Magnetic-conductivity effects on electromagnetic propagation in dispersive matter

The Chiral Magnetic Effect (CME) has been investigated as a new transport phenomenon in condensed matter. Such an effect appears in systems with chiral fermions and involves an electric current generated by a magnetic field by means of an"exotic"magnetic conductivity. This effect can also be connected with extensions of the usual Ohm's law either in magnetohydrodynamics or in Lorentz-violating scenarios. In this work, we study the classical propagation of electromagnetic waves in isotropic dispersive matter subject to a generalized Ohm's law. The latter involves currents linear in the magnetic field and implies scenarios inducing parity violation. We pay special attention to the case of a vanishing electric conductivity. For a diagonal magnetic conductivity, which includes the CME, the refractive index is modified such that it implies birefringence. For a nondiagonal magnetic conductivity, modified refractive indices exhibiting imaginary parts occur ascribing a conducting behavior to a usual dielectric medium. Our findings provide new insight into typical material properties associated with a magnetic conductivity.

The Chiral Magnetic Effect (CME) has been investigated as a new transport phenomenon in condensed matter. Such an effect appears in systems with chiral fermions and involves an electric current generated by a magnetic field by means of an "exotic" magnetic conductivity. This effect can also be connected with extensions of the usual Ohm's law either in magnetohydrodynamics or in Lorentz-violating scenarios. In this work, we study the classical propagation of electromagnetic waves in isotropic dispersive matter subject to a generalized Ohm's law. The latter involves currents linear in the magnetic field and implies scenarios inducing parity violation. We pay special attention to the case of a vanishing electric conductivity. For a diagonal magnetic conductivity, which includes the CME, the refractive index is modified such that it implies birefringence. For a nondiagonal magnetic conductivity, modified refractive indices exhibiting imaginary parts occur ascribing a conducting behavior to a usual dielectric medium. Our findings provide new insight into typical material properties associated with a magnetic conductivity.

I. INTRODUCTION
The Chiral Magnetic Effect (CME) is the macroscopic generation of an electric current in the presence of a magnetic field as the result of an asymmetry between the number density of left-and right-handed chiral fermions. It leads to a current that is linear in the magnetic field [1]. This quantum effect has been under extensive research in particle and field theory as well as nuclear and condensedmatter physics. It was investigated in quark-gluon plasmas with a chiral chemical potential under the influence of an external magnetic field [2,3] and was also derived in the context of high-energy physics by Vilenkin (in the 1980's) [4,5] who supposed an imbalance of fermion chirality in the presence of cosmic magnetic fields in the early Universe. The CME was studied in cosmology [6], as well, where it was applied to explain the origin of the very high magnetic field strengths (up to 10 15 G) observed in neutron stars [7,8]. An interesting question is the possible influence of the external axial-vector field V µ 5 on the magnitude of the anomalous current in the CME [9], which was also examined for the polarization tensor of a photon in a fermion plasma under the influence of V µ 5 [10]. Connections between the CME and matter subject to the electroweak interaction were established, too [11,12]. * pdiego.10@hotmail.com † manojr.ufma@gmail.com ‡ marco.schreck@ufma.br § urrutia@nucleares.unam.mx In condensed-matter systems, the CME plays an uttermost important role. It appears as a relevant effect in Weyl semimetals, where it is usually connected to the chiral anomaly associated with Weyl nodal points [13]. In such materials, massless fermions acquire a drift velocity along the magnetic field, whose direction is given by their chirality. Opposite chirality implies opposite velocities, creating a chiral-fermion imbalance that is proportional to the chiral magnetic current. The first experimental observation of the CME was reported in 2014 [14]. Several investigations have been carried out on the properties of this phenomenon, including the CME in the absence of Weyl nodes [15], anisotropic effects stemming from tilted Weyl cones [16], the CME and anomalous transport in Weyl semimetals [17], quantum oscillations arising from the CME [18], computation of the electromagnetic fields produced by an electric charge near a topological Weyl semimetal with two Weyl nodes [19], and chiral superconductivity [20].
An interesting relation between the CME and Maxwell-Carroll-Field-Jackiw electrodynamics has been established in the literature [21] by examining the connection between the CME and Lorentz-violating theories including axion electrodynamics. The possibility of Lorentz invariance violation was proposed in the context of physics at the Planck scale such as strings [22]. Presently, the Standard-Model Extension [23], where fixed background tensor fields are coupled to the dynamical fields, is usually employed to parametrize it. A violation in the photon sector can occur by means of a CPT -odd [24] or a CPT -even term [25]. A Lorentzviolating extension for the current density that resembles the macroscopic description of the CME can be found in Ref. [26], where some preliminary studies of aspects on Lorentz-violating electrodynamics in continuous matter were performed.
Additional motivation for studying conduction currents driven by magnetic fields comes from magnetohydrodynamics. After taking some simplifying assumptions for plasmas of colliding particles, the resistive Ohm's law for the one-fluid model is written as [27] (E + V × B) i = η ij J j . (1) Here V is the average velocity of the electrons and ions, while J is the total current density. The expression η ij J j corresponds to an effective collision term for electrons and ions and the resistivity η ij can, in general, be a tensor. Even if Eq. (1) is not a rigorous model, it is widely used, because it captures the most important deviations from the ideal magnetohydrodynamic model. Inverting Eq. (1) yields J i = σ ij E j + σ B ij B j , with σ ij being the inverse of η ij and σ B ij = σ ip V q ǫ pqj defining a magnetic conductivity. In the following, we will generalize this motivation by considering the magnetic conductivity tensor σ B ij to be completely independent of the electric conductivity σ ij .
The main purpose of this work is to analyze the possible phenomena of a magnetic conductivity in a usual dielectric medium. Particularly, we are interested in investigating the effects stemming from the magnetic conductivity on the propagation of electromagnetic waves in a dispersive dielectric continuous medium characterized by the parameters ǫ (electric permittivity), µ (magnetic permeability), and σ (Ohmic conductivity). In this sense, we propose some configurations for the magnetic conductivity.
We start from the Maxwell equations in a continuous medium, supplemented by the constitutive relations D = ǫE, B = µH, and the magnetic current, J i CME = σ B ij B j , arising from an extension of Ohm's law. We obtain the corresponding refractive indices and electric fields for the propagating modes in some scenarios. First, we address an isotropic and an anisotropic diagonal conductivity tensor, configurations which describe the chiral magnetic current observed in Weyl semimetals. After doing so, we examine more exotic configurations of antisymmetric and symmetric nondiagonal conductivity tensors. In what follows, we will take ǫ, µ, σ ∈ R.
The isotropic or anisotropic diagonal magnetic conductivity generate the scenario of a birefringent dielectric crystal, described by two different refractive indices for each wave vector. On the other hand, the nondiagonal anisotropic magnetic conductivity (symmetric or antisymmetric) creates the remarkable behavior of a conducting phase in the dielectric substrate.
This work is outlined as follows: in Sec. II we briefly review some basic aspects of electrodynamics in matter, focusing on the refractive index, constitutive relations, the generalized form of Ohm's law, and the magnetic current. In Sec. III we discuss the general effect of a magnetic conductivity on wave propagation in continuous dielectric matter. We analyze diagonal conductivity tensors and some special cases of an exotic conductivity. Section IV is dedicated to deriving the contributions to the charge and current density that result from the magnetic conductivity and are needed to guarantee the consistency of Maxwell's equations. Furthermore, in Sec. V we study what impact a magnetic conductivity has on certain phenomena such as the skin effect and reflection. Finally, we summarize our results in Sec. VI.

