\omega_{0}$. The frequencies $\omega^{a}_{1,2}$ define a region where
occurs an absorption peak followed by a strong reduction of the real part of
the refractive index $n$, as indicated in Figs. 6 and 7. In details, we
observe that:

  * •

For $0<\omega<\omega^{a}_{1}$: one has a total absorption range, since
$\mathrm{Re}[n]=0$, where there is no propagation. Hence, the energy velocity
is null in this frequency window.

  * •

For $\omega^{a}_{1}<\omega<\omega^{\prime}$: there occurs propagation together
with absorption, since $\mathrm{Re}[n]\neq 0$ and $\mathrm{Im}[n]\neq 0$. As a
consequence, there is a non-null energy velocity in this small region, as
indicated in Fig. 9. As the frequency approaches $\omega^{\prime}$, there
occurs an absorption peak followed by a rapid decreasing of $\mathrm{Re}[n]$
to zero, which marks the beginning of a second total absorption zone, where
$V_{E}$ vanishes. This behavior appears in terms of a diminishing $V_{E}$ for
$\omega$ approaching $\omega^{\prime}$ and $V_{E}=0$ thereafter. See Fig. 9.

  * •

For $\omega^{\prime}<\omega<\omega^{a}_{2}$: there is no propagation,
$\mathrm{Re}[n]=0$, as pointed in Figs. 6 and 7, in such a way only absorption
takes place and $V_{E}=0$. This interval magnitude represents the enlargement
of the absorption zone in the presence of the antisymmetric magnetic
conductivity, compared to the usual DD absorption window, which is now
incremented by $\Delta\omega=\omega^{a}_{2}-\omega^{\prime\prime}$.

  * •

For $\omega>\omega^{a}_{2}$: the absorption starts to diminish while the
propagation regime is partially restored. As a consequence, the energy
velocity assumes increasing magnitude, tending to its asymptotic value.

  * •

The entire plot of Fig. 9 reveals a scenario where the signal propagation is
severely constrained for frequencies below $\omega^{a}_{2}$, which is in
contrast with the signal propagation in the absorption windows of the Drude
isotropic conductivity case, represented by the energy velocity of Fig. 5.

Figure 9:  Energy velocity of Eq. (81a) in terms of the dimensionless
parameter $\omega/\omega_{0}$. The solid red line represents $V^{E}$ for
$\cos\theta=\pm 1$. The black solid curve illustrates the usual case (23),
where $b=0$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and $b/\omega_{0}=2.5$.
The vertical dashed lines indicate
$\omega/\omega_{0}\approx\\{0.59,1,1.8,2.11\\}$, from left to right,
accordingly with Eq. (82).

The characteristic frequencies can be determined by solving the sixth-order
equation in $\omega$,

$\displaystyle\gamma^{2}\left(\omega^{4}-\frac{\mu
b^{2}\cos^{2}\theta}{4\epsilon_{0}}\omega_{0}^{2}\right)+(\omega_{0}^{2}-\omega^{2})\times$
$\displaystyle\times\left[\omega^{2}\left(\omega_{0}^{2}-\omega^{2}+\omega_{p}^{2}\right)-\frac{\mu
b^{2}\cos^{2}\theta}{4\epsilon_{0}}\left(\omega_{0}^{2}-\omega^{2}-\gamma^{2}\right)\right]=0\;,$
(82)

from which the real and positive solutions yield $\omega^{a}_{1,2,3}$ when
$b\cos\theta\neq 0$, and $\omega^{\prime,\prime\prime}$ when $b\cos\theta=0$.
However, as $\omega^{a}_{3}\approx\omega^{\prime}$, we have chosen to depict
only $\omega^{\prime}$ in the plots.

##  IV The case of the symmetric conductivity tensor

We write the magnetic conductivity tensor in the symmetric form

$\displaystyle\sigma_{Bij}=\frac{1}{2}\left(a_{i}\,c_{j}+a_{j}\,c_{i}\right)\;,$
(83)

where $a_{i}$ and $c_{j}$ are components of the orthogonal background vectors
${\bf a}$ and ${\bf c}$, respectively. In this case, the permittivity tensor
(13) is

$\displaystyle\overline{\epsilon}_{ij}(\omega)=\epsilon(\omega)\delta_{ij}+\frac{i}{2\omega}\left[a_{i}\left({\bf
c}\times{\bf n}\right)_{j}+c_{i}\left({\bf a}\times{\bf
n}\right)_{j}\right].\;\;$ (84)

