# Overview of superconductivity in field-cooled magnetic materials.

Naoum Karchev  Department of Physics, University of Sofia, 1164 Sofia,
Bulgaria

###### Abstract

Considerable experimental skills have been accumulated in the preparation of
field-cooled (FC) magnetic materials. This stimulates the search for FC
magnetic materials that are superconductors.

The article overviews the recent proposed mechanism of superconductivity in
field-cooled magnetic materials. It is based on previously published results
for magnon-induced superconductivity in field-cooled spin-1/2 antiferromagnets
$[PRB96,214409]$ (arXiv:1712.02983), Sequence of superconducting states in
field cooled $FeCr_{2}S_{4}$ $[JPCM33,495604]$ (arXiv:2111.02765) and Partial
order induced superconductivity in $Fe^{2+}$ iron. $[EPL126,47001]$
(arXiv:1902.02290).

##  I INTRODUCTION

The study of the mechanisms of superconductivity and the methods to synthesize
superconductors is an exciting challenge in solid state physics. Since the
discovery of superconductivity (1911), these two activities have been a major
driver for the successful dissemination of superconductivity ideas in other
fields of physic and for application in technology.

One way to prepare superconducting state is to subject a suitable material to
strong hydrostatic pressure. The coexistence of ferromagnetism and
superconductivity in $UGe_{2}$ under pressure was reported in the paper
UGe2000 . The invention triggered a very intense experimental and theoretical
study of the phenomenon Huxley2001 ; Tateiwa2001b ; Motoyama2001 ; Huxley2002
; Pfleiderer2009 ; Aoki2019 .

The theoretical predictions are important, since they allow to focus
experimental research on specific chemical compositions and also to suggest
appropriate conditions for the synthesis. In his theoretical studies Ashcroft
predicted that metalized hydrogen Ashcroft68  or hydrogen rich alloys
Ashcroft04  can possess high temperature superconductivity. Ashcroft’s idea is
that the hydrogen is the lightest element, and if it can be compressed in a
solid state, it could become superconductor at a very high transition
temperature $T_{c}$ due to the strong electron-phonon coupling. Accurate
electronic structure and electron-phonon coupling calculations predicted high
$T_{c}$ for metallic hydrogen Gross10 ; McMahon12 ; Bernstein15 ; Duan15 ;
Pickard15 . Thanks the development of high-pressure techniques numerous
experiments discussed the prediction. An important discovery leading to room-
temperature superconductivity is the pressure-driven hydrogen sulfide with a
confirmed transition temperature of 203 K at 155 GPa Drozdov15 . The most
recent examples of a metal hydride are lanthanum hydride which has
$Tc=250-260K$ at $180-200GPa$ Drozdov19 ; Somayazulu19 ; Hong20  and
carbonaceous sulfur hydride with room-temperature $T_{c}=287.7K$ achieved at
$267GPa$ Snider20 . More than ten hydrogen-rich compounds under high pressure
have been found to be high temperature conventional superconductors: yttrium
superhydride Troyan21 ; Snider21 , thorium hydride Semenok20 , praseodymium
superhydride Zhou20 , barium superhydride Chen21  and others Sanna16 ;
Gorkov18 ; Meng19 ; RevHSc20 ; Pickard20 .

Another way to fabricate unconventional superconductor is by chemical
manipulation. The most famous example is copper-oxide superconductor. The
parent compound $La_{2}CuO_{4}$ is an antiferromagnetic Mott insulator with
Néel temperature $T_{N}=300K$. The parent compound can be doped by
substituting some of the trivalent $La$ by divalent $Sr$. The result is that
$x$ holes are added to the $Cu-O$ plane in $La_{2-x}Sr_{x}CuO_{4}$, which is
called hole doping. The hole-doping suppresses the antiferromagnetic order and
at $x=0.03-0.05$ hole concentration the system undergoes quantum
antiferromagnetic-paramagnetic transition. After suppression of the
antiferromagnetism, superconductivity appears, ranging from $x=0.06-0.25$
Bednorz86 . The electron doping is realized in the compound
$Nd_{2-x}Ce_{x}CuO_{4}$ Tokura89  when $x$ electrons are added. Details are
given in many review articles and books, for example Ginsberg89 ; Dagotto94 ;
Anderson97 ; Lee06 .

In the present Overview we discuss theoretically the emergence of
superconductivity in field cooled magnetic materials. As examples we consider
spin $s=1/2$ antiferromagnetic insulator, chromium spinel $FeCr_{2}S_{4}$ and
$F^{2+}$ ferromagnetic iron. We also discuss some perspective
antiferromagnetic compounds.

##  II FIELD COOLED MAGNETIC MATERIALS

The material is field cooled (FC) if, during the preparation, an external
magnetic field as high as $300Oe$ is applied upon cooling. If the applied
field is below $1Oe$ it is zero field cooled (ZFC). The magnetization-
temperature and magnetic susceptibility curves for (ZFC) and (FC) spinel show
a remarkable difference below Néel $T_{N}$ temperature spinel+ ; spinelFeCr2S4
; spinelCv1 ; spinel08 ; spinel++ ; spinel11b ; spinel11a ; spinel11c ;
spinel12a ; spinel12b ; spinel+1 ; spinel+2 ; spinel+3 . In the case of
vanadium spinel $MnV_{2}O_{4}$ the curves, which show the temperature
dependence of spontaneous magnetization $M$, are depicted in Fig.1.

Figure 1: Magnetization of ZFC and FC vanadium spinel as a function of
temperature (after V.O.Garlea et al. PRL 100, 066404 (2008))

The spinel $MnV_{2}O_{4}$ is a two-sublattices ferrimagnet, with site $A$
occupied by the $Mn^{2+}$ ion, which is in the $3d^{5}$ high-spin
configuration with quenched orbital angular momentum, that can be regarded as
a simple $s=5/2$ spin. The B site is occupied by the $V^{3+}$ ion, which takes
the $3d^{2}$ high-spin configuration in the triply degenerate $t_{2g}$
orbital, and has orbital degrees of freedom. The measurements show that the
set in of the magnetic order is at Neel temperature $T_{N}=56.5K$ spinel+ ,
and that the magnetization has a maximum near $T^{*}=53.5K$. Below this
temperature the magnetization sharply decreases and goes to zero when
temperature approaches zero. The ferrimagnetic phase of vanadium spinel is
divided into two phases: high temperature $(T^{*},T_{N})$, where the
magnetization-temperature curves of (ZFC) and (FC) materials coincide, and low
temperature $0<T<T^{*}$, where there is a pronounced difference between the
two curves.

The two phases are basic characteristic of field cooled magnetic materials. As
a second example we consider chromium spinel $FeCr_{2}S_{4}$. The
magnetization-temperature curves are depicted in Fig.2.

Figure 2: Magnetization of ZFC and FC chromium spinel as a function of
temperature (after Zhaorong Yang at al. Phys Rev. B 62, 13872 (2000))

PARTIAL ORDER AND PARTIAL ORDER TRANSITION Magnetic state is a partial order
state if only part of the electrons in the system give contribution to the
magnetic order. It is studied in exactly solvable models Larkin66 ; Diep87 ,
by means of Green’s function approach Diep97 , Monte Carlo method Diep87  and
modified spin-wave theory of magnetism Karchev15 .

The spinel is typical example of a system with partial order and partial order
transition. The vanadium spinel is a system that obtains its magnetic
properties from $Mn$ and $V$ magnetic moments. The true magnons in this
system, which are the transversal fluctuations corresponding to the total
magnetization, are complicated mixtures of the $Mn$ and $V$ transversal
fluctuations. The magnons interact with manganese and vanadium ions in a
different way, and the magnons fluctuations suppress the $Mn$ and $V$
sublattice magnetizations at different temperatures. As a result, the
ferrimagnetic phase is divided into two phases: in the low temperature phase
$0<T<T^{*}$ the sublattice $Mn$ magnetization and sublattice $V$ magnetization
contribute to the magnetization of the system, while at the high temperature
$(T^{*},T_{N})$, the vanadium sublattice magnetization is suppressed by magnon
fluctuations, and only the manganese ions have non-zero spontaneous
magnetization Karchev09 . This means that high temperature phase is partial
order one and $T^{*}$ is partial order transition temperature.

The magnetic orders of vanadium and manganese electrons are antiparallel.
Magnetic field applied, during preparation of the material, along the $Mn$
magnetic moment, remains $Mn$ electrons localized with saturated magnetic
order, while $V$ electrons are delocalized. Increasing the applied magnetic
field the magnetization of $Mn$ electrons remains unchanged, as long as the
compensation of the Zeeman splitting of $V$ electrons increases and
respectively the magnetic moment of $V$ electrons decreases. As a result
increasing the applied, during the preparation, magnetic field the total
magnetization of the field cooled vanadium spinel increases below $T^{*}$
temperature (see Fig.1).