II. BASIC ASPECTS ON ELECTRODYNAMICS IN MATTER
The electrodynamic properties of a continuous medium are characterized by its electric permittivity ǫ, magnetic permeability µ, and Ohmic conductivity σ. The static and dynamic behavior is described by the standard Maxwell equations, The response of the medium to applied electromagnetic fields is measured in terms of the polarization vector P and magnetization vector M, defined by linear constitutive relations: P = ǫ 0 χ E E and M = χ M H, where χ E and χ M are the electric and magnetic susceptibility, respectively, of an isotropic ponderable medium. Such relations allow us to define the electric displacement field D and the magnetic flux density B: We then rewrite Eqs. (3a) and (3b) as with the identity matrix 1 in three spatial dimensions. By using a plane-wave Ansatz for the electromagnetic fields, E = E 0 e i(k·r−ωt) and B = B 0 e i(k·r−ωt) , the Maxwell equations and J = σE yield: with k 2 = ω 2 µǭ(ω), wherē is the frequency-dependent electric permittivity of the medium. This complex permittivity leads to a complex refractive index: 1 where The imaginary part of the refractive index leads to a real exponential factor, e −ωn ′′ (k·r) withk ≡ k/ √ k 2 , in the plane-wave solutions for the electromagnetic fields. This term damps the amplitude of the wave along the propagation through matter, which is related to the absorption coefficient α = 2ωn ′′ , whose inverse value determines the penetration depth. Such a scenario is typical for a conducting medium.
In a linear, continuous, ponderable medium, general constitutive relations can be envisaged as a theoretical possibility [32], which gained great attention with the advent of topological insulators [28,29]. In systems of this kind, it holds that with additional material parameters α, β that are not independent and whose sum provides what is known as the "activity constant" of a medium. Such an extension also occurs in a Lorentz-violating anisotropic electrodynamics [26] where with dimensionless (3 × 3) matrices κ DE , κ DB , κ HE , and κ HB that are composed of a vacuum part and a matter part. In component form, the latter relations read These generalized scenarios lead to an unusual electrodynamics where the electric displacement field receives a contribution from the magnetic flux density and the magnetic field receives a contribution from the electric field. Such modified constitutive relations, D = D(E, B) and H = H(E, B) are observed, for example, for topological insulators [28]. 1 Note that the refractive index can be a complex function, in general. Thus, instead of employing the norm |k| in the definition of the refractive index, which is a nonnegative, real number, we use √ k 2 . Furthermore, we only consider refractive indices with a nonnegative real part, which is indicated explicitly by the plus sign in front of the square root. At this point, we introduce a generalized Ohm's law for the current density J, considering the contribution of the magnetic current, J CME = σ B · B, that is, composed by the usual Ohmic term involving the conductivity tensor σ ij and by an exotic term with the magnetic conductivity tensor σ B ij , which has found realization in some condensed-matter systems. It can also be proposed as an extension of Ohm's law in magnetohydrodynamics [27] as well as in Lorentz-violating scenarios [26].
The magnetic conductivity tensor, σ B ij , is even under time reversal (T ) and charge conjugation (C), but odd under parity transformations (P ); see Tab. I. Most notably, this tensor is even under time reversal symmetry, since J and B are T -odd, which is highly unusual for a conductivity. This behavior is analog to that of the phenomenological parameter µ (not to be confused with the magnetic permeability) observed in London's superconductivity model (J = −µ 2 A) [20]. The T -even character of σ B ij is typical of nondissipative and reversible processes [1,20]. Such a property is what distinguishes the magnetic conductivity from the usual Ohmic conductivity, which is T -odd, and it is compatible with dissipative and nonreversible phenomena. Table I shows a comparative analysis between the magnetic and Ohmic conductivity tensors under discrete symmetry transformations, revealing crucial differences when the former is subject to T , P , CP, and CT transformations.
Using the generalized Ohm's law of Eq. (11) and conventional isotropic constitutive relations, D i = ǫδ ij E j , H i = µ −1 δ ij B j , in the Maxwell equations, Eq. (5) keeps its general form: whereǭ defines the frequency-dependent extended permittivity tensor (EPT). Equation (12a) implies Notice that the latter equation yields k iǭij E j = 0, where we can interpretD i =ǭ ij E j as an extended displacement vector.
For a general anisotropic continuous scenario, we write k = ωn where n is a vector pointing along the direction of the wave vector and yielding the refractive index: 2 n = + √ n 2 . Hence, Eq. (13) becomes The latter can also be cast into the form where the tensor M ij reads andǭ ij is given by Eq. (12b). This set of equations has a nontrivial solution for the electric field if the determinant of the coefficient matrix M ij vanishes. This condition on the determinant provides the dispersion relations associated with wave propagation in the medium.
In the following, we will discard refractive indices with negative real parts associated with frequencies that have the same property. Sophisticated composites of different materials can be designed that have negative permittivity and permeability. These are called metamaterials and the real parts of their refractive indices must be endowed with a ± sign according to the materials at the interface: material-material (+), material-metamaterial (−), metamaterial-metamaterial (+). On the contrary, negative refractive indices are not known to occur in crystals found in nature, which our focus is on in this paper.

