Effects of CPT-odd terms of dimensions three and five on electromagnetic propagation in continuous matter

The dynamics of electromagnetic fields in continuous media is governed by the Maxwell equations, supplemented by constitutive relations Jackson ; Zangwill that describe the response of the medium to external, applied electromagnetic fields. In vacuo these relations simply read ${\bf{D}}=\epsilon_{0}{\bf{E}}$ and ${\bf{H}}=\mu_{0}^{-1}{\bf{B}}$, where $\epsilon_{0}$ and $\mu_{0}$ are the electric vacuum permittivity and magnetic permeability constant, respectively. The first constitutive relation takes into account the electric polarization in a dielectric medium, while the latter includes magnetization effects. For an isotropic medium, the constitutive relations are ${\bf{D}}=\epsilon{\bf{E}}$ and ${\bf{H}}=\mu^{-1}{\bf{B}}$ with scalar material parameters $\epsilon,\mu$ replacing $\epsilon_{0},\mu_{0}$ governing the vacuum properties. More involved constitutive relations appear in two main scenarios: (i) anisotropic media, where electric permittivity and magnetic permeability become tensors (cf. uniaxial and biaxial crystals Landau ; Bain ; Fowles ; Hecht ; Kurmanov ; Yakov , Weyl semimetals Halterman ; Zu , and magnetized materials Krupka1 ; Krupka2 ); (ii) novel effects in matter described by extended constitutive relations that introduce additional electric and magnetic responses, encoded as linear functions of the type $\mathbf{D}=\mathbf{D}(\mathbf{E},\mathbf{B})$ and $\mathbf{H}=\mathbf{H}(\mathbf{E},\mathbf{B})$, in general. This happens, for instance, in bi-isotropic media Aladadi ; Sihvola ; Sihvola2 ; Nieves , chiral materials Hillion , topological insulators Li1 ; Urrutia ; Urrutia2 ; Lakhtakia ; Winder ; Li , relativistic electron gases Carvalho , axion electrodynamics Sekine ; Tobar2 ; BorgesAxion , and Lorentz-violating electrodynamics Tobar1 ; Bailey , as well.

Generalizations of electrodynamics including higher-derivative terms have also been conceived in the literature. First studies of electrodynamics in the presence of higher derivatives are ascribed to Bopp Bopp in 1940, and to Podolsky Podolsky1 ; Podolsky2 in 1942. This model implements a second-order derivative term $\theta^{2}\partial_{\alpha}F^{\alpha\beta}\partial_{\sigma}F^{\sigma\beta}$ into the Maxwell Lagrangian in vacuo. The modified Maxwell-Podolsky equations, sometimes called Bopp-Podolsky equations, yield a photon mass term, proportional to the inverse of the Podolsky parameter $\theta$. Furthermore, this extension exhibits two dispersion relations, the usual one from Maxwell theory and a second one ascribed to a massive mode.

The constraint structure of this theory was investigated in Galvao and its quantization was performed in Barcelos . Further aspects of the Maxwell-Podolsky model were examined, including the problems of self-force and self-interaction Gratus ; Zayats , Green functions and classical solutions Lazar , multipole expansion for fields in the static regime Bonin , symmetrization/conservation of the energy-momentum tensor Fan , its consistency based on the BRST approach Dai , quantum field theoretic properties Bufalo ; Zambrano as well as other aspects Granado .

A further example for a generalization of Maxwell electrodynamics is provided by Lee-Wick electrodynamics Lee-Wick-1 ; Lee-Wick-2 , which introduces the modification $F^{\mu\nu}\square F_{\mu\nu}$ Turcati ; Turcati2 ; Turcati3 . The higher-derivative Lee-Wick term can arise as a quantum correction in models with a nonminimal coupling between the gauge and fermionic fields Borges .

