# Cold plasma modes in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics

Filipe S. Ribeiroa<EMAIL_ADDRESS><EMAIL_ADDRESS>Pedro D. S. Silvaa<EMAIL_ADDRESS><EMAIL_ADDRESS>Manoel M. Ferreira Jr.b<EMAIL_ADDRESS><EMAIL_ADDRESS>aPrograma de Pós-graduação em Física, Universidade
Federal do Maranhão, Campus Universitário do Bacanga, São Luís, Maranhão
65080-805, Brazil  bDepartamento de Física, Universidade Federal do Maranhão,
Campus Universitário do Bacanga, São Luís, Maranhão 65080-805, Brazil

###### Abstract

In this work, we study the propagation and absorption of plasma waves in the
chiral Maxwell-Carroll-Field-Jackiw (MCJF) electrodynamics. The Maxwell
equations are rewritten for a cold, uniform, and collisionless fluid plasma
model, allowing us to determine the new refractive indices and propagating
modes. The cases of propagation parallel and orthogonal to the magnetic field
are examined considering a purely timelike CFJ background that plays the role
of the magnetic conductivity chiral parameter. The collective electromagnetic
modes are associated with four distinct refractive indices associated with
right-circularly polarized and left-circularly polarized waves. For each
index, the propagation and absorption zones are illustrated for some specific
parameter values. In low-frequency regime, we have obtained modified helicons
with right- and left-circularly polarizations. The optical behavior is
investigated by means of the rotatory power (RP) and dichroism coefficient.
The existence of a negative refraction zone enhances the rotatory power. It is
also observed RP sign reversal, a feature of rotating plasmas.

###### pacs:

11.30.Cp, 41.20.Jb, 41.90.+e, 42.25.Lc

##  I Introduction

The study of electromagnetic (EM) waves propagation [1, 2] in cold magnetized
plasma is based on magneto-ionic theory [3, 4, 8, 7, 5, 9, 6], developed by E.
Appleton [10] and D. Hartree [11] between 1929 and 1932 to describe the radio
waves propagation in the ionosphere, in the context of the usual
electrodynamics [12]. EM waves in plasmas have been studied in other scenarios
recently, as in logarithmic nonlinear electrodynamics [13].

The chiral magnetic effect (CME) is the macroscopic generation of an electric
current in the presence of a magnetic field, stemming from an asymmetry
between the number density of left- and right-handed chiral fermions [14, 15,
16, 17, 18]. It has been extensively investigated in several distinct
contexts, such as quark-gluon plasmas [19, 20, 21], cosmology [22], neutron
stars [23, 24], and electroweak interactions [25]. The CME plays a very
relevant role in Weyl semimetals, where it is usually connected to the chiral
anomaly associated with Weyl nodal points [26], the absence of the Weyl nodes
[27], anisotropic effects stemming from tilted Weyl cones [28], the CME and
anomalous transport in Weyl semimetals [29], quantum oscillations arising from
the CME [30], computation of the electromagnetic fields produced by an
electric charge near a topological Weyl semimetal with two Weyl nodes [31],
renormalization evaluations for Weyl semimetals and Dirac materials [32], and
solutions of axion electrodynamics [33].

The CME current can be classically described by the axion Lagrangian [33, 34,
35, 36, 37, 38, 39],

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\theta(\mathbf{E}\cdot\mathbf{B)},$
(1)

where $\theta$ is the axion field. In this context, the Maxwell equations are

$\displaystyle\mathbf{\nabla}\cdot\mathbf{E}$
$\displaystyle=\rho-\mathbf{\nabla}\theta\cdot\mathbf{B},$ (2)
$\displaystyle\mathbf{\nabla}\times\mathbf{B}-\partial_{t}\mathbf{E}$
$\displaystyle=\mathbf{j}+(\partial_{t}\theta)\mathbf{B}+\mathbf{\nabla}\theta\times\mathbf{E},$
(3)

where the terms involving $\theta$ derivatives find association with condensed
matter effects [39]. Indeed, $\mathbf{\nabla}\theta\cdot\mathbf{B}$ represents
an anomalous charge density, while $\mathbf{\nabla}\theta\times\mathbf{B}$
appears in the anomalous Hall effect, and $(\partial_{t}\theta)\mathbf{B}$
plays the role of the chiral magnetic current. When we address a cold axion
dark matter, the associated de Broglie wavelength is large enough to assure
the inexistence of variation of the axion field in the typical dimension of
experimental devices. In this case, the axion field is supposed to not depend
on the space coordinates, $\mathbf{\nabla}\theta={\bf{0}}$, so that the
Maxwell equations (2) and (3) read

$\mathbf{\nabla}\cdot\mathbf{E}=\rho,\quad\mathbf{\nabla}\times\mathbf{B}-\partial_{t}\mathbf{E}=\mathbf{j}+(\partial_{t}\theta)\mathbf{B},$
(4)

where $(\partial_{t}\theta)\mathbf{B}$, the chiral magnetic current, may also
be addressed as a term of Maxwell-Carroll-Field-Jackiw (MCFJ) theory. A
classical electrodynamics scenario endowed with a chiral magnetic current has
been investigated considering symmetric and antisymmetric conductivity [40].
The latter case has also been addressed in Ref. [41].

The MCFJ model [42] is the CPT-odd part of the U(1) gauge sector of the
Standard Model Extension (SME) [43]. It is described by the Lagrangian density

$\displaystyle\mathrm{{\mathcal{L}}}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{4}\epsilon^{\mu\nu\alpha\beta}\left(k_{AF}\right)_{\mu}A_{\nu}F_{\alpha\beta}-A_{\mu}J^{\mu},$
(5)

with $\left(k_{AF}\right)_{\mu}$ being the 4-vector background which controls
the Lorentz violation. This theory has been investigated in multiple respects
[44], encompassing radiative evaluations [45, 46], topological defects
solutions [47], supersymmetric generalizations [48], classical solutions,
quantum aspects and unitarity analysis [49]. It may also be connected with the
CME in the sense that it provides a modified Ampère’s law,

$\nabla\times\mathbf{B}-\frac{\partial\mathbf{E}}{\partial
t}=\mathbf{J}+k_{AF}^{0}\mathbf{B}+\mathbf{k}_{AF}\times\mathbf{E},$ (6)

containing the magnetic current, $\mathbf{J}_{B}=k_{AF}^{0}\mathbf{B}$, with
the component $k_{AF}^{0}$ playing the role of the magnetic conductivity.

The SME photon sector is also composed of a CPT-even term constituted of a
rank 4 Lorentz-violating tensor [50], whose components may be properly
parametrized in terms of dimensionless $3\times 3$ matrices, $\kappa_{DE}$,
$\kappa_{DB}$, $\kappa_{HE}$, and $\kappa_{HB}$, which allow to write
generalized constitutive relations between the fields
$(\mathbf{D},\mathbf{E})$ and $(\mathbf{H,B})$,

$\begin{pmatrix}\mathbf{D}\\\
\mathbf{H}\end{pmatrix}=\begin{pmatrix}\epsilon\mathbb{1}+\kappa_{DE}&\kappa_{DB}\\\
&\\\
\kappa_{HE}&{\mu}^{-1}\mathbb{1}+\kappa_{HB}\end{pmatrix}\begin{pmatrix}\mathbf{E}\\\
\mathbf{B}\end{pmatrix}\,,$ (7)

similar to the ones that hold in continuous medium electrodynamics, see Eqs.
(8a) and (8b). Here, $\mathbf{D}$ is the electric displacement, while
$\mathbf{H}$ is the magnetic field. This CPT-even electrodynamics was
investigated in several contexts, involving consistency aspects [51], finite
temperature and boundary effects [52]. Lorentz-violating electrodynamics in
continuous matter [53, 54] has been a topic of interest in the latest years
due to its potential to describe interesting effects of the phenomenology of
new materials, such as Weyl semimetals [55]. A classical field theory
description of wave propagation, refractive indices, and optical effects in a
continuous medium described by the MCFJ electrodynamics (with usual
constitutive relations), including its Lorentz-violating higher-order
derivative version [56], was discussed in Ref. [57].

Chiral media are endowed with parity violation [58, 59, 60, 61], being
described by parity-odd models, as bi-isotropic [62] and bi-anisotropic
electrodynamics [64, 63, 65, 66, 67, 68], whose constitutive relations read

$\displaystyle\mathbf{D}$
$\displaystyle=\hat{\epsilon}\,\mathbf{E}+\hat{\alpha}\,\mathbf{B},$ (8a)
$\displaystyle\mathbf{H}$
$\displaystyle=\hat{\beta}\,\mathbf{E}+\hat{\zeta}\,\mathbf{B},$ (8b)

and $\hat{\epsilon}=[\epsilon_{ij}]$, $\hat{\alpha}=[\alpha_{ij}]$,
$\hat{\beta}=[\beta_{ij}]$, and $\hat{\zeta}=[\zeta_{ij}]$ represent, in
principle, $3\times 3$ complex matrices. The bi-isotropic relations involve
the diagonal isotropic tensors, ${\epsilon_{ij}}=\epsilon\delta_{ij}$,
${\alpha_{ij}}=\alpha\delta_{ij}$, ${\beta_{ij}}=\beta\delta_{ij}$. In chiral
scenarios, left-circularly polarized (LCP) and right-circularly polarized
(RCP) waves travel at distinct phase velocities, implying birefringence and
optical rotation [69]. This phenomenon stems from the natural optical activity
of the medium or can be induced by the action of external fields (e. g.,
Faraday effect [70, 71, 72]), and it is measured in terms of the rotation
angle per unit length or rotatory power (RP) [73]. Magneto-optical effects are
used to investigate features of new materials, such as topological insulators
[74, 75, 76, 77, 78, 79, 80] and graphene compounds [81].