##  III SUPERCONDUCTIVITY IN FIELD-COOLED SPIN-1/2 ANTIFERROMAGNETSKarchev17

We discuss a novel mechanism for insulator-metal transition and
superconductivity in field-cooled spin-$1/2$ antiferromagnets on bcc lattice.
Applying magnetic field on sublattice A and B electrons along the sublattice B
magnetization, during preparation of the material, we change the magnetic and
transport properties of the material. Sublattice B electrons are localized,
while sublattice A ones are delocalized. This remains true when applied field
is switched off. Theoretically a ”frozen” magnetic field should be included in
the $A$ fermion dispersion, which leads effectively to decreasing of Zeeman
splitting. The effective model is a spin-fermion model with Zeeman splitting
of itinerant electrons compensated by the applied field. The Hamiltonian of
the system is

$\displaystyle h=$ $\displaystyle-$ $\displaystyle t\sum\limits_{\ll
ij\gg_{A}}{\left({c_{i\sigma}^{+}c_{j\sigma}+h.c.}\right)}+U\sum\limits_{i\in
A}n_{i\uparrow}n_{i\downarrow}-\mu\sum\limits_{i\in
A}{n_{i}}-H\sum\limits_{i\in A}{S^{zA}_{i}}$ (1) $\displaystyle+$
$\displaystyle J\sum\limits_{\langle ij\rangle}{{\bf S_{i}^{A}}}\cdot{\bf
S_{j}^{B}}-J_{B}\sum\limits_{\ll ij\gg_{B}}{{\bf S_{i}^{B}}\cdot{\bf
S_{j}^{B}}},$

where $t>0$ is the hopping parameter, ${\bf S_{i}^{A}}$ is the spin of the
itinerant electrons at the sublattice $A$ site with components

$S^{\nu
A}_{i}=\frac{1}{2}\sum\limits_{\sigma\sigma^{\prime}}c^{+}_{i\sigma}\tau^{\nu}_{\sigma\sigma^{\prime}}c^{\phantom{+}}_{i\sigma^{\prime}},$
(2)

and $(\tau^{x},\tau^{y},\tau^{z})$ are the Pauli matrices, ${\bf S_{i}^{B}}$
is the spin of the localized electrons at the sublattice $B$ site, $\mu$ is
the chemical potential, $n_{i\sigma}=c^{+}_{i\sigma}c_{i\sigma}$ and
$n_{i}=n_{i\uparrow}+n_{i\downarrow}$. The sums are over all sites of a body
centered cubic lattice, $\langle i,j\rangle$ denotes the sum over the nearest
neighbors, while $\ll ij\gg_{A}$ and $\ll ij\gg_{B}$ are sums over all sites
of sublattice $A$ and $B$ respectively. The Heisenberg term describes
ferromagnetic Heisenberg exchange between sublattice B $(J_{B}>0)$ electrons,
while the term $J>0$ is the antiferromagnetic exchange constant between
localized and itinerant electrons. The term with the constant $U>0$ is the
Coulomb repulsion. The ”frozen” magnetic field $H$ accounts for the effect of
the applied, during the preparation, magnetic field on itinerant electrons.

###  III.1  Schwinger-Bosons and Slave Fermions: Insulator-Metal Transition

We represent the Fermi operators, the spin of the itinerant electrons and the
density operators of sublattice A electrons in terms of the Schwinger-bosons
($\varphi_{i,\sigma},\varphi_{i,\sigma}^{+}$) and slave fermions
($h_{i},h_{i}^{+},d_{i},d_{i}^{+}$). The Bose fields are doublets
$(\sigma=1,2)$ without charge, while fermions are spinless with charges 1
($d_{i}$) and -1 ($h_{i}$):

$\displaystyle
c_{i\uparrow}=h_{i}^{+}\varphi_{i1}+\varphi_{i2}^{+}d_{i},\qquad
c_{i\downarrow}=h_{i}^{+}\varphi_{i2}-\varphi_{i1}^{+}d_{i},$ $\displaystyle
n_{i}=1-h^{+}_{i}h_{i}+d^{+}_{i}d_{i},\quad
s^{\nu}_{i}=\frac{1}{2}\sum\limits_{\sigma\sigma^{\prime}}\varphi^{+}_{i\sigma}{\tau}^{\nu}_{\sigma\sigma^{\prime}}\varphi_{i\sigma^{\prime}},$
$\displaystyle
c_{i\uparrow}^{+}c_{i\uparrow}c_{i\downarrow}^{+}c_{i\downarrow}=d_{i}^{+}d_{i}$
(3)

$\varphi_{i1}^{+}\varphi_{i1}+\varphi_{i2}^{+}\varphi_{i2}+d_{i}^{+}d_{i}+h_{i}^{+}h_{i}=1$
(4)

To solve the constraint (Eq.4), one makes a change of variables, introducing
Bose doublets $\zeta_{i\sigma}$ and $\zeta^{+}_{i\sigma}\,$ Schmeltzer

$\displaystyle\zeta_{i\sigma}$ $\displaystyle=$
$\displaystyle\varphi_{i\sigma}\left(1-h^{+}_{i}h_{i}-d^{+}_{i}d_{i}\right)^{-\frac{1}{2}},$
$\displaystyle\zeta^{+}_{i\sigma}$ $\displaystyle=$
$\displaystyle\varphi^{+}_{i\sigma}\left(1-h^{+}_{i}h_{i}-d^{+}_{i}d_{i}\right)^{-\frac{1}{2}},$
(5)

where the new fields satisfy the constraint
$\zeta^{+}_{i\sigma}\zeta_{i\sigma}\,=\,1$.

In terms of the new fields one obtains the Hamiltonian of free fermions
($h^{+},h,d^{+},d$)Karchev17

$h_{0}=\sum\limits_{k\in
B_{r}}\left(\varepsilon^{d}_{k}d_{k}^{+}d_{k}+\varepsilon^{h}_{k}h_{k}^{+}h_{k}\right)$
(6)

with dispersions

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$
$\displaystyle-4t\varepsilon_{k}+U-\mu+2J-\frac{H}{2}$
$\displaystyle\varepsilon^{h}_{k}$ $\displaystyle=$ $\displaystyle
4t\varepsilon_{k}+\mu+2J-\frac{H}{2}$ (7) $\displaystyle\varepsilon_{k}$
$\displaystyle=$ $\displaystyle\left(\cos k_{x}+\cos k_{y}+\cos k_{z}\right)$

The ground state of the system with Hamiltonian Eq.(6), is labeled by the
density of electrons

$n=1-<h^{+}_{i}h_{i}>+<d^{+}_{i}d_{i}>$ (8)

(see equation (III.1)) and the zero temperature spontaneous dimensionless
magnetization, of the sublattice A electron

$m=\frac{1}{2}\left(1-<h^{+}_{i}h_{i}>-<d^{+}_{i}d_{i}>\right).$ (9)

At half-filling

$<h^{+}_{i}h_{i}>=<d^{+}_{i}d_{i}>.$ (10)

To solve this equation, for all values of the parameters $U$, $t$ and $H$, one
sets the chemical potential $\mu=U/2$. Utilizing this representation of $\mu$
we calculate the dispersion of $"d"$ and $"h"$ fermions (III.1) as a function
of the applied magnetic field.

It is convenient to introduce the critical magnetic field

$H_{cr1}=U+4J-24t.$ (11)

Fermions dispersion can then be rewritten in the form

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$ $\displaystyle
4t\left[-\varepsilon_{k}+3+\frac{H_{cr1}-H}{8t}\right]$
$\displaystyle\varepsilon^{h}_{k}$ $\displaystyle=$ $\displaystyle
4t\left[\varepsilon_{k}+3+\frac{H_{cr1}-H}{8t}\right]$ (12)

When the applied magnetic field is below the critical one $H<H_{cr1}$ the
Fermion dispersions are positive ($\varepsilon^{d}_{k}>0$,
$\varepsilon^{h}_{k}>0$) for all values of the wave vector $k$. The minimum of
the $d$-fermion dispersion is in the center of the Brillouin zone of a cubic
lattice $B_{r}$ $\textbf{k}=(0,0,0)$ and $\varepsilon^{d}_{0}=(H_{cr1}-H)/2$.
The minimum of the $h$-fermion dispersion is in the vertices of the Brillouin
zone $\textbf{k}^{*}=(\pm\pi,\pm\pi,\pm\pi)$ and
$\varepsilon^{h}_{k^{*}}=(H_{cr1}-H)/2$. This means that Fermions excitations
are with gap which is our definition for insulating state. If one applies
magnetic field below the critical one the prepared material is insulator.

When the applied field is above the critical one $H>H_{cr1}$, the solutions of
the equations

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$ $\displaystyle
4t\left[-\varepsilon_{k}+3+\frac{H_{cr1}-H}{8t}\right]=0$
$\displaystyle\varepsilon^{h}_{k}$ $\displaystyle=$ $\displaystyle
4t\left[\varepsilon_{k}+3+\frac{H_{cr1}-H}{8t}\right]=0$ (13)

define the Fermi surfaces of $"d"$ and $"h"$ quasiparticles. The resultant
material is metal. The system possesses a novel insulator-metal transition
when magnetic field is applied and the critical value is $H_{cr1}$ (11).