In the past years, higher-derivative contributions have also been examined in the context of Lorentz-violating theories. The possibility of Lorentz invariance violation (LV) was proposed in the context of physics at the Planck scale such as strings Kostelecky:1988zi . Presently, the Standard-Model Extension (SME) Colladay , where fixed background tensor fields are coupled to the dynamical Standard-Model fields, is usually employed as the main framework to parameterize it. Lorentz violation in the electromagnetic sector of the SME occurs by means of a CPT-odd or a CPT-even term KM ; Escobar ; Santos ; Belich . The CPT-odd part is represented by the Carroll-Field-Jackiw (CFJ) contribution CFJ ; CFJ2 ; CFJ3 ; CFJ4 ; CFJ5 ; CFJ6 ; CFJ7 , which has found applications in condensed-matter systems that violate parity (P) and time reversal (T) symmetry Qiu as well as those endowed with the chiral magnetic effect Fukushima ; Kharzeev ; Kharzeev1 ; Kharzeev2 and the anomalous Hall effect Zyuzin .

Nonminimal extensions of the SME were proposed including higher-derivative terms with mass dimensions greater than four (in natural units) Kostelecky ; Mewes ; Schreck . In this context, the Myers-Pospelov model Myers ; Marat was a pioneering proposal focusing on a dimension-five contribution. Recently, classical aspects of a modified, higher-derivative electrodynamics in vacuo were discussed in Leticia1 ; Leticia2 , including the derivation of the gauge propagator, the dispersion relations as well as an analysis of causality, unitary, and stability of the modes. Profound analyses were accomplished for the Maxwell-Podolsky electrodynamics modified by CPT-even, dimension-six terms Leticia2 and for a CPT-odd, dimension-five electrodynamics Leticia1 . Some results of Ref. Leticia1 were revisited and discussed in Ref. Passos2 . Nonminimal higher-derivative models have also been used to study the interaction energy between electromagnetic sources Borges-Ferrari and the thermodynamic properties of electrodynamic systems Filho-Maluf as well as in the context of Horǎva-Lifshitz electrodynamics Passos and radiative corrections Ferrari .

The plethora of nonminimal LV theories on the one hand and the optical properties of new materials Aladadi ; Shibata on the other hand is a strong motivation for investigating higher-derivative effects on the propagation of electromagnetic waves in dielectric substrates, including aspects of optical activity and dichroism. In this sense, the present work is devoted to analyzing the behavior of a continuous medium governed by a MCFJ-type electrodynamics in the absence and presence of higher-derivative terms.

This paper is outlined as follows. In Sec. II, we briefly review the covariant description of electrodynamics in macroscopic materials recapitulating the definition of birefringence (double refraction) and dichroism. We will be considering simple matter as opposed to designed materials with highly peculiar properties such as metamaterials Shelby ; Valanju ; Kshetrimayum . In Sec. III, we present aspects of the MCFJ electrodynamics in a ponderable medium, showing that the timelike CFJ background yields birefringence, while the spacelike one provides birefringence and dichroism. In Sec. IV, we discuss the higher-derivative MCFJ electrodynamics in continuous matter based on more involved scenarios. Finally, we present our main findings in Sec. V. Throughout the paper, we employ natural units with $\hbar=c=1$ unless otherwise stated. Furthermore, our signature choice for the Minkowski metric $\eta_{\mu\nu}$ is $(+,-,-,-)$.

In a continuous medium, the electromagnetic properties are described by the Maxwell equations Jackson ; Zangwill combined with the constitutive relations. For a general linear, homogeneous, and anisotropic medium, the constitutive relations can be written as

where $\epsilon$ and $\mu$ represent the electric permittivity and magnetic permeability tensors Yakov ; Aladadi ; Sihvola ; Sihvola2 ; Nieves , respectively. The tensor $\gamma$ measures the magnetic contribution to the electric displacement field $\mathbf{D}$, while $\tilde{\gamma}$ represents the electric contribution to the magnetic field $\mathbf{H}$. Regarding the structure of constitutive relations (1a) and (1b), interesting scenarios of electromagnetic behavior may occur, e.g., in anisotropic media Aladadi ; Bain ; Fowles ; Kurmanov ; Yakov ; Sihvola ; Sihvola2 ; Nieves , Weyl semimetals Halterman ; Zu ; Grushin:2012mt , magnetized ferrites Krupka1 ; Krupka2 , and in chiral media and topological insulators Urrutia ; Urrutia2 ; Lakhtakia . Besides Eq. (1), general constitutive relations for the current density, ${\bf{J}}={\bf{J}}({\bf{E}},{\bf{B}})$, can also be considered. As an example, a dielectric system endowed with a magnetic conductivity has recently been examined at the classical level Pedro , reporting interesting effects such as an induced electric conductivity, isotropic birefringence, and parity violation. A physical realization of the antisymmetric magnetic current examined in Pedro was addressed in Ref. Kaushik .