III. PROPAGATION BEHAVIOR UNDER CHIRAL AND EXOTIC MAGNETIC CONDUCTIVITY
In this section, we will investigate the effects stemming from the chiral and exotic magnetic conductivity, incorporated into the formalism used for describing the propagation of electromagnetic waves in anisotropic dispersive media. In order to investigate the magnetic current J i CME = σ B ij B j classically, we consider the conductivity tensor σ B ij originating from the manifestation of the magnetic field in Eqs. (12b) and (15a). Studies on the CME [1,2,20,21] have reported the generation of an electric current induced by a magnetic field, where e is the fermion charge, B the applied magnetic field, ∆µ ≡ µ R − µ L is also known as the chiral chemical potencial, and Σ is the chiral magnetic conductivity [6][7][8][9][10][11][12][13]. As written in Eq. (16), this effect is clearly represented by an isotropic diagonal magnetic conductivity, that is, in which We can write down the magnetic conductivity σ B ij in the following form where Σ is 1/3 of the trace of the σ B ij matrix and represents the isotropic part of this conductivity, while Σ ij stands for all off-diagonal components of σ B ij . Hence, the diagonal piece of the conductivity tensor is related to the CME and constitutes the first case to be analyzed. We further examine the generalization of the magnetic conductivity to exotic scenarios, where Σ ij comprises off-diagonal or anisotropic conductivity components. It is worthwhile to note that the anisotropic chiral magnetic effect [16] represents an interesting theoretical possibility to be proposed and investigated. At first, it can be induced by a diagonal anisotropic conductivity tensor, as will be examined in Sec. III B.

A. Isotropic diagonal chiral conductivity
First of all, we discuss the behavior of an isotropic magnetic conductivity, which is represented by a diagonal isotropic matrix; cf. Eq. (17a). By inserting the latter into Eq. (12b), one obtains where the last term of Eq. (20) represents the contribution from the chiral conductivity. Note that we are starting from an isotropic permittivity tensor, ǫδ ij , where all effective anisotropies manifest themselves via the way the magnetic conductivity is coupled to the fields. In this case, the tensor M ij given by Eq. (15b), has the form 2 Here we again take into account that the norm |n| is nonnegative.
To permit complex refractive indices, we consider √ n 2 instead of |n|. The plus sign again indicates that we discard refractive indices with negative real parts.
Requiring det[M ij ] = 0, we get which can be split into real and imaginary parts as follows: Equation (23a) yields two distinct refractive indices for each frequency ω, which is compatible with the physics of a conducting dielectric medium endowed with birefringence. This behavior already occurs for the isotropic conductivity tensor of Eq. (17a), revealing that birefringence comes from the way the chiral conductivity is coupled to the fields. Thus, when considered within a medium of Ohmic conductivity (σ = 0), the chiral conductivity modifies the refractive index of the conducting medium. It alters the phase velocity associated with its real part and the absorption (or attenuation) coefficient related to its imaginary part.