The RP is a probe to examine the optical behavior of several distinct systems,
for instance, crystals [82, 83], organic compounds [58, 84], graphene
phenomena at terahertz band [85], and gas of fast-spinning molecules [86]. The
optical rotation may depend on the frequency (RP dispersion) and undergo
reversion (anomalous RP dispersion) [88, 87, 89]. It also finds interesting
applications in chiral metamaterials [90, 91, 92], chiral semimetals [93, 94],
in the determination of the rotation direction of pulsars [95], and in
rotating plasmas, which constitutes a scenario where RP sign reversal also
takes place [96]. Recently, RP reversal was also reported in a bi-isotropic
dielectric in the presence of chiral magnetic current [97]. Furthermore, in
the presence of absorption, dichroism is another useful tool for the optical
characterization of matter. It occurs when LCP and RCP light waves are
absorbed by the medium at different degrees. It has been used to distinguish
between Dirac and Weyl semimetals [98], perform enantiomeric discrimination
[99, 100], and for developing graphene-based devices at terahertz frequencies
[101].

Another feature of chiral systems is the possible occurrence of negative
refraction and negative refractive index, which was first proposed by Veselago
in 1968 [102] and experimentally observed in 2000 [104, 103]. Later, other
experiments confirmed the negative refraction by using Snell’s law [105, 106].
This unusual property was achieved in constructed metamaterials with both
negative-electric permittivity and magnetic permeability [107, 108]. The
negative refractive index also appears in quark-gluon plasmas [109, 110],
magnetoelectric materials [111], metasurfaces [112], chiral bi-anisotropic
metamaterials [113, 114], and new materials, such as Dirac semimetals [115,
116]. In chiral plasmas described by generalized bi-isotropic constitutive
relations [117, 118], the negative refractive index can occur within some
frequency band and is not necessarily associated with simultaneously negative
electric permittivity and negative magnetic permeability, being attributed to
the chirality parameter introduced in the constitutive relations,

$D^{i}=\varepsilon_{ij}E^{j}+i\xi_{c}B^{i},\quad
H^{i}=\mu^{-1}B^{i}+i\xi_{c}E^{i},$ (9)

where $\varepsilon_{ij}$, $\mu$, and $\xi_{c}$ are the plasma electric
permittivity tensor, the magnetic permeability, and the constant chirality
parameter. Plasmas metamaterials have been investigated as new media endowed
with interesting properties, such as negative refraction and nonlinearities
[119, 120]. Rotating plasmas constitute a scenario in which the birefringence
is enhanced, with special properties on the RP and attenuation [121].
Therefore, plasmas provide a rich framework for theoretical and experimental
investigation on the optical behavior of chiral systems, as the one developed
in this work.

In this work, we are interested in examining the wave propagation in a
magnetized cold plasma ruled by the MCFJ model, a chiral route distinct from
the bi-isotropic/anisotropic electrodynamics of the relations (9). We carry
out our analysis considering the timelike Lorentz-violating background
component, which plays the role of the chiral magnetic conductivity. Such a
choice is analog to the coupling with a cold axion field, for which
$\mathbf{\nabla}\theta={\bf{0}}$, as stated in Eq. (4). The refractive indices
and dispersion relations are evaluated, representing altered collective
electromagnetic modes. For the propagation along the magnetic field axis, in
the low-frequency limit, there appear RCP and LCP helicons, due to the
presence of the chiral fator $V_{0}$. Optical effects, such as birefringence
and dichroism, are examined, which could be useful to trace analogies with
other material properties. We also find that the chiral conductivity yields
negative refraction in specific frequency bands, amplifying the rotatory power
and dichroism signals.

This paper is outlined as follows. In Sec. II, we briefly review some aspects
of the MCFJ model. In Sec. III, the main properties of propagation in usual
cold magnetized plasmas are presented. The dispersion relations, refractive
indices, and helicons for cold plasma in chiral electrodynamics are adressed
in Sec. IV. In Sec. V, we discuss the case of propagation orthogonal to the
magnetic field. The optical effects are examined in Sec. VI. Finally, we
summarize our results in Sec. VII.

##  II BASICS ON MCFJ ELECTRODYNAMICS

The Carroll-Field-Jackiw model was proposed as a gauge invariant CPT-odd
electrodynamics constrained by birefringence data of distant galaxies [42]. It
was later incorporated as the CPT-odd sector of the SME [43], and it has been
investigated in several respects [44, 45]. In matter, it is described by the
following Lagrangian density [57]:111We use natural units $h=c=1$ and the
Minkowski metric signature $g_{\mu\nu}=\mathrm{diag}\left(1,-1,-1,-1\right)$.

$\displaystyle\mathrm{{\mathcal{L}}}=-\frac{1}{4}G^{\mu\nu}F_{\mu\nu}-\frac{1}{4}\epsilon^{\mu\nu\alpha\beta}\left(k_{AF}\right)_{\mu}A_{\nu}F_{\alpha\beta}-A_{\mu}J^{\mu},$
(10)

yielding the MCFJ equation of motion,

$\partial_{\rho}G^{\rho\kappa}+\epsilon^{\beta\kappa\mu\nu}\left(k_{AF}\right)_{\beta}F_{\mu\nu}=J^{\kappa}\,.$
(11)

Here, $\left(k_{AF}\right)^{\mu}=\left(k_{AF}^{0},{\bf{k}}_{AF}\right)$ is a
constant 4-vector background responsible for the Lorentz violation, and

$\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\quad
G^{\mu\nu}=\frac{1}{2}\chi^{\mu\nu\alpha\beta}F_{\alpha\beta},$ (12)

are the usual $U(1)$ vacuum and continuous matter field strength,
respectively. The 4-rank tensor, $\chi^{\mu\nu\alpha\beta}$, describes the
medium constitutive tensor [124], whose components provide the electric and
magnetic responses of the medium. Indeed, the electric permittivity and
magnetic permeability tensor components are written as
$\epsilon_{ij}\equiv\chi^{0ij0}$ and
$\mu^{-1}_{lk}\equiv\frac{1}{4}\epsilon_{ijl}\chi^{ijmn}\epsilon_{mnk}$,
respectively. For isotropic polarization and magnetization, it holds
$\epsilon_{ij}=\epsilon\delta_{ij}$ and $\mu^{-1}_{ij}=\mu^{-1}\delta_{ij}$,
providing the usual isotropic constitutive relations,

$\displaystyle\mathbf{D}=\epsilon\mathbf{E},\quad\mathbf{H}=\mu^{-1}\mathbf{B}.$
(13)

A straightforward calculation from Eq. (11) yields

$\displaystyle\nabla\cdot\mathbf{D}$
$\displaystyle=J^{0}-\mathbf{k}_{AF}\cdot\mathbf{B},$ (14)
$\displaystyle\nabla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}$
$\displaystyle=\mathbf{J}+k_{AF}^{0}\mathbf{B}+\mathbf{k}_{AF}\times\mathbf{E},$
(15)

where $G^{i0}=D^{i}$ and $G^{ij}=-\epsilon_{ijk}H^{k}$. The homogeneous
Maxwell equations are given by

$\displaystyle\nabla\cdot\mathbf{B}$
$\displaystyle=0,\quad\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial
t}=\bf{0}.$ (16)

By using a plane-wave ansatz for the electromagnetic fields, the MCFJ
equations (14) (16) read,

$\displaystyle i\mathbf{k}\cdot\mathbf{{D}}+\mathbf{k}_{AF}\cdot\mathbf{{B}}$
$\displaystyle=J^{0},$ (17a) $\displaystyle
i\mathbf{k}\times\mathbf{{H}}+i\omega\mathbf{{D}}-k_{AF}^{0}\mathbf{{B}}-\mathbf{k}_{AF}\times\mathbf{{E}}$
$\displaystyle=\mathbf{{J}},$ (17b)
$\displaystyle\mathbf{k}\cdot\mathbf{{B}}=0,\quad\mathbf{k}\times\mathbf{{E}}-\omega\mathbf{{B}}$
$\displaystyle=\bf{0},$ (17c)

where $\mathbf{k}$ is the wave vector and $\omega$ is the (angular) wave
frequency.