The constitutive relations in Eq. (1) can be naturally encoded in the Lagrange density formalism via

with $\chi^{\mu\nu\alpha\beta}$ being the constitutive tensor that parameterizes the medium’s response to the applied electromagnetic fields Post . The constitutive tensor satisfies the following symmetry properties:

compatible with the symmetries of the field strength tensor. The Euler-Lagrange equation applied to the Lagrangian of Eq. (2a) (a complete derivation is presented in Appendix A) yields

The homogenous Maxwell equations are obtained from the Bianchi identity valid for the curvature $F_{\mu\nu}$ of the principal U(1) fiber bundle:

which is why the latter are not affected by the presence of the medium. A straightforward calculation from Eqs. (4), (5) leads to the well-known Maxwell equations in simple matter, namely:

where the constitutive relations for the electric displacement field ${\bf{D}}$ and magnetic field ${\bf{H}}$ are given in Eq. (1). The Maxwell equations of Eq. (6) and the constitutive relations given by Eq. (1) allow us to describe the dynamics of electromagnetic fields in simple matter.

In crystals and generic anisotropic media, the dispersion relations are given by the Fresnel equation Zangwill . The latter is obtained from algebraic manipulations of the Maxwell equations for continuous media whose properties are encoded in the permittivity tensor. In such a medium, the dielectric permittivity tensor is a function of the frequency $\omega$, the wave vector $\mathbf{k}$, and the (external) magnetic field $\mathbf{B}$ (or the magnetization, alternatively), $\epsilon_{ij}=\epsilon_{ij}(\omega,{\bf{k}},{\bf{B}})$, and can be expanded as Shibata

The first term, $\epsilon^{0}_{ij}$, is the usual permittivity tensor of a dielectric. The second term, $\alpha_{ijl}k_{l}$, is a signature of spatial inversion symmetry breaking. It implies an optical activity that becomes manifest via linear birefringence or a rotation of the oscillation plane of linearly polarized light Bain ; Fowles . The third term, $\beta_{ijl}B_{l}$, is associated with a fixed external magnetic field or magnetization and leads to a violation of time reversal symmetry. It gives rise to a magneto-optical activity via the Faraday or the Cotton-Mouton effect Shibata .

As already mentioned, a known consequence of the optical activity of a medium is linear birefringence, occurring when two circularly polarized modes of opposite chiralities, with refractive indices $n_{+}$ and $n_{-}$, respectively, have different phase velocities, $c/n_{+}$ and $c/n_{-}$. This property implies a rotation of the polarization plane of a linearly polarized wave. The latter phenomenon is quantified by the specific rotatory power $\delta$, which measures the rotation of the oscillation plane of linearly polarized light per unit traversed length in the medium. It is defined as

where $n_{+}$ and $n_{-}$ are associated with left and right-handed circularly polarized waves, respectively.

Another interesting effect occurring in anisotropic crystals, dichroism, takes place when one polarization component is more strongly absorbed than the other. Obviously, this property is linked to the imaginary part of the refractive index. The difference of absorption of left and right-handed circularly polarized modes Shibata is given by the dichroism coefficient [see Appendix B for the derivation of Eqs. (8), (9)]:

In the following, we examine two models of modified electrodynamics in continuous matter: the first one governed by the MCFJ Lagrangian and the second one by a higher-derivative MCFJ-type Lagrangian.

