# A Dirac-material-inspired non-linear electrodynamic model

M. J. Neves<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil
Patricio Gaete<EMAIL_ADDRESS>Departamento de Física and Centro
Científico-Tecnológico de Valparaíso-CCTVal, Universidad Técnica Federico
Santa María, Valparaíso, Chile  L. P. R. Ospedal<EMAIL_ADDRESS>Centro
Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de
Janeiro, Brasil, CEP 22290-180  J. A. Helayël-Neto<EMAIL_ADDRESS>Centro
Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de
Janeiro, Brasil, CEP 22290-180

###### Abstract

We propose and study the properties of a non-linear electrodynamics that
emerges inspired on the physics of Dirac materials. This new electrodynamic
model is an extension of the one-loop corrected non-linear effective
Lagrangian computed in the work of ref. [3]. In the particular regime of a
strong magnetic and a weak electric field, it reduces to the photonic non-
linear model worked out by the authors of ref. [3]. We pursue our
investigation of the proposed model by analyzing properties of the
permittivity and permeability tensors, the energy- momentum tensor and wave
propagation effects in presence of a uniform magnetic background. It is shown
that the electrodynamics here presented exhibits the vacuum birefringence
phenomenon. Subsequently, we calculate the lowest-order modifications to the
interaction energy, considering still the presence of a uniform external
magnetic field. Our analysis is carried out within the framework of the gauge-
invariant but path-dependent variables formalism. The calculation reveals a
screened Coulomb-like potential with an effective electric charge that runs
with the external magnetic field but, as expected for Dirac-type materials,
the screening disappears whenever the external magnetic field is switched off.

Dirac materials, Non-linear electrodynamics, Birefringence.

##  I Introduction

As well-known, over the last decades, the area of quantum condensed matter
physics and quantum field theory have become increasingly interconnected. This
success is based on the fact that phenomena appearing in different systems are
unified by common physical mechanisms [1, 2]. An illustrative example arises
when one considers Weyl and Dirac semimetals, where it was shown in ref. [3]
that quantum vacuum non-linear effects contribute to the experimentally
observed high field magnetization. These effects have their origin in the
well-known result due to Heisenberg and Euler [4].

In this context, it may be recalled that the subject of quantum vacuum non-
linearities has been of great interest since the pioneering work by Euler and
Heisenberg [4]. In fact, in the case of non-linear electrodynamic models that
appear as effective descriptions by summing up electron-positron quantum
effects, the polarizable medium is the Quantum Electrodynamics (QED) vacuum,
made up of virtual electrons and positrons. That is the case of the Euler-
Heisenberg effective model. Incidentally, it is of interest to notice that
Schwinger later confirmed this remarkable quantum prediction of light-by-light
scattering from QED [5].

In this connection, we call attention to the fact that one of the striking
physical effects of the Heisenberg and Euler result has been vacuum
birefringence. In other words, the quantum vacuum when is stressed by external
electromagnetic fields behaves as if it were a birefringent material medium,
which has been emphasized from different perspectives [6, 7, 8, 9, 10, 11].
However, this optical phenomenon has not yet been confirmed. Mention should be
made, at this point, to recent results from ATLAS and CMS collaborations at
the Large Hadron Collider (LHC), which have reported on the high energy gamma-
gamma pair emission from virtual gamma-gamma scattering in ultraperipheral Pb-
Pb collisions [12, 13, 14]. It is worthy to emphasize here that, in these
results, there is no modification of the optical properties of the vacuum
[15]. We further mention that the coming of laser facilities has given rise to
various proposals to probe quantum vacuum non-linearities [16, 17, 15].

Inspired by these observations, the purpose of this paper is to further
elaborate on the physical content of the non-linear one-loop corrected
effective electrodynamics discussed in ref. [3]. Of special interest will be
to consider the non-linear electrodynamics for Dirac materials in the familiar
language of standard quantum field theory. Along these lines, we establish a
connection between condensed matter and quantum field effective theories.
Evidently, these connections are of interest in providing unifications among
diverse models. We clarify that the model we present in this contribution is
not derived as an effective physics stemming from Dirac materials. What we
actually do is to write down a Lagrangian density that extends the effective
physics worked out in ref. [3] and recovers the latter in the special limit of
a strong magnetic and weak electric field. Once the Dirac-material-inspired
action is formulated, we obtain the field equations, conservation laws, as
well as the properties of the medium due to non-linear contributions. Next, we
study the propagation effects of this non-linear electrodynamics under an
external and uniform magnetic field. We add the prescription of linearization
expanding the Lagrangian density of the theory up to second order in the
propagating fields. Thereby, we are able to extract the birefringence
phenomenon in some Dirac materials. At last, to make contact with the
discussion of the Coulomb screening that occurs in Dirac materials [18], we
calculate the interaction energy in Dirac materials under an uniform magnetic
field via path-dependent variables formalism, and show that an effective
charge comes out that depends on the external magnetic field.

The organization of the paper is as follows: In Section II, we introduce a new
non-linear electrodynamics for Dirac materials and analyze aspects such as
equations of motion, the permittivity and permeability tensors and the
corresponding energy- momentum tensor. In Section III, we study wave
propagation effects under a uniform magnetic background. In Section IV, we
consider the birefringence phenomenon for this new electrodynamics. In Section
V, we compute the interaction energy for this electrodynamics under an uniform
magnetic field. Our discussion is accomplished using the gauge-invariant but
path-dependent variables formalism, which is an alternative to the usual
Wilson loop approach. Finally, some concluding comments and considerations are
cast in Section VI.

Throughout this work, we adopt natural units $\hbar=c=1$ with
$4\pi\epsilon_{0}=1$, and the Minkowski metric
$\eta^{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$. The electric and magnetic fields
have squared-energy mass dimension, where the conversion of Volt/m and Tesla
(T) to the natural system is given by $1\,\mbox{Volt/m}=2.27\times
10^{-24}\,\mbox{GeV}^{2}$ and $1\,\mbox{T}=6.8\times
10^{-16}\,\mbox{GeV}^{2}$, respectively.

##  II A non-linear electrodynamics for Dirac materials

As was shown in the work [3], the authors computed the non-linear one-loop
contribution for a Dirac material, given by the effective Lagrangian density:

$\displaystyle{\cal
L}_{DM}\simeq\frac{\Delta}{24\pi^{2}\lambda_{D}^{3}}\left[\,\frac{({\bf
e}\cdot{\bf b})^{2}}{|{\bf b}|}+{\bf b}^{2}\,\ln(|{\bf b}|)\,\right],$ (1)

in the regime of a strong magnetic field and a weak electric field. In
expression (1), the dimensionless EM fields ${\bf e}$ and ${\bf b}$ depends on
the fine structure constant $\alpha_{D}=e^{2}/v$ in the Dirac material :

$\displaystyle{\bf e}(\alpha_{D})=\frac{{\cal U}\,{\bf
E}}{E_{\star}(\alpha_{D})}\;,\;{\bf b}(\alpha_{D})=\frac{{\cal U}^{-1}\,{\bf
B}}{B_{\star}(\alpha_{D})}\;,$ (2)

where $E_{\star}(\alpha_{D})$ and $B_{\star}(\alpha_{D})$ are, respectively,
the critical Schwinger’s electric and magnetic fields in the material, such
that