The null determinant of the matrix $M_{ij}$ with permittivity (84) provides
$n$-polynomial equations,

$\displaystyle n^{2}\,-\,\mu\,\epsilon(\omega)\pm\frac{i\,\mu}{2\omega}\,{\bf
n}\cdot({\bf a}\times{\bf c})$ $\displaystyle=0\;.$ (85)

Writing ${\bf n}=n\,\hat{{\bf k}}$, the solutions of (85) are

$\displaystyle n_{\pm}(\omega)$
$\displaystyle=\mp\frac{i\,\mu}{4\omega}\,\hat{{\bf k}}\cdot({\bf a}\times{\bf
c})+\sqrt{\mu\epsilon(\omega)-\frac{\mu^{2}}{16\omega^{2}}[\hat{{\bf
k}}\cdot({\bf a}\times{\bf c})]^{2}}\;,$ (86)

or

$n_{\pm}=\Re[n_{\pm}]+i\,\Im[n_{\pm}]\;,$ (87)

with

$\displaystyle\Re[n_{\pm}]$
$\displaystyle=\sqrt{\frac{|g(\omega)|}{2}}S_{+}\;,$ (88a)
$\displaystyle\Im[n_{\pm}]$
$\displaystyle=\sqrt{\frac{|g(\omega)|}{2}}\,S_{-}\mp\frac{\mu}{4\omega}\,\hat{\bf{k}}\cdot({\bf{a}}\times{\bf
c})\;,$ (88b)

where

$\displaystyle S_{\pm}$
$\displaystyle=\sqrt{\sqrt{1+\mu^{2}\left[\frac{\epsilon^{\prime\prime}(\omega)}{g(\omega)}\right]^{2}}\pm\mbox{sgn}[g(\omega)]}\;,$
(89)

and

$\displaystyle
g(\omega)=\mu\epsilon^{\prime}(\omega)-\mu^{2}(\hat{\bf{k}}\cdot({\bf{a}}\times{\bf
c}))^{2}/(4\omega)^{2}.$ (90)

Notice that the expressions (88a) and (88b) can be obtained from the results
of the antisymmetric case (66b) and (66c) by implementing ${\bf
b}\rightarrow\pm({\bf a}\times{\bf c})/2$. The behavior of $n_{\pm}$ as
function of $\omega/\omega_{0}$ is illustrated in Figs. 10 and 11, where we
have used
$\hat{\bf{k}}\cdot({\bf{a}}\times{\bf{c}})=|{\bf{a}}\times{\bf{c}}|\cos\varphi$.
From Eq. (87), one notices that $\Re[n_{+}]=\Re[n_{-}]$ and
$\Im[n_{\pm}]|_{\varphi=0,\pi}=\Im[n_{\mp}]|_{\varphi=\pi,0}$, which indicates
a kind of correspondence symmetry between the absorptive terms of $n_{\pm}$
relative to the sense (positive or negative) of propagation.

Figure 10:  Refractive indices $n_{\pm}$ of Eq. (87). Solid (dotted) lines
indicate the real (imaginary) part of $n_{\pm}$. The magenta (red) curves
illustrate $n_{\pm}$ for $\cos\varphi=1$, respectively. The black lines
depicts the usual case where $|{\bf{a}}\times{\bf{c}}|=0$. Here, we have used:
$\mu=1$, $\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$,
$\gamma/\omega_{0}=0.01$, and $|{\bf{a}}\times{\bf{c}}|/\omega_{0}=2.5$. The
solid red and magenta curves lie on top of each other. The plot also reveals
the opening of a new absorption band for low frequencies, while the dominant
lossy region width is slightly augmented.  Figure 11: Refractive indices
$n_{\pm}$ of Eq. (87). Solid (dotted) lines indicate the real (imaginary) part
of $n_{\pm}$. The cyan (blue) lines represent $n_{\pm}$ for $\cos\varphi=-1$,
respectively. The black lines illustrates the usual case, where
$|{\bf{a}}\times{\bf{c}}|=0$. Here, we have used: $\mu=1$, $\epsilon_{0}=1$,
$\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$, and
$|{\bf{a}}\times{\bf{c}}|/\omega_{0}=2.5$. The solid blue and cyan curves are
overlapped. One notes the opening of a new absorption band for low
frequencies, while the dominant lossy window length is slightly expanded.