The equation (III.1) shows that when the magnetic field is zero the system is
insulator if Coulomb repulsion is strong. When a hydrostatic pressure is
applied the hopping parameters $t$ increases, and for $24t>U+4J$ the system is
metal. The point is that under a hydrostatic pressure all electrons in the
material delocalize, while when a magnetic field is applied the electrons in
the system are geometrically separated and sublattice A electrons are
delocalized, but sublattice B ones are localized. This is important novelty.

###  III.2  Magnon-Induced Superconductivity

There is a second critical value

$H_{cr2}=U+4J.$ (14)

When $H=H_{cr2}$ the material is metal ($H_{cr2}>H_{cr1}$) and Zeeman
splitting of sublattice A electrons is zero. The Fermion dispersions
$\varepsilon^{d}_{k}$ and $\varepsilon^{h}_{k}$ (III.1) adopt the form

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$
$\displaystyle-4t\varepsilon_{k}$ $\displaystyle\varepsilon^{h}_{k}$
$\displaystyle=$ $\displaystyle 4t\varepsilon_{k}$ (15)

With dispersions (III.2) spontaneous magnetization $m$ (9) of sublattice A
electrons is zero and they do not contribute the magnetization of the system.
At this critical point the system is in partial order state. Only sublattice B
electrons are magnetically ordered, while sublattice A electrons are
magnetically disordered.

When Zeeman splitting is zero we use the approximate representation for the
fermi operators at quantum partial-order point (QPOP)

$c_{i\uparrow}=d_{i},\qquad c_{i\downarrow}=h_{i}^{+}.$ (16)

Then we can write the Hamiltonian of the system at QPOP in terms of the
fermion operators $c_{i\sigma}^{+},\,c_{i\sigma}$ and Holstein-Primakoff (HP)
bose operators $a^{+}_{j},\,a_{j}$ used to represent the spin operators of
sublattice B localized electrons ${\bf S^{B}_{j}}(a^{+}_{j},a_{j})$. The
Hamiltonian is a sum of three terms

$\displaystyle h^{A}$ $\displaystyle=$ $\displaystyle-t\sum\limits_{\ll
ij\gg_{A}}{\left({c_{i\sigma}^{+}c_{j\sigma}+h.c.}\right)}$ $\displaystyle
h^{AB}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{s}{2}}J\sum\limits_{\langle
ij\rangle}\left(c_{i\downarrow}^{+}c_{i\uparrow}a_{j}+c_{i\uparrow}^{+}c_{i\downarrow}a_{j}^{+}\right)$
(17) $\displaystyle h^{B}$ $\displaystyle=$ $\displaystyle-
J_{B}\sum\limits_{\ll ij\gg_{B}}{{\bf S_{i}^{B}}\cdot{\bf S_{j}^{B}}}$

Magnon-Fermion Effective Theory

We introduce two sublattices. The Hamiltonian of the Magnon-Fermion Effective
Theory, in momentum space, reads

$h=\sum\limits_{k\in
B_{r}}\left[\varepsilon_{k}^{A}c_{k\sigma}^{+}c_{k\sigma}+\varepsilon_{k}^{B}a_{k}^{+}a_{k}\right]+\frac{4J\sqrt{2s}}{\sqrt{N}}\sum\limits_{kqp\in
B_{r}}\delta(p-q-k)\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\cos\frac{k_{z}}{2}\times\left(c_{p\downarrow}^{+}c_{q\uparrow}a_{k}+c_{q\uparrow}^{+}c_{p\downarrow}a_{k}^{+}\right),$
(18)

with fermi $\varepsilon_{k}^{A}$ and bose $\varepsilon_{k}^{B}$ dispersions

$\displaystyle\varepsilon_{k}^{A}$ $\displaystyle=$
$\displaystyle-4t\left(\cos k_{x}+\cos k_{y}+\cos k_{z}\right)$ (19)
$\displaystyle\varepsilon_{k}^{B}$ $\displaystyle=$ $\displaystyle
2J_{B}u\left(3-\cos k_{x}-\cos k_{y}-\cos k_{z}\right)$

where the bose dispersion is calculated in Hartree-Fock approximation and $u$
is HF parameter that renormalizes the sublattice B exchange constant $J_{B}$.
The two equivalent sublattices A and B of the body center cubic lattice are
simple cubic lattices. Therefor the wave vectors $p,q,k$ run over the first
Brillouin zone of a cubic lattice $B_{r}$ .

Let us average in the subspace of Bosons $(a^{+},a)$-to integrate the Bosons
in the path integral approach. In static approximation one obtains an
effective fermion theory with Hamiltonian

$h_{eff}=\sum\limits_{k\in
B_{r}}\varepsilon_{k}^{A}c_{k\sigma}^{+}c_{k\sigma}-\frac{1}{N}\sum\limits_{k_{i}p_{i}\in
B_{r}}\delta(k_{1}-k_{2}-p_{1}+p_{2})V_{k_{1}-k_{2}}c_{k_{1}\downarrow}^{+}c_{k_{2}\uparrow}c_{p_{2}\uparrow}^{+}c_{p_{1}\downarrow}$
(20)

and potential

$V_{k}=\frac{J^{2}(1+\cos{k_{x}})(1+\cos{k_{y}})(1+\cos{k_{z}})}{J_{B}u\left(3-\cos
k_{x}-\cos k_{y}-\cos k_{z}\right)}.$ (21)

Following standard procedure one obtains the effective Hamiltonian in the
Hartree-Fock approximation

$h^{HF}_{eff}=\sum\limits_{k\in
B_{r}}\left[\varepsilon_{k\sigma}c_{k\sigma}^{+}c_{k\sigma}+\Delta_{k}c_{-k\downarrow}^{+}c_{k\uparrow}+\Delta_{k}^{+}c_{k\uparrow}c_{-k\downarrow}\right],$
(22)

with gap function

$\Delta_{k}=\frac{1}{N}\sum\limits_{p\in
B_{r}}<c_{-p\uparrow}c_{p\downarrow}>V_{p-k}$ (23)

The Hamiltonian can be written in a diagonal form by means of Bogoliubov
excitations $\alpha^{+},\alpha,\beta^{+},\beta$ with dispersions

$E^{\alpha}_{k}=E^{\beta}_{k}=E_{k}=\sqrt{(\varepsilon_{k}^{A})^{2}+|\Delta_{k}|^{2}}$
(24)

In terms of the new excitations the gap equation reads

$\displaystyle\Delta_{k}=$ $\displaystyle-$
$\displaystyle\frac{1}{2N}\sum\limits_{p\in
B_{r}}V_{k+p}\frac{\Delta_{p}}{\sqrt{(\varepsilon^{A})^{2}+|\Delta_{p}|^{2}}}$
(25) $\displaystyle\times$
$\displaystyle\left(1-\frac{2}{e^{{E_{p}}/{T}}+1}\right),$

where $T$ is the temperature.

Having in mind that sublattices are simple cubic lattices and following the
classifications for spin-triplet gap functions $\Delta_{-k}=-\Delta_{k}$ RKS10
, we obtained that the gap function with $T_{1u}$ configuration

$\Delta_{k}=\Delta\left(\sin k_{x}+\sin k_{y}+\sin k_{z}\right)$ (26)

is a solution of the gap equation (25) for some values of the temperature. The
dimensionless gap $gap/J_{B}$, as a function of dimensionless temperature
$T/J_{B}$, is depicted in figure (3) for two different values of the parameter
$J/J_{B}=4$, $J/J_{B}=7$ and $t/J_{B}=0.5$.

The parameters are chosen having in mind that $J$ is nearest neighbor exchange
constant while $J_{B}$ next to nearest neighbor exchange constant, therefore
$J>J_{B}$. With value $J/J_{B}=7$ we overestimate the parameter to give better
understanding of the phenomenon.

Figure 3: (Color online) The temperature dependence of the dimensionless gap
$(gap/J_{B})$ for $t/J_{B}=0.5$ and two different values of the parameter
$J/J_{B}=7$ \- upper (red) graph, $J/J_{B}=4$ \- lower (black) graph. The
vertical dash (green) line marks the Néel temperature.

The figure (3) shows that the temperature dependence of the gap is quite
unusual. The gap is approximately constant when the temperature is below the
Néel temperature $T_{N}$, marked with vertical dash green line, weakly
increases when the temperature approaches $T_{N}$ and abruptly falls to zero
in paramagnetic phase. This is because the pairing of fermions, below the Néel
temperature, is mediated by gapless bosons-magnons. The potential $V_{k}$
depends on temperature since the Hartree-Fock parameter $u$ does. Near the
Néel temperature the parameter $u$ decreases Karchev17  and potential $V_{k}$
increases. Above Néel temperature the magnon opens a gap which rapidly
increases when the temperature increases. This suppresses the
superconductivity since the maximal value of the potential in paramagnetic
phase is one over the magnon gap, so that when the magnon gap increases
abruptly the potential decreases.