$\displaystyle
E_{\star}^{2}(\alpha_{D})=v^{2}\,B_{\star}^{2}(\alpha_{D})=\frac{\Delta}{\alpha_{D}\,\lambda_{D}^{3}}=\left(\frac{\Delta^{2}}{ev}\right)^{2}\;,$
(3)

and ${\cal U}$ is a symmetric matrix with $\det({\cal U})=1$. Thereby, the
terms ${\cal U}\,{\bf E}$ and ${\cal U}^{-1}\,{\bf B}$ in (2) are linear
combinations of ${\bf E}$ and ${\bf B}$, respectively. The physical parameters
from (1) are resumed here as : $\Delta$ is the material gap with energy
dimension and $\lambda_{D}=v/\Delta$ is the Dirac wavelength in the material.
The Dirac magneton is $\mu_{D}=e\,v^{2}/(2\Delta)$, that replaces the Bohr
magneton in the theory. All these quantities depends on the nature of the
material and are resumed, as example, for some materials in the table 1 where
we have used the natural units.

|  $\Delta(\mbox{meV})$ |  $\alpha_{D}/\alpha$ |  $E_{\star}(\mbox{eV}^{2})$ |   $B_{\star}(\mbox{eV}^{2})$
---|---|---|---|---
QED |  $5\times 10^{8}$ |  $1$ |  $2.95\times 10^{12}$ |   $3.0\times 10^{12}$
PbSnTe |  $31.5$ |  $580$ |  $6.58$ |   $3.8\times 10^{-3}$
BiSb |  $7.75$ |  $188$ |  $0.13$ |  $24.4$
TaAs |  $0$ |  $357$ |  $0$ |  $0$
Table 1: The comparison of the parameters for some materials and the QED. The
fine structure constant in Dirac material $(\alpha_{D})$ is defined in
relation to QED $(\alpha)$, i.e., $\alpha_{D}/\alpha=v^{-1}$.

We propose a general non-linear ED governed by the Lagrangian density

$\displaystyle{\cal L}_{DM}=-\frac{\alpha_{D}}{24\pi^{2}}\,{\cal
F}\,\ln\left[\,1-\frac{2\,{\cal F}}{\beta^{2}}+\frac{2\,{\cal
G}^{2}}{\beta^{3}\sqrt{-2\,{\cal F}}}\,\right]\;,$ (4)

where ${\cal F}$ and ${\cal G}$ are the Lorentz and gauge-invariants

$\displaystyle{\cal F}\\!$ $\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,F_{\mu\nu}^{2}=\frac{1}{2}\,\left(\,{\bf
E}^{2}-{\bf B}^{2}\,\right)\;,$ (5a) $\displaystyle{\cal G}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,F_{\mu\nu}\widetilde{F}^{\mu\nu}={\bf
E}\cdot{\bf B}\;,$ (5b)

where ${\bf E}$ and ${\bf B}$ are the electric and magnetic fields,
respectively. Using the approximation $|{\bf B}|\gg|{\bf E}|$, the Lagrangian
density (4) is reduced to the expression (1). The $\beta$-parameter is
identified as the Schwinger’s electric field in the material,

$\displaystyle\beta=\sqrt{\frac{\Delta}{\alpha_{D}\,\lambda_{D}^{3}}}=\frac{\Delta^{2}}{ev}\;.$
(6)

If the magnitude of the magnetic field is $|{\bf B}|>|{\bf E}|$, the non-
linear Lagrangian density (4) is real. Otherwise, if $|{\bf E}|>|{\bf B}|$,
this theory can present complex terms in the logarithmic argument. Notice also
that eq. (4) goes to zero in the limit $|{\bf E}|\rightarrow|{\bf B}|$.

We add the usual Maxwell Lagrangian to the non-linear electrodynamics in eq.
(4) :

$\displaystyle{\cal L}={\cal F}-\frac{\alpha_{D}}{24\pi^{2}}\,{\cal
F}\,\ln\left[\,1-\frac{2\,{\cal F}}{\beta^{2}}+\frac{2\,{\cal
G}^{2}}{\beta^{3}\sqrt{-2\,{\cal F}}}\,\right]\;,$ (7)

in which the usual Maxwell theory is recovered in the limit
$\beta\rightarrow\infty$. The action principle applied to eq. (7) yields the
field equation

$\partial_{\mu}\left[\,\xi({\cal F},{\cal G})\,F^{\mu\nu}+\chi({\cal F},{\cal
G})\,\widetilde{F}^{\mu\nu}\,\right]=0\;,$ (8)

where the coefficients $\xi({\cal F},{\cal G})$ and $\chi({\cal F},{\cal G})$
are functions of ${\bf E}$ and ${\bf B}$, according to

$\displaystyle\xi({\cal F},{\cal G})\\!$ $\displaystyle=$
$\displaystyle\\!1+\frac{\alpha_{D}}{48\pi^{2}}\frac{\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F}}{\sqrt{2}\,{\cal G}^{2}+\sqrt{-{\cal
F}}\beta(-2{\cal F}+\beta^{2})}$ (9a)
$\displaystyle-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]\;,\;\;\;$ $\displaystyle\chi({\cal F},{\cal G})\\!$
$\displaystyle=$ $\displaystyle\\!-\frac{\alpha_{D}}{6\pi^{2}}\frac{{\cal
F}\,{\cal G}}{2\,{\cal G}^{2}+\sqrt{-2{\cal F}}\beta(-{\cal F}+\beta^{2})}\;.$
(9b)

As in the usual case, the dual tensor
$\widetilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$
satisfies the Bianchi identity $\partial_{\mu}\widetilde{F}^{\mu\nu}=0$.

The modified Maxwell equations can be written in terms of the constitutive
vectors,

$\displaystyle\nabla\cdot{\bf D}\\!$ $\displaystyle=$ $\displaystyle\\!0\hskip
8.5359pt,\hskip 8.5359pt\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial
t}={\bf 0}\;,$ (10a) $\displaystyle\nabla\cdot{\bf B}\\!$ $\displaystyle=$
$\displaystyle\\!0\hskip 8.5359pt,\hskip 8.5359pt\nabla\times{\bf
H}-\frac{\partial{\bf D}}{\partial t}={\bf 0}\;,$ (10b)

where the components of ${\bf D}$ and ${\bf H}$ vectors are defined by
$D_{i}=\epsilon_{ij}({\bf E},{\bf B})\,E_{j}$ and $H_{i}=(\mu_{ij})^{-1}({\bf
E},{\bf B})\,B_{j}$, respectively. The permittivity symmetric tensor
$\epsilon_{ij}$ and the inverse of the permeability tensor $(\mu_{ij})^{-1}$
are read below :

