# Nonlinear electrodynamics for the vacuum of Dirac materials. Photon magnetic
properties and radiation pressures

A. W. Romero Jorge  adrian<EMAIL_ADDRESS>Instituto de Cibernética, Matemática y Física (ICIMAF),
Calle E esq a 15 Vedado 10400 La Habana Cuba  Frankfurt Institute for Advanced
Studies, Ruth Moufang Str. 1, 60438 Frankfurt, Germany  Helmholtz Research
Academy Hessen for FAIR (HFHF), GSI Helmholtz Center for Heavy Ion Physics.
Campus Frankfurt, 60438 Frankfurt, Germany  Institut für Theoretische Physik,
Johann Wolfgang Goethe University, Max-von-Laue-Str. 1, 60438 Frankfurt,
Germany  A. Pérez Martínez<EMAIL_ADDRESS>Instituto de Cibernética,
Matemática y Física (ICIMAF),
Calle E esq a 15 Vedado 10400 La Habana Cuba  E. Rodríguez Querts
<EMAIL_ADDRESS>Instituto de Cibernética, Matemática y Física (ICIMAF),
Calle E esq a 15 Vedado 10400 La Habana Cuba

(August 30, 2024)

###### Abstract

We investigate the magnetic properties of photons propagating through Dirac
materials in the presence of a magnetic field, taking into account both medium
and vacuum contributions. The vacuum photon propagation properties are
obtained through a second-order expansion of nonlinear Euler-Heisenberg
electrodynamics considering Dirac material parameters (effective structure
constant, band gap and Fermi velocity, respectively). Total magnetization and
effective magnetic moment are obtained. Observables such as energy density,
radiation pressure, and Poynting vector are getting by an average of
components of the energy-momentum tensor. All these quantities are expressed
in terms of Lagrangian derivatives and are valid for arbitrary values of the
external magnetic field. The weak and strong field limits are recovered for
all the quantities. We discuss some ideas of experiments that may contribute
to testing in Dirac materials the phenomenology of the strong magnetic field
in the Quantum Electrodynamic’s vacuum.

##  I Introduction

Maxwell’s theory predicts that plane-polarized light propagates in a vacuum:
empty space with the speed of light $c$. Quantum Electrodynamics (QED) has
another picture of the vacuum, according to this theory, the vacuum is not
really an empty space, but it is a ”sea” of virtual electron-positron pairs
whose negative energy states occupied and unoccupied the positive ones [1].
This picture can explain, how in the presence of an external magnetic field,
the speed of light propagating parallel to the magnetic field remains c while
if the propagation is perpendicular, light changes its plane of polarization.
The resulting incoming wave splits in two polarized modes moving with
different speeds (and frequencies). This phenomenon is a consequence of the
interaction of the magnetic field with the virtual electron-positron pairs it
is well-known as the Cotton-Mouton birefringence [2, 3].

Illuminated by the Quantum field theories vacuum concept becomes in the ground
state of the theories, being for QED the virtual pairs of electron-positrons,
and for Quantum Chromodynamics vacuum virtual quarks-antiquarks, etc. In
summary, the shift from Maxwell’s vacuum to Quantum Electrodynamics’ dynamic
’sea’ of virtual particle pairs redefines its conceptual meaning and trying to
understand its properties and complexity.

The vacuum of QED in the presence of a strong electromagnetic field behaves as
a medium, virtual pairs interact with magnetic fields. Thereby, the properties
of light propagating in a QED-vacuum may be studied using the tools of non-
linear optics, based on Maxwell’s theory with all the magnitudes described in
terms of a medium electric permittivity and magnetic permeability depending on
the non-linear of the external electric and magnetic field111Vacuum electric
permittivity and magnetic permeability are related with the speed of light
[4]. Besides birefringences222In the presence of strong electric field the
birefringence also appears, and it is called Kerr-birefringence, QED predicts
other exotic properties, Casimir effect, vacuum instability (close to the
critical field, $E_{c}=\frac{m^{2}c^{3}}{e\hbar}=1.3\times 10^{18}$ V/m,
electron-positron pairs creation appear [1, 5, 6]), anisotropy pressures, etc
[7]. All these vacuum phenomenology predicted 90 years ago are still waiting
for experimental confirmation.

Testing the vacuum properties of the QED requires large magnetic and/or
electric fields. So far, there are technological barriers that limit the
generation of high magnetic/electric fields in the laboratory. Magnetic fields
achieved in laboratories do not exceed a few teslas ($10^{4}$ G). For
instance, vacuum birefringence, even when manifests for any strength of the
magnetic field, at weak field limit, has a quadratic dependence of the ratio
between magnetic field B and the parameter $\xi$, i.e, $(B/\xi)^{2}$ with
$\xi=\frac{8\alpha^{2}}{45B_{c}}$ and $B_{c}=\frac{m^{2}c^{3}}{e\hbar}$
yielding a very tiny theoretical prediction of $\Delta n_{theo}=10^{-22}$,
turning into a challenge to measurement it. Precisely, the experiment
Polarization of Vacuum with Laser (PVLAS) [8] constructed to pursue vacuum
magnetic birefringence could not confirm the prediction, obtaining a higher
bound $\Delta n_{PVLAS}\sim 10^{4}\Delta n_{theo}$. PVLAS has blamed the
axions, as responsible for the difference between experiment and theoretical
results, and they concluded that there is new physics beyond QED.

But in the Universe, we can find entities like Neutron stars, whose magnetic
fields are close to the critical field, and even more in the case of magnetars
[9, 10, 11]. Short-time huge magnetic fields may also be found in heavy ion
colliders, which can reach the order of $10^{18}$ G [12]. In both scenarios,
neutron stars and heavy ions colliders have reported indirect vacuum magnetic
birefringence [10, 11]. The work [10] claims the first evidence of vacuum
birefringence by measuring the degree and angle of polarization of the photons
coming from the pulsar RX J1856.5-4 and comparing them with the theoretical
results, but the uncertainties in their measurements are large enough to fully
trust in this measurement.

These observations have boosted the search directly for the phenomenon based
on other experimental setups. One plausible alternative is the design of
experiments with scattering of the pulsating laser [13]. It is predictable
that with the developed technology of lasers, in the near future, the power of
lasers will reach an intensity around $I\sim 10^{33}$ Wm-2 that corresponds to
an electric (magnetic) field close to the critical field, which may allow
detection easier the effect of the birefringence of vacuum and/or to test the
pair’s creation from the vacuum.

Besides, another exciting and new alternative to probe vacuum properties would
be experiments designed with Dirac materials due to their critical fields are
$E_{c}=\frac{\Delta^{2}}{ev_{F}}\sim 10^{5}$ V/m,
$B_{c}=\frac{\Delta^{2}}{ev_{F}^{2}}\sim 10^{4}\,\text{G}=1$ T, accessible to
get in laboratories. A very hopeful work [14] has studied the contribution to
the magnetization of the vacuum for three different Dirac materials and the
possibility of testing in experiments.

Dirac materials, including Weyl metals, topological materials, and graphene
are characterized by electrons that behave as relativistic fermions with a
linear dispersion relation. This leads to unique optical, magnetic, and
transport properties for these materials. The pioneer theoretical work of the
Dirac materials is [15], but the boom of these materials only appeared with
the amazing finding of graphene by Novoselov [16]. This experiment led to an
onrush of experimental and theoretical works to look for other similar
materials and/or other properties and consequently, technological
applications.

The discovery of Dirac materials has been gratifying news for the Quantum
Field Theory because it has opened the window to test in top-table
experiments, theories, and phenomena that would require a high scale of
energies and expensive experiments in particle accelerators. Consequently, the
tools of Quantum Field theory have been extended to condensed matter physics.
On track of this direction the study of the photon propagation parallel to the
external magnetic field for electron-positron plasma [17, 18, 19] was extended
to the graphene-like system, and the study of Faraday and Quantum Hall effect
[20, 21, 22]. The study was done using previous calculations [17, 18, 19] of
the one-loop studies of 3D electron-positron plasma in a magnetized medium,
doing a ”dimensional reduction” to 2D and considering also the properties of
the graphene-like systems.