In principle, MCFJ electrodynamics has connections with systems of chiral fermions, in particular, the chiral magnetic effect (CME), the anomalous Hall effect (AHE), and the anomalous generation of charge, besides birefringence effects. In Ref. Qiu , the Maxwell equations in vacuo modified by the CFJ background were obtained, which yields the terms ascribed to the CME and AHE. In this section, we examine aspects of the MCFJ electrodynamics embedded in a continuous medium. The latter plays a role for condensed-matter systems such as Weyl semimetals Yan:2016euz . These novel materials are characterized by an even number of Weyl cones separated from each other in momentum space. In the vicinity of these cones, electrons behave as massless particles and have a certain Fermi velocity associated with them, whereupon their description via the Weyl equation is admissible. Having a microscopic realization of such a material at hand, it can be consistently described in the context of effective field theory via a $b$ term of the minimal SME fermion sector Colladay . The modified Dirac theory is frequently recast into the form Grushin:2012mt ; Behrends:2018qkj

where $\psi$ is a Dirac spinor field, $\overline{\psi}$ its Dirac conjugate, $m$ the electron mass, $\gamma^{\mu}$ the standard Dirac matrices, and $\gamma^{5}$ is the chiral Dirac matrix. The vector-valued background field $b_{\mu}$ is known to catch the essential properties of a certain class of these materials. Integrating out the fermion fields implies an action for the electromagnetic fields. The latter decomposes into a CPT-even part, which gives rise to a nontrivial permittivity and permeability of the system, as well as a CPT-odd part corresponding to a CFJ term. We will come back to this point below.

The MCFJ Lagrange density in matter has the form

with the intrinsic vector-valued field $V^{\mu}$. Furthermore, $\epsilon^{\beta\lambda\mu\nu}$ is the Levi-Civita symbol in Minkowski spacetime fulfilling $\epsilon^{\beta\lambda\mu\nu}=-\epsilon_{\beta\lambda\mu\nu}$ and $\epsilon_{0123}=1$. The latter Lagrange density yields the following modified inhomogeneous Maxwell equations:

where $\mathbf{V}$ is the spatial part of $V^{\mu}$ and the fields $\bf{D},\bf{H}$ fulfill the linear constitutive relations of Eq. (1). Notice that the presence of the tensor $G^{\mu\nu}$ renders the Lagrangian (11) different from the one of Ref. Qiu , meaning that the modified Maxwell equations of Eq. (12) apply to a ponderable medium. Concerning the discrete symmetries, C (charge conjugation), P, and T, it is worthwhile to recall that the CFJ term is CPT-odd and the free part of the Lagrangian in Eq. (11) can be written as:

In this sense, the pieces involving $V_{0}$ are P-odd and T-even, while the terms composed of $\bf{V}$ are P-even and T-odd, as properly shown in Tab. 1. Thus, these terms can induce an optical activity of the medium (in the form of birefringence or dichroism).

Another interesting aspect of MCFJ electrodynamics is that the term $V_{0}{\bf{B}}$ plays a significant role in a chiral magnetic current,

usually generated in chiral fermion systems Pedro ; Fukushima ; Kharzeev ; Kharzeev1 ; Kharzeev2 . Here, $\Delta\mu\equiv{\mu}_{R}-\mu_{L}$ is also known as the chiral chemical potential and fermions of electric charge $\pm e$ are considered. In Eq. (14), $\Sigma$ represents a chiral magnetic isotropic conductivity and plays a role equivalent to that of the timelike component $V_{0}$ in Eq. (12b), as pointed out in Ref. Qiu . For the present analysis, we consider the usual constitutive relations. We set $\gamma_{ij}=\tilde{\gamma}_{ij}=0$ in Eq. (1), implying

which can be restricted to the special scenario of an isotropic medium by choosing configurations proportional to the identity,

where $\epsilon$ and $\mu$ are the electric permittivity and magnetic permeability constants, respectively. This approach is equivalent to taking the constitutive relations ${\bf{D}}=\epsilon{\bf{E}}$, and ${\bf{H}}=\mu^{-1}{\bf{B}}$.