$\displaystyle\epsilon_{ij}({\bf E},{\bf B})\\!$ $\displaystyle=$
$\displaystyle\\!\left\\{1+\frac{\alpha_{D}}{48\pi^{2}}\frac{(\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F})}{\sqrt{2}\,{\cal
G}^{2}+\sqrt{-{\cal F}}\beta(-2{\cal F}+\beta^{2})}\right.$ (11a)
$\displaystyle\left.-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]\right\\}\,\delta_{ij}$
$\displaystyle-\frac{\alpha_{D}}{6\pi^{2}}\frac{{\cal
F}\,B_{i}\,B_{j}}{2\,{\cal G}^{2}+\sqrt{-2{\cal F}}\beta(-{\cal
F}+\beta^{2})}\;,$ $\displaystyle(\mu_{ij})^{-1}({\bf E},{\bf B})\\!$
$\displaystyle=$
$\displaystyle\\!\left\\{1+\frac{\alpha_{D}}{48\pi^{2}}\frac{(\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F})}{\sqrt{2}\,{\cal
G}^{2}+\sqrt{-{\cal F}}\beta(-2{\cal F}+\beta^{2})}\right.$ (11b)
$\displaystyle\left.-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]\right\\}\,\delta_{ij}$
$\displaystyle+\frac{\alpha_{D}}{6\pi^{2}}\frac{{\cal
F}\,E_{i}\,E_{j}}{2\,{\cal G}^{2}+\sqrt{-2{\cal F}}\beta(-{\cal
F}+\beta^{2})}\;.$

The inverse of the matrix (11b) yields the permeability tensor

$\displaystyle\mu_{ij}=\frac{1}{1+a}\left[\,\delta_{ij}-\frac{b\,E_{i}\,E_{j}}{1+a+b\,{\bf
E}^{2}}\,\right]\;,$ (12)

in which, for convenience, we define

$\displaystyle a\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\alpha_{D}}{48\pi^{2}}\frac{\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F}}{\sqrt{2}\,{\cal
G}^{2}+\beta\sqrt{-{\cal F}}(-2{\cal F}+\beta^{2})}$ (13a)
$\displaystyle-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]\;,$ $\displaystyle b\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\alpha_{D}}{6\pi^{2}}\frac{{\cal F}}{2\,{\cal
G}^{2}+\beta\sqrt{-2{\cal F}}(-{\cal F}+\beta^{2})}\;.$ (13b)

In the optical basis of the EM field, the permittivity and permeability
tensors can be diagonalized by an $SO(3)$ transformation in which the diagonal
elements are given by the corresponding eigenvalues :

$\displaystyle\lambda_{1\epsilon}$ $\displaystyle=$
$\displaystyle\lambda_{2\epsilon}=1+a\;,\;\lambda_{3\epsilon}=1+a-b\,{\bf
B}^{2}\;,$ (14a) $\displaystyle\lambda_{1\mu}$ $\displaystyle=$
$\displaystyle\lambda_{2\mu}=\frac{1}{1+a}\;,\;\lambda_{3\mu}=\frac{1}{1+a+b\,{\bf
E}^{2}}\;.$ (14b)

Thereby, the positive conditions for the eigenvalues $1+a>0$, $1+a-b\,{\bf
B}^{2}>0$ and $1+a+b\,{\bf E}^{2}>0$ insure that the previous tensors are
positive.

The energy-momentum tensor associated with the equations (10a) and (10b) reads
as below:

$\displaystyle\Theta^{\mu\rho}=\xi({\cal F},{\cal
G})\,F^{\mu\nu}F_{\nu}^{\;\;\rho}\,+\,\eta^{\mu\rho}\left[\,\chi({\cal
F},{\cal G})\,{\cal G}-{\cal L}\,\right]\;.$ (15)

that satisfies the conservation law $\partial_{\mu}\Theta^{\mu\rho}=0$. The
conserved components of the energy-momentum tensor are read

$\displaystyle\Theta^{00}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{2}\left({\bf E}^{\,2}+{\bf
B}^{2}\right)+\frac{\alpha_{D}}{48\pi^{2}}\frac{\left(\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F}\right){\bf E}^{\,2}}{\sqrt{2}\,{\cal
G}^{2}+\sqrt{-{\cal F}}\beta(-2{\cal F}+\beta^{2})}$ (16a)
$\displaystyle-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]{\bf E}^{2}$
$\displaystyle-\frac{\alpha_{D}}{6\pi^{2}}\frac{{\cal F}\,({\bf E}\cdot{\bf
B})^{2}}{2\,{\cal G}^{2}+\sqrt{-2{\cal F}}\beta(-{\cal F}+\beta^{2})}\;,$
$\displaystyle\Theta^{0i}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left\\{1+\frac{\alpha_{D}}{48\pi^{2}}\frac{\sqrt{2}\,{\cal
G}^{2}+4\beta\,\sqrt{-{\cal F}}\,{\cal F}}{\sqrt{2}\,{\cal G}^{2}+\sqrt{-{\cal
F}}\beta(-2{\cal F}+\beta^{2})}\right.$ (16b)
$\displaystyle\left.-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left[\,1-\frac{2\,{\cal
F}}{\beta^{2}}+\frac{2\,{\cal G}^{2}}{\beta^{3}\sqrt{-2\,{\cal
F}}}\,\right]\right\\}\left({\bf E}\times{\bf B}\right)^{i}.\;\;\;$

The result (16a) requires that $|{\bf B}|>|{\bf E}|$, thus the energy density
and the Poynting vector are real. The same condition must be imposed for the
eigenvalues (14a) and (14b) be real. All the results of the usual and linear
Maxwell ED are recovered in the limit $\beta\rightarrow\infty$.

In the electrostatic case, we consider ${\bf B}={\bf 0}$ in the previous
equations. For a point-like charge, $\rho({\bf r})=Q\,\delta({\bf r})$, and
the Gauss law yields ${\bf D}=Q\,\hat{{\bf r}}/r^{2}$. Using the permittivity
tensor for ${\bf B}={\bf 0}$, the electrostatic field $(E)$ satisfies the
following equation :

$\displaystyle
E\left[\,1+\frac{\alpha_{D}}{24\pi^{2}}\frac{E^{2}}{\beta^{2}-E^{2}}-\frac{\alpha_{D}}{24\pi^{2}}\,\ln\left(1-\frac{E^{2}}{\beta^{2}}\right)\,\right]=\frac{Q}{r^{2}}\;.\;\;\;$
(17)

Notice that the limit $\beta\rightarrow\infty$ reduces (17) to the usual
electrostatic field of a point-charge. We solve numerically the equation (17)
to obtain the plot (1). The figure shows the dimensionless variable $E/\beta$
(electric field for like-point charge over the $\beta$-parameter) versus
$x=r\,\sqrt{\beta}$ (radial distance times the squared root of
$\beta$-parameter) for the Dirac material PbSnTe. In this plot, we use the
fundamental charge in natural units $Q=e=0.085$, and
$\alpha_{D}=580/137=4.23$.

Figure 1: The dimensionless variable $E/\beta$ versus $x=r\sqrt{\beta}$ for
the Dirac material PbSnTe.