Dielectric nonconducting medium
In the case we start from a dielectric medium with zero Ohmic conductivity, σ = 0, Eq. (22a) provides two distinct real values for the refractive index, which characterizes a dispersive nonconducting behavior typical of a uniaxial crystal, where the propagation occurs with different velocities along the principal dielectric directions [31]. Therefore, the system will behave like a birefringent dispersive dielectric medium where electromagnetic waves propagate without undergoing attenuation (nonconducting or absorbing behavior). The corresponding refractive indices are given by with Υ Σ of Eq. (22b). The square of the latter leads back to Eq. (24). In the present configuration, it is important to point out that the chiral conductivity implies a typical conducting behavior for the medium only when it is defined simultaneously with the Ohmic conductivity (σ = 0, σ B = 0), as shown in the complex refractive index of Eq. (23a). When it is defined for a nonconducting dielectric (σ = 0, σ B = 0), the behavior remains that of a dispersive nonabsorbing medium, as shown by the complex refractive index in Eq. (24) in spite of the chiral conductivity. This is because N − = 0 when σ = 0. Alternatively, a refractive index can be determined from the frequency as a function of the wave vector, 7a)). The possible frequencies ω also follow from the requirement that det[M ij ] = 0 and are associated with particular modes of the electric field. In order to better examine the features of propagation, we implement n = √ k 2 /ω in Eq. (24) and obtain the dispersion equation with k ≡ √ k 2 . Solving for ω, one gets Equation (27) represents two distinct modes, ω + and ω − . While the frequency ω + is real for any value of k, the frequency ω − of the second mode can be imaginary if k < Σµ. In order to assure that ω − represents the frequency of a physical propagating mode, we should require that k > Σµ. Birefringence occurs when distinct polarization modes propagate with different phase velocities. In this medium of zero Ohmic conductivity, the phase velocities are yielding the following phase velocity difference: showing that the trace Σ of the isotropic chiral conductivity is really responsible for birefringence. Thus, a diagonal isotropic σ B ij generates a birefringent dispersive nonconducting behavior.
We can also analyze the effect that such a term implies on the group velocity, which has a singularity for small momenta indicating problems with classical causality due to v g(±) > 1. Causality is preserved for wave propagation in the largemomentum regime. One can also obtain the refractive indices for this medium (σ = 0) by diagonalizing the permittivity tensor of Eq. (20) and setting each eigenvalue equal to n 2 /µ. In general, this procedure provides the refractive indices only for the propagation along the directions of the principal axes ofǭ ij .
The eigenvalues ǫ a (a = 1, 2, 3), whereǭ ij e j a = ǫ a e i a , are given by: which can be associated with the following refractive indices: Surprisingly, the latter result reproduces Eq. (25)stemming from det[M ij ] = 0 and valid for an arbitrary direction, which means that the eigenvalues ǫ 2 and ǫ 3 only correspond to the refractive indices of the medium n + and n − , respectively. In the following, we explain this behavior. The proposed method of finding the refractive indices n through the equation n 2 = µǫ a (n), where ǫ a are the eigenvalues of the EPTǭ ij , only works under the forthcoming condition. For a general vector n, the related electric field E a , which satisfies the condition M ij E j a = 0 according to Eq. (14), must be such that n · E a = 0. In this case, diagonalizing M is equivalent to diagonalizingǭ and the result E a ∼ e a follows. Notice that we must have also n · e a = 0. This situation is clearly illustrated in the present case, where Eqs. (14), (20) yield the general condition n · E = 0. Here, the three eigenvectors of the generalized permittivity are with the unit vector m defining the planes of constant phase of the wave. Let us observe that the eigenvectors in Eqs. (33a), (33b) are independent of the respective refractive indices n ± , being just functions of the direction given by m. We note that e 1 · e * 2 = e 1 · e * 3 = e 2 · e * 3 = 0, whereupon these three eigenvectors are linearly independent. In particular, e 2 and e 3 are orthogonal to e 1 = m, thus yielding the correct refractive indices n ± of Eq. (24), according to the proposed method. In this case, the propagating modes of the electric field are correctly described by the eigenvectors e 2 and e 3 and the eigenvalue ǫ 1 = ǫ has to be rejected, because m · e 1 is nonzero.
To decide which refractive indices are physical, we can also look at the modes of the electric field. The latter are obtained from solving the homogeneous system of equations M ij E j = 0 for E with ω replaced by the dispersion relations ω(k) determined from the coefficient determinant. In the dielectric nonconducting medium under consideration, a particular frequency does not correspond to a physical mode when the electric field is longitudinal. In others words, it then points along the direction of the wave vector, i.e., E · k = |E||k| or k × E = 0. As [M, ǫ] = 0 for the particular isotropic configuration of Eq. (17a), the eigenvector of Eq. (33a) also corresponds to a mode of the electric field. As it is longitudinal, the associated solution for the permittivity ǫ 1 = ǫ cannot be physical and must be discarded.
Since the configuration studied is isotropic, we can choose m = (0, 0, m) without a restriction of generality. Equation (33b) then results in These can be interpreted as the polarization vectors of left-handed (L) and right-handed (R) polarized electromagnetic waves, corresponding to e 2 and e 3 , respectively. 3 Hence, we identify n L,R ≡ n ± . These polarizations are transverse, i.e., perpendicular to k. According to Tab. I, the magnetic conductivity σ B ij is odd under parity transformations. While parity violation does not show up in a single refractive index of Eq. (25) on its own, it becomes manifest in the distinct propagation properties of left-and right-handed polarized electromagnetic waves. This is the physical reason for birefringence.
The behavior found for this configuration is highly interesting. Parity violation could be expected to imply refractive indices that are angular-dependent. In other words, birefringence in a material is usually caused by the presence of at least a single optical axis. An optical axis indicates a preferred direction in the crystal, whereupon its refractive index cannot be isotropic, anymore. Hence, under usual circumstances, an occurrence of parity violation and birefringence seems to contradict isotropy of a crystal.
If a magnetic conductivity is present, we found that birefringence can also occur in an isotropic crystal. If birefringence occurs in a medium with a single optical axis, a light ray can split into an ordinary and an extraordinary ray. The ordinary one behaves according to Snell's law whereas the extraordinary one does not do so. The electric field associated with the extraordinary ray is no longer orthogonal to its wave vector. However, the polarizations of the electric field stated in Eq. (34) are orthogonal to the wave vector, whereupon an extraordinary ray cannot be identified in this setting. The two polarizations only split due to their distinct propagation velocities, cf. Eq. (30). Therefore, the polarization plane of a linearly polarized wave will rotate indicating an optically active material. This phenomena is quantified by defining the specific rotatory power δ, measuring the rotation of the plane of linearly polarized light per unit traversed length in the medium. It is given by δ ≡ (n R − n L )ω/2, which in our case yields thus providing a frequency-independent specific rotary power due to the chiral magnetic conductivity Σ.

B. Diagonal anisotropic chiral conductivity
As a next step, we will consider another particular case for the chiral conductivity, σ B ij , which is represented by a diagonal tensor that describes an anisotropic system [15]: with a set {Σ} = {Σ x , Σ y , Σ z } of distinct elements in the diagonal, Σ x = Σ y = Σ z . The general permittivity tensor is given by Eq. (12b), with the components shown as follows: The dispersion relations are obtained from det[M ij ] = 0 where the tensor M ij is defined in Eq. (15b), withǭ ij given by Eq. (37). Evaluating this condition leads to the dispersion equation with the function Ω = Ω(n) = µ 2 (Σ x Σ y n 2 3 + Σ x Σ z n 2 2 + Σ y Σ z n 2 1 ) . (38b) The latter is an involved expression, since it explicitly contains the components of the vector n = (n 1 , n 2 , n 3 ) instead of its modulus n as in the isotropic case investigated previously. To avoid problems with separating real and imaginary parts of refractive indices, we will assume that Ω ≥ 0. The following parametrization [29] for n permits a convenient examination of the physical contents of Eq. (38a): with angles θ ∈ [0, π] and φ ∈ [0, 2π). With the latter, Eq. (38a) is rewritten as where The dispersion equation (40) yields two solutions for n, revealing two values for any frequency.