In what follows, we implement the latter relations as well as ${\bf{J}}=\sigma{\bf{E}}$, where $\sigma$ is the Ohmic conductivity. Furthermore, we employ a plane-wave ansatz for the fields, $\mathbf{E}=\mathbf{E}_{0}e^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega t)}$, $\mathbf{B}=\mathbf{B}_{0}e^{\mathrm{i}(\mathbf{k}\cdot\mathbf{r}-\omega t)}$, and similarly for ${\bf{D}}$ and ${\bf{H}}$ in Eq. (12b). One then gets

which can be simplified by using Faraday’s law, yielding

as an effective electric permittivity tensor. Here, $\epsilon_{ijk}$ is the Levi-Civita symbol in three dimensions. The second term on the right-hand side of Eq. (18b), $\epsilon_{iaj}k_{a}V_{0}$, is analogous to the tensor $\alpha_{ijl}k_{l}$ of Eq. (7), which breaks parity invariance. The third term, $\epsilon_{iaj}\omega V_{a}$, breaks time reversal invariance. Both are responsible for the optical activity of the medium, becoming manifest in birefringence, as we shall see.

For a continuous medium, we write $\mathbf{k}=\omega\mathbf{n}$, where $\mathbf{n}$ is a vector pointing along the direction of the wave vector and yielding the refractive index $n=+\sqrt{\mathbf{n}^{2}}$. To permit complex refractive indices, we take $+\sqrt{\mathbf{n}^{2}}$ instead of $|\mathbf{n}|$. The plus sign indicates, in principle, that we discard refractive indices with negative real parts, whenever such could occur. Composites with negative real parts of their refractive indices are called metamaterials Shelby ; Valanju ; Kshetrimayum and such possibilities will not be taken into account. Then, Eq. (18a) becomes

The nontrivial solutions for the electric field are obtained by requiring that the determinant of the matrix $M_{ij}$ vanish, $\mathrm{det}[M_{ij}]=0$, which yields the dispersion relations that describe wave propagation in the medium. The matrix $M_{ij}$ is explicitly given by

with the $(3\times 3)$ identity matrix $\mathds{1}_{3}$. The dispersion equation follows from $\mathrm{det}[M_{ij}]=0$:

We note that via the choices

our Eq. (21) is equivalent to

Alternatively, we can introduce an effective metric of the form

and write the dispersion equation as

In vacuo, where $\epsilon=\mu=1$ and $\sigma=0$, the conventional four-momentum $p^{\mu}=(\omega,{\bf{k}})$ and the preferred spacetime direction $V^{\mu}=(V^{0},\bf{V})$ satisfy

as expected. Equation (26) is the well-known dispersion equation of the MCFJ model in vacuo CFJ ; CFJ2 . Therefore, we interpret Eq. (23) as the dispersion equation for a generalized MCFJ theory in media. The four-vector $\overline{p}^{\mu}$ of Eq. (22) plays the role of an effective four-momentum that formally satisfies an analogous dispersion equation as in vacuo when the preferred direction is replaced by $\overline{V}^{\mu}$. The possibility of expressing the dispersion equation in terms of the effective metric in Eq. (24) and the conventional four-momentum $p^{\mu}$ is a different way of understanding this result. The presence of a medium described by the material parameters $\epsilon,\mu$, and $\sigma$ leads to electromagnetic waves obeying an analogous dispersion equation as in vacuum, but with the Minkowski metric replaced by an effective metric.

Now, let us have another look at MCFJ theory in the context of Weyl semimetals. As described in Grushin:2012mt , integrating out the fermion fields of the effective field theory stated in Eq. (10) implies a modified electrodynamics described by the first two terms in Eq. (11). Considering the realization of a Weyl semimetal studied in the latter reference leads to a particular choice of $V^{\mu}$ with $V^{0}=0$ and $\mathbf{V}$ pointing along the third spatial axis (compare the modified inhomogeneous Maxwell equations of their Eqs. (31), (34) to our Eqs. (12a), (12b)). Although the CFJ term incorporates the most intriguing properties of such materials, we must also take into account that a Weyl semimetal (like any material) is characterized by a nontrivial permittivity (whereas the permeability is often simply to set 1). Thus, the optical response of such a material is very well described by a dispersion equation of the form of Eq. (III) (cf. Eq. (36) in Grushin:2012mt ) being a formidable motivation for considering theories such as Eq. (11). The author of the latter paper emphasizes that the presence of the CFJ term implies birefringence in Weyl semimetals.