The electric field is finite at the origin $(r\rightarrow 0)$, and it is given
by the critical electric field of the material, i. e.,
$E(r=0)=\beta=6.58\,\mbox{eV}^{2}$ (in the case of PbSnTe). This behavior is
similar for the BiSb. The energy density in the electrostatic case is

$u_{E}=\frac{1}{2}\,{\bf E}^{2}\left[1+\frac{\alpha_{D}}{12\pi^{2}}\frac{{\bf
E}^{2}}{\beta^{2}-{\bf
E}^{2}}-\frac{\alpha_{D}}{12\pi^{2}}\,\ln\left(1-\frac{{\bf
E}^{2}}{\beta^{2}}\right)\right]\,,\;\;\;$ (18)

that is positive if $\beta>|{\bf E}|$, and the total electrostatic energy is

$\displaystyle U=2\pi\int_{0}^{\infty}{\bf E}^{2}\,r^{2}\,dr\;.$ (19)

Using the expression of the electric field for a like-point charge in (17), we
solve numerically the integral (19) to obtain the finite electron’s self-
energy

$\displaystyle U\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!713.16\,\mbox{MeV}\;\;\mbox{for}\;\;\mbox{QED}\;,$ (20a)
$\displaystyle U\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!1.065\,\mbox{eV}\;\;\mbox{for}\;\;\mbox{PbSnTe}\;,$ (20b)
$\displaystyle U\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!0.15\,\mbox{eV}\;\;\mbox{for}\;\;\mbox{BiSb}\;.$ (20c)

##  III Wave propagation properties in a uniform magnetic background

We now examine wave propagation effects under a uniform magnetic field for the
non-linear ED (7). Following a similar procedure of ref. [19], we start off
our analysis by introducing a prescription for the external and uniform
magnetic field through the gauge 4-potential $A_{\mu}=a_{\mu}+A_{B\mu}$, where
$a_{\mu}$ is interpreted as the propagation $4$-potential, and $A_{B\mu}$ is
the potential associated with the external and uniform magnetic field ${\bf
B}_{0}$. Consequently, the EM field-strength tensor is also decomposed as
$F_{\mu\nu}=f_{\mu\nu}+F_{B\mu\nu}$, in which
$f^{\mu\nu}=\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}=\left(\,-e^{i}\,,\,-\epsilon^{ijk}\,b^{k}\,\right)$
denotes the EM field strength tensor of the propagating fields, whereas
$F_{B}^{\;\,\mu\nu}=\partial^{\mu}A_{B}^{\;\,\nu}-\partial^{\nu}A_{B}^{\;\,\mu}=\left(\,0\,,\,-\epsilon^{ijk}\,B_{0}^{\,\,k}\,\right)$
corresponds to the field strength of the magnetic background. Using this
prescription, the Lagrangian density (7) up to second order in the propagation
gauge field leads to

${\cal
L}_{model}^{(2)}=-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}-\frac{1}{4}\,c_{2}\,f_{\mu\nu}\,\widetilde{f}^{\mu\nu}+\,\frac{1}{8}\,Q_{B\mu\nu\kappa\lambda}\,f^{\mu\nu}\,f^{\kappa\lambda}\;,$
(21)

where
$\widetilde{f}_{\mu\nu}=\varepsilon_{\mu\nu\alpha\beta}f^{\alpha\beta}/2$ is
the propagation field strength dual tensor, the tensor
$Q_{B\mu\nu\kappa\lambda}$ evaluated at the background $(B)$ is defined by

$\displaystyle Q_{B\mu\nu\kappa\lambda}\\!$ $\displaystyle=$
$\displaystyle\\!d_{1}\,F_{B\mu\nu}F_{B\kappa\lambda}+d_{2}\,\widetilde{F}_{B\mu\nu}\widetilde{F}_{B\kappa\lambda}$
(22)
$\displaystyle+\,d_{3}\,F_{B\mu\nu}\widetilde{F}_{B\kappa\lambda}+d_{3}\,\widetilde{F}_{B\mu\nu}F_{B\kappa\lambda}\;,$

and
$\widetilde{F}_{B\mu\nu}=\varepsilon_{\mu\nu\alpha\beta}\,F_{B}^{\;\,\,\alpha\beta}/2$
is the dual tensor of the background. The coefficients of the expansion
$c_{i}\,(i=1,2)$ and $d_{i}\,(i=1,2,3)$ are given by

$\displaystyle c_{1}=\left.\frac{\partial{\cal L}}{\partial{\cal
F}}\right|_{{\bf B}_{0}},\left.c_{2}=\frac{\partial{\cal L}}{\partial{\cal
G}}\right|_{{\bf B}_{0}},\left.d_{1}=\frac{\partial^{2}{\cal L}}{\partial{\cal
F}^{2}}\right|_{{\bf B}_{0}},$
$\displaystyle\left.d_{2}=\frac{\partial^{2}{\cal L}}{\partial{\cal
G}^{2}}\right|_{{\bf B}_{0}},\left.d_{3}=\frac{\partial^{2}{\cal
L}}{\partial{\cal F}\partial{\cal G}}\right|_{{\bf B}_{0}},$ (23)

For the particular model in eq. (7), we obtain the coefficients in a magnetic
background :

$\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha_{D}}{24\pi^{2}}\frac{{\bf B}_{0}^{2}}{\beta^{2}+{\bf
B}_{0}^{2}}-\frac{\alpha_{D}}{24\pi^{2}}\ln\left(1+\frac{{\bf
B}_{0}^{2}}{\beta^{2}}\right)\;,$ $\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle 0\;,\;d_{1}=\frac{\alpha_{D}}{12\pi^{2}}\frac{{\bf
B}_{0}^{2}+2\beta^{2}}{({\bf B}_{0}^{2}+\beta^{2})^{2}}\;,$ $\displaystyle
d_{2}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{D}}{24\pi^{2}}\frac{|{\bf
B}_{0}|/\beta}{{\bf B}_{0}^{2}+\beta^{2}}\;,\;d_{3}=0\;.\hskip 19.91684pt$
(24)

Substituting the non-null coefficients of the expansion, the Lagrangian
density (21) has the following terms

$\displaystyle{\cal
L}^{(2)}=-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}+\frac{d_{1}}{8}\,(F_{B\mu\nu}f^{\mu\nu})^{2}+\frac{d_{2}}{8}\,(\widetilde{F}_{B\mu\nu}f^{\mu\nu})^{2}.\;\;\;$

The action principle applied to the Lagrangian density (III) yields the field
equation

$\partial^{\mu}\left[c_{1}\,f_{\mu\nu}-\frac{d_{1}}{2}\,F_{B\mu\nu}(F_{B\kappa\lambda}f^{\kappa\lambda})-\frac{d_{2}}{2}\,\widetilde{F}_{B\mu\nu}(\widetilde{F}_{B\kappa\lambda}f^{\kappa\lambda})\right]=0,$
(26)

and the dual tensor $\widetilde{f}^{\mu\nu}$ satisfies the Bianchi identity
$\partial_{\mu}\widetilde{f}^{\mu\nu}=0$. For the analysis of the propagation
effects, we consider the Lagrangian density (III). The correspondent field
equation are similar to eqs. (10a) and (10b), just exchanging $({\bf E},{\bf
B})$ by $({\bf e},{\bf b})$, where ${\bf D}=\epsilon_{ij}({\bf B}_{0})\,e_{j}$
and ${\bf H}=(\mu_{ij})^{-1}({\bf B}_{0})\,b_{j}$, in which the permittivity
and the permeability tensors are functions of the magnetic background
components :