It is important to underline that the applied, during preparation of the
material, magnetic field separates spatially electrons. Sublattice A ones are
delocalized and participate in the formation of Cooper pairs, while B ones are
localized and form the magnetic moment of the system. Thus the system
possesses coexistence of p-type superconductivity and magnetism. At zero
temperature the magnetic moment of sublattice B electrons is maximal. This is
in contrast with known systems that possess coexistence of superconductivity
and magnetism near quantum critical point.

The above result shows that a superconductor can be prepared from any
antiferromagnet .

##  IV SEQUENCE OF SUPERCONDUCTING STATES IN FIELD COOLED $FeCr_{2}S_{4}$
Karchev21

In the present section we investigate two sublattice $FeCr_{2}S_{4}$ spinel.
The sublattice A sites are occupied by $F^{2+}$ (s=2) iron ions, and
sublattice B sites are occupied by $Cr^{3+}$ (s=3/2) chromium ions. The
$Fe^{2+}$ and $Cr^{3+}$ ions are located at the center of tetrahedral and
octahedral $S^{2-}$ cages, respectively, and the three $Cr^{3+}$ electrons
occupy the lower energy $t_{2g}$ bands.

The shape of magnetization-temperature diagrams for all spinels is remarkable
and emblematic. In the case of $FeCr_{2}S_{4}$ spinel the curves, which show
the temperature dependence of spontaneous magnetization $M^{s}$, are depicted
in Fig.4.

Figure 4: (Color online) The temperature dependence of the spontaneous
magnetization $M^{s}$ of ZFC, FC1, FC2, FC3 and FC4 ($FeCr_{2}S_{4}$) spinel.
The ZFC, FC2 and FC3 curves are adapted from experimental results Yang00 ;
Tsurkan10 ; Yto11 ; Lin14 . The ZFC curve has unusual maximum at $T^{*}=160K$
attributed to magnetic domain dynamics in Tsurkan10 . Small dip of the curve
at approximately $10K$ is interpreted as onset of long-range orbital order. It
is reported JB14 ; Lin14  that below $50K$ the curve FC2 shows a transition to
noncollinear ferrimagnetism. In the present paper we consider that the red,
vertical line indicates the onset of magnetization order of one of the
chromium electrons at temperature ($160K$), the blue line ($60K$) -the second
electron and the green one ($10K$) the third. The curves, FC1 and FC4, are
phenomenological extrapolations based on our experience gained from the
experimental and theoretical study of many other ferrimagnets.

The ZFC, FC2 and FC3 curves are adapted from experimental results Yang00 ;
Tsurkan10 ; Yto11 ; Lin14  and the rest ones, FC1 and FC4, are
phenomenological extrapolations. It is important to underline the different
interpretations of the behavior of the system near the four characteristic
temperatures $T_{N}=170K$, $T^{*}=160K$, $T=60K$ and $T=10K$ from experimental
results, and those in the present paper. The Néel temperature is $T_{N}=170K$.
For ZFC spinel the temperature, at which the magnetization is maximum, is
$T^{*}=160K$. This unusual maximum is attributed to magnetic domain dynamics
in Tsurkan10 . In the same article small dip of the curve at approximately
$10K$ is interpreted as onset of long-range orbital order. In our
interpretation supported by calculations using Green function approach Diep97
and modified spin-wave theory Karchev08  the system undergoes a partial-order
transition from the high temperature $(T^{*},T_{N})$ phase, where only the
iron ions have non-zero spontaneous magnetization, to low temperature one
$(0,T^{*})$, where both the iron and chromium ions have non-zero spontaneous
magnetization. The exchange constant of chromium and iron spins is
antiferromagnetic, because of which the magnetization increases below $T_{N}$
and decreases below $T^{*}$. The subtle point is that the onset of magnetism
of the three $Cr^{3+}$ ($t_{2g}$) electrons is at different temperatures. One
of them starts to form magnetic order below $T^{*}$. The ZFC curve in (Fig.4),
shows small dip at $10K$ which indicates that another chromium electron starts
to contribute the magnetic order of the system at this temperature. The
contribution of the third $Cr^{3+}$ electron becomes more clear in the
experiment with FC spinel-curve FC2. It shows that applied during the
preparation magnetic field exceeds the Zeeman splitting energy of the chromium
electron and its magnetic order becomes parallel to the iron magnetic moment.
As a result, below $T^{*}$ the system undergoes partial-order transition, due
to the onset of magnetic order of chromium electron and the spontaneous
magnetization of the system increases and reaches the maximum at $60K$. It
undergoes a second partial-order transition because of the onset of the
magnetic order of the second $Cr^{3+}$ electron anti-parallel to the iron one
and the spontaneous magnetization of the system decreases. Alternatively it is
reported JB14 ; Lin14  that below $50K$ the system undergoes a transition to
non-collinear ferrimagnetism. Important consequence is the emergence of
multiferroic phase below $10K$. One expects that applying magnetic field,
during preparation, the non-collinear order is suppressed and collinear
ferrimagnetism is restored. At $10K$ the curve has neither dip nor increase,
which means that applied magnetic field compensates Zeeman splitting of the
third chromium electron. The FC1 curve in the middle is an extrapolation for
the case when the magnetic field applied during the preparation compensates
the Zeeman splitting energy of the $Cr^{3+}$ electron. Therefore, in the
temperature range $(60K-170K)$ only iron electrons contribute to the
magnetization of the system. The extrapolation is based on our experimental
and theoretical knowledge acquired in the study of other spinels. Increasing
the applied, during preparation, magnetic field one obtains the magnetization-
temperature curve FC3. It has two characteristic features: first the onset of
the magnetism of iron and the first chromium electron is at the same
temperature $T_{N}=170K$, second the applied magnetic field exceeds the Zeeman
splitting energy of the third chromium electron and its magnetic order becomes
parallel to the iron magnetic moment. As a result, below $T=10K$ the system
undergoes partial-order transition, due to onset of the magnetic order of the
third chromium electron and the spontaneous magnetization of the system
increases. The last curve FC4 is an extrapolation for the case when the
applied field compensates Zeeman splitting of the second $Cr^{3+}$ electron.

Based on the analysis of Fig.4, we consider a spin-fermion model of
$FeCr_{2}S_{4}$ spinel, with three bands describing $t_{2g}$ chromium
electrons and spin $s=2$ operators for localized $F^{2+}$ electrons. The iron-
chromium exchange constants are antiferromagnetic, different for the three
chromium electrons. Magnetic field in the Hamiltonian models the decrease of
the Zeeman splitting during preparation of the material. We study the
appearance and disappearance of superconducting states as a function of the
field. The Hamiltonian of the model is

$h=-t\sum\limits_{\ll ij\gg_{B}}\sum\limits_{\sigma,l}{\left({c_{i\sigma
l}^{+}c_{j\sigma l}+h.c.}\right)}-H\sum\limits_{i\in
B,\,l}{S^{zB}_{il}}+\sum\limits_{\langle ij\rangle,l}J_{l}\,{{\bf
S_{i}^{A}}}\cdot{\bf S_{jl}^{B}}-J^{A}\sum\limits_{\ll ij\gg_{A}}{{\bf
S_{i}^{A}}\cdot{\bf S_{j}^{A}}},$ (27)

where $S^{\nu
B}_{il}=\frac{1}{2}\sum\limits_{\sigma\sigma^{\prime}}c^{+}_{i\sigma
l}\tau^{\nu}_{\sigma\sigma^{\prime}}c^{\phantom{+}}_{i\sigma^{\prime}l}$, with
the Pauli matrices $(\tau^{x},\tau^{y},\tau^{z})$, is the spin of the $lth$ \-
$t_{2g}$ chromium electron ($l=1,2,3$) at the sublattice $B$ site , ${\bf
S}_{i}^{A}$ is the spin operator of the localized iron electrons at the
sublattice $A$ site. The sums are over all sites of a body centered cubic
lattice, $\langle i,j\rangle$ denotes the sum over the nearest neighbors, $\ll
ij\gg_{A}$ is a sum over all sites of sublattice $A$ and $\ll ij\gg_{B}$ is a
sum over all sites of sublattice $B$ . The Heisenberg term $(J^{A}>0)$
describes ferromagnetic exchange between iron spins, and $J_{l}>0$ are the
antiferromagnetic exchange constants between iron and chromium spins. $H>0$ is
the ”frozen” applied magnetic field in units of energy.

The compensation mechanism of the field induced superconductivity suggests
that the formation of Cooper pairs is possible when Zeeman splitting of
electrons is compensated by the applied magnetic field. We choose the exchange
constants well separated $J_{1}<J_{2}<J_{3}$, so that if the magnetic field
$H$ compensates the Zeeman splitting of one of $t_{2g}$ chromium electrons, it
is far from the compensation of Zeeman energy of the other two electrons.