In this paper we proceed similarly extrapolating to Dirac materials physics,
the method and results of the study of photon propagation transverse to the
external magnetic field. We use Euler Heisenberg (EH) non-linear
electrodynamics to study the vacuum electron positron-plasma. In particular,
we use an approximated version of EH Lagrangian which corresponds to an
expansion up to second order on the photon field [23] in the limit of
$\omega\ll 2mc^{2}$, where $m$ is the electron mass. From this Lagrangian, we
investigate the magnetic and dielectric properties of a photon traveling in
the Dirac vacuum as well as the observables, energy densities and pressures,
and Poynting vector, that result from the energy-momentum tensor. The
properties of the medium Dirac materials are considered for the study of the
contribution to total magnetization. We used previous calculations of
magnetization of electron-positron plasma and we have translated them into the
language of Dirac materials.

Concerning the treatment of the Dirac vacuum, our study has at least two
novelties that deserve to be highlighted. The first one is that all properties
of the photon are obtained in terms of the effective Lagrangian derivatives,
so they can extend to other non-linear electrodynamics. Besides, the
calculations are valid for arbitrary values of the magnetic field, making it
possible to recover the weak and strong limits.

Our work is also important not only for the interest that awakens the magnetic
properties of Dirac materials (including the vacuum properties) but also it
eventually testing the properties of the strong field QED regime. As we have
already commented, critical fields for the Dirac materials are experimentally
reachable. Hence, these materials could imitate the birefringence, anisotropic
pressures, and other exotic properties of the vacuum of QED at the regime of
the strong magnetic field.

The paper is organized as follows, in section I a brief overview of Dirac
materials characteristics is illustrated. Section II we present the second
order in the photon field of the Euler Heisenberg Lagrangian, in section III
we present the solution of the Maxwell equation and dispersion equation of
photon propagating in the vacuum Dirac. Section IV is devoted to studying the
magnetic properties and magnetic moment of the photon. In section V the
energy-momentum tensor is presented and the observables like energy density
pressures and Poynting vector are obtained. Finally, we give a conclusion
about the work and in the appendixes, some calculations are presented.

##  II A brief overview of Dirac materials characteristics

The crystalline structure of Dirac materials is characterized by a ”Dirac
points” or well, with a small band gap around them, the electrons behave like
relativistic particles with free electrons as Dirac fermions exhibiting an
extremely high Fermi speed, about $10^{6}$ meters per second and electrons
could be described by a linear relationship according to $\epsilon\approx
v_{F}p$ between the energy and momentum of its electrons in the conduction and
valence band for low energies [14, 24]. Dirac materials with zero mass (gap)
are known as Weyl fermions and the very well-known is graphene [25]. This one,
a highly conductive material with no energy gap in its Dirac points
$(\Delta=0)$, it is distinguished from conventional semiconductors by its
absence of this gap $(\Delta\neq 0)$, which limits its use in devices that
require precise control of electron flow, such as transistors and diodes.
Unlike semiconductors, which can turn current flow on and off by excitation
and the relaxation of electrons between the valence band and the conduction
band, graphene maintains a constant flow due to a lack of gap. This property
makes graphene ideal for applications that demand high electrical
conductivity, such as high-speed electronic devices [26], but limit their
useful in controllable electronic components.

To deal with this limitation, graphene-derived structures have been developed,
such as graphene ”nanoribbon” or within a substrate, which can have
controllable energy gaps and, therefore, be used in electronics applications
[27, 28, 29, 30]. These energy gaps can be controlled by manipulating the
width of the graphene ribbon and introducing defects or doping, among other
factors. By employing these techniques, small energy gaps are observed,
ranging approximately from 100 meV to 250 meV [31, 27, 28, 29, 30]. In our
study, we consider a non-zero gap graphene energy with an energy gap of 100
meV [32].

Formally, graphene is not a 2D Dirac semimetal actually, with the top of the
valence and bottom of the conduction band of a band insulator just touch,
before gapping up again after band inversion, a graphene-like semimetallic
state results. Besides, graphene and topological insulators which have 2D
Dirac surface states, there are 3D semimetal with a Dirac point in the bulk 3D
Brillouin zone linear dispersion in all three $k$ directions. The theoretical
prediction and the experimental proves in Bi-based materials did not take long
[33, 34, 35, 36]. These materials are also called (3D) Dirac semimetals.

Our study will be focused on these materials considering a simple model for
them. We know that the crystal structure of these materials is sufficiently
complicated with various nodes, but we consider in the first approximation
that $1)$ all nodes are equal. On the other hand, the value of the energy gap
of each material varies depending on the node and properties such as
temperature, but $2)$ we assume an energy gap independent of all the
parameters of the material, that is, a constant value of gap and $3)$ we
neglect the contribution to magnetization due to material impurities. In Table
4.1 we listed these values for the materials of our interest. These values are
based on experimental measurements or an average of gap values [14]. Each
material will be defined only by its Fermi speed and its energy gap, so there
can be several materials with the same characteristics. We are referring, for
example, to the graphene-type material that has its Fermi speed and energy gap
given, but it can be another material with the same characteristics.

With all these approximations, we will proceed to study the propagation of
photons in 3D Dirac materials extrapolating the results of QED, to Dirac
materials. It means shifting the values of the critical fields in Dirac
materials are defined as [14] [24]

$E_{c}(\Delta,v_{F})=v_{F}B_{c}(\Delta,v_{F})=\left(\dfrac{\Delta}{\alpha_{D}\lambda_{D}^{3}}\right)^{1/2}=\dfrac{\Delta^{2}}{e\hbar
v_{F}},$ (1)

where the speed of light has been replaced by the Fermi velocity of the
material ($c\rightarrow v_{F}\approx 10^{6}$ m/s), the fine structure constant
has been changed to an effective or Dirac fine structure constant
($\alpha\rightarrow\alpha_{D}$) defined as $\alpha_{D}=e^{2}/(\hbar v_{F})\sim
1-4$, $\lambda_{D}$ is the Dirac’s Compton wavelength defined by
$\lambda_{D}=\hbar v_{F}/\Delta$. Here $\Delta$ is the band gap or energy gap
of the material and is defined as $\Delta=m^{*}mv_{F}^{2}$, where $m^{*}\sim
0.01-0.5$ is the effective mass of electrons and holes [14][24].

|  $\Delta$(meV) | $\alpha_{D}/\alpha$ |  $B_{c}$ (T) | $b\leq 3\pi/\alpha_{D}$
---|---|---|---|---
QED | $10^{9}$ | 1 | $4.4\times 10^{9}$ | $1291$
$Pb_{1-x}Sn_{x}Te$ | $31.5$ | 580 | $5.6$ | $2.22$
$Bi_{1-x}Sb_{x}$ | $7.75$ | 188 | $0.036$ | $6.86$
$Ta_{3}As_{4}$ | 21 | 357 | $0.95$ | $3.61$
graphene | 100 | 301 | $15.4$ | $4.28$

Table I. Comparison of QED and four Dirac materials for different parameters
like band gap $(\Delta)$, Dirac structure constant ($\alpha_{D}$) over
$\alpha$, critical magnetic field ($B_{c}$), and bound of the dimensionless
magnetic field imposed to ensure the validity of non-linear electrodynamics
$b=B/B_{c}$. Some values of QED are listed as a reference and the Dirac
materials: $Pb_{1-x}Sn_{x}Te$ [37, 38, 14, 24], $Bi_{1-x}Sb_{x}$ [39, 38, 24,
14], $Ta_{3}As_{4}$ [40, 14, 24], and graphene[16, 21, 32].

Table I shows a comparison of QED and four materials that we use in our study:
graphene, tantalum arsenide ($Ta_{3}As_{4}$), bismuth antimony
($Bi_{1-x}Sb_{x}$), and lead-tin-tellurium ($Pb_{1-x}Sn_{x}Te$). Also, we have
listed the maximum allowed values of the magnetic field for each material to
get real corrections to phase velocity in the strong field limit
$v_{f}^{2}=1-\dfrac{\alpha_{D}}{3\pi}\dfrac{B_{e}}{B_{c}}>0$. This bound
ensures the validity of one-loop approximation and the effective non-linear
electrodynamics[41].