$\displaystyle\epsilon_{ij}({\bf B}_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\delta_{ij}+\frac{d_{2}}{c_{1}}\,B_{0i}\,B_{0j}\;,$ (27a)
$\displaystyle\mu_{ij}({\bf B}_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\delta_{ij}+\frac{d_{1}\,B_{0i}\,B_{0j}}{c_{1}-d_{1}\,{\bf
B}_{0}^{2}}\;.$ (27b)

The eigenvalues of the permittivity and permeability matrices are given by

$\displaystyle\lambda_{1\varepsilon}\\!$ $\displaystyle=$
$\displaystyle\\!\lambda_{2\varepsilon}=1\;,\;\lambda_{3\varepsilon}({\bf
B}_{0})=1+\frac{d_{2}}{c_{1}}\,{\bf B}_{0}^{2}\;,$ (28a)
$\displaystyle\lambda_{1\mu}\\!$ $\displaystyle=$
$\displaystyle\\!\lambda_{2\mu}=1\;,\;\lambda_{3\mu}({\bf
B}_{0})=\frac{c_{1}}{c_{1}-d_{1}\,{\bf B}_{0}^{2}}\;.$ (28b)

In the limit $\beta\rightarrow\infty$, the matrices (27a) and (27b) reduce to
identity matrix, and the eigenvalues go to one. Note that these limits are
also recovered by turning off the magnetic background field. The permittivity
matrix is positive for any background magnitude, while that the permeability
matrix is positive if we impose the conditions $c_{1}>0$ and
$c_{1}>d_{1}\,{\bf B}_{0}^{2}$. Under these conditions, the material medium
behaves like a paramagnetic material. Using the relations
$\epsilon_{ij}=\delta_{ij}+\chi_{Eij}$ and $\mu_{ij}=\delta_{ij}+\chi_{Mij}$,
the electric and magnetic susceptibility tensors are, respectively, read below
:

$\displaystyle\chi_{Eij}({\bf B}_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\frac{d_{2}}{c_{1}}\,B_{0i}\,B_{0j}\;,$ (29a)
$\displaystyle\chi_{Mij}({\bf B}_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\frac{d_{1}\,B_{0i}\,B_{0j}}{c_{1}-d_{1}\,{\bf
B}_{0}^{2}}\;.$ (29b)

In addition, the eigenvalues of the susceptibility matrices are given by

$\displaystyle\chi_{1E}\\!$ $\displaystyle=$
$\displaystyle\\!\chi_{2E}=0\;,\;\chi_{3E}({\bf
B}_{0})=\frac{d_{2}}{c_{1}}\,{\bf B}_{0}^{2}\;,$ (30a)
$\displaystyle\chi_{1M}\\!$ $\displaystyle=$
$\displaystyle\\!\chi_{2M}=0\;,\;\chi_{3M}({\bf
B}_{0})=\frac{1}{c_{1}-d_{1}\,{\bf B}_{0}^{2}}\;,$ (30b)

where the magnetic susceptibility is positive if we also impose the condition
$c_{1}>d_{1}\,{\bf B}_{0}^{2}$.

Contracting the field eq. (26) with $f_{\nu\alpha}$ and using the Bianchi
identity, we obtain the conservation law

$\displaystyle\partial_{\mu}\Theta^{\mu}_{\;\;\alpha}=0\;,$ (31)

with the following energy-momentum tensor

$\displaystyle\Theta^{\mu}_{\;\;\,\alpha}\\!$ $\displaystyle=$
$\displaystyle\\!c_{1}\,f^{\mu\nu}f_{\nu\alpha}-\frac{d_{1}}{2}\,(F_{B}^{\;\,\,\mu\nu}f_{\nu\alpha})(F_{B\kappa\lambda}f^{\kappa\lambda})$
(32)
$\displaystyle-\frac{d_{2}}{2}\,(\widetilde{F}_{B}^{\;\,\,\mu\nu}f_{\nu\alpha})(\widetilde{F}_{B\kappa\lambda}f^{\kappa\lambda})-\delta^{\mu}_{\;\;\,\alpha}\,{\cal
L}^{(2)}\;.$

Notice that the eq. (31) is due to the uniform magnetic background field. The
conserved components of the energy-momentum tensor are $\Theta^{\mu
0}=\left(\Theta^{00},\Theta^{i0}\right)$ in which the energy density and the
Poynting vector components can be written as

$\displaystyle\Theta^{00}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{2}\,\epsilon_{ij}({\bf
B}_{0})\,e_{i}\,e_{j}+\frac{1}{2}\,(\mu_{ij})^{-1}({\bf
B}_{0})\,b_{i}\,b_{j}\;,$ (33a) $\displaystyle\Theta^{i0}\\!$ $\displaystyle=$
$\displaystyle\\!c_{1}\,({\bf e}\times{\bf b})^{i}+d_{1}\,({\bf e}\cdot{\bf
E})\;({\bf e}\times{\bf B})^{i}$ (33b) $\displaystyle-\,d_{1}\,({\bf
b}\cdot{\bf B})\;({\bf e}\times{\bf B})^{i}-\,d_{2}\,({\bf e}\cdot{\bf
B})\;({\bf e}\times{\bf E})^{i}$ $\displaystyle-d_{2}\,({\bf b}\cdot{\bf
E})\;({\bf e}\times{\bf E})^{i}\,.$

Thereby, the energy density is so positive if it also satisfies the condition
$c_{1}>d_{1}{\bf B}_{0}^{2}$.

The plane wave solutions for the propagating fields ${\bf e}$ and ${\bf b}$
are given in the usual form :

$\displaystyle{\bf e}({\bf r},t)={\bf e}_{0}\,e^{i({\bf k}\cdot{\bf r}-\omega
t)}\;\,\,\mbox{and}\;\,\,{\bf b}({\bf r},t)={\bf b}_{0}\,e^{i({\bf k}\cdot{\bf
r}-\omega t)}\;,$ (34)

in which the electric ${\bf e}_{0}$ and magnetic ${\bf b}_{0}$ amplitudes are
related by the Faraday’s law as ${\bf b}_{0}=({\bf k}\times{\bf
e}_{0})/\omega$, where ${\bf k}$ denotes the wave vector, and $\omega$ is the
frequency. Substituting these solutions in the field equations, the wave
equation of the electric amplitude is

$\displaystyle M_{ij}\,e_{0j}=0\;,$ (35)

in which the matrix elements of $M_{ij}$ are written as

$\displaystyle M_{ij}\\!$ $\displaystyle=$ $\displaystyle\\!\left[1-{\bf
n}^{2}+\frac{d_{1}}{c_{1}}\left({\bf n}\times{\bf
B}_{0}\right)^{2}\right]\delta_{ij}+$ (36)
$\displaystyle+\left(1-\frac{d_{1}}{c_{1}}\,{\bf
B}_{0}^{2}\right)n_{i}\,n_{j}+\left(\frac{d_{2}}{c_{1}}-\frac{d_{1}}{c_{1}}\,{\bf
n}^{2}\right)B_{0i}\,B_{0j}$ $\displaystyle+\frac{d_{1}}{c_{1}}\left({\bf
B}_{0}\cdot{\bf n}\right)\left(B_{0i}\,n_{j}+B_{0j}\,n_{i}\right)\;,$

where we have defined ${\bf n}={\bf k}/\omega$. The null determinant
$\mbox{det}(M)=0$ yields the ${\bf n}$-polynomial equation