With this in mind, we can simplify our study. When the value of the ”frozen”
magnetic field is close to the Zeeman energy of one of $t_{2g}$ chromium
electrons, we can consider one band spin-fermion model of this electron
instead of model (27). The contribution of dropped fermions can be accounted
for by appropriate choice of the parameters. In this way we consider three
independent, one band spin-fermion models. In momentum space representation,
the Hamiltonians $h_{l}$ ($l=1,2,3$) have the form

$h_{l}=\sum\limits_{k\in B_{r}}\varepsilon_{k}a_{k}^{+}a_{k}+\sum\limits_{k\in
B_{r}\sigma}\varepsilon_{k\sigma l}c_{k\sigma l}^{+}c_{k\sigma
l}+\frac{4J_{l}\sqrt{2s}}{\sqrt{N}}\sum\limits_{kqp\in
B_{r}}\delta(p-q-k)\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\cos\frac{k_{z}}{2}\left(c_{p\downarrow
l}^{+}c_{q\uparrow l}a_{k}+c_{q\uparrow l}^{+}c_{p\downarrow
l}a_{k}^{+}\right),$

with bose dispersion $\varepsilon_{k}$ of spin ($s=2$) iron magnons

$\varepsilon_{k}=2sJ^{A}\left(3-\cos k_{x}-\cos k_{y}-\cos k_{z}\right),$ (28)

and fermi $\varepsilon_{k\sigma l}$ dispersions of chromium electrons

$\displaystyle\varepsilon_{k\uparrow l}$ $\displaystyle=$
$\displaystyle-2t\left(\cos k_{x}+\cos k_{y}+\cos
k_{z}\right)+\frac{8sJ_{l}-H}{2}$ $\displaystyle\varepsilon_{k\downarrow l}$
$\displaystyle=$ $\displaystyle-2t(\cos k_{x}+\cos k_{y}+\cos
k_{z})-\frac{8sJ_{l}-H}{2}.$

The bosons $(a_{k}^{+}a_{k})$ are introduced by means of Holstein-Primakoff
representation of the spin-2 operators of localized iron electrons.

To proceed we account for the spin fluctuations of iron, and in static
approximation obtain three effective four-fermion theories with Hamiltonians
$h^{eff}_{l}$,

$h^{eff}_{l}=\sum\limits_{k\in B_{r}\sigma}\varepsilon_{k\sigma l}c_{k\sigma
l}^{+}c_{k\sigma l}-\frac{1}{N}\sum\limits_{k_{i}p_{i}\in
B_{r}}\delta(k_{1}-k_{2}-p_{1}+p_{2})V_{k_{1}-k_{2}}^{l}c_{k_{1}\downarrow
l}^{+}c_{k_{2}\uparrow l}c_{p_{2}\uparrow l}^{+}c_{p_{1}\downarrow l}$ (30)

and potentials

$V_{k}^{l}=\frac{J^{2}_{l}(1+\cos k_{x})(1+\cos k_{y})(1+\cos
k_{z})}{J^{A}\left(3-\cos k_{x}-\cos k_{y}-\cos k_{z}\right)}$ (31)

The Hamiltonians in the Hartree-Fock approximation are

$h_{l}^{HF}=\sum\limits_{k\in B_{r}}\left[\varepsilon_{k\sigma l}c_{k\sigma
l}^{+}c_{k\sigma l}+\Delta_{kl}c_{-k\downarrow l}^{+}c_{k\uparrow
l}+\Delta_{kl}^{+}c_{k\uparrow l}c_{-k\downarrow l}\right],$ (32)

with gap functions

$\Delta_{kl}=\frac{1}{N}\sum\limits_{p\in B_{r}}<c_{-p\uparrow
l}c_{p\downarrow l}>V_{p-k}^{l}$ (33)

In terms of Bogoliubov excitations
$\alpha_{l}^{+},\alpha_{l},\beta_{l}^{+},\beta_{l}$, with dispersions

$\displaystyle E^{\alpha}_{kl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\varepsilon_{k\uparrow
l}-\varepsilon_{k\downarrow l}+\sqrt{(\varepsilon_{k\uparrow
l}+\varepsilon_{k\downarrow l})^{2}+4|\Delta_{kl}|^{2}}\right]$ (34)
$\displaystyle E^{\beta}_{kl}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[-\varepsilon_{k\uparrow
l}+\varepsilon_{k\downarrow l}+\sqrt{(\varepsilon_{k\uparrow
l}+\varepsilon_{k\downarrow l})^{2}+4|\Delta_{kl}|^{2}}\right].$

the gap equations have the form

$\displaystyle\Delta_{kl}=$ $\displaystyle-$
$\displaystyle\frac{1}{N}\sum\limits_{p\in
B_{r}}V_{k+p}^{l}\frac{\Delta_{pl}}{\sqrt{(\varepsilon_{p\uparrow
l}+\varepsilon_{p\downarrow l})^{2}+4|\Delta_{pl}|^{2}}}$ (35)
$\displaystyle\times$
$\displaystyle\left(1-<\alpha^{+}_{pl}\alpha_{pl}>-<\beta^{+}_{pl}\beta_{pl}>\right),$

where $<\alpha^{+}_{pl}\alpha_{pl}>$ and $<\beta^{+}_{pl}\beta_{pl}>$ are
fermi functions for Bogoliubov fermions.

Having in mind that sublattices are simple cubic lattices and following the
classifications for spin-triplet gap functions $\Delta_{-kl}=-\Delta_{kl}$, we
obtained that the gap functions with $T_{1u}$ configuration RKS10

$\Delta_{kl}=\Delta_{l}\left(\sin k_{x}+\sin k_{y}+\sin k_{z})\right)$ (36)

are solutions of the gap equations for some values of the applied, during the
preparation, magnetic field and temperature. The dimensionless gaps
$\Delta_{l}/J^{A}$ at zero temperature, as a function of $H/H_{1}$ where
$H_{1}=8sJ_{1}$, are depicted in Fig.(5) for parameters $J_{1}/J^{A}=2$,
$J_{2}/J_{1}=1.4$, $J_{3}/J_{1}=1.8$ and $t/J^{A}=1$.

Figure 5: (Color online)Sequence of superconducting states in field cooled
$FeCr_{2}S_{4}$. The first state is realized near $H=H_{1}=2sJ_{1}$, the
second one near $H=H_{2}=2sJ_{2}=1.4H_{1}$, and the third state near
$H=H_{3}=2sJ_{3}=1.8H_{1}$.

The applied magnetic field $H_{1}$ compensates the Zeeman splitting of
$t_{2g}$ chromium electrons with minimum Zeeman energy (IV). The Fig.(5) shows
that near this value the above mentioned electrons form Cooper pairs and
superconductivity emerges. Increasing the magnetic field we restore the Zeeman
splitting with opposite sign and suppress the superconductivity. Further
increasing the magnetic field, we reach $H_{2}=2sJ_{2}$ that compensates the
Zeeman splitting of another $t_{2g}$ electrons. Now, the Cooper pairs are
formed by the second group electrons and the superconductivity is restored
near $H/H_{1}=H_{2}/H_{1}=J_{2}/J_{1}=1.4$. This process continues until the
magnetic field, applied during preparation, becomes equal to Zeeman energy of
the third group of chromium electrons $H_{3}=2sJ_{3}$ and third superconductor
state emerges near $H_{3}/H_{1}=1.8$. In that way we can create a sequence of
superconducting states in field cooled $FeCr_{2}S_{4}$. Actually there are
three different superconductors prepared applying, during preparation,
different magnetic fields. In these compounds superconductivity coexist with
the saturated magnetism of iron ions.

In summary, we have predicted the possibility to synthesize three different
superconductors. The difficult moment is to applied a field which compensates
the Zeeman splitting of one of the chromium electrons. An useful guidance to
do that are the curves depicted in figure (4). The curves FC1, FC2 and FC4
illustrate the three cases when one of chromium electrons is with compensated
Zeeman splitting. They are well separated from the others which permits the
exact choice of the applied, during preparation, magnetic field.

The spinel $FeCr_{2}S_{4}$ is well studied compound, but superconductivity has
not been observed. The explanation is very simple. The focus of research is on
colossal magnetoresistance effect (CMR) RCK1997 ; Yang00  and on the existence
of multiferroic phase. The investigation of CMR requires measurements of
resistivity as a function of temperature, but they are realized for ZFC
materials only. In the rare cases, when measuring the resistivity of FC
compounds, there is no clarity about the applied magnetic field. This can be
achieved by simultaneous study spontaneous magnetization and resistivity as
functions of temperature, which would lead to correct choice of field.

The experiments with spinel stimulate the study in detail some perspective
antiferromagnets. Closest to the $FeCr_{2}S_{4}$ spinels are calcium manganese
oxides ($CaMnO_{3}$). They are G-type antiferromagnetic insulators with Néel
temperature $T_{N}=350K$. As in the case with spinels, the electrons of the
$t_{2g}$ triplet contribute magnetic order. Experiments must be performed to
show the degeneracy or non-degeneracy of the $t_{2g}$ states, which will
ultimately determine how to prepare the superconductor.