In order to further understand some properties of MCFJ electrodynamics in a continuous dielectric medium with magnetic properties, we address two main scenarios: (i) a timelike and (ii) a spacelike background field $V^{\mu}$. We choose a non-Ohmic dielectric as a substrate, which implies $\sigma\mapsto 0$ in Eq. (21).

After analyzing the properties of MCFJ theory in a material (see Eq. (11)), the next logical step is to construct and investigate an extension involving additional four-derivatives. Such extensions are naturally contained in the nonminimal (nongravitational) SME Kostelecky ; Mewes ; Schreck , which is a comprehensive framework for the parameterization of Lorentz and CPT violation in effective field theory in Minkowski spacetime. For the past two decades it has been the foundation of various experiments testing the fundamental spacetime symmetries Kostelecky:2008ts . No signal of Lorentz violation in vacuo has been found, so far. However, Lorentz violation can be considered as an intrinsic property of material media, which is why the SME is more than suitable as a base for representing certain material properties within a field theory setting and to even propose novel materials with unusual characteristics.

The electromagnetic sector of the nonminimal SME gives rise to a modified electrodynamics and is given by

The CPT-odd and CPT-even operators, $(\hat{k}_{AF})_{\kappa}$ and $(\hat{k}_{F})^{\kappa\lambda\mu\nu}$, respectively, are the analogs of $(k_{AF})_{\kappa}$ and $(k_{F})^{\kappa\lambda\mu\nu}$ of the minimal SME. However, they involve nonminimal coefficients contracted with additional four-derivatives in the form of the following infinite operator series:

where $d$ is the mass dimension of the tensor field operator that a certain coefficient is contracted with. Besides, $(4-d)$ is the mass dimension of the associated controlling coefficients $(k_{AF}^{(d)})_{\kappa}^{\phantom{1}\alpha_{1}\dots\alpha_{(d-3)}}$ and $(k_{F}^{(d)})^{\kappa\lambda\mu\nu\alpha_{1}\dots\alpha_{(d-4)}}$. The Lorentz indices $\alpha_{i}$ are contracted with additional spacetime derivatives.

We are interested in the CPT-odd dimension-five ($d=5$) extension, which is represented by a CFJ-like term of the form

For our investigation, we will use the parameterization

with the Lorentz-violating four-vector, $U_{\kappa}$, and the Minkowski metric tensor, $\eta^{\mu\nu}$. Using Eq. (57), the higher-derivative term becomes

where we have introduced the d’Alembertian $\square=\eta^{\alpha_{1}\alpha_{2}}\partial_{\alpha_{1}}\partial_{\alpha_{2}}$. The resulting higher-derivative Lagrangian,

involves LV parameterized by the background vector, $U^{\mu}=(U^{0},{\bf{U}})$. Some classical aspects of this model were examined in Refs. Leticia1 ; Marat .

In order to study the effects of this higher-derivative term on electromagnetic propagation in continuous matter, we take as a starting point the Lagrangian (59), but employ the field strength tensor $G^{\mu\nu}$ in its kinetic term, as it occurs in Eq. (2a). Thus, the Lagrangian of this new model is

where the tensor $G^{\mu\nu}$ is written in terms of the constitutive tensor $\chi^{\mu\nu\alpha\beta}$, defined in Eqs. (2b), (3). The latter provides a generalization of the electrodynamics of Eq. (59) in matter. One may expect a connection between this theory and a generalization of the modified Dirac theory given by Eq. (10) where additional derivatives are included in the second contribution. However, it is beyond the scope of the current paper to demonstrate such a connection explicitly. Thus, by using Eq. (59) we can take into consideration an additional energy-momentum dependence that goes beyond that of the CFJ term in matter.

The Lagrangian of Eq. (60) involves a third-order derivative of the four-potential, which requires an associated Euler-Lagrange equation endowed with derivatives for field derivatives that are of the same order. In principle, the derivative order can be decreased by rewriting Eq. (60) in the form