In the presence of anisotropy, the permittivity and permeability are
represented by rank 2 tensors, $\varepsilon_{ij}$ and $\mu_{ij}$, which may
also depend on the frequency (for a dispersive medium). For an anisotropic
medium, the constitutive relations (13) are replaced by [1, 2],

$D^{i}=\varepsilon_{ij}(\omega)E^{j},\ \ \ B^{i}=\mu_{ij}(\omega)H^{j}.$ (18)

For nonmagnetic media with isotropic magnetic permeability, it holds
$\mu_{ij}(\omega)=\mu_{0}$, where $\mu_{0}$ is the vacuum permeability.
Considering the constitutive relations (18), the modified Ampère-Maxwell’s
law, Eq.(17b), and Faraday’s law, Eq.(17c), in the absence of sources, we
obtain a modified wave equation for the electric field,

$k^{i}\left(k^{j}{E^{j}}\right)-k^{2}{E}^{i}=-\omega^{2}\mu_{0}\bar{\varepsilon}_{ij}\left(\omega\right){E}^{j},$
(19)

where we define the extended permittivity tensor,

$\bar{\varepsilon}_{ij}(\omega)={\varepsilon}_{ij}(\omega)+i\frac{k_{AF}^{0}}{\omega^{2}}\epsilon_{ikj}k^{k}+i\epsilon_{ikj}\frac{k_{AF}^{k}}{\omega}.$
(20)

Using the definition to the refractive index, $\mathbf{n}=\mathbf{k}/\omega$,
the modified wave equation becomes

$M_{ij}{E}^{j}=0,$ (21)

with $M_{ij}$ given by

$M_{ij}=n^{2}\delta_{ij}-n_{i}n_{j}-\frac{\varepsilon_{ij}}{\varepsilon_{0}}-\frac{i}{\omega}\left(V_{0}\epsilon_{ikj}n^{k}+\epsilon_{ikj}V^{k}\right),$
(22)

in which $\varepsilon_{0}$ is the vacuum electric permittivity, and

$V_{0}=k_{AF}^{0}/\varepsilon_{0},\quad V^{k}=k_{AF}^{k}/\varepsilon_{0}.$
(23)

appear as the components of a redefined background,
$V^{\mu}=\left(V_{0},V^{i}\right)$. The nontrivial solutions for the electric
field require a vanishing determinant of the matrix $M_{ij}$, $\det M_{ij}=0$,
which provides the dispersion relations that describe the wave propagation in
the medium.

In this work, we will study plasma waves propagation for a chiral (parity-odd)
medium, which means restraining our investigation to the case of a purely
timelike Lorentz-violating background vector,
$\left(k_{AF}\right)^{\mu}=\left(k_{AF}^{0},\textbf{0}\right)$, which also
plays the role of chiral magnetic conductivity. This choice is physically
meaningful since it represents cold dark matter, for which the space variation
of the axion field can be neglected,
$\mathbf{\nabla}\theta=\mathbf{k}_{AF}=0$. In this scenario, the wave equation
(21) becomes

$\left[n^{2}\delta_{ij}-n^{i}n^{j}-\frac{{\varepsilon}_{ij}}{\varepsilon_{0}}-i\frac{V_{0}}{\omega}\epsilon_{ikj}n^{k}\right]{E}^{j}=0.$
(24)

##  III The usual magnetized cold plasma

In this work we will adopt the fluid theory approach in the cold plasma limit
[3, 4, 8, 7, 5]:

$\displaystyle\frac{\partial n}{\partial
t}+\mathbf{\nabla}\cdot\left(n\mathbf{u}\right)=0,$ (25)
$\displaystyle\frac{\partial\mathbf{u}}{\partial
t}+\mathbf{u}\cdot\mathbf{\nabla
u}=\frac{q}{m}\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}_{0}\right),$ (26)

where $n$ is the electron number density, $\mathbf{u}$ is the electron fluid
velocity field, $q$ and $m$ are the electron charge and mass, respectively,
and $\mathbf{B}_{0}$ is the equilibrium magnetic field. For simplicity, the
ions are supposed to be infinitely massive, which is appropriate for high-
frequency waves. Furthermore, thermal and collisional effects are also
disregarded. The linearized version of the magnetized cold plasmas [8]
consider fluctuations around average quantities, $n_{0}$ and $\mathbf{B}_{0}$,
which are constant in time and space. Thus, the plasma quantities read

$\displaystyle n$ $\displaystyle=n_{0}+\delta n,$ (27a)
$\displaystyle\mathbf{u}$ $\displaystyle=\delta\mathbf{u},$ (27b)
$\displaystyle\mathbf{E}$ $\displaystyle=\delta\mathbf{E}$ (27c)
$\displaystyle\mathbf{B}$ $\displaystyle=\mathbf{B}_{0}+\delta\mathbf{B},$
(27d)

with $\delta n$, $\delta\mathbf{u}$, $\delta\mathbf{E}$ and $\delta\mathbf{B}$
being first-order plane wave magnitude perturbations. Following the usual
procedure [3, 4, 7, 5], assuming $\mathbf{B}_{0}=B_{0}\hat{z}$, we write the
corresponding dielectric tensor,

$\varepsilon_{ij}(\omega)=\varepsilon_{0}\begin{bmatrix}S&-iD&0\\\ iD&S&0\\\
0&0&P\end{bmatrix},$ (28)

where

$S=1-\frac{\omega_{p}^{2}}{\left(\omega^{2}-\omega_{c}^{2}\right)},\
D=\frac{\omega_{c}\omega_{p}^{2}}{\omega\left(\omega^{2}-\omega_{c}^{2}\right)},\
P=1-\frac{\omega_{p}^{2}}{\omega^{2}},$ (29)

and

$\omega_{p}=\frac{n_{0}q^{2}}{m\epsilon_{0}},\quad\omega_{c}=\frac{|q|B_{0}}{m},$
(30)

are the plasma and cyclotron frequencies, respectively.

In cold magnetized plasmas, it is usual to investigate modes that propagate
parallel and perpendicular to the magnetic field. For longitudinal propagation
to the magnetic field, $\mathbf{k}\parallel\mathbf{B}_{0}$, two distinct
refractive indices are obtained,

$n_{\pm}=\sqrt{1-\frac{\omega_{p}^{2}}{\omega\left(\omega\pm\omega_{c}\right)}},$
(31)

which provide left-circularly polarized and right-circularly polarized modes,
respectively,

$\mathbf{{E}}_{LCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\
i\end{bmatrix},\quad\mathbf{{E}}_{RCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\
-i\end{bmatrix},$ (32)

for the propagating modes associated to $n_{\pm}$, respectively. This is the
standard result of wave propagation in the usual magnetized cold plasma. We
recall that a cutoff happens whenever the refractive index, $n$, goes to zero.
On the other hand, a resonance occurs if $n$ tends to infinity. From the
indices (31), we obtain the following cutoff frequencies:

$\displaystyle\omega_{\pm}$
$\displaystyle=\frac{1}{2}\left(\sqrt{\omega_{c}^{2}+4\omega_{p}^{2}}\mp\omega_{c}\right),$
(33)

where $\omega_{\pm}$ is related to $n_{\pm}$, respectively.

As for perpendicular propagation to the magnetic field,
$\mathbf{k}\perp\mathbf{B}_{0}$, $\mathbf{k}=(k_{x},k_{y},0)$, two refractive
indices are obtained. The one corresponding to the transversal mode, with
$\delta\mathbf{E}=(0,0,\delta E_{z})\perp\mathbf{k}$, is

$n_{T}^{2}=P,$ (34)

while the extraordinary mode [8],

$n_{O}^{2}=\frac{\left(S+D\right)\left(S-D\right)}{S},$ (35)

is longitudinal, that is, $\delta\mathbf{E}=(\delta E_{x},\delta E_{y},0)$.
The parameters $P$, $S$ and $D$ are given in Eq. (29). The refractive index
$n_{T}$ provides a linearly polarized mode, whereas $n_{O}$, in general, is
related to an elliptically polarized mode.

A very usual effect in magnetized plasmas is the circular birefringence222In
plasmas, the birefringence is usually a consequence of the Faraday effect,
occurring due to the presence of the external field $\mathbf{B}_{0}$, which
generates distinct phase velocities for the propagating modes [71]., which
causes the rotation of the plane of polarization of a linearly polarized wave
that propagates within the medium. Thus the linearly polarized wave emerges
from the medium with an electric field whose polarization is rotated relative
to its initial linear configuration. Such a phenomenon can be properly
explained by decomposing the initial wave into two circularly polarized waves
(RCP and LCP) that travel with different phase velocities. In this case, the
rotation angle of the electric field can be expressed as the difference
between the refractive indices associated with the RCP and LCP waves [69, 73]:

$\theta=\frac{\pi
L}{\lambda_{0}}\left(\mathrm{Re}\left[n_{RCP}\right]-\mathrm{Re}\left[n_{LCP}\right]\right),$
(36)

where $\lambda_{0}$ is the vacuum wavelength of the incident wave. The
rotation power $\delta=\theta/L$ (phase difference per unit length), is given
as

$\delta=-\frac{\omega}{2}\left(\mathrm{Re}\left[n_{LCP}\right]-\mathrm{Re}\left[n_{RCP}\right]\right).$
(37)

For parallel propagation in a cold magnetized plasma,
$\mathbf{k}\parallel\mathbf{B}_{0}$, the refractive indices (31) provide the
following rotatory power:

$\delta=-\frac{\omega}{2}\mathrm{Re}\left(\sqrt{1-\frac{\omega_{p}^{2}}{\omega\left(\omega+\omega_{c}\right)}}-\sqrt{1-\frac{\omega_{p}^{2}}{\omega\left(\omega-\omega_{c}\right)}}\right).$
(38)

The behavior of the RP (38) in terms of the frequency $\omega$ is depicted in
Fig. 1. One notices that there is a divergence at $\omega_{c}$, being positive
for $\omega<\omega_{c}$ and negative for $\omega>\omega_{c}$. It tends to zero
at the high-frequency limit $\omega>>\left(\omega_{p},\omega_{c}\right)$,
where it decays as

$\delta\approx-\frac{\omega_{p}^{2}\omega_{c}}{2\omega^{2}}.$ (39)

Associated with the imaginary part of the refractive index, one can also
examine dichroism, an optical effect that occurs when circularly polarized
waves are absorbed by the medium at different degrees [57, 59, 60]. Thus
dichroism coefficient refers to the difference in absorption of LCP and RCP
waves, being given by:

$\delta_{d}=-\frac{\omega}{2}\left(\mathrm{Im}[n_{LCP}]-\mathrm{Im}[n_{RCP}]\right).$
(40)

which, for the refractive indices (31), implies

$\delta_{d}=-\frac{\omega}{2}\mathrm{Im}\left(\sqrt{1-\frac{\omega_{p}^{2}}{\omega\left(\omega+\omega_{c}\right)}}-\sqrt{1-\frac{\omega_{p}^{2}}{\omega\left(\omega-\omega_{c}\right)}}\right).$
(41)

Figure 1: Rotatory power (38) in terms of $\omega$. Here,
$\omega_{c}=\omega_{p}$ (red line) and $\omega_{c}=2\omega_{p}$ (blue line),
with the choice $\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

Such a quantity is plotted in Fig. 2, which shows singularity at the cyclotron
frequency $\omega_{c}$. For $\omega_{c}=\omega_{p}$ (red curve), the dichroism
coefficient (41) is negative for $\omega<\omega_{+}^{red}$, positive for
$2\omega_{c}<\omega<\omega_{-}^{red}$ and null for other frequencies. The case
for $\omega_{c}=\omega_{p}/2$ (blue curve) differs in the fact that
$\omega_{+}^{blue}$ is greater than $\omega_{c}$, showing that (41) is now
negative for $\omega<\omega_{c}$.

