# Interplay between antiferromagnetic order and spin polarization in
ferromagnetic metal/electron-doped cuprate superconductor junctions

Pok-Man Chiu,1 C. S. Liu,2,1 and W. C. Wu1 1Department of Physics, National
Taiwan Normal Univesity, Taipei 11650, Taiwan
2Department of Physics, Yanshan University, Qinhuangdao 066004, China

###### Abstract

Recently we proposed a theory of point-contact spectroscopy and argued that
the splitting of zero-bias conductance peak (ZBCP) in electron-doped cuprate
superconductor point-contact spectroscopy is due to the coexistence of
antiferromagnetic (AF) and $d$-wave superconducting orders [Phys. Rev. B 76,
220504(R) (2007)]. Here we extend the theory to study the tunneling in the
ferromagnetic metal/electron-doped cuprate superconductor (FM/EDSC) junctions.
In addition to the AF order, the effects of spin polarization, Fermi-wave
vector mismatch (FWM) between the FM and EDSC regions, and effective barrier
are investigated. It is shown that there exits midgap surface state (MSS)
contribution to the conductance to which Andreev reflections are largely
modified due to the interplay between the exchange field of ferromagnetic
metal and the AF order in EDSC. Low-energy anomalous conductance enhancement
can occur which could further test the existence of AF order in EDSC. Finally,
we propose a more accurate formula in determining the spin polarization value
in combination with the point-contact conductance data.

###### pacs:

74.20.-z, 74.25.Ha, 74.45.+c, 74.50.+r

##  I Introduction

Using point contact technique to measure the spin polarization in
ferromagnetic metal/conventional superconductor (FM/CS) junctions was
pioneeringly done by Soulen et al. Soulen98  and Upadhyay et al. Upadhyay98
in 1998. Their works showed that determining the spin polarization at Fermi
surface is essentially not an easy task. That leads to some definitions of
spin polarization such as “tunneling polarization” proposed by Tedrow and
Meservey Tedrow94  and “point-contact polarization” proposed by Soulen et al.
Soulen98 . One year later, Zhu et al. Zhu99 ; Zhu00  and Kashiwaya et al.
Kashiwaya99  have utilized the ideas to study the spin-polarized quasiparticle
transport in ferromagnet/$d$-wave superconductor junctions. Zhu et al. Zhu99 ;
Zhu00  predicted that conductance resonances occur in a normal-metal-
ferromagnet/$d$-wave superconductor junction and in a following paper, they
further studied the junctions by solving the Bogoliubov-de Gennes (BdG)
equations within an extended Hubbard model which included the proximity
effect, the spin-flip interfacial scattering at the interface, and the local
magnetic moment. They have reported that the proximity can induce order
parameter oscillation in the ferromagnetic region. In contrast, Kashiwaya et
al. Kashiwaya99  focused on the spin current and spin filtering effects at the
magnetic interface. In the works of Zutic and Valls Zutic99 ; Zutic00 , they
first considered the effect of Fermi-wave vector mismatch (FWM) and have
pointed out that if one neglects FWM, the effect of spin polarization
invariably leads to the suppression of Andreev reflection (AR). Among many
other junction studies, Dong et al. Dong01  studied a little different
junction which forms a four layer sandwich, i.e., FM/I/$d+is$/$d$-wave
junctions, by taking into account the roughness of the interfacial barrier and
broken time-reversal symmetry states.

The pioneering works of Soulen et al. and Upadhyay et al. have inspired
several experimental studies Ji01 ; Strijkers01 ; Kant02 ; Raychaudhuri03 ;
Perez-Willard04 ; Woods04 ; Mukhopadhyay07 ; Chalsani07  as well. Especially
normal and ferromagnetic metal/conventional superconductor or $s$-wave
superconductor (FM/$s$-wave SC) junctions have been intensely studied
experimentally and theoretical modelings (Blonder-Tinkham-Klapwijk (BTK)
formula Blonder82  or its extension) had a good fitting with the conductance
data. Recently Linder and Sudbø Linder07  presented a theoretical study of
FM/$s$-wave SC junction that investigated the possibility of induced triplet
pairing state in the ferromagnetic metal side. They have also used the BTK
approach but allowed for arbitrary magnetization strength and direction in the
ferromagnet, arbitrary spin-active barrier, arbitrary FWM, and different
effective masses in the two sides of the junction. As is expected, there is no
retroreflection process when an exchange field is present. However, they
pointed out that retroreflection can occur under some conditions Linder07 .

If one replaces the conventional superconductor by the high-temperature or
$d$-wave superconductor into the junction, it will occur several novel
phenomena due to its $d$-wave pairing symmetry, complex band structure, and
rich magnetic properties. Of particular interest, in the electron-doped side
of cuprate superconductors (EDSC), it is strongly suggested that
antiferromagnetic (AF) order may coexist with the $d$-wave superconducting
order, especially in the underdoped and optimally-doped regimes Liu07 . In
this paper, we shall explore the possible novel phenomena in the FM/EDSC
junction case, taking into account the interplay between antiferromagnetic
order and spin polarization. The ideas and models developed in FM/CS junctions
in the literature will be applied to the current FM/EDSC junction cases.

This paper is organized as follows. In Sec. II, the basic formulation is
given. We set up the condition of the junction and generalize the BdG
equations to include AF order parameter. As the formal process, we utilize
WKBJ approximation to obtain the more simple Andreev-like equations, which are
then solved to determine the four spin-dependent reflection coefficients
(detailed derivations are given in Appendix A). Formulas of charge and spin
conductances are derived . Sec. III are our main results and discussions. In
Sec. III.A, the condition of midgap surface states was derived (details are
given in Appendix B). In Sec. III.B, the effect of FWM was studied. In Secs.
III.C and III.D, we discuss the effects of spin-polarization and generalized
effective barrier, respectively. It is shown that anomalous conductance
enhancement can occur at low energies which could provide a further test for
the existence of AF order in EDSC. In Sec. III.E, a more general formula for
determining the spin polarization is proposed in terms of the experimental
zero-bias conductance data. Finally in Sec. IV, a brief conclusion is given.

##  II Formalism

Our formulation is given based on the following assumptions. We consider a
point contact or planar FM/I/EDSC junction where the superconductor overlayer
is coated with a clean, size-quantized, ferromagnetic-metal overlayer of
thickness $d$, that is much shorter than the mean free path $l$ of normal
electrons. The interface is assumed to be perfectly flat and infinitely large.
Considering $l\rightarrow\infty$ limit, the discontinuity of all parameters at
the interface can be neglected, except for the SC order parameter to which the
proximity effect is ignored Hu94 . When SC and AF orders coexist,
quasiparticle (QP) excitations of an inhomogeneous superconductor can have a
coupled electron-hole character associated with the coupled $\mathbf{k}$ and
${\bf k}+{\bf Q}$ [$\mathbf{Q}=(\pi,\pi)$] subspaces. QP states are thus
governed by the generalized BdG equations Gennes66 ; Liu07

$\displaystyle Eu_{1\sigma}({\bf{x}})$ $\displaystyle=$
$\displaystyle\hat{H}_{\sigma}u_{1\sigma}({\bf{x}})+\int{d{\bf{x}^{\prime}}\Delta({\bf{s}},{\bf{r}})v_{1\bar{\sigma}}({\bf{x}^{\prime}})+\Phi
u_{2\sigma}({\bf{x}})}$ $\displaystyle Ev_{1\bar{\sigma}}({\bf{x}})$
$\displaystyle=$
$\displaystyle\int{d{\bf{x}^{\prime}}\Delta^{*}({\bf{s}},{\bf{r}})u_{1\sigma}({\bf{x}^{\prime}})-\hat{H}_{\sigma}v_{1\bar{\sigma}}({\bf{x}})+\Phi
v_{2\bar{\sigma}}({\bf{x}})}$ $\displaystyle Eu_{2\sigma}({\bf{x}})$
$\displaystyle=$ $\displaystyle\Phi
u_{1\sigma}({\bf{x}})+\hat{H}_{\sigma}u_{2\sigma}({\bf{x}})-\int{d{\bf{x}^{\prime}}\Delta({\bf{s}},{\bf{r}})v_{2\bar{\sigma}}({\bf{x}^{\prime}})}$
$\displaystyle Ev_{2\bar{\sigma}}({\bf{x}})$ $\displaystyle=$
$\displaystyle\Phi
v_{1\bar{\sigma}}({\bf{x}})-\int{d{\bf{x}^{\prime}}\Delta^{*}({\bf{s}},{\bf{r}})u_{2\sigma}({\bf{x}^{\prime}})-\hat{H}_{\sigma}v_{2\bar{\sigma}}({\bf{x}})},$

where ${\bf{s}}\equiv{\bf{x}}-{\bf{x}^{\prime}}$,
${\bf{r}}\equiv({\bf{x}}+{\bf{x}^{\prime}})/2$,
$\hat{H}_{{}_{\sigma}}\equiv-\hbar^{2}\nabla^{2}/2m-E_{F}^{F,S}-\sigma J$ with
$J$ the exchange energy and $\sigma=\uparrow$ ($\downarrow$) for up (down)
spin ($\bar{\sigma}=-\sigma$), and $\Phi$ is the AF order parameter.
$\Delta({\bf{s}},{\bf{r}})$ is the Cooper pair order parameter in terms of
relative and center-of-mass coordinates. In the FM region, we define
$E_{F}^{F}\equiv\hbar^{2}q_{F}^{2}/2m=(\hbar^{2}q_{F\uparrow}^{2}/2m+\hbar^{2}q_{F\downarrow}^{2}/2m)/2$
as the spin averaged value. It differs from the value in the superconductor,
$E_{F}^{S}\equiv\hbar^{2}k_{F}^{2}/2m$, to which a FWM can occur between the
FM and EDSC regions Zutic00 . In (LABEL:eq:BdG), the wave functions $u_{1}$
and $v_{1}$ are considered related to the $\mathbf{k}$ subspace, while $u_{2}$
and $v_{2}$ are related to the ${\bf k}+{\bf Q}$ subspace. Comparing with the
first and second lines of Eq. (LABEL:eq:BdG), minus signs associated with the
$\Delta({\bf{s}},{\bf{r}})$ term in the third and fourth lines occur due to
the symmetry requirement, $\Delta({\bf k}+{\bf Q})=-\Delta({\bf{k}})$, for a
$d_{x^{2}-y^{2}}$-wave superconductor in ${\bf k}$ space. At Fermi level, the
$d_{x^{2}-y^{2}}$-wave SC gap is
$\Delta({\bf{\hat{k}}}_{F})\equiv\Delta_{0}\sin 2\theta$ with $\Delta_{0}$ the
gap magnitude and $\theta$ the azimuthal angle relative to the $x$-axis.

Figure 1: Schematic plot showing all possible reflection and transmission
processes for an up-spin electron incident into the FM/I/EDSC junction. An AF
order is assumed to exist in the EDSC. For convenience for a $d$-wave
superconductor, $\mathbf{k_{x}}$ axis is chosen to be along the [110]
direction. The right-bottom inset shows a given Fermi wave vector
$\mathbf{k}_{F}=\left(k_{F},k_{y},k_{z}\right)$ and its coupled AF wave vector
$\mathbf{k}_{F}+\mathbf{Q}\equiv\mathbf{k}_{F\mathbf{Q}}=\left(-k_{F},k_{y},k_{z}\right)$.
Both vectors are tied to the Fermi surface, which is approximated by a square
(thick line). NR, AR, AF-NR, and AF-AR stand for normal reflection, Andreev
reflection, antiferromagnetic-normal reflection, and antiferromagnetic-Andreev
reflection respectively. Their corresponding reflection angles are also shown.
For the case of an incident down-spin electron, all spin indices just reverse.

In a $d$-wave superconductor, it’s useful to consider a junction to which the
superconductor surface is allied along the [110] direction. A thin insulating
layer exists between the ferromagnetic metal and the superconductor (see Fig.
1) to which the barrier potential is assumed to take a delta function,
$V(x)=H\delta\left(x\right)$. Considering that an up-spin electron is injected
into the FM/I/EDSC junction from the ferromagnetic metal side, there are four
possible reflections as follows: (a) Normal reflection (NR): reflected as
electrons. (b) Andreev reflection (AR): reflected as holes, due to electron
and hole coupling in the $\mathbf{k}$ subspace. (c) Antiferromagnetic-Normal
reflection (AF-NR): reflected as electrons, due to the coupling of
$\mathbf{k}$ and ${\bf k}+{\bf Q}$ subspaces. (d) Antiferromagnetic-Andreev
reflection (AF-AR): reflected as holes, due to electron and hole coupling in
the ${\bf k}+{\bf Q}$ subspace (see Fig. 1).