##  V SUPERCONDUCTIVITY IN FIELD-COOLED $Fe^{2+}$ iron Karchev19

An important advancement in the theory of ferromagnetic metals are the
principles formulated by Zener. The third Zener’s principle Zener51  is that
the spin of an incomplete d shell is strongly coupled to the spin of the
conduction electrons. This coupling tends to align the spins of the incomplete
d shells in a ferromagnetic manner. Guided by the Zener’s principle one can
formulate the theory of ferromagnetic metals in terms of d electrons only. To
match the experimental results for saturation magnetization in units of
$\mu_{B}$ per lattice site $M=2.217$, one has to consider a model with six d
electrons. Five of them are localized and parallel oriented while the sixth
one, in accordance with Pauli principle, is antiparallel aligned with respect
to localized electrons.

We consider spin-fermion model of ferromagnetism and conductivity of $Fe^{2+}$
iron. The five localized electrons are described by spin $s=5/2$ spin
operators, while the sixth electrons are fermions. The spin-fermion exchange
is antiferromagnetic. To explain the magnetism and conductivity of the iron we
invoke the Mott theory of the insulator-metal transition Mott68 . It states
that if the kinetic energy of the electron is high enough compared with
Coulomb repulsion, doubly occupied states can be realized and respectively
empty states. Then the hopping of the electrons realizes the electric
transport and the material is metal. At the same time the doubly occupied and
empty states are spin singlets, so that their existence effectively decreases
the value of the magnetic moment of the sixth electrons. One can fix the
parameters of fermion model so that the total magnetic moment per atom in
units of Bohr magneton to match the experimental. measurement.

###  V.1 Ferromagnetism and conductivity of $Fe^{2+}$ iron

The Hamiltonian of the 3d electrons of iron in $Fe^{2+}$ state is

$\displaystyle h$ $\displaystyle=h_{1}\,+\,h_{2}\,+\,h_{3}$ $\displaystyle
h_{1}$
$\displaystyle=-t\sum\limits_{<ij>}\left(c_{i\sigma}^{+}c_{j\sigma}+h.c.\right)+U\sum\limits_{i}n_{i\uparrow}n_{i\downarrow}-\mu\sum\limits_{i}{n_{i}}$
$\displaystyle h_{2}$ $\displaystyle=J\sum\limits_{i}{\bf S}_{i}^{it}\cdot{\bf
S}_{i}^{l}$ $\displaystyle h_{3}$ $\displaystyle=-J^{l}\sum\limits_{<ij>}{\bf
S}_{i}^{l}\cdot{\bf S}_{j}^{l}.$

The spin operators ${\bf S}_{i}^{l}$ are the spin-$5/2$ operators of the
localized electrons, $c_{i\sigma}^{+}$ and $c_{i\sigma}$
($\sigma=\uparrow,\downarrow$) are creation and annihilation operators for
spin-1/2 Fermi operators of itinerant electrons, ${\bf S}_{i}^{it}$ are the
spin operators of the itinerant electrons, $U>0$ is the Coulomb repulsion and
$\mu$ is the chemical potential. The parameters $J^{l}$ and $J$ are positive
and describe the ferromagnetic exchange between localized electrons ($J^{l}$)
and the antiferromagnetic spin fermion exchange ($J$) . The parameter $J$ is
characteristic of intra-atom interaction, while $J^{l}$ features the exchange
between two sites (two atoms). This is why $J$ is much larger then $J^{l}$.
Sums are over all sites of a body centered cubic (bcc) lattice.

To study the effects of hopping $t$ and Coulomb repulsion $U$ we represent the
fermi operators $c_{i\sigma}^{+},c_{i\sigma}$, spin operators ${\bf
S}_{i}^{it}$ and density operators $n_{i\sigma}=c^{+}_{i\sigma}c_{i\sigma}$ of
itinerant electrons by means of the Schwinger-bosons
($\varphi_{i,\sigma},\varphi_{i,\sigma}^{+}$) and slave fermions
($h_{i},h_{i}^{+},d_{i},d_{i}^{+}$) . In this representation Coulomb term is
quadratic and one can study its impact exactly. The spin operators of
localized electrons ${\bf S_{i}^{l}}(a^{+}_{j},a_{j})$ are represented by
means of Holstein-Primakoff representation where $a^{+}_{j},\,a_{j}$ are Bose
fields.

The Hamiltonian of free ($h_{i},h_{i}^{+},d_{i},d_{i}^{+}$) is

$h_{0}=\sum\limits_{k}\left(\varepsilon^{d}_{k}d_{k}^{+}d_{k}+\varepsilon^{h}_{k}h_{k}^{+}h_{k}\right)$
(38)

At half filling $\mu=U/2$ and dispersions adopt the form

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$
$\displaystyle-t\varepsilon_{k}+\frac{U}{2}+\frac{sJ}{2}$
$\displaystyle\varepsilon^{h}_{k}$ $\displaystyle=$ $\displaystyle
t\varepsilon_{k}+\frac{U}{2}+\frac{sJ}{2}$ (39) $\displaystyle\varepsilon_{k}$
$\displaystyle=$
$\displaystyle-8\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\cos\frac{k_{z}}{2},$

where $k$ runs over the first Brillouin zone of a bcc lattice.

At zero temperature the spin fluctuations are suppressed and the saturated
magnetization is $M=s-m$. The experimental result for $M$ is $M=2.217$ in
units of Born magneton. For $s=5/2$,the magnetic moment of the itinerant
electron is $m=0.283<0.5$. The magnetization of the sixth electron (9) is
$m<0.5$ if charge carriers fermions $h$ and $d$ have Fermi surfaces
($<h^{+}_{i}h_{i}>+<d^{+}_{i}d_{i}>\neq 0$). Therefore the material is metal.
The theoretical calculations show that $m=0.283$ if the parameter
$(Js+U)/(2t)=1.78$.

We continue examining a field-cooled iron. If, during the preparation, we
apply magnetic field along the magnetic order of localized 3d electrons the
magnetization of these electrons arrive at their saturation, while the Zeeman
splitting of itinerants electron decreases. The nontrivial point is that this
is true and after switching off the magnetic field when the process is over
spinelFeCr2S4 ; spinel+ ; spinelCv1 ; spinel08 ; spinel++ ; spinel11b ;
spinel11a ; spinel11c . To account for the effect we include ”frozen” magnetic
field adding a term $-H\sum\limits_{i}{S^{it\,z}_{i}}$ in the Hamiltonian
(V.1). Then the dispersions of the slave fermions ($d,h$) (V.1) adopted the
form:

$\displaystyle\varepsilon^{d}_{k}$ $\displaystyle=$
$\displaystyle-t\varepsilon_{k}+\frac{U}{2}+\frac{sJ}{2}-\frac{H}{2}$
$\displaystyle\varepsilon^{h}_{k}$ $\displaystyle=$ $\displaystyle
t\varepsilon_{k}+\frac{U}{2}+\frac{sJ}{2}-\frac{H}{2},$ (40)

where $s=5/2$. We consider a state prepared with applied magnetic field
$H=H_{c}=U+sJ$. Then magnetic moment of itinerant electron is $m=0$. By means
of this field one prepares a quantum partial order (QPO) state, when the
itinerant sixth 3d electron does not contribute the magnetization of the
system and magnetic order is formed by the five localized 3d electrons. In QPO
state the itinerant electrons do not form transversal spin fluctuations, and
we can represent them by means of creation and annihilation operators in
Hamiltonian (1). The Hamiltonian of iron in QPO state is

$\displaystyle h^{QPO}$ $\displaystyle=$
$\displaystyle-t\sum\limits_{<ij>}{\left({c_{i\sigma}^{+}c_{j\sigma}+h.c.}\right)}-J^{l}\sum\limits_{<ij>}{{\bf
S_{i}^{l}}\cdot{\bf S_{j}^{l}}}$ (41) $\displaystyle+$
$\displaystyle\sqrt{\frac{s}{2}}J\sum\limits_{i}\left(c_{i\downarrow}^{+}c_{i\uparrow}a_{i}+c_{i\uparrow}^{+}c_{i\downarrow}a_{i}^{+}\right),$

where $a_{i},a_{i}^{+}$ are Bose operators from the Holstein-Primakoff
representation of spin operators of localized electrons $\bf S_{i}^{l}$. In
the spin-fermion interaction they are accounted for in linear approximation.

To study the superconductivity of iron, induced by spin fluctuations, we
integrate out the bosons and obtain an effective four-fermion interaction.
Then we proceed in the standard way to find the gap function equation.