Figure 2: Dichroism coefficient (41) in terms of $\omega$. Here,
$\omega_{c}=\omega_{p}$ (red line) and $\omega_{c}=\omega_{p}/2$ (blue line),
with $\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

##  IV Wave propagation in chiral plasma along the magnetic field

Collective modes of chiral systems were examined in the context of the Weyl
materials by means of the chiral kinetic theory [122]. Using a similar
formalism, low-energy collective modes (pseudomagnetic helicons) were
predicted for Dirac and Weyl matter [123]. In this section, we derive the
collective electromagnetic modes of the cold chiral plasma ruled by the
permittivity (20). In this sense, we start from the wave equation (24) and use
the expression of the magnetized plasma dielectric permittivity, given in Eq.
(28), obtaining a linear homogeneous system,

$\begin{bmatrix}n^{2}-n_{x}^{2}-S&iD-
n_{x}n_{y}+i\left(V_{0}/\omega\right)n_{z}&-n_{x}n_{z}-i\left(V_{0}/\omega\right)n_{y}\\\
-iD-
n_{x}n_{y}-i\left(V_{0}/\omega\right)n_{z}&n^{2}-n_{y}^{2}-S&-n_{y}n_{z}+i\left(V_{0}/\omega\right)n_{x}\\\
-n_{x}n_{z}+i\left(V_{0}/\omega\right)n_{y}&-n_{y}n_{z}-i\left(V_{0}/\omega\right)n_{x}&n^{2}-n_{z}^{2}-P\end{bmatrix}\begin{bmatrix}\delta
E_{x}\\\ \delta E_{y}\\\ \delta E_{z}\end{bmatrix}=0.$ (42)

Let us consider, for simplicity, the case the refractive index is parallel to
the magnetic field, $\mathbf{n}=n\hat{z}$, such that one obtains

$\begin{bmatrix}n^{2}-S&iD+i\left(V_{0}/\omega\right)n&0\\\
-iD-i\left(V_{0}/\omega\right)n&n^{2}-S&0\\\
0&0&-P\end{bmatrix}\begin{bmatrix}\delta E_{x}\\\ \delta E_{y}\\\ \delta
E_{z}\end{bmatrix}=0,$ (43)

for which $\mathrm{det}[M_{ij}]=0$ provides the dispersion relations

$P\left(\omega^{2}\left(n^{2}-S\right)^{2}-(\omega D+nV_{0})^{2}\right)=0.$
(44)

Longitudinal waves, $\mathbf{n}\parallel\delta\mathbf{E}$ or
$\delta\mathbf{E}=(0,0,\delta E_{z})$, may emerge, when $P=0$, with
nonpropagating vibration at the plasma frequency, $\omega=\omega_{p}$. Under
an electromagnetic perspective, this longitudinal oscillation is a plasmon. In
a solid-state context, the collective mode of electrons vibrating
(longitudinally) under the action of the electromagnetic field is also called
plasmons.

For transverse waves, $\mathbf{n}\perp\delta\mathbf{E}$ or
$\delta\mathbf{E}=(\delta E_{x},\delta E_{y},0)$, the dispersion relation (44)
simplifies as

$\left(n^{2}-S\right)^{2}-\left(D+n\left(V_{0}/\omega\right)\right)^{2}=0,$
(45)

also written as a fourth-order equation in $n$,

$n^{4}-\left(2S+\left(V_{0}/\omega\right)^{2}\right)n^{2}-2D\left(V_{0}/\omega\right)n+\left(S^{2}-D^{2}\right)=0.$
(46)

Taking into account the relations (29), the dispersion relation (46) provides
the following refractive indices for electromagnetic modes of the model:

$\displaystyle n_{R,M}$
$\displaystyle=-\frac{V_{0}}{2\omega}\pm\sqrt{1+\left(\frac{V_{0}}{2\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega(\omega-\omega_{c})}},$
(47) $\displaystyle n_{L,E}$
$\displaystyle=\frac{V_{0}}{2\omega}\pm\sqrt{1+\left(\frac{V_{0}}{2\omega}\right)^{2}-\frac{\omega_{p}^{2}}{\omega(\omega+\omega_{c})}}.$
(48)

In general, the indices $n_{R},n_{L},n_{E},n_{M}$ may be real, imaginary, or
complex (presenting both pieces) at some frequency ranges. As well-known, the
real part is associated with propagation, while the complex piece is concerned
with absorption. Furthermore, these indices may have positive or negative real
pieces. The indices $n_{L}$ and $n_{M}$ are always positive and negative,
respectively, the latter one being a negative refractive index. On the other
hand, the indices $n_{R}$ and $n_{E}$ can be positive or negative, depending
on the frequency zone examined, in such a way the associated modes can
manifest negative refraction behavior (in a suitable frequency band).

The propagating modes associated with the refractive indices in Eq. (47)) and
Eq. (48) are obtained by inserting each one in Eq. (43) and carrying out the
corresponding eigenvector (with a null eigenvalue). The emerging electric
field are the $\mathbf{{E}}_{LCP}$ and $\mathbf{{E}}_{RCP}$, given in Eq.
(32), where $n_{R}$, $n_{M}$ are associated with the RCP mode, and $n_{L}$,
$n_{E}$ are related to the LCP mode,

$n_{L},n_{E}\ \mapsto\
\mathbf{{E}}_{LCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\ i\end{bmatrix},$ (49)
$n_{R},n_{M}\ \mapsto\
\mathbf{{E}}_{RCP}=\frac{i}{\sqrt{2}}\begin{bmatrix}1\\\ -i\end{bmatrix}.$
(50)

From the indices $n_{R}$, $n_{E}$, given by Eqs. (47) and (48), we obtain the
same cutoff frequencies (33) of the standard case: in fact, $\omega_{-}$ is
related to the refractive index $n_{R}$, and $\omega_{+}$ is associated with
the refractive index $n_{E}$. In contrast, the refractive indices $n_{L}$ and
$n_{M}$ have no real root. The behavior of the refractive indices in Eqs. (47)
and (48) will be examined in the following.

###  IV.1 About the index $n_{R}$

We initiate discussing some properties of the index $n_{R}$. The behavior of
$n_{R}$ in terms of the dimensionless parameter $\omega/\omega_{c}$ is
illustrated in Fig. 3, which displays the real imaginary pieces of the
refractive index $n_{R}$. We point out:

  1. (i)

It takes on a finite value when $\omega\rightarrow 0$, given by

$n_{R}\left(0\right)=\frac{1}{V_{0}}\left(\frac{\omega_{p}^{2}}{\omega_{c}}\right),$
(51)

describing a modified helicon in relation to the standard scenario, where the
usual magnetized index $n_{-}$ goes to infinity near the origin.

  2. (ii)

For $0<\omega<\omega_{c}$, $n_{R}$ is positive since the square root in (47)
is real, positive, and larger than the negative piece before it. Such a
positivity also holds for the usual index $n_{-}$. See the black line in this
frequency zone in Fig. 3.

  3. (iii)

For $\omega\rightarrow\omega_{c}$, $n_{R}\rightarrow\infty$, and there occurs
a resonance at the cyclotron frequency.

  4. (iv)

For $\omega_{c}<\omega<\omega_{r}$, there appears a negative refractive index
zone with absorption, where $\mathrm{Re}[n_{R}]<0$ and $\mathrm{Im}[n_{R}]\neq
0$, as shown in Fig. 3. The frequency $\omega_{r}$ is the root of the radicand
in Eq. (47),

$R_{-}\left(\omega\right)=1+\frac{V_{0}^{2}}{4\omega^{2}}-\frac{\omega_{p}^{2}}{\omega\left(\omega-\omega_{c}\right)},$
(52)

which yields a cubic equation in $\omega$.

  5. (v)

For $\omega_{r}<\omega<\omega_{-}$, one finds a negative refractive index zone
without absorption, that is, $\mathrm{Re}[n_{R}]<0$ and
$\mathrm{Im}[n_{R}]=0$.

  6. (vi)

For $\omega>\omega_{-}$, the quantity $n_{R}$ is always positive,
corresponding to a propagating zone, with $n_{R}\rightarrow 1$ in the high-
frequency limit.