In addition to the effect of AF order, AR is largely modified due to the
exchange field of ferromagnetic metal when electron is not normally incident
into the EDSC region. Owing to the momentum conserved parallel to the
interface, Snell’s law Jackson75 ; Zutic00 ; Kashiwaya99  requires that

$\displaystyle
q_{F\sigma}\sin\theta_{N\sigma}=q_{F\bar{\sigma}}\sin\theta_{A\bar{\sigma}}=$
$\displaystyle k_{F}\sin\theta_{S\sigma},$ (2)

where $\theta_{N\sigma}$, $\theta_{A\bar{\sigma}}$, and $\theta_{S\sigma}$ are
the angles of NR, AR, and transmission into the SC respectively (see Fig. 1).
Incident angle $\theta_{N\sigma}$ is typically not equal to the AR angle
$\theta_{A\bar{\sigma}}$ except when $J=0$ or for normal incidence. Assuming
that there is no FWM and $q_{F\downarrow}<k_{F}<q_{F\uparrow}$, ranges of six
normal and Andreev reflection angles are
$0<\theta_{N\uparrow}<\sin^{-1}(k_{F}/q_{F\uparrow})\equiv\theta_{c2}$,
$0<\theta_{A\uparrow}<\sin^{-1}(q_{F\downarrow}/q_{F\uparrow})\equiv\theta_{c1}$,
and
$0<\theta_{S\uparrow},\theta_{S\downarrow}<\sin^{-1}(q_{F\downarrow}/k_{F})$,
while $\theta_{A\downarrow}$ and $\theta_{N\downarrow}$ can be any angles. For
AF reflections, the angles $\theta_{A\sigma}^{\rm AF}=\pi-\theta_{A\sigma}$
and $\theta_{N\sigma}^{\rm AF}=\theta_{N\sigma}$ respectively. It is noted
that when $\theta_{N\uparrow}$ is within the range
$\theta_{c1}<\theta_{N\uparrow}<\theta_{c2}$, $x$ component of the wave
vector, $\sqrt{q_{F\downarrow}^{2}-k_{F}^{2}\sin^{2}\theta_{S\uparrow}}$,
becomes purely imaginary for the AR process Zutic00 ; Kashiwaya99 . Although
spin down electron as a propagating wave is impossible for AR, it can still
transmit into the superconductor side.

As emphasized by Kashiwaya et al. Kashiwaya99 , one can define two types of
conductance in a FM, namely the charge and spin conductances. As a matter of
fact, the normalized angle and spin dependent tunneling charge conductance is
given by

$\displaystyle
C_{q\sigma}=1-\left|{R_{N\sigma}}\right|^{2}+a_{\sigma}\left|{R_{A\bar{\sigma}}}\right|^{2}+\left|{R^{AF}_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R^{AF}_{A\bar{\sigma}}}\right|^{2},$
(3)

where $a_{\downarrow}\equiv 1$ and $a_{\uparrow}\equiv
L_{\downarrow}\lambda_{2\downarrow}/L_{\uparrow}\lambda_{1\uparrow}$ with
$\lambda_{1\uparrow}=\cos\theta_{N\uparrow}/\cos\theta_{S\uparrow}$,
$\lambda_{2\downarrow}=\cos\theta_{A\downarrow}/\cos\theta_{S\uparrow}$, and
$L_{\sigma}=\sqrt{(q_{F}/k_{F})(1-\sigma J/E_{F}^{F})}$. Detailed derivations
of all four reflection coefficients ($R_{N\sigma}$, $R_{A\bar{\sigma}}$,
$R^{AF}_{N\sigma}$, and $R^{AF}_{A\bar{\sigma}}$) are given in Appendix A.
Similarly, the normalized angle and spin dependent spin conductance is given
by

$\displaystyle
C_{s\sigma}=1-\left|{R_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R_{A\bar{\sigma}}}\right|^{2}+\left|{R^{AF}_{N\sigma}}\right|^{2}-a_{\sigma}\left|{R^{AF}_{A\bar{\sigma}}}\right|^{2}.$
(4)

Comparing with the results of charge conductance in (3), due to the spin
imbalance induced by the exchange field, different sign of $R_{A\bar{\sigma}}$
terms occurs in the spin conductances. Consequently normalized total charge
(spin) conductance is given by

$\displaystyle G_{q(s)}(E)=G_{q(s)\uparrow}(E)\pm G_{q(s)\downarrow}(E),$ (5)

where $+$ ($-$) sign is for charge (spin) channel and

$\displaystyle
G_{q(s)\sigma}(E)=\frac{1}{{G^{N}_{q(s)}}}\int_{\alpha}^{\beta}{d\theta_{N\sigma}\cos\theta_{N\sigma}C_{q(s)\sigma}(E,\theta_{N\sigma})P_{\sigma}}.$
(6)

The lower and upper integration limits of $\alpha$ and $\beta$ are restricted
by Snell’s law (as discussed before) or experimental setup. In practice,
integration over two separate ranges of incident angle, i.e.,
$0<\left|{\theta_{N\sigma}}\right|<\theta_{c1}$ and
$\theta_{c1}<\left|{\theta_{N\sigma}}\right|<\theta_{c2}$ should be carried
and results are added up to the total conductance. In (6), the normal-state
charge (spin) conductance

$\displaystyle
G^{N}_{q(s)}=\int_{-\pi/2}^{\pi/2}{d\theta_{N\sigma}\cos\theta_{N\sigma}[C_{N\uparrow}P_{\uparrow}\pm
C_{N\downarrow}P_{\downarrow}]},$ (7)

where

$\displaystyle
C_{N\sigma}(\theta_{N\sigma})=\frac{{4\lambda_{1}L_{\sigma}}}{{\left|{1+\lambda_{1}L_{\sigma}+2iZ}\right|^{2}}}$
(8)

with $Z=mH/\hbar^{2}k_{F}$ the barrier. In both (6) and (7), we have
introduced a factor $P_{\sigma}=(E_{F}^{F}+\sigma J)/2E_{F}^{F}$ which can be
interpreted as the probability of spin-$\sigma$ incident electron as a
function of the exchange energy Kashiwaya99 ; Zutic00 ; Linder07 . When $J=0$,
$P_{\uparrow}=P_{\downarrow}=1/2$.

In addition to the conductances, the normalized total charge (spin) current
can be given by

$\displaystyle\begin{array}[]{l}I_{q(s)}=I_{q(s)\uparrow}\pm
I_{q(s)\downarrow},\\\ \end{array}$ (10)

where

$\displaystyle I_{q(s)\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{I^{N}_{q(s)}}\int_{-\infty}^{\infty}{dE}\int_{\alpha}^{\beta}d\theta_{N\sigma}\cos\theta_{N\sigma}C_{q(s)\sigma}(E,\theta_{N\sigma})P_{\sigma}q_{F\sigma}$

with

$\displaystyle{I^{N}_{q(s)}}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}{dE}\int_{-\pi/2}^{\pi/2}{d\theta_{N\sigma}\cos\theta_{N\sigma}[C_{N\uparrow}P_{\uparrow}q_{F\uparrow}\pm
C_{N\downarrow}P_{\downarrow}q_{F\downarrow}]}.$

Charge and spin currents and their conversion are important probes for spin-
related phenomena such as those in spin Hall effect.

##  III Results and Discussions

Both charge and spin conductances are important probes for tunneling in spin-
polarized junctions. In this paper, we will focus on the charge conductance
however. Moreover, for simplicity, all the results presented are for normal
incidence ($\theta_{N\sigma}=0$).

###  III.1 Midgap Surface States

Detailed derivations of the midgap surface states (MSS) in the current FM/EDSC
junction are give in Appendix B. Basically it is an extension of Hu’s Hu94
and Liu and Wu’s Liu07  works. The boundary condition that leads to the MSS is
the wave function $\psi_{N\sigma}(x=-d)=0$ for a free boundary at $x=-d$.
Consequently one obtains the following condition for the MSS (see Appendix B):

$\displaystyle e^{-2ik_{1\sigma}d}E_{+}+e^{2ik_{1\sigma}d}E_{-}=2\Phi,$ (13)

where $E_{\pm}\equiv E\pm\varepsilon^{\prime}_{\sigma}$ with
$\varepsilon^{\prime}_{\sigma}=\sqrt{(E+\sigma J)^{2}-\Delta^{2}-\Phi^{2}}$
incident spin-$\sigma$ electron is assumed to have wave vector $k_{1\sigma}$
along the $x$ direction.

In case of $J=0$, the result is reduced to our previous case without spin
polarization Liu07 . In case of $J=\Phi=0$, the result is reduced to Hu’s case
Hu94 , i.e.,

$\displaystyle
e^{4ik_{1}d}=-\frac{{E+{\varepsilon^{\prime}}}}{{E-{\varepsilon^{\prime}}}},$
(14)

where $\varepsilon^{\prime}\equiv\sqrt{E^{2}-\Delta^{2}}$. The most crucial
result of the above is that there exists a zero-energy state which is
responsible for the ZBCP widely observed in hole-doped $d_{x^{2}-y^{2}}$-wave
cuprate superconductors Hu94 . When $J=0$ but $\Phi\neq 0$, zero-energy state
no longer exists such that the energy of the existing state is always finite
($E=\Phi$ in the limit of $d=0$). This leads to the splitting of the ZBCP.
When $J$ is also finite, there will be further effect caused by spin
polarization although the splitting peak remains at $E=\Phi$ in the limit of
$d=0$. It is interesting to note that beyond the quasiclassical approximation,
a more accurate calculation for the surface bound-state energies in
$d_{x^{2}-y^{2}}$-wave and other unconventional cuprate superconductors was
reported by Walker et al. Walker97 .

###  III.2 Effect of Fermi-Wave-Vector Mismatch

Tunneling conductances are in general strongly modified by the effect of
Fermi-wave-vector mismatch (FWM) Zutic00 ; Linder07 . In our case, due to the
presence of the AF order, the conductance spectra are somewhat different from
those obtained by Žutić and Valls Zutic00  and Linder et al. Linder07 . Here
we introduce a parameter

$\displaystyle L_{0}\equiv{q_{F}\over k_{F}}$ (15)

to account for the effect of FWM. Both $L_{0}$ greater and smaller than one
cases are considered. As shown in Figs. 2-4 for the normalized charge
conductance $G_{q}$, the effect of FWM is typically strong when $L_{0}<1$,
while it has relatively minor effect when $L_{0}>1$ (That is, $G_{q}$ changes
little from the no FWM $L_{0}=1$ case.). As first pointed out by Blonder and
Tinkham Blonder83 , FWM can be interpreted as a type of barrier which could
enhance the conductance near zero bias.

Figure 2: Effect of FWM on normalized charge conductance spectra $G_{q}$ for
various wave-vector mismatch value $L_{0}$ with fixed barrier $Z=0$, AF order
$\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$. Figure 3: Effect of FWM
on normalized charge conductance spectra $G_{q}$ for various wave-vector
mismatch value $L_{0}$ with fixed barrier $Z=1$, AF order $\Phi=0$, and spin
polarization $X=0$. This can be considered as the case of hole-doped cuprate
superconductors without AF order and in the absence of spin polarization.

In our previous paper Liu07 , it was shown that ZBCP of a
$d_{x^{2}-y^{2}}$-wave superconductor can be split by the AF order $\Phi$. No
spin-active barrier Kashiwaya99 ; Linder07 , external magnetic field, and spin
polarization effects were considered in our previous case though. Previously
Žutić and Valls Zutic00  had given a detailed analysis of the FWM effect on
the conductance in ferromagnet/s-wave and d-wave superconductor junctions.
Here we show how FWM influences the conductance in the current case and point
out the key physics. Fig. 2 plots $G_{q}$ for various $L_{0}$ with barrier
$Z=0$, AF order $\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$ [see Eq.
(18) for the definition of $X$]. One sees that the effect of FWM is most
noticeable at large FWM ($L_{0}=0.2$ case) to which a ZBCP is developed, while
the spectra are humdrum when $L_{0}\geq 1$. Since no barrier ($Z=0$) is
considered, no effect of AF order and spin polarization is seen in terms of
peak splitting. Note that normalized zero-bias conductance is not equal to 2
due to the presence of AF order and spin polarization. In order to compare
with the case of hole-doped high-$T_{c}$ superconductors (without AF order),
Fig. 3 plots $G_{q}$ at different values of $L_{0}$ with $\Phi=X=0$ and $Z=1$.
One sees that ZBCP is largely enhanced by the FWM effect (see $L_{0}=0.2$
case). Thus FWM can significantly enhance the number of midgap surface states
near zero-bias voltage.

Figure 4: Effect of FWM on normalized charge conductance spectra $G_{q}$ for
various wave-vector mismatch value $L_{0}$ with fixed barrier $Z=1$, AF order
$\Phi=0.5\Delta_{0}$, and spin polarization $X=0.5$. FWM causes the reduction
of conductance at zero bias, while enhances the splitting peak associated with
the AF order.

Aiming to electron-doped cuprate superconductors, Fig. 4 shows the effect of
FWM on the splitting peak when the AF order is present ($\Phi=0.5\Delta_{0}$).
Here barrier $Z=1$ and spin polarization $X=0.5$. In contrast to the case of
$\Phi=0$ in Fig. 3, FWM actually reduces the number of midgap surface states
near zero bias. At the same time, it enhances the strength of the splitting
peak associated with the AF order. Following the idea of Blonder and Tinkham
Blonder83  such that FWM can be interpreted as a type of barrier, the
enhancement of ZBCP in Fig. 3 and the reduction of zero-bias conductance in
Fig. 4 is a natural outcome at large FWM.

In principle, the effect of FWM should be included when a serious calculation
is performed for spin-polarized conductances.