In the very high-frequency limit, the refractive indices become
$n_{\pm}\rightarrow\sqrt{\mu\epsilon_{0}}$. On the other hand, in the low-
frequency regime, one obtains

$\displaystyle
n_{\pm}\simeq\frac{i\mu|{\bf{a}}\times{\bf{c}}|}{4\omega}\left(\mp\cos\varphi+|\cos\varphi|\right)\;.$
(91)

Hence $n_{+}|_{\varphi=0}=n_{-}|_{\varphi=\pi}=0$ and
$n_{+}|_{\varphi=\pi}=n_{-}|_{\varphi=0}\simeq
i\mu|{\bf{a}}\times{\bf{c}}|/(2\omega)$, which explains the divergence in the
origin observed in Figs. 10 and 11.

###  IV.1 Propagation modes

To evaluate the propagating modes, we choose an appropriate coordinate system
where the magnetic conductivity vectors are given by
${\bf{a}}=(a_{1},0,a_{3})$, ${\bf{c}}=(0,c_{2},0)$, such that

$\displaystyle{\bf{a}}\times{\bf{c}}=(-a_{3}c_{2},0,a_{1}c_{2})\;.$ (92)

For propagation along the ${\cal Z}$-axis, ${\bf{n}}=(0,0,n)$, the matrix
$M_{ij}$ is

$\displaystyle[M_{ij}]$
$\displaystyle=\left[\begin{array}[]{ccc}n^{2}-\mu{\epsilon}-i\frac{\mu}{2\omega}a_{1}c_{2}n&0&0\\\
\\\ 0&n^{2}-\mu{\epsilon}+i\frac{\mu}{2\omega}a_{1}c_{2}&0\\\ \\\
-i\frac{\mu}{2\omega}a_{3}c_{2}n&0&-\mu{\epsilon}\end{array}\right],$ (98)

where ${\epsilon=\epsilon(\omega)}$ is the complex electric permittivity.
Thus, by implementing Eq. (85) in Eq. (98), the relation $M_{ij}E_{j}=0$
provides the following polarization vectors:

$\displaystyle{\bf{E}}_{+}=\frac{1}{\sqrt{1+|A|^{2}}}\begin{pmatrix}1\\\ 0\\\
-iA\end{pmatrix}\,,\quad{\bf{E}}_{-}=\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix},$
(99)

associated with $n_{\pm}$, respectivelly, with
$A={a_{3}c_{2}}/{(2\omega{\epsilon}n)}$. We observe that ${\bf{E}}_{\pm}$
represent linearly polarized vectors, with ${\bf{E}}_{+}$ endowed with a
longitudinal component. Since the the real peace of $n_{\pm}$ is the same one,
there is no birefringence. On the other hand, the imaginary pieces are
different, meaning that the propagating modes are absorbed at different
degrees, which can be measured by the absorption difference per unit length,

$\displaystyle\frac{\Delta_{d}}{l}=\frac{2\pi}{\lambda_{0}}\,\left[\,\mathrm{Im}(n_{+})-\mathrm{Im}(n_{-})\,\right]\;,$
(100)

with $\lambda_{0}$ being the vacuum wavelength of incident light. Hence, for
the indices of Eq. (87) and propagation along the $z$-axis, the absorption
difference is

$\displaystyle\frac{\Delta_{d}}{l}=-\frac{\mu a_{1}\,c_{2}}{2}\;.$ (101)

###  IV.2 The group and energy velocities of the symmetric case

Backing to Eq. (85) in terms of ${\bf k}$ and $\omega$, we rewrite it as

$\displaystyle k^{2}\,\pm\,\frac{i\,\mu}{2}\,{\bf k}\cdot({\bf a}\times{\bf
c})\,-\,\mu\,\omega^{2}\,\epsilon(\omega)$ $\displaystyle=0\;,$ (102)

whose solutions are

$k_{\pm}(\omega)=k^{\prime}_{\pm}+ik^{\prime\prime}_{\pm}\;,$ (103)

with

$\displaystyle k^{\prime}_{\pm}(\omega)$ $\displaystyle=$
$\displaystyle\omega\,\sqrt{\frac{|g(\omega)|}{2}}\,S_{+}\;,$ (104a)
$\displaystyle k^{\prime\prime}_{\pm}(\omega)$ $\displaystyle=$
$\displaystyle\omega\,\sqrt{\frac{|g(\omega)|}{2}}\,S_{-}\mp\frac{\mu}{4}\,\hat{\bf{k}}\cdot({\bf{a}}\times{\bf
c})\;,$ (104b)

and $S_{\pm}$ of Eq. (89). Evaluating the derivative of Eq. (102) relative to
$k^{i}$-component, one finds the following group velocities:

$\displaystyle{\bf v}_{g}^{\pm}=\frac{\partial\omega}{\partial{\bf
k}}=\frac{\displaystyle\frac{{\bf k}}{\mu\omega}\pm i\,\frac{({\bf a\times
c})}{4\omega}}{\displaystyle\left(\epsilon^{\prime}+\frac{\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}\right)+i\left(\epsilon^{\prime\prime}+\frac{\omega}{2}\frac{\partial\epsilon^{\prime\prime}}{\partial\omega}\right)}\;.$
(105)

Using Eq. (103) the group velocity (105) takes on the form,

$\displaystyle{\bf v}_{g}^{\pm}$ $\displaystyle=\frac{\displaystyle\frac{{\bf
k}^{\prime}}{\mu\omega}+i\,\frac{\omega\epsilon^{\prime\prime}\,\hat{{\bf
k}}}{2k^{\prime}}}{\displaystyle\left(\epsilon^{\prime}+\frac{\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}\right)+i\left(\epsilon^{\prime\prime}+\frac{\omega}{2}\frac{\partial\epsilon^{\prime\prime}}{\partial\omega}\right)}\;.$
(106)

whose real and imaginary pieces are

$\displaystyle\Re[v^{\pm}_{g}]$
$\displaystyle=\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|g(\omega)|}{2}}\left(S_{+}C+S_{-}D\right)\;,$
(107a) $\displaystyle\Im[v^{\pm}_{g}]$
$\displaystyle=\frac{(\mu\epsilon_{0})^{-1}}{C^{2}+D^{2}}\sqrt{\frac{|g(\omega)|}{2}}\left(S_{-}C-S_{+}D\right)\;.$
(107b)

Figure 12 shows the real and imaginary parts of the group velocity given in
Eq. (107) in terms of the dimensionless parameter $\omega/\omega_{0}$. We have
considered the previous case where ${\bf a}=(a_{1},0,a_{3})$ and ${\bf
c}=(0,c_{2},0)$, such that ${\bf a}\times{\bf c}=(-a_{3}c_{2},0,a_{1}c_{2})$,
and $\hat{{\bf k}}=\hat{{\bf z}}$. For high frequencies
$(\omega\gg\omega_{0})$, one has $\Re[v_{g}]\rightarrow
1/\sqrt{\mu_{0}\epsilon_{0}}=1$ and $\Im[v_{g}]=0$.

Considering $a_{3}=0$, the propagating modes represented by Eq. (99) turn out
perpendicular to the direction of propagation, satisfying the conditions (18a)
and (18b) for the evaluation of the energy velocity. In this sense, Eqs.
(104a) and (104b) yield

$\displaystyle V_{E}^{\pm}$
$\displaystyle=\sqrt{\frac{|g(\omega)|}{2}}\frac{S_{+}}{U^{S}_{\pm}}\;,$
(108a) with $\displaystyle U^{S}_{\pm}$
$\displaystyle=\frac{\mu\epsilon^{\prime}}{2}+\frac{\mu\omega}{2}\frac{\partial\epsilon^{\prime}}{\partial\omega}+\frac{|g(\omega)|}{4}(S_{+}^{2}+S_{-}^{2})+$
$\displaystyle\phantom{=}+\frac{\mu^{2}|{\bf{a}}\times{\bf{c}}|^{2}}{32\omega^{2}}\cos^{2}\varphi\pm\frac{|{\bf{a}}\times{\bf{c}}|}{4\omega}\cos\varphi\sqrt{\frac{|g(\omega)|}{2}}S_{-}\;.$
(108b)

The behavior of the energy velocity is represented in Fig. 13 in terms of
$\omega/\omega_{0}$. As it happened in the antisymmetric case, see Sec. III,
the frequencies $\omega^{s}_{1,2}$ define the region where occurs an
absorption peak followed by an abrupt and strong reduction of the real part of
the refractive indices. Also, we observe the existence of two lossy windows
where the energy velocity is null. In details:

  * •

For $0<\omega<\omega^{s}_{1}$: there does not occur propagation, since
$\mathrm{Re}[n_{\pm}]=0$ and the energy velocity is null, $V_{E}^{\pm}=0$.