##  III NLED for Dirac materials, Lagrangian

The Lagrangian of QED in the presence of an electromagnetic field is given by
the expression

$\mathcal{L}(\bar{\psi},\psi,A_{\mu})=\bar{\psi}({\ooalign{${\partial}$\cr$/$\cr}}-e\not{A}_{\mu}-m)\psi+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$
(2)

where $\bar{\psi}$, $\psi$ are the fermion spinors,
${\ooalign{$\partial$\cr$/$\cr}}=\gamma^{\nu}\partial_{\nu}$,
$\not{A}_{\mu}=\gamma^{\mu}A_{\mu}$, $\gamma_{\mu}$ are the Dirac matrices,
$A_{\mu}$ is the cuatri-potential vector of the electromagnetic field and
$F_{\mu\nu}$ is the electromagnetic tensor.

If we integrate the electron contribution to the phase amplitude we define the
amplitude as $Z=\int{\mathcal{D}}A_{\mu}e^{i\int
d^{4}x{\mathcal{L}}_{eff}(A_{\mu})}$ and all orders of the one-loop
approximation of the photon-photon interaction processes are enclosed in the
so-called Heisenberg-Euler Lagrangian density [1][42],

$\displaystyle\mathcal{L}^{EH}=-\mathcal{F}-\frac{1}{8\pi^{2}}\int_{0}^{i\infty}\frac{ds}{s^{3}}e^{-m^{2}s}$
$\displaystyle\times\left[(es)^{2}\tilde{a}\tilde{b}\coth(e\tilde{a}s)\cot(e\tilde{b}s)-\frac{(es)^{2}}{3}\left(\tilde{a}^{2}-\tilde{b}^{2}\right)-1\right],$
(3)

which is gauge invariant and finite after renormalization (see appendix in[43,
6]) where
$\tilde{a}=\left[\left(\mathcal{F}^{2}+\mathcal{G}^{2}\right)^{1/2}+\mathcal{F}\right]^{1/2}$
y $\tilde{b}=\left[\left(\mathcal{F}^{2}+\mathcal{G}^{2}\right)^{1/2}-\right.$
$\mathcal{F}]^{1/2}$, $e$ and $m$ are the electron charge and mass
respectively. $\mathcal{F},\mathcal{G}$ are secular invariants derived from
the gauge and Lorentz invariants of the generic electromagnetic fields $({\bf
E,B})$ defined as

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}F_{\mu\nu}=\frac{1}{2}(-\epsilon_{0}E^{2}+\frac{B^{2}}{\mu_{0}}),$
(4) $\displaystyle\mathcal{G}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}\tilde{F^{\mu\nu}}=\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}\boldsymbol{(-E.B)},$
(5)

where $\mu_{0}$ and $\epsilon_{0}$ are electrical permittivity and magnetic
permeability. Along the paper, by simplicity, we take
$\mu_{0}=\epsilon_{0}=1$. $E_{i}$ and $B_{i}$ are the components of
electromagnetic tensors defined as $E_{i}=F_{0i}$,
$B_{i}=-\frac{1}{2}\epsilon_{ijk}F^{jk}$ with $i=1,2,3$,
$\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$ is the dual
tensor, and $\epsilon^{\mu\nu\alpha}$ and $\epsilon^{\mu\nu\alpha\beta}$ are
the totally antisymmetric Levi-Civita tensors of rank 3 and 4, respectively.
We consider only the presence of an external magnetic field, which means that
$\mathcal{F}=B^{2}/2$ and $\mathcal{G}=0$ ($E_{e}=0$).

As we are interested in studying the photon propagation transverse to a
constant external magnetic field using the EH non-linear electrodynamics we
use the prescription introduced in [23] for potential fields and
electromagnetic tensor. We assume that $A_{\mu}=A_{\mu}^{ext}+a_{\mu}$, where
$a_{\mu}$ is the cuatri-potential of the photon and $A^{ext}_{\mu}$, is the
potential associated with the external and constant magnetic field
$\bf{B}_{e}$. The electromagnetic field tensor becomes in
$\mathcal{F}_{\mu\nu}=f_{\mu\nu}+\mathcal{F}^{B}_{\mu\nu}$,
$f_{\mu\nu}=\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu},$ with
$F^{B}_{\mu\nu}=\partial^{\mu}A^{ext,\nu}-\partial^{\nu}A^{ext,\mu}$ and
$\bf{B}_{tot}=B_{e}+B_{w}$ and $\bf{E}=E_{w}$. Then, an expansion of the EH
Lagrangian up to the second order of the photon field can be done (see details
for NLED for QED in [23]). The coefficients of the expansion are Lagrangian
derivatives function of the external magnetic field
$\mathcal{L}_{\mathcal{F}}=\frac{\partial\mathcal{L}}{\partial\mathcal{F}}|_{f=0}$,
$\mathcal{L}_{\mathcal{FF}}=\frac{\partial^{2}\mathcal{L}}{\partial{\mathcal{F}}^{2}}|_{f=0}$
and
$\mathcal{L}_{\mathcal{GG}}=\frac{\partial^{2}\mathcal{L}}{\partial{\mathcal{G}}^{2}}|_{f=0}$
(details of calculation can be seen in [23]-[41]).

This procedure can be adapted to study the propagation of photons in Dirac
materials in the presence of the external magnetic field. In that case, the
effective Lagrangian of a photon interacting with the magnetic field would
take the form

$\displaystyle\mathcal{L}_{D}^{\rm(ph-B)}=-\frac{1}{4}(1-{\mathcal{L}}_{\mathcal{F}})f^{\mu\nu}f_{\mu\nu}+\frac{\mathcal{L}_{\mathcal{FF}}}{8}(f^{\mu\nu}F_{\mu\nu})^{2}+\frac{\mathcal{L}_{\mathcal{GG}}}{8}(f^{\mu\nu}\tilde{F}_{\mu\nu})^{2},$
(6)

where $\mathcal{L}^{(\rm ph-B_{e})}$ only contains information about the
interaction of photons with magnetic fields, and we have neglected quartic
terms related to photon and magnetic field self-interaction [23].

Writing the effective Lagrangian in terms of the photon fields $({\bf E}_{w}$,
${\bf B}_{w})$ it has the expression

$\displaystyle\mathcal{L}_{D}^{\rm(ph-B)}$
$\displaystyle=\frac{(1-\mathcal{L}_{\mathcal{F}})}{2}(E_{w}^{2}-B_{w}^{2})+\frac{\mathcal{L}_{\mathcal{FF}}}{2}({\bf
B_{e}}\cdot{\bf B}_{w})^{2}+\frac{\mathcal{L}_{\mathcal{GG}}}{2}({\bf
B_{e}}\cdot{\bf E}_{w})^{2}.$ (7)

In the limit $E\rightarrow 0$, i.e. $\tilde{b}\rightarrow 0$ the scalars
$\mathcal{L}_{\mathcal{F}}$, $\mathcal{L}_{\mathcal{FF}}$, and
$\mathcal{L}_{\mathcal{GG}}$ becomes in regularized integrals dependent on
arbitrary values of the external magnetic field [23, 41, 44, 45]. They are
quantum corrections proportional to fine structure constant $\alpha_{D}$.
Their explicit forms are

$\displaystyle\mathcal{L}_{\mathcal{F}}=\frac{\partial\mathcal{L}^{EH}}{\partial\mathcal{F}}$
$\displaystyle=-\frac{\alpha_{D}}{2\pi\mu_{0}}\left(\frac{1}{3}+2h_{D}^{2}-8\zeta^{\prime}(-1,h_{D})+4h_{D}{\rm
ln}\Gamma(h_{D})-2h_{D}{\rm ln}h_{D}+\frac{2}{3}{\rm ln}h_{D}-2h_{D}{\rm
ln}2\pi\right),$
$\displaystyle\mathcal{L}_{\mathcal{FF}}=\frac{\partial^{2}\mathcal{L}^{EH}}{\partial^{2}\mathcal{F}}$
$\displaystyle=\frac{\alpha_{D}}{2\pi\mu_{0}B^{2}}\left(\frac{2}{3}+4h_{D}^{2}\psi(1+h_{D})-2h_{D}-4h_{D}^{2}-4h_{D}{\rm
ln}\Gamma(h_{D})+2h_{D}{\rm ln}2\pi-2h_{D}{\rm ln}h_{D}\right),$
$\displaystyle\mathcal{L}_{\mathcal{GG}}=\frac{\partial^{2}\mathcal{L}^{EH}}{\partial^{2}\mathcal{G}}$
$\displaystyle=\frac{\alpha_{D}}{2\pi\mu_{0}B^{2}}\left(-\frac{1}{3}-\frac{2}{3}\left(\psi(1+h_{D})-2h_{D}^{2}+(3h_{D})^{-1}\right)+8\zeta^{\prime}(-1,h_{D})\right.$
$\displaystyle\left.-4h_{D}{\rm ln}\Gamma(h_{D})+2h_{D}{\rm ln}2\pi+2h_{D}{\rm
ln}h_{D}\right),$ (8)

where $h_{D}=\frac{B_{c}}{2B_{e}}$, $\psi$ denotes the PolyGamma function
(first derivative of ${\rm ln}\,\Gamma$) and $\zeta^{\prime}$ is the first
derivative of the Hurwitz zeta function with respect to the first argument.