The sums in Eq.(V.1) are over all sites of a body centered cubic (bcc)
lattice. The first Brillouin zone of a bcc lattice is quite complicated and it
is difficult to integrate over wave vectors. To circumvent this problem we
introduce two equivalent simple cubic sub-lattices. Following the standard
procedure one obtains the effective Hamiltonian in the Hartree-Fock
approximation

$\displaystyle h^{f^{4}}_{HF}$ $\displaystyle=$
$\displaystyle\sum\limits_{k}\left[\Delta^{A}_{k}c^{A+}_{\downarrow-k}c^{A+}_{\uparrow
k}+\Delta^{A+}_{k}c^{A}_{\uparrow k}c^{A}_{\downarrow-k}\right.$ (42)
$\displaystyle+$
$\displaystyle\left.\Delta^{B}_{k}c^{B+}_{\downarrow-k}c^{B+}_{\uparrow
k}+\Delta^{B+}_{k}c^{B}_{\uparrow k}c^{B}_{\downarrow-k}\right],$

where the gap functions of the wave vector $k$ are defined by the equations

$\displaystyle\Delta^{A}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{p}<c^{A}_{\uparrow-p}c^{A}_{\downarrow
p}>V_{p-k}$ $\displaystyle\Delta^{B}_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum\limits_{p}<c^{B}_{\uparrow-p}c^{B}_{\downarrow
p}>V_{p-k}.$

The potential $V(p-k)=V(q)$

$V({\bf
q})=\frac{J}{16J^{l}}\frac{1}{1-\cos^{2}(\frac{q_{x}}{2})\cos^{2}(\frac{q_{y}}{2})\cos^{2}(\frac{q_{z}}{2})}$
(44)

is the fermion binding potential result of transversal spin fluctuations of
the 3d localized electrons. The wave vectors $k$ and $p$ run over the first
Brillouin zone of a simple cubic lattice.

The two sublattices are equivalent, therefore the Hamiltonian should be
invariant under the replacement $A\leftrightarrows B$. This is true if

$\Delta^{A}=\Delta^{B}=\Delta.$ (45)

We set $\Delta$ in equation (42) and by means of the Bogoliubov transformation
we rewrite the Hamiltonian in a diagonal form

$h^{f^{4}}_{HF}=\sum\limits_{k}E_{k}\left[f^{A+}_{k}f^{A}_{k}+f^{B+}_{k}f^{B}_{k}-\rho^{A+}_{k}\rho^{A}_{k}-\rho^{B+}_{k}\rho^{B}_{k}\right],$
(46)

where

$E_{k}=\sqrt{\varepsilon^{2}_{k}+\Delta^{2}_{k}}$ (47)

Bearing in mind Bogoliubov transformation we calculate the gap equation at
zero temperature

$\Delta_{k}=\frac{1}{2}\frac{1}{N}\sum\limits_{p}\frac{\Delta_{p}}{\sqrt{\varepsilon^{2}_{p}+\Delta^{2}_{p}}}V_{(}p-k)$
(48)

with

$\Delta_{k}=\Delta\left(\sin k_{x}+\sin k_{y}+\sin k_{z}\right).$ (49)

Figure 6:  The dimensionless $gap/J$ ($\Delta/J$) as a function of
dimensionless hoping parameter $t/J$ for different values of the dimensionless
parameter $J^{l}/J$, where $J^{l}$ is the exchange between the localized
d-electrons and $J$ is the exchange between the localized and itinerant
d-electrons. The figure shows that superconductivity is suppressed when $t/J$
and $J^{l}/J$ increase.

The dimensionless gap/J ($\Delta/J$) as a function of dimensionless hoping
parameter ($t/J$) is depicted in figure (6) for different values of the
parameter ($J^{l}/J$). The figure shows that superconductivity is suppressed
when $t/J$ and $J^{l}/J$ increase. It is important to repeat that $J$ is an
intra-atomic exchange while $t$ and $J^{l}$ are exchanges between atoms, hence
$J$ is much larger than $t$ and $J^{l}$. Therefore the small values of $t/J$
and $J^{l}/J$ are physical relevant. The figures show that this is the case of
the factual superconductivity.

Another ferromagnet which can be treated in analogy of bcc $Fe^{2+}$ is the
fcc cobalt.

##  VI DO NOT CONFUSE WITH JACCARINO-PETER COMPENSATION MECHANISM OF
SUPERCONDUCTIVITY JP62

The magnetic field induced superconductivity (FISC) is one more issue of
special interest. Experimentally, (FISC) in the $H_{c2}-T$ phase diagram was
observed in $Eu_{x}Sn_{1-x}Mo_{6}S_{8}$ Fischer75 . The domain of
superconductivity in $Eu_{0.75}Sn_{0.25}Mo_{6}S_{8}$ extends from 4 to 22 T at
$T=0$ and from $T=0$ to $T=1K$ at $H=12T$ Fischer84 . The magnetic-field
induced superconductivity was attributed to Jaccarino-Peter (JP) compensation
mechanismJP62 . The idea is that in a ferromagnetic metal the conduction
electrons are in an effective field due to the exchange interaction with the
localized spins. It is in general so large as to inhibit the occurrence of
superconductivity. For some systems the exchange interaction have a negative
sign. This allows for the conduction electron polarization to be canceled by
an external magnetic field so that if, in addition these metals possess
phonon-induced attractive electron-electron interaction, superconductivity
occurs in the compensation region. In more complicated cases superconductivity
can occur in two domains: one extends from zero applied magnetic field to
small field which suppress bose condensation of Cooper pairs and respectively
superconductivity, and the other at the high field in the compensation region
Fischer84 . The experiments show that the compensation field is not affected
by superconductivity.

A great deal of interest has been centered on the heavy fermions in cerium and
uranium systems. The heavy-fermion system $CePb_{3}$ at zero field is an
antiferromagnet. In Lin85  the authors report magnetic field of 14 T induces
the system into the superconducting state below 0.20 K. Similarly, at 0.48 K,
15 T magnetic field drives the sample superconducting. The (FISC) in these
compounds is considered to be due to the Jaccarino-Peter mechanism, extended
to antiferromagnetic superconductors Shimahara02 .

$URhGe$ displays ferromagnetism with magnetic moment oriented along the c
-axis, and spin-triplet superconductivity at a lower temperature Hardy05 . In
an external magnetic field along the b-axis perpendicular to c-axis,
superconductivity disappears at about $H=2T$. However, at higher magnetic
fields, in the range from $8T$ to $13.5T$, it reappears again Sheikin05 .

Finally, magnetic-field-induced superconductivity has been observed in organic
superconductors Uji01 ; Balicas01 ; Konoike04 .

It is important to emphasize, that magnetic-field induced superconductivity
disappears when the field is switched off.

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##  Appendix A Consistency between Fourier space and configuration space
across HOD models

The BAO analysis for quantifying the systematic error due to HOD dependence
has been conducted for both configuration and Fourier spaces. Hence, it is
reasonable not only to perform each analysis separately but also to provide
some comparison of both results to assess whether they are consistent with
each other or not. It is important as well, to evaluate if the
consistency/inconsistency of the results has something to do with a particular
HOD model.

We focus first on the efficiency of CV on our BAO fits results. Fig.
LABEL:Fig:CV_reduction_factor shows the CV improvement factor of the
dispersion on the BAO parameters obtained from the BAO fits with respect to
the standard non-CV fits. We observe that the improvement in the errors is
higher for mocks with higher number density compared to mocks with lower
number density such as HMQ${}_{i}^{(3\sigma)}$ ($i=1,2,...,6$). This is in
agreement with what was argued in [94]. We also found that on average, the
dispersion in the CV results is a factor of $\sim$1.5 times lower for
$\alpha_{\text{iso}}$ and $\epsilon$, compared to the non-CV results. Here we
use $\epsilon$ for direct comparison with the original work presented in [94]
and due to the fact that the improvement is not so visible in
$\alpha_{\text{AP}}$ compared to $\epsilon$. Indeed, our findings in terms of
the values of the improvement factors are consistent with the results from
[94]. We observe that other BAO-related parameters give different improvement
factors. For example, while it has been found that $\alpha_{\parallel}$ gives
the highest improvement rate ($\sim$1.6 times in Fourier space, and $\sim$1.5
in configuration space), $\alpha_{\perp}$ shows no major improvement after CV,
with a dispersion of around 1.2 times lower compared to the non-CV results. In
principle, we expect these factors as such given the fact that the Zeldovich
approximation is more accurate at larger scales, where $\alpha_{\parallel}$ is
more sensitive to, due to the Kaiser effect that causes the coherent infall of
galaxies onto still collapsing structures. Overall, we found consistent
results between both Fourier space and configuration space in terms of the
error improvement factor in the dispersion of the BAO parameters due to the CV
technique.