###  III.3 Effect of Spin Polarization

In the literature, there exists different definitions of spin polarization.
One example is the “tunneling polarization” proposed by Tedrow and Meservey
Tedrow94 . In point contact experiment, the more suitable definition is the
so-called “contact polarization” Soulen98

$\displaystyle
P_{c}=\frac{{N_{\uparrow}(E_{F})v_{F\uparrow}-N_{\downarrow}(E_{F})v_{F\downarrow}}}{{N_{\uparrow}(E_{F})v_{F\uparrow}+N_{\downarrow}(E_{F})v_{F\downarrow}}},$
(16)

where $v_{F\sigma}$ and $N_{\sigma}(E_{F})$ are respectively the Fermi
velocity and DOS at Fermi level for spin-$\sigma$ electron. Since
$I_{\sigma}\propto N_{\sigma}(E_{F})v_{F\sigma}$, Eq. (16) is identical to

$\displaystyle
P_{c}=\frac{{I_{\uparrow}-I_{\downarrow}}}{{I_{\uparrow}+I_{\downarrow}}}.$
(17)

However, the most natural definition of spin polarization is

$\displaystyle
X\equiv\frac{{N_{\uparrow}(E_{F})-N_{\downarrow}(E_{F})}}{{N_{\uparrow}(E_{F})+N_{\downarrow}(E_{F})}}.$
(18)

In ballistic point contact situation, the electron density of states in the
presence of an exchange field can be written as
$N_{\sigma}(E_{F})=q_{F\sigma}^{2}A/4\pi$, where $A$ is the area of the
interface. Thus $X=J/E_{F}^{F}$ with
$E_{F}^{F}\equiv\hbar^{2}q_{F}^{2}/2m=(\hbar^{2}q_{F\uparrow}^{2}/2m+\hbar^{2}q_{F\downarrow}^{2}/2m)/2$
Chalsani07 . In Sec. III.E, we will show that spin polarization $X$ can be
determined by a general formula in combination with the experimental
conductance data.

Figure 5: Effect of spin polarization on normalized charge conductance spectra
$G_{q}$ for various spin polarization value $X$ with fixed barrier $Z=0$, AF
order $\Phi=0.5\Delta_{0}$, and without FWM ($L_{0}=1$). Figure 6: Effect of
spin polarization on normalized charge conductance spectra $G_{q}$ for various
spin polarization value $X$ with fixed barrier $Z=1$, AF order $\Phi=0$, and
without FWM ($L_{0}=1$). This is considered an example of the hole-doped
cuprate superconductor without AF order and FWM. Figure 7: Effect of spin
polarization on normalized charge conductance spectra $G_{q}$ for various spin
polarization value $X$ with fixed barrier $Z=1$, AF order
$\Phi=0.5\Delta_{0}$, and without FWM ($L_{0}=1$). Low-energy anomalous
conductance enhancement arises due to AF contributions (see text for details).

Note that current quasiparticle wave function of BdG equations has four
components which involve two components associated with the AF order. In the
limit of $Z=0$ and without spin polarization ($X=0$), normalized charge
conductance has value 2 as expected (see Fig. 5). With a finite AF order
($\Phi=0.5\Delta_{0}$), the resulting effective gap magnitude is about
$\tilde{\Delta}\simeq 1.12\Delta_{0}$ (see Fig. 5). In general, at
$E<\tilde{\Delta}$, effect of spin polarization is to suppress the
conductance. When FWM is absent ($L_{0}=1$) together with $Z=0$, normal
reflection has no contribution and Andreev reflection actually dominates the
tunneling process for $E<\tilde{\Delta}$ Blonder82 . In our current case,
Andreev reflection involves contributions from both $R_{A}$ and $R_{A}^{AF}$
channels.

The most interesting results occur when the barrier $Z$ is finite. When the AF
order $\Phi=0$ (as for the case of hole-doped cuprate superconductors) to
which $R_{N}^{AF}=R_{A}^{AF}=0$, ZBCP appears whose (normalized) strength is
largely suppressed due to the strong spin polarization effect (see Fig. 6).
However, as seen in Fig. 7, when AF order is finite ($\Phi=0.5\Delta_{0}$), in
contrast, the strengths of both the zero-bias conductance and the splitting
peak turn out to get enhanced by the strong spin polarization effect. This
“anomalous conductance enhancement” phenomenon is in drastic contrast as
compared to the ZBCP associated with $\Phi=0$ case (Fig. 6). These somewhat
surprising results arise due to a significant increase of $|R_{N}^{AF}|$ and
at the same time, a significant decrease of $|R_{A}^{AF}|$ for large $X$ cases
– a consequence of the interplay between AF order and spin polarization. Since
$|R_{N}^{AF}|$ contributes positively to the conductance, while $|R_{A}^{AF}|$
contributes negatively to the conductance [see Eq. (3)], resultantly they
cause the anomalous conductance enhancement at low energies ($E\leq\Phi$). It
should be emphasized that this low-energy conductance enhancement is not due
to the spin-flip effect which is not considered in this paper. At higher
energies, $E>\Phi$, the conductances behave more normally such that they get
suppressed due to the spin polarization effect. Anomalous conductance
enhancement at low energies can serve as a test to see whether there is an
significant AF order in electron-doped cuprate superconductors.

Interface barrier and band structure are in general having strong effect on
spin polarization. Kant et al. have built an“extended interface” model to
illustrate the decay of spin polarization Kant02 . Besides, Mazin had a
detailed discussion on the definition of spin polarization and band structure
effects in spin polarization Mazin99 .

###  III.4 Effect of Effective Barrier

In the study of the tunneling transition in Cu-Nb point contacts, Blonder and
Tinkham Blonder83  pointed out that barrier is not the only source for normal
reflection and in a more realistic system, one should consider “impedance” or
FWM as well which results in normal reflection even with no barrier present.
They proposed an effective barrier $Z_{\rm eff}=[Z^{2}+(1-r)^{2}/4r]^{1/2}$
where $r$ is the Fermi velocity ratio. They showed that effective barrier has
an obvious effect on the conductance when $E<\Delta_{0}$, as shown in Fig. 2
of Ref. Blonder83 . Here we generalize their idea to consider a spin, FWM, and
angle dependent effective barrier $Z_{\rm eff}$ Blonder83 ; Zutic00 :

$\displaystyle Z_{\rm
eff}\equiv[Z^{2}+(1-L_{\sigma})^{2}/4L_{\sigma}]^{1/2}/\cos\theta_{S\sigma},$
(19)

where $L_{\sigma}=q_{F\sigma}/k_{F}$ corresponds to the spin-dependent FWM. It
is noted that we are not considering the spin-active barrier which has spin
filtering effects and can lead to the ZBCP splitting Kashiwaya99 ; Linder07 .
Instead we propose a possible alternative mechanism to account for the decay
of spin polarization. Based on the generalized effective barrier, spin-up and
-down particles experience different strength of effective barrier that causes
spin-up and -down currents to decrease at different speed as compared to the
current in the absence of barrier. Consequently, $Z_{\rm eff}$ can modify the
values of $I_{\uparrow}-I_{\downarrow}$ (and thus $P_{c}$) dramatically. With
this strong effect at work, the decay of spin polarization should not be
dominant by the spin-flitting process in the point contact spin polarization
case.

Figure 8: Effect of effective barrier $Z_{\rm eff}$ on normalized charge
conductance spectra $G_{q}$ for various values of FWM $L_{0}$. AF order
$\Phi=0.5\Delta_{0}$ and spin polarization $X=0.5$. The bare barrier $Z$ is
set to zero, while $Z_{\rm eff}$ is given by Eq. (19).

As seen in Eq. (19), $Z_{\rm eff}$ can differ significantly from $Z$,
especially when $Z$ is small. Essentially their difference can be measured by
spin-polarized tunneling experiments. In Fig. 8, we compare the effects of $Z$
and $Z_{\rm eff}$ on the conductance with bare barrier $Z$ set to zero and
vary the FWM $L_{0}$ value. For $Z=0$ and $\theta_{S\sigma}=0$, $Z_{\rm
eff}=[(1-L_{\sigma})^{2}/4L_{\sigma}]^{1/2}$ [see (19)]. In our case, we have
also included AF order and spin polarization. The difference is most
noticeable when FWM is large ($L_{0}=0.2$ case). Since $Z=0$, AF order and
spin polarization have little effect at small FWM. However, when FWM is large,
AF order and spin polarization can have a strong effect such that a splitting
peak can develop at $E\approx\Phi=0.5\Delta_{0}$ with the effective barrier
$Z_{\rm eff}$ (see Fig. 8). This supports Blonder and Tinkham’s idea of
“impedance” mismatch which enhances the normal reflection.

###  III.5 A General Formula for Determining the Spin Polarization

Based on the phenomenon of Andreev reflection, Soulen et al. Soulen98
proposed a formula for determining the point contact spin polarization $P_{c}$
[see Eqs. (16) and (17)] when the normalized zero-bias conductance data is
compared. Their original form was

$\displaystyle G(0)/G_{N}=2(1-P_{c}),$ (20)

which is valid only when FWM is absent Strijkers01 . Since Andreev reflection
could be strongly modified due to the FWM effect, it’s useful to replace Eq.
(20) by

$\displaystyle
G(0)/G_{N}=\left[1+\left|{R_{A}}\right|^{2}-\left|{R_{A}^{AF}}\right|^{2}\right](1-P_{c}),$
(21)

where $R_{A}$ and $R^{AF}_{A}$ are the AR and AF-AR coefficients respectively.
Eq. (21) can be reduced to Eq. (20) when the exchange energy $J$ is set to
zero in $R_{A}$ and the AF order $\Phi$ is set to zero in $R^{AF}_{A}$. Note
also that the parameter $X$ should be set to zero when the “contact
polarization” $P_{c}$ is determined under the idea of Soulen et al.

Here we propose a more general formula for determining the spin polarization:

$\displaystyle G(0)/G_{N}=A_{\uparrow}+A_{\downarrow},$ (22)

where

$\displaystyle
A_{\uparrow}=\int_{\alpha}^{\beta}d\theta_{N\sigma}\cos\theta_{N\sigma}(1+a_{\uparrow}|{R_{A\downarrow}}|^{2}-a_{\uparrow}|{R_{A\downarrow}^{AF}}|^{2})P_{\uparrow}$
(23)

and

$\displaystyle
A_{\downarrow}=\int_{\alpha}^{\beta}{d\theta_{N\sigma}\cos\theta_{N\sigma}(1+|{R_{A\uparrow}}|^{2}-|{R_{A\uparrow}^{AF}}|^{2})}P_{\downarrow}.$
(24)

Here $R_{A\sigma}=R_{A\sigma}(L_{0},X,\Phi,\theta_{N\sigma})$ and
$R_{A\sigma}^{AF}=R_{A\sigma}^{AF}(L_{0},X,\Phi,\theta_{N\sigma})$ with $E=0$.
Eq. (22) is a natural result of our earlier formalism. It is regarded as the
generalization of Eq. (20) of Soulen et al., which includes the effects of
FWM, spin polarization, AF order, as well as the incident angle.

##  IV Conclusions

Tunneling experiment provides a useful tool for probing the properties of a
superconductor such as the magnitude and symmetry of the superconducting order
parameter, quasiparticle density of states, and any existing competing orders.
In fact, tunneling experiment is also a powerful probe for investigating the
spin-charge separation in connection with the spin-injection techniques. This
involves both charge imbalance and spin imbalance studies.

In this paper, we have presented a detailed study of the tunneling conductance
spectra of a ferromagnetic metal/electron-doped superconductor junctions,
taking into account an AF order existing in the the electron-doped
superconductor. Interesting result, such as low-energy anomalous conductance
enhancement, occurs as a result of the interplay between AF order and spin
polarization (see Fig. 7). These results in turn provide a further opportunity
to test whether there is an significant AF order in electron-doped cuprate
superconductors.

###### Acknowledgements.

This work is supported by National Science Council of Taiwan (Grant No.
96-2112-M-003-008) and National Natural Science Foundation of China (Grant No.
10347149). We also acknowledge the support from the National Center for
Theoretical Sciences, Taiwan.