Let us remark that all this treatment used for transverse propagation photons
to the constant magnetic field can be straightforwardly extended to the study
of photon propagation in a pure background electric field and/or in the
background of an orthogonal electric and magnetic field, with
$h_{D}\rightarrow\frac{m^{2}}{2e\sqrt{2\mathcal{F}}}$.

##  IV Dispersion equation of photon propagating in a Dirac vacuum

Let us study the motion equation and the dispersion equation of the photon
using the minimum action principle over the Lagrangian (6). The modified
Maxwell equation in terms of constitutive vectors $\mathbf{D}_{w}$ and
$\mathbf{H}_{w}$ are

$\displaystyle\frac{\partial\mathbf{D}_{w}}{\partial
t}=-\nabla\times\mathbf{H}_{w},\quad\nabla\cdot\mathbf{D}_{w}=0,$ (9)

with

$\displaystyle D_{w,i}$ $\displaystyle=\frac{\partial}{\partial
E_{w,i}}[\mathcal{L}^{\rm(ph-B)}]=\epsilon_{ij}E_{w,j}=E_{w,i}+P_{w,i},$ (10)
$\displaystyle H_{w,i}$ $\displaystyle=-\frac{\partial}{\partial
B_{w,i}}[\mathcal{L}^{\rm(ph-B)}]=(\mu^{-1})_{ij}{B}_{w,j}=B_{w,i}-M_{w,i},$
(11)

and a second pair for ${\bf E}_{w}$ and ${\bf B}_{w}$

$\displaystyle\nabla\cdot\mathbf{B}_{w}=0,\quad\frac{\partial\bf{B}_{w}}{\partial
t}=-\nabla\times\mathbf{\bf{E}}_{w},$ (12)

with $i=1,2,3$. Similar to an optical medium, ${\bf P}_{w}$, and ${\bf M}_{w}$
are the resulting polarization and magnetization of the photon probe due to
the magnetized vacuum. Note that the presence of an arbitrary strength
external magnetic field will impact not only the electric permittivity and
magnetic permeability tensors, $\epsilon(B)$ and $\mu^{-1}(B)$, but also
Maxwell equation solutions for the photon fields in ${\bf D}_{w}$ and ${\bf
H}_{w}$ thus affecting its propagation in a magnetized vacuum with explicit
expressions for the $\epsilon$ and $\mu^{-1}$. They can be obtained in terms
of Lagrangian derivatives as

$\displaystyle\epsilon_{\bot}=\mu_{\bot}=1-\mathcal{L}_{\mathcal{F}},\quad\quad\epsilon_{\parallel}=(1-\mathcal{L}_{\mathcal{F}}+2{\mathcal{F}}\mathcal{L}_{\mathcal{GG}}),$
(13)
$\displaystyle\mu_{\parallel}=(1-\mathcal{L}_{\mathcal{F}}-2{\mathcal{F}}\mathcal{L}_{\mathcal{FF}}),$
(14)

Considering the photon propagation in $\boldsymbol{\hat{y}}-$direction of a
plane wave as ${\bf E}_{w}={\bf E}_{0}\rm
e^{-i(\mathbf{k_{\perp}}\cdot\mathbf{y}-\omega t)}$, from the Maxwell
equations we can obtain the dispersion equation of photon as

$(\epsilon_{ijk}\epsilon_{lab}k_{j}(\mu^{-1})_{kl}k_{a}+\omega^{2}\epsilon_{ib})E_{w\,b}=0,$
(15)

where $i,j,k,a,b,l=1,2,3$. The solution of Eq. (15) describes two physical
transverse polarization modes of the photon field, mode (2) where the
$E_{w}\parallel B_{e}$ and mode (3) $B_{w}\parallel B_{e}$. The dispersion
relation for each mode has the form

$\displaystyle\omega^{(2)}\simeq\mid{\bf
k}_{\bot}\mid(1-\frac{{\mathcal{L}}_{\mathcal{GG}}B_{e}^{2}}{2}),\quad\quad\omega^{(3)}\simeq\mid{\bf
k}_{\bot}\mid(1-\frac{{\mathcal{L}}_{\mathcal{FF}}B_{e}^{2}}{2}),$ (16)

in agreement with [46, 47, 48]. The appearance of Cotton-Mouton birefringence
[2] brings the existence of two refraction indexes associated with the two
different polarization modes: $n_{\parallel}$ for mode (2) and $n_{\bot}$ for
mode (3).

Figure 1: Phase velocity for the second mode of propagation of the photon as a
function of the dimensionless magnetic field for a strong field limit. The
non-linear effect decreases the phase velocity with the $B_{e}$ increase. We
have plotted curves for four Dirac materials: $Bi_{1-x}Sb_{x}$ (Dashed Red),
graphene (Blue), $Ta_{3}As_{4}$ (Dashed Orange), and $Pb_{1-x}Sn_{x}Te$
(Black). As we can see the lowest velocity is obtained for $Pb_{1-x}Sn_{x}Te$
due to its effective fine structure constant is the highest.

$n_{\parallel,\bot}=\frac{\mid\bf{k_{\bot}}\mid}{\omega^{(2,3)}}=\sqrt{\frac{\epsilon_{\parallel,\bot}}{\mu_{\bot,\parallel}}}.$
(17)

The difference between the refraction index $\Delta n=n_{\parallel}-n_{\bot}$
takes the form

$\Delta
n=\frac{(\mathcal{L}_{\mathcal{GG}}-\mathcal{L}_{\mathcal{FF}})B_{e}^{2}}{2}.$
(18)

In the weak field limit, it is reduced to $\Delta n^{\rm
WF}_{CM}=3/4\xi_{D}B_{e}^{2}$, instead of the strong limit $\Delta n^{\rm
SF}_{\rm CM}=\frac{\alpha_{D}}{3\pi}(\frac{B_{e}}{B_{c}}-1)$. Since the phase
velocity $v_{\parallel,\bot}=1/n_{\parallel,\bot}$, in the strong field limit
the condition,
$v_{\parallel}^{SF}=1-\frac{\alpha_{D}}{3\pi}(\frac{B_{e}}{B_{c}})>0$ fixes
the validity of one-loop approximation up to values of the magnetic field [41]
$b\equiv B_{e}/B_{c}\leq 3\pi/\alpha_{D}$.

The phase velocity as a function of the magnetic field is plotted for a second
mode for four Dirac materials in (1) . We have depicted curves for
$Bi_{1-x}Sb_{x}$, graphene, $Ta_{3}As_{4}$, and $Pb_{1-x}Sn_{x}Te$. The
graphic shows that $Pb_{1-x}Sn_{x}Te$ with the high effective fine structure
constant has the lowest velocity.