Figure 3: Index of refraction $n_{R}$ in terms of the frequency $\omega$. The
dashed blue (black) line corresponds to the imaginary piece of $n_{R}$
($n_{-}$), while the solid blue (black) line represents the real piece of
$n_{R}$ ($n_{-}$). Here $\omega_{c}=\omega_{p}$, $V_{0}=2\omega_{p}$, and
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

The frequency zone in which $\mathrm{Im}[n_{R}]\neq 0$, that is,
$\omega_{c}<\omega<\omega_{r}$, corresponds to the absorption zone for the
metamaterial (negative refractive index) RCP wave, as already mentioned
before. The frequency ranges in which $\mathrm{Im}[n_{R}]=0$ define the
propagation zone for the RCP wave.

###  IV.2 About the index $n_{L}$

The index $n_{L}$, given in Eq. (48), has no real root, presenting the
following features:

  1. (i)

For $\omega\rightarrow 0$, $n_{L}$ $\rightarrow+\infty$. Then, the presence of
the term $V_{0}$ turns the refractive index real and positively divergent at
the origin, differing from the usual index $n_{+}$ behavior, see Eq. (31),
which is complex and divergent, $\mathrm{Im}[n_{+}]\rightarrow\infty$, at the
origin. As it will be clear in Eq. (60) of Sec. E, this index also supports
helicons, or more specifically, LCP helicons.

  2. (ii)

For $\omega>0$, it is necessary to analyze the radicand in Eq. (48),

$R_{+}\left(\omega\right)=1+\frac{V_{0}^{2}}{4\omega^{2}}-\frac{\omega_{p}^{2}}{\omega\left(\omega+\omega_{c}\right)},$
(53)

since it can be positive or negative, which determines the absence or presence
of an absorption zone, respectively. Note that for $\omega>\omega_{+}$ the
term $1-\omega_{p}^{2}/\omega\left(\omega+\omega_{c}\right)$ is greater than
zero ($\omega_{+}$ is the root of such a term), such that $R_{+}$ is positive.
Therefore, the possibility of $R_{+}$ being negative occurs only in the range
$0<\omega<\omega_{+}$, for which the term
$1-\omega_{p}^{2}/\omega\left(\omega+\omega_{c}\right)$ is less than zero.
Hence, this positivity for $R_{+}$ is stated by the condition,

$\frac{V_{0}^{2}}{4\omega^{2}}>\left|1-\frac{4\omega_{p}^{2}}{\omega(\omega+\omega_{c})}\right|_{\omega<\omega_{+}},$
(54)

for which $R_{+}$ is always positive and the refractive index $n_{L}$ is real
for any $\omega>0$. This corresponds to a propagating mode for the entire
frequency domain. The behavior of $n_{L}$ in terms of the dimensionless
parameter $\omega/\omega_{c}$, considering the condition (54), that is,
$R_{+}>0$, is shown in Fig. 4.

  3. (iii)

On the other hand, for

$\frac{V_{0}^{2}}{4\omega^{2}}<\left|1-\frac{4\omega_{p}^{2}}{\omega(\omega+\omega_{c})}\right|_{\omega<\omega_{+}},$
(55)

one has $R_{+}<0$ and $n_{L}$ becomes complex, $\mathrm{Im}[n_{L}]\neq 0$,
determining the opening of an absorption zone located within the interval
$\omega_{i}<\omega<\omega_{f}$, as shown in Fig. 5. The frequencies
$\omega_{i}$ and $\omega_{f}$ are positive and real roots of $R_{+}$, a cubic
equation in the frequency.

Figure 4: Refractive index $n_{L}$ (blue lines) for the condition (54),
$R_{+}>0$. Refractive index $n_{+}$ (black lines) of Eq. (31). The dashed
(solid) lines correspond to the imaginary (real) pieces of $n_{L}$ and
$n_{+}$. Here $\omega_{c}=\omega_{p}$, $V_{0}=2\omega_{p}$, and $\omega_{c}=1$
$\mathrm{rad}$ $s^{-1}$. Figure 5: Refractive index $n_{L}$ (blue lines) for
the condition (55), $R_{+}<0$. Refractive index $n_{+}$ (black lines) of Eq.
(31). The dashed (solid) lines correspond to the imaginary (real) pieces of
$n_{L}$ and $n_{+}$. Here $\omega_{c}=\omega_{p}$, $V_{0}=0.7\omega_{p}$, and
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

###  IV.3 About the index $n_{E}$

The quantity $n_{E}$ is a refractive index that only exists as a positive
quantity due to the presence of the chiral Lorentz-violating term. In the case
we set $V_{0}=0$, the second relation in Eq. (48) yields
$\mathrm{Re}[n_{E}]<0$ (negative index of refraction). For $V_{0}\neq 0$, the
index $n_{E}$ presents a small positivity range, $\mathrm{Re}[n_{E}]>0$, which
provides propagation for the associated LCP wave. We present below some
aspects of $n_{E}$:

  1. (i)

For $\omega\rightarrow 0$, the index $n_{E}$ tends to a finite value at
origin,

$n_{E}\left(0\right)=\frac{1}{V_{0}}\left[\frac{\omega_{p}^{2}}{\omega_{c}}\right],$
(56)

which is inversely proportional to the magnitude of the chiral factor,
$V_{0}$. Such a distinct low-frequency behavior also implies a LCP helicon,
inexistent in the usual case.

  2. (ii)

Since the radicand of $n_{E}$ is the same one of $n_{L}$, see Eq. (48), it
holds here the same procedure applied for $n_{L}$. For values of $V_{0}$ that
satisfy the condition (54), $R_{+}>0$, $n_{E}$ is always real,
$\mathrm{Im}[n_{E}]=0$, being positive within the interval
$0<\omega<\omega_{+}$, and negative for $\omega>\omega_{+}$, since
$\sqrt{R_{+}}>V_{0}/2\omega$ at this range. The real and imaginary parts of
$n_{E}$ are represented in Fig. 6.

Figure 6: Red line: plot of the index $n_{E}$ for the condition (54),
$R_{+}>0$. Black line: plot of the index $-n_{+}$ of Eq. (31). Dashed (solid)
lines represent the imaginary (real) pieces of $n_{E}$ and $-n_{+}$. Here, we
have used $\omega_{c}=\omega_{p}$ and $V_{0}=2\omega_{p}$, with the choice
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

  3. (iii)

Considering the condition (55), $n_{E}$ becomes complex and exhibits an
absorption zone, $\mathrm{Im}[n_{E}]\neq 0$, in the interval
$\omega_{i}<\omega<\omega_{f}$, with $\omega_{i},\omega_{f}<\omega_{+}$, as
shown in Fig. 7. Such a figure depicts the real and imaginary pieces of
$n_{E}$ [under the condition (55)].

Figure 7: Red line: plot of the index $n_{E}$ for the condition (55),
$R_{+}<0$. Black line: plot of the index $-n_{+}$ of Eq. (31). Dashed (solid)
lines represent the imaginary (real) pieces of $n_{E}$ and $-n_{+}$. Here, we
have set $\omega_{c}=\omega_{p}$, $V_{0}=0.7\omega_{p}$, and $\omega_{c}=1$
$\mathrm{rad}$ $s^{-1}$.

###  IV.4 About the index $n_{M}$

The additional index $n_{M}$, given in Eq. (47), is always negative (negative
refraction) and has no real root. The behavior of $n_{M}$ in terms of the
dimensionless parameter $\omega/\omega_{c}$ is shown in Fig. 8. We notice the
following features:

  1. (i)

For $0<\omega<\omega_{c}$, $n_{M}$ is real and negative since the square root
in (47) is real. This is the same behavior of the index $-n_{-}$. See the
black line in Fig. (8).

  2. (ii)

For $\omega\rightarrow\omega_{c}$, $n_{M}\rightarrow-\infty$, and there occurs
a resonance at the cyclotron frequency.

  3. (iii)

For $\omega_{c}<\omega<\omega_{r}$, there appears an absorption zone for
metamaterial, $\mathrm{Re}[n_{M}]<0$ and $\mathrm{Im}[n_{M}]\neq 0$, while the
index $-n_{-}$ is purely imaginary, $\mathrm{Re}[n_{M}]=0$ and
$\mathrm{Im}[n_{M}]\neq 0$, as shown in Fig. 8. The frequency $\omega_{r}$ is
the root of $R_{-}$.

  4. (iv)

For $\omega>\omega_{r}$, the quantity $n_{M}$ is always negative,
corresponding to a negative propagation zone, with $n_{M}\rightarrow-1$ in the
high-frequency limit.

Figure 8: Blue line: plot of the index $n_{M}$. Black line: plot of the real
piece of $-n_{-}$ of Eq. (31). Dashed (solid) lines represent the imaginary
(real) pieces of $n_{M}$ and $-n_{-}$. Here, we have used:
$\omega_{c}=\omega_{p}$, $V_{0}=2\omega_{p}$, and $\omega_{c}=1$
$\mathrm{rad}$ $s^{-1}$.