$\displaystyle\left[1-{\bf n}^{2}+\frac{d_{1}}{c_{1}}\,({\bf n}\times{\bf
B}_{0})^{2}\right]\times$
$\displaystyle\times\left[\left(1+\frac{d_{2}}{c_{1}}\,{\bf
B}_{0}^{2}\right)(1-{\bf n}^{2})+\frac{d_{2}}{c_{1}}\,({\bf n}\times{\bf
B}_{0})^{2}\right]=0\;,$ (37)

whose solutions provide two possible refractive indices

$\displaystyle n_{1}\\!$ $\displaystyle=$
$\displaystyle\\!\sqrt{\frac{c_{1}}{c_{1}-d_{1}\,({\bf B}_{0}\times\hat{{\bf
k}})^{2}}}\;,$ (38a) $\displaystyle n_{2}\\!$ $\displaystyle=$
$\displaystyle\\!\sqrt{\frac{c_{1}+d_{2}\,{\bf B}_{0}^{2}}{c_{1}+d_{2}\,({\bf
B}_{0}\cdot\hat{{\bf k}})^{2}}}\;.$ (38b)

Using the coefficients from (III), the refractive indices are functions of the
angle $\theta$ between the magnetic background field and wave propagation
direction $\hat{{\bf k}}$. In the limit $\beta\rightarrow\infty$, both the
refractive index go to one. Moreover, in the regime of a strong magnetic field
$(|{\bf B}_{0}|\gg\beta)$, the first solution is $n_{1}\simeq 1$, and the
second refractive index depends only on the $\theta$-angle

$\displaystyle n_{2}\simeq|\sec\theta|\;.$ (39)

For the TaAs Dirac material listed in the table (1), $\beta\rightarrow 0$, and
the refractive index have the same results, $n_{1}=1$ and
$n_{2}=|\sec\theta|$, respectively. In the case of the QED, we consider
$\alpha_{D}=\alpha=0.0073$, and in this approximation, the refractive index
are given by

$\displaystyle n_{1}\\!$ $\displaystyle\simeq$
$\displaystyle\\!1+\frac{\alpha}{24\pi^{2}}\frac{{\bf
B}_{0}^{2}+2\beta^{2}}{({\bf B}_{0}^{2}+\beta^{2})^{2}}\,(\hat{{\bf
k}}\times{\bf B}_{0})^{2}\;,$ (40a) $\displaystyle n_{2}\\!$
$\displaystyle\simeq$ $\displaystyle\\!1+\frac{\alpha}{24\pi^{2}}\frac{|{\bf
B}_{0}|}{\beta}\,\frac{(\hat{{\bf k}}\times{\bf B}_{0})^{2}}{{\bf
B}_{0}^{2}+\beta^{2}}\;.$ (40b)

In the regime of a strong magnetic field, the refraction index in the QED are
reduced to the results :

$\displaystyle n_{1}\\!$ $\displaystyle\simeq$
$\displaystyle\\!1+\frac{\alpha}{24\pi^{2}}\,\sin^{2}\theta\;,$ (41a)
$\displaystyle n_{2}\\!$ $\displaystyle\simeq$
$\displaystyle\\!1+\frac{\alpha}{24\pi^{2}}\frac{|{\bf
B}_{0}|}{\beta}\,\sin^{2}\theta\;.$ (41b)

The first solution does not depend on the magnitude of the magnetic
background, and it is function of the $\theta$-angle. This conclusion shows
that the correspondent dispersion relations does not depend on the magnitude
of the magnetic background field in the regime of a strong background, and it
depends only on the $\theta$-angle.

##  IV Birefringence phenomenon in a uniform magnetic background

We now consider the birefringence phenomenon, which rises from the wave
propagation associated with different amplitudes. In this case, we consider
the magnetic background on the ${z}$-direction, i.e., ${\bf
B}_{0}=B_{0}\,\hat{{\bf z}}$, and in the first situation, the plane wave
solution for the electric field has an amplitude parallel to the magnetic
background, where ${\bf e}({\bf r},t)=e_{03}\,\hat{{\bf z}}\,e^{i({\bf
k}\cdot{\bf r}-\omega t)}$. Using this solution, the wave equation can be
written as

$\displaystyle\left[\,\frac{{\bf
k}^{2}}{\omega^{2}}-\epsilon_{33}(B_{0})\,\mu_{22}(B_{0})\,\right]e_{03}=0\;,$
(42)

in which the correspondent refractive index is

$\displaystyle
n_{\parallel}(B_{0})=\sqrt{\mu_{22}(B_{0})\,\epsilon_{33}(B_{0})}=\sqrt{\,1+\frac{d_{2}}{c_{1}}\,B_{0}^{2}\,}\;.$
(43)

The second situation happens when the plane wave has amplitude perpendicular
to the magnetic background, such that ${\bf e}({\bf r},t)=e_{02}\,\hat{{\bf
y}}\,e^{i({\bf k}\cdot{\bf r}-\omega t)}$. In this case, the wave equation is

$\displaystyle\left[\,\frac{{\bf
k}^{2}}{\omega^{2}}-\epsilon_{22}(B_{0})\,\mu_{33}(B_{0})\,\right]e_{02}=0\;,$
(44)

with the refractive index

$\displaystyle
n_{\perp}(B_{0})=\sqrt{\mu_{33}(B_{0})\,\epsilon_{22}(B_{0})}=\sqrt{\,\frac{c_{1}}{c_{1}-d_{1}\,B_{0}^{2}}\,}\;.$
(45)

The birefringence phenomena rises from the difference

$\displaystyle\Delta n(B_{0})=n_{\parallel}(B_{0})-n_{\perp}(B_{0})\neq 0\;.$
(46)

Therefore, the birefringence does not occur if we constraint the condition

$\displaystyle\mu_{22}(B_{0})\,\epsilon_{33}(B_{0})=\epsilon_{22}(B_{0})\,\mu_{33}(B_{0})\;,$
(47)

and using the $22$ and $33$ components from eqs. (27a) and (27b), we obtain
the relation :

$\displaystyle d_{2}-d_{1}=\frac{d_{1}\,d_{2}}{c_{1}}\,B_{0}^{2}\;.$ (48)

In the limit $\beta\rightarrow\infty$, the birefringence disappears in eq.
(46). For $\beta\gg B_{0}$, this difference is residual in
$(B_{0}/\beta)^{2}$,

$\displaystyle\Delta
n(B_{0})\simeq-\frac{\alpha_{D}}{12\pi^{2}}\left(\frac{B_{0}}{\beta}\right)^{2}\;.$
(49)

The plot of $\Delta n$ versus the dimensionless variable $x=B_{0}/\beta$ is
illustrated in the fig. (2), for the Dirac materials from the table 1 : BiSb
(blue line) and PbSnTe (red line). The lines intercept the horizontal axis
where the birefringence is null in accord with the condition (48). Using the
coefficients (III) and the physical parameters in the table 1, we obtain the
magnitude of the magnetic background field for the QED and the Dirac materials
in which the birefringence is null :

$\displaystyle B_{0}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!4.0\;\mbox{MeV}^{2}=5.6\,\mbox{GT}\hskip
5.69046pt\mbox{for}\hskip 5.69046pt\mbox{QED}\;,$ (50a) $\displaystyle
B_{0}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!8.90\;\mbox{eV}^{2}=12\,\mbox{mT}\hskip
5.69046pt\mbox{for}\hskip 5.69046pt\mbox{PbSnTe}\;,$ (50b) $\displaystyle
B_{0}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!0.18\;\mbox{eV}^{2}=0.25\,\mbox{mT}\hskip
5.69046pt\mbox{for}\hskip 5.69046pt\mbox{BiSb}\;.$ (50c)

Figure 2: The variation of the refractive index $(\Delta n)$ versus the
dimensionless variable $x=B_{0}/\beta$ for the Dirac materials BiSb (blue
line) and PbSnTe (red line).