The dispersion equation in weak magnetic field limit [46, 49, 23, 50] is
obtained as

$\displaystyle\omega^{WF,(2)}\simeq\mid\boldsymbol{k_{\perp}}\mid\left(1-\frac{7}{4}\xi_{D}B_{e}^{2}\right),\quad\omega^{WF,(3)}\simeq\mid\boldsymbol{k_{\perp}}\mid(1-\xi_{D}B_{e}^{2}),$
(19)

while for strong magnetic field limit, the result is

$\displaystyle\omega^{SF,(2)}\simeq\mid\boldsymbol{k_{\perp}}\mid\left(1-\frac{\alpha_{D}}{3\pi}\frac{B_{e}}{B_{c}}\right),\quad\omega^{SF,\,(3)}\simeq\mid\boldsymbol{k_{\perp}}\mid\left(1-\frac{\alpha_{D}}{3\pi}\right).$
(20)

##  V Magnetic properties of photon propagation in Dirac materials.
Magnetization

We devote this section to studying the magnetic properties of the Dirac
materials considering the contribution of the photon propagating in the
material in the presence of the magnetic field. We start from the grand
partition function defined as $Z=Tr\\{e^{kT(H-\mu N)}\\},$ where $k$ is the
Boltzmann constant, $T$ is the temperature, $H$ is the Hamiltonian, $\mu$ is
the chemical potential, and N is the number of particle density. The partition
function in the path integral language corresponds to
$Z=\int{\mathcal{D}}\bar{\psi}\psi e^{ikT\int_{0}^{1}d\tau\int
d^{3}x{\mathcal{L}}(A_{\mu},\psi)},$ where we use euclidean space-time with
$\tau=it$ a variable in the interval $0,kT$.

In one-loop approximation, the thermodynamical potential of the electron-
positron plasma in the presence of an electromagnetic field reads as,

$\Omega=kT\,Tr\\{lnZ\\}=ikT\,ln\det{\\{G^{-1}(x,x^{\prime})\\}},$ (21)

where $G^{-1}$ is the inverse of the Green function of electrons in the
presence of the electromagnetic field. As we show in the appendix, we can
separate from the total thermodynamical potential the vacuum, dependent only
on the electromagnetic field, and the medium contribution dependent on
temperature, chemical potential, and magnetic field, allowing the study
separately both contributions as

$\Omega=\Omega_{vac}+\Omega_{med},$ (22)

where $\Omega_{vac}$, after renormalization (see appendix A) is just the
Euler-Heisenberg effective Lagrangian Eq. (3). Using its approximation up to
second order of the photon field in vacuum Dirac material the effective
potential writes as

$\Omega_{D}^{\rm(ph-B)}\equiv\langle\frac{(1-\mathcal{L}_{\mathcal{F}})}{2}(E_{w}^{2}-B_{w}^{2})+\frac{\mathcal{L}_{\mathcal{FF}}}{2}({\bf
B_{e}}\cdot{\bf B}_{w})^{2}+\frac{\mathcal{L}_{\mathcal{GG}}}{2}({\bf
B_{e}}\cdot{\bf E}_{w})^{2}\rangle,$ (23)

where $\langle...\rangle\equiv\int d^{3}x$ denotes the average over
coordinates of the photon fields.

In turn, $\Omega_{med}$ is the thermodynamical potential of the electrons-hole
Dirac material plasma [21] and its explicit form before the integrate over the
$p_{3}$ momentum component

$\Omega_{med}=-\Omega_{0}\sum_{n}^{\infty}a_{n}B_{e}\int
dp_{3}ln(1+e^{(\epsilon_{n}-\mu)\beta})(1+e^{(\epsilon_{n}+\mu)\beta}),$ (24)

where $\Omega_{0}$=$e\Delta^{2}/(4\pi^{2})$ and
$\epsilon^{2}_{n}=p_{3}^{2}v_{F}^{2}+\Delta^{2}+2eB_{e}nv_{F}$, with $n$ the
Landau levels.

Usually, when we study electron-positron plasma in the presence of a magnetic
field, the vacuum contribution is ignored because it is relevant close to the
critical field. However, for Dirac materials, the critical fields are
reachable in laboratories, so studying the relevance and the presence of this
contribution is very timely for backing up the QED vacuum properties.

Figure 2: Magnetization of the vacuum as a function of external magnetic field
strength over the critical field for each Dirac Material: $Bi_{1-x}Sb_{x}$,
$Pb_{1-x}Sn_{x}Te$, $Ta_{3}As_{4}$, and graphene. The value of $E_{w}$ is
normalized using the electrical critical field, $E_{\omega}/E_{c}=0.3$ used in
[14] and $E_{\omega}\approx B_{\omega}$. The curves are plotted up to the
maximum value of the magnetic field according to Table I.

Thereby we will calculate the total magnetization of the Dirac material
considering the photon and medium contribution. The photon magnetization reads
as $\mathcal{M}^{ph-B_{e}}(B_{e})=-\frac{\partial\Omega^{(\rm
ph-B)}}{\partial{\bf B}_{e}}$ and it has the form

${\mathbf{\mathcal{M}}}^{ph-B_{e}}=\langle\mathcal{L}_{\mathcal{GG}}({\bf
E}_{w}\cdot{\bf B}_{e}){\bf E}_{w,3}+\mathcal{L}_{\mathcal{FF}}({\bf
B}_{w}\cdot{\bf B}_{e}){\bf B}_{w,3}\rangle,$ (25)

it depends on the polarization mode, which is positive for both modes,
behaving the vacuum as a paramagnetic ”medium”. The magnetization for each
mode in the weak and strong limit [23] has the form

$\displaystyle{\mathcal{M}}^{(2),WF}_{\parallel}=\frac{7\xi_{D}}{2}B_{e}\langle
E_{w}^{2}\rangle\hat{z},\quad\quad{\mathcal{M}}^{(3),WF}_{\parallel}=2\xi_{D}B_{e}\langle
B_{w}^{2}\rangle\hat{z},$ (26)
$\displaystyle{\mathcal{M}}^{(2),SF}_{\parallel}=\frac{\alpha_{D}}{3\pi}\frac{\langle
E_{w}^{2}\rangle}{B_{c}}\hat{z},\quad\quad{\mathcal{M}}^{(3),SF}_{\parallel}=\frac{\alpha_{D}}{3\pi}\frac{\langle
B_{w}^{2}\rangle}{B_{e}}\hat{z}.$ (27)

Note, that in [14] the weak field is delt for the external magnetic and
electric field. In that case mode (3) becomes diamagnetic while, mode (2)
continues being positive.

To get electron magnetization we will consider the thermodynamical potential
in the approximation of the degenerate electron gas, ($T<<\mu$) due to typical
experimental temperatures of the Dirac materials being around $10$ K $<1$ meV
[40, 38] and electron densities are high, with chemical potentials around
$10^{2}$ meV. Then, $\Omega_{med}(\mu,B_{e})$ [51] reads as

$\Omega_{med}(\mu,B_{e})=-\Omega_{0}\sum_{n}^{n_{\mu}}a_{n}B_{e}\left(\mu^{\prime}p^{\prime}_{f}-\left(1+2n\frac{B_{e}}{B_{c}}\right)ln\left(\frac{\mu+p_{f}}{\Delta_{n}}\right)\right),$
(28)

where $\mu^{\prime}=\mu/\Delta$, $p^{\prime}_{f}=p_{f}/\Delta$,
$\Delta_{n}^{2}=\Delta^{2}+2eB_{e}nv_{F}$,
$p_{f}=1/v_{F}\sqrt{\mu^{2}-\Delta_{n}^{2}}$ is the Fermi momentum and
$n_{\mu}$ is the maximum number of Landau levels taken from the integer part
of $n_{\mu}=\frac{\mu^{2}-\Delta^{2}}{2eB_{e}}$.

The magnetization $\mathcal{M}^{med}=-\frac{\partial\Omega}{\partial B_{e}}$,
yields

$\mathcal{M}^{med}=\Omega_{0}\sum_{n}^{n_{\mu}}a_{n}\left(\mu^{\prime}p_{f}^{\prime}-\left(1+4n\frac{B_{e}}{B_{c}}\right)ln\left(\frac{\mu+p_{f}}{\Delta_{n}}\right)\right).$
(29)

To illustrate the behavior of the magnetization for Dirac materials
$Bi_{1-x}Sb_{x}$, $Pb_{1-x}Sn_{x}Te$, $Ta_{3}As_{4}$, and graphene, we have
plotted it as a function of the magnetic field and the chemical potential in
Fig (3). We can appreciate similar paramagnetic behavior for all Dirac
materials, and the Hass-van Alphen effect due to the quantization on Landau
levels. One notes that magnetization increases with the gap and it tends to a
constant value when the magnetic field reaches the corresponding critical
field for each material. The magnetization is strongly dependent on the
increase of the chemical potential. The magnetization of each material depends
on the band gap and Fermi velocity as a consequence of our simple model but
also depends on other geometrical factors and chemical composition that we do
not consider.