###  IV.5 Low-frequency modes

Considering the low-frequency regime,

$\displaystyle\omega\ll\omega_{p},\quad\omega_{c}\ll\omega_{p},\quad\omega\ll\omega_{c},$
(57)

the magnetized plasma RCP refractive index (31) provides a helicon mode,
described by

$\displaystyle n_{-}$
$\displaystyle=\omega_{p}\sqrt{\frac{1}{\omega\omega_{c}}}.$ (58)

Helicons are RCP modes that propagate at very low frequencies and along the
magnetic field axis. See Ref. [5] (Chapter 9), and Ref. [6] (Chapter 8) for
basic details. For the electromagnetic modes obtained in Eq. (47) and Eq.
(48), the corresponding helicons indices are

$\displaystyle\bar{n}_{R,E}$
$\displaystyle=\frac{\omega_{p}^{2}}{\omega_{c}V_{0}},$ (59)
$\displaystyle\bar{n}_{L,M}$
$\displaystyle=-\frac{\omega_{p}^{2}}{\omega_{c}V_{0}}\pm\frac{V_{0}}{\omega},$
(60)

where we have used the “bar” notation to indicate the helicons quantities. In
this chiral context, we observe the existence of both RCP and LCP helicons,
that both propagate in the low-energy regime, due to the presence of the
timelike component, $V_{0}$. The helicon modes given by $\bar{n}_{R,E}$ have
constant and nondispersive indices, which depend on the inverse of $V_{0}$. On
the other hand, negative-refraction dispersive helicons are associated with
$\bar{n}_{M}$, and also with $\bar{n}_{L}$ for
$\omega<\omega_{c}V_{0}^{2}/\omega_{p}^{2}$. Note that the term $V_{0}/\omega$
appears as a channel of distinction between the helicons associated with
$\bar{n}_{R,E}$ and $\bar{n}_{L,M}$, as expected.

The cold plasma usual helicon mode is recovered if one also considers, besides
the relations (57), the condition $V_{0}\ll\omega$. In this case, the
expansion of $n_{R,E}$ of Eq. (47) and $n_{L,M}$ of Eq. (48) yields

$\displaystyle\tilde{n}_{R,M}$
$\displaystyle=\pm\omega_{p}\sqrt{\frac{1}{\omega\omega_{c}}},$ (61)

which is the same result as the usual case [see Eq. (58)], and
$\tilde{n}_{L,E}$ being purely imaginary. Here, the “tilde” notation indicates
helicons in the specific case where $\omega_{c}\ll\omega_{p}$,
$\omega\ll\omega_{c}$, and $V_{0}\ll\omega$.

###  IV.6 Dispersion relations behavior

The wave dispersion associated with each refractive index is usually
visualized in plots $\omega\times k$. In the following, we work with
dimensionless plots, $(\omega/\omega_{c})\times(k/\omega_{c})$.

The dispersion relations associated with $n_{R}$ and $n_{M}$ are depicted in
Fig. 9 for $\omega_{c}=\omega_{p}$. The propagation occurs for
$0<\omega<\omega_{c}$ and $\omega>\omega_{-}$, while absorption takes place in
$\omega_{c}<\omega<\omega_{r}$. The range $\omega_{r}<\omega<\omega_{-}$
corresponds to negative refraction propagation zone ($k<0$) for $n_{R}$. The
refractive index $n_{M}$ is negative for $k<0$ and $\omega>0$.

Figure 9: Plot of the dispersion relations related to refractive indices
$n_{R}$ (solid red line) and $n_{M}$ (solid blue line). The dashed black line
corresponds to the indices of the usual case ($\pm n_{-}$). The highlighted
area in red (gray) indicates the absorption zone for $n_{R,M}$ ($\pm n_{-}$).
Here, we have used $\omega_{c}=\omega_{p}$ and $V_{0}=2\omega_{p}$, with
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

Figure 10 depicts the dispersion relations related to $n_{L}$ and $n_{E}$. The
wave associated with $n_{L}$ propagates for all frequencies. For $n_{E}$, the
conventional propagation zone occurs in $0<\omega<\omega_{+}$. For
$\omega>\omega_{+}$, there occurs a propagation zone with negative refraction.
For the standard indices, $\pm n_{+}$, the absorption zone is
$0<\omega<\omega_{+}$.

Figure 10: Plot of the dispersion relations related to refractive indices
$n_{L}$ (solid red line) and $n_{E}$ (solid blue line). The dashed line
corresponds to the indices $\pm n_{+}$ of the usual case. The highlighted gray
area indicates the absorption zone for $\pm n_{+}$, where now also occurs
propagation. Here, we have used $\omega_{c}=\omega_{p}$ and
$V_{0}=\omega_{p}$, with $\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

Furthermore, Fig. 11 shows the dispersion relations for $n_{L}$ and $n_{E}$ in
the case there is a modified absorption zone for
$\omega_{i}<\omega<\omega_{f}$, while the free propagation occurs for
$0<\omega<\omega_{i}$ and $\omega>\omega_{f}$. The frequencies $\omega_{i}$,
$\omega_{f}$, and $\omega_{r}$ define the limits for unusual propagation
zones. As already discussed, these frequencies are obtained from the radicands
(52) and (53).

Figure 11: Plot of the dispersion relations related to refractive indices
$n_{L}$ (solid red line) and $n_{E}$ (solid blue line). The dashed line
corresponds to the usual case with indices $\pm n_{+}$. The highlighted areas
in red (gray) indicate the absorption zone for $n_{L,E}$ ($\pm n_{+}$). Here,
we have used $\omega_{c}=\omega_{p}$ and $V_{0}=0.7\omega_{c}$, with
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

##  V WAVE PROPAGATION ORTHOGONAL TO THE MAGNETIC FIELD

For investigating wave propagation orthogonally to the magnetic field
direction, we rewrite Eq. (42) for $\mathbf{n}=\left(n_{x},n_{y},0\right)$,
that is,

$\begin{bmatrix}n^{2}-n_{x}^{2}-S&iD-
n_{x}n_{y}&-i\left(V_{0}/\omega\right)n_{y}\\\ -iD-
n_{x}n_{y}&n^{2}-n_{y}^{2}-S&+i\left(V_{0}/\omega\right)n_{x}\\\
+i\left(V_{0}/\omega\right)n_{y}&-i\left(V_{0}/\omega\right)n_{x}&n^{2}-P\end{bmatrix}\begin{bmatrix}\delta
E_{x}\\\ \delta E_{y}\\\ \delta E_{z}\end{bmatrix}=0.$ (62)

Using the parametrization $\mathbf{n}=n\left(\cos\phi,\sin\phi,0\right)$, it
becomes

$\begin{bmatrix}n^{2}-n^{2}\cos^{2}\phi-S&iD-n^{2}\sin\phi\cos\phi&-i\left(V_{0}/\omega\right)n\sin\phi\\\
-iD-n^{2}\sin\phi\cos\phi&n^{2}-n^{2}\sin^{2}\phi-S&+i\left(V_{0}/\omega\right)n\cos\phi\\\
+i\left(V_{0}/\omega\right)n\sin\phi&-i\left(V_{0}/\omega\right)n\cos\phi&n^{2}-P\end{bmatrix}\begin{bmatrix}\delta
E_{x}\\\ \delta E_{y}\\\ \delta E_{z}\end{bmatrix}=0,$ (63)

whose null determinant yields the angle-independent dispersion relation,

$\frac{n^{2}SV_{0}^{2}}{\omega^{2}}-\left(n^{2}-P\right)\left(D^{2}+S\left(n^{2}-S\right)\right)=0,$
(64)

providing two refractive indices given by

$n_{O\pm}^{2}=\frac{P+S}{2}-\frac{D^{2}}{2S}+\frac{V_{0}^{2}}{2\omega^{2}}\pm\frac{\Gamma}{2S},$
(65)

where

$\Gamma=\sqrt{\left(D^{2}-PS-S^{2}-\frac{SV_{0}^{2}}{\omega^{2}}\right)^{2}-4SP\left(S^{2}-D^{2}\right)}.$
(66)

From Eq. (63), the indices in Eq. (65) are related to the following
propagating modes:

$\mathbf{{E}}=C\begin{bmatrix}\zeta\\\ 1\\\
i\frac{SV_{0}n_{O\pm}}{\gamma\omega\left(n_{O\pm}^{2}-P\right)}\end{bmatrix},$
(67)

with

$C=\frac{\omega\left(n_{O\pm}^{2}-P\right)\left|\gamma\right|}{\sqrt{2\omega^{2}\left(n_{O\pm}^{2}-P\right)^{2}\left|\gamma\right|^{2}+\left(SV_{0}n_{O\pm}\right)^{2}}},$
(68) $\zeta=-\frac{S\sin\phi-iD\cos\phi}{\gamma},$ (69)
$\gamma=S\cos\phi+iD\sin\phi.$ (70)

The refractive index $n_{O+}$ has a cutoff at the plasma frequency
$\omega_{p}$, while $n_{O-}$ has two cutoff frequencies, $\omega_{\pm}$, given
in Eq. (33).

###  V.1 About the index $n_{O+}$

The refractive index $n_{O+}$ has to be compared to the index $n_{T}$, given
in Eq. (34), associated with the usual transversal mode, since in the limit
$V_{0}\rightarrow 0$ it recovers the refractive index $n_{T}$. We present
below some aspects of $n_{O+}$:

  1. (i)

For $0<\omega<\omega_{p}$, there is a propagation zone in which $n_{O+}$ is
real. This behavior is markedly different from the usual case, for which there
corresponds to an absorption zone (in this range). See the dashed black line
in Fig. 12.

  2. (ii)

For $\omega\rightarrow\omega_{p}$, $n_{O+}$ has an unusual discontinuity, as
shown in Fig. 12 (see the red curve).

  3. (iii)

For $\omega>\omega_{p}$, the index $n_{O+}$ is always real, corresponding to a
propagation zone.