Otherwise, the condition (46) is kept for values of $B_{0}$ different from the
results (50a)-(50c).

##  V Interaction energy for an electrodynamics in Dirac materials under a
uniform magnetic field

As already expressed, here we calculate the interaction energy between static
point-like sources for an electrodynamics in Dirac materials under a uniform
magnetic field, along the lines of refs. [20, 21, 22, 23, 24]. To do that, we
compute the expectation value of the energy operator $H$ in the physical state
$\left|\Phi\right\rangle$, which we write as $\langle H\rangle_{\Phi}$.

In this perspective, our aim here is to evaluate to the lowest-order the
interaction energy. For this purpose, we linearize the effective theory (7) in
a similar manner to that which led to the eq. (III). Furthermore, in what
follows, we confine our attention to the case of a pure and uniform magnetic
background (${\bf E}={\bf 0}$), as in the section III. In such a case, we back
to the effective Lagrangian density in eq. (III).

We now carry out a Hamiltonian analysis of this theory. The canonical momenta
following from eq. (III) are

${\Pi^{\mu}}=-{c_{1}}{f^{0\mu}}+\frac{{{d_{1}}}}{2}F_{B}^{\;\,\,\rho\sigma}{f_{\rho\sigma}}F_{B}^{\;\,\,0\mu}+\frac{{{d_{2}}}}{2}\,\tilde{F}_{B}^{\;\,\,\rho\sigma}f_{\rho\sigma}\,\tilde{F}_{B}^{\;\,\,0\mu},$
(51)

which produces the usual primary constraint $\Pi^{0}=0$, and
${\Pi_{i}}={c_{1}}\left({{\delta_{ij}}+\frac{{{d_{2}}}}{{{c_{1}}}}\,{B_{0i}}\,{B_{0j}}}\right){e_{j}}$
.

Making use of this last equation, we readily verify that the electric field
associated to the $a^{\mu}$-field takes the form

${e_{i}}=\frac{1}{{{c_{1}}\det D}}\left({{\delta_{ij}}\det
D-\frac{{{d_{2}}}}{{{c_{1}}}}\,{B_{0i}}{B_{0j}}}\right){\Pi_{j}}\,.$ (52)

Here $\det D=1+\frac{{{d_{2}}}}{{{c_{1}}}}\,{{\bf B}_{0}^{2}}$, where ${\bf
B}_{0}$ represents the external (background) magnetic field around which the
$a^{\mu}$-field fluctuates.

The canonical Hamiltonian is now obtained in the usual way by a Legendre
transformation to obtain

$\displaystyle{H_{C}}$ $\displaystyle=$
$\displaystyle\int{{d^{3}}x}\left[\,{{\Pi_{i}}{\partial^{i}}{a_{0}}+\frac{{{{\bf\Pi}^{2}}}}{{2\,{c_{1}}}}+\frac{{{c_{1}}}}{2}\,{{\bf
b}^{2}}-\frac{{{d_{1}}}}{2}\,{{\left({{\bf B}_{0}\cdot{\bf
b}}\right)}^{2}}}\,\right]$ (53) $\displaystyle-$
$\displaystyle\int{{d^{3}}x}\,\frac{{{d_{2}}}}{{2\,c_{1}^{2}\det
D}}\,{\left({{\bf B}_{0}\cdot{\bf\Pi}}\right)^{2}}.$

It can be quickly checked that the preservation in time of the primary
constraint, ${\Pi}_{0}$, leads to the secondary constraint
${\Gamma_{1}}\equiv{\partial_{i}}{\Pi^{i}}=0$ (Gauss constraint). In this
case, there are two constraints, which are first-class ones. According to the
general theory, we obtain the extended Hamiltonian, which generates
translations in time, by adding all the first-class constraints with arbitrary
coefficients to the canonical Hamiltonian. Accordingly, we write

$\displaystyle
H=H_{C}+\int{d^{3}}x\left[\,{c_{0}\left(x\right)\Pi_{0}\left(x\right)+c_{1}\left(x\right)\Gamma_{1}\left(x\right)}\,\right],$
(54)

where $c_{0}\left(x\right)$ and $c_{1}\left(x\right)$ are arbitrary functions
of the space-time coordinates. Since $\Pi^{0}=0$ always, neither $a^{0}$ nor
$\Pi^{0}$ are of interest in describing the system and may be excluded from
the theory. Thus the variables $a^{0}$ and $\Pi^{0}$ play no role in the
theory. If a new arbitrary coefficient $c(x)=c_{1}(x)-a_{0}(x)$ is introduced
the Hamiltonian may be rewritten in the form

$\displaystyle H$ $\displaystyle=$
$\displaystyle\int{{d^{3}}x}\left[\,{c\left(x\right){\partial^{i}}{\Pi_{i}}+\frac{{{{\bf\Pi}^{2}}}}{{2\,{c_{1}}}}+\frac{{{c_{1}}}}{2}\,{{\bf
b}^{2}}-\frac{{{d_{1}}}}{2}{{\left({{\bf B}_{0}\cdot{\bf
b}}\right)}^{2}}}\,\right]$ (55) $\displaystyle-$
$\displaystyle\int{{d^{3}}x}\,\frac{{{d_{2}}}}{{2\,c_{1}^{2}\det
D}}\,{\left({{\bf B}_{0}\cdot{\bf\Pi}}\right)^{2}}.$

Since there is one first constraint, ${\Gamma_{1}}$, we choose a gauge-fixing
condition that will make the full set of constraints to become second class.
To do this, we adopt the gauge discussed previously [20], that is,

$\Gamma_{2}\left(x\right)\equiv\int\limits_{C_{\zeta
x}}{dz^{\nu}}\,a_{\nu}\left(z\right)\equiv\int\limits_{0}^{1}{d\lambda\,x^{i}}a_{i}\left({\lambda
x}\right)=0.$ (56)

Here $\lambda$ $(0\leq\lambda\leq 1)$ is the parameter describing the space-
like straight path $x^{i}=\zeta^{i}+\lambda\left({x-\zeta}\right)^{i}$, and
$\zeta^{i}$ is a fixed point (reference point). There is no essential loss of
generality if we restrict our considerations to $\zeta^{i}=0$. With such a
choice, the fundamental Dirac bracket is given by