Note in both figures the oscillations (characteristic behavior of magnetized
quantum gases) that are due to the so-called Hass-van Alphen effect associated
with the transitions of fermions between Landau levels.

Figure 3: (Left) Magnetization of the medium as the dimensionless magnetic
field $b$ for a fix value of $\mu/\Delta=1.5$. (Right) Magnetization of the
medium versus dimensionless chemical potential $\mu/\Delta$ for a fixed value
of the dimensionless magnetic field $B_{e}/B_{c}=2$. In both figures have been
depicted the magnetization for studied Dirac materials: $Bi_{1-x}Sb_{x}$,
$Pb_{1-x}Sn_{x}Te$, $Ta_{3}As_{4}$, and graphene.  Figure 4: Second mode (2)
magnetization for vacuum ($M^{vac}$) and medium($M^{med}$) as a function of
the external magnetic field normalized to the critical magnetic field of each
material. We used $\mu/\Delta=1.5$ for $Bi_{1-x}Sb_{x}$ (Black),
$Pb_{1-x}Sn_{x}Te$ (Purple), $Ta_{3}As_{4}$ (Orange), and graphene (Blue).
Continue colors represent the medium magnetization, while the discontinue
lines represent the vacuum magnetization.

Let’s compare the vacuum and medium magnetization showing in Fig (4) the
magnetization for vacuum ($M^{vac}$) and medium($M^{med}$) of the materials
$Bi_{1-x}Sb_{x}$, $Pb_{1-x}Sn_{x}Te$, $Ta_{3}As_{4}$, and graphene, as a
function of the magnetic field. Obviously, we can see that the leading
contribution to the magnetization comes from electrons for each material.
Depending on the Dirac material parameters this magnetization is 2 or 3 orders
higher than the corresponding photon or vacuum one (Fig. 2). The contribution
of the vacuum to the total magnetization is very low at the weak magnetic
field scale however, at high values of the magnetic field this magnetization
increases becoming almost constant.

##  VI Effective photon magnetic moment

Let us study the effective magnetic moment of a photon probe propagating in a
magnetized Dirac vacuum defined from the variation of the magnetic energy of
the vacuum with respect to the magnetic field. We translate to Dirac material
language the results obtained in [23] The explicit expression for the two
modes $\mid\boldsymbol{\mu}_{ph}^{(2,3)}\mid$ reads as

$\displaystyle\mid\boldsymbol{\mu}_{ph}^{(2)}\mid=\dfrac{\alpha_{D}}{16\pi}\dfrac{1}{b^{3}}\left\\{3-12\zeta^{(1,1)}\left(-1,\dfrac{1}{2b}\right)+3\psi\left(\dfrac{1}{2b}\right)\right\\}$
$\displaystyle+b\left[-3+\log\Gamma\left(\dfrac{1}{2b}\right)\left(\dfrac{\pi}{b}\right)^{2}\right.+\left.\left.\psi^{(1)}\left(1+\dfrac{1}{2b}\right)+2b^{2}\right]\right\\}\dfrac{\mid\bf{k}_{\bot}\mid}{B_{c}},$
(30)

$\displaystyle\mid\boldsymbol{\mu}_{ph}^{(3)}\mid=\dfrac{\alpha_{D}}{8\pi}\dfrac{1}{b^{4}}\left\\{-\psi^{(1)}\left(1+\dfrac{1}{2b}\right)\right.+b\left[4-4\psi\left(1+\dfrac{1}{2b}\right)+2\psi\left(\dfrac{1}{2b}\right)\right]$
$\displaystyle+b^{2}\left[4-2\log(2\pi)+\left.\left.4\log\left(\Gamma\left(\dfrac{1}{2b}\right)\left(\dfrac{\pi}{b}\right)^{1/2}\right)\right]\right\\}\dfrac{\mid{\bf{k}_{\bot}}\mid}{B_{c}},\right.$
(31)

Figure 5: Photon effective magnetic moment as a function of magnetic field
strength and $B_{e}/B_{c}$. The left figure shows the magnetic moment for two
Dirac materials: $Pb_{1-x}Sn_{x}Te$ (red) and $Bi_{1-x}Sb_{x}$ (blue). In
black we have plotted the corresponding one for QED for both modes. For each
case, the modes (2) and (3) are depicted with continuous and dashed lines. The
right figure shows for $Pb_{1-x}Sn_{x}Te$ the comparison of the magnetic
moment for an arbitrary value of the magnetic field with the corresponding two
limits weak (orange and green) and strong (gray) field limits for both modes
respectively.

$\psi^{(1)}=\partial_{h}\psi[h]$ being $\psi$ the PolyGamma o Digamma
function, (first derivative of ${\rm ln}\,\Gamma$).
$\zeta^{(1,1)}[s,h]=\partial_{h}\zeta^{\prime}$ with
$\zeta^{\prime}=\partial_{s}\zeta[s,h]$ and $\zeta[s,h]$ is the Hurwitz zeta
function [52]. Considering the form of $\psi$ and $\zeta^{\prime}$ in weak and
strong field limits (see details in [23]). For modes (2) and (3) yields
$\mid\mu_{ph}^{WF\,(2)}\mid=\frac{\alpha}{4\pi}\frac{28}{45}\frac{B_{e}}{B_{c}^{2}}\mid{\bf
k}_{\bot}\mid$ and
$\mid\mu_{ph}^{WF\,(3)}\mid=\frac{4}{7}\mid\mu_{ph}^{WF\,(2)}\mid$. While for
strong field limit, only mode (2) contributes to the effective magnetic moment
and it goes to a constant value,
$\mid\mu_{ph}^{SF\,\,(2)}\mid=\frac{\alpha_{D}}{3\pi}\frac{\langle
E_{w}^{2}\rangle}{B_{c}}$ and $\mid\mu_{ph}^{SF\,\,(3)}\mid=0.$ Note that the
effective photon magnetic moment $\mid\mu_{ph}^{SF\,\,(2)}\mid$ is a decimal
of the electron magnetic moment, contrasting to the QED value, which is two
orders lower.

In Fig. (5) left we plot the photon effective magnetic moment as a function of
$b$ for light polarization modes (2) and (3) of Dirac materials. For
comparison, we have depicted the magnetic moment for light propagating in a
magnetized vacuum of QED. One can see that for the Dirac material, the
effective magnetic moment for $Pb_{1-x}Sn_{x}Te$ is one order below the QED
magnetic moment while for $Bi_{1-x}Sb_{x}$ is 3 orders lower than the value of
QED. In the right side of fig. (5) we plot for $Pb_{1-x}Sn_{x}Te$ the
effective magnetic moment considering the results for arbitrary values of the
magnetic field as well as we have plotted it for weak and strong limit for
both modes. Then, for a magnetic field $B_{e}\gtrsim 2B_{c}$ the effective
magnetic moment of photons polarized on mode (2) tends asymptotically to a
constant value [53]\- [54]\- [46]. Mode (3) slowly decreases with the magnetic
field strength, being zero its strong field limit value.