Dielectric nonconducting medium
In the special limit of zero Ohmic conductivity, σ = 0, all the complex pieces of Eq. (38a) or Eqs. (40) vanish and the refractive index of Eq. (41) is real. Some complex term could occur in Eq. (41) if the discriminant is negative: ∆ 0 = D 2 0 + 4ω 2 G 0 < 0, where G 0 = −ǫ 2 µ 2 ω 2 and D 0 = 2ǫµω 2 +Ω. But ∆ 0 =Ω 2 + 4Ωǫµω 2 > 0 and therefore, this possibility is not realized. In fact, for σ = 0, Eqs. (38a) and (41) become whereupon In case we apply the prescription stated in Eq. (39) to Eq. (41), the refractive indices are given by It is interesting to observe that by using the parametrization of Eq. (39) in each refractive index of Eq. (43) one obtains an explicit solution for n ± . Each equation is of second order and the refractive indices are where one easily verifies that their squares reproduce the results in Eq. (44). Hence, for σ = 0, σ B ij = 0, we have a dielectric nonconducting and dispersive medium. One can verify that Eq. (42) provides three different expressions for the following three situations: n = (0, n 2 , n 3 ), n = (n 1 , 0, n 3 ), and n = (n 1 , n 2 , 0). For each one of these three choices, Eq. (43) yields two values for n, resulting in birefringence.
Diagonalizing the EPT of Eq. (37) for zero Ohmic conductivity (σ = 0) and dropping the parametrization given by Eq. (39), one obtains ǫ 1 = ǫ and with In this case, when the two eigenvalues of the EPT, given in Eq. (46a), are multiplied by µ, we obtain µǫ 2,3 = µǫ ± , being equal to the values for n 2 ± calculated from the condition det[M ij ] = 0 in Eq. (43). This numerical coincidence provides a nice example of the fact that even though we satisfy the condition n 2 ± = µǫ ± , this equality does not imply that the respective eigenvectors of the EPT would necessarily correspond to the propagation modes of the electric field. The corresponding eigenvectors are where with m defined in Eq. (39). Note that The above calculation shows that k · e a = 0 for all a = 1, 2, 3 in such a way that these eigenvectors cannot describe the propagation modes E ± of the electric field. In order to clearly illustrate this point, we calculate such propagation modes for the particular case where k = (k 1 , k 2 , 0) = ω n (m 1 , m 2 , 0) and compare those with the corresponding eigenvectors of the EPT. The results for the propagation modes are with The refractive indices satisfy the quadratic equation which is written in terms of the unit vector m defining the planes of constant phase of the wave. The dispersion relations ω ± = ω ± (k) are The functions u ± and v ± satisfy the useful relations which are just a consequence of Eq. (51). We observe that Due to γ| k3=0 = √ κ, from Eqs. (47b), (47c) we read that in this case the eigenvectors of the EPT are which do not coincide with E ± of Eq. (50). The propagating modes of Eq. (50) are not orthogonal to the wave vector k. This is a characteristic of wave propagation in anisotropic media, which nevertheless must always fulfill k · D = 0 in the absence of sources.
In order to have a direct physical interpretation of the propagating modes in Eq. (50), we consider the particular case of propagation along the y-axis, i.e., we take m = (0, 1, 0). Now we have n 2 ± = u ± and √ κ ± = ωn ± √ Σ x Σ z . After substituting in Eq. (50), we obtain the normalized electric fieldŝ which describe elliptically polarized modes with distinct refractive indices The latter are the corresponding particular cases of Eq. (45). The modes associated withÊ ± in Eq. Making cyclic changes among {x, y, z}, we identify n + = n L and n − = n R , thus characterizing an optically active medium. Note that in this particular case we have k · E ± = 0, so we expect E ± to be proportional to the eigenvectors e ± in Eq. (55a), as can be readily verified.
Still, a main difference arises: in the former case, the two refractive indices are independent of the direction defined by the wave vector k (see Eq. (24)), while in the latter case a direction dependence emerges due to the appearance ofΩ in Eq. (44). Nevertheless,Ω is a quadratic function of the components of the wave vector k, which yields a parity-invariant individual refractive index. As in the isotropic case, parity violation manifests itself in the different refractive indices of the left-and right-handed modes. Another remarkable feature of the diagonal anisotropic case, which will also appear in the instances to be discussed as follows, is the presence of a non-Hermitian EPT. Although this opens up interesting new possibilities, we defer this study to future research.
Finally, it is intriguing to notice that a diagonal chiral conductivity tensor inserted into the Maxwell equations is not enough to establish a conducting behavior for a dielectric medium although this is the chiral conductivity configuration most used and cited in the literature.