Figure 12: Red line: plot of the index $n_{O+}$. Black line: plot of the index
$n_{T}$. Dashed (solid) lines represent the imaginary (real) pieces of
$n_{O+}$ and $n_{T}$. Here, we have used: $\omega_{c}=\omega_{p}$,
$V_{0}=2\omega_{p}$, and $\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

###  V.2 About the index $n_{O-}$

The refractive index $n_{O-}$, given in Eq. (65), is a modification of the
index (35), associated with the usual extraordinary mode. The index $n_{O-}$
shares the same usual resonance frequency,

$\omega_{cp}=\sqrt{\omega_{c}^{2}+\omega_{p}^{2}.}$ (71)

We point out:

  1. (i)

For $0<\omega<\omega_{+}$, $n_{O-}$ is real and positive, corresponding to a
propagation zone. It contrasts with the usual case, where $n_{O}$ is imaginary
in this range (absorption zone). See the black line in Fig. 13.

  2. (ii)

For $\omega_{+}<\omega<\omega_{p}$, there occurs an absorption zone, where
$\mathrm{Re}[n_{O-}]=0$ and $\mathrm{Im}[n_{O-}]\neq 0$. In the standard case,
there is a propagation zone in this range.

  3. (iii)

For $\omega\rightarrow\omega_{p}$, $n_{O-}$ has a discontinuity, as shown in
Fig. 13. For $\omega_{p}<\omega<\omega_{cp}$, the index $n_{O-}$ is real and
there appears a propagation zone. The same positivity occurs in the usual
case.

  4. (iv)

For $\omega\rightarrow\omega_{cp}$, the index $n_{O-}\rightarrow+\infty$ and
there occurs a resonance. For $\omega_{cp}<\omega<\omega_{-}$,
$\mathrm{Re}[n_{O-}]=0$ and $\mathrm{Im}[n_{O-}]\neq 0$, and one has an
absorption zone, the same behavior of the usual case in this range.

  5. (v)

For $\omega>\omega_{-}$, the quantity $n_{O-}$ is always positive,
corresponding to a propagation zone.

Figure 13: Red line: plot of the index $n_{O-}$. Black line: plot of the index
$n_{O}$. Dashed (solid) lines represent the imaginary (real) pieces of
$n_{O-}$ and $n_{O-}$. Here, we have used $\omega_{c}=\omega_{p}$,
$V_{0}=2\omega_{p}$, and $\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

##  VI Birefringence, rotatory power and dichroism

The phase velocity in terms of the refractive index $n$ is defined (in natural
units) as $v_{phase}=1/n$. Hence, the corresponding phase velocities,
$v_{R}=1/\left(n_{R}\right)$, $v_{L}=1/\left(n_{L}\right)$,
$v_{E}=1/\left(n_{E}\right)$, $v_{M}=1/\left(n_{M}\right)$, can be defined
with the indices $n_{R}$, $n_{L}$, $n_{E}$, $n_{M}$ of Eqs. (47) and (48).
Accordingly with the previous analysis of the refractive indices, in general,
the RCP and LCP modes propagate at different phase velocities for each
frequency value, generating circular birefringence in the propagation band,
expressed in terms of the rotatory power (37). On the other hand, in the
absorption zones, there occurs dichroism, measured in terms of the coefficient
of Eq. (40).

###  VI.1 Rotatory power

In order to write the rotatory power, we need to consider the refractive
indices $n_{L}$, $n_{E}$, associated with the LCP wave, and the indices
$n_{R}$, $n_{M}$, associated to the RCP wave. It allows, in principle, to
determine four distinct RPs at the propagation zones, some of which we examine
in this section.

We start by writing the rotation power defined in terms of real pieces of the
refractive indices $n_{L}$ and $n_{R}$,

$\delta_{LR}=-\frac{\omega}{2}\left(\mathrm{Re}[n_{L}]-\mathrm{Re}[n_{R}]\right),$
(72)

or explicitly,

$\delta_{LR}=-\frac{\omega}{2}\mathrm{Re}\left[V_{0}/\omega+\sqrt{R_{+}}-\sqrt{R_{-}}\right],$
(73)

where $R_{+}$ and $R_{-}$ are given in Eqs. (52) and (53). We find a positive
frequency,

$\hat{\omega}=\sqrt{\omega_{c}^{2}+\omega_{p}^{2}/2-\frac{\omega_{p}^{2}\sqrt{4\omega_{c}^{2}+V_{0}^{2}}}{2V_{0}}},$
(74)

where the RP (73) undergoes a sign reversal. In Fig. 14, we illustrate the
behavior of RP for the condition (54). For the interval
$0<\omega<\hat{\omega}$, the RP is negative, and for
$\hat{\omega}<\omega<\omega_{c}$, it is positive. The RP reversion that occurs
at $\omega=\hat{\omega}$ is not usual in cold plasmas theory. However, it is
reported in graphene systems [85], rotating plasmas [96], and bi-isotropic
dielectrics supporting chiral magnetic current [97]. For $\omega>\omega_{c}$,
the RP is always negative. Nevertheless, it is necessary to pay attention to
the interval $\omega_{c}<\omega<\omega_{r}$, where the refractive index
$n_{R}$ has an imaginary piece and the RCP wave is absorbed. At
$\omega=\omega_{r}$, the real piece of $n_{R}$ undergoes a sharp change (see
Fig. 3), which also appears in the RP profile of Fig. 14.

Figure 14: The solid blue line represents the rotatory power (73) defined by
the refractive index $n_{L}$ and $n_{R}$, for the condition (54). The dashed
black line corresponds to the usual rotatory power (38). Here, we have used
$\omega_{c}=\omega_{p}$, $V_{0}=\omega_{p}$, and $\omega_{c}=1$ $\mathrm{rad}$
$s^{-1}$.

We can safely claim that both modes associated with the $n_{L}$ and $n_{R}$
propagate for $\omega>\omega_{-}$, range in which the RP magnitude decreases
monotonically with $\omega$, approaching to its asymptotic value, $-V_{0}/2$
(see Fig. 14). Assuming the limit where
$\omega>>\left(\omega_{p},\omega_{c}\right)$, we can write

$\displaystyle n_{L,R}$ $\displaystyle\approx
1\pm\frac{V_{0}}{2\omega}+\frac{V_{0}^{2}}{8\omega^{2}}-\frac{\omega_{p}^{2}}{2\omega\left(\omega\pm\omega_{c}\right)},$
(75)

so that the rotatory power is

$\delta_{LR}\approx-\frac{V_{0}}{2}-\frac{\omega_{p}^{2}\omega_{c}}{2\omega^{2}}.$
(76)

Note that taking the limit $V_{0}\rightarrow 0$, the usual Faraday effect RP
(39) is recovered for the high-frequency regime. It is also interesting to
point out that the Faraday effect disappears for a null magnetic field,
$\omega_{c}=0$. However, the birefringence still remains, due to the presence
of the chiral term, which yields the following RP:

$\delta\approx-V_{0}/2.$ (77)

For the condition (55), the RP (73) also exhibits a sign reversal and a very
similar profile to the one of Fig. 14, in such a way that it will not be
depicted here.

Considering now the refractive indices $n_{E}$ and $n_{R}$, the rotatory power
is

$\delta_{ER}=-\frac{\omega}{2}\left(\mathrm{Re}[n_{E}]-\mathrm{Re}[n_{R}]\right),$
(78)

or,

$\delta_{ER}=-\frac{\omega}{2}\mathrm{Re}\left[V_{0}/\omega-\sqrt{R_{+}}-\sqrt{R_{-}}\right].$
(79)

Recalling that the LCP wave associated with $n_{E}$ has a conventional free
propagation for $\omega<\omega_{+}$ and propagation with negative refractive
index ($n_{E}<0$) for $\omega>\omega_{+}$ (with $\omega_{+}<\omega_{c}$), the
RP magnitude is enhanced in the latter zone. This behavior is depicted in Fig.
15, which shows the RP (79) for $n_{E}$ given by the condition (54),
$R_{+}>0$. The RP is positive for $\omega<\omega_{c}$ and negative for
$\omega_{c}<\omega<\omega^{\prime\prime}$, becoming positive again for
$\omega>\omega^{\prime\prime}$, where $\omega^{\prime\prime}$ is the reversal
frequency. For $n_{E}$ given by the condition condition (55), the RP is
depicted in Fig. 16, revealing a small reversion at
$\omega^{\prime\prime}<\omega_{c}$. Note that the increasing RP with $\omega$,
depicted in Figs. 15 and 16, is due to the negative behavior of the index
$n_{E}$ for $\omega>\omega_{+}$, that is, enhancement associated with the
negative refraction.

In the asymptotic limit, where $\omega>>\left(\omega_{p},\omega_{c}\right)$,
the RP (79) goes as

$\delta_{ER}\approx\omega-\frac{V_{0}}{2},$ (80)

presentig a predominant linear behavior in $\omega$, as it appears in Figs. 15
and 16. It is also worth mentioning that the limit $V_{0}\rightarrow 0$,
implying $\delta\approx\omega$, does not stand for a valid result for usual
magnetized plasma, since the RP (79) is not defined for achiral cold plasmas.

Figure 15: Solid blue lines: plot of the rotatory power (79) associated to the
refractive indices $n_{E}$ and $n_{R}$ for the condition (54). The dashed line
represents the usual rotatory power (38). Here, we have used
$\omega_{c}=\omega_{p}$, $V_{0}=\omega_{p}$, and $\omega_{c}=1$ $\mathrm{rad}$
$s^{-1}$. The inset plot highlights the behavior of $\delta$ around
$\omega=\omega^{\prime\prime}$. Figure 16: Solid red lines: rotatory power
(79) associated to the refractive indices $n_{E}$ and $n_{R}$ for the
condition (55). The dashed line represents the usual rotatory power (38).
Here, we have used $\omega_{c}=\omega_{p}$, $V_{0}=0.7\omega_{p}$, and
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$. The inset plot highlights the behavior
of $\delta$ around $\omega=\omega^{\prime\prime}$.