##  VII Photon Energy-momentum tensor: energy density, Pointing vector and
radiation pressures

In [23] we widely discussed the EMT calculated by the Hilbert method from the
effective Lagrangian Eq. (7). The resulting EMT is symmetric, gauge invariant,
and conserved. Extrapolating its form to Dirac material yields

$\displaystyle t^{\gamma\rho}$
$\displaystyle=(1-\mathcal{L}_{\mathcal{F}})f^{\gamma}_{\lambda}f^{\lambda\rho}+\frac{\mathcal{L}_{\mathcal{FF}}}{2}f^{\mu\nu}F_{\mu\nu}(F^{\gamma\alpha}f_{\alpha}^{\rho}+F^{\rho\alpha}f_{\alpha}^{\gamma})$
$\displaystyle+\frac{\mathcal{L}_{\mathcal{GG}}}{2}f^{\mu\nu}\tilde{F}_{\mu\nu}(\tilde{F}^{\gamma\alpha}f_{\alpha}^{\rho}+\tilde{F}^{\rho\alpha}f_{\alpha}^{\gamma})+\frac{\eta^{\gamma\rho}}{4}\left((1-\mathcal{L}_{\mathcal{F}})f_{\mu\nu}f^{\mu\nu}\right.$
$\displaystyle+\frac{\mathcal{L}_{\mathcal{FF}}}{2}f^{\mu\nu}F_{\mu\nu}f^{\alpha\beta}F_{\alpha\beta}+\frac{\mathcal{L}_{\mathcal{GG}}}{2}\left.f^{\mu\nu}\tilde{F}_{\mu\nu}f^{\alpha\beta}\tilde{F}_{\alpha\beta}\right)$
$\displaystyle+\mathcal{L}_{\mathcal{F}}(F^{\gamma\alpha}f_{\alpha}^{\rho}+F^{\rho\alpha}f_{\alpha}^{\gamma})+\frac{\eta^{\gamma\rho}}{2}\mathcal{L}_{\mathcal{F}}f^{\mu\nu}F_{\mu\nu}.$
(32)

The external magnetic field makes EMT fully anisotropic [23]. The average of
the diagonal part of the tensor corresponds to energy density and anisotropic
pressures while the non-diagonal components account for the Poynting vector.
The energy density $\langle t^{00}\rangle$ for both modes is positive [24] and
the explicit expression for photon energies for both modes are

$\displaystyle\mathcal{E^{D}}_{w}^{(2)}$
$\displaystyle\simeq(1-\mathcal{L}_{\mathcal{F}}+{\frac{3}{2}}\mathcal{L}_{\mathcal{GG}}B_{e}^{2})\langle
E_{w}^{2}\rangle,\quad\quad\mathcal{E^{D}}_{w}^{(3)}\simeq(1-\mathcal{L}_{\mathcal{F}}-{\frac{1}{2}}\mathcal{L}_{\mathcal{FF}}B_{e}^{2})\langle
E^{D\,\,(2)}_{w}\rangle.$ (33)

However, as the Hamiltonian of the effective theory is

$\displaystyle\mathcal{H^{D}}=D_{w}E_{w}-\mathcal{L}_{D}^{\rm(ph-B)}=\frac{(1-\mathcal{L}_{\mathcal{F}})}{2}(E_{w}^{2}-B_{w}^{2})-\frac{\mathcal{L}_{\mathcal{FF}}}{2}({\bf
B}\cdot{\bf B}_{w})^{2}+\frac{\mathcal{L}_{\mathcal{GG}}}{2}({\bf B}\cdot{\bf
E}_{w})^{2}.$ (34)

only the energy density for mode for mode (3) becomes in the eigenvalue of the
Hamiltonian.

The anisotropic pressures take the form

$\displaystyle p^{D\,\,(2)}_{1,w}$
$\displaystyle=(1-\frac{1}{2}\mathcal{L}_{\mathcal{GG}}B_{e}^{2})\langle
E_{w}^{2}\rangle,\quad\quad
p^{D\,\,(3)}_{1,w}=(1-\frac{1}{2}\mathcal{L}_{\mathcal{FF}}B_{e}^{2})\langle
E_{w}^{2}\rangle,$ $\displaystyle p^{D\,\,(2)}_{2,w}$
$\displaystyle=(1-\mathcal{L}_{\mathcal{F}}+\frac{\mathcal{L}_{\mathcal{GG}}B^{2}}{2})\langle
E_{w}^{2}\rangle,\quad\quad
p^{D\,\,(3)}_{2,w}=(1-\mathcal{L}_{\mathcal{F}}-\frac{3}{2}\mathcal{L}_{\mathcal{FF}}B^{2})\langle
E_{w}^{2}\rangle,$ $\displaystyle p^{D\,\,(2)}_{w,\parallel}$
$\displaystyle\simeq(1-\frac{3}{2}\mathcal{L}_{\mathcal{GG}}B_{e}^{2})\langle
E_{w}^{2}\rangle,\quad\quad
p^{D\,\,(3)}_{w,\parallel}\simeq(1+\frac{1}{2}\mathcal{L}_{\mathcal{FF}}B_{e}^{2})\langle
E_{w}^{2}\rangle.$ (35)

while the Poynting vector has the form

$\displaystyle\mathcal{P^{D}}_{w}^{(2)}$
$\displaystyle\simeq(1-\mathcal{L}_{\mathcal{F}}+\mathcal{L}_{\mathcal{GG}}B_{e}^{2})\langle
E_{w}^{2}\rangle),\quad\quad\mathcal{P^{D}}_{w}^{(3)}\simeq(1-\mathcal{L}_{\mathcal{F}}-\mathcal{L}_{\mathcal{FF}}B_{e}^{2})\langle
E_{w}^{2}\rangle,$

we can rewrite in terms of magnetization the above quantities. In particular
the Poynting vector takes the form

$\mathcal{P^{D}}_{w}^{(2,3)}=P_{0}\pm\frac{1}{2}\mathcal{M}_{w}^{(2,3)}B_{e},$
(37)

where $P_{0}$ contains the non-linear correction that emerges from the scalar
invariant $\mathcal{F}$ with $P_{0}=(1-\mathcal{L}_{\mathcal{F}})\left\langle
E_{w}^{2}\right\rangle$ while the second term comes from the pseudo-scalar
invariant $\mathcal{G}$ and it is proportional to the magnetization.

Figure 6: Proposed experiment for measuring the pressure of the radiation over
the Dirac material with (left) and without (right) external magnetic field.

Note, that only non-linear isotropic vacuum fulfills that the radiation
pressure calculated by the Poynting vector to be equal to the target pressure
in ($\boldsymbol{{\hat{y}}}$-direction) $p_{w}^{2}$ Eq (VII)
$p_{w}^{2}=\mathcal{P}_{w}=P_{0}$. In that case, the non-linearity of
Lagrangian depends only on the scalar invariant $\mathcal{F}$. On the
contrary, the presence of the magnetic field leads to a non-linear effective
Lagrangian also dependent on the pseudo-scalar invariant $\mathcal{G}$ and
leads to anisotropies, the rotation symmetry is broken, entailing also that
$p_{w}^{2}\neq\mathcal{P}_{w}$. Then, the radiation pressure behaves
differently depending on their propagation mode. For mode (2) both are higher
than the corresponding “classical pressures” while for mode (3) it becomes
lower than the classical value. At first glance, the behavior of mode (2)
seems to be contrary to intuition. Although a photon propagating in a
magnetized vacuum has a lower velocity than if it propagates in an “empty
vacuum”, the contribution of magnetic energy enters the radiation pressure as
an additive term increasing the pressure with respect to the classical one,
see Eq.(37-VII). Instead, for mode (3) this contribution appears subtracting,
yielding thus a lower pressure value, see Eq.(VII).

Let us imagine an experimental setup composed of an emitter of polarized
photon beam transversely to an external magnetic field $\mathbf{B}$ in the
direction $x_{3}$. The photon beam is pointing to the Dirac material $\Delta
l$ located in the plane $xz$ (parallel to the magnetic field), Fig. (6). The
radiation pressure Eq.(37) is exerted over the sheet by the photon beam. The
measurement of the non-linear radiation pressures would depend on
$\mathcal{L}_{\mathcal{GG}}$ and $\mathcal{L}_{\mathcal{FF}}$ for a chosen
polarization beam. Of course, with a similar experimental setup is also
possible to measure the different phases and phase velocities (birefringence)
of the photon beam that crosses the sheet in comparison with one propagates
without a sheet both propagating transverse in the magnetic field. The
birefringence will be characterized by the difference between
$\mathcal{L}_{\mathcal{GG}}$ and $\mathcal{L}_{\mathcal{FF}}$ as was pointed
out in Eq (18).

###  VII.1 Conclusions

In this paper, we have extrapolated our previous studies related to photon
propagation in a constant magnetic field in the QED vacuum and medium to the
context of Dirac materials. For the treatment of Dirac vacuum, we started from
the expansion up to second order of the Euler-Heisenberg effective nonlinear
Lagrangian for the photon fields, considering photon low energy $\omega\ll
2mc^{2}$ ($\omega\ll 2\Delta$) and replacing the fine structure constant
$\alpha=\frac{e^{2}}{4\pi\hbar c}$ by $\alpha_{D}$ that depends on Fermi
velocity $v_{F}$ instead of $c$.