  * •

For $\omega^{s}_{1}<\omega<\omega^{\prime}$: propagation and absorption occur
simultaneously and $V_{E}^{\pm}\neq 0$, since the real and imaginary pieces of
$n_{\pm}$ are non-null, see Figs. 10 and 11. Whereupon the resonance frequency
$\omega^{\prime}$, the pieces $\mathrm{Re}[n_{\pm}]$ undergo a rapid
deacreasing, implying a null energy velocity.

  * •

For $\omega^{\prime}<\omega<\omega^{s}_{2}$: only absorption occurs, since
$\mathrm{Re}[n_{\pm}]=0$ and $V_{E}^{\pm}=0$ (see Figs. 11 and 13).

  * •

Finally, for $\omega>\omega^{s}_{2}$: the propagation of both modes $n_{\pm}$
is restored, while the absorption effect diminishes rapidly to zero. The
energy velocity increases monotonically to its asymptotic value.

To obtain the characteristic frequencies, we need to solve the following the
sixth-order equation:

$\displaystyle\gamma^{2}\left(\omega^{4}-s\,\omega_{0}^{2}\right)+(\omega_{0}^{2}-\omega^{2})\times$
$\displaystyle\times\left[\omega^{2}\left(\omega_{0}^{2}-\omega^{2}+\omega_{p}^{2}\right)-s\left(\omega_{0}^{2}-\omega^{2}-\gamma^{2}\right)\right]=0\;,\hskip
14.22636pt$ (109)

with

$\displaystyle
s=\frac{\mu}{16\epsilon_{0}}\,|{\bf{a}}\times{\bf{c}}|^{2}\cos^{2}\varphi\;,$
(110)

in the place of the factor ${\mu b^{2}\cos^{2}\theta}/{4\epsilon_{0}}$ in Eq.
(82). Eq. (109) provides the roots $\omega^{s}_{1,2,3}$, when
$|{\bf{a}}\times{\bf{c}}|\cos\varphi\neq 0$, and
$\omega^{\prime,\prime\prime}$ when $|{\bf{a}}\times{\bf{c}}|\cos\varphi=0$.
We mention that $\omega^{s}_{3}\approx\omega^{\prime}$, then we have depicted
only $\omega^{\prime}$ in the previous plots.

Figure 12:  Group velocity of Eq. (107) in terms of the dimensionless
parameter $\omega/\omega_{0}$ for $\cos\varphi=\pm 1$. Solid (dashed) lines
represent the real (imaginary) parts of $v_{g}$. The solid blue line
represents $V_{E}^{\pm}$ for $\cos\theta=\pm 1$. Here, we have used: $\mu=1$,
$\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$, $\gamma/\omega_{0}=0.01$,
$|{\bf{a}}\times{\bf{c}}|/\omega_{0}=2.5$ (red), and
$|{\bf{a}}\times{\bf{c}}|/\omega_{0}=0$ (black). The vertical dashed lines
indicate $\omega/\omega_{0}\approx\\{0.33,1,1.8,1.87\\}$, from left to right,
accordingly with Eq. (109).  Figure 13:  Energy velocity $V_{E}^{\pm}$ of Eq.
(108a) in terms of the dimensionless parameter $\omega/\omega_{0}$. The solid
blue line represents $V_{E}^{\pm}$ for $\cos\theta=\pm 1$. The black solid
curve depicts the usual case (null magnetic conductivity) of Eq. (23). Here,
we have used: $\mu=1$, $\epsilon_{0}=1$, $\omega_{p}/\omega_{0}=1.5$,
$\gamma/\omega_{0}=0.01$, and $|{\bf{a}}\times{\bf{c}}|/\omega_{0}=2.5$. The
vertical dashed lines indicate
$\omega/\omega_{0}\approx\\{0.33,1,1.8,1.87\\}$, from left to right,
accordingly with Eq. (109).