###  VI.2 Dichroism coefficients

As well known, absorption depends on the magnitude of the imaginary parts of
the refractive indices. When one mode is more absorbed than the other, there
occurs dichroism. Considering the refractive indices $n_{L}$ and $n_{R}$, the
circular dichroism coefficient is

$\delta_{dLR}=-\frac{\omega}{2}\left(\mathrm{Im}[n_{L}]-\mathrm{Im}[n_{R}]\right).$
(81)

Considering the condition (54), only $n_{R}$ has imaginary part (localized in
the interval $\omega_{c}<\omega<\omega_{-}$), while $n_{L}$ is real for
$\omega>0$. In this case, the dichroism coefficient is given by

$\delta_{dLR}=\begin{cases}\text{$0,$}&\quad\text{for
$0<\omega<\omega_{c},$}\\\ \text{$\sqrt{R_{-}},$}&\quad\text{for
$\omega_{c}<\omega<\omega_{r},$}\\\ \text{$0,$}&\quad\text{for
$\omega>\omega_{r},$}\end{cases}$ (82)

being non-null only in the range $\omega_{c}<\omega<\omega_{-}$, as properly
shown in Fig 17.

Figure 17: Plot of the dichroism coefficient (82)(red solid lines) associated
to the refractive indices $n_{L}$ and $n_{R}$, under the condition (54). The
black dashed line represents the usual dichroism coefficient (41). Here
$\omega_{c}=\omega_{p}$, $V_{0}=\left(3/2\right)\omega_{c}$, and
$\omega_{c}=1$ $\mathrm{rad}$ $s^{-1}$.

Considering the condition (55), both $n_{R}$ and $n_{L}$ have non-null
imaginary parts in the intervals $\omega_{c}<\omega<\omega_{r}$ and
$\omega_{i}<\omega<\omega_{f}$, respectively. The dichroism coefficient is
null for $0<\omega<\omega_{i}$, $\omega_{f}<\omega<\omega_{c}$, and
$\omega>\omega_{r}$, being non-null only for

$\delta_{dLR}=\begin{cases}\text{$-\frac{\omega}{2}\sqrt{R_{+}}$},&\quad\text{for
$\omega_{i}<\omega<\omega_{f}$},\\\
\text{$+\frac{\omega}{2}\sqrt{R_{-}}$},&\quad\text{for
$\omega_{c}<\omega<\omega_{r}$},\\\ \end{cases}$ (83)

whose general behavior is exhibited in Fig. 18.

Figure 18: Plot of the dichroism coefficient (83)(solid red lines) associated
to the refractive indices $n_{L}$ and $n_{R}$, under the condition (55). The
dashed line represents the usual dichroism coefficient (41). Here, we have set
$\omega_{c}=\omega_{p}$, $V_{0}=0.7\omega_{p}$, and $\omega_{c}=1$
$\mathrm{rad}$ $s^{-1}$.

For the refractive indices $n_{E}$ and $n_{R}$, the circular dichroism
coefficient is

$\delta_{dER}=-\frac{\omega}{2}\left(\mathrm{Im}[n_{E}]-\mathrm{Im}[n_{R}]\right).$
(84)

If we consider $n_{E}$ under the condition (54), the same behavior of Fig. 17
is obtained, since $n_{E}$ is always real, not contributing to the dichroism.
On the other hand, regarding the condition (55), both $n_{R}$ e $n_{E}$ have
nonzero imaginary parts in the intervals $\omega_{c}<\omega<\omega_{r}$ and
$\omega_{i}<\omega<\omega_{f}$, respectively. In this case, we have

$\delta_{dER}=\begin{cases}\text{$0$},&\quad\text{for
$0<\omega<\omega_{i}$},\\\
\text{$+\frac{\omega}{2}\sqrt{R_{+}}$},&\quad\text{for
$\omega_{i}<\omega<\omega_{f}$},\\\ \text{$0$},&\quad\text{for
$\omega_{f}<\omega<\omega_{c}$},\\\
\text{$+\frac{\omega}{2}\sqrt{R_{-}}$},&\quad\text{for
$\omega_{c}<\omega<\omega_{r}$},\\\ \text{$0$},&\quad\text{for
$\omega>\omega_{r}$}.\end{cases}$ (85)

The general behavior of the dichroism coefficient (85) is illustrated in Fig.
19.

Figure 19: Plot of the dichroism coefficients (85) associated to the
refractive indices $n_{E}$ and $n_{R}$ [for the condition (55)]. The dashed
line represents the usual dichroism coefficient (41). Here, we have used
$\omega_{c}=\omega_{p}$, $V_{0}=0.7\omega_{p}$, and $\omega_{c}=1$
$\mathrm{rad}$ $s^{-1}$.

##  VII Final remarks

In this work, we have examined the propagation of electromagnetic waves in a
cold magnetized plasma in the context of the chiral MCFJ electrodynamics,
describing the implied optical effects as well. We have adopted a MCFJ
timelike background vector in order to represent the chirality factor that
breaks the parity. Starting from the modified Maxwell equations and employing
the usual methods, we obtained four modified refractive indices given by Eqs.
(47) and (48), associated with circularly polarized propagating modes. Such
indices were analyzed in detail in the Secs. IV.1-IV.4, where some of them
exhibited significant modifications, as the index $n_{R}$, see Fig. 3. It
presents a negative refraction behavior in the range
$\omega_{c}<\omega<\omega_{-}$, in which it occurs propagation with absorption
for $\omega_{c}<\omega<\omega_{r}$ and free (metamaterial) propagation for
$\omega_{r}<\omega<\omega_{-}$. The usual counterpart index presents only pure
absorption in this range. The low-frequency limit was investigated, there
appearing propagating RCP and LCP helicons due to the presence of the chiral
factor, $V_{0}$. Wave propagation orthogonally to the magnetic field was also
investigated in Sec. V, providing refractive indices and propagating modes
modified by $V_{0}$.

Optical effects of this system, involving birefringence and dichroism, were
discussed in Sec. VI, considering the refractive index $n_{L}$ and $n_{R}$ and
$n_{E}$. In Sec.VI.1, the RP $\delta_{LR}$ was introduced, see Eq. (73),
exhibiting sign reversion at $\omega=\hat{\omega}$, for the conditions (54)
and (55). The RP $\delta_{ER}$ also exhibits sign change at
$\omega=\omega^{\prime\prime}>\omega_{c}$ for the condition (54), and
$\omega=\omega^{\prime\prime}<\omega_{c}$ under the condition (55), as shown
in Figs. 15 and 16, respectively. The RP also increases with the frequency for
$\omega>\omega_{c}$. The reported RP reversal is not usual in cold plasmas,
being reported in graphene systems [85], rotating plasmas [96], Weyl metals
and semimetals with low electron density with chiral conductivity [93, 94],
and bi-isotropic dielectrics with magnetic chiral conductivity [97]. Comparing
our results with the rotating plasma scenario of Ref. [96], there appear
differences. In the rotating plasma, the RP undergoes reversal and decays as
$1/\omega^{2}$ for high frequencies. In the present case, the rotatory power
tends to the asymptotical value $-V_{0}$, see Eq. (77), or increases with
$\omega$ when it involves the negative refraction index, see Eq. (80). These
distinct RP properties may provide a channel to optically characterize chiral
cold plasmas, being of experimental interest.

Besides the nonconventional effect of reversion, the RP can also be enhanced
when it is defined in the negative refraction zone. Such an enhancement occurs
for $\delta_{ER}$, given in Eq. (79), for $\omega>\omega_{+}$ (zone in which
$n_{E}$ is negative), being a topic of interest in metamaterial plasmas [117,
118, 119, 120]. Dichroism was examined in Sec.VI.2, where the coefficients
$\delta_{dLR}$ and $\delta_{dER}$ have been shown to be non-null only in the
range $\omega_{c}<\omega<\omega_{r}$, for the condition (54)—see Figs. 17, and
in the intervals $\omega_{c}<\omega<\omega_{r}$,
$\omega_{i}<\omega<\omega_{f}$, for the condition (55), in accordance with
Figs. 18 and 19.

For a cold-axion dark matter, $\mathbf{\nabla}\theta={\bf{0}}$, it holds
$\mathbf{\nabla}\times\mathbf{B}-\partial_{t}\mathbf{E}=\mathbf{j}+V_{0}\mathbf{B}$,
with $V_{0}=\partial_{t}\theta$, related to time dependence of the axion
field. If one considers $V_{0}$ constant, one has an effective chiral
electrodynamics in the background of an axion field, as discussed in Sec. XI
of Ref. [125]. This is the scenario addressed in the present manuscript. On
the other hand, a plasma interacting with a vibrating cold axion dark matter
with frequency $\omega_{a}$ is an interesting problem that may open nice
connections between cold plasmas and axion/dark matter systems, including the
possibility of experiments involving dielectric haloscopes [126] and tunable
plasmas haloscopes [127, 128].

###### 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 FAPEMA Universal/01187/18, CNPq/Produtividade 311220/2019-3 and
CNPq/Universal/422527/2021-1. P.D.S.S is supported by FAPEMA BPD-12562/22.
Furthermore, we are indebted to CAPES/Finance Code 001 and FAPEMA/POS-
GRAD-02575/21.

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