We have obtained the dispersion law, the phase velocity and refraction index
of photons propagating in the Dirac vacuum, electric permittivity, and the
magnetic susceptibility that the photon feels. The results are in agreement
with previous ones obtained in [24].

Magnetic properties in the Dirac vacuum as well as the magnetic moment have
also been studied and they depend on the polarization mode of the photon. The
photon magnetization is positive, showing that it is paramagnetic.

Magnetization of electron degenerate gas has been obtained and translated to
the language of Dirc materials using our previous results obtained for the
plasma electron-positron plasma in the presence of a magnetic field.

The magnetization of a Dirac material (due to electrons) not only depends on
$\alpha_{D}$ but also on the energy gap. For zero temperature and high
density, the magnetization preserves the paramagnetic character. The magnetic
properties have been illustrated for four Dirac materials: $Bi_{1-x}Sb_{x}$,
$Pb_{1-x}Sn_{x}Te$, $Ta_{3}As_{5}$ and graphene-like, and the properties are
determined by the value of the energy gap of each material. Although the
medium magnetization (due to electrons) is higher than the vacuum
magnetization (due to photons) for two or three orders of magnitude depending
on the material energy gap, with the increasing of the magnetic field the
vacuum magnetization increases becoming constant. Its contribution changes the
magnetization in a quantity that could be detected at values of magnetic field
reachable at the laboratory. This conclusion reinforces the importance of
taking into account the vacuum properties to study Dirac materials.

We have discussed the energy density, pointing vector, and radiation pressures
of the photon propagating transverse to the magnetic field starting from the
energy-momentum tensor calculated ”a la Hilbert” for an effective non-linear
electrodynamics for Dirac material. These quantities are dependent on the
polarization mode. We have discussed that in our framework the pressure target
in the direction y is not coincide with the Poynting vector. That is a
consequence of the symmetry breaking that the magnetic field produces.

Although our model for Dirac materials from the point of view of material
science is simplified, has all the wealth of quantum theories and thereby
could be used to study other phenomena that appear in QED and their analogies
in Dirac materials. On the other hand, the properties studies are valid for
all values of the magnetic field reproducing the weak and strong magnetic
field limit. Besides, the study could be extended to other non-linear
Lagrangian contribute to extract some general properties of Dirac materials.

We are hopeful that Dirac materials may be a medium where the vacuum
properties of QED may be tested.

##  VIII Acknowledgments

The authors thank to Angeles Perez Garcia for her contribution to our previous
work. A.R.J, A.P.M, and E.R.Q. were supported by the project of No.
NA211LH500-002 from AENTA-CITMA, Cuba.

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##  Appendix A Calculation of the thermodynamical potential for the vacuum and
medium

The thermodynamical potential of electron-positron plasma in the presence of
an external magnetic field in $x_{3}$ direction is given by

$\Omega(B_{e},\mu,T)=\frac{1}{\beta}Tr\\{lnZ\\}=ikTln\det{\\{G^{-1}(x,x^{\prime})\\}},$
(38)

where trace and logarithm are functional operations. In the momentum space, it
has the form

$G^{-1}_{n}(\overline{p})=\overline{p}\cdot\gamma-m,$ (39)

with the notation ${\overline{p}}=(ip^{4},0,\sqrt{2eBl},p^{3})$ over Landau
number $n=0,1,2,...$ in Euclidean space, $\gamma$ are the Gamma-matrices.

In the momentum space, the thermodynamical potential has the form [55, 56]

$\Omega(B_{e},\mu,T)=-\frac{eB_{e}}{\beta}\left[\sum_{\sigma=\pm
1}\sum_{n=0}^{\infty}\sum_{p_{4}}\int\limits_{-\infty}^{\infty}\frac{dp_{3}}{(2\pi)^{2}}\ln\det
G^{-1}_{n}(\overline{p}^{*})\right],$ (40)

here $\sigma=\pm 1$ is the eigenvalues of the spin, $p_{4}=i\omega_{m}$,
$\omega_{m}$ is the Matsubara frequencies $\omega_{m}=kT(2m+1)\pi$ with
$m=0,1,2...$ and, $p_{3}$ is the momentum in $x_{3}$-direction.

Performing the sum over Matsubara frequencies and calculating the determinants
in Eq. (40) we obtain

$\Omega(B_{e},\mu,T)=-\frac{1}{2}\frac{eB_{e}}{4\pi^{2}}\int_{-\infty}^{\infty}dp_{3}\sum_{\sigma
l}|E_{\sigma
n}|+\frac{eB_{e}}{4\pi^{2}}\int_{-\infty}^{\infty}dp_{3}\sum_{\sigma,n}\frac{1}{\beta}\ln(1+e^{-\beta|E_{\sigma
n}-\mu|})(1+e^{-\beta|E_{\sigma n}+\mu|}),$ (41)

where $E_{\sigma n}=\sqrt{p_{3}^{2}+m^{2}+eB_{e}(2n+1+\sigma)}$, the first
term of Eq. (41) corresponds to the $\Omega_{vac}$ the second $\Omega_{med}$.

$\Omega_{vac}$ has an ultraviolet divergence and has to be renormalized. To do
it we use the integral representation of the integral of $E$ leads,

$E=\int ds(1-e^{-E^{2}s})s^{-3/2},$ (42)

Now we can perform the Gaussian integral on $p_{3}$ obtaining

$\Omega_{vac}=-\frac{eB_{e}}{8\pi^{2}}\sum_{\sigma=\pm
1}\int_{\epsilon}^{\infty}\frac{ds}{s^{2}}\left(2e^{-(m^{2}+eB_{e}(2n+\sigma+1))s}\right),$
(43)

doing the sum over the spin $\sigma$ we obtain the integral

$\Omega_{vac}=\frac{eB_{e}}{8\pi^{2}}\sum_{n=1}^{\infty}\int_{\epsilon}^{\infty}\frac{ds}{s^{2}}\left(2e^{-(m^{2}+2eB_{e}n)s}-1\right).$
(44)

Let’s do the sum over Landau levels using

$\sum_{n=1}^{\infty}x^{n}=\frac{1}{1-x},$ (45)

$\Omega_{vac}$ yields

$\Omega_{vac}=\frac{eB}{8\pi^{2}}\int_{\epsilon}^{\infty}ds\frac{e^{-m^{2}s}}{s^{2}}\left(\frac{1}{1-e^{-2eB_{e}s}}-1\right),$
(46)

in terms of hyperbolic trigonometric function, we have

$\Omega_{vac}=\frac{eB_{e}}{8\pi^{2}}\int_{\epsilon}^{\infty}\frac{ds}{s^{2}}\left(e^{-m^{2}s}coth(2eB_{e}s)\right),$
(47)

$coth(2eB_{e}s)$ provokes that the integral has an ultraviolet divergence.
Expanding $coth(2eBs)$ in powers of small s we obtain

$coth(2eB_{e}s)\sim\left(1+\frac{(eB_{e}s)^{2}}{3}\right),$ (48)

The prescription to regularize the integral consists of adding and subtracting
divergencies terms Eq. (48) obtaining

$\Omega_{vac}=-\frac{1}{8\pi^{2}}\int_{\epsilon}^{\infty}\frac{ds}{s^{3}}\left(2e^{-m^{2}s}((eB_{e}s)coth(eB_{e}s)-1-\frac{(eB_{e}s)^{2}}{3}\right).$
(49)

In the case of we study the vacuum in the presence of a constant electric
field the thermodynamical potential is obtained substituting in Eq(49)
$B_{e}=iE_{e}$

$\Omega_{vac}(E_{e})=-\frac{i}{8\pi^{2}}\int_{\epsilon}^{\infty}\frac{ds}{s^{3}}\left(2e^{-m^{2}s}((eE_{e}s)cot(eE_{e}s)-1-\frac{(eE_{e}s)^{2}}{3}\right).$
(50)

Considering previous results it is possible to get straightforward the
$\Omega_{vac}$ [57] dependent on the electromagnetic field, external electric
and magnetic field, and obtain the Lagrangian Eq (3).