C. Nondiagonal antisymmetric conductivity
Now we analyze the case where the magnetic conductivity is described by an exotic configuration given by an antisymmetric tensor, σ B ij , parameterized in terms of a three-vector, b = (b 1 , b 2 , b 3 ), as: with the Levi-Civita symbol ǫ ijk in three dimensions. Notice that this case corresponds to situation in magnetohydrodynamics when the resistivity is isotropic in Eq. (1) and the vector b is proportional to the plasma velocity. Inserting Eq. (58) into Eq. (12b), leads to the EPT By employing Eq. (59) in Eq. (15b), one obtains: for which det[M ij ] = 0 implies: In contrast to Eqs. (40), (42), the latter dispersion equation involves the square of a quadratic polynomial in the components of n. Hence, the solution for the refractive index is doubly degenerate and there is only a single refractive index with nonnegative real part. Implementing b · n = bn cos θ with b = |b|, Eq. (61) provides the fol-lowing refractive index: Note the presence of the minus sign between the two contributions in Υ b in contrast to Υ Σ of Eq. (22b) and Υ {Σ} in Eq. (45b). We assume that µǫ ≥ Ξ 2 b . Decomposing the latter into its real and imaginary part implies where In this case, we obtain an anisotropic complex refractive index that captures the effects of the exotic conductivity, as shown in Eq. (63a), which is compatible with a conducting medium. The imaginary part takes the role of the absorption coefficient for the electromagnetic wave, which undergoes attenuation while it propagates. Therefore, an electromagnetic wave cannot propagate through such a medium, since absorption damps its intensity. When considered in a dielectric conducting medium, (ǫ, σ, σ B ), the magnetic conductivity modifies the real and imaginary part of the refractive index. This causes a modification of the absorption coefficient given byα = ωα ′ − + µb cos θ, where α ′ − is given by Eq. (63b). Note that for a diagonal magnetic conductivity there were two distinct refractive indices with positive real part; see Eqs. (25), (45). Thus, the occurrence of a single refractive index is quite unexpected in the context of a parity-violating theory. In contrast to the cases studied before, a magnetic conductivity given by Eq. (58) does not imply birefringence. These results suggest that an antisymmetric magnetic conductivity leads to an EPT ǫ ij of a form that permits a single refractive index with positive real part only. In the forthcoming section, we will calculate the propagation modes to get a definite prediction.
A parity transformation in three spatial dimensions implies cos θ → cos(π −θ) = − cos θ for the polar angle (and φ → π + φ for the azimuthal angle, which does not occur in the refractive index). As the real part of the refractive index in Eq. (62) only contains squares of cos θ, it is invariant under parity transformations. However, at least the imaginary part of n exhibits parity-violating properties.

Dielectric nonconducting medium
In case we start from a dielectric with zero Ohmic conductivity (ǫ = 0, σ = 0, σ B = 0), Eq. (62) reduces to: which is compatible with the behavior of a conducting medium. Therefore, the off-diagonal chiral conductivity of Eq. (58) ascribes a conducting behavior to the material even for a pure dielectric substrate (ǫ = 0, σ = 0).

Propagation modes
For a phenomenological analysis of the physics, let us choose a convenient coordinate system, without a restriction of generality. Since n and b define a plane, to be labeled the y-z-plane, we take the z-axis along the direction of n so that Recall that θ is the angle between n and b. This choice of coordinates leads to a very simple expression for the matrix of Eq. (60): which immediately yields the dispersion equation with nb 3 = n · b. The latter corresponds exactly to Eq. (61) when σ = 0. Equation (67) has a single solution with nonnegative real part, given by Eq. (64).
Recalling that √ 2Υ b = µǫ − Ξ 2 b , we distinguish between two cases according to choice of the sign inside the square root of the above relation. In the first case, when µǫ ≤ Ξ 2 b , the refractive index is purely imaginary with a positive imaginary part, which damps propagation. The second alternative, µǫ > Ξ 2 b , yields the propagation modes that are to be described now. The condition M ij E j = 0 requires leaving E 1 completely arbitrary. We take advantage of this freedom to chose two orthogonal vectors satisfying the condition in Eq. (68). We find Here we parameterized the complex refractive index as We can easily verify that E * + · E − = 0 and also that the y and z-components of the propagation modes satisfy Eq. (68). Thus, we have recovered two orthogonal modes whose propagation is described by the same refractive index. This is an unexpected result, which nevertheless is analogous to the simplest isotropic case (without the presence of a magnetic conductivity) where two linear polarization modes associated with the same refractive index occur. Notice that we have k · E ± = 0 in this case, which prevents an interpretation of the fields in terms of standard elliptical polarizations with components defined in the two-dimensional subspace orthogonal to k.
Since in the coordinate system defined in Eq. (65) we only have access to the inversion z → −z, the parity transformation n → −n can be better studied in a rotated frame where now the vector b defines the new zaxis endowing the system with axial symmetry. The new frame is obtained by rotating the former one with an angle θ = arccos(b 3 /|b|) with respect to an axis perpendicular to the n-b plane. The latter can be associated with an arbitrary plane having a constant azimuthal angle φ in spherical coordinates. CallingĒ i the components of the electric field in the rotated frame, we derive the following expressions: The corresponding magnetic fields are obtained using B ± = n × E ± .
As n lies in a plane, we restrict the parity transformation to the replacement rule of the polar angle: θ → π − θ. Then, a parity transformation of a vector in two dimensions corresponds to its reflection at one of the two axes modulo a rotation by π. Thus, the expected behavior under a parity transformation in the plane is that only one of the two vector components flips its sign. The vector n is transformed in a way that the n 3 -component is reflected at the n 2 -axis. If parity was conserved, the sign of the second componentsĒ 2 ± should change under a parity transformation, while the third componentsĒ 3 ± should remain invariant. However, one can verify thatĒ 2 ± andĒ 3 ± do not behave in this manner. On the contrary, the terms involving b spoil the behavior expected, which is a signal for parity violation. Note that the refractive indices of Eqs. (62a), (64) also change under parity transformations, as Ξ b → −Ξ b .

D. Nondiagonal symmetric conductivity tensor
Now we examine the case where the magnetic conductivity is given by a traceless symmetric tensor, in accordance with the following parametrization: where a i and c i are the components of two orthogonal background vectors a and c, i.e.
The evaluation of det[M ij ] = 0 yields the forthcoming dispersion equation: In contrast to the antisymmetric magnetic conductivity of Eq. (58), the new configuration of Eq. (72) does not imply a single doubly-degenerate refractive index (cf. Eq. (61)). In contrast, we obtain two distinct refractive indices in the current scenario. Using n · (a × c) = n|a||c| cos ϕ in Eq. (76) results in with the presence of new imaginary terms stemming from the exotic conductivity and modifying the absorption coefficient. What is analogous to the antisymmetric configuration of Eq. (58) is the structure of a single one of the two refractive indices present, i.e., the dependence of the refractive index on an angle between k and a three-vector (b for the antisymmetric case and a × c for the current scenario). Also, there is again a relative minus sign between the two contributions in Eq. (77d). Therefore, we assume µǫ ≥ Ξ 2 a,c . A crucial difference is that two angles play a role for the current configuration: the angle ϕ between k and a × c for one mode and the complementary angle π − ϕ for the other mode. This also means that both modes interchange their role when ϕ exceeds π/2. As the modes differ in their imaginary parts only, birefringence does not occur. It is the attenuation only that differs for both modes.

Dielectric nonconducting medium
In this case we start from a medium with zero Ohmic conductivity (ǫ = 0, σ = 0, σ B = 0). Equation (77) is then reduced to which exhibits an exotic absorbing behavior for nonconductive matter, with attenuation coefficient 2α = µ|a × c| cos ϕ provided µǫ ≥ Ξ 2 a,c . In this case, the modification is proportional to |a × c|, which is why it vanishes for parallel background vectors. It confirms that such an effect is associated with the nondiagonal elements of σ B ij instead of its trace.
Note that this nondiagonal magnetic conductivity also provides a conducting behavior for a dielectric medium, in much the same way as observed for the nondiagonal antisymmetric case (cf. Eq. (64)).

Propagation modes
In order to examine the propagating modes, we will rewrite the matrix given in Eq. (75a) for a simplified coordinate system where a = (0, a, 0) and c = (0, 0, c), that is with The explicit form of the matrix in Eq. (75a) simplifies to providing the following dispersion equation: Since A − n 2 = −µǫ, the dispersion relations are with n 1 = n cos ϕ and corresponding to the σ → 0 limit of Eqs. (76). We observe that the refractive indices depend on the direction of n, which is defined in terms of the spherical angles θ and φ according to Eq. (39). Recalling that ϕ is the angle between n and a × c we have that cos ϕ = sin θ cos φ.
Taking the plus sign in Eq. (82) and using Eq. (80) we obtain Therefore, the electric field for the plus propagating mode is: with an appropriately chosen amplitude E Thus the electric field is: with another amplitude E (0) − . We observe that n·E ± = 0. Equations (84), (86) represent the propagation modes for the case of a symmetric exotic magnetic conductivity with the corresponding magnetic fields given by B ± = n × E ± .

IV. CONSISTENCY OF MAXWELL'S EQUATIONS
For completeness, in this section we extend the current J in Eq. (2a) to include an external-source contribution J e such that now J = J e + σ ij E i + σ B ij B i . In the previous sections, we discussed wave propagation outside sources which effectively meant to take J e = 0. Moreover, the different scenarios we have considered were defined by fixing the current J via specific choices of the electric and magnetic conductivities. Also, the propagation properties of the fields were obtained just by using Faraday's and Ampère's law, incorporated into Eq. (12a), with no reference to Gauss's law given by the first of Eqs. (2a).
A natural question that arises is the identification of the particular contributions to the charge density ρ that are consistent with the arbitrary choice of currents, in such a way that charge conservation ∂ t ρ + ∇ · J = 0 is preserved. To this end, we work in momentum space with the standard conventions ∇ → ik and ∂ t → −iω. The main point to recognize is that Ampère's law yields which together with Gauss's law ik · D = ρ gives the identification for ω = 0, which is precisely the charge conservation condition in momentum space. In what follows, we summarize the expressions for the charge densities corresponding to our previous choices of the magnetic conductivity, setting σ ij = 0, and recalling that the Maxwell equations retain the general form of Eqs. (2a) and (2b). For the isotropic case of Sec. III A with J = J e + Σ B, the charge density is just ρ = ρ e = k · J e /ω, since k · B = 0. The antisymmetric case studied in Sec. III C with where we used some of the Maxwell equations in order to get rid of the spatial derivatives arising from Eq. (88). Finally, the symmetric case of Sec. III D, where we have assumed a · c = 0, starts from and implies The alternative second form of Eq. (90) was motivated by an attempt to use the remaining Maxwell equations in favor of rewriting Eq. (88) without spatial derivatives, which, unfortunately, was not possible in this case.

V. SOME CLASSICAL EFFECTS
Towards the end of the paper, we intend to understand the impact that a magnetic conductivity has on certain phenomena in electrodynamics that are a consequence of material parameters such as refractive indices. As modified refractive indices for particular choices of a magnetic conductivity have already been determined earlier, we will now benefit from these findings.

A. Skin depth effect
When an electromagnetic wave falls on the surface of a conductor, its amplitude will partially penetrate the material due to the attenuation coefficient, while another part will be reflected. The characteristic penetration length into the conducting medium defines the skin depth [29,30]: where Im[n] is the imaginary part of the complex refractive index. In the usual scenario for a simple conductor, the general skin depth reads: for a good (Ohmic) conductor and with n ′′ given by Eq. (7b). Consequently, the skin depth decreases for high frequencies.
We can now write down the skin depth for the particular symmetric (S) and antisymmetric (AS) scenarios of Sec. III C and Sec. III D, respectively, in the case when the conducting behavior is directly associated with the magnetic conductivity only, i.e., when σ = 0. From Eqs. (64) and (78), we arrive at Therefore, such a skin depth effect does not exhibit a frequency dependence, which means that the penetration length is the same for all frequency bands. This is an unusual characteristic for conductors.
B. Reflection coefficient at the surface of conducting matter Consider a system composed of an ordinary dielectric characterized by a refractive index n 1 = √ µ 1 ǫ 1 and a conducting phase of matter with a complex refractive index n 2 = n ′ 2 + in ′′ 2 described by the parameters ǫ 2 , µ 2 and the Ohmic conductivity σ. For a wave that propagates from the dielectric and enters the surface of the conductor, the reflection coefficient for normal incidence is given by [29]: Considering n 1 ≪ n ′ 2 one can rewrite Eq. (95) in the general form: In standard electrodynamics, for a good conductor (σ/(ωµ 2 ) ≫ 1) one gets n ′ 2 = n ′′ 2 = µ 2 σ/(2ω). Then the reflection coefficient R from Eq. (96) yields: Setting µ 1 = µ 2 we obtain the known Hagen-Rubens formula [29]: Now we will derive the version of the latter relation for dielectric media (σ = 0) endowed with an exotic magnetic conductivity σ B ij . In the scenario of an antisymmetric σ B ij , the refractive index is modified according to Eq. (64), so that By inserting Eq. (99) into Eq. (96), one obtains: for real n ′ 2 . This result is a Hagen-Rubens-like formula for the case when there is a contribution from the antisymmetric magnetic conductivity only. It is also very different from the reflection coefficient for an ordinary dielectric, since the magnetic conductivity introduces a frequencydependent term in R. Hence the exotic conductivity results in a conducting matter phase in the limit σ → 0.
In the scenario of a symmetric σ B ij , one obtains a similar result:

VI. FINAL REMARKS
Electrodynamics in matter is well described by the Maxwell equations and the constitutive relations. Extensions of the Maxwell equations including the possibility of a chiral magnetic current is a topical issue [14][15][16][17][18][19][20][21]. In this work, we have extended the scenario of electric currents generated by magnetic fields by studying some basic classical properties of a magnetic conductivity implemented into the Maxwell equations through the extension of Ohm's law given in Eq. (11).
The main purpose was to examine the propagation of electromagnetic waves in dispersive dielectric materials, paying attention to the conduction properties induced upon non-Ohmic materials (σ=0). To this end, we proposed some particular realizations for the magneticconductivity tensor σ B ij : (i) a diagonal isotropic and a diagonal anisotropic tensor (which include the chiral magnetic effect) in Secs. III A, III B (ii) a nondiagonal antisymmetric tensor in Sec. III C, and (iii) a traceless nondiagonal symmetric tensor in Sec. III D. Let us point out that the trace of the magnetic-conductivity tensor σ B ij is related to the chiral magnetic effect and that the offdiagonal components of σ B ij describe generalizations of such an effect. All these configurations induce parity violation, since the associated current entering Ampère's law is linear in B.
We have verified that a diagonal isotropic tensor σ B ij = Σδ ij modifies the refractive index of a dispersive dielectric medium yielding two distinct complex values n ± . Most notably, these results are independent of the propagation direction, implying what we could call an "isotropic birefringence." In the σ = 0 case, the resulting refractive indices are real and the propagation modes correspond to left-and right-handed circular polarizations. As such, these media can be characterized as optically active having a frequency-independent specific rotary power δ = −µΣ/2. This is a consequence of the EPT being Hermitian and the electric field being orthogonal to the wave vector k. In this case, parity violation manifests itself only in the fact that n + is different from n − .
The diagonal anisotropic case also exhibits birefringence, but this time the complex refractive indices are direction-dependent. Nevertheless, the functions depend on the squares of the momentum componentes k i for i = 1, 2, 3 and are insensitive to parity transformations. Thus, the violation of this symmetry is again manifest only in the different values of n ± . The EPT is not Hermitian and k · E = 0, in general, thus preventing the description of polarization in terms of left-and righthanded modes.
Focusing on the σ = 0 case, we find again real refractive indices. In this situation we have explored in detail the case of propagation along n = (n 1 , n 2 , 0) showing explicitly that the eigenvectors of the EPT do not correspond to the polarization modes of the electric field. A further particular case of propagation along n = (0, n 2 , 0) restores the orthogonality between k and E and ends up with elliptically polarized propagation modes. The case of circular polarization is recovered when taking the isotropic limit Σ i = Σ, which also reproduces the previous expressions for the refractive indices. As expected, in this specific situation the eigenvectors of the EPT and the propagation modes coincide. We remark that both cases of a diagonal chiral conductivity tensor do not induce a conducting behavior in a non-Ohmic dielectric medium.
We have also examined a nondiagonal (antisymmetric and symmetric) σ B ij . In these cases, the off-diagonal components of σ B ij provide complex refractive indices even for a vanishing Ohmic conductivity (σ = 0), which leads to the remarkable behavior of a conducting phase in the dielectric substrate. The magnetic conductivity in these cases implies nonzero absorption coefficients that damp the intensity of electromagnetic waves propagating through the medium. In both cases, the EPT is non-Hermitian, k · E = 0, and the refractive indices are direction-dependent. The latter exhibit contributions that obviously induce parity violation.
A rather unexpected feature in the context of a parityviolating theory occurs for the antisymmetric case, which is the absence of birefringence. That is to say, we obtain only one doubly-degenerate value for the refractive index, which nevertheless supports two orthogonal polarization modes, as shown in general when σ = 0. On the contrary, the symmetric case exhibits birefringence.
For the nondiagonal cases (antisymmetric or symmetric) we have calculated the associated skin depth, which in the limit σ → 0 becomes a constant for all frequency bands of the electromagnetic wave that enters the medium. We also have derived a generalization of the Hagen-Rubens relation for nonconducting media (σ = 0), with contributions stemming from the magnetic conductivity only. In this case, the reflection coefficient is frequency-dependent, which is a property that does not occur at the interface of two ordinary dielectric substrates.
It is interesting to observe that when σ = 0, the equation determining the refractive indices in all four cases considered boils down to the form where B specifies each case according to: for the isotropic diagonal case of Sec. III A, for the anisotropic diagonal case of Sec. III B, for the antisymmetric case of Sec. III C and finally, for the symmetric case of Sec. III D.
A comment in relation to the consistency of our calculation is now in order. A current density J linear in the magnetic field has been the only additional input we have introduced into the calculation of the propagation properties of a wave, without making any statement on the corresponding charge density required by current conservation. The point to be recalled is that when ω = 0, Ampère's law directly yields charge conservation via the use of Gauss's law, as shown in Sec. IV. Thus, for completeness, we have identified in this section the charge densities corresponding to some of the currents we have introduced before. Perhaps an unexpected feature is that some of the parameters defining such currents (b and a × c, for example) produce additional contributions to the charge density proportional to the external currents that we have set equal to zero in this analysis.
Finally, we conclude that this work presents a classical perspective of some novel effects that a magnetic conductivity can provide for the propagation of electromagnetic waves in dispersive media.