$\displaystyle\left\\{{a_{i}\left({\bf x}\right),\Pi^{j}\left({\bf
y}\right)}\right\\}^{\ast}\\!$ $\displaystyle=$
$\displaystyle\\!\delta_{i}^{\;\,j}\,\delta^{\left(3\right)}\left({{\bf
x}-{\bf y}}\right)$ (57)
$\displaystyle-\partial_{i}^{x}\int\limits_{0}^{1}{d\lambda\,x^{j}}\delta^{\left(3\right)}\left({\lambda{\bf
x}-{\bf y}}\right).$

We now recall that the physical states $\left|\Phi\right\rangle$ are gauge-
invariant. In this case, we may therefore write the stringy gauge-invariant
state

$\displaystyle\left|\Phi\right\rangle\\!$ $\displaystyle\equiv$
$\displaystyle\\!\left|\,{\overline{\Psi}\left({\bf y}\right)\Psi\left({{{\bf
y}^{\prime}}}\right)}\,\right\rangle$ (58) $\displaystyle=$
$\displaystyle\overline{\Psi}\left({\bf y}\right)\exp\left[{iq\int_{{{\bf
y}^{\prime}}}^{\bf y}{d{z^{i}}{a_{i}}\left(z\right)}}\right]\Psi\left({{{\bf
y}^{\prime}}}\right)\left|0\right\rangle,$

where the line integral is along a space-like path on a fixed time slice, $q$
is the fermion charge and $\left|0\right\rangle$ is the physical vacuum state.

At this point, we have all the elements necessary for the calculation of the
expectation value ${\left\langle H\right\rangle_{\Phi}}$. From the foregoing
discussion we readily see that ${\left\langle H\right\rangle_{\Phi}}$ can
therefore be written as follows

${\left\langle H\right\rangle_{\Phi}}={\left\langle
H\right\rangle_{0}}+\left\langle H\right\rangle_{\Phi}^{\left(1\right)},$ (59)

where ${\left\langle H\right\rangle_{0}}=\left\langle
0\right|H\left|0\right\rangle$, whereas the $\left\langle
H\right\rangle_{0}^{\left(1\right)}$ term is given by

$\left\langle
H\right\rangle_{\Phi}^{\left(1\right)}=\left\langle\Phi\right|\int{{d^{3}}x}\left[{\frac{{{{\bf\Pi}^{2}}}}{{2\,{c_{1}}}}-\frac{{{d_{2}}}}{{2\,c_{1}^{2}\det
D}}{{\left({{\bf
B}_{0}\cdot{\bf\Pi}}\right)}^{2}}}\right]\left|\Phi\right\rangle,$ (60)

Following our earlier procedure [23, 24], the static potential profile for two
opposite charges located at ${\bf y}$ and ${\bf y}^{\prime}$ then reads

$V(L)=-\frac{{{q_{eff}^{2}({\bf B}_{0})}}}{{4\pi}}\frac{1}{L}\;,$ (61)

where the effective charge is

$\displaystyle q_{eff}({\bf B}_{0})=\frac{|q|}{\sqrt{c_{1}+d_{2}\,{{\bf
B}_{0}^{2}}}}\;,$ (62)

and $L\equiv|{\bf y}-{{\bf y}^{\prime}}|$ is the distance that separates the
two charges. We see, therefore, that at the lowest-order in our effective
model, the introduction of the non-linearities induce redefinition in the
charges. Whenever $|{\bf B}_{0}|\rightarrow 0$, the effective charge $q_{eff}$
reduces to the usual $q$-charge. This result agrees with the absence of
screening in the Coulomb potential that is a characteristic of Dirac
materials. Under an intense magnetic background, $q_{eff}$ goes to zero, and
consequently, the potential (61) signals a situation of free particles. The
dimensionless quantity $\tilde{q}=q_{eff}/|q|$ is plotted as function of
$B_{0}/\beta$ in fig. (3), for the QED case (black line) and the Dirac
materials BiSb (blue line) and PbSnTe (red line).

Figure 3: The dimensionless effective charge $\tilde{q}$ as function of the
ratio $B_{0}/\beta$ (magnetic background field over critical field) for the
QED (black line), and for the Dirac materials BiSb (blue line) and PbSnTe (red
line).

The QED curve is approximately horizontal since the contribution of the fine
structure constant is very weak in this case. When the fine structure constant
increases (that are the cases of the BiSb and PbSnTe), there arises a peak
showed in the blue and red lines, respectively. In the case of the PbSnTe, the
peak is localized at magnetic background $B_{0}=4.70\,\mbox{eV}^{2}$, where
the effective charge is maximum at $q_{eff}=0.3035$ for two electrons
interacting under the potential (61).

##  VI Concluding comments and considerations

A non-linear electrodynamics (ED) similar to a non-linear logarithmic model is
proposed in this paper as a possible formulation from which there follows the
effective Lagrangian density in ref. [3], obtained from the one-loop radiative
corrections for Dirac materials. The non-linear ED under consideration reduces
to the situation presented in ref. [3] in the regime of a strong magnetic
field related to electric field. The non-linear model introduces a critical
electric field as a parameter that depends on the characteristics of the
particular Dirac material. The usual Maxwell electrodynamics is recovered in
the limit in which the critical electric field goes to infinity. Using the
non-linear Lagrangian density, we obtain the non-linear field equations, the
permittivity and permeability tensors, and the correspondent energy momentum
tensor of this theory. Afterwards, we study the aspects of the model when the
system is submitted to an external (and uniform) magnetic field. We expand the
non-linear Lagrangian up to fourth order in the propagating electromagnetic
field. Thereby, the permittivity and permeability tensors are obtained in
terms of the background magnetic field components. The refractive index are
calculated as function of the critical field ($\beta$-parameter) of this non-
linear theory, of the magnetic background, and also it depends on the angle
$(\theta)$ in that the propagation direction does with magnetic background. In
the regime of a strong magnetic background, one of the solutions for the
refractive index depends only on the $\theta$-angle. We investigate the
birefringence phenomenon for the wave electric field propagating in this non-
linear theory under a uniform magnetic background. We calculate numerically
the magnetic background values for which birefringence does not take place; to
do that, we must use physical characteristics of the particular Dirac
material.

To conclude, we apply the path-dependent formalism to the linearized
Lagrangian to obtain the interaction energy between two static point-like
charges. There results a Coulomb-like potential with an effective charge that
runs with the magnetic background field. For the PbSnTe Dirac material, the
effective charge has a smooth peak at a magnetic field of
$4.70\,\mbox{eV}^{2}$, as illustrated in fig. 3.

Taking a forward-looking perspective, as a possible path to further exploit
the effective photonic model in discussion here, we believe that a natural
follow-up would be the inspection of the regime of vortex formation and the
consequent study of the properties of the magnetic vortices governed by the
set of the non-linear electromagnetic field equations. As a consequence of the
non-linearity, charged vortices might naturally arise and investigating the
dynamics of fermions in presence of such vortices may be a research question
of interest in the context of Dirac materials.

##  VII Acknowledgments

L.P.R. Ospedal is grateful to FAPERJ for his post-doctoral fellowship. P.
Gaete was partially supported by ANID PIA/APOYO AFB220004.

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