##  V Final Remarks

In this paper, we have discussed the properties of wave propagation and
absorption in a Drude-Lorentz dielectric in the presence of magnetic
conductivity. The analysis was performed for three configurations of the
magnetic conductivity tensor: isotropic, antisymmetric, and symmetric. For
each case, we have obtained the dispersion relation and refractive indices as
functions of the frequency and of the Drude-Lorentz parameters, with their
properties properly scrutinized for all configurations. The involved
refractive indices were examined in the range of low- and high-frequencies,
being always compared to Drude’s usual dielectric behavior. The group and the
energy velocities were evaluated for all cases, being the latter one used as
tool to examine signal propagation.

In lossy scenarios, as the group velocity becomes complex and loses physical
interpretation, the signal propagation passes to be described by the energy
velocity, which is real. In the high-frequency limit, one obtained
$\Re[n]=\sqrt{\mu\,\epsilon_{0}}$, while the energy velocity is reduced to
$V_{E}=1/\sqrt{\mu\epsilon_{0}}$ (for all the cases examined). The absorption
goes to zero in this frequency regime, $\Im[n]\rightarrow 0$. Near the origin
and at intermediary frequencies, however, the refractive indices behave in
some different ways.

The first case studied was the DD with isotropic magnetic conductivity,
$\Sigma\neq 0$, examined in Sec. II. The associated refractive indices show
distinct features in relation to the conventional DD. Near the origin, it
occurs $\Re[n_{+}]\rightarrow\infty$, approaching the behavior of a magnetized
plasma index, while one has $\Re[n_{-}]\rightarrow 0$ (for $\omega\rightarrow
0$). These latter properties also manifest in a simple dielectric,
$\epsilon^{\prime}=cte$, $\epsilon^{\prime\prime}=\sigma/\omega$, with
$\Sigma=cte$ [40]. The absorption window now has its length reduced by the
magnetic conductivity. Moreover, the lossy interval begins to support signal
propagation, $V_{E}^{\pm}\neq 0$, because the refractive indices $n_{\pm}$
develop non-null real pieces in the range
$\omega^{i}_{1}<\omega<\omega^{\prime\prime}$, in which the signal propagation
is favoured. See Figs. 1, 2 and 5 for cheking details. Thus, the frequency
window where $V_{E}=0$ is nearly supressed in the presence of the magnetic
conductivity, $\Sigma\neq 0$.

For the antisymmetric and symmetric cases of Secs. III and IV, respectively,
we observe that the frequency window, where the energy velocity is null, is
slightly “enhanced” due to the presence of the magnetic conductivity
($b\cos\theta\neq 0$, and $|{\bf{a}}\times{\bf{c}}|\cos\varphi\neq 0$). Such a
feature occurs because the non-null magnetic conductivity enlarges the
frequency window where absorption effects can take place, see Figs. 6 and 7
for the antisymmetric case, and Figs. 10 and 11 for the symmetric case. In
both configurations, the energy velocity becomes predominantly null for
frequencies approximately smaller than $2\omega_{0}$, a window in which the
signal propagation is severely constrained. See the plots of Fig. 9 and Fig.
13. Such a behavior states a sharp difference in relation to the DD with
isotropic conductivity of Sec. II, which presents enhanced progapation in this
window, as already mentioned. The properties of the DD refractive indices and
DD with magnetic conductivity are displayed in Table 1.

As it is known in the literature, in non-absorbing media the energy velocity
and group velocity are equal, $v_{g}=v_{E}$, equivalence that disappears in
lossy media where it holds $v_{g}\neq v_{E}$. In the present work all the
analyzed scenarios are endowed with absorption and
$\epsilon^{\prime\prime}=0$, which justifies the attainment of $v_{g}\neq
v_{E}$ in all sections of the work .

Table 1: Comparison of propagation properties of each modified scenario. |  refractive indices at origin |  absorption zone |  negative refraction |  signal propagation
---|---|---|---|---
|  |  |  |
Drude dielectric (DD) |  $n\neq 0$, $n\in\mathbb{R}$ |  intermediary one, whereupon the resonance |  no |  yes, except in the window $\omega^{i}_{1}<\omega<\omega^{\prime\prime}$ of Fig. 5
|  |  |  |
DD with $\Sigma$ |  $n_{+}\rightarrow\infty$ $n_{-}\rightarrow 0$ $\Im[n_{\pm}]=0$ |  one intermediary reduced window |  no yes |  yes, favoured for $\omega\lesssim 2\omega_{0}$
|  |  |  |
DD/antisymmetric $\sigma^{B}_{ij}$ |  $\Re[n_{\pm}]=0$ for $\theta=0$ $\Im[n]\rightarrow 0$ for $\theta=0$ $\Im[n]\rightarrow\infty$ for $\theta=\pi$ |  two zones: i) near the origin; ii) whereupon the resonance (enlarged) |  no |  severely constrained for $\omega\lesssim 2\omega_{0}$
|  |  |  |
DD/symmetric $\sigma^{B}_{ij}$ |  $\Re[n_{\pm}]=0$ $\Im[n_{+},n_{-}]\rightarrow 0$ for $\varphi=0,\pi$ $\Im[n_{-},n_{+}]\rightarrow\infty$ for $\varphi=0,\pi$ |  two zones: i) near the origin; ii) whereupon the resonance (enlarged) |  no |  severely constrained for $\omega\lesssim 2\omega_{0}$

###### Acknowledgements.

The authors express their gratitude to FAPEMA, CNPq, and CAPES (Brazilian
research agencies) for their invaluable financial support. M.M.F. is supported
by CNPq/Produtividade 311220/2019-3 and CNPq/Universal/422527/2021-1. P.D.S.S
is grateful to grant CNPq/PDJ 150584/23. Furthermore, we are indebted to
CAPES/Finance Code 001 and FAPEMA/POS-GRAD-02575/21.

## References

  * [1] A. Zangwill, Modern Electrodynamics. New York (USA): Cambridge University Press, 2012.
  * [2] J. D. Jackson, Classical Electrodynamics, 3rd edition. New York (USA): John Wiley & Sons, 1999.
  * [3] L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Course of Theoretical Physics, Volume 8, 2nd ed. (Pergamon Press, New York, 1984).
  * [4] R. P. Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press (1985).
  * [5] Paul Drude, Zur Elektronentheorie der Metalle, Annalen der Physik, 306 (3): 566–613 (1900).
  * [6] Paul Drude, Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte, Annalen der Physik. 308 (11): 369–402 (1900).
  * [7] Neil Ashcroft and N. David Mermin, Solid State Physics, New York: Holt, Rinehart and Winston (1976).
  * [8] M. Fox, Optical properties of Solids, New York: Oxford University Press (2001).
  * [9] F. Wooten, Optical properties of Solids, New York: Academic Press (1972).
  * [10] D. E. Kharzeev, The chiral magnetic effect and anomaly induced transport, Prog. Part. Nucl. Phys. 75, 133 (2014).
  * [11] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effects in highenergy nuclear collisions – A status report, Prog. Part. Nucl. Phys. 88, 1 (2016).
  * [12] D. Kharzeev, K. Landsteiner, A. Schmitt and H.U. Yee, Strongly Interacting Matter in Magnetic Fields, Lect. Notes Phys. 871 (Springer-Verlag,Berlin Heidelberg, 2013).
  * [13] M. Dvornikov and V.B. Semikoz, Influence of the turbulent motion on the chiral magnetic effect in the early universe, Phys. Rev. D 95, 043538 (2017).
  * [14] J. Schober, A. Brandenburg and I. Rogachevskii, Chiral fermion asymmetry in high-energy plasma simulations, Geophys. Astrophys. Fluid Dynamics 114, 106 (2020).
  * [15] M. Dvornikov and V. B. Semikoz, Instability of magnetic fields in electroweak plasma driven by neutrino asymmetries, JCAP 05, 002 (2014).
  * [16] M. Dvornikov, Chiral magnetic effect in the presence of an external axial-vector field, Phys. Rev. D 98, 036016 (2018).
  * [17] M. Dvornikov, Electric current induced by an external magnetic field in the presence of electroweak matter, EPJ Web Conf. 191, 05008 (2018).
  * [18] A. A. Burkov, Chiral anomaly and transport in Weyl metals, J. Phys. Condens. Matter 27, 113201 (2015).
  * [19] Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu, and T. Valla, Chiral magnetic effect in ZrTe5, Nat. Phys. 12, 550 (2016).
  * [20] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs, Phys. Rev. X 5, 031023 (2015).
  * [21] E. Barnes, J. J. Heremans, and Djordje Minic, Electromagnetic Signatures of the Chiral Anomaly in Weyl Semimetals, Phys. Rev. Lett. 117, 217204 (2016).
  * [22] P. D. S. Silva, M. M. Ferreira Jr., M. Schreck, and L. F. Urrutia, Magnetic-conductivity effects on electromagnetic propagation in dispersive matter, Phys. Rev. D 102, 076001 (2020).
  * [23] J. Askne and B. Lind, Energy of Electromagnetic Waves in the Presence of Absorption and Dispersion, Phys. Rev. A 2, 6 (1970).
  * [24] S. Kaushik, D.E. Kharzeev, and E.J. Philip, Transverse chiral magnetic photocurrent induced by linearly polarized light in symmetric Weyl semimetals, Phys. Rev. Research 2, 042011(R) (2020).
  * [25] P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Group velocity in lossy periodic structured media, Phys. Rev. A 82, 053825 (2010).
  * [26] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, New York, 1960).
  * [27] R. Loudon, The propagation of electromagnetic energy through an absorbing dielectric, J. Phys. A: Gen. Phys.3, 233 (1970).
  * [28] J. Neufeld, Wave propagation and group velocity in absorbing media, Phys. Lett. A Phys. 29, 68 (1969).
  * [29] G. C. Sherman and K. E. Oughstun, Energy-velocity description of pulse propagation in absorbing, dispersive dielectrics, J. Opt. Soc. Am. B 12, 229-247 (1995).
  * [30] M.V. Davidovich, Electromagnetic energy density and velocity in a medium with anomalous positive dispersion, Tech. Phys. Lett. 32, 982–986 (2006).
  * [31] R. Ruppin, Electromagnetic energy density in a dispersive and absorptive material, Phys. Lett. A 299, 309-312 (2002).
  * [32] F. D. Nunes, T. C. Vasconcelos, M. Bezerra, and J. Weiner, Electromagnetic energy density in dispersive and dissipative media, J. Opt. Soc. Am. B, 28 6 (2011).
  * [33] A. Moradi and Pi‑G. Luan, Electromagnetic energy density in hyperbolic metamaterials, Sci. Rep. 12, 10760 (2022).
  * [34] A. N. Darinskii, Group and energy velocities of electromagnetic waves in bianisotropic superlattices, Phys. Rev. A 108, 013510 (2023).
  * [35] T. J. Cui and J. A. Kong, Time-domain electromagnetic energy in a frequency-dispersive left-handed medium, Phys. Rev. B 70, 205106 (2004).
  * [36] I. Semchenko, A. Balmakou, S. Khakhomov, and S. Tretyakov, Stored and absorbed energy of fields in lossy chiral single-component metamaterials, Phys. Rev. B 97, 014432 (2018).
  * [37] J. Chen, Y. Dai and Y. Xuanyuan, A possible way to experimentally examine validity of the expressions of dissipated energy density, Optik 242, 165756 (2021).
  * [38] J. Chen and J. She, Analysis on energy density difference between linearly and circularly polarized electromagnetic waves,  Eur. Phys. J. Plus 137, 502 (2022).
  * [39] V. Gerasik and M. Stastna, Complex group velocity and energy transport in absorbing media, Phys. Rev. E 81, 056602 (2010).
  * [40] P. D. S. Silva, M. J. Neves and M. M. Ferreira Jr., Group velocity and energy propagation in a dielectric medium supporting magnetic current, arXiv:2305.08153v1(2023).
  * [41] F. S. Ribeiro, P. D. S. Silva, and M. M. Ferreira Jr., Cold plasma modes in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics, Phys. Rev. D 107, 096018 (2023).
  * [42] P. Hosur, and X-L. Qi, Tunable circular dichroism due to the chiral anomaly in Weyl semimetals, Phys. Rev. B 91, 081106(R) (2015).
  * [43] M. Nieto-Vesperinas, Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism, Phys. Rev. A 92, 023813 (2015).
  * [44] Y. Tang and A. E. Cohen, Enhanced Enantioselectivity in Excitation of Chiral Molecules by Superchiral Light, Science 332, 333 (2011).
  * [45] M. Amin, O. Siddiqui, and M. Farhat, Linear and circular dichroism in graphene-based reflectors for polarization control, Phys. Rev. Applied 13, 024046 (2020).
  * [46] J. Shibata, A. Takeuchi, H. Kohno, and G. Tatara, Theory of electromagnetic wave propagation in ferromagnetic Rashba conductor, J. App. Phys. 123, 063902 (2018).