##  Appendix A Reflection Coefficients

Under the WKBJ approximation Bardeen69 ; Bar-Sagi72 ; Hu75 ; Bruder90 ;
Blonder82 ; Hu94 ; Tanaka95 ; Kashiwaya96 , the wave functions in the
generalized BdG equations (LABEL:eq:BdG) can be approximated by

$\displaystyle\left({\begin{array}[]{*{20}c}{u_{1\sigma}}\\\
{v_{1\bar{\sigma}}}\\\ {u_{2\sigma}}\\\ {v_{2\bar{\sigma}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{e^{i{\bf{k}}_{F}\cdot{\bf{r}}}\tilde{u}_{1\sigma}}\\\
{e^{i{\bf{k}}_{F}\cdot{\bf{r}}}\tilde{v}_{1\bar{\sigma}}}\\\
{e^{i{\bf{k}}_{F{\bf Q}}\cdot{\bf{r}}}\tilde{u}_{2\sigma}}\\\
{e^{i{\bf{k}}_{F{\bf Q}}\cdot{\bf{r}}}\tilde{v}_{2\bar{\sigma}}}\\\
\end{array}}\right).$ (33)

Thus one obtains a set of Andreev equations in the $x$ direction,

$\displaystyle E\tilde{u}_{1\sigma}(x)$ $\displaystyle=$ $\displaystyle
H_{\sigma}\tilde{u}_{1\sigma}(x)+\Delta({\bf{\hat{k}}}_{F})\tilde{v}_{1\bar{\sigma}}(x)+\Phi\tilde{u}_{2\sigma}(x)$
$\displaystyle E\tilde{v}_{1\bar{\sigma}}(x)$ $\displaystyle=$
$\displaystyle\Delta^{*}({\bf{\hat{k}}}_{F})\tilde{u}_{1\sigma}(x)-H_{\sigma}\tilde{v}_{1\bar{\sigma}}(x)+\Phi\tilde{v}_{2\bar{\sigma}}(x)$
$\displaystyle E\tilde{u}_{2\sigma}(x)$ $\displaystyle=$
$\displaystyle\Phi\tilde{u}_{1\sigma}(x)-H_{\sigma}\tilde{u}_{2\sigma}(x)+\Delta({\bf{\hat{k}}}_{F{\bf
Q}})\tilde{v}_{2\bar{\sigma}}(x)$ $\displaystyle
E\tilde{v}_{2\bar{\sigma}}(x)$ $\displaystyle=$
$\displaystyle\Phi\tilde{v}_{1\bar{\sigma}}(x)+\Delta^{*}({\bf{\hat{k}}}_{F{\bf
Q}})\tilde{u}_{2\sigma}(x)+H_{\sigma}\tilde{v}_{2\bar{\sigma}}(x),$

where $H_{\sigma}=-\frac{{i\hbar^{2}k_{F}}}{m}\frac{d}{{dx}}-\sigma J$ and $x$
is the coordinate normal to the interface. The $d_{x^{2}-y^{2}}$-wave SC gap
$\Delta({\bf{\hat{k}}}_{F})=-\Delta({\bf{\hat{k}}}_{F{\bf
Q}})\equiv\Delta_{0}\sin 2\theta$ with $\theta$ the azimuthal angle relative
to the $x$-axis. In obtaining Eq. (LABEL:eq:BdG2), the Fourier transform of
the Cooper pair order parameter $\Delta({\bf s},{\bf r})$ from relative
coordinate ${\bf s}$ to ${\bf k}$ space is assumed to take the form,
$\Delta({\bf{k}},{\bf{r}})=\Delta({\bf\hat{k}}_{F})\Theta(x)$, with
$\Theta\left(x\right)$ the Heaviside step function Hu94 ; Kashiwaya96 .

Solving Eq. (LABEL:eq:BdG2), one obtains four eigenvectors which build up the
spin-$\sigma$ wave function in the superconductor region ($x>0$) Zutic00 ,

$\displaystyle\psi_{S\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}\Delta\\\
{E_{-}}\\\ 0\\\ \Phi\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{E_{+}}\\\
\Delta\\\ \Phi\\\ 0\\\ \end{array}}\right)}\right]e^{ik^{+}x}$ (43)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{E_{-}}\\\
{-\Delta}\\\ \Phi\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-\Delta}\\\
{E_{+}}\\\ 0\\\ \Phi\\\ \end{array}}\right)}\right]e^{-ik^{-}x}.$ (52)

Here $E_{\pm}\equiv E\pm\varepsilon_{\sigma}$ with
$\varepsilon_{\sigma}=\sqrt{E^{2}-\Delta^{2}-\Phi^{2}}$,
$\Delta\equiv\Delta({\bf{\hat{k}}}_{F})$,
$k^{+}=k^{-}=k_{F}\cos\theta_{S\sigma}$, and $c_{i\sigma}$ are coefficients of
the corresponding waves. As pointed out by Blonder et al. Blonder82 , there is
no need to normalize the coefficients as it just complicates the calculation.
If we set $\Phi=J=0$ and normalize the coefficients, it will reduce to the
case for a typical N/I/S junction Blonder82 ; Tanaka95 ; Kashiwaya96 .

Since we consider that there is an AF order in the EDSC side, an incident
electron from the FM side will have four possible reflections Liu07 . The
spin-$\sigma$ wave function in the FM side $(x<0)$ with incident angle
$\theta_{N\sigma}$ can thus be written as Blonder82 ; Kashiwaya96 ;
Kashiwaya99

$\displaystyle\Psi_{N\sigma}(x)=\left({\begin{array}[]{*{20}c}{e^{iq_{F\sigma}\cos\theta_{N\sigma}x}+R_{N\sigma}e^{-iq_{F\sigma}\cos\theta_{N\sigma}x}}\\\
{R_{A\bar{\sigma}}e^{iq_{F\bar{\sigma}}\cos\theta_{A\bar{\sigma}}x}}\\\
{R^{AF}_{N\sigma}e^{iq_{F\sigma}\cos\theta_{N\sigma}x}}\\\
{R^{AF}_{A\bar{\sigma}}e^{-iq_{F\bar{\sigma}}\cos\theta_{A\bar{\sigma}}x}}\\\
\end{array}}\right),$ (57)

where $R_{N\sigma}$, $R_{A\bar{\sigma}}$, $R^{AF}_{N\sigma}$, and
$R^{AF}_{A\bar{\sigma}}$ are amplitudes of NR, AR, AF-NR, and AF-AR
respectively. Applying the following boundary conditions:

$\displaystyle\psi_{N\sigma}\left(x\right)|_{x=0^{-}}$ $\displaystyle=$
$\displaystyle\psi_{S\sigma}\left(x\right)|_{x=0^{+}}$ (58)
$\displaystyle\frac{2mH}{\hbar^{2}}\psi_{S\sigma}\left(x\right)|_{x=0^{+}}$
$\displaystyle=$
$\displaystyle\frac{d\psi_{S\sigma}\left(x\right)}{dx}|_{x=0^{+}}-\frac{d\psi_{N\sigma}\left(x\right)}{dx}|_{x=0^{-}},$

the four reflection amplitudes (coefficients) are solved to be

$\displaystyle R_{N\sigma}$ $\displaystyle=$
$\displaystyle\frac{{E_{-}(1-L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})B}}{{(1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})D}}$
$\displaystyle-$
$\displaystyle\frac{{\Delta(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}+2iZ_{\theta})A}}{{(1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta})D}}$
$\displaystyle-$
$\displaystyle\frac{{1-L_{\sigma}\lambda_{1\sigma}-2iZ_{\theta}}}{{1+L_{\sigma}\lambda_{1\sigma}+2iZ_{\theta}}}$
$\displaystyle R_{A\bar{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{{\Delta(1+L_{\sigma}\lambda_{1\sigma}-2iZ_{\theta})B}}{{(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})D}}$
$\displaystyle+$
$\displaystyle\frac{{E_{-}(1-L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})A}}{{(1+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}-2iZ_{\theta})D}}$
$\displaystyle R^{AF}_{N\sigma}$ $\displaystyle=$ $\displaystyle\frac{{\Phi
B}}{D}$ $\displaystyle R^{AF}_{A\bar{\sigma}}$ $\displaystyle=$
$\displaystyle\frac{{\Phi A}}{D},$ (59)

where

$\displaystyle A$ $\displaystyle=$ $\displaystyle 2\Delta
L_{\sigma}\lambda_{1\sigma}[1-L_{\sigma}L_{\bar{\sigma}}\lambda_{1\sigma}\lambda_{2\bar{\sigma}}+4Z_{\theta}^{2}$
$\displaystyle+$ $\displaystyle
2iZ_{\theta}(L_{\sigma}\lambda_{1\sigma}+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}})]$
$\displaystyle B$ $\displaystyle=$ $\displaystyle
2L_{\sigma}\lambda_{1\sigma}[2L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}E+\varepsilon(1+L_{\bar{\sigma}}^{2}\lambda_{2\bar{\sigma}}^{2})]$
$\displaystyle D$ $\displaystyle=$
$\displaystyle\Delta^{2}[(1-L_{\sigma}L_{\bar{\sigma}}\lambda_{1\sigma}\lambda_{2\bar{\sigma}}+4Z_{\theta}^{2})^{2}$
(60) $\displaystyle+$ $\displaystyle
4Z_{\theta}^{2}(L_{\sigma}\lambda_{1\sigma}+L_{\bar{\sigma}}\lambda_{2\bar{\sigma}})^{2}]$
$\displaystyle+$ $\displaystyle[2L_{\sigma}\lambda_{1\sigma}E+4\varepsilon
Z_{\theta}^{2}+\varepsilon(1+L_{\sigma}^{2}\lambda_{1\sigma}^{2})]$
$\displaystyle\times$
$\displaystyle[2L_{\bar{\sigma}}\lambda_{2\bar{\sigma}}E+4\varepsilon
Z_{\theta}^{2}+\varepsilon(1+L_{\bar{\sigma}}^{2}\lambda_{2\bar{\sigma}}^{2})].$

Moreover $Z_{\theta}=Z/\cos\theta_{S\sigma}$ with the barrier
$Z=mH/\hbar^{2}k_{F}$,
$\lambda_{1\sigma}=\cos\theta_{N\sigma}/\cos\theta_{S\sigma}$,
$\lambda_{2\bar{\sigma}}=\cos\theta_{A\bar{\sigma}}/\cos\theta_{S\sigma}$, and
${\rm{}}L_{\sigma}=\sqrt{q_{F}/k_{F}-\sigma(q_{F}/k_{F})(J/E_{FN})}$. It is
interesting to note in (59) that $R^{AF}_{N\sigma}$ and
$R^{AF}_{A\bar{\sigma}}$ are proportional to the AF order $\Phi$, as is
expected.

##  Appendix B Midgap Surface States

Following Ref. Hu94 , we first assume that

$\displaystyle\left(\begin{array}[]{c}\tilde{u}_{l\sigma}\\\
\tilde{v}_{l\sigma}\end{array}\right)=e^{-\gamma_{\sigma}x}\left(\begin{array}[]{c}\hat{u}_{l\sigma}\\\
\hat{v}_{l\sigma}\end{array}\right),$ (65)

where $\gamma_{\sigma}$ is the attenuation constant for
$|{E({\bf{q}}_{F\sigma})}|<\sqrt{|{\Delta({\bf{\hat{k}}}_{F})}|^{2}+\Phi^{2}}$.
With (65), Eq. (LABEL:eq:BdG2) becomes

$\displaystyle E\left({\begin{array}[]{*{20}c}{\hat{u}_{1\sigma}}\\\
{\hat{v}_{1\bar{\sigma}}}\\\ {\hat{u}_{2\sigma}}\\\
{\hat{v}_{2\bar{\sigma}}}\\\
\end{array}}\right)=\left({\begin{array}[]{*{20}c}{h}&{\Delta}&\Phi&0\\\
{\Delta}&{-h}&0&\Phi\\\ \Phi&0&{-h}&{-\Delta}\\\ 0&\Phi&{-\Delta}&{h}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}{\hat{u}_{1\sigma}}\\\
{\hat{v}_{1\bar{\sigma}}}\\\ {\hat{u}_{2\sigma}}\\\
{\hat{v}_{2\bar{\sigma}}}\\\ \end{array}}\right)$ (78)

for the superconducting overlayer ($x>0$). Here
$h={\varepsilon^{\prime}_{\sigma}}-\sigma J$ with
${\varepsilon^{\prime}_{\sigma}}=i\hbar^{2}m^{-1}\gamma_{\sigma}q_{F}\cos\theta_{N\sigma}$.
The wave-vector components parallel to the interface are conserved for all
possible processes.

Solving Eq. (78), one obtains double degenerate eigenvalues
$E=\pm\sqrt{\Delta^{2}+\Phi^{2}+{\varepsilon^{\prime}_{\sigma}}^{2}}-\sigma
J$, where $+$ ($-$) corresponds to the electron- (hole-) like QP excitation.
Similar to the wave function (52), superposition of the four eigenstates makes
up the formal wave function for the superconductor overlayer ($x>0$)

$\displaystyle\psi_{S\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}\Delta\\\
{E_{-}}\\\ 0\\\ \Phi\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{E_{+}}\\\
\Delta\\\ \Phi\\\ 0\\\
\end{array}}\right)}\right]e^{-\gamma_{\sigma}x}e^{ik^{+}x}$ (87)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{E_{-}}\\\
{-\Delta}\\\ \Phi\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-\Delta}\\\
{E_{+}}\\\ 0\\\ \Phi\\\
\end{array}}\right)}\right]e^{-\gamma_{\sigma}x}e^{-ik^{-}x}.$ (96) (97)

Here $E_{\pm}\equiv E\pm\varepsilon^{\prime}_{\sigma}$ with
$\varepsilon^{\prime}_{\sigma}=\sqrt{(E+\sigma J)^{2}-\Delta^{2}-\Phi^{2}}$,
$c_{i}$ are coefficients of the corresponding waves, and
$k^{+}=k^{-}=k_{F}\cos\theta_{S\sigma}$. At the interface, the wave functions
of FM and superconductor meet ideal continuity
$\psi_{N\sigma}(x=0)=\psi_{S\sigma}(x=0)$. After some algebra, the formal wave
function for the FM overlayer is obtained to be ($x<0$):

$\displaystyle\psi_{N\sigma}(x)=\left[{c_{1\sigma}\left({\begin{array}[]{*{20}c}{e^{ik_{1\sigma}x}\Delta}\\\
{e^{-ik_{1\sigma}x}E_{-}}\\\ 0\\\ {e^{ik_{1\sigma}x}\Phi}\\\
\end{array}}\right)+c_{2\sigma}\left({\begin{array}[]{*{20}c}{e^{ik_{1\sigma}x}E_{+}}\\\
{e^{-ik_{1\sigma}x}\Delta}\\\ {e^{-ik_{1\sigma}x}\Phi}\\\ 0\\\
\end{array}}\right)}\right]e^{ik^{+}x}$ (104)
$\displaystyle+\left[{c_{3\sigma}\left({\begin{array}[]{*{20}c}{e^{-ik_{1\sigma}x}E_{-}}\\\
{-e^{ik_{1\sigma}x}\Delta}\\\ {e^{ik_{1\sigma}x}\Phi}\\\ 0\\\
\end{array}}\right)+c_{4\sigma}\left({\begin{array}[]{*{20}c}{-e^{-ik_{1\sigma}x}\Delta}\\\
{e^{ik_{1\sigma}x}E_{+}}\\\ 0\\\ {e^{-ik_{1\sigma}x}\Phi}\\\
\end{array}}\right)}\right]e^{-ik^{-}x},$ (112) (113)

where it is assumed that incident spin-$\sigma$ electron has the wave vector
$k_{1\sigma}$ along the $x$ direction. Considering the free boundary at
$x=-d$, $\psi_{N\sigma}(x=-d)=0$, one thus obtains the condition for the
surface bound states:

$\displaystyle e^{-2ik_{1\sigma}d}E_{+}+e^{2ik_{1\sigma}d}E_{-}=2\Phi.$ (114)

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# Photon frequency variation in non-linear electro-magnetism

Alessandro D.A.M. Spallicci 111email<EMAIL_ADDRESS>Laboratoire
de Physique et Chimie de l’Environnement et de l’Espace (LPC2E) UMR 7328
Centre National de la Recherche Scientifique (CNRS), Université d’Orléans
(UO), Centre National d’Études Spatiales (CNES)
3A Avenue de la Recherche Scientifique, 45071 Orléans, France Observatoire des
Sciences de l’Univers en region Centre (OSUC) UMS 3116
Université d’Orléans (UO), Centre National de la Recherche Scientifique (CNRS)
Observatoire de Paris (OP), Université Paris Sciences & Lettres (PSL)
1A rue de la Férollerie, 45071 Orléans, France UFR Sciences et Techniques,
Université d’Orléans, Rue de Chartres, 45100 Orléans, France Abedennour Dib
222email<EMAIL_ADDRESS>Laboratoire de Physique et Chimie de
l’Environnement et de l’Espace (LPC2E) UMR 7328
Centre National de la Recherche Scientifique (CNRS), Université d’Orléans
(UO), Centre National d’Études Spatiales (CNES)
3A Avenue de la Recherche Scientifique, 45071 Orléans, France Observatoire des
Sciences de l’Univers en region Centre (OSUC) UMS 3116
Université d’Orléans (UO), Centre National de la Recherche Scientifique (CNRS)
Observatoire de Paris (OP), Université Paris Sciences & Lettres (PSL)
1A rue de la Férollerie, 45071 Orléans, France UFR Sciences et Techniques,
Université d’Orléans, Rue de Chartres, 45100 Orléans, France José A. Helayël-
Neto 333email<EMAIL_ADDRESS>Departamento de Astrofísica, Cosmologia e
Interações Fundamentais (COSMO), Centro Brasileiro de Pesquisas Físicas (CBPF)
Rua Xavier Sigaud 150, 22290-180 Urca, Rio de Janeiro, RJ, Brazil

(6 April 2024)

###### Abstract

We set a generalised non-linear Lagrangian, encompassing Born-Infeld and
Heisenberg-Euler theories among others. The Lagrangian reduces to the Maxwell
Lagrangian at lowest order. The field is composed by a propagating light-wave
in an electro-magnetic background. The wave exhibits energy variation when the
background is space-time dependent. In the photon description, this implies a
red or a blue shift, like what we obtained in massive theories, as the de
Broglie-Proca or effective mass theories as the Standard-Model Extension under
Lorentz symmetry violation. The two results, photon energy-conservation and
the frequency shift are instead new for non-linear electro-magnetism. We
conclude by discussing how these static frequency shifts when added to the
expansion red shift allow new interpretations in cosmology or for atomic
spectra. We finally consider the consequences on the Poincaré symmetry.

Non-linear electro-magnetism, Light propagation, Energy non-conservation,
Photon mass

###### pacs:

##  I Introduction.

Non-Linear Electro-Magnetism (NLEM) was proposed first by Born and Infeld (BI)
Born and Infeld (1934, 1935) for regularising point charges and by Heisenberg
and Euler (HE) von Heisenberg and Euler (1936) for dealing with strong fields.
Both theories are used for second order Quantum Electro-Dynamics (QED) Fouché
et al. (2016). While the HE model is nowadays thoroughly derivable through QED
loop calculations, it is still a very useful tool to also study vacuum
birefringence and it is actively used by various collaborations to analyse
through strong magnetic field based experiments vacuum polarisation, constancy
of the speed of light and the nature of vacuum Ejlli et al. (2020); Agil et
al. (2022, 2023); Robertson et al. (2021); Kraych et al. (2024). The QED
predicted value of birefringence was only a factor seven away from the
experimental confirmation Ejlli et al. (2020). Meanwhile, the BI model found a
renewed interest from the community as it naturally emerges as a low-energy
limit of certain string theories A. A. Tseytlin (2000).

For a review including more recent NLEM theories, see Sorokin (2022). The non-
linear effects of electro-magnetism are investigated experimentally also
through interferometry Schellstede et al. (2015) and colliders for photon
splitting Akhmadaliev et al. (2002), photon-photon interactions d’Enterria and
da Silveira (2013); Pike et al. (2014); Aaboud et al. (2017), analysed with
respect to the BI theory Ellis et al. (2017, 2022).

In the Standard-Model (SM) Extension (SME) under the Lorentz Symmetry
Violation (LSV), an effective mass emerges and its value is related to the LSV
vector or tensor Bonetti et al. (2017a); Bonetti et al. (2018). There occurs
also an energy variation of the light-wave propagating in an Electro-Magnetic
(EM) and LSV backgrounds Bonetti et al. (2018). The variation of the light-
wave energy contains a term that couples to the LSV background and to an EM
constant background.

In massive theories as de Broglie-Proca (dBP) de Broglie (1923, 1936); Proca
(1936, 1937); de Broglie (1940), the variation of the energy contains a term
which again does not require the EM background to be space-time dependent.
Even if the field is constant, the associated potential is not, and this
condition suffices. Energy varies also in Maxwell’s theory if the background
is space-time dependent Spallicci et al. (2022a).

Herein, we turn to Generalised NLEM (GNLEM) and work out the frequency shift,
while in future work, we deal with the emergence of a mass in GNLEM. Photon
masses are differently tested in the solar wind Retinò et al. (2016) and
through Fast Radio Bursts Bonetti et al. (2016); Bonetti et al. (2017b).
Herein, we work in absence of LSV and of any pre-defined mass and ask under
which conditions in vacuo the photon energy would not be conserved.

As we did for the dBP and SME cases, we use the correspondence wave-particle,
down to a single photon Aspect and Grangier (1987) and derive the energy-
momentum tensor variation of the photon and therefore a frequency shift. We
consider the EM background non-dynamical and we do not describe its equations
of motion. We figure the photon as a small perturbation of the EM background.
In future work, we will tackle the issue of a dynamical background, imposing
that its energy-momentum tensor variation equates that of the photon.

Finally, the frequency shift is towards the blue or the red. This $z_{\rm S}$
shift is static by nature and it is not associated to the universe expansion.
But if added to the expansion red shift, it spins-off new interpretations in
cosmology, thanks to the recasting of the observed $z$ shift. We applied the
recasting for examining alternatives to the accelerated expansion. Indeed, by
combining the static and the expansion shifts, the red shift and luminosity
distances of Supernovae Ia (SNeIa) get in agreement, without evoking dark
energy J. A. Helayël-Neto and Spallicci (2019); Spallicci et al. (2021);
Spallicci et al. (2022a); Sarracino et al. (2022). The findings of the James
Webb Space telescope appear to be in tension with $\Lambda$CDM cosmology Gupta
(2023) and again applying the recasting might alleviate or cancel the tension.

The GNLEM photon energy variation has a large range of orders of magnitude.
The uncertainty is due to a very difficult assessment, requiring not only the
knowledge of the magnetic fields (strength and direction) for each of the
galaxies and of the intergalactic spaces crossed by the photon, but also
knowledge of their respective alignments.

##  II Generalised Non-Linear Electro-Magnetism

We build a general Lagrangian ${\cal L}={\cal L}({\cal F},{\cal G})$ [SI
units], as polynomial, function of integer powers of the quadratic fields
$({\cal F}$ and its dual ${\cal G})$.

For the sake of generality, we consider the Lagrangian density to be built up
in terms of both the invariants ${\cal F},{\cal G}$. The choice of using also
the parity violating ${\cal G}$ term is adopted for the purpose of generality.
It is nonetheless important to underline that the Lagrangian and the field
equations we shall be working with, i.e. Eqs. (14, 17), do not break the
parity invariance of electrodynamics because we are dealing with a purely
photonic model, and parity violations occurs in the Fermionic interaction
sector. To consider parity preserving phenomena we could simply consider only
even powers of G at the cost of generality. We also would like to point out
that the ${\cal G}$-terms - and not only the ${\cal F}$-terms - may be induced
by some fundamental physics, as it is case with the BI Lagrangian, which
naturally appears in the low-energy (field-theoretical) limit of string
theories A. A. Tseytlin (2000).

For $\vec{E}$ and $\vec{B}$, the electric and magnetic field, respectively;
$F^{\sigma\tau}$ and ${\tilde{F}}^{\sigma\tau}=G^{\sigma\tau}$ the electro-
magnetic field tensor and its dual, respectively; ${A}^{\sigma}$ the
4-potential for $\phi$ time and ${\vec{A}}$ space components; the vacuum
permeability $\mu_{0}$ and the speed of light $c$;
$\epsilon_{\sigma\tau\kappa\lambda}$ the Levi Civita pseudo-tensor, we have

$\displaystyle{\cal
F}\\!=\\!{\displaystyle-\frac{1}{4\mu_{0}}}{F}^{2}\\!=\\!{\displaystyle-\frac{1}{4\mu_{0}}}{F}_{\sigma\tau}{F}^{\sigma\tau}\\!=\\!{\displaystyle\frac{1}{2\mu_{0}}}\left({\displaystyle\frac{{\vec{E}^{2}}}{c^{2}}}-\vec{B}^{2}\right)~{},$
(1) $\displaystyle{\cal
G}\\!=\\!-{\displaystyle\frac{1}{4\mu_{0}}}{F}_{\sigma\tau}{\tilde{F}}^{\sigma\tau}\\!=\\!-{\displaystyle\frac{1}{4\mu_{0}}}{F}_{\sigma\tau}G^{\sigma\tau}\\!=\\!{\displaystyle\frac{1}{\mu_{0}}}{\displaystyle\frac{{\vec{E}}}{c}}\cdot{\vec{B}}~{},$
(2) $\displaystyle
F_{\sigma\tau}=\partial_{\sigma}{A}_{\tau}-\partial_{\tau}{A}_{\sigma}~{},~{}~{}~{}F^{\sigma\tau}=\partial^{\sigma}{A}^{\tau}-\partial^{\tau}{A}^{\sigma}~{},$
(3)
$\displaystyle{\tilde{F}}_{\sigma\tau}\\!=\\!G_{\sigma\tau}\\!=\\!{\displaystyle\frac{1}{2}}\epsilon_{\sigma\tau\kappa\lambda}F^{\kappa\lambda}~{},~{}{\tilde{F}}^{\sigma\tau}\\!=\\!G^{\sigma\tau}\\!=\\!{\displaystyle\frac{1}{2}}\epsilon^{\sigma\tau\kappa\lambda}F_{\kappa\lambda}~{},$
(4)
$\displaystyle{A}^{\sigma}=\left({\displaystyle\frac{\phi}{c}},{\vec{A}}\right)~{},~{}~{}~{}{A}_{\sigma}=\left({\displaystyle\frac{\phi}{c}},-{\vec{A}}\right)~{}.$
(5)

We set our scenario where the total $F$ ($G$) fields are composed by the
background $F_{\rm B}$ ($G_{\rm B}$) and the photon $f$ ($g$) fields

$\displaystyle\\!\\!\\!\\!F^{\mu\nu}=F_{\rm
B}^{\mu\nu}+f^{\mu\nu},~{}~{}~{}G^{\mu\nu}=G_{\rm
B}^{\mu\nu}+g^{\mu\nu}={\tilde{F}}_{\rm B}^{\mu\nu}+{\tilde{f}}^{\mu\nu}~{}.$
(6)

The Minkowski metric $\eta$ signature is (+ - - -). The Greek (Latin) indices
run over the four space-time (three space) dimensions. We drop momentarily
$\mu_{0}$ for the sake of space to pick it up again for the main results.

##  III The photon energy-momentum tensor

We rework Eqs. (1,2) by applying the position (6)

$\displaystyle{\cal F}\\!=\\!\underbrace{-\frac{1}{4}F^{2}_{\rm B}}_{{\cal
F}_{\rm B}}-\frac{1}{2}F_{\rm B}f\underbrace{-\frac{1}{4}f^{2}}_{{\cal F}_{\rm
f}}\\!=\\!{\cal F}_{\rm B}\underbrace{-\frac{1}{2}F_{\rm B}f+{\cal F}_{\rm
f}}_{\delta{\cal F}}\\!=\\!{\cal F}_{\rm B}+\delta{\cal F},$ (7)
$\displaystyle{\cal G}\\!=\\!\underbrace{-\frac{1}{4}F_{\rm B}{\tilde{F}}_{\rm
B}}_{{\cal G}_{\rm B}}-\frac{1}{2}{\tilde{F}}_{\rm
B}f\underbrace{\\!-\frac{1}{4}f{\tilde{f}}}_{{\cal G}_{\rm f}}\\!=\\!{\cal
G}_{\rm B}\underbrace{\\!-\\!\frac{1}{2}{\tilde{F}}_{\rm B}f\\!+\\!{\cal
G}_{\rm f}}_{\delta{\cal G}}\\!=\\!{\cal G}_{\rm B}\\!+\\!\delta{\cal G}~{},$
(8)

having used the identity $F_{\rm B}{\tilde{f}}={\tilde{F}}_{\rm B}f$. We
develop $\cal L$ in series on the background, where $f={\tilde{f}}=0$, up to
the fourth order

$\displaystyle{\cal L}({\cal F},{\cal G})={\cal L}({\cal F}_{\rm
B}+\delta{\cal F},{\cal G}_{\rm B}+\delta{\cal G})={\cal L}({\cal F}_{\rm
B},{\cal G}_{\rm B})+\left.\frac{1}{n!}\left(\delta{\cal
F}\frac{\partial}{\partial{\cal F}}+\delta{\cal
G}\frac{\partial}{\partial{\cal G}}\right)^{n}{\cal L}\right|_{\rm
B}\simeq{\cal L}({\cal F}_{\rm B},{\cal G}_{\rm B})+\left.\delta{\cal
F}\frac{\partial{\cal L}}{\partial{\cal F}}\right|_{\rm B}+\left.\delta{\cal
G}\frac{\partial{\cal L}}{\partial{\cal G}}\right|_{\rm B}+$
$\displaystyle\left.\frac{\delta{\cal F}^{2}}{2}\frac{\partial^{2}{\cal
L}}{\partial{\cal F}^{2}}\right|_{\rm B}+\left.\delta{\cal F}\delta{\cal
G}\frac{\partial^{2}{\cal L}}{\partial{\cal F}\partial{\cal G}}\right|_{\rm
B}+\left.\frac{\delta{\cal G}^{2}}{2}\frac{\partial^{2}{\cal L}}{\partial{\cal
G}^{2}}\right|_{\rm B}+\left.\frac{\delta{\cal
F}^{3}}{6}\frac{\partial^{3}{\cal L}}{\partial{\cal F}^{3}}\right|_{\rm
B}+\left.\frac{\delta{\cal F}^{2}\delta{\cal G}}{2}\frac{\partial^{3}{\cal
L}}{\partial{\cal F}^{2}\partial{\cal G}}\right|_{\rm
B}+\left.\frac{\delta{\cal F}\delta{\cal G}^{2}}{2}\frac{\partial^{3}{\cal
L}}{\partial{\cal F}\partial{\cal G}^{2}}\right|_{\rm
B}+\left.\frac{\delta{\cal G}^{3}}{6}\frac{\partial^{3}{\cal L}}{\partial{\cal
G}^{3}}\right|_{\rm B}+$ $\displaystyle\left.\frac{\delta{\cal
F}^{4}}{24}\frac{\partial^{4}{\cal L}}{\partial{\cal F}^{4}}\right|_{\rm
B}+\left.\frac{\delta{\cal F}^{3}\delta{\cal G}}{6}\frac{\partial^{4}{\cal
L}}{\partial{\cal F}^{3}\partial{\cal G}}\right|_{\rm
B}+\left.\frac{\delta{\cal F}^{2}\delta{\cal G}^{2}}{3}\frac{\partial^{4}{\cal
L}}{\partial{\cal F}^{2}\partial{\cal G}^{2}}\right|_{\rm
B}+\left.\frac{\delta{\cal F}\delta{\cal G}^{3}}{6}\frac{\partial^{4}{\cal
L}}{\partial{\cal F}\partial{\cal G}^{3}}\right|_{\rm
B}+\left.\frac{\delta{\cal G}^{4}}{24}\frac{\partial^{4}{\cal
L}}{\partial{\cal G}^{4}}\right|_{\rm B}~{},$ (9)

and after some lengthy computation, we explicit the fields up to fourth order

$\displaystyle{\cal L}_{4}$ $\displaystyle=-C_{1}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)-C_{2}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)+\frac{D_{1}}{2}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{2}+D_{2}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)+\frac{D_{3}}{2}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{2}-$
$\displaystyle\frac{M_{1}}{6}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{3}-\frac{M_{3}}{2}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{2}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)-\frac{M_{4}}{2}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{2}-\frac{M_{2}}{6}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{3}+$
$\displaystyle\frac{N_{1}}{24}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{4}+\frac{N_{3}}{6}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{3}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)+\frac{N_{4}}{3}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)^{2}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{2}+$
$\displaystyle\frac{N_{5}}{6}\left(\frac{1}{2}F_{\rm
B}f+\frac{1}{4}f^{2}\right)\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{3}+\frac{N_{2}}{24}\left(\frac{1}{2}{\tilde{F}}_{\rm
B}f+\frac{1}{4}f{\tilde{f}}\right)^{4}~{},$ (10)

having defined

$\displaystyle\left.{\displaystyle\frac{\partial{\cal L}}{\partial{\cal
F}}}\right|_{\rm B}=C_{1}~{},\left.{\displaystyle\frac{\partial{\cal
L}}{\partial{\cal G}}}\right|_{\rm
B}=C_{2}~{},\left.{\displaystyle\frac{\partial^{2}{\cal L}}{\partial{\cal
F}^{2}}}\right|_{\rm B}=D_{1}~{},\left.{\displaystyle\frac{\partial^{2}{\cal
L}}{\partial{\cal F}\partial{\cal G}}}\right|_{\rm
B}=D_{2}~{},\left.{\displaystyle\frac{\partial^{2}{\cal L}}{\partial{\cal
G}^{2}}}\right|_{\rm B}=D_{3}~{},\left.{\displaystyle\frac{\partial^{3}{\cal
L}}{\partial{\cal F}^{3}}}\right|_{\rm
B}=M_{1}~{},\left.{\displaystyle\frac{\partial^{3}{\cal L}}{\partial{\cal
F}^{2}\partial{\cal G}}}\right|_{\rm
B}=M_{3}~{},\left.{\displaystyle\frac{\partial^{3}{\cal L}}{\partial{\cal
F}\partial{\cal G}^{2}}}\right|_{\rm B}=M_{4}~{},$
$\displaystyle\left.{\displaystyle\frac{\partial^{3}{\cal L}}{\partial{\cal
G}^{3}}}\right|_{\rm B}=M_{2}~{},\left.{\displaystyle\frac{\partial^{4}{\cal
L}}{\partial{\cal F}^{4}}}\right|_{\rm
B}=N_{1}~{},\left.{\displaystyle\frac{\partial^{4}{\cal L}}{\partial{\cal
F}^{3}\partial{\cal G}}}\right|_{\rm
B}=N_{3}~{}\left.{\displaystyle\frac{\partial^{4}{\cal L}}{\partial{\cal
F}^{2}\partial{\cal G}^{2}}}\right|_{\rm
B}=N_{4}~{},\left.{\displaystyle\frac{\partial^{4}{\cal L}}{\partial{\cal
F}\partial{\cal G}^{3}}}\right|_{\rm
B}=N_{5}~{},\left.{\displaystyle\frac{\partial^{4}{\cal L}}{\partial{\cal
G}^{4}}}\right|_{\rm B}=N_{2}~{}.$

The linearisation of the Lagrangian shows that interactions are absent at
first order; at second, there is interaction between the photon and background
field; at third, we know that a photon may split in two photons and two merge
into one; at fourth, the photon-photon interaction produces two new photons.
We return to the index notation and write

$\displaystyle{\cal L}_{4}=$ $\displaystyle-\frac{1}{2}\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}\right)f_{\mu\nu}-\frac{1}{4}C_{1}f^{\mu\nu}f_{\mu\nu}-\frac{1}{4}C_{2}f^{\mu\nu}{\tilde{f}}_{\mu\nu}+$
$\displaystyle\frac{1}{8}\left(K_{\rm B}^{\mu\nu\kappa\lambda}+2T_{\rm
B}^{\mu\nu\kappa\lambda}\right)f_{\mu\nu}f_{\kappa\lambda}+\frac{1}{8}R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\mu\nu}f_{\kappa\lambda}f_{\rho\sigma}+$
$\displaystyle\frac{1}{16}S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\mu\nu}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}~{},$
(11)

having posed the following definitions

$K^{\mu\nu\kappa\lambda}_{\rm B}=D_{1}F_{\rm B}^{\mu\nu}F_{\rm
B}^{\kappa\lambda}+D_{3}{\tilde{F}}_{\rm B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}~{},~{}~{}T^{\mu\nu\kappa\lambda}_{\rm
B}=D_{2}{\tilde{F}}_{\rm B}^{\mu\nu}{F}_{\rm B}^{\kappa\lambda}~{},$

where $K^{\mu\nu\kappa\lambda}$ and $T^{\mu\nu\kappa\lambda}_{B}$ are anti-
symmetric in [$\mu\nu$] and [$\kappa\lambda$] but symmetric under the exchange
of the pairs [$\mu\nu$] - [$\kappa\lambda$];

$\displaystyle R^{\mu\nu\kappa\lambda\rho\sigma}_{\rm B}=$ $\displaystyle
D_{1}F_{\rm
B}^{\mu\nu}\eta^{\kappa\rho}\eta^{\lambda\sigma}\\!+\\!\frac{1}{2}D_{3}{\tilde{F}}_{\rm
B}^{\mu\nu}\epsilon^{\kappa\lambda\rho\sigma}\\!+\\!\frac{1}{2}D_{2}F_{\rm
B}^{\mu\nu}\epsilon^{\kappa\lambda\rho\sigma}\\!+\\!$ $\displaystyle
D_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}\eta^{\kappa\rho}\eta^{\lambda\sigma}\\!-\\!\frac{1}{6}M_{1}F_{\rm
B}^{\mu\nu}F_{\rm B}^{\kappa\lambda}F_{\rm
B}^{\rho\sigma}\\!-\\!\frac{1}{6}M_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}{\tilde{F}}_{\rm B}^{\kappa\lambda}{\tilde{F}}_{\rm
B}^{\rho\sigma}\\!-\\!$ $\displaystyle\frac{1}{2}M_{3}F_{\rm B}^{\mu\nu}F_{\rm
B}^{\kappa\lambda}{\tilde{F}}_{\rm
B}^{\rho\sigma}\\!-\\!\frac{1}{2}M_{4}F_{\rm B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}{\tilde{F}}_{\rm B}^{\rho\sigma}~{},$ (12) $\displaystyle
S^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}_{B}=$
$\displaystyle\frac{1}{2}D_{1}\eta^{\mu\kappa}\eta^{\nu\lambda}\eta^{\rho\omega}\eta^{\sigma\tau}\\!+\\!\frac{1}{8}D_{3}\epsilon^{\mu\nu\kappa\lambda}\epsilon^{\rho\sigma\omega\tau}\\!+\\!$
$\displaystyle\frac{1}{2}D_{2}\eta^{\mu\kappa}\eta^{\nu\lambda}\epsilon^{\rho\sigma\omega\tau}\\!-\\!\frac{1}{2}M_{1}F_{\rm
B}^{\mu\nu}F_{\rm B}^{\kappa\lambda}\eta^{\rho\omega}\eta^{\sigma\tau}\\!-\\!$
$\displaystyle\frac{1}{4}M_{2}{\tilde{F}}_{\rm B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}\epsilon^{\rho\sigma\omega\tau}\\!-\\!\frac{1}{4}M_{3}F_{\rm
B}^{\mu\nu}F_{\rm B}^{\kappa\lambda}\epsilon^{\rho\sigma\omega\tau}\\!-\\!$
$\displaystyle M_{3}F_{\rm B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}\eta^{\rho\omega}\eta^{\sigma\tau}\\!-\\!\frac{1}{2}M_{4}F_{\rm
B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}\epsilon^{\rho\sigma\omega\tau}\\!-\\!$
$\displaystyle\frac{1}{2}M_{4}{\tilde{F}}_{\rm B}^{\rho\sigma}{\tilde{F}}_{\rm
B}^{\omega\tau}\eta^{\mu\kappa}\eta^{\nu\lambda}\\!+\\!\frac{1}{24}N_{1}F_{\rm
B}^{\mu\nu}F_{\rm B}^{\kappa\lambda}F_{\rm B}^{\rho\sigma}F_{\rm
B}^{\omega\tau}\\!+\\!$ $\displaystyle\frac{1}{24}N_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}{\tilde{F}}_{\rm B}^{\kappa\lambda}{\tilde{F}}_{\rm
B}^{\rho\sigma}{\tilde{F}}_{\rm B}^{\omega\tau}\\!+\\!\frac{1}{6}N_{3}F_{\rm
B}^{\mu\nu}F_{\rm B}^{\kappa\lambda}F_{\rm B}^{\rho\sigma}{\tilde{F}}_{\rm
B}^{\omega\tau}\\!+\\!$ $\displaystyle\frac{1}{4}N_{4}F_{\rm B}^{\mu\nu}F_{\rm
B}^{\kappa\lambda}{\tilde{F}}_{\rm B}^{\rho\sigma}{\tilde{F}}_{\rm
B}^{\omega\tau}\\!+\\!\frac{1}{6}N_{5}F_{\rm B}^{\mu\nu}{\tilde{F}}_{\rm
B}^{\kappa\lambda}{\tilde{F}}_{\rm B}^{\rho\sigma}{\tilde{F}}_{\rm
B}^{\omega\tau}~{}.$ (13)

The parameters $C_{i},D_{i},M_{i},N_{i}$ depend only on the background field.
If the latter is constant, the parameters will be constant. We pose
$K^{\mu\nu\kappa\lambda}_{\rm B}+2T^{\mu\nu\kappa\lambda}_{\rm
B}={\hat{Q}}^{\mu\nu\kappa\lambda}_{\rm B}$.

$R^{\mu\nu\kappa\lambda\rho\sigma}_{\rm B}$ is contracted with $f_{\mu\nu}$,
$f_{\kappa\lambda}$ and $f_{\rho\sigma}$; the sole surviving components is
anti-symmetric in [$\mu\nu$], [$\kappa\lambda$], and [$\rho\sigma$], and named
${\tilde{R}}^{\mu\nu\kappa\lambda\rho\sigma}_{\rm B}$. Similarly, it occurs
for ${\tilde{S}}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}_{\rm B}$.

The background-dependent tensorial coefficients, $R_{\rm B}$ and $S_{\rm B}$,
in the 3- and 4-photon vertices of the Lagrangian density, Eq. (11),
respectively, present negative mass dimensions: $R_{\rm B}$ with dimension
(-2) and $S_{\rm B}$, with dimension (-4) - imply that the model, in its
quantum-field theoretical version, is non-renormalisable. This is not an issue
as we deal with a classical effective field theory that holds below a well
defined cut-off, fixed by the physics we investigate. In this case, the
effective field theory stretches at most up to the GeV scale, since it is a
purely photonic model, without available energy to excite the weak boson
mediators.

The field equations are obtained by the variation of the Lagrangian with
respect to the 4-potential $a_{\mu}$ of the photon field
$f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$.

We observe that $R$ and $S$ are contracted with $f$

$\displaystyle{\cal L}_{4}=$ $\displaystyle-\frac{1}{2}\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}\right)f_{\mu\nu}-\frac{1}{4}C_{1}f^{\mu\nu}f_{\mu\nu}-\frac{1}{4}C_{2}f^{\mu\nu}{\tilde{f}}_{\mu\nu}$
$\displaystyle+\frac{1}{8}{\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\mu\nu}f_{\kappa\lambda}+\frac{1}{8}R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\mu\nu}f_{\kappa\lambda}f_{\rho\sigma}+$
$\displaystyle\frac{1}{16}S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\mu\nu}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}~{},$
(14)

and thereby, we have

$\displaystyle{\cal L}_{4}=$ $\displaystyle-\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm B}^{\mu\nu}\right)\partial_{\mu}\delta
a_{\nu}-C_{1}f^{\mu\nu}\partial_{\mu}\delta
a_{\nu}-C_{2}{\tilde{f}}^{\mu\nu}\partial_{\mu}\delta a_{\nu}+$
$\displaystyle\frac{1}{2}{\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\kappa\lambda}\partial_{\mu}\delta
a_{\nu}+R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\kappa\lambda}f_{\rho\sigma}\partial_{\mu}\delta
a_{\nu}+$ $\displaystyle S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}\partial_{\mu}\delta
a_{\nu}~{}.$ (15)

Integrating by parts, we get the field equations

$\displaystyle\partial_{\mu}\left(C_{1}f^{\mu\nu}+C_{2}{\tilde{f}}^{\mu\nu}\right)\delta
a_{\nu}-\frac{1}{2}\partial_{\mu}\left({\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\kappa\lambda}\right)\delta
a_{\nu}$ $\displaystyle-$ $\displaystyle\partial_{\mu}\left(R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\kappa\lambda}f_{\rho\sigma}\right)\delta
a_{\nu}-\partial_{\mu}\left(S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}\right)\delta
a_{\nu}=$ $\displaystyle\partial_{\mu}\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm B}^{\mu\nu}\right)\delta a_{\nu}~{},$ (16)

that is

$\displaystyle\partial_{\mu}\left(C_{1}f^{\mu\nu}+C_{2}{\tilde{f}}^{\mu\nu}\right)-\frac{1}{2}\partial_{\mu}\left({\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\kappa\lambda}\right)-\partial_{\mu}\left(R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\kappa\lambda}f_{\rho\sigma}\right)$
$\displaystyle-$ $\displaystyle\partial_{\mu}\left(S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}\right)=\partial_{\mu}\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm B}^{\mu\nu}\right)~{}.$ (17)

Adding an external current and multiplying each term by $f_{\nu\alpha}$, we
obtain

$\displaystyle\partial_{\mu}\left(C_{1}f^{\mu\nu}+C_{2}{\tilde{f}}^{\mu\nu}\right)f_{\nu\alpha}-\frac{1}{2}\partial_{\mu}\left({\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\kappa\lambda}\right)f_{\nu\alpha}-$
$\displaystyle\partial_{\mu}\left(R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\kappa\lambda}f_{\rho\sigma}\right)f_{\nu\alpha}-\partial_{\mu}\left(S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}\right)f_{\nu\alpha}=$
$\displaystyle\partial_{\mu}\left(C_{1}F_{\rm
B}^{\mu\nu}+C_{2}{\tilde{F}}_{\rm
B}^{\mu\nu}\right)f_{\nu\alpha}+j^{\nu}f_{\nu\alpha}~{}.$ (18)

After considerable manipulation, we have the density of the photon energy-
momentum tensor (EMT), which for zero external current, is given by

$\displaystyle{\theta^{\mu}}_{\alpha}=$ $\displaystyle
C_{1}f^{\mu\nu}f_{\nu\alpha}-\frac{1}{2}{\hat{Q}}^{\mu\nu\kappa\lambda}_{B}f_{\kappa\lambda}f_{\nu\alpha}-\frac{3}{4}R_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma}f_{\kappa\lambda}f_{\rho\sigma}f_{\nu\alpha}$
$\displaystyle-$ $\displaystyle S_{\rm
B}^{\mu\nu\kappa\lambda\rho\sigma\omega\tau}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}f_{\nu\alpha}+{\delta^{\mu}}_{\alpha}\left(\frac{1}{4}C_{1}f^{\beta\nu}f_{\beta\nu}-\frac{1}{8}{\hat{Q}}^{\beta\nu\kappa\lambda}_{B}f_{\beta\nu}f_{\kappa\lambda}\right.$
$\displaystyle-$ $\displaystyle\frac{1}{8}R_{\rm
B}^{\beta\nu\kappa\lambda\rho\sigma}f_{\beta\nu}f_{\kappa\lambda}f_{\rho\sigma}-\left.\frac{1}{16}S_{\rm
B}^{\beta\nu\kappa\lambda\rho\sigma\omega\tau}f_{\mu\nu}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\tau}\right)~{}.$
(19)

We recall the physical meaning of the photon energy-momentum tensor density
[Jm-3], in SI units: $\theta^{0}_{\ 0}$ = energy density, $\theta^{0}_{\ k}$ =
energy flux divided by $c$ along the $k$ direction, $\theta^{k}_{\ 0}$ =
momentum density through the orthogonal surface to $k$, multiplied by c. The
density of the photon EMT variation in SI units [Jm-4] and reinserting
$\mu_{0}$ is given by

$\displaystyle\partial_{\alpha}{\theta^{\alpha}}_{\tau}=\underbrace{-\frac{1}{\mu_{0}}\partial_{\alpha}C_{1}{F}^{\alpha\nu}f_{\nu\tau}}_{\text{Maxwellian
term}}\underbrace{-\frac{1}{\mu_{0}}\partial_{\alpha}C_{2}\tilde{F}^{\alpha\nu}f_{\nu\tau}}_{\text{non-
Maxwellian linear term}}$
$\displaystyle\underbrace{+\frac{1}{4\mu_{0}}\left(\partial_{\tau}C_{1}\right)f^{\alpha\nu}f_{\alpha\nu}+\frac{1}{4\mu_{0}}\left(\partial_{\tau}C_{2}\right)\tilde{f}^{\alpha\nu}f_{\alpha\nu}-\frac{1}{8\mu_{0}^{2}}\left(\partial_{\tau}{\hat{Q}}^{\alpha\nu\kappa\lambda}\right)f_{\alpha\nu}f_{\kappa\lambda}}_{\text{second
order non-linear terms}}$
$\displaystyle\underbrace{-\frac{1}{8\mu_{0}^{3}}\left(\partial_{\tau}R^{\alpha\nu\kappa\lambda\rho\sigma}\right)f_{\alpha\nu}f_{\kappa\lambda}f_{\rho\sigma}-\frac{1}{16\mu_{0}^{4}}\left(\partial_{\tau}S^{\alpha\nu\kappa\lambda\rho\sigma\omega\xi}\right)f_{\alpha\nu}f_{\kappa\lambda}f_{\rho\sigma}f_{\omega\xi}}_{\text{third
and fourth non-linear terms}}~{}.$ (20)

The energy density for the photon field can be obtained by taking the (0,0)
component of the EMT, named $\cal E$, which at the second order, in SI units,
gives

$\displaystyle{\cal
E}=\frac{C_{1}}{2\mu_{0}}\left(\frac{\vec{e}^{~{}2}}{c^{2}}+\vec{b}^{2}\right)+\frac{D_{1}}{2{\mu_{0}}^{2}}\left[\left(\frac{\vec{E}\cdot\vec{e}}{c^{2}}\right)^{2}-\left(\vec{B}\cdot\vec{b}\right)^{2}\right]~{}.$
(21)

The energy density time variation is

$\displaystyle\frac{\partial\mathcal{E}}{\partial t}=$
$\displaystyle\frac{1}{2\mu_{0}}\left(\frac{\vec{e}^{~{}2}}{c^{2}}+\vec{b}^{2}\right)\partial_{t}C_{1}+\frac{1}{2{\mu_{0}}^{2}}\left[\left(\frac{\vec{E}\cdot\vec{e}}{c^{2}}\right)^{2}-\left(\vec{B}\cdot\vec{b}\right)^{2}\right]\partial_{t}D_{1}$
$\displaystyle+\frac{D_{1}}{2{\mu_{0}}^{2}}\left(\frac{\vec{e}^{2}}{c^{4}}\partial_{t}\vec{E}^{2}-\vec{b}^{2}\partial_{t}\vec{B}^{2}\right)~{}.$
(22)

Due to the experimental correspondence wave-single photon Aspect and Grangier
(1987), these two quantities represent also respectively the frequency and the
frequency shift. We now take into account two specific scenarii, namely the
Euler-Heisenberg and the Born-Infeld cases.

##  IV The case of Euler-Heisenberg and Born-Infeld theories

The non-conservation of photon energy already appears at second order for
which the EH and BI Lagrangians, represented by our generalised formalism,
look as Sorokin (2022)

$\displaystyle{\cal L}_{\rm
EH}=-\frac{1}{4\mu_{0}}{F}_{\sigma\tau}{F}^{\sigma\tau}+\beta\left[\left({F}_{\sigma\tau}{F}^{\sigma\tau}\right)^{2}+7\left({G}_{\sigma\tau}{F}^{\sigma\tau}\right)^{2}\right]~{},$
(23) $\displaystyle{\cal L}_{\rm
BI}=-\frac{1}{4\mu_{0}}{F}_{\sigma\tau}{F}^{\sigma\tau}+\frac{1}{32b}\left[\left({F}_{\sigma\tau}{F}^{\sigma\tau}\right)^{2}+\left({G}_{\sigma\tau}{F}^{\sigma\tau}\right)^{2}\right]~{},$
(24)

where $b$ is a scale parameter and $\beta$ is a constant depending on the
electron mass and fine structure constant
$\beta={2\alpha^{2}}/{45{m_{e}}^{4}}$.

For the Euler-Heisenberg case, we get the coefficients
$C_{1}=1+2\beta\mathcal{F}_{B},C_{2}=14\beta\mathcal{G}_{B},D_{1}=2\beta,D_{3}=14\beta,D_{2}=0$,
and the energy density is

${\cal E}_{\rm
EH}=\frac{1}{2\mu_{0}}\left[1+\frac{\beta}{\mu_{0}}\left(\frac{\vec{E}^{~{}2}}{c^{2}}-\vec{B}^{2}\right)\right]\left(\frac{\vec{e}^{~{}2}}{c^{2}}+\vec{b}^{2}\right)+\frac{\beta}{{\mu_{0}}^{2}}\left(\left[\frac{\vec{E}\cdot\vec{e}}{c^{2}}\right)^{2}-\left(\vec{B}\cdot\vec{b}\right)^{2}\right]~{}.$
(25)

We repeat the same procedure for the Born-Infeld model. Starting by its
coefficients
$C_{1}=1+1/16b\mathcal{F}_{B},C_{2}=1/16b\mathcal{G}_{B},D_{1}=D_{3}=1/16b,D_{2}=0$,
the energy density is given by

${\cal E}_{\rm
BI}=\frac{1}{2\mu_{0}}\left[1+\frac{1}{16\mu_{0}b}\left(\frac{\vec{E}^{~{}2}}{c^{2}}-\vec{B}^{2}\right)\right]\left(\frac{\vec{e}^{~{}2}}{c^{2}}+\vec{b}^{2}\right)+\frac{1}{{16\mu_{0}}^{2}b}\left[\left(\frac{\vec{E}\cdot\vec{e}}{c^{2}}\right)^{2}-\left(\vec{B}\cdot\vec{b}\right)^{2}\right]~{}.$
(26)

Now, we compare the two energy density variations

$\displaystyle\frac{\partial\mathcal{E}_{EH}}{\partial
t}=\frac{\beta}{2\mu_{0}^{2}}\left(\frac{\vec{b}^{2}}{c^{2}}\partial_{t}\vec{E}^{2}-\frac{\vec{e}^{~{}2}}{c^{2}}\partial_{t}\vec{B}^{2}\right)+\frac{3\beta}{2{\mu_{0}}^{2}}\left(\frac{\vec{e}^{2}}{c^{4}}\partial_{t}\vec{E}^{2}-\vec{b}^{2}\partial_{t}\vec{B}^{2}\right)~{},$
(27)

$\displaystyle\frac{\partial\mathcal{E}_{BI}}{\partial t}=$
$\displaystyle\frac{1}{64b\mu_{0}^{2}}\left(\frac{\vec{b}^{2}}{c^{2}}\partial_{t}\vec{E}^{2}-\frac{\vec{e}^{~{}2}}{c^{2}}\partial_{t}\vec{B}^{2}\right)+\frac{3}{64b{\mu_{0}}^{2}}\left(\frac{\vec{e}^{2}}{c^{4}}\partial_{t}\vec{E}^{2}-\vec{b}^{2}\partial_{t}\vec{B}^{2}\right)~{}.$
(28)

Differentiating BI from EH frequency shifts is not an easy undertaking.
Furthermore, in a laboratory setting when using an interferometric cavity,
other components than $\theta^{0}_{~{}0}$ have to computed. The discrimination
of frequency changes from velocity changes, cavity wall displacements and
accumulated phase changes, require that the entire EMT is to be computed.

##  V Discussions and perspectives

###  V.1 Discussion

We call into question the issue of the Poincaré Symmetry Violation (PSV).The
fundamental symmetries of space-time (isotropy and homogeneity) can be
expressed in the language of group theory. These symmetries are demonstrated
by the generators of the Poincaré group which adds translation transformations
to the Lorentz sub-group of transformations, namely rotations in space and
boosts. A Poincaré symmetry Violation implies that the full Poincaré group
breaks down.

In the case of a space-time constant electro-magnetic background, the
continuity equation, Eq. (20), shows that the energy-momentum tensor is
conserved. This is a consequence, via Noether’s theorem, of the Lagrangian
density, Eq. (15) not exhibiting an explicit space-time dependence. The
Lorentz sub-group of the Poincaré symmetry is violated in the active sense.
Active (particle) transformations are transformations of the physical fields,
whereas passive (observer) transformations are merely relabellings of points.
Active transformations act on the physical system under consideration without
touching the background. Conversely, the passive viewpoint applies when both
the physical system and the background are transformed. Active and passive
transformations coincide when the metric components rest the same, but differ
in presence of a tensorial anisotropic field Duffy . If fundamental theories
must have the same symmetries as space-time itself, differing between active
and passive becomes unnecessary. In the case of a space-time varying
background fields yield non-vanishing components of the Q-, R- and S-tensors
present in Eq. (15). So, symmetry under space- time rotations is broken, which
means Lorentz symmetry violation in the active sense. This can be verified by
explicitly computing the canonical angular momentum current tensor density of
the photon in this system through Noether’s theorem

$\displaystyle\mathcal{M}^{\alpha\mu\nu}=$ $\displaystyle
i\left(x^{\mu}T^{\alpha\nu}-x^{\nu}T^{\alpha\mu}\right)-\mathcal{S}^{\alpha\mu\nu}~{},$
(29)

where $T$ is the canonical energy-momentum tensor, and

$\displaystyle\mathcal{S}^{\alpha\mu\nu}=\mathcal{J}^{\alpha\mu}a^{\nu}-\mathcal{J}^{\alpha\nu}a^{\mu}+C_{1}(f^{\alpha\mu}a^{\nu}-f^{\alpha\nu}a^{\mu})+$
$\displaystyle
C_{2}(\tilde{f}^{\alpha\mu}a^{\nu}-\tilde{f}^{\alpha\nu}a^{\mu})+\frac{1}{2}\left(Q^{\alpha\nu\rho\sigma}f_{\rho\sigma}a^{\mu}-Q^{\alpha\mu\rho\sigma}f_{\rho\sigma}a^{\nu}\right)$
(30)

is the spin density tensor, being $\mathcal{J}^{\alpha\mu}=\left(C_{1}F_{\rm
B}^{\alpha\mu}+C_{2}{\tilde{F}}_{\rm B}^{\alpha\mu}\right)$.

The presence of vector and tensor anti-symmetric quantities in the background
- space-time variable or not - implies that the energy-momentum tensor is no
longer symmetric, Eq. (19). In this situation, the Poynting vector,
$\theta^{~{}i0}$, and the momentum density carried by the propagating field,
$\theta^{~{}0i}$, are different. We do not report here, but it can be shown
that, from the time variation of the momentum density $\theta^{~{}0i}$ \-
which indicates the breaking of space translational symmetry - the effects of
the (variable) background contribute asymmetrically to the usual Lorentz force
density.

The fact that we are dealing with a Poincaré symmetry violating model brings
into question multiple issues, but one in particular: the Casimir
operators444The two Casimir operators are $P^{2}=P_{\mu}P^{\mu}$ and
$W^{2}=W_{\mu}W^{\mu}$, where $P$ is momentum, and $W$ is the Pauli-Lubanski
pseudo-vector. In the rest frame, we get $P^{2}=mc^{2}$, that is mass for the
first Casimir operator. Instead $W^{2}=P_{\mu}P^{\mu}$. has to do with spin
for massive particles and helicity for massless particles. of the group.
Indeed, the Poincaré group possesses two invariant operators, which are
associated to the mass and the helicity/spin of the states that carry its
unitary irreducible representations. In the case described above, it is safe
to assume that at least one of these quantities is no longer an invariant,
thereby putting into question the formal definition of mass and/or spin. Could
a Poincaré symmetry violation lead to emerging non-constant masses? What would
a variable spin imply physically? How would this fundamentally affect the
structure of space-time?

###  V.2 Perspectives

Herein, we remind the discussion we have presented in J. A. Helayël-Neto and
Spallicci (2019); Spallicci et al. (2021); Spallicci et al. (2022a); Sarracino
et al. (2022), where the reader can find the different cosmological models and
the detailed calculations for massive and the SME formalisms.

The sizing of the frequency shift is not straightforward, as many factors
enter into play. First, it depends on the amount of distance crossed by the
photon. Second, the photon crosses several EM fields for the source to the
observer. The magnetic field in the Milky Way has a strength of around 0.5 nT
and it has regular and fluctuating components of comparable strengths. In the
external spiral galaxies the fields resemble presumably that in our own
Galaxy. In elliptical galaxies, it is supposed that only the fluctuating
components survive. For the Inter-Galactic Medium, reliable conclusion cannot
be drawn neither, but supposing a strength of nT order of magnitude, it is the
safest hypothesis we can make Ferrière (2015); Han (2017) The inter-stellar
and inter-galactic media are good electric conductors, such that magnetic
fields are frozen in the plasma. Thereby, the electric field is given by
${\vec{E}}\propto{\vec{v}}_{p}\times{\vec{B}}$, where $v_{p}$ is the plasma
velocity. In general, $v_{p}\ll c$, thus $E\ll B$ and thereby potentially
negligible. This assumption may not hold locally though, and photons may pass
through intense electric fields. Third, the orientation of the
(inter-)galactic fields with respect to the photon propagation vector. Fourth,
the effect and the alignment of the EM fields encountered by the photon. An EM
field may determine a shift towards the blue, while another towards the red.
Fifth, we the evaluation value of the EM field of a photon is a fundamental
issue.

For the SME, we had some preliminary assessments J. A. Helayël-Neto and
Spallicci (2019). The static shift varies in the range 0.01 - 10% of the
expansion shift. Applied to SNeIa, the static shift evidently differs from one
SNIa to the other, as the above mentioned variables are many. But this is a
great advantage as it allows a great freedom to apply the model to any SNIa in
the luminosity-red shift plot.

But we can reverse our approach and ask which frequency shift is necessary to
explain specific phenomena. In Spallicci et al. (2021); Spallicci et al.
(2022a); Sarracino et al. (2022), we found agreement between SNe luminosity
and red shift distances by recasting the observed red shift as a combination
of the expansion shift and a static shift due to a massive theory (physical
mass as in dBP or effective as in the SME). Thereby, we could get rid of the
necessity of the accelerated expansion. Acceleration fits in the Friedmann-
Lemaître-Robertson-Walker cosmology, which is not more than an idealised
solution of general relativity, having disregarded other interactions. Dark
energy is not an observational result, but it is a plausible interpretation,
among others. The real result of the observations is the detection of non-
linear dependence of the registered energy current density of SNe Ia with
respect to red shift.

Let $z_{\rm C}$ be the red-shift due to expansion of the universe and $z_{\rm
S}$ be the shift due to GNLEM. We have then $1+z=(1+z_{\rm S})(1+z_{\rm
C})\simeq 1+z_{\rm S}+1+z_{\rm C}+z_{\rm S}z_{\rm C}$.

Supposing that the $z_{\rm S}$ shifts, additional to the red-shift due to
expansion, be negligible for astrophysics at large scale, they would remain of
relevance for the foundations of EM.

Another thread concerns the testing of these effects. The averaged value of
the expansion is 70 km/s per Mpc, which corresponds to $2.3\times 10^{-18}$
m/s per metre Capozziello et al. (2020); Spallicci et al. (2022b). Translated
as static frequency shift, it becomes $7.7\times 10^{-27}\Delta f/f$ per
metre. This is to be considered as upper limit, as we intend to add an effect
to the expansion, and not to deny the latter. But the state of art of
interferometry, for which the measurement of these shifts is absolutely a
challenge.

But leaving aside, these additional shifts, we can ask ourselves another
question. Can we test expansion? The formidable technical challenge of
measuring the Hubble-(Humason-)Lemaître constant of $2.3\times 10^{-18}$ m/s
per metre is not the only hurdle. We have to face the prediction that
expansion does not occur at small (below galactic) scale. Therefore, a non-
null result in terrestrial or spatial laboratories would be of great
significance, as it would prove that frequency shifts of static nature do
exist.

The static frequency shift has recently received attention for the James Webb
Space Telescope data Gupta (2023). We emphasise that our GNLEM shift, as the
massive and SME theories or even the Maxwellian theory shifts, are computed
out of established theories and not out of ad hoc assumptions, as those
reviewed in López-Corredoira and Marmet (2022); Marmet (2018).

A further perspective is offered by the reinterpretation of the rotation
curves of stars and gasses in galaxies through frequency shifts. If the
Doppler shift generated by baryonic matter rotation were accompanied by a
shift due to the magnetic field, acting as background, recurring to dark
matter to explain the non-Keplerianity of the curves would be undermined.

These frequency shifts, if applicable to a quantum system, should semi-
classically lead to small divergences from the expected spectral emissions of
atoms Carley and Kiessling (2006); Carley et al. (2019).

Spin helicity could be not conserved in curved space-time even when dealing
with Maxwellian electro-magnetism Galaverni and G. Gionti S.J. (2021). In the
frame of GNLEM and generally speaking of Extended Theories of Electro-
Magnetism, when exhibiting Poincaré symmetry violations, helicity might be
affected too. These violations are also sought in $\beta$ radiative processes
Gronau et al. (2002).

Another avenue is constituted by phenomenology in condensed matter systems and
especially supraconductivity. Condensed maatter models deal with translational
symmetry breaking as well as emerging masses. In fact, they are highly propice
systems to study Lorentz symmetry violations and topological theories à la
Chern-Simons. It could be of interest to analyse whether a split between
photons and background were to produce new insights.

##  VI Results

We have shown that GNLEM theories predict a static frequency shift due to the
non-conservation of the energy-momentum tensor, occurring when the background
is space-time dependent, that we have computed for the Euler-Heisenberg and
Born-Infeld theories. The GNLEM shift, like the other static shifts coming
from massive photon and SME frameworks, can be added to the expansion shift
providing options for reading astrophysical data or for reading laboratory
measurements. We have argued on the asymmetry of the energy-momentum tensor
and on the possible emergence of a mass due to the Poincaré symmetry breaking
Gupta et al. (2023), discussing the nature of the latter for non-linear
theories and the physical implications of said violations.

## Acknowledgements

ADAMS is grateful to UERJ (M. A. Lopés Capri, V. E. Lemos Rodino hosts for
2019-2020) and the French Consulate in Rio de Janeiro for his nomination as
’Chaire Française’. Part of this work was carried out at the D. Poisson in
Tours Inst. (M. Chernodub, host) in 2021-2022 during a CNRS funded half-
sabbatical. All authors thank the comments by L. Ospedal Prestas Rosas.

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