We show scatter plots for both $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$
for some HOD models in Fig. LABEL:fig:scatter_plots_CS_vs_FS. We can draw
several conclusions from these scatter plots. First of all, we found that the
CV results were less scattered with respect to the fiducial values, compared
to the non-CV BAO fits. Some exceptions can be observed as in the case of the
1st-Gen mocks, where the BAO fits from the correlation function multipoles
prefer a higher value of $\alpha_{\text{iso}}$ after CV, which turns out to
drive our final systematics error budget for configuration space. We observe
that while the mean values are consistent between the two analyses, sometimes
the results for individual realizations are not, as is the case for the
HMQ${}_{1}^{(3\sigma)}$ model. This in principle is due to sample variance
noise which is reduced after considering several realizations. Scatter plots
such as the ones for $\alpha_{\text{iso}}$ in the case of the SFHOD, HMQ, and
mHMQ+cf+Sh models show us that while CV does not particularly improve the
consistency of the results between Fourier space and configuration space, it
helps to correctly shift the BAO fits towards consistent values of
$\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$ by reducing sample variance
noise. Thus, in general, whereas CV shows to reduce the dispersion between the
$\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$ measurements there is no clear
trend about CV improving the consistency between configuration space and
Fourier space. Part of this can be due to the fact that we are using
analytical covariance matrices tuned to the clustering of the non-CV two-point
measurements for both CV and non-CV BAO fits. Then, this mismatch might
introduce some degree of inconsistency between the results of both spaces,
even though the dispersion between the BAO scaling parameters is reduced.
Nevertheless, overall we found consistency between the results from Fourier
space and configuration space analyses. We found no extreme outliers that
point out inconsistencies between the two spaces. At most, models with very
low number density show more scatter but the mean values found over the 25
realizations are still consistent.

##  Appendix B Extended set of results

Our full set of results comprehends 22 HOD models. 25 realization mocks are
used for each HOD model and we apply reconstruction to each mock. Then, we
calculate two-point measurements before and after reconstruction in both
Fourier space and configuration space. All these analyses give a total of
around 2200 BAO fits used to estimate the systematics in our BAO analysis due
to HOD dependence. As we are using 25 mocks for each HOD we present the
results of this work by quoting the mean values of the 25 fits. Table
LABEL:Table:BAO_fits_PS shows the results for the BAO fits times a factor of
$10^{3}$. This means that our differences in the recovery of the correct value
of the BAO parameters are at the sub-sub-percent level. We found that most of
our shifts with respect to 1 in $\alpha_{\text{iso}}$ are reduced with the CV
two-point measurements. We can observe that this comes together with a
reduction in the dispersion $\sigma(\overline{\alpha}_{\text{iso}})$. Such
reduction is more pronounced for mocks with less shot-noise as CV becomes more
efficient with a higher number density. This is related to the factors
described in Fig. LABEL:Fig:CV_reduction_factor. We found that our
$\langle\chi^{2}\rangle$ per DoF values are less than $\sim$1.1, which
indicates that our analytical covariance matrices for the power spectrum
multipoles derived from thecov based on the clustering of 25 mocks are
appropriated for our analysis. We then use the same analytical covariance
matrices for our fits to the CV two-point measurements. Hence, we observe a
drop in our $\langle\chi^{2}\rangle$ values as the dispersion on our BAO
parameters with respect to the fiducial values is reduced. We show the
analogous for the configuration space analysis in Table
LABEL:Table:BAO_fits_CF. We observe a similar behavior to our results in
Fourier space, except for 1st-Gen where our measurements after CV increase the
bias in $\alpha_{\text{iso}}$. This ended up driving our HOD systematics for
the configuration space case. However, we also found reasonable
$\langle\chi^{2}\rangle$ values. For BAO fits to CV two-point measurements we
also observed a drop in $\langle\chi^{2}\rangle$, although smaller compared to
the Fourier space case. This might have to do with the fact that the reduction
factors due to CV are smaller in the configuration space case. This is related
to CV being less efficient for configuration space analysis as the noise
mitigation is less scale-dependent compared to the Fourier spaces, leading to
less sensitivity at large scales.

We also show in this section the complete version of the simplified heatmaps
presented in Section 5.2. Fig. LABEL:Fig:Heatmaps_cv_pk_alpha_iso represents
the comparison for all the permutations of pairs using our 22 HOD models.
While the statistical error for $\alpha_{\text{iso}}$ is about 1%, we found at
most differences of around 0.08% for HOD mocks at $z=1.1$. In the case of the
HOD mocks centered at $z=0.8$, we observe that the models
HMQ${}_{i}^{(3\sigma)}$ ($i=1,2,3$) are fairly consistent with each other. A
similar trend is observed for the models with complex galactic conformity
($i=4,5,6$). On the other hand, we observed mild differences between the
models HMQ${}_{i}^{(3\sigma)}$ with and without complex galactic conformity.
However, these differences are of about 0.1% at most for
$\alpha_{\text{iso}}$. We remind the reader that we do not compare HOD models
based on mocks built from simulations centered at different redshifts since we
do not want to introduce simulation noise differences beyond the HOD model
itself. We did not find shifts in $\alpha_{\text{iso}}$ above 3-$\sigma$ in
Fourier space. Similar results are found for $\alpha_{\text{AP}}$ as shown in
Fig. LABEL:Fig:Heatmaps_cv_pk_alpha_ap. In such case, the highest significance
found in $\Delta\alpha_{\text{AP}}$ is 2-$\sigma$ of less. We found shifts of
at most 0.26% in $\alpha_{\text{AP}}$, but still this is very small compared
to the statistical error of DESI 2024.

The full heatmaps for $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$ in the
configuration space analysis are shown in Fig.
LABEL:Fig:Heatmaps_cv_xi_alpha_iso and Fig. LABEL:Fig:Heatmaps_cv_xi_alpha_ap,
respectively. We found higher shifts in $\alpha_{\text{iso}}$ between pairs of
HOD models compared to the Fourier Space case. The highest shifts come from
comparing 1st-Gen with other HOD models. The maximum shift found is 0.17%,
which is 1/6 of the statistical error for DESI 2024. The 1st-Gen mocks show
significant differences with various HOD models at more than 3-$\sigma$. We
quote the maximum shift found at more than 3-$\sigma$ as our systematic error
for $\alpha_{\text{iso}}$. We do not find significant shifts for the models at
$z=0.8$. In a similar fashion to the Fourier space analysis, we did not find
any significant shift in $\alpha_{\text{AP}}$. Therefore, our quoted
systematic error comes from the dispersion over all shifts. It is worth
mentioning that we also tested the effect of the shot-noise differences in our
analysis. We produced sub-sampled versions of our HOD mocks to have the same
number density for all of them ($\sim 10^{-3}h^{3}/\text{Mpc}^{-3}$). We found
that this sub-sampling leads to higher shifts in the BAO parameters and the AP
parameter as well as higher significance in the shifts. However, these biases
are introduced by increasing the shot-noise and reducing the efficiency of the
CV technique.

Finally, we show the measurements of the BAO feature in both Fourier space and
configuration space after removing the smoothed component for each HOD model
in Fig. LABEL:Fig:post_recon_data. The smoothed component is calculated after
performing the BAO fits and it is then subtracted to the measurements. While
we do not show explicit error bars to facilitate the comparison between
measurements of various HOD models, we found that the measurements are
consistent within the error bars when the covariance matrices are taken into
consideration.

##  Appendix C Author Affiliations

1Department of Physics, The University of Texas at Dallas, Richardson, TX
75080, USA

2Laboratoire de Physique Subatomique et de Cosmologie, 53 Avenue des Martyrs,
38000 Grenoble, France

3Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

4IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

5SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

6Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA

7University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720,
USA

8University of Michigan, Ann Arbor, MI 48109, USA

9Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA

10Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA

11Physics Department, Yale University, P.O. Box 208120, New Haven, CT 06511,
USA

12Institute of Cosmology and Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK

13School of Mathematics and Physics, University of Queensland, 4072, Australia

14Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA

15Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA

16The Ohio State University, Columbus, 43210 OH, USA

17Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA

18Leinweber Center for Theoretical Physics, University of Michigan, 450 Church
Street, Ann Arbor, Michigan 48109-1040, USA

19Department of Physics & Astronomy, University of Rochester, 206 Bausch and
Lomb Hall, P.O. Box 270171, Rochester, NY 14627-0171, USA

20Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

21Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

22Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK

23Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México

24NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

25Department of Physics & Astronomy and Pittsburgh Particle Physics,
Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941
O’Hara Street, Pittsburgh, PA 15260, USA

26Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China

27Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Menlo Park, CA 94305, USA

28Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia

29Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia

30Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

31Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain

32Departament de Física Quàntica i Astrofísica, Universitat de Barcelona,
Martí i Franquès 1, E08028 Barcelona, Spain

33Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c.
Martí i Franquès, 1, 08028 Barcelona, Spain.

34Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

35Department of Astrophysical Sciences, Princeton University, Princeton NJ
08544, USA

36Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA

37Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France

38Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

39Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

40Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain

41Department of Physics and Astronomy, Siena College, 515 Loudon Road,
Loudonville, NY 12211, USA

42Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
U.K

43Department of Physics & Astronomy, University of Wyoming, 1000 E.
University, Dept. 3905, Laramie, WY 82071, USA

44National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China

45Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México

46Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202.
Magdalena Contreras, 10720. Ciudad de México, México

47Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

48Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada

49Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada

50Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA

51Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße 1, 85748
Garching, Germany

52Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea

53Centre for Astrophysics & Supercomputing, Swinburne University of
Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia

54CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

55Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA